Subgroups — C₂₃×#19763 #19167389

Order 5888.

756 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3384	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#3295	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄×SD₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
4	D₄×D₈×C₂₃	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#1715	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1688	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1677	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1671	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×#1629	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#1627	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂₃×#1622	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂₃×#1582	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂₃×#1580	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×SD₁₆×C₂₃	—	2	1	no	—
17	C₂²×D₈×C₂₃	—	2	1	no	—
18	C₂₃×#554	C₂²	1	1	yes	characteristic, complemented
19	D₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic, complemented
20	D₄²×C₂₃	C₂²	1	1	yes	characteristic
21	D₄²×C₂₃	C₂²	1	1	yes	characteristic
22	C₂₃×#501	—	2	1	no	—
23	C₂₃×#498	C₂²	1	1	yes	characteristic, complemented
24	C₂₃×#493	C₂²	1	1	yes	characteristic
25	C₂₃×#492	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×#481	—	2	1	no	—
30	C₂₃×#478	—	2	1	no	—
31	C₂₃×#476	—	2	1	no	—
32	C₂₃×#476	C₂²	1	1	yes	characteristic
33	C₂₃×#476	C₂²	1	1	yes	characteristic, complemented
34	C₂₃×#474	C₂²	1	1	yes	characteristic, complemented
35	C₂₃×#466	—	2	1	no	—
36	C₂₃×#465	—	2	1	no	—
37	C₂₃×#465	—	2	1	no	—
38	C₂₃×#461	C₂²	1	1	yes	characteristic
39	C₂₃×#460	C₂²	1	1	yes	characteristic, complemented
40	C₂₃×#460	C₂²	1	1	yes	characteristic, complemented
41	C₂₃×#459	C₂²	1	1	yes	characteristic
42	C₂₃×#459	C₂²	1	1	yes	characteristic
43	C₂₃×#450	—	2	1	no	—
44	C₂₃×#450	—	2	1	no	—
45	C₂₃×#448	—	2	1	no	—
46	C₂₃×#447	—	2	1	no	—
47	C₂₃×#447	—	2	1	no	—
48	C₂₃×#446	C₂²	1	1	yes	characteristic, complemented
49	C₂₃×#442	—	2	1	no	—
50	C₄×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic
51	C₄×D₈×C₂₃	C₂²	1	1	yes	characteristic
52	C₈×D₄×C₂₃	C₂²	1	1	yes	characteristic
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×#334	C₂²	1	1	yes	characteristic, complemented
57	C₂₃×#334	C₂²	1	1	yes	characteristic, complemented
58	C₂₃×#333	C₂²	1	1	yes	characteristic, complemented
59	C₂₃×#332	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	C₂²×Q₈×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	C₂×SD₁₆×C₂₃	—	2	1	no	—
73	C₂×SD₁₆×C₂₃	—	2	1	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	—	4	1	no	—
78	C₂×SD₁₆×C₂₃	—	4	4	no	—
79	C₂×SD₁₆×C₂₃	D₄	1	1	yes	characteristic
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	—	4	4	no	—
