Subgroups — C₂₃×#19756 #19167382

Order 5888.

808 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3358	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×SD₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄×D₈×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#3274	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1721	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1685	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1681	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1668	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1667	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1639	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1627	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#1621	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×SD₁₆⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₂₃×AGL(1,ℤ/8)	—	2	1	no	—
17	C₂²×SD₁₆×C₂₃	—	2	1	no	—
18	C₂²×D₈×C₂₃	—	2	1	no	—
19	C₂₃×#550	C₂²	1	1	yes	characteristic, complemented
20	D₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic, complemented
21	D₄²×C₂₃	C₂²	1	1	yes	characteristic
22	D₄²×C₂₃	C₂²	1	1	yes	characteristic
23	C₂₃×#493	C₂²	1	1	yes	characteristic
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	—	2	1	no	—
28	C₂₃×#486	C₂²	1	1	yes	characteristic, complemented
29	C₂₃×#486	C₂²	1	1	yes	characteristic, complemented
30	C₂₃×#474	—	2	1	no	—
31	C₂₃×#474	C₂²	1	1	yes	characteristic, complemented
32	C₂₃×#469	—	2	1	no	—
33	C₂₃×#466	—	2	1	no	—
34	C₂₃×#465	—	2	1	no	—
35	C₂₃×#461	C₂²	1	1	yes	characteristic
36	C₂₃×#460	C₂²	1	1	yes	characteristic, complemented
37	C₂₃×#460	C₂²	1	1	yes	characteristic, complemented
38	C₂₃×#459	—	2	1	no	—
39	C₂₃×#459	C₂²	1	1	yes	characteristic
40	C₂₃×#459	C₂²	1	1	yes	characteristic
41	C₂₃×#459	C₂²	1	1	yes	characteristic
42	C₂₃×#459	C₂²	1	1	yes	characteristic, complemented
43	C₂₃×#450	—	2	1	no	—
44	C₂₃×#449	—	2	1	no	—
45	C₂₃×#448	—	2	1	no	—
46	C₂₃×#447	—	2	1	no	—
47	C₂₃×#447	—	2	1	no	—
48	C₂₃×#447	—	2	1	no	—
49	C₈×Q₈×C₂₃	C₂²	1	1	yes	characteristic, complemented
50	C₄×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic
51	C₄×D₈×C₂₃	—	2	1	no	—
52	C₄×D₈×C₂₃	C₂²	1	1	yes	characteristic
53	C₂₃×#436	C₂²	1	1	yes	characteristic, complemented
54	C₈×D₄×C₂₃	C₂²	1	1	yes	characteristic
55	C₈×D₄×C₂₃	C₂²	1	1	yes	characteristic, complemented
56	C₂₃×#335	C₂²	1	1	yes	characteristic, complemented
57	C₂₃×#335	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×Q₈⋊C₈	C₂²	1	1	yes	characteristic, complemented
63	C₂₃×D₄⋊C₈	C₂²	1	1	yes	characteristic
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×Q₈×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	C₂₃×AGL(1,ℤ/8)	—	4	4	no	—
74	C₂×SD₁₆×C₂₃	—	2	1	no	—
75	C₂×SD₁₆×C₂₃	—	2	1	no	—
76	C₂×SD₁₆×C₂₃	—	2	1	no	—
