Subgroups — C₂₃×#19781 #19167407

Order 5888.

616 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3380	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#3358	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#3333	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#3299	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1719	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1703	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1682	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1677	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1676	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1644	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1637	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#1631	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂₃×#1622	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂₃×#1581	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂₃×#1579	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂₃×#554	C₂²	1	1	yes	characteristic
17	C₂₃×#554	C₂²	1	1	yes	characteristic
18	C₂₃×#550	C₂²	1	1	yes	characteristic
19	D₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
20	C₂₃×#500	C₂²	1	1	yes	characteristic
21	C₂₃×#499	—	2	1	no	—
22	C₂₃×#492	C₂²	1	1	yes	characteristic
23	C₂₃×#487	C₂²	1	1	yes	characteristic
24	C₂₃×#487	C₂²	1	1	yes	characteristic
25	C₂₃×#486	—	2	1	no	—
26	C₂₃×#486	C₂²	1	1	yes	characteristic
27	C₂₃×#486	C₂²	1	1	yes	characteristic
28	C₂₃×#484	—	2	1	no	—
29	C₂₃×#480	—	2	1	no	—
30	C₂₃×#478	—	2	1	no	—
31	C₂₃×#478	—	2	1	no	—
32	C₂₃×#477	C₂²	1	1	yes	characteristic
33	C₂₃×#477	C₂²	1	1	yes	characteristic
34	C₂₃×#476	C₂²	1	1	yes	characteristic
35	C₂₃×#476	C₂²	1	1	yes	characteristic
36	C₂₃×#474	—	2	1	no	—
37	C₂₃×#474	C₂²	1	1	yes	characteristic
38	C₂₃×#468	—	2	1	no	—
39	C₂₃×#467	—	2	1	no	—
40	C₂₃×#462	C₂²	1	1	yes	characteristic
41	C₂₃×#460	C₂²	1	1	yes	characteristic
42	C₂₃×#459	—	2	1	no	—
43	C₂₃×#459	C₂²	1	1	yes	characteristic
44	C₂₃×#452	—	2	1	no	—
45	C₂₃×#451	—	2	1	no	—
46	C₈×Q₈×C₂₃	C₂²	1	1	yes	characteristic
47	C₈×Q₈×C₂₃	C₂²	1	1	yes	characteristic
48	C₄×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
49	C₄×D₈×C₂₃	—	2	1	no	—
50	C₄×D₈×C₂₃	—	2	1	no	—
51	C₄×D₈×C₂₃	C₂²	1	1	yes	characteristic
52	C₂₃×#436	C₂²	1	1	yes	characteristic
53	C₈×D₄×C₂₃	C₂²	1	1	yes	characteristic
54	C₂₃×#417	—	2	1	no	—
55	C₂×C₂₃×#103	—	2	1	no	—
56	C₂₃×#335	C₂²	1	1	yes	characteristic
57	C₂₃×#334	C₂²	1	1	yes	characteristic