85	C₂×D₈×C₂₃	D₄	1	1	yes	characteristic
86	C₂²×C₈×C₂₃	—	2	1	no	—
87	C₂²×C₈×C₂₃	—	2	1	no	—
88	C₂²×C₈×C₂₃	—	2	1	no	—
89	C₂₃×C₄⋊Q₈	—	2	1	no	—
90	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
91	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
92	C₂₃×#127	C₂³	1	1	yes	characteristic
93	C₂₃×#127	C₂³	1	1	yes	characteristic
94	C₂₃×#125	—	2	1	no	—
95	C₂₃×C₂²⋊Q₈	—	2	1	no	—
96	C₂₃×C₂²⋊Q₈	—	2	1	no	—
97	C₂₃×C₂²⋊Q₈	—	4	1	no	—
98	C₂₃×C₂²⋊Q₈	—	4	1	no	—
99	C₂₃×#121	—	2	1	no	—
100	C₂₃×#121	—	2	1	no	—
101	C₂₃×#121	—	2	1	no	—
102	C₂₃×#121	—	2	1	no	—
103	C₂₃×#120	—	4	1	no	—
104	C₂₃×#120	—	4	1	no	—
105	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
106	C₄×D₄×C₂₃	—	2	1	no	—
107	C₄×D₄×C₂₃	—	2	1	no	—
108	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
109	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
110	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
111	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
112	C₂₃×#107	—	2	1	no	—
113	C₂₃×#107	D₄	1	1	yes	characteristic
114	C₂₃×#106	—	2	1	no	—
115	C₂₃×#106	—	2	1	no	—
116	C₂₃×#106	—	2	1	no	—
117	C₂₃×#106	—	2	1	no	—
118	C₂₃×#106	D₄	1	1	yes	characteristic
119	C₂₃×C₄⋊C₈	—	2	1	no	—
120	C₂₃×C₄⋊C₈	C₂³	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×#103	—	2	1	no	—
125	C₂₃×#103	D₄	1	1	yes	characteristic
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×C₂²⋊C₈	—	2	1	no	—
140	C₂₃×C₂²⋊C₈	—	2	1	no	—
141	C₂₃×C₂²⋊C₈	—	2	1	no	—
142	C₂₃×#97	—	2	1	no	—
143	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
144	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
145	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
146	C₂⁴×C₂₃	—	4	1	no	—
147	C₂⁴×C₂₃	—	4	1	no	—
148	C₂×Q₈×C₂₃	—	2	1	no	—
149	C₂×Q₈×C₂₃	—	2	1	no	—
150	C₂×Q₈×C₂₃	—	4	1	no	—
151	C₂×Q₈×C₂₃	—	4	2	no	—
152	C₂×Q₈×C₂₃	—	8	1	no	—
153	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
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	—	4	1	no	—
161	C₂×D₄×C₂₃	—	4	1	no	—
162	C₂×D₄×C₂₃	—	4	1	no	—
163	C₂×D₄×C₂₃	—	4	1	no	—
164	C₂×D₄×C₂₃	—	4	2	no	—
165	C₂×D₄×C₂₃	—	4	2	no	—
166	C₂×D₄×C₂₃	—	4	2	no	—
167	C₂×D₄×C₂₃	—	4	2	no	—
168	C₂×D₄×C₂₃	—	8	1	no	—
169	C₂×D₄×C₂₃	—	8	1	no	—
170	C₂×D₄×C₂₃	—	8	1	no	—
171	C₂×D₄×C₂₃	—	8	1	no	—
172	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
173	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
174	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
175	C₂²×C₄×C₂₃	—	2	1	no	—
176	C₂²×C₄×C₂₃	—	2	1	no	—
177	C₂²×C₄×C₂₃	—	2	1	no	—
178	C₂²×C₄×C₂₃	—	4	1	no	—
179	C₂²×C₄×C₂₃	—	4	1	no	—
180	SD₁₆×C₂₃	—	2	2	no	—
181	SD₁₆×C₂₃	—	4	2	no	—
182	SD₁₆×C₂₃	—	4	2	no	—
183	SD₁₆×C₂₃	—	4	2	no	—
184	SD₁₆×C₂₃	—	8	1	no	—
185	SD₁₆×C₂₃	—	8	1	no	—
186	SD₁₆×C₂₃	—	8	2	no	—
187	D₈×C₂₃	—	4	2	no	—
188	D₈×C₂₃	—	4	2	no	—
189	D₈×C₂₃	—	4	2	no	—
190	D₈×C₂₃	—	8	1	no	—