77	C₂×SD₁₆×C₂₃	—	2	1	no	—
78	C₂×SD₁₆×C₂₃	—	2	1	no	—
79	C₂×SD₁₆×C₂₃	—	4	4	no	—
80	C₂×SD₁₆×C₂₃	D₄	1	1	yes	characteristic
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	—	2	1	no	—
89	C₂×D₈×C₂₃	—	4	1	no	—
90	C₂×D₈×C₂₃	—	4	4	no	—
91	C₂×D₈×C₂₃	D₄	1	1	yes	characteristic
92	C₂×M₁₆×C₂₃	—	2	1	no	—
93	C₂²×C₈×C₂₃	—	2	1	no	—
94	C₂²×C₈×C₂₃	—	2	1	no	—
95	C₂₃×C₄⋊Q₈	—	2	1	no	—
96	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
97	C₂₃×#127	—	2	1	no	—
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×C₂²⋊Q₈	—	2	1	no	—
102	C₂₃×C₂²⋊Q₈	—	4	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	—	4	1	no	—
109	C₂₃×#120	—	4	1	no	—
110	C₂₃×#120	—	4	1	no	—
111	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
112	C₄×D₄×C₂₃	—	2	1	no	—
113	C₄×D₄×C₂₃	—	2	1	no	—
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	C₂₃×#107	—	2	1	no	—
118	C₂₃×#107	D₄	1	1	yes	characteristic
119	C₂₃×#106	D₄	1	1	yes	characteristic
120	C₂₃×C₄⋊C₈	—	2	1	no	—
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×#103	D₄	1	1	yes	characteristic
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	—	2	1	no	—
136	C₂₃×#102	—	4	1	no	—
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	C₄×C₈×C₂₃	—	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	#41×C₂₃	—	4	2	no	—
150	#41×C₂₃	—	8	1	no	—
151	C₂×Q₈×C₂₃	—	2	1	no	—
152	C₂×Q₈×C₂₃	—	4	1	no	—
153	C₂×Q₈×C₂₃	—	4	2	no	—
154	C₂×Q₈×C₂₃	—	8	1	no	—
155	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
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	—	2	1	no	—
163	C₂×D₄×C₂₃	—	2	1	no	—
164	C₂×D₄×C₂₃	—	2	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	1	no	—
169	C₂×D₄×C₂₃	—	4	1	no	—
170	C₂×D₄×C₂₃	—	4	1	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	2	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	C₂×D₄	1	1	yes	characteristic
180	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
181	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
182	C₂²×C₄×C₂₃	—	2	1	no	—
183	C₂²×C₄×C₂₃	—	2	1	no	—
184	C₂²×C₄×C₂₃	—	2	1	no	—
185	C₂²×C₄×C₂₃	—	4	1	no	—
186	C₂²×C₄×C₂₃	—	4	1	no	—
187	SD₁₆×C₂₃	—	4	2	no	—
188	SD₁₆×C₂₃	—	4	2	no	—
189	SD₁₆×C₂₃	—	4	2	no	—
190	SD₁₆×C₂₃	—	4	2	no	—
191	SD₁₆×C₂₃	—	8	1	no	—
192	SD₁₆×C₂₃	D₈	1	4	yes	complemented
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	—	8	1	no	—
201	D₈×C₂₃	—	8	1	no	—
202	D₈×C₂₃	—	8	2	no	—
203	M₁₆×C₂₃	—	4	2	no	—
204	C₂×C₈×C₂₃	—	2	1	no	—
205	C₂×C₈×C₂₃	—	2	1	no	—
206	C₂×C₈×C₂₃	—	2	1	no	—
207	C₂×C₈×C₂₃	—	2	1	no	—
208	C₂×C₈×C₂₃	—	2	1	no	—
209	C₂×C₈×C₂₃	—	4	2	no	—
210	C₂×C₈×C₂₃	—	4	2	no	—
211	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
212	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
213	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
214	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
