Subgroups — C₂₃×#19583 #19167209

Order 5888.

580 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×#3248	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#3239	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#3209	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#2994	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#1700	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1699	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1690	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1689	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1680	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1679	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1670	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#1669	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂₃×#1573	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂₃×#1552	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×C₂₃×#332	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂₃×#533	C₂²	1	1	yes	characteristic
17	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
18	C₂₃×#529	C₂²	1	1	yes	characteristic
19	C₂₃×#501	—	2	1	no	—
20	C₂₃×#494	C₂²	1	1	yes	characteristic
21	C₂₃×#494	C₂²	1	1	yes	characteristic
22	C₂₃×#489	—	2	1	no	—
23	C₂₃×#487	C₂²	1	1	yes	characteristic
24	C₂₃×#487	C₂²	1	1	yes	characteristic
25	C₂₃×#484	—	2	1	no	—
26	C₂₃×#481	—	2	1	no	—
27	C₂₃×#479	—	2	1	no	—
28	C₂₃×#475	C₂²	1	1	yes	characteristic
29	C₂₃×#475	C₂²	1	1	yes	characteristic
30	C₂₃×#475	C₂²	1	1	yes	characteristic
31	C₂₃×#474	C₂²	1	1	yes	characteristic
32	C₂₃×#474	C₂²	1	1	yes	characteristic
33	C₂₃×#474	C₂²	1	1	yes	characteristic
34	C₂₃×#467	—	2	1	no	—
35	C₂₃×#465	—	2	1	no	—
36	C₂₃×#464	—	2	1	no	—
37	C₂₃×#462	—	2	1	no	—
38	C₂₃×#462	—	2	1	no	—
39	C₂₃×#462	C₂²	1	1	yes	characteristic
40	C₂₃×#461	C₂²	1	1	yes	characteristic
41	C₂₃×#451	—	2	1	no	—
42	C₂₃×#451	—	2	1	no	—
43	C₂₃×#446	C₂²	1	1	yes	characteristic
44	C₈×Q₈×C₂₃	C₂²	1	1	yes	characteristic
45	C₂₃×#436	C₂²	1	1	yes	characteristic
46	C₂₃×#435	—	2	1	no	—
47	C₈×D₄×C₂₃	—	2	1	no	—
48	C₈×D₄×C₂₃	C₂²	1	1	yes	characteristic
49	C₂₃×#433	—	2	1	no	—
50	C₂₃×#432	C₂²	1	1	yes	characteristic
51	C₂₃×#431	C₂²	1	1	yes	characteristic
52	C₂₃×#424	—	2	1	no	—
53	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
54	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
55	C₂₃×#335	C₂²	1	1	yes	characteristic
56	C₂₃×#335	C₂²	1	1	yes	characteristic
57	C₂₃×#334	C₂²	1	1	yes	characteristic
58	C₂₃×#334	C₂²	1	1	yes	characteristic
59	C₂₃×#332	C₂²	1	1	yes	characteristic
60	C₂₃×#332	C₂²	1	1	yes	characteristic
61	C₂₃×#332	C₂²	1	1	yes	characteristic
62	C₂₃×#332	C₂²	1	1	yes	characteristic