191	D₈×C₂₃	SD₁₆	1	4	yes	complemented
192	C₂×C₈×C₂₃	—	2	1	no	—
193	C₂×C₈×C₂₃	—	2	1	no	—
194	C₂×C₈×C₂₃	—	2	1	no	—
195	C₂×C₈×C₂₃	—	2	1	no	—
196	C₂×C₈×C₂₃	—	2	1	no	—
197	C₂×C₈×C₂₃	—	4	2	no	—
198	C₂×C₈×C₂₃	—	4	2	no	—
199	C₂×C₈×C₂₃	—	8	1	no	—
200	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
201	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
202	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
203	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
204	C₄⋊C₄×C₂₃	—	2	1	no	—
205	C₄⋊C₄×C₂₃	—	2	1	no	—
206	C₄⋊C₄×C₂₃	—	2	1	no	—
207	C₄⋊C₄×C₂₃	—	2	1	no	—
208	C₄⋊C₄×C₂₃	—	2	1	no	—
209	C₄⋊C₄×C₂₃	—	4	1	no	—
210	C₄⋊C₄×C₂₃	—	4	1	no	—
211	C₄⋊C₄×C₂₃	—	4	1	no	—
212	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
213	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
214	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
215	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
216	C₂²⋊C₄×C₂₃	—	2	1	no	—
217	C₂²⋊C₄×C₂₃	—	2	1	no	—
218	C₂²⋊C₄×C₂₃	—	2	1	no	—
219	C₂²⋊C₄×C₂₃	—	4	1	no	—
220	C₂²⋊C₄×C₂₃	—	4	1	no	—
221	C₂²⋊C₄×C₂₃	—	4	1	no	—
222	C₂²⋊C₄×C₂₃	—	4	1	no	—
223	C₄²×C₂₃	—	2	1	no	—
224	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
225	#19763	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
226	C₂³×C₂₃	—	2	1	no	—
227	C₂³×C₂₃	—	2	1	no	—
228	C₂³×C₂₃	—	2	1	no	—
229	C₂³×C₂₃	—	4	1	no	—
230	C₂³×C₂₃	—	4	1	no	—
231	C₂³×C₂₃	—	8	2	no	—
232	C₂³×C₂₃	—	8	2	no	—
233	C₂³×C₂₃	—	8	2	no	—
234	C₂³×C₂₃	—	8	2	no	—
235	C₂³×C₂₃	—	16	1	no	—
236	C₂³×C₂₃	—	16	1	no	—
237	Q₈×C₂₃	—	2	1	no	—
238	Q₈×C₂₃	—	2	1	no	—
239	Q₈×C₂₃	—	4	1	no	—
240	Q₈×C₂₃	—	4	1	no	—
241	Q₈×C₂₃	—	8	1	no	—
242	Q₈×C₂₃	—	8	2	no	—
243	D₄×C₂₃	—	2	1	no	—
244	D₄×C₂₃	—	2	1	no	—
245	D₄×C₂₃	—	2	1	no	—
246	D₄×C₂₃	—	4	1	no	—
247	D₄×C₂₃	—	4	1	no	—
248	D₄×C₂₃	—	4	1	no	—
249	D₄×C₂₃	—	4	1	no	—
250	D₄×C₂₃	—	4	1	no	—
251	D₄×C₂₃	—	4	1	no	—
252	D₄×C₂₃	—	4	1	no	—
253	D₄×C₂₃	—	4	1	no	—
254	D₄×C₂₃	—	8	1	no	—
255	D₄×C₂₃	—	8	1	no	—
256	D₄×C₂₃	—	8	2	no	—
257	D₄×C₂₃	—	8	2	no	—
258	D₄×C₂₃	—	8	2	no	—
259	D₄×C₂₃	—	8	2	no	—
260	D₄×C₂₃	C₂×SD₁₆	1	2	yes	—
261	D₄×C₂₃	C₂×SD₁₆	1	2	yes	complemented
262	D₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
263	C₂×C₄×C₂₃	—	2	1	no	—
264	C₂×C₄×C₂₃	—	2	1	no	—
265	C₂×C₄×C₂₃	—	2	1	no	—
266	C₂×C₄×C₂₃	—	2	1	no	—
267	C₂×C₄×C₂₃	—	2	1	no	—
268	C₂×C₄×C₂₃	—	4	1	no	—
269	C₂×C₄×C₂₃	—	4	1	no	—
270	C₂×C₄×C₂₃	—	4	1	no	—
271	C₂×C₄×C₂₃	—	4	1	no	—
272	C₂×C₄×C₂₃	—	4	1	no	—
273	C₂×C₄×C₂₃	—	4	1	no	—
274	C₂×C₄×C₂₃	—	4	1	no	—
275	C₂×C₄×C₂₃	—	4	1	no	—
276	C₂×C₄×C₂₃	—	8	2	no	—
277	C₂×C₄×C₂₃	—	16	1	no	—
278	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
279	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
280	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
281	C₈×C₂₃	—	2	1	no	—
282	C₈×C₂₃	—	2	1	no	—
283	C₈×C₂₃	—	2	1	no	—
