Subgroups — C₂₃×#19755 #19167381

Order 5888.

842 conjugacy classes of subgroups.

The subgroup lattice has 710 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3294	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×D₈×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄×D₈×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#3272	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1721	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1687	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1681	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1667	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1665	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1639	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1625	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#1619	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂₃×#1619	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂₃×#1581	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂₃×#1580	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₂₃×AGL(1,ℤ/8)	—	2	1	no	—
17	C₂²×D₈×C₂₃	—	2	1	no	—
18	C₂²×D₈×C₂₃	—	2	1	no	—
19	C₂₃×#547	C₂²	1	1	yes	characteristic, complemented
20	D₄²×C₂₃	C₂²	1	1	yes	characteristic
21	D₄²×C₂₃	C₂²	1	1	yes	characteristic, complemented
22	D₄²×C₂₃	C₂²	1	1	yes	characteristic, complemented
23	C₂₃×#493	C₂²	1	1	yes	characteristic, complemented
24	C₂₃×#493	C₂²	1	1	yes	characteristic, complemented
25	C₂₃×#493	C₂²	1	1	yes	characteristic, complemented
26	C₂₃×#492	C₂²	1	1	yes	characteristic, complemented
27	C₂₃×#486	C₂²	1	1	yes	characteristic, complemented
28	C₂₃×#486	C₂²	1	1	yes	characteristic, complemented
29	C₂₃×#480	—	2	1	no	—
30	C₂₃×#474	C₂²	1	1	yes	characteristic, complemented
31	C₂₃×#468	—	2	1	no	—
32	C₂₃×#466	—	2	1	no	—
33	C₂₃×#466	—	2	1	no	—
34	C₂₃×#466	—	2	1	no	—
35	C₂₃×#461	C₂²	1	1	yes	characteristic, complemented
36	C₂₃×#459	C₂²	1	1	yes	characteristic
37	C₂₃×#459	C₂²	1	1	yes	characteristic
38	C₂₃×#459	C₂²	1	1	yes	characteristic, complemented
39	C₂₃×#459	C₂²	1	1	yes	characteristic, complemented
40	C₂₃×#459	C₂²	1	1	yes	characteristic, complemented
41	C₂₃×#459	C₂²	1	1	yes	characteristic, complemented
42	C₂₃×#450	—	2	1	no	—
43	C₂₃×#449	—	2	1	no	—
44	C₂₃×#447	—	2	1	no	—
45	C₂₃×#447	—	2	1	no	—
46	C₂₃×#447	—	2	1	no	—
47	C₂₃×#447	—	2	1	no	—
48	C₂₃×#447	—	2	1	no	—
49	C₄×D₈×C₂₃	C₂²	1	1	yes	characteristic
50	C₄×D₈×C₂₃	C₂²	1	1	yes	characteristic
51	C₂₃×#436	C₂²	1	1	yes	characteristic, complemented
52	C₈×D₄×C₂₃	C₂²	1	1	yes	characteristic, complemented
53	C₈×D₄×C₂₃	C₂²	1	1	yes	characteristic, complemented
54	C₈×D₄×C₂₃	C₂²	1	1	yes	characteristic, complemented
55	C₂×C₂₃×#102	—	2	1	no	—
56	C₂₃×#335	C₂²	1	1	yes	characteristic, complemented
57	C₂₃×#334	C₂²	1	1	yes	characteristic, complemented
58	C₂₃×#332	C₂²	1	1	yes	characteristic, complemented
59	C₂₃×#331	C₂²	1	1	yes	characteristic, complemented
60	C₂₃×#331	C₂²	1	1	yes	characteristic, complemented
61	C₂₃×#331	C₂²	1	1	yes	characteristic, complemented
62	C₂₃×D₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
63	C₂₃×D₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
64	C₂₃×D₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
