Subgroups — C₂₃×#19850 #19167476

Order 5888.

584 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3396	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#3375	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3365	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#3363	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1717	C₂	1	1	yes	maximal, characteristic
6	C₂₃×#1702	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1684	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1678	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1668	C₂	1	1	yes	maximal, characteristic
10	C₂₃×#1640	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1630	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#1626	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂₃×#1623	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂₃×#1584	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂₃×#1579	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂₃×#559	C₂²	1	1	yes	characteristic
17	C₂₃×#559	C₂²	1	1	yes	characteristic
18	Q₈²×C₂₃	C₂²	1	1	yes	characteristic
19	C₂₃×#557	C₂²	1	1	yes	characteristic
20	C₂₃×#495	—	2	1	no	—
21	C₂₃×#493	C₂²	1	1	yes	characteristic
22	C₂₃×#492	C₂²	1	1	yes	characteristic
23	C₂₃×#489	—	2	1	no	—
24	C₂₃×#488	—	2	1	no	—
25	C₂₃×#487	—	2	1	no	—
26	C₂₃×#487	C₂²	1	1	yes	characteristic
27	C₂₃×#486	C₂²	1	1	yes	characteristic
28	C₂₃×#486	C₂²	1	1	yes	characteristic
29	C₂₃×#486	C₂²	1	1	yes	characteristic
30	C₂₃×#478	—	2	1	no	—
31	C₂₃×#477	—	2	1	no	—
32	C₂₃×#477	C₂²	1	1	yes	characteristic
33	C₂₃×#477	C₂²	1	1	yes	characteristic
34	C₂₃×#475	C₂²	1	1	yes	characteristic
35	C₂₃×#474	—	2	1	no	—
36	C₂₃×#464	—	2	1	no	—
37	C₂₃×#464	—	2	1	no	—
38	C₂₃×#463	—	2	1	no	—
39	C₂₃×#463	—	2	1	no	—
40	C₂₃×#462	—	2	1	no	—
41	C₂₃×#462	C₂²	1	1	yes	characteristic
42	C₂₃×#460	C₂²	1	1	yes	characteristic
43	C₂₃×#460	C₂²	1	1	yes	characteristic
44	C₂₃×#460	C₂²	1	1	yes	characteristic
45	C₂₃×#460	C₂²	1	1	yes	characteristic
46	C₂₃×#446	C₂²	1	1	yes	characteristic
47	C₈×Q₈×C₂₃	C₂²	1	1	yes	characteristic
48	C₈×Q₈×C₂₃	C₂²	1	1	yes	characteristic
49	C₈×Q₈×C₂₃	C₂²	1	1	yes	characteristic
50	C₂₃×#440	—	2	1	no	—
51	C₄×Q₁₆×C₂₃	—	2	1	no	—
52	C₄×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
53	C₄×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
54	C₄×SD₁₆×C₂₃	—	2	1	no	—
55	C₄×D₈×C₂₃	—	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×#332	C₂²	1	1	yes	characteristic
60	C₂₃×#331	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×Q₈⋊C₈	C₂²	1	1	yes	characteristic
65	C₂₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