284	C₈×C₂₃	—	4	1	no	—
285	C₈×C₂₃	—	4	1	no	—
286	C₈×C₂₃	—	4	1	no	—
287	C₈×C₂₃	—	4	1	no	—
288	C₈×C₂₃	—	4	1	no	—
289	C₈×C₂₃	C₂×SD₁₆	1	2	yes	complemented
290	#3384	C₂×C₂₃	1	1	yes	characteristic, complemented
291	#3295	C₂×C₂₃	1	1	yes	characteristic, complemented
292	D₄×SD₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
293	D₄×D₈	C₂×C₂₃	1	1	yes	characteristic
294	#1715	C₂×C₂₃	1	1	yes	characteristic, complemented
295	#1688	C₂×C₂₃	1	1	yes	characteristic, complemented
296	#1677	C₂×C₂₃	1	1	yes	characteristic, complemented
297	#1671	C₂×C₂₃	1	1	yes	characteristic, complemented
298	#1665	C₂×C₂₃	1	1	yes	characteristic, complemented
299	#1639	C₂×C₂₃	1	1	yes	characteristic, complemented
300	#1629	C₂×C₂₃	1	1	yes	characteristic, complemented
301	#1627	C₂×C₂₃	1	1	yes	characteristic, complemented
302	#1622	C₂×C₂₃	1	1	yes	characteristic, complemented
303	#1582	C₂×C₂₃	1	1	yes	characteristic, complemented
304	#1580	C₂×C₂₃	1	1	yes	characteristic, complemented
305	C₂²×C₂₃	—	2	2	no	—
306	C₂²×C₂₃	—	2	2	no	—
307	C₂²×C₂₃	—	2	2	no	—
308	C₂²×C₂₃	—	4	1	no	—
309	C₂²×C₂₃	—	4	1	no	—
310	C₂²×C₂₃	—	4	1	no	—
311	C₂²×C₂₃	—	4	1	no	—
312	C₂²×C₂₃	—	4	1	no	—
313	C₂²×C₂₃	—	4	1	no	—
314	C₂²×C₂₃	—	8	1	no	—
315	C₂²×C₂₃	—	8	1	no	—
316	C₂²×C₂₃	—	8	1	no	—
317	C₂²×C₂₃	—	8	1	no	—
318	C₂²×C₂₃	—	8	1	no	—
319	C₂²×C₂₃	—	8	1	no	—
320	C₂²×C₂₃	—	16	4	no	—
321	C₂²×C₂₃	—	16	4	no	—
322	C₂²×C₂₃	D₄²	1	1	yes	characteristic
323	C₄×C₂₃	—	2	1	no	—
324	C₄×C₂₃	—	4	1	no	—
325	C₄×C₂₃	—	4	1	no	—
326	C₄×C₂₃	—	4	1	no	—
327	C₄×C₂₃	—	4	1	no	—
328	C₄×C₂₃	—	4	1	no	—
329	C₄×C₂₃	—	8	1	no	—
330	C₄×C₂₃	—	8	1	no	—
331	C₄×C₂₃	C₂²×SD₁₆	1	1	yes	characteristic
332	C₄×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
333	C₄×C₂₃	#576	1	1	yes	characteristic
334	C₄×C₂₃	#577	1	1	yes	characteristic
335	C₂²×SD₁₆	—	2	1	no	—
336	C₂²×D₈	—	2	1	no	—
337	#554	C₂²×C₂₃	1	1	yes	characteristic, complemented
338	D₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
339	D₄²	C₂²×C₂₃	1	1	yes	characteristic
340	D₄²	C₂²×C₂₃	1	1	yes	characteristic
341	#501	—	2	1	no	—
342	#498	C₂²×C₂₃	1	1	yes	characteristic, complemented
343	#493	C₂²×C₂₃	1	1	yes	characteristic
344	#492	C₂²×C₂₃	1	1	yes	characteristic, complemented
345	#492	C₂²×C₂₃	1	1	yes	characteristic, complemented
346	#486	C₂²×C₂₃	1	1	yes	characteristic, complemented
347	#486	C₂²×C₂₃	1	1	yes	characteristic, complemented
348	#481	—	2	1	no	—
349	#478	—	2	1	no	—
350	#476	—	2	1	no	—
351	#476	C₂²×C₂₃	1	1	yes	characteristic
352	#476	C₂²×C₂₃	1	1	yes	characteristic, complemented
353	#474	C₂²×C₂₃	1	1	yes	characteristic, complemented
354	#466	—	2	1	no	—
355	#465	—	2	1	no	—
356	#465	—	2	1	no	—
357	#461	C₂²×C₂₃	1	1	yes	characteristic
358	#460	C₂²×C₂₃	1	1	yes	characteristic, complemented
359	#460	C₂²×C₂₃	1	1	yes	characteristic, complemented