65	C₂₃×D₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
66	C₂₃×#322	C₂²	1	1	yes	characteristic, complemented
67	C₂×#41×C₂₃	—	2	1	no	—
68	C₂²×D₄×C₂₃	—	2	1	no	—
69	C₂²×D₄×C₂₃	—	2	1	no	—
70	C₂²×D₄×C₂₃	—	2	1	no	—
71	C₂²×D₄×C₂₃	—	2	1	no	—
72	C₂²×D₄×C₂₃	—	2	1	no	—
73	C₂²×D₄×C₂₃	—	2	1	no	—
74	C₂₃×AGL(1,ℤ/8)	—	4	4	no	—
75	C₂×SD₁₆×C₂₃	—	2	1	no	—
76	C₂×SD₁₆×C₂₃	—	2	1	no	—
77	C₂×SD₁₆×C₂₃	—	2	1	no	—
78	C₂×D₈×C₂₃	—	2	1	no	—
79	C₂×D₈×C₂₃	—	2	1	no	—
80	C₂×D₈×C₂₃	—	2	1	no	—
81	C₂×D₈×C₂₃	—	2	1	no	—
82	C₂×D₈×C₂₃	—	2	1	no	—
83	C₂×D₈×C₂₃	—	2	1	no	—
84	C₂×D₈×C₂₃	—	2	1	no	—
85	C₂×D₈×C₂₃	—	2	1	no	—
86	C₂×D₈×C₂₃	—	2	1	no	—
87	C₂×D₈×C₂₃	—	2	1	no	—
88	C₂×D₈×C₂₃	—	4	1	no	—
89	C₂×D₈×C₂₃	—	4	4	no	—
90	C₂×D₈×C₂₃	—	4	4	no	—
91	C₂×D₈×C₂₃	D₄	1	1	yes	characteristic
92	C₂×D₈×C₂₃	D₄	1	1	yes	characteristic
93	C₂×M₁₆×C₂₃	—	2	1	no	—
94	C₂²×C₈×C₂₃	—	2	1	no	—
95	C₂²×C₈×C₂₃	—	2	1	no	—
96	C₂²×C₈×C₂₃	—	2	1	no	—
97	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
98	C₂₃×#127	C₂³	1	1	yes	characteristic
99	C₂₃×#127	C₂³	1	1	yes	characteristic
100	C₂₃×#127	C₂³	1	1	yes	characteristic
101	C₂₃×#123	—	4	1	no	—
102	C₂₃×C₂²⋊Q₈	—	2	1	no	—
103	C₂₃×#121	—	2	1	no	—
104	C₂₃×#121	—	2	1	no	—
105	C₂₃×#121	—	2	1	no	—
106	C₂₃×#121	—	2	1	no	—
107	C₂₃×#121	—	2	1	no	—
108	C₂₃×#121	—	2	1	no	—
109	C₂₃×#121	—	2	1	no	—
110	C₂₃×#120	—	4	1	no	—
111	C₂₃×#120	—	4	1	no	—
112	C₂₃×#120	—	4	1	no	—
113	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
114	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
115	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
116	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
117	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
118	C₂₃×#107	—	2	1	no	—
119	C₂₃×#107	D₄	1	1	yes	characteristic
120	C₂₃×#107	D₄	1	1	yes	characteristic
121	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
122	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
123	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
124	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
125	C₂₃×#103	—	2	1	no	—
126	C₂₃×#102	—	2	1	no	—
127	C₂₃×#102	—	2	1	no	—
128	C₂₃×#102	—	2	1	no	—
129	C₂₃×#102	—	2	1	no	—
130	C₂₃×#102	—	2	1	no	—
131	C₂₃×#102	—	2	1	no	—
132	C₂₃×#102	—	2	1	no	—
133	C₂₃×#102	—	2	1	no	—
134	C₂₃×#102	—	2	1	no	—
135	C₂₃×#102	—	4	1	no	—
136	C₂₃×#102	D₄	1	1	yes	characteristic
137	C₂₃×#102	D₄	1	1	yes	characteristic
138	C₂₃×#102	D₄	1	1	yes	characteristic
139	C₂₃×#102	D₄	1	1	yes	characteristic
140	C₂₃×C₂²⋊C₈	—	2	1	no	—
141	C₂₃×C₂²⋊C₈	—	2	1	no	—
142	C₂₃×C₂²⋊C₈	—	2	1	no	—
143	C₂₃×C₂²⋊C₈	—	2	1	no	—
144	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
145	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
146	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
147	C₂⁴×C₂₃	—	4	1	no	—
148	C₂⁴×C₂₃	—	4	1	no	—
149	C₂⁴×C₂₃	—	4	1	no	—
150	#41×C₂₃	—	4	2	no	—
151	#41×C₂₃	—	8	1	no	—
152	C₂×Q₈×C₂₃	—	2	1	no	—
153	C₂×D₄×C₂₃	—	2	1	no	—
154	C₂×D₄×C₂₃	—	2	1	no	—
155	C₂×D₄×C₂₃	—	2	1	no	—
156	C₂×D₄×C₂₃	—	2	1	no	—
157	C₂×D₄×C₂₃	—	2	1	no	—
158	C₂×D₄×C₂₃	—	2	1	no	—
159	C₂×D₄×C₂₃	—	2	1	no	—
160	C₂×D₄×C₂₃	—	2	1	no	—
161	C₂×D₄×C₂₃	—	2	1	no	—
162	C₂×D₄×C₂₃	—	4	1	no	—
163	C₂×D₄×C₂₃	—	4	1	no	—
164	C₂×D₄×C₂₃	—	4	1	no	—
165	C₂×D₄×C₂₃	—	4	1	no	—