63	C₂₃×#332	C₂²	1	1	yes	characteristic
64	C₂₃×#332	C₂²	1	1	yes	characteristic
65	C₂₃×#332	C₂²	1	1	yes	characteristic
66	C₂₃×#332	C₂²	1	1	yes	characteristic
67	C₂²×Q₈×C₂₃	—	2	1	no	—
68	C₂×Q₁₆×C₂₃	—	2	1	no	—
69	C₂×Q₁₆×C₂₃	—	2	1	no	—
70	C₂×Q₁₆×C₂₃	—	2	1	no	—
71	C₂×Q₁₆×C₂₃	—	2	1	no	—
72	C₂×Q₁₆×C₂₃	—	2	1	no	—
73	C₂×SD₁₆×C₂₃	—	2	1	no	—
74	C₂×M₁₆×C₂₃	—	2	1	no	—
75	C₂²×C₈×C₂₃	—	2	1	no	—
76	C₂²×C₈×C₂₃	—	2	1	no	—
77	C₂²×C₈×C₂₃	—	2	1	no	—
78	C₂₃×C₄⋊Q₈	—	2	1	no	—
79	C₂₃×C₄⋊Q₈	—	2	1	no	—
80	C₂₃×C₄⋊Q₈	—	2	1	no	—
81	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
82	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
83	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
84	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
85	C₂₃×#126	—	2	1	no	—
86	C₂₃×#125	—	2	1	no	—
87	C₂₃×#125	D₄	1	1	yes	characteristic
88	C₂₃×#124	D₄	1	1	yes	characteristic
89	C₂₃×#123	—	2	1	no	—
90	C₂₃×C₂²⋊Q₈	—	2	1	no	—
91	C₂₃×C₂²⋊Q₈	—	2	1	no	—
92	C₂₃×C₂²⋊Q₈	D₄	1	1	yes	characteristic
93	C₂₃×#121	D₄	1	1	yes	characteristic
94	C₄×Q₈×C₂₃	—	2	1	no	—
95	C₄×Q₈×C₂₃	—	2	1	no	—
96	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
97	C₄×D₄×C₂₃	—	2	1	no	—
98	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
99	C₂₃×#117	—	2	1	no	—
100	C₂₃×#117	—	2	1	no	—
101	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
102	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
103	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
104	C₂₃×#107	—	2	1	no	—
105	C₂₃×#107	—	2	1	no	—
106	C₂₃×#107	—	2	1	no	—
107	C₂₃×#106	—	2	1	no	—
108	C₂₃×#106	—	2	1	no	—
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	D₄	1	1	yes	characteristic
115	C₂₃×C₄⋊C₈	D₄	1	1	yes	characteristic
116	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
117	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
118	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
119	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
120	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
121	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
122	C₂₃×#103	—	2	1	no	—
123	C₂₃×#103	—	2	1	no	—
124	C₂₃×#103	—	2	1	no	—
125	C₂₃×#103	—	2	1	no	—
126	C₂₃×#103	—	2	1	no	—
127	C₂₃×#103	—	2	1	no	—
128	C₂₃×#103	—	2	1	no	—
129	C₂₃×#103	—	2	1	no	—
130	C₂₃×#103	—	2	1	no	—
131	C₂₃×#102	—	2	1	no	—
132	C₂₃×#102	—	2	1	no	—
133	C₂₃×#102	—	2	1	no	—
134	C₂₃×C₂²⋊C₈	—	2	1	no	—
135	C₂₃×C₂²⋊C₈	—	2	1	no	—
136	C₂₃×C₂²⋊C₈	—	2	1	no	—
137	C₂₃×C₂²⋊C₈	D₄	1	1	yes	characteristic
138	C₂₃×C₂²⋊C₈	D₄	1	1	yes	characteristic
139	C₂₃×#97	—	2	1	no	—
140	C₂₃×#97	—	2	1	no	—
141	C₄×C₈×C₂₃	—	2	1	no	—
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	—	2	1	no	—
147	C₂×Q₈×C₂₃	—	2	1	no	—