58	C₂₃×#333	C₂²	1	1	yes	characteristic
59	C₂₃×#333	C₂²	1	1	yes	characteristic
60	C₂₃×#332	C₂²	1	1	yes	characteristic
61	C₂₃×#331	C₂²	1	1	yes	characteristic
62	C₂₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
63	C₂₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
64	C₂₃×D₄⋊C₈	C₂²	1	1	yes	characteristic
65	C₂₃×D₄⋊C₈	C₂²	1	1	yes	characteristic
66	C₂₃×#322	C₂²	1	1	yes	characteristic
67	C₂×#41×C₂₃	—	2	1	no	—
68	C₂²×Q₈×C₂₃	—	2	1	no	—
69	C₂×Q₁₆×C₂₃	—	2	1	no	—
70	C₂×Q₁₆×C₂₃	D₄	1	1	yes	characteristic
71	C₂×SD₁₆×C₂₃	—	2	1	no	—
72	C₂×SD₁₆×C₂₃	—	4	1	no	—
73	C₂×D₈×C₂₃	—	2	1	no	—
74	C₂×D₈×C₂₃	D₄	1	1	yes	characteristic
75	C₂×M₁₆×C₂₃	—	2	1	no	—
76	C₂²×C₈×C₂₃	—	2	1	no	—
77	C₂₃×C₄⋊Q₈	—	2	1	no	—
78	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
79	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
80	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
81	C₂₃×#127	—	2	1	no	—
82	C₂₃×#127	C₂³	1	1	yes	characteristic
83	C₂₃×#125	—	2	1	no	—
84	C₂₃×#125	—	2	1	no	—
85	C₂₃×C₂²⋊Q₈	—	2	1	no	—
86	C₂₃×C₂²⋊Q₈	—	2	1	no	—
87	C₂₃×C₂²⋊Q₈	—	2	1	no	—
88	C₂₃×C₂²⋊Q₈	—	4	1	no	—
89	C₂₃×C₂²⋊Q₈	—	4	1	no	—
90	C₂₃×C₂²⋊Q₈	—	4	1	no	—
91	C₂₃×#121	—	2	1	no	—
92	C₂₃×#121	—	4	1	no	—
93	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
94	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
95	C₄×D₄×C₂₃	—	2	1	no	—
96	C₄×D₄×C₂₃	—	2	1	no	—
97	C₄×D₄×C₂₃	—	2	1	no	—
98	C₄×D₄×C₂₃	—	2	1	no	—
99	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
100	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
101	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
102	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
103	C₂₃×#107	—	2	1	no	—
104	C₂₃×#107	—	2	1	no	—
105	C₂₃×#107	—	2	1	no	—
106	C₂₃×#107	—	2	1	no	—
107	C₂₃×#107	—	2	1	no	—
108	C₂₃×#107	D₄	1	1	yes	characteristic
109	C₂₃×#107	D₄	1	1	yes	characteristic
110	C₂₃×#106	—	2	1	no	—
111	C₂₃×#106	—	2	1	no	—
112	C₂₃×C₄⋊C₈	—	2	1	no	—
113	C₂₃×C₄⋊C₈	—	2	1	no	—
114	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
115	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
116	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
117	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
118	C₂₃×#103	—	2	1	no	—
119	C₂₃×#103	—	2	1	no	—
120	C₂₃×#103	—	2	1	no	—
121	C₂₃×#103	—	2	1	no	—
122	C₂₃×#103	—	2	1	no	—
123	C₂₃×#103	—	4	1	no	—
124	C₂₃×#103	D₄	1	1	yes	characteristic
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	D₄	1	1	yes	characteristic