166	C₂×D₄×C₂₃	—	4	1	no	—
167	C₂×D₄×C₂₃	—	4	1	no	—
168	C₂×D₄×C₂₃	—	4	2	no	—
169	C₂×D₄×C₂₃	—	4	2	no	—
170	C₂×D₄×C₂₃	—	4	2	no	—
171	C₂×D₄×C₂₃	—	4	2	no	—
172	C₂×D₄×C₂₃	—	4	2	no	—
173	C₂×D₄×C₂₃	—	4	2	no	—
174	C₂×D₄×C₂₃	—	4	1	no	—
175	C₂×D₄×C₂₃	—	8	1	no	—
176	C₂×D₄×C₂₃	—	8	1	no	—
177	C₂×D₄×C₂₃	—	8	1	no	—
178	C₂×D₄×C₂₃	—	8	1	no	—
179	C₂×D₄×C₂₃	—	8	1	no	—
180	C₂×D₄×C₂₃	—	8	1	no	—
181	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
182	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
183	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
184	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
185	C₂²×C₄×C₂₃	—	2	1	no	—
186	C₂²×C₄×C₂₃	—	2	1	no	—
187	C₂²×C₄×C₂₃	—	2	1	no	—
188	C₂²×C₄×C₂₃	—	2	1	no	—
189	C₂²×C₄×C₂₃	—	4	1	no	—
190	SD₁₆×C₂₃	—	4	2	no	—
191	SD₁₆×C₂₃	—	4	2	no	—
192	SD₁₆×C₂₃	—	8	1	no	—
193	D₈×C₂₃	—	2	2	no	—
194	D₈×C₂₃	—	4	2	no	—
195	D₈×C₂₃	—	4	2	no	—
196	D₈×C₂₃	—	4	2	no	—
197	D₈×C₂₃	—	4	2	no	—
198	D₈×C₂₃	—	4	2	no	—
199	D₈×C₂₃	—	4	2	no	—
200	D₈×C₂₃	—	4	2	no	—
201	D₈×C₂₃	—	8	1	no	—
202	D₈×C₂₃	—	8	1	no	—
203	D₈×C₂₃	—	8	1	no	—
204	D₈×C₂₃	—	8	2	no	—
205	D₈×C₂₃	D₈	1	4	yes	complemented
206	M₁₆×C₂₃	—	4	2	no	—
207	C₂×C₈×C₂₃	—	2	1	no	—
208	C₂×C₈×C₂₃	—	2	1	no	—
209	C₂×C₈×C₂₃	—	2	1	no	—
210	C₂×C₈×C₂₃	—	2	1	no	—
211	C₂×C₈×C₂₃	—	2	1	no	—
212	C₂×C₈×C₂₃	—	4	2	no	—
213	C₂×C₈×C₂₃	—	4	2	no	—
214	C₂×C₈×C₂₃	—	8	1	no	—
215	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
216	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
217	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
218	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
219	C₄⋊C₄×C₂₃	—	2	1	no	—
220	C₄⋊C₄×C₂₃	—	2	1	no	—
221	C₄⋊C₄×C₂₃	—	4	1	no	—
222	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
223	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
224	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
225	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
226	C₂²⋊C₄×C₂₃	—	2	1	no	—
227	C₂²⋊C₄×C₂₃	—	2	1	no	—
228	C₂²⋊C₄×C₂₃	—	2	1	no	—
229	C₂²⋊C₄×C₂₃	—	2	1	no	—
230	C₂²⋊C₄×C₂₃	—	4	1	no	—
231	C₂²⋊C₄×C₂₃	—	4	1	no	—
232	C₂²⋊C₄×C₂₃	—	4	1	no	—
233	C₂²⋊C₄×C₂₃	—	4	1	no	—
234	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
235	#19755	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
236	C₂³×C₂₃	—	2	1	no	—
237	C₂³×C₂₃	—	2	1	no	—
238	C₂³×C₂₃	—	2	1	no	—
239	C₂³×C₂₃	—	2	1	no	—
240	C₂³×C₂₃	—	4	1	no	—
241	C₂³×C₂₃	—	4	1	no	—
242	C₂³×C₂₃	—	4	1	no	—
243	C₂³×C₂₃	—	8	2	no	—
244	C₂³×C₂₃	—	8	2	no	—
245	C₂³×C₂₃	—	8	2	no	—
246	C₂³×C₂₃	—	8	2	no	—
247	C₂³×C₂₃	—	8	2	no	—
248	C₂³×C₂₃	—	8	2	no	—
249	C₂³×C₂₃	—	16	1	no	—
250	C₂³×C₂₃	—	16	1	no	—
251	C₂³×C₂₃	—	16	1	no	—
252	Q₈×C₂₃	—	4	1	no	—
253	Q₈×C₂₃	—	4	1	no	—
254	D₄×C₂₃	—	2	1	no	—
255	D₄×C₂₃	—	2	1	no	—
256	D₄×C₂₃	—	2	1	no	—
257	D₄×C₂₃	—	2	1	no	—
258	D₄×C₂₃	—	2	1	no	—
259	D₄×C₂₃	—	4	1	no	—
260	D₄×C₂₃	—	4	1	no	—
261	D₄×C₂₃	—	4	1	no	—
262	D₄×C₂₃	—	4	1	no	—
263	D₄×C₂₃	—	4	1	no	—
264	D₄×C₂₃	—	4	1	no	—
265	D₄×C₂₃	—	4	1	no	—
266	D₄×C₂₃	—	4	1	no	—
267	D₄×C₂₃	—	4	1	no	—
268	D₄×C₂₃	—	4	1	no	—
269	D₄×C₂₃	—	4	1	no	—
270	D₄×C₂₃	—	4	1	no	—
271	D₄×C₂₃	—	8	1	no	—
272	D₄×C₂₃	—	8	1	no	—
273	D₄×C₂₃	—	8	1	no	—
274	D₄×C₂₃	—	8	2	no	—
275	D₄×C₂₃	—	8	2	no	—
276	D₄×C₂₃	—	8	2	no	—
277	D₄×C₂₃	—	8	2	no	—
