Subgroups — C₂₃×#19861 #19167487

Order 5888.

596 conjugacy classes of subgroups.

The subgroup lattice has 572 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×#3357	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3335	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#3303	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#1723	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1691	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1684	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1675	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1675	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1643	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1635	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#1634	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂₃×#1632	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	C₂₃×#555	C₂²	1	1	yes	characteristic
17	C₂₃×#554	C₂²	1	1	yes	characteristic
18	C₂₃×#554	C₂²	1	1	yes	characteristic
19	C₂₃×#547	C₂²	1	1	yes	characteristic
20	C₂₃×#500	C₂²	1	1	yes	characteristic
21	C₂₃×#500	C₂²	1	1	yes	characteristic
22	C₂₃×#499	—	2	1	no	—
23	C₂₃×#498	C₂²	1	1	yes	characteristic
24	C₂₃×#494	C₂²	1	1	yes	characteristic
25	C₂₃×#487	C₂²	1	1	yes	characteristic
26	C₂₃×#487	C₂²	1	1	yes	characteristic
27	C₂₃×#485	—	2	1	no	—
28	C₂₃×#485	—	2	1	no	—
29	C₂₃×#483	—	2	1	no	—
30	C₂₃×#481	—	2	1	no	—
31	C₂₃×#480	—	2	1	no	—
32	C₂₃×#478	—	2	1	no	—
33	C₂₃×#478	—	2	1	no	—
34	C₂₃×#477	C₂²	1	1	yes	characteristic
35	C₂₃×#476	C₂²	1	1	yes	characteristic
36	C₂₃×#476	C₂²	1	1	yes	characteristic
37	C₂₃×#476	C₂²	1	1	yes	characteristic
38	C₂₃×#476	C₂²	1	1	yes	characteristic
39	C₂₃×#476	C₂²	1	1	yes	characteristic
40	C₂₃×#475	C₂²	1	1	yes	characteristic
41	C₂₃×#468	—	2	1	no	—
42	C₂₃×#465	—	2	1	no	—
43	C₂₃×#465	—	2	1	no	—
44	C₂₃×#461	C₂²	1	1	yes	characteristic
45	C₂₃×#449	—	2	1	no	—
46	C₈×Q₈×C₂₃	C₂²	1	1	yes	characteristic
47	C₄×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic
48	C₄×D₈×C₂₃	—	2	1	no	—
49	C₄×D₈×C₂₃	C₂²	1	1	yes	characteristic
50	C₂₃×#436	C₂²	1	1	yes	characteristic
51	C₈×D₄×C₂₃	C₂²	1	1	yes	characteristic
52	C₈×D₄×C₂₃	C₂²	1	1	yes	characteristic
53	C₂₃×#428	—	2	1	no	—
54	C₂×C₂₃×#106	—	2	1	no	—
55	C₂₃×#416	—	2	1	no	—
56	C₂₃×#335	C₂²	1	1	yes	characteristic
57	C₂₃×#335	C₂²	1	1	yes	characteristic
58	C₂₃×#333	C₂²	1	1	yes	characteristic
59	C₂₃×#333	C₂²	1	1	yes	characteristic
60	C₂₃×#333	C₂²	1	1	yes	characteristic
61	C₂₃×#332	C₂²	1	1	yes	characteristic
62	C₂₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
63	C₂₃×D₄⋊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	C₂×Q₁₆×C₂₃	—	4	1	no	—
69	C₂×SD₁₆×C₂₃	—	2	1	no	—
70	C₂×SD₁₆×C₂₃	D₄	1	1	yes	characteristic
71	C₂×D₈×C₂₃	D₄	1	1	yes	characteristic
72	C₂×M₁₆×C₂₃	—	2	1	no	—
73	C₂²×C₈×C₂₃	—	2	1	no	—