66	C₂₃×#322	C₂²	1	1	yes	characteristic
67	C₂×Q₁₆×C₂₃	—	2	1	no	—
68	C₂×Q₁₆×C₂₃	D₄	1	1	yes	characteristic
69	C₂×Q₁₆×C₂₃	D₄	1	1	yes	characteristic
70	C₂×SD₁₆×C₂₃	—	2	1	no	—
71	C₂×SD₁₆×C₂₃	—	2	1	no	—
72	C₂×SD₁₆×C₂₃	—	2	1	no	—
73	C₂×SD₁₆×C₂₃	—	2	1	no	—
74	C₂×SD₁₆×C₂₃	—	4	1	no	—
75	C₂×D₈×C₂₃	—	2	1	no	—
76	C₂×D₈×C₂₃	—	2	1	no	—
77	C₂₃×C₄⋊Q₈	—	2	1	no	—
78	C₂₃×C₄⋊Q₈	—	2	1	no	—
79	C₂₃×C₄⋊Q₈	—	2	1	no	—
80	C₂₃×C₄⋊Q₈	—	4	1	no	—
81	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
82	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
83	C₂₃×#127	C₂³	1	1	yes	characteristic
84	C₂₃×#127	C₂³	1	1	yes	characteristic
85	C₂₃×#125	—	2	1	no	—
86	C₂₃×#125	—	4	1	no	—
87	C₂₃×#124	—	2	1	no	—
88	C₂₃×#124	—	2	1	no	—
89	C₂₃×#124	—	2	1	no	—
90	C₂₃×#124	—	2	1	no	—
91	C₂₃×#123	—	4	1	no	—
92	C₂₃×#123	—	4	1	no	—
93	C₄×Q₈×C₂₃	—	2	1	no	—
94	C₄×Q₈×C₂₃	—	2	1	no	—
95	C₄×Q₈×C₂₃	—	2	1	no	—
96	C₄×Q₈×C₂₃	—	2	1	no	—
97	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
98	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
99	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
100	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
101	C₄×D₄×C₂₃	—	2	1	no	—
102	C₄×D₄×C₂₃	—	2	1	no	—
103	C₄×D₄×C₂₃	—	2	1	no	—
104	C₄×D₄×C₂₃	—	2	1	no	—
105	C₂₃×#107	—	2	1	no	—
106	C₂₃×#107	—	2	1	no	—
107	C₂₃×#107	D₄	1	1	yes	characteristic
108	C₂₃×#107	D₄	1	1	yes	characteristic
109	C₂₃×#106	—	2	1	no	—
110	C₂₃×C₄⋊C₈	—	2	1	no	—
111	C₂₃×C₄⋊C₈	—	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	—	2	1	no	—
124	C₂₃×#103	D₄	1	1	yes	characteristic
125	C₂₃×#103	D₄	1	1	yes	characteristic
126	C₂₃×#103	D₄	1	1	yes	characteristic
127	C₂₃×#103	D₄	1	1	yes	characteristic
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×#97	—	2	1	no	—
138	C₄×C₈×C₂₃	—	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	C₂×Q₈×C₂₃	—	2	1	no	—
145	C₂×Q₈×C₂₃	—	2	1	no	—
146	C₂×Q₈×C₂₃	—	4	1	no	—
147	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
148	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
149	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
150	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
151	C₂×D₄×C₂₃	—	2	1	no	—
152	C₂×D₄×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	C₂²×C₄×C₂₃	—	4	1	no	—
158	C₂²×C₄×C₂₃	—	4	1	no	—
159	Q₁₆×C₂₃	—	2	2	no	—
160	Q₁₆×C₂₃	—	4	1	no	—
161	Q₁₆×C₂₃	—	4	2	no	—
162	SD₁₆×C₂₃	—	4	2	no	—
163	SD₁₆×C₂₃	—	4	2	no	—
164	SD₁₆×C₂₃	—	8	1	no	—
165	SD₁₆×C₂₃	—	8	1	no	—
166	SD₁₆×C₂₃	—	8	2	no	—
167	D₈×C₂₃	—	4	2	no	—
168	D₈×C₂₃	—	8	1	no	—
169	C₂×C₈×C₂₃	—	2	1	no	—
170	C₂×C₈×C₂₃	—	2	1	no	—
171	C₂×C₈×C₂₃	—	2	1	no	—