278	D₄×C₂₃	—	8	2	no	—
279	D₄×C₂₃	—	8	2	no	—
280	D₄×C₂₃	—	8	2	no	—
281	D₄×C₂₃	C₂×D₈	1	2	yes	—
282	D₄×C₂₃	C₂×D₈	1	2	yes	complemented
283	D₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	complemented
284	C₂×C₄×C₂₃	—	2	1	no	—
285	C₂×C₄×C₂₃	—	2	1	no	—
286	C₂×C₄×C₂₃	—	2	1	no	—
287	C₂×C₄×C₂₃	—	2	1	no	—
288	C₂×C₄×C₂₃	—	4	1	no	—
289	C₂×C₄×C₂₃	—	4	1	no	—
290	C₂×C₄×C₂₃	—	4	1	no	—
291	C₂×C₄×C₂₃	—	4	1	no	—
292	C₂×C₄×C₂₃	—	4	1	no	—
293	C₂×C₄×C₂₃	—	4	1	no	—
294	C₂×C₄×C₂₃	—	4	1	no	—
295	C₂×C₄×C₂₃	—	4	1	no	—
296	C₂×C₄×C₂₃	—	4	1	no	—
297	C₂×C₄×C₂₃	—	8	2	no	—
298	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
299	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
300	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
301	C₈×C₂₃	—	2	1	no	—
302	C₈×C₂₃	—	2	1	no	—
303	C₈×C₂₃	—	2	1	no	—
304	C₈×C₂₃	—	4	1	no	—
305	C₈×C₂₃	—	4	1	no	—
306	C₈×C₂₃	—	4	1	no	—
307	C₈×C₂₃	—	4	1	no	—
308	C₈×C₂₃	—	4	1	no	—
309	C₈×C₂₃	C₂×D₈	1	2	yes	complemented
310	#3294	C₂×C₂₃	1	1	yes	characteristic, complemented
311	D₄×D₈	C₂×C₂₃	1	1	yes	characteristic, complemented
312	D₄×D₈	C₂×C₂₃	1	1	yes	characteristic, complemented
313	#3272	C₂×C₂₃	1	1	yes	characteristic, complemented
314	#1721	C₂×C₂₃	1	1	yes	characteristic, complemented
315	#1687	C₂×C₂₃	1	1	yes	characteristic, complemented
316	#1681	C₂×C₂₃	1	1	yes	characteristic, complemented
317	#1667	C₂×C₂₃	1	1	yes	characteristic, complemented
318	#1665	C₂×C₂₃	1	1	yes	characteristic, complemented
319	#1639	C₂×C₂₃	1	1	yes	characteristic, complemented
320	#1625	C₂×C₂₃	1	1	yes	characteristic, complemented
321	#1619	C₂×C₂₃	1	1	yes	characteristic, complemented
322	#1619	C₂×C₂₃	1	1	yes	characteristic, complemented
323	#1581	C₂×C₂₃	1	1	yes	characteristic, complemented
324	#1580	C₂×C₂₃	1	1	yes	characteristic, complemented
325	C₂²×C₂₃	—	2	2	no	—
326	C₂²×C₂₃	—	2	2	no	—
327	C₂²×C₂₃	—	2	2	no	—
328	C₂²×C₂₃	—	4	1	no	—
329	C₂²×C₂₃	—	4	1	no	—
330	C₂²×C₂₃	—	4	1	no	—
331	C₂²×C₂₃	—	4	1	no	—
332	C₂²×C₂₃	—	4	1	no	—
333	C₂²×C₂₃	—	4	1	no	—
334	C₂²×C₂₃	—	4	1	no	—
335	C₂²×C₂₃	—	4	1	no	—
336	C₂²×C₂₃	—	4	1	no	—
337	C₂²×C₂₃	—	8	1	no	—
338	C₂²×C₂₃	—	8	1	no	—
339	C₂²×C₂₃	—	8	1	no	—
340	C₂²×C₂₃	—	8	1	no	—
341	C₂²×C₂₃	—	8	1	no	—
342	C₂²×C₂₃	—	8	1	no	—
343	C₂²×C₂₃	—	8	1	no	—
344	C₂²×C₂₃	—	8	1	no	—
345	C₂²×C₂₃	—	8	1	no	—
346	C₂²×C₂₃	—	16	4	no	—
347	C₂²×C₂₃	—	16	4	no	—
348	C₂²×C₂₃	—	16	4	no	—
349	C₂²×C₂₃	D₄²	1	1	yes	characteristic
350	C₄×C₂₃	—	2	1	no	—
351	C₄×C₂₃	—	4	1	no	—
352	C₄×C₂₃	—	4	1	no	—
353	C₄×C₂₃	—	4	1	no	—
354	C₄×C₂₃	—	4	1	no	—
355	C₄×C₂₃	—	8	1	no	—
356	C₄×C₂₃	C₂²×D₈	1	1	yes	characteristic
357	C₄×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
358	C₄×C₂₃	#576	1	1	yes	characteristic
359	C₄×C₂₃	#576	1	1	yes	characteristic
360	C₂×AGL(1,ℤ/8)	—	2	1	no	—
361	C₂²×D₈	—	2	1	no	—
362	C₂²×D₈	—	2	1	no	—
363	#547	C₂²×C₂₃	1	1	yes	characteristic, complemented
364	D₄²	C₂²×C₂₃	1	1	yes	characteristic
365	D₄²	C₂²×C₂₃	1	1	yes	characteristic, complemented
366	D₄²	C₂²×C₂₃	1	1	yes	characteristic, complemented
367	#493	C₂²×C₂₃	1	1	yes	characteristic, complemented
368	#493	C₂²×C₂₃	1	1	yes	characteristic, complemented
369	#493	C₂²×C₂₃	1	1	yes	characteristic, complemented
370	#492	C₂²×C₂₃	1	1	yes	characteristic, complemented