136	C₂₃×#102	D₄	1	1	yes	characteristic
137	C₂₃×C₂²⋊C₈	—	2	1	no	—
138	C₂₃×C₂²⋊C₈	—	2	1	no	—
139	C₄×C₈×C₂₃	—	2	1	no	—
140	C₄×C₈×C₂₃	—	2	1	no	—
141	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
142	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
143	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
144	#41×C₂₃	—	4	2	no	—
145	#41×C₂₃	—	8	1	no	—
146	C₂×Q₈×C₂₃	—	2	1	no	—
147	C₂×Q₈×C₂₃	—	2	1	no	—
148	C₂×Q₈×C₂₃	—	2	1	no	—
149	C₂×Q₈×C₂₃	—	4	1	no	—
150	C₂×Q₈×C₂₃	—	4	2	no	—
151	C₂×Q₈×C₂₃	—	8	1	no	—
152	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
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	—	4	1	no	—
158	C₂×D₄×C₂₃	—	4	1	no	—
159	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
160	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
161	C₂²×C₄×C₂₃	—	2	1	no	—
162	C₂²×C₄×C₂₃	—	2	1	no	—
163	C₂²×C₄×C₂₃	—	4	1	no	—
164	C₂²×C₄×C₂₃	—	4	1	no	—
165	C₂²×C₄×C₂₃	—	4	1	no	—
166	C₂²×C₄×C₂₃	—	4	1	no	—
167	Q₁₆×C₂₃	—	2	2	no	—
168	Q₁₆×C₂₃	—	8	1	no	—
169	SD₁₆×C₂₃	—	8	1	no	—
170	SD₁₆×C₂₃	—	8	2	no	—
171	D₈×C₂₃	—	2	2	no	—
172	D₈×C₂₃	—	4	2	no	—
173	M₁₆×C₂₃	—	8	1	no	—
174	C₂×C₈×C₂₃	—	2	1	no	—
175	C₂×C₈×C₂₃	—	2	1	no	—
176	C₂×C₈×C₂₃	—	2	1	no	—
177	C₂×C₈×C₂₃	—	2	1	no	—
178	C₂×C₈×C₂₃	—	2	1	no	—
179	C₂×C₈×C₂₃	—	8	1	no	—
180	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
181	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
182	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
183	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
184	C₄⋊C₄×C₂₃	—	2	1	no	—
185	C₄⋊C₄×C₂₃	—	2	1	no	—
186	C₄⋊C₄×C₂₃	—	2	1	no	—
187	C₄⋊C₄×C₂₃	—	2	1	no	—
188	C₄⋊C₄×C₂₃	—	2	1	no	—
189	C₄⋊C₄×C₂₃	—	2	1	no	—
190	C₄⋊C₄×C₂₃	—	2	1	no	—
191	C₄⋊C₄×C₂₃	—	2	1	no	—
192	C₄⋊C₄×C₂₃	—	4	1	no	—
193	C₄⋊C₄×C₂₃	—	4	1	no	—
194	C₄⋊C₄×C₂₃	—	4	1	no	—
195	C₄⋊C₄×C₂₃	—	4	1	no	—
196	C₄⋊C₄×C₂₃	—	4	1	no	—
197	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
198	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
199	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
200	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
201	C₂²⋊C₄×C₂₃	—	2	1	no	—
202	C₂²⋊C₄×C₂₃	—	2	1	no	—
203	C₂²⋊C₄×C₂₃	—	4	1	no	—
204	C₂²⋊C₄×C₂₃	—	4	1	no	—
205	C₂²⋊C₄×C₂₃	—	4	1	no	—
206	C₂²⋊C₄×C₂₃	—	4	1	no	—
207	C₄²×C₂₃	—	2	1	no	—
208	C₄²×C₂₃	—	2	1	no	—
209	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
210	#19781	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
211	C₂³×C₂₃	—	2	1	no	—
212	C₂³×C₂₃	—	2	1	no	—
213	C₂³×C₂₃	—	4	1	no	—