172	C₂×C₈×C₂₃	—	2	1	no	—
173	C₂×C₈×C₂₃	—	2	1	no	—
174	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
175	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
176	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
177	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
178	C₄⋊C₄×C₂₃	—	2	1	no	—
179	C₄⋊C₄×C₂₃	—	2	1	no	—
180	C₄⋊C₄×C₂₃	—	2	1	no	—
181	C₄⋊C₄×C₂₃	—	2	1	no	—
182	C₄⋊C₄×C₂₃	—	2	1	no	—
183	C₄⋊C₄×C₂₃	—	2	1	no	—
184	C₄⋊C₄×C₂₃	—	2	1	no	—
185	C₄⋊C₄×C₂₃	—	2	1	no	—
186	C₄⋊C₄×C₂₃	—	4	1	no	—
187	C₄⋊C₄×C₂₃	—	4	1	no	—
188	C₄⋊C₄×C₂₃	—	4	1	no	—
189	C₄⋊C₄×C₂₃	—	4	1	no	—
190	C₄⋊C₄×C₂₃	—	4	1	no	—
191	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
192	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
193	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
194	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
195	C₂²⋊C₄×C₂₃	—	4	1	no	—
196	C₂²⋊C₄×C₂₃	—	4	1	no	—
197	C₂²⋊C₄×C₂₃	—	4	1	no	—
198	C₂²⋊C₄×C₂₃	—	4	1	no	—
199	C₂²⋊C₄×C₂₃	—	4	1	no	—
200	C₂²⋊C₄×C₂₃	—	4	1	no	—
201	C₄²×C₂₃	—	2	1	no	—
202	C₄²×C₂₃	—	2	1	no	—
203	C₄²×C₂₃	—	2	1	no	—
204	C₄²×C₂₃	—	2	1	no	—
205	C₄²×C₂₃	—	4	1	no	—
206	C₄²×C₂₃	—	4	1	no	—
207	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
208	#19850	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
209	C₂³×C₂₃	—	4	1	no	—
210	C₂³×C₂₃	—	4	1	no	—
211	Q₈×C₂₃	—	2	1	no	—
212	Q₈×C₂₃	—	2	1	no	—
213	Q₈×C₂₃	—	2	1	no	—
214	Q₈×C₂₃	—	2	1	no	—
215	Q₈×C₂₃	—	2	1	no	—
216	Q₈×C₂₃	—	2	1	no	—
217	Q₈×C₂₃	—	2	1	no	—
218	Q₈×C₂₃	—	4	1	no	—
219	Q₈×C₂₃	—	4	1	no	—
220	Q₈×C₂₃	—	8	1	no	—
221	Q₈×C₂₃	—	16	1	no	—
222	Q₈×C₂₃	#135	1	2	yes	—
223	D₄×C₂₃	—	4	1	no	—
224	D₄×C₂₃	—	4	1	no	—
225	D₄×C₂₃	—	4	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	—	4	1	no	—
231	D₄×C₂₃	—	8	1	no	—
232	D₄×C₂₃	—	8	1	no	—
233	C₂×C₄×C₂₃	—	2	1	no	—
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	—	2	1	no	—
241	C₂×C₄×C₂₃	—	4	1	no	—
242	C₂×C₄×C₂₃	—	4	1	no	—
243	C₂×C₄×C₂₃	—	16	1	no	—
244	C₂×C₄×C₂₃	—	16	1	no	—
245	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
246	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
247	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
248	C₈×C₂₃	—	2	1	no	—
249	C₈×C₂₃	—	2	1	no	—
250	C₈×C₂₃	—	2	1	no	—
251	C₈×C₂₃	—	2	1	no	—
252	C₈×C₂₃	—	4	1	no	—
253	C₈×C₂₃	—	4	1	no	—
254	C₈×C₂₃	—	4	1	no	—
255	C₈×C₂₃	—	4	1	no	—
256	C₈×C₂₃	—	4	1	no	—
257	#3396	C₂×C₂₃	1	1	yes	characteristic
258	#3375	C₂×C₂₃	1	1	yes	characteristic, complemented
259	#3365	C₂×C₂₃	1	1	yes	characteristic, complemented
260	#3363	C₂×C₂₃	1	1	yes	characteristic, complemented
261	#1717	C₂×C₂₃	1	1	yes	characteristic