215	C₄⋊C₄×C₂₃	—	2	1	no	—
216	C₄⋊C₄×C₂₃	—	2	1	no	—
217	C₄⋊C₄×C₂₃	—	2	1	no	—
218	C₄⋊C₄×C₂₃	—	4	1	no	—
219	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
220	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
221	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
222	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
223	C₂²⋊C₄×C₂₃	—	2	1	no	—
224	C₂²⋊C₄×C₂₃	—	2	1	no	—
225	C₂²⋊C₄×C₂₃	—	2	1	no	—
226	C₂²⋊C₄×C₂₃	—	4	1	no	—
227	C₂²⋊C₄×C₂₃	—	4	1	no	—
228	C₂²⋊C₄×C₂₃	—	4	1	no	—
229	C₂²⋊C₄×C₂₃	—	4	1	no	—
230	C₄²×C₂₃	—	2	1	no	—
231	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
232	#19756	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
233	C₂³×C₂₃	—	2	1	no	—
234	C₂³×C₂₃	—	2	1	no	—
235	C₂³×C₂₃	—	2	1	no	—
236	C₂³×C₂₃	—	4	1	no	—
237	C₂³×C₂₃	—	4	1	no	—
238	C₂³×C₂₃	—	4	1	no	—
239	C₂³×C₂₃	—	8	2	no	—
240	C₂³×C₂₃	—	8	2	no	—
241	C₂³×C₂₃	—	8	2	no	—
242	C₂³×C₂₃	—	8	2	no	—
243	C₂³×C₂₃	—	16	1	no	—
244	C₂³×C₂₃	—	16	1	no	—
245	Q₈×C₂₃	—	2	1	no	—
246	Q₈×C₂₃	—	4	1	no	—
247	Q₈×C₂₃	—	4	1	no	—
248	Q₈×C₂₃	—	8	2	no	—
249	Q₈×C₂₃	C₂×D₈	1	2	yes	complemented
250	D₄×C₂₃	—	2	1	no	—
251	D₄×C₂₃	—	2	1	no	—
252	D₄×C₂₃	—	2	1	no	—
253	D₄×C₂₃	—	2	1	no	—
254	D₄×C₂₃	—	4	1	no	—
255	D₄×C₂₃	—	4	1	no	—
256	D₄×C₂₃	—	4	1	no	—
257	D₄×C₂₃	—	4	1	no	—
258	D₄×C₂₃	—	4	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	—	8	2	no	—
267	D₄×C₂₃	—	8	1	no	—
268	D₄×C₂₃	—	8	1	no	—
269	D₄×C₂₃	—	8	1	no	—
270	D₄×C₂₃	—	8	2	no	—
271	D₄×C₂₃	—	8	2	no	—
272	D₄×C₂₃	—	8	2	no	—
273	D₄×C₂₃	—	8	2	no	—
274	D₄×C₂₃	—	16	1	no	—
275	D₄×C₂₃	C₂×D₈	1	2	yes	—
276	D₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	complemented
277	C₂×C₄×C₂₃	—	2	1	no	—
278	C₂×C₄×C₂₃	—	2	1	no	—
279	C₂×C₄×C₂₃	—	2	1	no	—
280	C₂×C₄×C₂₃	—	2	1	no	—
281	C₂×C₄×C₂₃	—	2	1	no	—
282	C₂×C₄×C₂₃	—	4	1	no	—
283	C₂×C₄×C₂₃	—	4	1	no	—
284	C₂×C₄×C₂₃	—	4	1	no	—
285	C₂×C₄×C₂₃	—	4	1	no	—
286	C₂×C₄×C₂₃	—	4	1	no	—
287	C₂×C₄×C₂₃	—	4	1	no	—
288	C₂×C₄×C₂₃	—	4	1	no	—
289	C₂×C₄×C₂₃	—	8	2	no	—
290	C₂×C₄×C₂₃	—	8	2	no	—
291	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
292	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
293	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
294	C₈×C₂₃	—	2	1	no	—
295	C₈×C₂₃	—	2	1	no	—
296	C₈×C₂₃	—	2	1	no	—
297	C₈×C₂₃	—	4	1	no	—
298	C₈×C₂₃	—	4	1	no	—
299	C₈×C₂₃	—	4	1	no	—
300	C₈×C₂₃	—	4	1	no	—
301	C₈×C₂₃	—	4	1	no	—
302	C₈×C₂₃	C₂×D₈	1	2	yes	complemented
303	#3358	C₂×C₂₃	1	1	yes	characteristic, complemented
304	D₄×SD₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
305	D₄×D₈	C₂×C₂₃	1	1	yes	characteristic, complemented
306	#3274	C₂×C₂₃	1	1	yes	characteristic, complemented
307	#1721	C₂×C₂₃	1	1	yes	characteristic, complemented
308	#1685	C₂×C₂₃	1	1	yes	characteristic, complemented
309	#1681	C₂×C₂₃	1	1	yes	characteristic, complemented