214	Q₈×C₂₃	—	2	1	no	—
215	Q₈×C₂₃	—	2	1	no	—
216	Q₈×C₂₃	—	2	1	no	—
217	Q₈×C₂₃	—	4	1	no	—
218	Q₈×C₂₃	—	4	1	no	—
219	Q₈×C₂₃	—	8	2	no	—
220	Q₈×C₂₃	—	8	1	no	—
221	Q₈×C₂₃	—	8	1	no	—
222	Q₈×C₂₃	#135	1	2	yes	—
223	D₄×C₂₃	—	2	1	no	—
224	D₄×C₂₃	—	2	1	no	—
225	D₄×C₂₃	—	2	1	no	—
226	D₄×C₂₃	—	4	1	no	—
227	D₄×C₂₃	—	4	1	no	—
228	D₄×C₂₃	—	4	1	no	—
229	D₄×C₂₃	—	4	1	no	—
230	D₄×C₂₃	—	8	2	no	—
231	D₄×C₂₃	—	8	1	no	—
232	D₄×C₂₃	—	16	1	no	—
233	D₄×C₂₃	#137	1	2	yes	—
234	C₂×C₄×C₂₃	—	2	1	no	—
235	C₂×C₄×C₂₃	—	2	1	no	—
236	C₂×C₄×C₂₃	—	2	1	no	—
237	C₂×C₄×C₂₃	—	2	1	no	—
238	C₂×C₄×C₂₃	—	2	1	no	—
239	C₂×C₄×C₂₃	—	2	1	no	—
240	C₂×C₄×C₂₃	—	4	1	no	—
241	C₂×C₄×C₂₃	—	4	1	no	—
242	C₂×C₄×C₂₃	—	4	1	no	—
243	C₂×C₄×C₂₃	—	4	1	no	—
244	C₂×C₄×C₂₃	—	4	1	no	—
245	C₂×C₄×C₂₃	—	4	1	no	—
246	C₂×C₄×C₂₃	—	4	1	no	—
247	C₂×C₄×C₂₃	—	8	2	no	—
248	C₂×C₄×C₂₃	—	8	2	no	—
249	C₂×C₄×C₂₃	—	16	1	no	—
250	C₂×C₄×C₂₃	—	16	1	no	—
251	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
252	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
253	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
254	C₈×C₂₃	—	2	1	no	—
255	C₈×C₂₃	—	2	1	no	—
256	C₈×C₂₃	—	2	1	no	—
257	C₈×C₂₃	—	2	1	no	—
258	C₈×C₂₃	—	4	1	no	—
259	C₈×C₂₃	—	4	1	no	—
260	C₈×C₂₃	—	4	1	no	—
261	C₈×C₂₃	—	4	1	no	—
262	C₈×C₂₃	—	4	1	no	—
263	#3380	C₂×C₂₃	1	1	yes	characteristic, complemented
264	#3358	C₂×C₂₃	1	1	yes	characteristic
265	#3333	C₂×C₂₃	1	1	yes	characteristic, complemented
266	#3299	C₂×C₂₃	1	1	yes	characteristic, complemented
267	#1719	C₂×C₂₃	1	1	yes	characteristic, complemented
268	#1703	C₂×C₂₃	1	1	yes	characteristic, complemented
269	#1682	C₂×C₂₃	1	1	yes	characteristic, complemented
270	#1677	C₂×C₂₃	1	1	yes	characteristic, complemented
271	#1676	C₂×C₂₃	1	1	yes	characteristic, complemented
272	#1644	C₂×C₂₃	1	1	yes	characteristic, complemented
273	#1637	C₂×C₂₃	1	1	yes	characteristic, complemented
274	#1631	C₂×C₂₃	1	1	yes	characteristic, complemented
275	#1622	C₂×C₂₃	1	1	yes	characteristic, complemented
276	#1581	C₂×C₂₃	1	1	yes	characteristic, complemented
277	#1579	C₂×C₂₃	1	1	yes	characteristic, complemented
278	C₂²×C₂₃	—	2	2	no	—
279	C₂²×C₂₃	—	4	1	no	—
280	C₂²×C₂₃	—	4	1	no	—
281	C₂²×C₂₃	—	4	1	no	—
282	C₂²×C₂₃	—	4	1	no	—
283	C₂²×C₂₃	—	4	1	no	—
284	C₂²×C₂₃	—	8	1	no	—
285	C₂²×C₂₃	—	8	1	no	—
286	C₂²×C₂₃	—	8	1	no	—
287	C₂²×C₂₃	D₄²	1	1	yes	characteristic
288	C₄×C₂₃	—	2	2	no	—
289	C₄×C₂₃	—	2	1	no	—
290	C₄×C₂₃	—	4	1	no	—
291	C₄×C₂₃	—	4	1	no	—
292	C₄×C₂₃	—	4	1	no	—