74	C₂²×C₈×C₂₃	—	2	1	no	—
75	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
76	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
77	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
78	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
79	C₂₃×#125	—	2	1	no	—
80	C₂₃×#125	—	2	1	no	—
81	C₂₃×#124	—	4	1	no	—
82	C₂₃×#123	—	4	1	no	—
83	C₂₃×C₂²⋊Q₈	—	2	1	no	—
84	C₂₃×C₂²⋊Q₈	—	2	1	no	—
85	C₂₃×C₂²⋊Q₈	—	2	1	no	—
86	C₂₃×C₂²⋊Q₈	—	4	1	no	—
87	C₂₃×C₂²⋊Q₈	—	4	1	no	—
88	C₂₃×#121	—	2	1	no	—
89	C₂₃×#121	—	2	1	no	—
90	C₂₃×#121	—	2	1	no	—
91	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
92	C₄×D₄×C₂₃	—	2	1	no	—
93	C₄×D₄×C₂₃	—	2	1	no	—
94	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
95	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
96	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
97	C₂₃×#117	—	2	1	no	—
98	C₂₃×#117	—	2	1	no	—
99	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
100	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
101	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
102	C₂₃×#107	—	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	D₄	1	1	yes	characteristic
108	C₂₃×#106	—	2	1	no	—
109	C₂₃×#106	—	2	1	no	—
110	C₂₃×#106	—	2	1	no	—
111	C₂₃×#106	—	2	1	no	—
112	C₂₃×#106	—	2	1	no	—
113	C₂₃×#106	—	2	1	no	—
114	C₂₃×#106	—	2	1	no	—
115	C₂₃×#106	—	2	1	no	—
116	C₂₃×#106	D₄	1	1	yes	characteristic
117	C₂₃×C₄⋊C₈	—	2	1	no	—
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	—	4	1	no	—
128	C₂₃×#103	D₄	1	1	yes	characteristic
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	D₄	1	1	yes	characteristic
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	C₂₃×C₂²⋊C₈	—	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	—	2	1	no	—
150	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
151	C₂×D₄×C₂₃	—	4	1	no	—
152	C₂×D₄×C₂₃	—	4	1	no	—
153	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
154	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
155	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
156	C₂²×C₄×C₂₃	—	2	1	no	—
157	C₂²×C₄×C₂₃	—	2	1	no	—
158	C₂²×C₄×C₂₃	—	2	1	no	—
159	C₂²×C₄×C₂₃	—	4	1	no	—
160	C₂²×C₄×C₂₃	—	4	1	no	—
161	C₂²×C₄×C₂₃	—	4	1	no	—
162	Q₁₆×C₂₃	—	8	2	no	—
163	SD₁₆×C₂₃	—	2	2	no	—
164	SD₁₆×C₂₃	—	8	1	no	—
165	D₈×C₂₃	—	2	2	no	—
166	M₁₆×C₂₃	—	4	2	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	—	4	2	no	—
173	C₂×C₈×C₂₃	—	8	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	—	2	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	—	4	1	no	—
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	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
196	C₂²⋊C₄×C₂₃	—	2	1	no	—
197	C₂²⋊C₄×C₂₃	—	2	1	no	—
198	C₂²⋊C₄×C₂₃	—	2	1	no	—
199	C₂²⋊C₄×C₂₃	—	4	1	no	—
200	C₂²⋊C₄×C₂₃	—	4	1	no	—
201	C₂²⋊C₄×C₂₃	—	4	1	no	—