310	#1668	C₂×C₂₃	1	1	yes	characteristic, complemented
311	#1667	C₂×C₂₃	1	1	yes	characteristic, complemented
312	#1639	C₂×C₂₃	1	1	yes	characteristic, complemented
313	#1627	C₂×C₂₃	1	1	yes	characteristic, complemented
314	#1621	C₂×C₂₃	1	1	yes	characteristic, complemented
315	#1619	C₂×C₂₃	1	1	yes	characteristic, complemented
316	#1581	C₂×C₂₃	1	1	yes	characteristic, complemented
317	SD₁₆⋊C₈	C₂×C₂₃	1	1	yes	characteristic, complemented
318	C₂²×C₂₃	—	2	2	no	—
319	C₂²×C₂₃	—	2	2	no	—
320	C₂²×C₂₃	—	4	1	no	—
321	C₂²×C₂₃	—	4	1	no	—
322	C₂²×C₂₃	—	4	1	no	—
323	C₂²×C₂₃	—	4	1	no	—
324	C₂²×C₂₃	—	4	1	no	—
325	C₂²×C₂₃	—	4	1	no	—
326	C₂²×C₂₃	—	4	1	no	—
327	C₂²×C₂₃	—	8	1	no	—
328	C₂²×C₂₃	—	8	1	no	—
329	C₂²×C₂₃	—	8	1	no	—
330	C₂²×C₂₃	—	8	1	no	—
331	C₂²×C₂₃	—	8	1	no	—
332	C₂²×C₂₃	—	8	1	no	—
333	C₂²×C₂₃	—	8	1	no	—
334	C₂²×C₂₃	—	8	1	no	—
335	C₂²×C₂₃	—	8	1	no	—
336	C₂²×C₂₃	—	16	4	no	—
337	C₂²×C₂₃	—	16	4	no	—
338	C₂²×C₂₃	D₄²	1	1	yes	characteristic
339	C₄×C₂₃	—	2	2	no	—
340	C₄×C₂₃	—	2	1	no	—
341	C₄×C₂₃	—	4	1	no	—
342	C₄×C₂₃	—	4	1	no	—
343	C₄×C₂₃	—	4	1	no	—
344	C₄×C₂₃	—	4	1	no	—
345	C₄×C₂₃	—	8	1	no	—
346	C₄×C₂₃	C₂²×D₈	1	1	yes	characteristic
347	C₄×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
348	C₄×C₂₃	#576	1	1	yes	characteristic
349	C₄×C₂₃	#577	1	1	yes	characteristic
350	C₂×AGL(1,ℤ/8)	—	2	1	no	—
351	C₂²×SD₁₆	—	2	1	no	—
352	C₂²×D₈	—	2	1	no	—
353	#550	C₂²×C₂₃	1	1	yes	characteristic, complemented
354	D₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
355	D₄²	C₂²×C₂₃	1	1	yes	characteristic
356	D₄²	C₂²×C₂₃	1	1	yes	characteristic
357	#493	C₂²×C₂₃	1	1	yes	characteristic
358	#493	C₂²×C₂₃	1	1	yes	characteristic, complemented
359	#493	C₂²×C₂₃	1	1	yes	characteristic, complemented
360	#492	C₂²×C₂₃	1	1	yes	characteristic, complemented
361	#486	—	2	1	no	—
362	#486	C₂²×C₂₃	1	1	yes	characteristic, complemented
363	#486	C₂²×C₂₃	1	1	yes	characteristic, complemented
364	#474	—	2	1	no	—
365	#474	C₂²×C₂₃	1	1	yes	characteristic, complemented
366	#469	—	2	1	no	—
367	#466	—	2	1	no	—
368	#465	—	2	1	no	—
369	#461	C₂²×C₂₃	1	1	yes	characteristic
370	#460	C₂²×C₂₃	1	1	yes	characteristic, complemented
371	#460	C₂²×C₂₃	1	1	yes	characteristic, complemented
372	#459	—	2	1	no	—
373	#459	C₂²×C₂₃	1	1	yes	characteristic
374	#459	C₂²×C₂₃	1	1	yes	characteristic
375	#459	C₂²×C₂₃	1	1	yes	characteristic
376	#459	C₂²×C₂₃	1	1	yes	characteristic, complemented
377	#450	—	2	1	no	—
378	#449	—	2	1	no	—
379	#448	—	2	1	no	—
380	#447	—	2	1	no	—
381	#447	—	2	1	no	—
382	#447	—	2	1	no	—
383	C₈×Q₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
384	C₄×SD₁₆	C₂²×C₂₃	1	1	yes	characteristic
385	C₄×D₈	—	2	1	no	—
386	C₄×D₈	C₂²×C₂₃	1	1	yes	characteristic
387	#436	C₂²×C₂₃	1	1	yes	characteristic, complemented
388	C₈×D₄	C₂²×C₂₃	1	1	yes	characteristic