360	#459	C₂²×C₂₃	1	1	yes	characteristic
361	#459	C₂²×C₂₃	1	1	yes	characteristic
362	#450	—	2	1	no	—
363	#450	—	2	1	no	—
364	#448	—	2	1	no	—
365	#447	—	2	1	no	—
366	#447	—	2	1	no	—
367	#446	C₂²×C₂₃	1	1	yes	characteristic, complemented
368	#442	—	2	1	no	—
369	C₄×SD₁₆	C₂²×C₂₃	1	1	yes	characteristic
370	C₄×D₈	C₂²×C₂₃	1	1	yes	characteristic
371	C₈×D₄	C₂²×C₂₃	1	1	yes	characteristic
372	C₈×D₄	C₂²×C₂₃	1	1	yes	characteristic, complemented
373	C₈×D₄	C₂²×C₂₃	1	1	yes	characteristic, complemented
374	C₂×#102	—	2	1	no	—
375	#334	C₂²×C₂₃	1	1	yes	characteristic, complemented
376	#334	C₂²×C₂₃	1	1	yes	characteristic, complemented
377	#333	C₂²×C₂₃	1	1	yes	characteristic, complemented
378	#332	C₂²×C₂₃	1	1	yes	characteristic, complemented
379	#331	C₂²×C₂₃	1	1	yes	characteristic, complemented
380	#331	C₂²×C₂₃	1	1	yes	characteristic, complemented
381	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
382	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
383	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
384	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
385	#322	C₂²×C₂₃	1	1	yes	characteristic, complemented
386	C₂×C₂₃	—	4	2	no	—
387	C₂×C₂₃	—	4	2	no	—
388	C₂×C₂₃	—	4	2	no	—
389	C₂×C₂₃	—	16	1	no	—
390	C₂×C₂₃	—	16	1	no	—
391	C₂×C₂₃	#3264	1	1	yes	characteristic
392	C₂×C₂₃	D₄×SD₁₆	1	1	yes	characteristic
393	C₂×C₂₃	#3288	1	1	yes	characteristic
394	C₂²×Q₈	—	2	1	no	—
395	C₂²×D₄	—	2	1	no	—
396	C₂²×D₄	—	2	1	no	—
397	C₂²×D₄	—	2	1	no	—
398	C₂²×D₄	—	2	1	no	—
399	C₂×SD₁₆	—	2	1	no	—
400	C₂×SD₁₆	—	2	1	no	—
401	C₂×SD₁₆	—	2	1	no	—
402	C₂×SD₁₆	—	2	1	no	—
403	C₂×SD₁₆	—	2	1	no	—
404	C₂×SD₁₆	—	4	1	no	—
405	C₂×SD₁₆	—	4	4	no	—
406	C₂×SD₁₆	D₄×C₂₃	1	1	yes	characteristic
407	C₂×D₈	—	2	1	no	—
408	C₂×D₈	—	2	1	no	—
409	C₂×D₈	—	2	1	no	—
410	C₂×D₈	—	2	1	no	—
411	C₂×D₈	—	4	4	no	—
412	C₂×D₈	D₄×C₂₃	1	1	yes	characteristic
413	C₂²×C₈	—	2	1	no	—
414	C₂²×C₈	—	2	1	no	—
415	C₂²×C₈	—	2	1	no	—
416	C₄⋊Q₈	—	2	1	no	—
417	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
418	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
419	#127	C₂³×C₂₃	1	1	yes	characteristic
420	#127	C₂³×C₂₃	1	1	yes	characteristic
421	#125	—	2	1	no	—
422	C₂²⋊Q₈	—	2	1	no	—
423	C₂²⋊Q₈	—	2	1	no	—
424	C₂²⋊Q₈	—	4	1	no	—
425	C₂²⋊Q₈	—	4	1	no	—
426	#121	—	2	1	no	—
427	#121	—	2	1	no	—
428	#121	—	2	1	no	—
429	#121	—	2	1	no	—
430	#120	—	4	1	no	—
431	#120	—	4	1	no	—
432	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
433	C₄×D₄	—	2	1	no	—
434	C₄×D₄	—	2	1	no	—
435	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
436	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
437	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
438	C₂×C₄⋊C₄	—	2	1	no	—
439	#107	—	2	1	no	—
440	#107	D₄×C₂₃	1	1	yes	characteristic
441	#106	—	2	1	no	—
442	#106	—	2	1	no	—
443	#106	—	2	1	no	—
