Subgroups — C₂₃×#3211 #15670830

Order 2944.

410 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#529	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂₃×#124	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#495	C₂	1	2	yes	maximal, complemented
4	C₂₃×#488	C₂	1	2	yes	maximal, complemented
5	C₂₃×#464	C₂	1	2	yes	maximal, complemented
6	C₂₃×#463	C₂	1	2	yes	maximal, complemented
7	C₂₃×#451	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#450	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#448	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#447	C₂	1	1	yes	maximal, characteristic
11	C₂₃×#431	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×Q₈×C₂₃	C₂²	1	1	yes	characteristic
13	C₂²×D₄×C₂₃	C₂²	1	1	yes	characteristic
14	C₂×Q₁₆×C₂₃	C₂²	1	2	yes	complemented
15	C₂×SD₁₆×C₂₃	C₂²	1	2	yes	—
16	C₂×SD₁₆×C₂₃	C₂²	1	2	yes	complemented
17	C₂×D₈×C₂₃	C₂²	1	2	yes	—
18	C₂₃×#126	—	2	1	no	—
19	C₂₃×#125	C₂²	1	1	yes	characteristic, complemented
20	C₂₃×#124	—	2	1	no	—
21	C₂₃×#124	—	2	1	no	—
22	C₂₃×#124	C₂²	1	2	yes	—
23	C₂₃×#124	C₂²	1	2	yes	—
24	C₂₃×#124	C₂²	1	1	yes	characteristic, complemented
25	C₂₃×#123	—	2	1	no	—
26	C₂₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
27	C₂₃×#121	C₂²	1	1	yes	characteristic
28	C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic, complemented
29	C₄×D₄×C₂₃	—	2	1	no	—
30	C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic, complemented
31	C₂₃×#117	—	2	1	no	—
32	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
33	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
34	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
35	C₂₃×C₄⋊C₈	C₂²	1	2	yes	complemented
36	C₂₃×#103	C₂²	1	2	yes	complemented
37	C₂₃×#103	C₂²	1	2	yes	complemented
38	C₂₃×#102	C₂²	1	2	yes	—
39	C₂₃×#102	C₂²	1	2	yes	—
40	C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
41	C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
42	C₄×C₈×C₂₃	C₂²	1	2	yes	complemented
43	C₂⁴×C₂₃	—	2	1	no	—
44	C₂×Q₈×C₂₃	—	2	1	no	—
45	C₂×Q₈×C₂₃	—	4	2	no	—
46	C₂×Q₈×C₂₃	C₂³	1	2	yes	—
47	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
48	C₂×D₄×C₂₃	—	2	1	no	—
49	C₂×D₄×C₂₃	—	2	1	no	—
50	C₂×D₄×C₂₃	—	4	2	no	—
51	C₂×D₄×C₂₃	C₂³	1	2	yes	—
52	C₂×D₄×C₂₃	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₂₃	—	2	1	no	—
54	C₂²×C₄×C₂₃	—	2	1	no	—
55	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
56	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
57	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
58	Q₁₆×C₂₃	—	4	2	no	—
59	SD₁₆×C₂₃	—	4	2	no	—
60	SD₁₆×C₂₃	—	4	2	no	—
61	D₈×C₂₃	—	4	2	no	—
62	C₂×C₈×C₂₃	C₂³	1	2	yes	—
63	C₂×C₈×C₂₃	C₂³	1	2	yes	—
64	C₄⋊C₄×C₂₃	—	2	1	no	—
65	C₄⋊C₄×C₂₃	—	2	1	no	—
66	C₄⋊C₄×C₂₃	—	2	1	no	—