389	C₈×D₄	C₂²×C₂₃	1	1	yes	characteristic, complemented
390	#335	C₂²×C₂₃	1	1	yes	characteristic, complemented
391	#335	C₂²×C₂₃	1	1	yes	characteristic, complemented
392	#332	C₂²×C₂₃	1	1	yes	characteristic, complemented
393	#331	C₂²×C₂₃	1	1	yes	characteristic, complemented
394	#331	C₂²×C₂₃	1	1	yes	characteristic, complemented
395	#331	C₂²×C₂₃	1	1	yes	characteristic, complemented
396	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
397	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
398	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
399	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
400	#322	C₂²×C₂₃	1	1	yes	characteristic, complemented
401	C₂×C₂₃	—	4	2	no	—
402	C₂×C₂₃	—	4	2	no	—
403	C₂×C₂₃	—	8	1	no	—
404	C₂×C₂₃	—	16	1	no	—
405	C₂×C₂₃	—	16	1	no	—
406	C₂×C₂₃	—	16	1	no	—
407	C₂×C₂₃	#3265	1	1	yes	characteristic
408	C₂×C₂₃	D₄×D₈	1	1	yes	characteristic
409	C₂×C₂₃	#3288	1	1	yes	characteristic
410	C₂×#41	—	2	1	no	—
411	C₂²×Q₈	—	2	1	no	—
412	C₂²×D₄	—	2	1	no	—
413	C₂²×D₄	—	2	1	no	—
414	C₂²×D₄	—	2	1	no	—
415	C₂²×D₄	—	2	1	no	—
416	AGL(1,ℤ/8)	—	4	4	no	—
417	C₂×SD₁₆	—	2	1	no	—
418	C₂×SD₁₆	—	2	1	no	—
419	C₂×SD₁₆	—	2	1	no	—
420	C₂×SD₁₆	—	2	1	no	—
421	C₂×SD₁₆	—	2	1	no	—
422	C₂×SD₁₆	—	4	4	no	—
423	C₂×SD₁₆	D₄×C₂₃	1	1	yes	characteristic
424	C₂×D₈	—	2	1	no	—
425	C₂×D₈	—	2	1	no	—
426	C₂×D₈	—	2	1	no	—
427	C₂×D₈	—	2	1	no	—
428	C₂×D₈	—	2	1	no	—
429	C₂×D₈	—	2	1	no	—
430	C₂×D₈	—	2	1	no	—
431	C₂×D₈	—	2	1	no	—
432	C₂×D₈	—	4	1	no	—
433	C₂×D₈	—	4	4	no	—
434	C₂×D₈	D₄×C₂₃	1	1	yes	characteristic
435	C₂×M₁₆	—	2	1	no	—
436	C₂²×C₈	—	2	1	no	—
437	C₂²×C₈	—	2	1	no	—
438	C₄⋊Q₈	—	2	1	no	—
439	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
440	#127	—	2	1	no	—
441	#127	C₂³×C₂₃	1	1	yes	characteristic
442	#127	C₂³×C₂₃	1	1	yes	characteristic
443	#127	C₂³×C₂₃	1	1	yes	characteristic
444	C₂²⋊Q₈	—	2	1	no	—
445	C₂²⋊Q₈	—	4	1	no	—
446	#121	—	2	1	no	—
447	#121	—	2	1	no	—
448	#121	—	2	1	no	—
449	#121	—	2	1	no	—
450	#121	—	2	1	no	—
451	#121	—	4	1	no	—
452	#120	—	4	1	no	—
453	#120	—	4	1	no	—
454	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
455	C₄×D₄	—	2	1	no	—
456	C₄×D₄	—	2	1	no	—
457	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
458	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
459	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
460	#107	—	2	1	no	—
461	#107	D₄×C₂₃	1	1	yes	characteristic
462	#106	D₄×C₂₃	1	1	yes	characteristic
463	C₄⋊C₈	—	2	1	no	—
464	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
465	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
466	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
467	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
468	#103	—	2	1	no	—
469	#103	D₄×C₂₃	1	1	yes	characteristic
470	#102	—	2	1	no	—
471	#102	—	2	1	no	—
472	#102	—	2	1	no	—
473	#102	—	2	1	no	—