371	#486	C₂²×C₂₃	1	1	yes	characteristic, complemented
372	#486	C₂²×C₂₃	1	1	yes	characteristic, complemented
373	#480	—	2	1	no	—
374	#474	C₂²×C₂₃	1	1	yes	characteristic, complemented
375	#468	—	2	1	no	—
376	#466	—	2	1	no	—
377	#466	—	2	1	no	—
378	#466	—	2	1	no	—
379	#461	C₂²×C₂₃	1	1	yes	characteristic, complemented
380	#459	C₂²×C₂₃	1	1	yes	characteristic
381	#459	C₂²×C₂₃	1	1	yes	characteristic
382	#459	C₂²×C₂₃	1	1	yes	characteristic, complemented
383	#459	C₂²×C₂₃	1	1	yes	characteristic, complemented
384	#459	C₂²×C₂₃	1	1	yes	characteristic, complemented
385	#459	C₂²×C₂₃	1	1	yes	characteristic, complemented
386	#450	—	2	1	no	—
387	#449	—	2	1	no	—
388	#447	—	2	1	no	—
389	#447	—	2	1	no	—
390	#447	—	2	1	no	—
391	#447	—	2	1	no	—
392	#447	—	2	1	no	—
393	C₄×D₈	C₂²×C₂₃	1	1	yes	characteristic
394	C₄×D₈	C₂²×C₂₃	1	1	yes	characteristic
395	#436	C₂²×C₂₃	1	1	yes	characteristic, complemented
396	C₈×D₄	C₂²×C₂₃	1	1	yes	characteristic, complemented
397	C₈×D₄	C₂²×C₂₃	1	1	yes	characteristic, complemented
398	C₈×D₄	C₂²×C₂₃	1	1	yes	characteristic, complemented
399	C₂×#102	—	2	1	no	—
400	#335	C₂²×C₂₃	1	1	yes	characteristic, complemented
401	#334	C₂²×C₂₃	1	1	yes	characteristic, complemented
402	#332	C₂²×C₂₃	1	1	yes	characteristic, complemented
403	#331	C₂²×C₂₃	1	1	yes	characteristic, complemented
404	#331	C₂²×C₂₃	1	1	yes	characteristic, complemented
405	#331	C₂²×C₂₃	1	1	yes	characteristic, complemented
406	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
407	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
408	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
409	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
410	#322	C₂²×C₂₃	1	1	yes	characteristic, complemented
411	C₂×C₂₃	—	4	2	no	—
412	C₂×C₂₃	—	4	2	no	—
413	C₂×C₂₃	—	4	2	no	—
414	C₂×C₂₃	—	8	1	no	—
415	C₂×C₂₃	—	16	1	no	—
416	C₂×C₂₃	—	16	1	no	—
417	C₂×C₂₃	—	16	1	no	—
418	C₂×C₂₃	#3264	1	1	yes	characteristic
419	C₂×C₂₃	D₄×D₈	1	1	yes	characteristic
420	C₂×C₂₃	#3288	1	1	yes	characteristic
421	C₂×#41	—	2	1	no	—
422	C₂²×D₄	—	2	1	no	—
423	C₂²×D₄	—	2	1	no	—
424	C₂²×D₄	—	2	1	no	—
425	C₂²×D₄	—	2	1	no	—
426	C₂²×D₄	—	2	1	no	—
427	C₂²×D₄	—	2	1	no	—
428	AGL(1,ℤ/8)	—	4	4	no	—
429	C₂×SD₁₆	—	2	1	no	—
430	C₂×SD₁₆	—	2	1	no	—
431	C₂×SD₁₆	—	2	1	no	—
432	C₂×D₈	—	2	1	no	—
433	C₂×D₈	—	2	1	no	—
434	C₂×D₈	—	2	1	no	—
435	C₂×D₈	—	2	1	no	—
436	C₂×D₈	—	2	1	no	—
437	C₂×D₈	—	2	1	no	—
438	C₂×D₈	—	2	1	no	—
439	C₂×D₈	—	2	1	no	—
440	C₂×D₈	—	2	1	no	—
441	C₂×D₈	—	2	1	no	—
442	C₂×D₈	—	4	1	no	—
443	C₂×D₈	—	4	4	no	—
444	C₂×D₈	—	4	4	no	—
445	C₂×D₈	D₄×C₂₃	1	1	yes	characteristic
446	C₂×D₈	D₄×C₂₃	1	1	yes	characteristic
447	C₂×M₁₆	—	2	1	no	—
448	C₂²×C₈	—	2	1	no	—
449	C₂²×C₈	—	2	1	no	—
450	C₂²×C₈	—	2	1	no	—
451	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
452	#127	C₂³×C₂₃	1	1	yes	characteristic
453	#127	C₂³×C₂₃	1	1	yes	characteristic
454	#127	C₂³×C₂₃	1	1	yes	characteristic
455	#123	—	4	1	no	—
456	C₂²⋊Q₈	—	2	1	no	—
457	#121	—	2	1	no	—
458	#121	—	2	1	no	—
459	#121	—	2	1	no	—
460	#121	—	2	1	no	—
461	#121	—	2	1	no	—
462	#121	—	2	1	no	—
463	#121	—	2	1	no	—
464	#120	—	4	1	no	—
465	#120	—	4	1	no	—
466	#120	—	4	1	no	—
467	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