202	C₂²⋊C₄×C₂₃	—	4	1	no	—
203	C₂²⋊C₄×C₂₃	—	4	1	no	—
204	C₄²×C₂₃	—	2	1	no	—
205	C₄²×C₂₃	—	4	1	no	—
206	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
207	#19861	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
208	C₂³×C₂₃	—	2	1	no	—
209	C₂³×C₂₃	—	2	1	no	—
210	C₂³×C₂₃	—	2	1	no	—
211	Q₈×C₂₃	—	2	1	no	—
212	Q₈×C₂₃	—	2	1	no	—
213	Q₈×C₂₃	—	4	1	no	—
214	Q₈×C₂₃	—	4	1	no	—
215	Q₈×C₂₃	—	8	1	no	—
216	Q₈×C₂₃	—	8	1	no	—
217	Q₈×C₂₃	—	8	1	no	—
218	D₄×C₂₃	—	2	1	no	—
219	D₄×C₂₃	—	2	1	no	—
220	D₄×C₂₃	—	2	1	no	—
221	D₄×C₂₃	—	2	1	no	—
222	D₄×C₂₃	—	2	1	no	—
223	D₄×C₂₃	—	8	2	no	—
224	D₄×C₂₃	—	16	1	no	—
225	D₄×C₂₃	#137	1	2	yes	—
226	C₂×C₄×C₂₃	—	2	1	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	—	4	1	no	—
232	C₂×C₄×C₂₃	—	4	1	no	—
233	C₂×C₄×C₂₃	—	4	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	—	4	1	no	—
241	C₂×C₄×C₂₃	—	8	2	no	—
242	C₂×C₄×C₂₃	—	16	1	no	—
243	C₂×C₄×C₂₃	—	16	1	no	—
244	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
245	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
246	C₂×C₄×C₂₃	H(5,𝔽₂)	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	—	4	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	#135	1	2	yes	—
256	#3380	C₂×C₂₃	1	1	yes	characteristic, complemented
257	#3357	C₂×C₂₃	1	1	yes	characteristic, complemented
258	#3335	C₂×C₂₃	1	1	yes	characteristic, complemented
259	#3303	C₂×C₂₃	1	1	yes	characteristic
260	#1723	C₂×C₂₃	1	1	yes	characteristic, complemented
261	#1691	C₂×C₂₃	1	1	yes	characteristic, complemented
262	#1684	C₂×C₂₃	1	1	yes	characteristic, complemented
263	#1675	C₂×C₂₃	1	1	yes	characteristic, complemented
264	#1675	C₂×C₂₃	1	1	yes	characteristic, complemented
265	#1643	C₂×C₂₃	1	1	yes	characteristic, complemented
266	#1635	C₂×C₂₃	1	1	yes	characteristic, complemented
267	#1634	C₂×C₂₃	1	1	yes	characteristic, complemented
268	#1632	C₂×C₂₃	1	1	yes	characteristic, complemented
269	#1581	C₂×C₂₃	1	1	yes	characteristic, complemented
270	SD₁₆⋊C₈	C₂×C₂₃	1	1	yes	characteristic, complemented
271	C₂²×C₂₃	—	2	2	no	—
272	C₂²×C₂₃	—	4	1	no	—
273	C₂²×C₂₃	—	4	1	no	—
274	C₂²×C₂₃	—	4	1	no	—
275	C₂²×C₂₃	—	4	1	no	—
276	C₂²×C₂₃	—	4	1	no	—
277	C₂²×C₂₃	—	4	1	no	—
278	C₂²×C₂₃	—	4	1	no	—
279	C₂²×C₂₃	—	4	1	no	—
280	C₂²×C₂₃	D₄²	1	1	yes	characteristic
281	C₄×C₂₃	—	2	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	—	8	1	no	—
288	C₄×C₂₃	—	8	1	no	—
289	C₄×C₂₃	—	8	1	no	—
290	C₄×C₂₃	—	8	1	no	—
291	C₄×C₂₃	C₂×#135	1	1	yes	characteristic
292	C₄×C₂₃	C₂×#137	1	1	yes	characteristic
293	C₄×C₂₃	#577	1	1	yes	characteristic
294	C₄×C₂₃	#578	1	1	yes	characteristic
295	#555	C₂²×C₂₃	1	1	yes	characteristic
296	#554	C₂²×C₂₃	1	1	yes	characteristic
297	#554	C₂²×C₂₃	1	1	yes	characteristic
298	#547	C₂²×C₂₃	1	1	yes	characteristic
299	#500	C₂²×C₂₃	1	1	yes	characteristic