474	#102	—	2	1	no	—
475	#102	—	2	1	no	—
476	#102	—	2	1	no	—
477	#102	—	2	1	no	—
478	#102	—	2	1	no	—
479	#102	—	4	1	no	—
480	#102	D₄×C₂₃	1	1	yes	characteristic
481	#102	D₄×C₂₃	1	1	yes	characteristic
482	#102	D₄×C₂₃	1	1	yes	characteristic
483	C₂²⋊C₈	—	2	1	no	—
484	C₂²⋊C₈	—	2	1	no	—
485	C₂²⋊C₈	—	2	1	no	—
486	C₄×C₈	—	2	1	no	—
487	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
488	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
489	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
490	C₂₃	#19756	1	1	yes	characteristic, direct product factor, Sylow
491	C₂⁴	—	4	1	no	—
492	C₂⁴	—	4	1	no	—
493	#41	—	4	2	no	—
494	#41	—	8	1	no	—
495	C₂×Q₈	—	2	1	no	—
496	C₂×Q₈	—	4	1	no	—
497	C₂×Q₈	—	4	2	no	—
498	C₂×Q₈	—	8	1	no	—
499	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
500	C₂×D₄	—	2	1	no	—
501	C₂×D₄	—	2	1	no	—
502	C₂×D₄	—	2	1	no	—
503	C₂×D₄	—	2	1	no	—
504	C₂×D₄	—	2	1	no	—
505	C₂×D₄	—	2	1	no	—
506	C₂×D₄	—	2	1	no	—
507	C₂×D₄	—	2	1	no	—
508	C₂×D₄	—	2	1	no	—
509	C₂×D₄	—	4	1	no	—
510	C₂×D₄	—	4	1	no	—
511	C₂×D₄	—	4	1	no	—
512	C₂×D₄	—	4	1	no	—
513	C₂×D₄	—	4	1	no	—
514	C₂×D₄	—	4	1	no	—
515	C₂×D₄	—	4	2	no	—
516	C₂×D₄	—	4	2	no	—
517	C₂×D₄	—	4	2	no	—
518	C₂×D₄	—	4	2	no	—
519	C₂×D₄	—	8	1	no	—
520	C₂×D₄	—	8	1	no	—
521	C₂×D₄	—	8	1	no	—
522	C₂×D₄	—	8	1	no	—
523	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
524	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
525	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
526	C₂²×C₄	—	2	1	no	—
527	C₂²×C₄	—	2	1	no	—
528	C₂²×C₄	—	2	1	no	—
529	C₂²×C₄	—	4	1	no	—
530	C₂²×C₄	—	4	1	no	—
531	SD₁₆	—	4	2	no	—
532	SD₁₆	—	4	2	no	—
533	SD₁₆	—	4	2	no	—
534	SD₁₆	—	4	2	no	—
535	SD₁₆	—	8	1	no	—
536	SD₁₆	D₈×C₂₃	1	4	yes	complemented
537	D₈	—	2	2	no	—
538	D₈	—	4	2	no	—
539	D₈	—	4	2	no	—
540	D₈	—	4	2	no	—
541	D₈	—	4	2	no	—
542	D₈	—	4	2	no	—
543	D₈	—	4	2	no	—
544	D₈	—	8	1	no	—
545	D₈	—	8	1	no	—
546	D₈	—	8	2	no	—
547	M₁₆	—	4	2	no	—
548	C₂×C₈	—	2	1	no	—
549	C₂×C₈	—	2	1	no	—
550	C₂×C₈	—	2	1	no	—
551	C₂×C₈	—	2	1	no	—
552	C₂×C₈	—	2	1	no	—
553	C₂×C₈	—	4	2	no	—
554	C₂×C₈	—	4	2	no	—
555	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
556	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
557	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
558	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
559	C₄⋊C₄	—	2	1	no	—
560	C₄⋊C₄	—	2	1	no	—
561	C₄⋊C₄	—	2	1	no	—
562	C₄⋊C₄	—	4	1	no	—
563	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
564	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
565	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
566	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
567	C₂²⋊C₄	—	2	1	no	—
568	C₂²⋊C₄	—	2	1	no	—
569	C₂²⋊C₄	—	2	1	no	—
570	C₂²⋊C₄	—	4	1	no	—
571	C₂²⋊C₄	—	4	1	no	—
572	C₂²⋊C₄	—	4	1	no	—