468	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
469	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
470	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
471	C₂×C₄⋊C₄	—	2	1	no	—
472	#107	—	2	1	no	—
473	#107	D₄×C₂₃	1	1	yes	characteristic
474	#107	D₄×C₂₃	1	1	yes	characteristic
475	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
476	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
477	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
478	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
479	#103	—	2	1	no	—
480	#102	—	2	1	no	—
481	#102	—	2	1	no	—
482	#102	—	2	1	no	—
483	#102	—	2	1	no	—
484	#102	—	2	1	no	—
485	#102	—	2	1	no	—
486	#102	—	2	1	no	—
487	#102	—	2	1	no	—
488	#102	—	2	1	no	—
489	#102	—	4	1	no	—
490	#102	D₄×C₂₃	1	1	yes	characteristic
491	#102	D₄×C₂₃	1	1	yes	characteristic
492	#102	D₄×C₂₃	1	1	yes	characteristic
493	#102	D₄×C₂₃	1	1	yes	characteristic
494	C₂²⋊C₈	—	2	1	no	—
495	C₂²⋊C₈	—	2	1	no	—
496	C₂²⋊C₈	—	2	1	no	—
497	C₂²⋊C₈	—	2	1	no	—
498	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
499	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
500	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
501	C₂₃	#19755	1	1	yes	characteristic, direct product factor, Sylow
502	C₂⁴	—	4	1	no	—
503	C₂⁴	—	4	1	no	—
504	C₂⁴	—	4	1	no	—
505	#41	—	4	2	no	—
506	#41	—	8	1	no	—
507	C₂×Q₈	—	2	1	no	—
508	C₂×D₄	—	2	1	no	—
509	C₂×D₄	—	2	1	no	—
510	C₂×D₄	—	2	1	no	—
511	C₂×D₄	—	2	1	no	—
512	C₂×D₄	—	2	1	no	—
513	C₂×D₄	—	2	1	no	—
514	C₂×D₄	—	2	1	no	—
515	C₂×D₄	—	2	1	no	—
516	C₂×D₄	—	2	1	no	—
517	C₂×D₄	—	4	1	no	—
518	C₂×D₄	—	4	1	no	—
519	C₂×D₄	—	4	1	no	—
520	C₂×D₄	—	4	1	no	—
521	C₂×D₄	—	4	1	no	—
522	C₂×D₄	—	4	1	no	—
523	C₂×D₄	—	4	2	no	—
524	C₂×D₄	—	4	2	no	—
525	C₂×D₄	—	4	2	no	—
526	C₂×D₄	—	4	2	no	—
527	C₂×D₄	—	4	2	no	—
528	C₂×D₄	—	4	2	no	—
529	C₂×D₄	—	4	1	no	—
530	C₂×D₄	—	8	1	no	—
531	C₂×D₄	—	8	1	no	—
532	C₂×D₄	—	8	1	no	—
533	C₂×D₄	—	8	1	no	—
534	C₂×D₄	—	8	1	no	—
535	C₂×D₄	—	8	1	no	—
536	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
537	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
538	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
539	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
540	C₂²×C₄	—	2	1	no	—
541	C₂²×C₄	—	2	1	no	—
542	C₂²×C₄	—	2	1	no	—
543	C₂²×C₄	—	2	1	no	—
544	C₂²×C₄	—	4	1	no	—
545	SD₁₆	—	4	2	no	—
546	SD₁₆	—	4	2	no	—
547	SD₁₆	—	8	1	no	—
548	D₈	—	2	2	no	—
549	D₈	—	4	2	no	—
550	D₈	—	4	2	no	—
551	D₈	—	4	2	no	—
552	D₈	—	4	2	no	—
553	D₈	—	4	2	no	—
554	D₈	—	4	2	no	—
555	D₈	—	4	2	no	—
556	D₈	—	8	1	no	—
557	D₈	—	8	1	no	—
558	D₈	—	8	1	no	—
559	D₈	—	8	2	no	—
560	D₈	D₈×C₂₃	1	4	yes	complemented
561	M₁₆	—	4	2	no	—
562	C₂×C₈	—	2	1	no	—
563	C₂×C₈	—	2	1	no	—
564	C₂×C₈	—	2	1	no	—
565	C₂×C₈	—	2	1	no	—
566	C₂×C₈	—	2	1	no	—
567	C₂×C₈	—	4	2	no	—
568	C₂×C₈	—	4	2	no	—
569	C₂×C₈	—	8	1	no	—
570	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
571	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
572	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
573	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
574	C₄⋊C₄	—	2	1	no	—
575	C₄⋊C₄	—	2	1	no	—
576	C₄⋊C₄	—	4	1	no	—
577	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
578	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