148	C₂×Q₈×C₂₃	—	4	1	no	—
149	C₂×Q₈×C₂₃	—	8	1	no	—
150	C₂×Q₈×C₂₃	—	8	1	no	—
151	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
152	C₂×D₄×C₂₃	—	2	1	no	—
153	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
154	C₂²×C₄×C₂₃	—	2	1	no	—
155	C₂²×C₄×C₂₃	—	4	1	no	—
156	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
157	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
158	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
159	Q₁₆×C₂₃	—	8	1	no	—
160	Q₁₆×C₂₃	—	8	1	no	—
161	Q₁₆×C₂₃	—	8	1	no	—
162	Q₁₆×C₂₃	—	8	1	no	—
163	Q₁₆×C₂₃	—	8	1	no	—
164	SD₁₆×C₂₃	—	8	1	no	—
165	M₁₆×C₂₃	—	8	1	no	—
166	C₂×C₈×C₂₃	—	2	1	no	—
167	C₂×C₈×C₂₃	—	2	1	no	—
168	C₂×C₈×C₂₃	—	2	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	—	8	1	no	—
173	C₂×C₈×C₂₃	—	8	1	no	—
174	C₂×C₈×C₂₃	—	8	1	no	—
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	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
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	—	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	—	4	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	C₂×D₄	1	1	yes	characteristic
196	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
197	C₂²⋊C₄×C₂₃	—	2	1	no	—
198	C₂²⋊C₄×C₂₃	—	2	1	no	—
199	C₂²⋊C₄×C₂₃	—	2	1	no	—
200	C₂²⋊C₄×C₂₃	—	2	1	no	—
201	C₂²⋊C₄×C₂₃	—	2	1	no	—
202	C₂²⋊C₄×C₂₃	—	4	1	no	—
203	C₄²×C₂₃	—	2	1	no	—
204	C₄²×C₂₃	—	4	1	no	—
205	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
206	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
207	C₄²×C₂₃	#41	1	1	yes	characteristic
208	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
209	#19583	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
210	C₂³×C₂₃	—	2	1	no	—
211	C₂³×C₂₃	#121	1	1	yes	characteristic
212	Q₈×C₂₃	—	2	1	no	—
213	Q₈×C₂₃	—	2	1	no	—
214	Q₈×C₂₃	—	4	1	no	—
215	Q₈×C₂₃	—	4	1	no	—
216	Q₈×C₂₃	—	4	1	no	—
217	Q₈×C₂₃	—	4	1	no	—
218	Q₈×C₂₃	—	4	1	no	—
219	Q₈×C₂₃	—	4	1	no	—
220	Q₈×C₂₃	—	4	1	no	—
221	Q₈×C₂₃	—	4	1	no	—
222	Q₈×C₂₃	—	8	2	no	—
223	D₄×C₂₃	—	2	1	no	—
224	D₄×C₂₃	—	2	1	no	—
225	D₄×C₂₃	—	4	2	no	—
226	C₂×C₄×C₂₃	—	2	2	no	—
227	C₂×C₄×C₂₃	—	2	1	no	—
228	C₂×C₄×C₂₃	—	2	1	no	—
229	C₂×C₄×C₂₃	—	2	1	no	—
230	C₂×C₄×C₂₃	—	2	1	no	—
231	C₂×C₄×C₂₃	—	2	1	no	—
232	C₂×C₄×C₂₃	—	2	1	no	—
233	C₂×C₄×C₂₃	—	2	1	no	—
234	C₂×C₄×C₂₃	—	4	1	no	—
235	C₂×C₄×C₂₃	—	4	1	no	—
236	C₂×C₄×C₂₃	—	4	1	no	—
237	C₂×C₄×C₂₃	—	4	1	no	—
238	C₂×C₄×C₂₃	—	4	1	no	—
239	C₂×C₄×C₂₃	—	4	1	no	—
240	C₂×C₄×C₂₃	—	8	2	no	—
241	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
242	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
243	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
244	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