293	C₄×C₂₃	—	4	1	no	—
294	C₄×C₂₃	—	4	1	no	—
295	C₄×C₂₃	—	8	1	no	—
296	C₄×C₂₃	—	8	1	no	—
297	C₄×C₂₃	—	8	1	no	—
298	C₄×C₂₃	C₂×#135	1	1	yes	characteristic
299	C₄×C₂₃	C₂×#137	1	1	yes	characteristic
300	C₄×C₂₃	#576	1	1	yes	characteristic
301	C₄×C₂₃	#578	1	1	yes	characteristic
302	#554	C₂²×C₂₃	1	1	yes	characteristic
303	#554	C₂²×C₂₃	1	1	yes	characteristic
304	#550	C₂²×C₂₃	1	1	yes	characteristic
305	D₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
306	#500	C₂²×C₂₃	1	1	yes	characteristic
307	#499	—	2	1	no	—
308	#492	C₂²×C₂₃	1	1	yes	characteristic
309	#487	C₂²×C₂₃	1	1	yes	characteristic
310	#487	C₂²×C₂₃	1	1	yes	characteristic
311	#486	—	2	1	no	—
312	#486	C₂²×C₂₃	1	1	yes	characteristic
313	#486	C₂²×C₂₃	1	1	yes	characteristic
314	#484	—	2	1	no	—
315	#480	—	2	1	no	—
316	#478	—	2	1	no	—
317	#478	—	2	1	no	—
318	#477	C₂²×C₂₃	1	1	yes	characteristic
319	#477	C₂²×C₂₃	1	1	yes	characteristic
320	#476	C₂²×C₂₃	1	1	yes	characteristic
321	#476	C₂²×C₂₃	1	1	yes	characteristic
322	#474	—	2	1	no	—
323	#474	C₂²×C₂₃	1	1	yes	characteristic
324	#468	—	2	1	no	—
325	#467	—	2	1	no	—
326	#462	C₂²×C₂₃	1	1	yes	characteristic
327	#460	C₂²×C₂₃	1	1	yes	characteristic
328	#459	—	2	1	no	—
329	#459	C₂²×C₂₃	1	1	yes	characteristic
330	#452	—	2	1	no	—
331	#451	—	2	1	no	—
332	C₈×Q₈	C₂²×C₂₃	1	1	yes	characteristic
333	C₈×Q₈	C₂²×C₂₃	1	1	yes	characteristic
334	C₄×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic
335	C₄×D₈	—	2	1	no	—
336	C₄×D₈	—	2	1	no	—
337	C₄×D₈	C₂²×C₂₃	1	1	yes	characteristic
338	#436	C₂²×C₂₃	1	1	yes	characteristic
339	C₈×D₄	C₂²×C₂₃	1	1	yes	characteristic
340	#417	—	2	1	no	—
341	C₂×#103	—	2	1	no	—
342	#335	C₂²×C₂₃	1	1	yes	characteristic
343	#334	C₂²×C₂₃	1	1	yes	characteristic
344	#333	C₂²×C₂₃	1	1	yes	characteristic
345	#333	C₂²×C₂₃	1	1	yes	characteristic
346	#332	C₂²×C₂₃	1	1	yes	characteristic
347	#331	C₂²×C₂₃	1	1	yes	characteristic
348	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
349	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
350	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
351	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
352	#322	C₂²×C₂₃	1	1	yes	characteristic
353	C₂×C₂₃	—	4	2	no	—
354	C₂×C₂₃	—	8	1	no	—
355	C₂×C₂₃	—	16	1	no	—
356	C₂×C₂₃	#3270	1	1	yes	characteristic
357	C₂×C₂₃	#3280	1	1	yes	characteristic
358	C₂×C₂₃	#3288	1	1	yes	characteristic
359	C₂×#41	—	2	1	no	—
360	C₂²×Q₈	—	2	1	no	—
361	C₂×Q₁₆	—	2	1	no	—
362	C₂×Q₁₆	D₄×C₂₃	1	1	yes	characteristic
363	C₂×SD₁₆	—	2	1	no	—