579	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
580	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
581	C₂²⋊C₄	—	2	1	no	—
582	C₂²⋊C₄	—	2	1	no	—
583	C₂²⋊C₄	—	2	1	no	—
584	C₂²⋊C₄	—	2	1	no	—
585	C₂²⋊C₄	—	4	1	no	—
586	C₂²⋊C₄	—	4	1	no	—
587	C₂²⋊C₄	—	4	1	no	—
588	C₂²⋊C₄	—	4	1	no	—
589	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
590	C₂³	—	2	1	no	—
591	C₂³	—	2	1	no	—
592	C₂³	—	2	1	no	—
593	C₂³	—	2	1	no	—
594	C₂³	—	4	1	no	—
595	C₂³	—	4	1	no	—
596	C₂³	—	4	1	no	—
597	C₂³	—	8	2	no	—
598	C₂³	—	8	2	no	—
599	C₂³	—	8	2	no	—
600	C₂³	—	8	2	no	—
601	C₂³	—	8	2	no	—
602	C₂³	—	8	2	no	—
603	C₂³	—	16	1	no	—
604	C₂³	—	16	1	no	—
605	C₂³	—	16	1	no	—
606	Q₈	—	4	1	no	—
607	Q₈	—	4	1	no	—
608	D₄	—	2	1	no	—
609	D₄	—	2	1	no	—
610	D₄	—	2	1	no	—
611	D₄	—	2	1	no	—
612	D₄	—	2	1	no	—
613	D₄	—	4	1	no	—
614	D₄	—	4	1	no	—
615	D₄	—	4	1	no	—
616	D₄	—	4	1	no	—
617	D₄	—	4	1	no	—
618	D₄	—	4	1	no	—
619	D₄	—	4	1	no	—
620	D₄	—	4	1	no	—
621	D₄	—	4	1	no	—
622	D₄	—	4	1	no	—
623	D₄	—	4	1	no	—
624	D₄	—	4	1	no	—
625	D₄	—	8	1	no	—
626	D₄	—	8	1	no	—
627	D₄	—	8	1	no	—
628	D₄	—	8	2	no	—
629	D₄	—	8	2	no	—
630	D₄	—	8	2	no	—
631	D₄	—	8	2	no	—
632	D₄	—	8	2	no	—
633	D₄	—	8	2	no	—
634	D₄	—	8	2	no	—
635	D₄	C₂×D₈×C₂₃	1	2	yes	—
636	D₄	C₂×D₈×C₂₃	1	2	yes	complemented
637	D₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	complemented
638	C₂×C₄	—	2	1	no	—
639	C₂×C₄	—	2	1	no	—
640	C₂×C₄	—	2	1	no	—
641	C₂×C₄	—	2	1	no	—
642	C₂×C₄	—	4	1	no	—
643	C₂×C₄	—	4	1	no	—
644	C₂×C₄	—	4	1	no	—
645	C₂×C₄	—	4	1	no	—
646	C₂×C₄	—	4	1	no	—
647	C₂×C₄	—	4	1	no	—
648	C₂×C₄	—	4	1	no	—
649	C₂×C₄	—	4	1	no	—
650	C₂×C₄	—	4	1	no	—
651	C₂×C₄	—	8	2	no	—
652	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
653	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
654	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
655	C₈	—	2	1	no	—
656	C₈	—	2	1	no	—
657	C₈	—	2	1	no	—
658	C₈	—	4	1	no	—
659	C₈	—	4	1	no	—
660	C₈	—	4	1	no	—
661	C₈	—	4	1	no	—
662	C₈	—	4	1	no	—
663	C₈	C₂×D₈×C₂₃	1	2	yes	complemented
664	C₂²	—	2	2	no	—
665	C₂²	—	2	2	no	—
666	C₂²	—	2	2	no	—
667	C₂²	—	4	1	no	—
668	C₂²	—	4	1	no	—
669	C₂²	—	4	1	no	—
670	C₂²	—	4	1	no	—
671	C₂²	—	4	1	no	—
672	C₂²	—	4	1	no	—
673	C₂²	—	4	1	no	—
674	C₂²	—	4	1	no	—
675	C₂²	—	4	1	no	—
676	C₂²	—	8	1	no	—
677	C₂²	—	8	1	no	—
678	C₂²	—	8	1	no	—
679	C₂²	—	8	1	no	—
680	C₂²	—	8	1	no	—
681	C₂²	—	8	1	no	—
682	C₂²	—	8	1	no	—
683	C₂²	—	8	1	no	—
684	C₂²	—	8	1	no	—
685	C₂²	—	16	4	no	—
686	C₂²	—	16	4	no	—
687	C₂²	—	16	4	no	—
688	C₂²	D₄²×C₂₃	1	1	yes	characteristic
689	C₄	—	2	1	no	—
690	C₄	—	4	1	no	—
691	C₄	—	4	1	no	—
692	C₄	—	4	1	no	—
693	C₄	—	4	1	no	—
694	C₄	—	8	1	no	—
695	C₄	C₂²×D₈×C₂₃	1	1	yes	characteristic
696	C₄	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
697	C₄	C₂₃×#576	1	1	yes	characteristic
698	C₄	C₂₃×#576	1	1	yes	characteristic
699	C₂	—	4	2	no	—
700	C₂	—	4	2	no	—
701	C₂	—	4	2	no	—
702	C₂	—	8	1	no	—
703	C₂	—	16	1	no	—
704	C₂	—	16	1	no	—
705	C₂	—	16	1	no	—
706	C₂	C₂₃×#3264	1	1	yes	characteristic
707	C₂	D₄×D₈×C₂₃	1	1	yes	characteristic
708	C₂	C₂₃×#3288	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₄²×C₂₃ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂₃×#3294	C₂	semidirect