67	C₄⋊C₄×C₂₃	—	2	1	no	—
68	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
69	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
70	C₂²⋊C₄×C₂₃	—	2	1	no	—
71	C₂²⋊C₄×C₂₃	—	2	2	no	—
72	C₂²⋊C₄×C₂₃	—	2	2	no	—
73	C₂²⋊C₄×C₂₃	—	2	2	no	—
74	C₂²⋊C₄×C₂₃	—	2	2	no	—
75	C₂²⋊C₄×C₂₃	—	2	1	no	—
76	C₂²⋊C₄×C₂₃	—	2	1	no	—
77	C₂²⋊C₄×C₂₃	—	2	1	no	—
78	C₂²⋊C₄×C₂₃	—	2	1	no	—
79	C₄²×C₂₃	—	2	1	no	—
80	C₄²×C₂₃	D₄	1	1	yes	characteristic
81	C₄²×C₂₃	D₄	1	1	yes	characteristic
82	C₄²×C₂₃	C₂³	1	1	yes	characteristic
83	C₄²×C₂₃	C₂³	1	1	yes	characteristic
84	C₂³×C₂₃	—	2	1	no	—
85	C₂³×C₂₃	—	2	2	no	—
86	C₂³×C₂₃	—	4	2	no	—
87	C₂³×C₂₃	—	4	2	no	—
88	C₂³×C₂₃	—	4	2	no	—
89	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
90	Q₈×C₂₃	—	2	2	no	—
91	Q₈×C₂₃	—	2	2	no	—
92	Q₈×C₂₃	—	2	2	no	—
93	Q₈×C₂₃	—	4	2	no	—
94	D₄×C₂₃	—	2	2	no	—
95	D₄×C₂₃	—	2	2	no	—
96	D₄×C₂₃	—	2	2	no	—
97	D₄×C₂₃	—	4	2	no	—
98	D₄×C₂₃	—	4	2	no	—
99	C₂×C₄×C₂₃	—	2	2	no	—
100	C₂×C₄×C₂₃	—	2	2	no	—
101	C₂×C₄×C₂₃	—	2	1	no	—
102	C₂×C₄×C₂₃	—	2	1	no	—
103	C₂×C₄×C₂₃	—	2	2	no	—
104	C₂×C₄×C₂₃	—	2	1	no	—
105	C₂×C₄×C₂₃	—	4	1	no	—
106	C₂×C₄×C₂₃	—	4	2	no	—
107	C₂×C₄×C₂₃	—	4	2	no	—
108	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
111	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
112	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
113	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
114	C₈×C₂₃	—	2	2	no	—
115	C₈×C₂₃	—	2	2	no	—
116	#3211	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
117	C₂²×C₂₃	—	2	1	no	—
118	C₂²×C₂₃	—	2	1	no	—
119	C₂²×C₂₃	—	4	2	no	—
120	C₂²×C₂₃	—	4	1	no	—
121	C₂²×C₂₃	—	4	1	no	—
122	C₂²×C₂₃	—	4	2	no	—
123	C₂²×C₂₃	—	4	1	no	—
124	C₂²×C₂₃	—	4	2	no	—
125	C₂²×C₂₃	—	8	4	no	—
126	C₂²×C₂₃	#135	1	2	yes	—
127	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
128	C₄×C₂₃	—	2	1	no	—
129	C₄×C₂₃	—	2	1	no	—
130	C₄×C₂₃	—	2	1	no	—
131	C₄×C₂₃	—	2	1	no	—
132	C₄×C₂₃	—	2	1	no	—
133	C₄×C₂₃	—	4	2	no	—
134	C₄×C₂₃	—	4	1	no	—
135	C₄×C₂₃	—	4	1	no	—
136	C₄×C₂₃	—	4	1	no	—
137	C₄×C₂₃	H(5,𝔽₂)	1	2	yes	—
138	#529	C₂×C₂₃	1	1	yes	characteristic, complemented
139	C₂×#124	C₂×C₂₃	1	1	yes	characteristic, complemented
140	#495	C₂×C₂₃	1	2	yes	complemented
141	#488	C₂×C₂₃	1	2	yes	complemented
142	#464	C₂×C₂₃	1	2	yes	complemented
143	#463	C₂×C₂₃	1	2	yes	complemented
144	#451	C₂×C₂₃	1	1	yes	characteristic, complemented
145	#450	C₂×C₂₃	1	1	yes	characteristic, complemented
146	#448	C₂×C₂₃	1	1	yes	characteristic, complemented
147	#447	C₂×C₂₃	1	1	yes	characteristic
148	#431	C₂×C₂₃	1	1	yes	characteristic, complemented
149	C₂×C₂₃	—	2	2	no	—
150	C₂×C₂₃	—	8	1	no	—
151	C₂×C₂₃	—	8	2	no	—
152	C₂×C₂₃	#535	1	1	yes	characteristic
153	C₂×C₂₃	C₂×#135	1	1	yes	characteristic
154	C₂×C₂₃	#575	1	1	yes	characteristic