364	C₂×SD₁₆	—	4	1	no	—
365	C₂×D₈	—	2	1	no	—
366	C₂×D₈	D₄×C₂₃	1	1	yes	characteristic
367	C₂×M₁₆	—	2	1	no	—
368	C₂²×C₈	—	2	1	no	—
369	C₄⋊Q₈	—	2	1	no	—
370	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
371	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
372	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
373	#127	—	2	1	no	—
374	#127	C₂³×C₂₃	1	1	yes	characteristic
375	#125	—	2	1	no	—
376	#125	—	2	1	no	—
377	C₂²⋊Q₈	—	2	1	no	—
378	C₂²⋊Q₈	—	2	1	no	—
379	C₂²⋊Q₈	—	2	1	no	—
380	C₂²⋊Q₈	—	4	1	no	—
381	C₂²⋊Q₈	—	4	1	no	—
382	C₂²⋊Q₈	—	4	1	no	—
383	#121	—	2	1	no	—
384	#121	—	4	1	no	—
385	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
386	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
387	C₄×D₄	—	2	1	no	—
388	C₄×D₄	—	2	1	no	—
389	C₄×D₄	—	2	1	no	—
390	C₄×D₄	—	2	1	no	—
391	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
392	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
393	C₂×C₄⋊C₄	—	2	1	no	—
394	C₂×C₄⋊C₄	—	2	1	no	—
395	#107	—	2	1	no	—
396	#107	—	2	1	no	—
397	#107	—	2	1	no	—
398	#107	—	2	1	no	—
399	#107	—	2	1	no	—
400	#107	D₄×C₂₃	1	1	yes	characteristic
401	#107	D₄×C₂₃	1	1	yes	characteristic
402	#106	—	2	1	no	—
403	#106	—	2	1	no	—
404	C₄⋊C₈	—	2	1	no	—
405	C₄⋊C₈	—	2	1	no	—
406	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
407	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
408	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
409	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
410	#103	—	2	1	no	—
411	#103	—	2	1	no	—
412	#103	—	2	1	no	—
413	#103	—	2	1	no	—
414	#103	—	2	1	no	—
415	#103	—	4	1	no	—
416	#103	D₄×C₂₃	1	1	yes	characteristic
417	#103	D₄×C₂₃	1	1	yes	characteristic
418	#102	—	2	1	no	—
419	#102	—	2	1	no	—
420	#102	—	2	1	no	—
421	#102	—	2	1	no	—
422	#102	—	2	1	no	—
423	#102	—	2	1	no	—
424	#102	—	2	1	no	—
425	#102	—	2	1	no	—
426	#102	—	2	1	no	—
427	#102	D₄×C₂₃	1	1	yes	characteristic
428	#102	D₄×C₂₃	1	1	yes	characteristic
429	C₂²⋊C₈	—	2	1	no	—
430	C₂²⋊C₈	—	2	1	no	—
431	C₄×C₈	—	2	1	no	—
432	C₄×C₈	—	2	1	no	—
433	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
434	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
435	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
436	C₂₃	#19781	1	1	yes	characteristic, direct product factor, Sylow
437	#41	—	4	2	no	—
438	#41	—	8	1	no	—
439	C₂×Q₈	—	2	1	no	—
440	C₂×Q₈	—	2	1	no	—
441	C₂×Q₈	—	2	1	no	—
442	C₂×Q₈	—	4	1	no	—
443	C₂×Q₈	—	4	2	no	—
444	C₂×Q₈	—	8	1	no	—