2. D₄×D₈×C₂₃	C₂	semidirect
3. D₄×D₈×C₂₃	C₂	semidirect
4. C₂₃×#3272	C₂	semidirect
5. C₂₃×#1721	C₂	semidirect
6. C₂₃×#1687	C₂	semidirect
7. C₂₃×#1681	C₂	semidirect
8. C₂₃×#1667	C₂	semidirect
9. C₂₃×#1665	C₂	semidirect
10. C₂₃×#1639	C₂	semidirect
11. C₂₃×#1625	C₂	semidirect
12. C₂₃×#1619	C₂	semidirect
13. C₂₃×#1619	C₂	semidirect
14. C₂₃×#1581	C₂	semidirect
15. C₂₃×#1580	C₂	semidirect
19. C₂₃×#547	C₂²	semidirect
21. D₄²×C₂₃	C₂²	semidirect
22. D₄²×C₂₃	C₂²	semidirect
23. C₂₃×#493	C₂²	semidirect
24. C₂₃×#493	C₂²	semidirect
25. C₂₃×#493	C₂²	semidirect
26. C₂₃×#492	C₂²	semidirect
27. C₂₃×#486	C₂²	semidirect
28. C₂₃×#486	C₂²	semidirect
30. C₂₃×#474	C₂²	semidirect
35. C₂₃×#461	C₂²	semidirect
38. C₂₃×#459	C₂²	semidirect
39. C₂₃×#459	C₂²	semidirect
40. C₂₃×#459	C₂²	semidirect
41. C₂₃×#459	C₂²	semidirect
51. C₂₃×#436	C₂²	semidirect
52. C₈×D₄×C₂₃	C₂²	semidirect
53. C₈×D₄×C₂₃	C₂²	semidirect
54. C₈×D₄×C₂₃	C₂²	semidirect
56. C₂₃×#335	C₂²	semidirect
57. C₂₃×#334	C₂²	semidirect
58. C₂₃×#332	C₂²	semidirect
59. C₂₃×#331	C₂²	semidirect
60. C₂₃×#331	C₂²	semidirect
61. C₂₃×#331	C₂²	semidirect
62. C₂₃×D₄⋊C₈	C₂²	semidirect
63. C₂₃×D₄⋊C₈	C₂²	semidirect
64. C₂₃×D₄⋊C₈	C₂²	semidirect
65. C₂₃×D₄⋊C₈	C₂²	semidirect
66. C₂₃×#322	C₂²	semidirect
205. D₈×C₂₃	D₈	semidirect
235. #19755	C₂₃	direct (N × H)
282. D₄×C₂₃	C₂×D₈	semidirect
283. D₄×C₂₃	AGL(1,ℤ/8)	semidirect
309. C₈×C₂₃	C₂×D₈	semidirect
310. #3294	C₂×C₂₃	semidirect
311. D₄×D₈	C₂×C₂₃	semidirect
312. D₄×D₈	C₂×C₂₃	semidirect
313. #3272	C₂×C₂₃	semidirect
314. #1721	C₂×C₂₃	semidirect
315. #1687	C₂×C₂₃	semidirect
316. #1681	C₂×C₂₃	semidirect
317. #1667	C₂×C₂₃	semidirect
318. #1665	C₂×C₂₃	semidirect
319. #1639	C₂×C₂₃	semidirect
320. #1625	C₂×C₂₃	semidirect
321. #1619	C₂×C₂₃	semidirect
322. #1619	C₂×C₂₃	semidirect
323. #1581	C₂×C₂₃	semidirect
324. #1580	C₂×C₂₃	semidirect
363. #547	C₂²×C₂₃	semidirect
365. D₄²	C₂²×C₂₃	semidirect
366. D₄²	C₂²×C₂₃	semidirect
367. #493	C₂²×C₂₃	semidirect
368. #493	C₂²×C₂₃	semidirect
369. #493	C₂²×C₂₃	semidirect
370. #492	C₂²×C₂₃	semidirect
371. #486	C₂²×C₂₃	semidirect
372. #486	C₂²×C₂₃	semidirect
374. #474	C₂²×C₂₃	semidirect
379. #461	C₂²×C₂₃	semidirect
382. #459	C₂²×C₂₃	semidirect
383. #459	C₂²×C₂₃	semidirect
384. #459	C₂²×C₂₃	semidirect
385. #459	C₂²×C₂₃	semidirect
395. #436	C₂²×C₂₃	semidirect
396. C₈×D₄	C₂²×C₂₃	semidirect
397. C₈×D₄	C₂²×C₂₃	semidirect
398. C₈×D₄	C₂²×C₂₃	semidirect
400. #335	C₂²×C₂₃	semidirect
401. #334	C₂²×C₂₃	semidirect
402. #332	C₂²×C₂₃	semidirect
403. #331	C₂²×C₂₃	semidirect
404. #331	C₂²×C₂₃	semidirect
405. #331	C₂²×C₂₃	semidirect
406. D₄⋊C₈	C₂²×C₂₃	semidirect
407. D₄⋊C₈	C₂²×C₂₃	semidirect
408. D₄⋊C₈	C₂²×C₂₃	semidirect
409. D₄⋊C₈	C₂²×C₂₃	semidirect
410. #322	C₂²×C₂₃	semidirect
501. C₂₃	#19755	direct (N × H)
560. D₈	D₈×C₂₃	semidirect
636. D₄	C₂×D₈×C₂₃	semidirect
637. D₄	C₂₃×AGL(1,ℤ/8)	semidirect
663. C₈	C₂×D₈×C₂₃	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂₃×#3294	2
D₄×D₈×C₂₃	2
C₂₃×#3272	2
C₂₃×#1721	2
C₂₃×#1687	2
C₂₃×#1681	2
C₂₃×#1667	2
C₂₃×#1665	2
C₂₃×#1639	2
C₂₃×#1625	2
C₂₃×#1619	2
C₂₃×#1581	2
C₂₃×#1580	2
#19755	23

Subgroups — C23×#19755 #19167381

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#19755 #19167381