155	C₂²×Q₈	C₂²×C₂₃	1	1	yes	characteristic
156	C₂²×D₄	C₂²×C₂₃	1	1	yes	characteristic
157	C₂×Q₁₆	C₂²×C₂₃	1	2	yes	complemented
158	C₂×SD₁₆	C₂²×C₂₃	1	2	yes	—
159	C₂×SD₁₆	C₂²×C₂₃	1	2	yes	complemented
160	C₂×D₈	C₂²×C₂₃	1	2	yes	—
161	#126	—	2	1	no	—
162	#125	C₂²×C₂₃	1	1	yes	characteristic, complemented
163	#124	—	2	1	no	—
164	#124	—	2	1	no	—
165	#124	C₂²×C₂₃	1	2	yes	—
166	#124	C₂²×C₂₃	1	2	yes	—
167	#124	C₂²×C₂₃	1	1	yes	characteristic, complemented
168	#123	—	2	1	no	—
169	C₂²⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
170	#121	C₂²×C₂₃	1	1	yes	characteristic
171	C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
172	C₄×D₄	—	2	1	no	—
173	C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic, complemented
174	#117	—	2	1	no	—
175	C₂×C₂²⋊C₄	—	2	1	no	—
176	C₂×C₂²⋊C₄	—	2	1	no	—
177	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
178	C₄⋊C₈	C₂²×C₂₃	1	2	yes	complemented
179	#103	C₂²×C₂₃	1	2	yes	complemented
180	#103	C₂²×C₂₃	1	2	yes	complemented
181	#102	C₂²×C₂₃	1	2	yes	—
182	#102	C₂²×C₂₃	1	2	yes	—
183	C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
184	C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
185	C₄×C₈	C₂²×C₂₃	1	2	yes	complemented
186	C₂₃	#3211	1	1	yes	characteristic, direct product factor, Sylow
187	C₂⁴	—	2	1	no	—
188	C₂×Q₈	—	2	1	no	—
189	C₂×Q₈	—	4	2	no	—
190	C₂×Q₈	C₂³×C₂₃	1	2	yes	—
191	C₂×Q₈	C₂³×C₂₃	1	1	yes	characteristic
192	C₂×D₄	—	2	1	no	—
193	C₂×D₄	—	2	1	no	—
194	C₂×D₄	—	4	2	no	—
195	C₂×D₄	C₂³×C₂₃	1	2	yes	—
196	C₂×D₄	C₂³×C₂₃	1	1	yes	characteristic
197	C₂²×C₄	—	2	1	no	—
198	C₂²×C₄	—	2	1	no	—
199	C₂²×C₄	D₄×C₂₃	1	1	yes	characteristic
200	C₂²×C₄	D₄×C₂₃	1	1	yes	characteristic
201	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
202	Q₁₆	—	4	2	no	—
203	SD₁₆	—	4	2	no	—
204	SD₁₆	—	4	2	no	—
205	D₈	—	4	2	no	—
206	C₂×C₈	C₂³×C₂₃	1	2	yes	—
207	C₂×C₈	C₂³×C₂₃	1	2	yes	—
208	C₄⋊C₄	—	2	1	no	—
209	C₄⋊C₄	—	2	1	no	—
210	C₄⋊C₄	—	2	1	no	—
211	C₄⋊C₄	—	2	1	no	—
212	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
213	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
214	C₂²⋊C₄	—	2	1	no	—
215	C₂²⋊C₄	—	2	2	no	—
216	C₂²⋊C₄	—	2	2	no	—
217	C₂²⋊C₄	—	2	2	no	—
218	C₂²⋊C₄	—	2	2	no	—
219	C₂²⋊C₄	—	2	1	no	—
220	C₂²⋊C₄	—	2	1	no	—
221	C₂²⋊C₄	—	2	1	no	—
222	C₂²⋊C₄	—	2	1	no	—
223	C₄²	—	2	1	no	—
224	C₄²	D₄×C₂₃	1	1	yes	characteristic
225	C₄²	D₄×C₂₃	1	1	yes	characteristic
226	C₄²	C₂³×C₂₃	1	1	yes	characteristic
227	C₄²	C₂³×C₂₃	1	1	yes	characteristic
228	C₂³	—	2	1	no	—
229	C₂³	—	2	2	no	—
230	C₂³	—	4	2	no	—
231	C₂³	—	4	2	no	—
232	C₂³	—	4	2	no	—
233	C₂³	C₂×D₄×C₂₃	1	1	yes	characteristic
234	Q₈	—	2	2	no	—
235	Q₈	—	2	2	no	—
236	Q₈	—	2	2	no	—
237	Q₈	—	4	2	no	—
238	D₄	—	2	2	no	—
239	D₄	—	2	2	no	—
240	D₄	—	2	2	no	—
241	D₄	—	4	2	no	—