444	#106	—	2	1	no	—
445	#106	D₄×C₂₃	1	1	yes	characteristic
446	C₄⋊C₈	—	2	1	no	—
447	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
448	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
449	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
450	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
451	#103	—	2	1	no	—
452	#103	D₄×C₂₃	1	1	yes	characteristic
453	#102	—	2	1	no	—
454	#102	—	2	1	no	—
455	#102	—	2	1	no	—
456	#102	—	2	1	no	—
457	#102	—	2	1	no	—
458	#102	—	2	1	no	—
459	#102	—	2	1	no	—
460	#102	—	2	1	no	—
461	#102	—	2	1	no	—
462	#102	—	4	1	no	—
463	#102	D₄×C₂₃	1	1	yes	characteristic
464	#102	D₄×C₂₃	1	1	yes	characteristic
465	#102	D₄×C₂₃	1	1	yes	characteristic
466	C₂²⋊C₈	—	2	1	no	—
467	C₂²⋊C₈	—	2	1	no	—
468	C₂²⋊C₈	—	2	1	no	—
469	#97	—	2	1	no	—
470	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
471	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
472	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
473	C₂₃	#19763	1	1	yes	characteristic, direct product factor, Sylow
474	C₂⁴	—	4	1	no	—
475	C₂⁴	—	4	1	no	—
476	C₂×Q₈	—	2	1	no	—
477	C₂×Q₈	—	2	1	no	—
478	C₂×Q₈	—	4	1	no	—
479	C₂×Q₈	—	4	2	no	—
480	C₂×Q₈	—	8	1	no	—
481	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
482	C₂×D₄	—	2	1	no	—
483	C₂×D₄	—	2	1	no	—
484	C₂×D₄	—	2	1	no	—
485	C₂×D₄	—	2	1	no	—
486	C₂×D₄	—	2	1	no	—
487	C₂×D₄	—	2	1	no	—
488	C₂×D₄	—	4	1	no	—
489	C₂×D₄	—	4	1	no	—
490	C₂×D₄	—	4	1	no	—
491	C₂×D₄	—	4	1	no	—
492	C₂×D₄	—	4	2	no	—
493	C₂×D₄	—	4	2	no	—
494	C₂×D₄	—	4	2	no	—
495	C₂×D₄	—	4	2	no	—
496	C₂×D₄	—	8	1	no	—
497	C₂×D₄	—	8	1	no	—
498	C₂×D₄	—	8	1	no	—
499	C₂×D₄	—	8	1	no	—
500	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
501	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
502	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
503	C₂²×C₄	—	2	1	no	—
504	C₂²×C₄	—	2	1	no	—
505	C₂²×C₄	—	2	1	no	—
506	C₂²×C₄	—	4	1	no	—
507	C₂²×C₄	—	4	1	no	—
508	SD₁₆	—	2	2	no	—
509	SD₁₆	—	4	2	no	—
510	SD₁₆	—	4	2	no	—
511	SD₁₆	—	4	2	no	—
512	SD₁₆	—	8	1	no	—
513	SD₁₆	—	8	1	no	—
514	SD₁₆	—	8	2	no	—
515	D₈	—	4	2	no	—
516	D₈	—	4	2	no	—
517	D₈	—	4	2	no	—
518	D₈	—	8	1	no	—
519	D₈	SD₁₆×C₂₃	1	4	yes	complemented
520	C₂×C₈	—	2	1	no	—
521	C₂×C₈	—	2	1	no	—
522	C₂×C₈	—	2	1	no	—
523	C₂×C₈	—	2	1	no	—
524	C₂×C₈	—	2	1	no	—
525	C₂×C₈	—	4	2	no	—
526	C₂×C₈	—	4	2	no	—
527	C₂×C₈	—	8	1	no	—
528	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
529	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
530	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
531	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
532	C₄⋊C₄	—	2	1	no	—
533	C₄⋊C₄	—	2	1	no	—
534	C₄⋊C₄	—	2	1	no	—
535	C₄⋊C₄	—	2	1	no	—
536	C₄⋊C₄	—	2	1	no	—
537	C₄⋊C₄	—	4	1	no	—
538	C₄⋊C₄	—	4	1	no	—
539	C₄⋊C₄	—	4	1	no	—
540	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