300	#500	C₂²×C₂₃	1	1	yes	characteristic
301	#499	—	2	1	no	—
302	#498	C₂²×C₂₃	1	1	yes	characteristic
303	#494	C₂²×C₂₃	1	1	yes	characteristic
304	#487	C₂²×C₂₃	1	1	yes	characteristic
305	#487	C₂²×C₂₃	1	1	yes	characteristic
306	#485	—	2	1	no	—
307	#485	—	2	1	no	—
308	#483	—	2	1	no	—
309	#481	—	2	1	no	—
310	#480	—	2	1	no	—
311	#478	—	2	1	no	—
312	#478	—	2	1	no	—
313	#477	C₂²×C₂₃	1	1	yes	characteristic
314	#476	C₂²×C₂₃	1	1	yes	characteristic
315	#476	C₂²×C₂₃	1	1	yes	characteristic
316	#476	C₂²×C₂₃	1	1	yes	characteristic
317	#476	C₂²×C₂₃	1	1	yes	characteristic
318	#476	C₂²×C₂₃	1	1	yes	characteristic
319	#475	C₂²×C₂₃	1	1	yes	characteristic
320	#468	—	2	1	no	—
321	#465	—	2	1	no	—
322	#465	—	2	1	no	—
323	#461	C₂²×C₂₃	1	1	yes	characteristic
324	#449	—	2	1	no	—
325	C₈×Q₈	C₂²×C₂₃	1	1	yes	characteristic
326	C₄×SD₁₆	C₂²×C₂₃	1	1	yes	characteristic
327	C₄×D₈	—	2	1	no	—
328	C₄×D₈	C₂²×C₂₃	1	1	yes	characteristic
329	#436	C₂²×C₂₃	1	1	yes	characteristic
330	C₈×D₄	C₂²×C₂₃	1	1	yes	characteristic
331	C₈×D₄	C₂²×C₂₃	1	1	yes	characteristic
332	#428	—	2	1	no	—
333	C₂×#106	—	2	1	no	—
334	#416	—	2	1	no	—
335	#335	C₂²×C₂₃	1	1	yes	characteristic
336	#335	C₂²×C₂₃	1	1	yes	characteristic
337	#333	C₂²×C₂₃	1	1	yes	characteristic
338	#333	C₂²×C₂₃	1	1	yes	characteristic
339	#333	C₂²×C₂₃	1	1	yes	characteristic
340	#332	C₂²×C₂₃	1	1	yes	characteristic
341	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
342	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
343	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
344	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
345	#322	C₂²×C₂₃	1	1	yes	characteristic
346	C₂×C₂₃	—	4	2	no	—
347	C₂×C₂₃	—	8	1	no	—
348	C₂×C₂₃	—	8	1	no	—
349	C₂×C₂₃	#3266	1	1	yes	characteristic
350	C₂×C₂₃	#3283	1	1	yes	characteristic
351	C₂×C₂₃	#3288	1	1	yes	characteristic
352	C₂×#41	—	2	1	no	—
353	C₂×Q₁₆	—	4	1	no	—
354	C₂×SD₁₆	—	2	1	no	—
355	C₂×SD₁₆	D₄×C₂₃	1	1	yes	characteristic
356	C₂×D₈	D₄×C₂₃	1	1	yes	characteristic
357	C₂×M₁₆	—	2	1	no	—
358	C₂²×C₈	—	2	1	no	—
359	C₂²×C₈	—	2	1	no	—
360	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
361	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
362	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
363	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
364	#125	—	2	1	no	—
365	#125	—	2	1	no	—
366	#124	—	4	1	no	—
367	#123	—	4	1	no	—
368	C₂²⋊Q₈	—	2	1	no	—
369	C₂²⋊Q₈	—	2	1	no	—
370	C₂²⋊Q₈	—	2	1	no	—
371	C₂²⋊Q₈	—	4	1	no	—
372	C₂²⋊Q₈	—	4	1	no	—
373	#121	—	2	1	no	—
374	#121	—	2	1	no	—
375	#121	—	2	1	no	—
376	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
377	C₄×D₄	—	2	1	no	—
378	C₄×D₄	—	2	1	no	—
379	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
380	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
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×C₄⋊C₄	—	2	1	no	—