445	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
446	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
447	C₂×D₄	—	2	1	no	—
448	C₂×D₄	—	2	1	no	—
449	C₂×D₄	—	2	1	no	—
450	C₂×D₄	—	4	1	no	—
451	C₂×D₄	—	4	1	no	—
452	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
453	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
454	C₂²×C₄	—	2	1	no	—
455	C₂²×C₄	—	2	1	no	—
456	C₂²×C₄	—	4	1	no	—
457	C₂²×C₄	—	4	1	no	—
458	C₂²×C₄	—	4	1	no	—
459	C₂²×C₄	—	4	1	no	—
460	Q₁₆	—	2	2	no	—
461	Q₁₆	—	8	1	no	—
462	SD₁₆	—	8	1	no	—
463	SD₁₆	—	8	2	no	—
464	D₈	—	2	2	no	—
465	D₈	—	4	2	no	—
466	M₁₆	—	8	1	no	—
467	C₂×C₈	—	2	1	no	—
468	C₂×C₈	—	2	1	no	—
469	C₂×C₈	—	2	1	no	—
470	C₂×C₈	—	2	1	no	—
471	C₂×C₈	—	2	1	no	—
472	C₂×C₈	—	8	1	no	—
473	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
474	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
475	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
476	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
477	C₄⋊C₄	—	2	1	no	—
478	C₄⋊C₄	—	2	1	no	—
479	C₄⋊C₄	—	2	1	no	—
480	C₄⋊C₄	—	2	1	no	—
481	C₄⋊C₄	—	2	1	no	—
482	C₄⋊C₄	—	2	1	no	—
483	C₄⋊C₄	—	2	1	no	—
484	C₄⋊C₄	—	2	1	no	—
485	C₄⋊C₄	—	4	1	no	—
486	C₄⋊C₄	—	4	1	no	—
487	C₄⋊C₄	—	4	1	no	—
488	C₄⋊C₄	—	4	1	no	—
489	C₄⋊C₄	—	4	1	no	—
490	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
491	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
492	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
493	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
494	C₂²⋊C₄	—	2	1	no	—
495	C₂²⋊C₄	—	2	1	no	—
496	C₂²⋊C₄	—	4	1	no	—
497	C₂²⋊C₄	—	4	1	no	—
498	C₂²⋊C₄	—	4	1	no	—
499	C₂²⋊C₄	—	4	1	no	—
500	C₄²	—	2	1	no	—
501	C₄²	—	2	1	no	—
502	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
503	C₂³	—	2	1	no	—
504	C₂³	—	2	1	no	—
505	C₂³	—	4	1	no	—
506	Q₈	—	2	1	no	—
507	Q₈	—	2	1	no	—
508	Q₈	—	2	1	no	—
509	Q₈	—	4	1	no	—
510	Q₈	—	4	1	no	—
511	Q₈	—	8	2	no	—
512	Q₈	—	8	1	no	—
513	Q₈	—	8	1	no	—
514	Q₈	C₂₃×#135	1	2	yes	—
515	D₄	—	2	1	no	—
516	D₄	—	2	1	no	—
517	D₄	—	2	1	no	—
518	D₄	—	4	1	no	—
519	D₄	—	4	1	no	—
520	D₄	—	4	1	no	—
521	D₄	—	4	1	no	—
522	D₄	—	8	2	no	—
523	D₄	—	8	1	no	—
524	D₄	—	16	1	no	—
525	D₄	C₂₃×#137	1	2	yes	—
526	C₂×C₄	—	2	1	no	—
527	C₂×C₄	—	2	1	no	—
528	C₂×C₄	—	2	1	no	—
529	C₂×C₄	—	2	1	no	—
530	C₂×C₄	—	2	1	no	—
531	C₂×C₄	—	2	1	no	—
532	C₂×C₄	—	4	1	no	—
533	C₂×C₄	—	4	1	no	—
534	C₂×C₄	—	4	1	no	—
535	C₂×C₄	—	4	1	no	—
536	C₂×C₄	—	4	1	no	—
537	C₂×C₄	—	4	1	no	—
538	C₂×C₄	—	4	1	no	—
539	C₂×C₄	—	8	2	no	—