541	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
542	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
543	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
544	C₂²⋊C₄	—	2	1	no	—
545	C₂²⋊C₄	—	2	1	no	—
546	C₂²⋊C₄	—	2	1	no	—
547	C₂²⋊C₄	—	4	1	no	—
548	C₂²⋊C₄	—	4	1	no	—
549	C₂²⋊C₄	—	4	1	no	—
550	C₂²⋊C₄	—	4	1	no	—
551	C₄²	—	2	1	no	—
552	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
553	C₂³	—	2	1	no	—
554	C₂³	—	2	1	no	—
555	C₂³	—	2	1	no	—
556	C₂³	—	4	1	no	—
557	C₂³	—	4	1	no	—
558	C₂³	—	8	2	no	—
559	C₂³	—	8	2	no	—
560	C₂³	—	8	2	no	—
561	C₂³	—	8	2	no	—
562	C₂³	—	16	1	no	—
563	C₂³	—	16	1	no	—
564	Q₈	—	2	1	no	—
565	Q₈	—	2	1	no	—
566	Q₈	—	4	1	no	—
567	Q₈	—	4	1	no	—
568	Q₈	—	8	1	no	—
569	Q₈	—	8	2	no	—
570	D₄	—	2	1	no	—
571	D₄	—	2	1	no	—
572	D₄	—	2	1	no	—
573	D₄	—	4	1	no	—
574	D₄	—	4	1	no	—
575	D₄	—	4	1	no	—
576	D₄	—	4	1	no	—
577	D₄	—	4	1	no	—
578	D₄	—	4	1	no	—
579	D₄	—	4	1	no	—
580	D₄	—	4	1	no	—
581	D₄	—	8	1	no	—
582	D₄	—	8	1	no	—
583	D₄	—	8	2	no	—
584	D₄	—	8	2	no	—
585	D₄	—	8	2	no	—
586	D₄	—	8	2	no	—
587	D₄	C₂×SD₁₆×C₂₃	1	2	yes	—
588	D₄	C₂×SD₁₆×C₂₃	1	2	yes	complemented
589	D₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
590	C₂×C₄	—	2	1	no	—
591	C₂×C₄	—	2	1	no	—
592	C₂×C₄	—	2	1	no	—
593	C₂×C₄	—	2	1	no	—
594	C₂×C₄	—	2	1	no	—
595	C₂×C₄	—	4	1	no	—
596	C₂×C₄	—	4	1	no	—
597	C₂×C₄	—	4	1	no	—
598	C₂×C₄	—	4	1	no	—
599	C₂×C₄	—	4	1	no	—
600	C₂×C₄	—	4	1	no	—
601	C₂×C₄	—	4	1	no	—
602	C₂×C₄	—	4	1	no	—
603	C₂×C₄	—	8	2	no	—
604	C₂×C₄	—	16	1	no	—
605	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
606	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
607	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
608	C₈	—	2	1	no	—
609	C₈	—	2	1	no	—
610	C₈	—	2	1	no	—
611	C₈	—	4	1	no	—
612	C₈	—	4	1	no	—
613	C₈	—	4	1	no	—
614	C₈	—	4	1	no	—
615	C₈	—	4	1	no	—
616	C₈	C₂×SD₁₆×C₂₃	1	2	yes	complemented
617	C₂²	—	2	2	no	—
618	C₂²	—	2	2	no	—
619	C₂²	—	2	2	no	—
620	C₂²	—	4	1	no	—
621	C₂²	—	4	1	no	—
622	C₂²	—	4	1	no	—
623	C₂²	—	4	1	no	—
624	C₂²	—	4	1	no	—
625	C₂²	—	4	1	no	—
626	C₂²	—	8	1	no	—
627	C₂²	—	8	1	no	—
628	C₂²	—	8	1	no	—
629	C₂²	—	8	1	no	—
630	C₂²	—	8	1	no	—
631	C₂²	—	8	1	no	—
632	C₂²	—	16	4	no	—
633	C₂²	—	16	4	no	—
634	C₂²	D₄²×C₂₃	1	1	yes	characteristic
635	C₄	—	2	1	no	—
636	C₄	—	4	1	no	—
637	C₄	—	4	1	no	—
638	C₄	—	4	1	no	—
639	C₄	—	4	1	no	—
640	C₄	—	4	1	no	—
641	C₄	—	8	1	no	—
642	C₄	—	8	1	no	—
643	C₄	C₂²×SD₁₆×C₂₃	1	1	yes	characteristic
644	C₄	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
645	C₄	C₂₃×#576	1	1	yes	characteristic
646	C₄	C₂₃×#577	1	1	yes	characteristic
647	C₂	—	4	2	no	—
648	C₂	—	4	2	no	—
649	C₂	—	4	2	no	—
650	C₂	—	16	1	no	—
651	C₂	—	16	1	no	—
652	C₂	C₂₃×#3264	1	1	yes	characteristic
653	C₂	D₄×SD₁₆×C₂₃	1	1	yes	characteristic