387	#107	—	2	1	no	—
388	#107	—	2	1	no	—
389	#107	—	2	1	no	—
390	#107	—	2	1	no	—
391	#107	—	2	1	no	—
392	#107	D₄×C₂₃	1	1	yes	characteristic
393	#106	—	2	1	no	—
394	#106	—	2	1	no	—
395	#106	—	2	1	no	—
396	#106	—	2	1	no	—
397	#106	—	2	1	no	—
398	#106	—	2	1	no	—
399	#106	—	2	1	no	—
400	#106	—	2	1	no	—
401	#106	D₄×C₂₃	1	1	yes	characteristic
402	C₄⋊C₈	—	2	1	no	—
403	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
404	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
405	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
406	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
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	—	4	1	no	—
413	#103	D₄×C₂₃	1	1	yes	characteristic
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	D₄×C₂₃	1	1	yes	characteristic
420	#102	D₄×C₂₃	1	1	yes	characteristic
421	#102	D₄×C₂₃	1	1	yes	characteristic
422	C₂²⋊C₈	—	2	1	no	—
423	C₂²⋊C₈	—	2	1	no	—
424	C₂²⋊C₈	—	2	1	no	—
425	C₄×C₈	—	2	1	no	—
426	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
427	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
428	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
429	C₂₃	#19861	1	1	yes	characteristic, direct product factor, Sylow
430	#41	—	4	2	no	—
431	#41	—	8	1	no	—
432	C₂×Q₈	—	2	1	no	—
433	C₂×Q₈	—	2	1	no	—
434	C₂×Q₈	—	2	1	no	—
435	C₂×Q₈	—	2	1	no	—
436	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
437	C₂×D₄	—	4	1	no	—
438	C₂×D₄	—	4	1	no	—
439	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
440	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
441	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
442	C₂²×C₄	—	2	1	no	—
443	C₂²×C₄	—	2	1	no	—
444	C₂²×C₄	—	2	1	no	—
445	C₂²×C₄	—	4	1	no	—
446	C₂²×C₄	—	4	1	no	—
447	C₂²×C₄	—	4	1	no	—
448	Q₁₆	—	8	2	no	—
449	SD₁₆	—	2	2	no	—
450	SD₁₆	—	8	1	no	—
451	D₈	—	2	2	no	—
452	M₁₆	—	4	2	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	—	4	2	no	—
459	C₂×C₈	—	8	1	no	—
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	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
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	—	4	1	no	—
474	C₄⋊C₄	—	4	1	no	—
475	C₄⋊C₄	—	4	1	no	—
476	C₄⋊C₄	—	4	1	no	—
477	C₄⋊C₄	—	4	1	no	—
478	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
479	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
480	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
481	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
482	C₂²⋊C₄	—	2	1	no	—
483	C₂²⋊C₄	—	2	1	no	—
484	C₂²⋊C₄	—	2	1	no	—
485	C₂²⋊C₄	—	4	1	no	—
486	C₂²⋊C₄	—	4	1	no	—
487	C₂²⋊C₄	—	4	1	no	—
488	C₂²⋊C₄	—	4	1	no	—
489	C₂²⋊C₄	—	4	1	no	—
490	C₄²	—	2	1	no	—
491	C₄²	—	4	1	no	—
492	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
493	C₂³	—	2	1	no	—
494	C₂³	—	2	1	no	—
495	C₂³	—	2	1	no	—