245	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
246	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
247	C₈×C₂₃	—	2	1	no	—
248	C₈×C₂₃	—	2	1	no	—
249	C₈×C₂₃	—	2	1	no	—
250	C₈×C₂₃	—	2	1	no	—
251	C₈×C₂₃	—	4	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	#3248	C₂×C₂₃	1	1	yes	characteristic
258	#3239	C₂×C₂₃	1	1	yes	characteristic
259	#3209	C₂×C₂₃	1	1	yes	characteristic, complemented
260	#2994	C₂×C₂₃	1	1	yes	characteristic
261	#1700	C₂×C₂₃	1	1	yes	characteristic, complemented
262	#1699	C₂×C₂₃	1	1	yes	characteristic, complemented
263	#1690	C₂×C₂₃	1	1	yes	characteristic, complemented
264	#1689	C₂×C₂₃	1	1	yes	characteristic, complemented
265	#1680	C₂×C₂₃	1	1	yes	characteristic, complemented
266	#1679	C₂×C₂₃	1	1	yes	characteristic, complemented
267	#1670	C₂×C₂₃	1	1	yes	characteristic, complemented
268	#1669	C₂×C₂₃	1	1	yes	characteristic, complemented
269	#1573	C₂×C₂₃	1	1	yes	characteristic, complemented
270	#1552	C₂×C₂₃	1	1	yes	characteristic, complemented
271	C₂×#332	C₂×C₂₃	1	1	yes	characteristic, complemented
272	C₂²×C₂₃	—	2	1	no	—
273	C₂²×C₂₃	—	2	1	no	—
274	C₂²×C₂₃	—	4	1	no	—
275	C₂²×C₂₃	—	4	1	no	—
276	C₂²×C₂₃	—	4	1	no	—
277	C₂²×C₂₃	#473	1	2	yes	—
278	C₂²×C₂₃	C₂×#121	1	1	yes	characteristic
279	C₄×C₂₃	—	2	1	no	—
280	C₄×C₂₃	—	2	1	no	—
281	C₄×C₂₃	—	2	1	no	—
282	C₄×C₂₃	—	2	1	no	—
283	C₄×C₂₃	—	4	1	no	—
284	C₄×C₂₃	—	4	1	no	—
285	C₄×C₂₃	—	4	1	no	—
286	C₄×C₂₃	—	8	1	no	—
287	C₄×C₂₃	—	8	1	no	—
288	C₄×C₂₃	—	8	1	no	—
289	C₄×C₂₃	—	8	1	no	—
290	C₄×C₂₃	#576	1	1	yes	characteristic
291	C₄×C₂₃	#577	1	1	yes	characteristic
292	C₄×C₂₃	#577	1	1	yes	characteristic
293	C₄×C₂₃	#578	1	1	yes	characteristic
294	#533	C₂²×C₂₃	1	1	yes	characteristic
295	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
296	#529	C₂²×C₂₃	1	1	yes	characteristic
297	#501	—	2	1	no	—
298	#494	C₂²×C₂₃	1	1	yes	characteristic
299	#494	C₂²×C₂₃	1	1	yes	characteristic
300	#489	—	2	1	no	—
301	#487	C₂²×C₂₃	1	1	yes	characteristic
302	#487	C₂²×C₂₃	1	1	yes	characteristic
303	#484	—	2	1	no	—
304	#481	—	2	1	no	—
305	#479	—	2	1	no	—
306	#475	C₂²×C₂₃	1	1	yes	characteristic
307	#475	C₂²×C₂₃	1	1	yes	characteristic
308	#475	C₂²×C₂₃	1	1	yes	characteristic
309	#474	C₂²×C₂₃	1	1	yes	characteristic
310	#474	C₂²×C₂₃	1	1	yes	characteristic
311	#474	C₂²×C₂₃	1	1	yes	characteristic
312	#467	—	2	1	no	—
313	#465	—	2	1	no	—
314	#464	—	2	1	no	—
315	#462	—	2	1	no	—
316	#462	—	2	1	no	—
317	#462	C₂²×C₂₃	1	1	yes	characteristic
318	#461	C₂²×C₂₃	1	1	yes	characteristic
319	#451	—	2	1	no	—
320	#451	—	2	1	no	—
321	#446	C₂²×C₂₃	1	1	yes	characteristic
322	C₈×Q₈	C₂²×C₂₃	1	1	yes	characteristic
323	#436	C₂²×C₂₃	1	1	yes	characteristic
324	#435	—	2	1	no	—
325	C₈×D₄	—	2	1	no	—