262	#1702	C₂×C₂₃	1	1	yes	characteristic, complemented
263	#1684	C₂×C₂₃	1	1	yes	characteristic, complemented
264	#1678	C₂×C₂₃	1	1	yes	characteristic, complemented
265	#1668	C₂×C₂₃	1	1	yes	characteristic
266	#1640	C₂×C₂₃	1	1	yes	characteristic, complemented
267	#1630	C₂×C₂₃	1	1	yes	characteristic, complemented
268	#1626	C₂×C₂₃	1	1	yes	characteristic, complemented
269	#1623	C₂×C₂₃	1	1	yes	characteristic, complemented
270	#1584	C₂×C₂₃	1	1	yes	characteristic, complemented
271	#1579	C₂×C₂₃	1	1	yes	characteristic, complemented
272	C₂²×C₂₃	—	8	1	no	—
273	C₂²×C₂₃	—	8	1	no	—
274	C₂²×C₂₃	—	8	1	no	—
275	C₂²×C₂₃	—	8	1	no	—
276	C₂²×C₂₃	—	8	1	no	—
277	C₂²×C₂₃	—	8	1	no	—
278	C₂²×C₂₃	D₄²	1	1	yes	characteristic
279	C₄×C₂₃	—	2	2	no	—
280	C₄×C₂₃	—	2	1	no	—
281	C₄×C₂₃	—	4	1	no	—
282	C₄×C₂₃	—	4	1	no	—
283	C₄×C₂₃	—	4	1	no	—
284	C₄×C₂₃	—	4	1	no	—
285	C₄×C₂₃	—	4	1	no	—
286	C₄×C₂₃	—	4	1	no	—
287	C₄×C₂₃	—	4	1	no	—
288	C₄×C₂₃	—	8	1	no	—
289	C₄×C₂₃	—	8	1	no	—
290	C₄×C₂₃	C₂×#135	1	1	yes	characteristic
291	C₄×C₂₃	#575	1	1	yes	characteristic
292	C₄×C₂₃	#577	1	1	yes	characteristic
293	C₄×C₂₃	#578	1	1	yes	characteristic
294	#559	C₂²×C₂₃	1	1	yes	characteristic
295	#559	C₂²×C₂₃	1	1	yes	characteristic
296	Q₈²	C₂²×C₂₃	1	1	yes	characteristic
297	#557	C₂²×C₂₃	1	1	yes	characteristic
298	#495	—	2	1	no	—
299	#493	C₂²×C₂₃	1	1	yes	characteristic
300	#492	C₂²×C₂₃	1	1	yes	characteristic
301	#489	—	2	1	no	—
302	#488	—	2	1	no	—
303	#487	—	2	1	no	—
304	#487	C₂²×C₂₃	1	1	yes	characteristic
305	#486	C₂²×C₂₃	1	1	yes	characteristic
306	#486	C₂²×C₂₃	1	1	yes	characteristic
307	#486	C₂²×C₂₃	1	1	yes	characteristic
308	#478	—	2	1	no	—
309	#477	—	2	1	no	—
310	#477	C₂²×C₂₃	1	1	yes	characteristic
311	#477	C₂²×C₂₃	1	1	yes	characteristic
312	#475	C₂²×C₂₃	1	1	yes	characteristic
313	#474	—	2	1	no	—
314	#464	—	2	1	no	—
315	#464	—	2	1	no	—
316	#463	—	2	1	no	—
317	#463	—	2	1	no	—
318	#462	—	2	1	no	—
319	#462	C₂²×C₂₃	1	1	yes	characteristic
320	#460	C₂²×C₂₃	1	1	yes	characteristic
321	#460	C₂²×C₂₃	1	1	yes	characteristic
322	#460	C₂²×C₂₃	1	1	yes	characteristic
323	#460	C₂²×C₂₃	1	1	yes	characteristic
324	#446	C₂²×C₂₃	1	1	yes	characteristic
325	C₈×Q₈	C₂²×C₂₃	1	1	yes	characteristic
326	C₈×Q₈	C₂²×C₂₃	1	1	yes	characteristic
327	C₈×Q₈	C₂²×C₂₃	1	1	yes	characteristic
328	#440	—	2	1	no	—
329	C₄×Q₁₆	—	2	1	no	—
330	C₄×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic
331	C₄×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic
332	C₄×SD₁₆	—	2	1	no	—
333	C₄×D₈	—	2	1	no	—
334	#335	C₂²×C₂₃	1	1	yes	characteristic
335	#334	C₂²×C₂₃	1	1	yes	characteristic