496	Q₈	—	2	1	no	—
497	Q₈	—	2	1	no	—
498	Q₈	—	4	1	no	—
499	Q₈	—	4	1	no	—
500	Q₈	—	8	1	no	—
501	Q₈	—	8	1	no	—
502	Q₈	—	8	1	no	—
503	D₄	—	2	1	no	—
504	D₄	—	2	1	no	—
505	D₄	—	2	1	no	—
506	D₄	—	2	1	no	—
507	D₄	—	2	1	no	—
508	D₄	—	8	2	no	—
509	D₄	—	16	1	no	—
510	D₄	C₂₃×#137	1	2	yes	—
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	—	4	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	—	4	1	no	—
524	C₂×C₄	—	4	1	no	—
525	C₂×C₄	—	4	1	no	—
526	C₂×C₄	—	8	2	no	—
527	C₂×C₄	—	16	1	no	—
528	C₂×C₄	—	16	1	no	—
529	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
530	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
531	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
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	C₈	C₂₃×#135	1	2	yes	—
541	C₂²	—	2	2	no	—
542	C₂²	—	4	1	no	—
543	C₂²	—	4	1	no	—
544	C₂²	—	4	1	no	—
545	C₂²	—	4	1	no	—
546	C₂²	—	4	1	no	—
547	C₂²	—	4	1	no	—
548	C₂²	—	4	1	no	—
549	C₂²	—	4	1	no	—
550	C₂²	D₄²×C₂₃	1	1	yes	characteristic
551	C₄	—	2	1	no	—
552	C₄	—	4	1	no	—
553	C₄	—	4	1	no	—
554	C₄	—	4	1	no	—
555	C₄	—	4	1	no	—
556	C₄	—	4	1	no	—
557	C₄	—	8	1	no	—
558	C₄	—	8	1	no	—
559	C₄	—	8	1	no	—
560	C₄	—	8	1	no	—
561	C₄	C₂×C₂₃×#135	1	1	yes	characteristic
562	C₄	C₂×C₂₃×#137	1	1	yes	characteristic
563	C₄	C₂₃×#577	1	1	yes	characteristic
564	C₄	C₂₃×#578	1	1	yes	characteristic
565	C₂	—	4	2	no	—
566	C₂	—	8	1	no	—
567	C₂	—	8	1	no	—
568	C₂	C₂₃×#3266	1	1	yes	characteristic
569	C₂	C₂₃×#3283	1	1	yes	characteristic
570	C₂	C₂₃×#3288	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
prime radical	C₂₃×#3303	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
2. C₂₃×#3357	C₂	semidirect
3. C₂₃×#3335	C₂	semidirect
5. C₂₃×#1723	C₂	semidirect
6. C₂₃×#1691	C₂	semidirect
7. C₂₃×#1684	C₂	semidirect
8. C₂₃×#1675	C₂	semidirect
9. C₂₃×#1675	C₂	semidirect
10. C₂₃×#1643	C₂	semidirect
11. C₂₃×#1635	C₂	semidirect
12. C₂₃×#1634	C₂	semidirect
13. C₂₃×#1632	C₂	semidirect
14. C₂₃×#1581	C₂	semidirect
15. C₂₃×SD₁₆⋊C₈	C₂	semidirect
207. #19861	C₂₃	direct (N × H)
256. #3380	C₂×C₂₃	semidirect
257. #3357	C₂×C₂₃	semidirect
258. #3335	C₂×C₂₃	semidirect
260. #1723	C₂×C₂₃	semidirect
261. #1691	C₂×C₂₃	semidirect
262. #1684	C₂×C₂₃	semidirect
263. #1675	C₂×C₂₃	semidirect
264. #1675	C₂×C₂₃	semidirect
265. #1643	C₂×C₂₃	semidirect
266. #1635	C₂×C₂₃	semidirect
267. #1634	C₂×C₂₃	semidirect
268. #1632	C₂×C₂₃	semidirect
269. #1581	C₂×C₂₃	semidirect
270. SD₁₆⋊C₈	C₂×C₂₃	semidirect
429. C₂₃	#19861	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×#3357	2
C₂₃×#3335	2
C₂₃×#3303	2
C₂₃×#1723	2
C₂₃×#1691	2
C₂₃×#1684	2
C₂₃×#1675	2
C₂₃×#1643	2
C₂₃×#1635	2
C₂₃×#1634	2
C₂₃×#1632	2
C₂₃×#1581	2
C₂₃×SD₁₆⋊C₈	2
#19861	23

Subgroups — C23×#19861 #19167487

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#19861 #19167487