573	C₂²⋊C₄	—	4	1	no	—
574	C₄²	—	2	1	no	—
575	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
576	C₂³	—	2	1	no	—
577	C₂³	—	2	1	no	—
578	C₂³	—	2	1	no	—
579	C₂³	—	4	1	no	—
580	C₂³	—	4	1	no	—
581	C₂³	—	4	1	no	—
582	C₂³	—	8	2	no	—
583	C₂³	—	8	2	no	—
584	C₂³	—	8	2	no	—
585	C₂³	—	8	2	no	—
586	C₂³	—	16	1	no	—
587	C₂³	—	16	1	no	—
588	Q₈	—	2	1	no	—
589	Q₈	—	4	1	no	—
590	Q₈	—	4	1	no	—
591	Q₈	—	8	2	no	—
592	Q₈	C₂×D₈×C₂₃	1	2	yes	complemented
593	D₄	—	2	1	no	—
594	D₄	—	2	1	no	—
595	D₄	—	2	1	no	—
596	D₄	—	2	1	no	—
597	D₄	—	4	1	no	—
598	D₄	—	4	1	no	—
599	D₄	—	4	1	no	—
600	D₄	—	4	1	no	—
601	D₄	—	4	1	no	—
602	D₄	—	4	1	no	—
603	D₄	—	4	1	no	—
604	D₄	—	4	1	no	—
605	D₄	—	4	1	no	—
606	D₄	—	4	1	no	—
607	D₄	—	4	1	no	—
608	D₄	—	4	1	no	—
609	D₄	—	8	2	no	—
610	D₄	—	8	1	no	—
611	D₄	—	8	1	no	—
612	D₄	—	8	1	no	—
613	D₄	—	8	2	no	—
614	D₄	—	8	2	no	—
615	D₄	—	8	2	no	—
616	D₄	—	8	2	no	—
617	D₄	—	16	1	no	—
618	D₄	C₂×D₈×C₂₃	1	2	yes	—
619	D₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	complemented
620	C₂×C₄	—	2	1	no	—
621	C₂×C₄	—	2	1	no	—
622	C₂×C₄	—	2	1	no	—
623	C₂×C₄	—	2	1	no	—
624	C₂×C₄	—	2	1	no	—
625	C₂×C₄	—	4	1	no	—
626	C₂×C₄	—	4	1	no	—
627	C₂×C₄	—	4	1	no	—
628	C₂×C₄	—	4	1	no	—
629	C₂×C₄	—	4	1	no	—
630	C₂×C₄	—	4	1	no	—
631	C₂×C₄	—	4	1	no	—
632	C₂×C₄	—	8	2	no	—
633	C₂×C₄	—	8	2	no	—
634	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
635	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
636	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
637	C₈	—	2	1	no	—
638	C₈	—	2	1	no	—
639	C₈	—	2	1	no	—
640	C₈	—	4	1	no	—
641	C₈	—	4	1	no	—
642	C₈	—	4	1	no	—
643	C₈	—	4	1	no	—
644	C₈	—	4	1	no	—
645	C₈	C₂×D₈×C₂₃	1	2	yes	complemented
646	C₂²	—	2	2	no	—
647	C₂²	—	2	2	no	—
648	C₂²	—	4	1	no	—
649	C₂²	—	4	1	no	—
650	C₂²	—	4	1	no	—
651	C₂²	—	4	1	no	—
652	C₂²	—	4	1	no	—
653	C₂²	—	4	1	no	—
654	C₂²	—	4	1	no	—
655	C₂²	—	8	1	no	—
656	C₂²	—	8	1	no	—
657	C₂²	—	8	1	no	—
658	C₂²	—	8	1	no	—
659	C₂²	—	8	1	no	—
660	C₂²	—	8	1	no	—
661	C₂²	—	8	1	no	—
662	C₂²	—	8	1	no	—
663	C₂²	—	8	1	no	—
664	C₂²	—	16	4	no	—
665	C₂²	—	16	4	no	—
666	C₂²	D₄²×C₂₃	1	1	yes	characteristic
667	C₄	—	2	2	no	—
668	C₄	—	2	1	no	—
669	C₄	—	4	1	no	—
670	C₄	—	4	1	no	—
671	C₄	—	4	1	no	—
672	C₄	—	4	1	no	—
673	C₄	—	8	1	no	—
674	C₄	C₂²×D₈×C₂₃	1	1	yes	characteristic
675	C₄	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
676	C₄	C₂₃×#576	1	1	yes	characteristic
677	C₄	C₂₃×#577	1	1	yes	characteristic
678	C₂	—	4	2	no	—
679	C₂	—	4	2	no	—
680	C₂	—	8	1	no	—
681	C₂	—	16	1	no	—
682	C₂	—	16	1	no	—
683	C₂	—	16	1	no	—
684	C₂	C₂₃×#3265	1	1	yes	characteristic
685	C₂	D₄×D₈×C₂₃	1	1	yes	characteristic
686	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×#3358	C₂	semidirect
2. D₄×SD₁₆×C₂₃	C₂	semidirect
3. D₄×D₈×C₂₃	C₂	semidirect
4. C₂₃×#3274	C₂	semidirect
5. C₂₃×#1721	C₂	semidirect