654	C₂	C₂₃×#3288	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
prime radical	D₄×D₈×C₂₃	C₂
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×#3384	C₂	semidirect
2. C₂₃×#3295	C₂	semidirect
3. D₄×SD₁₆×C₂₃	C₂	semidirect
5. C₂₃×#1715	C₂	semidirect
6. C₂₃×#1688	C₂	semidirect
7. C₂₃×#1677	C₂	semidirect
8. C₂₃×#1671	C₂	semidirect
9. C₂₃×#1665	C₂	semidirect
10. C₂₃×#1639	C₂	semidirect
11. C₂₃×#1629	C₂	semidirect
12. C₂₃×#1627	C₂	semidirect
13. C₂₃×#1622	C₂	semidirect
14. C₂₃×#1582	C₂	semidirect
15. C₂₃×#1580	C₂	semidirect
18. C₂₃×#554	C₂²	semidirect
19. D₄×Q₈×C₂₃	C₂²	semidirect
23. C₂₃×#498	C₂²	semidirect
25. C₂₃×#492	C₂²	semidirect
26. C₂₃×#492	C₂²	semidirect
27. C₂₃×#486	C₂²	semidirect
28. C₂₃×#486	C₂²	semidirect
33. C₂₃×#476	C₂²	semidirect
34. C₂₃×#474	C₂²	semidirect
39. C₂₃×#460	C₂²	semidirect
40. C₂₃×#460	C₂²	semidirect
48. C₂₃×#446	C₂²	semidirect
53. C₈×D₄×C₂₃	C₂²	semidirect
54. C₈×D₄×C₂₃	C₂²	semidirect
56. C₂₃×#334	C₂²	semidirect
57. C₂₃×#334	C₂²	semidirect
58. C₂₃×#333	C₂²	semidirect
59. C₂₃×#332	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
191. D₈×C₂₃	SD₁₆	semidirect
225. #19763	C₂₃	direct (N × H)
261. D₄×C₂₃	C₂×SD₁₆	semidirect
289. C₈×C₂₃	C₂×SD₁₆	semidirect
290. #3384	C₂×C₂₃	semidirect
291. #3295	C₂×C₂₃	semidirect
292. D₄×SD₁₆	C₂×C₂₃	semidirect
294. #1715	C₂×C₂₃	semidirect
295. #1688	C₂×C₂₃	semidirect
296. #1677	C₂×C₂₃	semidirect
297. #1671	C₂×C₂₃	semidirect
298. #1665	C₂×C₂₃	semidirect
299. #1639	C₂×C₂₃	semidirect
300. #1629	C₂×C₂₃	semidirect
301. #1627	C₂×C₂₃	semidirect
302. #1622	C₂×C₂₃	semidirect
303. #1582	C₂×C₂₃	semidirect
304. #1580	C₂×C₂₃	semidirect
337. #554	C₂²×C₂₃	semidirect
338. D₄×Q₈	C₂²×C₂₃	semidirect
342. #498	C₂²×C₂₃	semidirect
344. #492	C₂²×C₂₃	semidirect
345. #492	C₂²×C₂₃	semidirect
346. #486	C₂²×C₂₃	semidirect
347. #486	C₂²×C₂₃	semidirect
352. #476	C₂²×C₂₃	semidirect
353. #474	C₂²×C₂₃	semidirect
358. #460	C₂²×C₂₃	semidirect
359. #460	C₂²×C₂₃	semidirect
367. #446	C₂²×C₂₃	semidirect
372. C₈×D₄	C₂²×C₂₃	semidirect
373. C₈×D₄	C₂²×C₂₃	semidirect
375. #334	C₂²×C₂₃	semidirect
376. #334	C₂²×C₂₃	semidirect
377. #333	C₂²×C₂₃	semidirect
378. #332	C₂²×C₂₃	semidirect
379. #331	C₂²×C₂₃	semidirect
380. #331	C₂²×C₂₃	semidirect
381. Q₈⋊C₈	C₂²×C₂₃	semidirect
383. D₄⋊C₈	C₂²×C₂₃	semidirect
384. D₄⋊C₈	C₂²×C₂₃	semidirect
385. #322	C₂²×C₂₃	semidirect
473. C₂₃	#19763	direct (N × H)
519. D₈	SD₁₆×C₂₃	semidirect
588. D₄	C₂×SD₁₆×C₂₃	semidirect
616. C₈	C₂×SD₁₆×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#3384	2
C₂₃×#3295	2
D₄×SD₁₆×C₂₃	2
D₄×D₈×C₂₃	2
C₂₃×#1715	2
C₂₃×#1688	2
C₂₃×#1677	2
C₂₃×#1671	2
C₂₃×#1665	2
C₂₃×#1639	2
C₂₃×#1629	2
C₂₃×#1627	2
C₂₃×#1622	2
C₂₃×#1582	2
C₂₃×#1580	2
#19763	23

Subgroups — C23×#19763 #19167389

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#19763 #19167389