540	C₂×C₄	—	8	2	no	—
541	C₂×C₄	—	16	1	no	—
542	C₂×C₄	—	16	1	no	—
543	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
544	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
545	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
546	C₈	—	2	1	no	—
547	C₈	—	2	1	no	—
548	C₈	—	2	1	no	—
549	C₈	—	2	1	no	—
550	C₈	—	4	1	no	—
551	C₈	—	4	1	no	—
552	C₈	—	4	1	no	—
553	C₈	—	4	1	no	—
554	C₈	—	4	1	no	—
555	C₂²	—	2	2	no	—
556	C₂²	—	4	1	no	—
557	C₂²	—	4	1	no	—
558	C₂²	—	4	1	no	—
559	C₂²	—	4	1	no	—
560	C₂²	—	4	1	no	—
561	C₂²	—	8	1	no	—
562	C₂²	—	8	1	no	—
563	C₂²	—	8	1	no	—
564	C₂²	D₄²×C₂₃	1	1	yes	characteristic
565	C₄	—	2	2	no	—
566	C₄	—	2	1	no	—
567	C₄	—	4	1	no	—
568	C₄	—	4	1	no	—
569	C₄	—	4	1	no	—
570	C₄	—	4	1	no	—
571	C₄	—	4	1	no	—
572	C₄	—	8	1	no	—
573	C₄	—	8	1	no	—
574	C₄	—	8	1	no	—
575	C₄	C₂×C₂₃×#135	1	1	yes	characteristic
576	C₄	C₂×C₂₃×#137	1	1	yes	characteristic
577	C₄	C₂₃×#576	1	1	yes	characteristic
578	C₄	C₂₃×#578	1	1	yes	characteristic
579	C₂	—	4	2	no	—
580	C₂	—	8	1	no	—
581	C₂	—	16	1	no	—
582	C₂	C₂₃×#3270	1	1	yes	characteristic
583	C₂	C₂₃×#3280	1	1	yes	characteristic
584	C₂	C₂₃×#3288	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
prime radical	C₂₃×#3358	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×#3380	C₂	semidirect
3. C₂₃×#3333	C₂	semidirect
4. C₂₃×#3299	C₂	semidirect
5. C₂₃×#1719	C₂	semidirect
6. C₂₃×#1703	C₂	semidirect
7. C₂₃×#1682	C₂	semidirect
8. C₂₃×#1677	C₂	semidirect
9. C₂₃×#1676	C₂	semidirect
10. C₂₃×#1644	C₂	semidirect
11. C₂₃×#1637	C₂	semidirect
12. C₂₃×#1631	C₂	semidirect
13. C₂₃×#1622	C₂	semidirect
14. C₂₃×#1581	C₂	semidirect
15. C₂₃×#1579	C₂	semidirect
210. #19781	C₂₃	direct (N × H)
263. #3380	C₂×C₂₃	semidirect
265. #3333	C₂×C₂₃	semidirect
266. #3299	C₂×C₂₃	semidirect
267. #1719	C₂×C₂₃	semidirect
268. #1703	C₂×C₂₃	semidirect
269. #1682	C₂×C₂₃	semidirect
270. #1677	C₂×C₂₃	semidirect
271. #1676	C₂×C₂₃	semidirect
272. #1644	C₂×C₂₃	semidirect
273. #1637	C₂×C₂₃	semidirect
274. #1631	C₂×C₂₃	semidirect
275. #1622	C₂×C₂₃	semidirect
276. #1581	C₂×C₂₃	semidirect
277. #1579	C₂×C₂₃	semidirect
436. C₂₃	#19781	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂₃×#3380	2
C₂₃×#3358	2
C₂₃×#3333	2
C₂₃×#3299	2
C₂₃×#1719	2
C₂₃×#1703	2
C₂₃×#1682	2
C₂₃×#1677	2
C₂₃×#1676	2
C₂₃×#1644	2
C₂₃×#1637	2
C₂₃×#1631	2
C₂₃×#1622	2
C₂₃×#1581	2
C₂₃×#1579	2
#19781	23

Subgroups — C23×#19781 #19167407

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#19781 #19167407