6. C₂₃×#1685	C₂	semidirect
7. C₂₃×#1681	C₂	semidirect
8. C₂₃×#1668	C₂	semidirect
9. C₂₃×#1667	C₂	semidirect
10. C₂₃×#1639	C₂	semidirect
11. C₂₃×#1627	C₂	semidirect
12. C₂₃×#1621	C₂	semidirect
13. C₂₃×#1619	C₂	semidirect
14. C₂₃×#1581	C₂	semidirect
15. C₂₃×SD₁₆⋊C₈	C₂	semidirect
19. C₂₃×#550	C₂²	semidirect
20. D₄×Q₈×C₂₃	C₂²	semidirect
24. C₂₃×#493	C₂²	semidirect
25. C₂₃×#493	C₂²	semidirect
26. C₂₃×#492	C₂²	semidirect
28. C₂₃×#486	C₂²	semidirect
29. C₂₃×#486	C₂²	semidirect
31. C₂₃×#474	C₂²	semidirect
36. C₂₃×#460	C₂²	semidirect
37. C₂₃×#460	C₂²	semidirect
42. C₂₃×#459	C₂²	semidirect
49. C₈×Q₈×C₂₃	C₂²	semidirect
53. C₂₃×#436	C₂²	semidirect
55. C₈×D₄×C₂₃	C₂²	semidirect
56. C₂₃×#335	C₂²	semidirect
57. C₂₃×#335	C₂²	semidirect
58. C₂₃×#332	C₂²	semidirect
59. C₂₃×#331	C₂²	semidirect
60. C₂₃×#331	C₂²	semidirect
61. C₂₃×#331	C₂²	semidirect
62. C₂₃×Q₈⋊C₈	C₂²	semidirect
64. C₂₃×D₄⋊C₈	C₂²	semidirect
65. C₂₃×D₄⋊C₈	C₂²	semidirect
66. C₂₃×#322	C₂²	semidirect
192. SD₁₆×C₂₃	D₈	semidirect
232. #19756	C₂₃	direct (N × H)
249. Q₈×C₂₃	C₂×D₈	semidirect
276. D₄×C₂₃	AGL(1,ℤ/8)	semidirect
302. C₈×C₂₃	C₂×D₈	semidirect
303. #3358	C₂×C₂₃	semidirect
304. D₄×SD₁₆	C₂×C₂₃	semidirect
305. D₄×D₈	C₂×C₂₃	semidirect
306. #3274	C₂×C₂₃	semidirect
307. #1721	C₂×C₂₃	semidirect
308. #1685	C₂×C₂₃	semidirect
309. #1681	C₂×C₂₃	semidirect
310. #1668	C₂×C₂₃	semidirect
311. #1667	C₂×C₂₃	semidirect
312. #1639	C₂×C₂₃	semidirect
313. #1627	C₂×C₂₃	semidirect
314. #1621	C₂×C₂₃	semidirect
315. #1619	C₂×C₂₃	semidirect
316. #1581	C₂×C₂₃	semidirect
317. SD₁₆⋊C₈	C₂×C₂₃	semidirect
353. #550	C₂²×C₂₃	semidirect
354. D₄×Q₈	C₂²×C₂₃	semidirect
358. #493	C₂²×C₂₃	semidirect
359. #493	C₂²×C₂₃	semidirect
360. #492	C₂²×C₂₃	semidirect
362. #486	C₂²×C₂₃	semidirect
363. #486	C₂²×C₂₃	semidirect
365. #474	C₂²×C₂₃	semidirect
370. #460	C₂²×C₂₃	semidirect
371. #460	C₂²×C₂₃	semidirect
376. #459	C₂²×C₂₃	semidirect
383. C₈×Q₈	C₂²×C₂₃	semidirect
387. #436	C₂²×C₂₃	semidirect
389. C₈×D₄	C₂²×C₂₃	semidirect
390. #335	C₂²×C₂₃	semidirect
391. #335	C₂²×C₂₃	semidirect
392. #332	C₂²×C₂₃	semidirect
393. #331	C₂²×C₂₃	semidirect
394. #331	C₂²×C₂₃	semidirect
395. #331	C₂²×C₂₃	semidirect
396. Q₈⋊C₈	C₂²×C₂₃	semidirect
398. D₄⋊C₈	C₂²×C₂₃	semidirect
399. D₄⋊C₈	C₂²×C₂₃	semidirect
400. #322	C₂²×C₂₃	semidirect
490. C₂₃	#19756	direct (N × H)
536. SD₁₆	D₈×C₂₃	semidirect
592. Q₈	C₂×D₈×C₂₃	semidirect
619. D₄	C₂₃×AGL(1,ℤ/8)	semidirect
645. 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×#3358	2
D₄×SD₁₆×C₂₃	2
D₄×D₈×C₂₃	2
C₂₃×#3274	2
C₂₃×#1721	2
C₂₃×#1685	2
C₂₃×#1681	2
C₂₃×#1668	2
C₂₃×#1667	2
C₂₃×#1639	2
C₂₃×#1627	2
C₂₃×#1621	2
C₂₃×#1619	2
C₂₃×#1581	2
C₂₃×SD₁₆⋊C₈	2
#19756	23

Subgroups — C23×#19756 #19167382

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#19756 #19167382