242	D₄	—	4	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	4	1	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
253	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
254	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
255	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
256	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
257	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
258	C₈	—	2	2	no	—
259	C₈	—	2	2	no	—
260	C₂²	—	2	1	no	—
261	C₂²	—	2	1	no	—
262	C₂²	—	4	1	no	—
263	C₂²	—	4	2	no	—
264	C₂²	—	4	1	no	—
265	C₂²	—	4	2	no	—
266	C₂²	—	4	2	no	—
267	C₂²	—	4	1	no	—
268	C₂²	—	8	4	no	—
269	C₂²	C₂₃×#135	1	2	yes	—
270	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
271	C₄	—	2	1	no	—
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	4	1	no	—
277	C₄	—	4	1	no	—
278	C₄	—	4	2	no	—
279	C₄	—	4	1	no	—
280	C₄	C₂₃×H(5,𝔽₂)	1	2	yes	—
281	C₂	—	2	2	no	—
282	C₂	—	8	2	no	—
283	C₂	—	8	1	no	—
284	C₂	C₂₃×#535	1	1	yes	characteristic
285	C₂	C₂×C₂₃×#135	1	1	yes	characteristic
286	C₂	C₂₃×#575	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₂₃
prime radical	C₂₃×#447	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ 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
1. C₂₃×#529	C₂	semidirect
2. C₂×C₂₃×#124	C₂	semidirect
3. C₂₃×#495	C₂	semidirect
4. C₂₃×#488	C₂	semidirect
5. C₂₃×#464	C₂	semidirect
6. C₂₃×#463	C₂	semidirect
7. C₂₃×#451	C₂	semidirect
8. C₂₃×#450	C₂	semidirect
9. C₂₃×#448	C₂	semidirect
11. C₂₃×#431	C₂	semidirect
14. C₂×Q₁₆×C₂₃	C₂²	semidirect
16. C₂×SD₁₆×C₂₃	C₂²	semidirect
19. C₂₃×#125	C₂²	semidirect
24. C₂₃×#124	C₂²	semidirect
28. C₄×Q₈×C₂₃	C₂²	semidirect
30. C₄×D₄×C₂₃	C₂²	semidirect
35. C₂₃×C₄⋊C₈	C₂²	semidirect
36. C₂₃×#103	C₂²	semidirect
37. C₂₃×#103	C₂²	semidirect
42. C₄×C₈×C₂₃	C₂²	semidirect
116. #3211	C₂₃	direct (N × H)
138. #529	C₂×C₂₃	semidirect
139. C₂×#124	C₂×C₂₃	semidirect
140. #495	C₂×C₂₃	semidirect
141. #488	C₂×C₂₃	semidirect
142. #464	C₂×C₂₃	semidirect
143. #463	C₂×C₂₃	semidirect
144. #451	C₂×C₂₃	semidirect
145. #450	C₂×C₂₃	semidirect
146. #448	C₂×C₂₃	semidirect
148. #431	C₂×C₂₃	semidirect
157. C₂×Q₁₆	C₂²×C₂₃	semidirect
159. C₂×SD₁₆	C₂²×C₂₃	semidirect
162. #125	C₂²×C₂₃	semidirect
167. #124	C₂²×C₂₃	semidirect
171. C₄×Q₈	C₂²×C₂₃	semidirect
173. C₄×D₄	C₂²×C₂₃	semidirect
178. C₄⋊C₈	C₂²×C₂₃	semidirect
179. #103	C₂²×C₂₃	semidirect
180. #103	C₂²×C₂₃	semidirect
185. C₄×C₈	C₂²×C₂₃	semidirect
186. C₂₃	#3211	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×#529	2
C₂×C₂₃×#124	2
C₂₃×#495	2
C₂₃×#488	2
C₂₃×#464	2
C₂₃×#463	2
C₂₃×#451	2
C₂₃×#450	2
C₂₃×#448	2
C₂₃×#447	2
C₂₃×#431	2
#3211	23

Subgroups — C23×#3211 #15670830

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#3211 #15670830