326	C₈×D₄	C₂²×C₂₃	1	1	yes	characteristic
327	#433	—	2	1	no	—
328	#432	C₂²×C₂₃	1	1	yes	characteristic
329	#431	C₂²×C₂₃	1	1	yes	characteristic
330	#424	—	2	1	no	—
331	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
332	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
333	#335	C₂²×C₂₃	1	1	yes	characteristic
334	#335	C₂²×C₂₃	1	1	yes	characteristic
335	#334	C₂²×C₂₃	1	1	yes	characteristic
336	#334	C₂²×C₂₃	1	1	yes	characteristic
337	#332	C₂²×C₂₃	1	1	yes	characteristic
338	#332	C₂²×C₂₃	1	1	yes	characteristic
339	#332	C₂²×C₂₃	1	1	yes	characteristic
340	#332	C₂²×C₂₃	1	1	yes	characteristic
341	#332	C₂²×C₂₃	1	1	yes	characteristic
342	#332	C₂²×C₂₃	1	1	yes	characteristic
343	#332	C₂²×C₂₃	1	1	yes	characteristic
344	#332	C₂²×C₂₃	1	1	yes	characteristic
345	C₂×C₂₃	—	2	2	no	—
346	C₂×C₂₃	—	8	1	no	—
347	C₂×C₂₃	#3045	1	1	yes	characteristic
348	C₂×C₂₃	#3063	1	1	yes	characteristic
349	C₂×C₂₃	C₂×#473	1	1	yes	characteristic
350	C₂²×Q₈	—	2	1	no	—
351	C₂×Q₁₆	—	2	1	no	—
352	C₂×Q₁₆	—	2	1	no	—
353	C₂×Q₁₆	—	2	1	no	—
354	C₂×Q₁₆	—	2	1	no	—
355	C₂×Q₁₆	—	2	1	no	—
356	C₂×SD₁₆	—	2	1	no	—
357	C₂×M₁₆	—	2	1	no	—
358	C₂²×C₈	—	2	1	no	—
359	C₂²×C₈	—	2	1	no	—
360	C₂²×C₈	—	2	1	no	—
361	C₄⋊Q₈	—	2	1	no	—
362	C₄⋊Q₈	—	2	1	no	—
363	C₄⋊Q₈	—	2	1	no	—
364	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
365	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
366	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
367	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
368	#126	—	2	1	no	—
369	#125	—	2	1	no	—
370	#125	D₄×C₂₃	1	1	yes	characteristic
371	#124	D₄×C₂₃	1	1	yes	characteristic
372	#123	—	2	1	no	—
373	C₂²⋊Q₈	—	2	1	no	—
374	C₂²⋊Q₈	—	2	1	no	—
375	C₂²⋊Q₈	D₄×C₂₃	1	1	yes	characteristic
376	#121	D₄×C₂₃	1	1	yes	characteristic
377	C₄×Q₈	—	2	1	no	—
378	C₄×Q₈	—	2	1	no	—
379	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
380	C₄×D₄	—	2	1	no	—
381	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
382	#117	—	2	1	no	—
383	#117	—	2	1	no	—
384	C₂×C₄⋊C₄	—	2	1	no	—
385	C₂×C₄⋊C₄	—	2	1	no	—
386	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
387	#107	—	2	1	no	—
388	#107	—	2	1	no	—
389	#107	—	2	1	no	—
390	#106	—	2	1	no	—
391	#106	—	2	1	no	—
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	D₄×C₂₃	1	1	yes	characteristic
398	C₄⋊C₈	D₄×C₂₃	1	1	yes	characteristic
399	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
400	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
401	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
402	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
403	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
404	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