336	#333	C₂²×C₂₃	1	1	yes	characteristic
337	#332	C₂²×C₂₃	1	1	yes	characteristic
338	#331	C₂²×C₂₃	1	1	yes	characteristic
339	#331	C₂²×C₂₃	1	1	yes	characteristic
340	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
341	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
342	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
343	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
344	#322	C₂²×C₂₃	1	1	yes	characteristic
345	C₂×C₂₃	—	16	1	no	—
346	C₂×C₂₃	—	16	1	no	—
347	C₂×C₂₃	#3271	1	1	yes	characteristic
348	C₂×C₂₃	#3285	1	1	yes	characteristic
349	C₂×C₂₃	#3288	1	1	yes	characteristic
350	C₂×Q₁₆	—	2	1	no	—
351	C₂×Q₁₆	D₄×C₂₃	1	1	yes	characteristic
352	C₂×Q₁₆	D₄×C₂₃	1	1	yes	characteristic
353	C₂×SD₁₆	—	2	1	no	—
354	C₂×SD₁₆	—	2	1	no	—
355	C₂×SD₁₆	—	2	1	no	—
356	C₂×SD₁₆	—	2	1	no	—
357	C₂×SD₁₆	—	4	1	no	—
358	C₂×D₈	—	2	1	no	—
359	C₂×D₈	—	2	1	no	—
360	C₄⋊Q₈	—	2	1	no	—
361	C₄⋊Q₈	—	2	1	no	—
362	C₄⋊Q₈	—	2	1	no	—
363	C₄⋊Q₈	—	4	1	no	—
364	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
365	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
366	#127	C₂³×C₂₃	1	1	yes	characteristic
367	#127	C₂³×C₂₃	1	1	yes	characteristic
368	#125	—	2	1	no	—
369	#125	—	4	1	no	—
370	#124	—	2	1	no	—
371	#124	—	2	1	no	—
372	#124	—	2	1	no	—
373	#124	—	2	1	no	—
374	#123	—	4	1	no	—
375	#123	—	4	1	no	—
376	C₄×Q₈	—	2	1	no	—
377	C₄×Q₈	—	2	1	no	—
378	C₄×Q₈	—	2	1	no	—
379	C₄×Q₈	—	2	1	no	—
380	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
381	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
382	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
383	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
384	C₄×D₄	—	2	1	no	—
385	C₄×D₄	—	2	1	no	—
386	C₄×D₄	—	2	1	no	—
387	C₄×D₄	—	2	1	no	—
388	#107	—	2	1	no	—
389	#107	—	2	1	no	—
390	#107	D₄×C₂₃	1	1	yes	characteristic
391	#107	D₄×C₂₃	1	1	yes	characteristic
392	#106	—	2	1	no	—
393	C₄⋊C₈	—	2	1	no	—
394	C₄⋊C₈	—	2	1	no	—
395	C₄⋊C₈	—	2	1	no	—
396	C₄⋊C₈	—	2	1	no	—
397	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
398	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
399	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
400	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
401	#103	—	2	1	no	—
402	#103	—	2	1	no	—
403	#103	—	2	1	no	—
404	#103	—	2	1	no	—
405	#103	—	2	1	no	—
406	#103	—	2	1	no	—
407	#103	D₄×C₂₃	1	1	yes	characteristic
408	#103	D₄×C₂₃	1	1	yes	characteristic
409	#103	D₄×C₂₃	1	1	yes	characteristic
410	#103	D₄×C₂₃	1	1	yes	characteristic
411	#102	—	2	1	no	—
412	#102	—	2	1	no	—
413	#102	—	2	1	no	—
414	#102	—	2	1	no	—
415	#102	—	2	1	no	—
416	#102	—	2	1	no	—
417	#102	—	2	1	no	—
418	#102	—	2	1	no	—
419	#102	—	4	1	no	—
420	#97	—	2	1	no	—
421	C₄×C₈	—	2	1	no	—
422	C₄×C₈	—	2	1	no	—
423	C₄×C₈	—	2	1	no	—
424	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
425	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
426	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
427	C₂₃	#19850	1	1	yes	characteristic, direct product factor, Sylow
428	C₂×Q₈	—	2	1	no	—
429	C₂×Q₈	—	2	1	no	—
430	C₂×Q₈	—	4	1	no	—
431	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
432	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
433	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
434	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
435	C₂×D₄	—	2	1	no	—
436	C₂×D₄	—	2	1	no	—
437	C₂×D₄	—	2	1	no	—
438	C₂×D₄	—	2	1	no	—
439	C₂×D₄	—	2	1	no	—
440	C₂×D₄	—	2	1	no	—
441	C₂²×C₄	—	4	1	no	—
442	C₂²×C₄	—	4	1	no	—
443	Q₁₆	—	2	2	no	—
444	Q₁₆	—	4	1	no	—
445	Q₁₆	—	4	2	no	—
446	SD₁₆	—	4	2	no	—
447	SD₁₆	—	4	2	no	—
448	SD₁₆	—	8	1	no	—
449	SD₁₆	—	8	1	no	—
450	SD₁₆	—	8	2	no	—
451	D₈	—	4	2	no	—
452	D₈	—	8	1	no	—
453	C₂×C₈	—	2	1	no	—
454	C₂×C₈	—	2	1	no	—
455	C₂×C₈	—	2	1	no	—
456	C₂×C₈	—	2	1	no	—
457	C₂×C₈	—	2	1	no	—
458	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
459	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
460	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
461	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
462	C₄⋊C₄	—	2	1	no	—
463	C₄⋊C₄	—	2	1	no	—
464	C₄⋊C₄	—	2	1	no	—
465	C₄⋊C₄	—	2	1	no	—
466	C₄⋊C₄	—	2	1	no	—
467	C₄⋊C₄	—	2	1	no	—
468	C₄⋊C₄	—	2	1	no	—
469	C₄⋊C₄	—	2	1	no	—
470	C₄⋊C₄	—	4	1	no	—
471	C₄⋊C₄	—	4	1	no	—
472	C₄⋊C₄	—	4	1	no	—
473	C₄⋊C₄	—	4	1	no	—
474	C₄⋊C₄	—	4	1	no	—
475	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
476	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
477	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
478	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
479	C₂²⋊C₄	—	4	1	no	—
480	C₂²⋊C₄	—	4	1	no	—
481	C₂²⋊C₄	—	4	1	no	—
482	C₂²⋊C₄	—	4	1	no	—
483	C₂²⋊C₄	—	4	1	no	—
484	C₂²⋊C₄	—	4	1	no	—
485	C₄²	—	2	1	no	—
486	C₄²	—	2	1	no	—
487	C₄²	—	2	1	no	—
488	C₄²	—	2	1	no	—
489	C₄²	—	4	1	no	—
490	C₄²	—	4	1	no	—
491	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
492	C₂³	—	4	1	no	—
493	C₂³	—	4	1	no	—
494	Q₈	—	2	1	no	—
495	Q₈	—	2	1	no	—
496	Q₈	—	2	1	no	—
497	Q₈	—	2	1	no	—
498	Q₈	—	2	1	no	—
499	Q₈	—	2	1	no	—
500	Q₈	—	2	1	no	—
501	Q₈	—	4	1	no	—
502	Q₈	—	4	1	no	—
503	Q₈	—	8	1	no	—
504	Q₈	—	16	1	no	—
505	Q₈	C₂₃×#135	1	2	yes	—
506	D₄	—	4	1	no	—
507	D₄	—	4	1	no	—
508	D₄	—	4	1	no	—
509	D₄	—	4	1	no	—
510	D₄	—	4	1	no	—