405	#103	—	2	1	no	—
406	#103	—	2	1	no	—
407	#103	—	2	1	no	—
408	#103	—	2	1	no	—
409	#103	—	2	1	no	—
410	#103	—	2	1	no	—
411	#103	—	2	1	no	—
412	#103	—	2	1	no	—
413	#103	—	2	1	no	—
414	#102	—	2	1	no	—
415	#102	—	2	1	no	—
416	#102	—	2	1	no	—
417	C₂²⋊C₈	—	2	1	no	—
418	C₂²⋊C₈	—	2	1	no	—
419	C₂²⋊C₈	—	2	1	no	—
420	C₂²⋊C₈	D₄×C₂₃	1	1	yes	characteristic
421	C₂²⋊C₈	D₄×C₂₃	1	1	yes	characteristic
422	#97	—	2	1	no	—
423	#97	—	2	1	no	—
424	C₄×C₈	—	2	1	no	—
425	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
426	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
427	C₂₃	#19583	1	1	yes	characteristic, direct product factor, Sylow
428	C₂×Q₈	—	2	1	no	—
429	C₂×Q₈	—	2	1	no	—
430	C₂×Q₈	—	2	1	no	—
431	C₂×Q₈	—	2	1	no	—
432	C₂×Q₈	—	4	1	no	—
433	C₂×Q₈	—	8	1	no	—
434	C₂×Q₈	—	8	1	no	—
435	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
436	C₂×D₄	—	2	1	no	—
437	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
438	C₂²×C₄	—	2	1	no	—
439	C₂²×C₄	—	4	1	no	—
440	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
441	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
442	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
443	Q₁₆	—	8	1	no	—
444	Q₁₆	—	8	1	no	—
445	Q₁₆	—	8	1	no	—
446	Q₁₆	—	8	1	no	—
447	Q₁₆	—	8	1	no	—
448	SD₁₆	—	8	1	no	—
449	M₁₆	—	8	1	no	—
450	C₂×C₈	—	2	1	no	—
451	C₂×C₈	—	2	1	no	—
452	C₂×C₈	—	2	1	no	—
453	C₂×C₈	—	2	1	no	—
454	C₂×C₈	—	2	1	no	—
455	C₂×C₈	—	2	1	no	—
456	C₂×C₈	—	8	1	no	—
457	C₂×C₈	—	8	1	no	—
458	C₂×C₈	—	8	1	no	—
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	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
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	—	2	1	no	—
471	C₄⋊C₄	—	2	1	no	—
472	C₄⋊C₄	—	2	1	no	—
473	C₄⋊C₄	—	2	1	no	—
474	C₄⋊C₄	—	2	1	no	—
475	C₄⋊C₄	—	4	1	no	—
476	C₄⋊C₄	—	4	1	no	—
477	C₄⋊C₄	—	4	1	no	—
478	C₄⋊C₄	—	4	1	no	—
479	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
480	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
481	C₂²⋊C₄	—	2	1	no	—
482	C₂²⋊C₄	—	2	1	no	—
483	C₂²⋊C₄	—	2	1	no	—
484	C₂²⋊C₄	—	2	1	no	—
485	C₂²⋊C₄	—	2	1	no	—
486	C₂²⋊C₄	—	4	1	no	—
487	C₄²	—	2	1	no	—
488	C₄²	—	4	1	no	—
489	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
490	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
491	C₄²	#41×C₂₃	1	1	yes	characteristic
492	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
493	C₂³	—	2	1	no	—
494	C₂³	C₂₃×#121	1	1	yes	characteristic
495	Q₈	—	2	1	no	—
496	Q₈	—	2	1	no	—
497	Q₈	—	4	1	no	—
498	Q₈	—	4	1	no	—
499	Q₈	—	4	1	no	—
500	Q₈	—	4	1	no	—
501	Q₈	—	4	1	no	—
502	Q₈	—	4	1	no	—
503	Q₈	—	4	1	no	—
504	Q₈	—	4	1	no	—
505	Q₈	—	8	2	no	—
506	D₄	—	2	1	no	—
507	D₄	—	2	1	no	—
508	D₄	—	4	2	no	—
509	C₂×C₄	—	2	2	no	—
510	C₂×C₄	—	2	1	no	—
511	C₂×C₄	—	2	1	no	—
512	C₂×C₄	—	2	1	no	—
513	C₂×C₄	—	2	1	no	—
514	C₂×C₄	—	2	1	no	—
515	C₂×C₄	—	2	1	no	—
516	C₂×C₄	—	2	1	no	—
517	C₂×C₄	—	4	1	no	—
518	C₂×C₄	—	4	1	no	—
519	C₂×C₄	—	4	1	no	—
520	C₂×C₄	—	4	1	no	—
521	C₂×C₄	—	4	1	no	—
522	C₂×C₄	—	4	1	no	—
523	C₂×C₄	—	8	2	no	—
524	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
525	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
526	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
527	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
528	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
529	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
530	C₈	—	2	1	no	—
531	C₈	—	2	1	no	—
532	C₈	—	2	1	no	—
533	C₈	—	2	1	no	—
534	C₈	—	4	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	—	2	1	no	—
541	C₂²	—	2	1	no	—
542	C₂²	—	4	1	no	—
543	C₂²	—	4	1	no	—
544	C₂²	—	4	1	no	—
545	C₂²	C₂₃×#473	1	2	yes	—
546	C₂²	C₂×C₂₃×#121	1	1	yes	characteristic
547	C₄	—	2	1	no	—
548	C₄	—	2	1	no	—
549	C₄	—	2	1	no	—
550	C₄	—	2	1	no	—
551	C₄	—	4	1	no	—
552	C₄	—	4	1	no	—
553	C₄	—	4	1	no	—
554	C₄	—	8	1	no	—
555	C₄	—	8	1	no	—
556	C₄	—	8	1	no	—
557	C₄	—	8	1	no	—
558	C₄	C₂₃×#576	1	1	yes	characteristic
559	C₄	C₂₃×#577	1	1	yes	characteristic
560	C₄	C₂₃×#577	1	1	yes	characteristic
561	C₄	C₂₃×#578	1	1	yes	characteristic
562	C₂	—	2	2	no	—
563	C₂	—	8	1	no	—
564	C₂	C₂₃×#3045	1	1	yes	characteristic
565	C₂	C₂₃×#3063	1	1	yes	characteristic
566	C₂	C₂×C₂₃×#473	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
prime radical	C₂₃×#121	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂₃×#3209	C₂	semidirect
5. C₂₃×#1700	C₂	semidirect
6. C₂₃×#1699	C₂	semidirect
7. C₂₃×#1690	C₂	semidirect
8. C₂₃×#1689	C₂	semidirect
9. C₂₃×#1680	C₂	semidirect
10. C₂₃×#1679	C₂	semidirect
11. C₂₃×#1670	C₂	semidirect
12. C₂₃×#1669	C₂	semidirect
13. C₂₃×#1573	C₂	semidirect
14. C₂₃×#1552	C₂	semidirect
15. C₂×C₂₃×#332	C₂	semidirect
209. #19583	C₂₃	direct (N × H)
259. #3209	C₂×C₂₃	semidirect
261. #1700	C₂×C₂₃	semidirect
262. #1699	C₂×C₂₃	semidirect
263. #1690	C₂×C₂₃	semidirect
264. #1689	C₂×C₂₃	semidirect
265. #1680	C₂×C₂₃	semidirect
266. #1679	C₂×C₂₃	semidirect
267. #1670	C₂×C₂₃	semidirect
268. #1669	C₂×C₂₃	semidirect
269. #1573	C₂×C₂₃	semidirect
270. #1552	C₂×C₂₃	semidirect
271. C₂×#332	C₂×C₂₃	semidirect
427. C₂₃	#19583	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×#3248	2
C₂₃×#3239	2
C₂₃×#3209	2
C₂₃×#2994	2
C₂₃×#1700	2
C₂₃×#1699	2
C₂₃×#1690	2
C₂₃×#1689	2
C₂₃×#1680	2
C₂₃×#1679	2
C₂₃×#1670	2
C₂₃×#1669	2
C₂₃×#1573	2
C₂₃×#1552	2
C₂×C₂₃×#332	2
#19583	23

Subgroups — C23×#19583 #19167209

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#19583 #19167209