511	D₄	—	4	1	no	—
512	D₄	—	4	1	no	—
513	D₄	—	4	1	no	—
514	D₄	—	8	1	no	—
515	D₄	—	8	1	no	—
516	C₂×C₄	—	2	1	no	—
517	C₂×C₄	—	2	1	no	—
518	C₂×C₄	—	2	1	no	—
519	C₂×C₄	—	2	1	no	—
520	C₂×C₄	—	2	1	no	—
521	C₂×C₄	—	2	1	no	—
522	C₂×C₄	—	2	1	no	—
523	C₂×C₄	—	2	1	no	—
524	C₂×C₄	—	4	1	no	—
525	C₂×C₄	—	4	1	no	—
526	C₂×C₄	—	16	1	no	—
527	C₂×C₄	—	16	1	no	—
528	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
529	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
530	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
531	C₈	—	2	1	no	—
532	C₈	—	2	1	no	—
533	C₈	—	2	1	no	—
534	C₈	—	2	1	no	—
535	C₈	—	4	1	no	—
536	C₈	—	4	1	no	—
537	C₈	—	4	1	no	—
538	C₈	—	4	1	no	—
539	C₈	—	4	1	no	—
540	C₂²	—	8	1	no	—
541	C₂²	—	8	1	no	—
542	C₂²	—	8	1	no	—
543	C₂²	—	8	1	no	—
544	C₂²	—	8	1	no	—
545	C₂²	—	8	1	no	—
546	C₂²	D₄²×C₂₃	1	1	yes	characteristic
547	C₄	—	2	2	no	—
548	C₄	—	2	1	no	—
549	C₄	—	4	1	no	—
550	C₄	—	4	1	no	—
551	C₄	—	4	1	no	—
552	C₄	—	4	1	no	—
553	C₄	—	4	1	no	—
554	C₄	—	4	1	no	—
555	C₄	—	4	1	no	—
556	C₄	—	8	1	no	—
557	C₄	—	8	1	no	—
558	C₄	C₂×C₂₃×#135	1	1	yes	characteristic
559	C₄	C₂₃×#575	1	1	yes	characteristic
560	C₄	C₂₃×#577	1	1	yes	characteristic
561	C₄	C₂₃×#578	1	1	yes	characteristic
562	C₂	—	16	1	no	—
563	C₂	—	16	1	no	—
564	C₂	C₂₃×#3271	1	1	yes	characteristic
565	C₂	C₂₃×#3285	1	1	yes	characteristic
566	C₂	C₂₃×#3288	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
prime radical	C₂₃×#493	C₂²
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
2. C₂₃×#3375	C₂	semidirect
3. C₂₃×#3365	C₂	semidirect
4. C₂₃×#3363	C₂	semidirect
6. C₂₃×#1702	C₂	semidirect
7. C₂₃×#1684	C₂	semidirect
8. C₂₃×#1678	C₂	semidirect
10. C₂₃×#1640	C₂	semidirect
11. C₂₃×#1630	C₂	semidirect
12. C₂₃×#1626	C₂	semidirect
13. C₂₃×#1623	C₂	semidirect
14. C₂₃×#1584	C₂	semidirect
15. C₂₃×#1579	C₂	semidirect
208. #19850	C₂₃	direct (N × H)
258. #3375	C₂×C₂₃	semidirect
259. #3365	C₂×C₂₃	semidirect
260. #3363	C₂×C₂₃	semidirect
262. #1702	C₂×C₂₃	semidirect
263. #1684	C₂×C₂₃	semidirect
264. #1678	C₂×C₂₃	semidirect
266. #1640	C₂×C₂₃	semidirect
267. #1630	C₂×C₂₃	semidirect
268. #1626	C₂×C₂₃	semidirect
269. #1623	C₂×C₂₃	semidirect
270. #1584	C₂×C₂₃	semidirect
271. #1579	C₂×C₂₃	semidirect
427. C₂₃	#19850	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×#3396	2
C₂₃×#3375	2
C₂₃×#3365	2
C₂₃×#3363	2
C₂₃×#1717	2
C₂₃×#1702	2
C₂₃×#1684	2
C₂₃×#1678	2
C₂₃×#1668	2
C₂₃×#1640	2
C₂₃×#1630	2
C₂₃×#1626	2
C₂₃×#1623	2
C₂₃×#1584	2
C₂₃×#1579	2
#19850	23

Subgroups — C23×#19850 #19167476

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#19850 #19167476