Subgroups — C₂₃×#3091 #15670710

Order 2944.

420 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×#525	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#117	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#485	C₂	1	2	yes	maximal, complemented
4	C₂₃×#484	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#483	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#482	C₂	1	2	yes	maximal, complemented
7	C₂₃×#481	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#480	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#427	C₂	1	2	yes	maximal, complemented
10	C₂₃×#416	C₂	1	2	yes	maximal, complemented
11	C₂×C₂₃×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₂₃	C₂²	1	1	yes	characteristic
13	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₂₃	C₂²	1	2	yes	—
15	C₂₃×#123	—	2	1	no	—
16	C₂₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
17	C₂₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
18	C₂₃×#121	C₂²	1	1	yes	characteristic
19	C₂₃×#121	C₂²	1	1	yes	characteristic
20	C₂₃×#120	—	2	1	no	—
21	C₄×D₄×C₂₃	—	2	1	no	—
22	C₄×D₄×C₂₃	—	2	1	no	—
23	C₂₃×#117	—	2	1	no	—
24	C₂₃×#117	—	2	1	no	—
25	C₂₃×#117	C₂²	1	2	yes	—
26	C₂₃×#117	C₂²	1	1	yes	characteristic
27	C₂₃×#117	C₂²	1	2	yes	—
28	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
30	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
31	C₂×C₄²×C₂₃	—	2	1	no	—
32	C₂₃×#107	C₂²	1	2	yes	complemented
33	C₂₃×#107	C₂²	1	2	yes	complemented
34	C₂₃×#106	C₂²	1	2	yes	complemented
35	C₂₃×#106	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	complemented
39	C₂₃×#102	C₂²	1	2	yes	complemented
40	C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
41	C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
42	C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
43	C₂⁴×C₂₃	D₄	1	1	yes	characteristic
44	#41×C₂₃	—	2	4	no	—
45	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
46	C₂×D₄×C₂₃	—	2	1	no	—
47	C₂×D₄×C₂₃	—	2	1	no	—
48	C₂×D₄×C₂₃	—	2	1	no	—
49	C₂×D₄×C₂₃	C₂³	1	1	yes	characteristic
50	C₂²×C₄×C₂₃	—	2	4	no	—
51	C₂²×C₄×C₂₃	—	2	1	no	—
52	C₂²×C₄×C₂₃	—	2	1	no	—
53	C₂²×C₄×C₂₃	—	2	1	no	—
54	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
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	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
59	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
60	C₂×C₈×C₂₃	—	2	4	no	—
61	C₂×C₈×C₂₃	C₂³	1	2	yes	—
62	C₂×C₈×C₂₃	C₂³	1	2	yes	—
63	C₄⋊C₄×C₂₃	—	2	1	no	—
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	C₂³	1	2	yes	—
68	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
69	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
70	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
71	C₂²⋊C₄×C₂₃	—	2	2	no	—
72	C₂²⋊C₄×C₂₃	—	2	2	no	—
73	C₂²⋊C₄×C₂₃	—	2	1	no	—
74	C₂²⋊C₄×C₂₃	—	2	1	no	—
75	C₂²⋊C₄×C₂₃	—	2	1	no	—
76	C₂²⋊C₄×C₂₃	—	2	1	no	—
77	C₂²⋊C₄×C₂₃	—	2	1	no	—
78	C₄²×C₂₃	—	2	2	no	—
79	C₄²×C₂₃	—	2	2	no	—
80	C₄²×C₂₃	—	2	1	no	—
81	C₂³×C₂₃	—	2	2	no	—
82	C₂³×C₂₃	—	2	2	no	—
83	C₂³×C₂₃	—	2	1	no	—
84	C₂³×C₂₃	—	4	1	no	—
85	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
86	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
87	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
88	Q₈×C₂₃	—	2	2	no	—
89	D₄×C₂₃	—	2	2	no	—
90	D₄×C₂₃	—	4	2	no	—
91	D₄×C₂₃	—	4	2	no	—
92	D₄×C₂₃	—	4	2	no	—
93	C₂×C₄×C₂₃	—	2	2	no	—
94	C₂×C₄×C₂₃	—	2	2	no	—
95	C₂×C₄×C₂₃	—	2	2	no	—
96	C₂×C₄×C₂₃	—	2	2	no	—
97	C₂×C₄×C₂₃	—	2	2	no	—
98	C₂×C₄×C₂₃	—	2	2	no	—
99	C₂×C₄×C₂₃	—	2	2	no	—
100	C₂×C₄×C₂₃	—	2	1	no	—
101	C₂×C₄×C₂₃	—	2	1	no	—
102	C₂×C₄×C₂₃	—	2	1	no	—
103	C₂×C₄×C₂₃	—	4	2	no	—
104	C₂×C₄×C₂₃	—	4	2	no	—
105	C₂×C₄×C₂₃	—	4	2	no	—
106	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
108	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄×C₂₃	#41	1	4	yes	—
110	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
111	C₈×C₂₃	—	2	2	no	—
112	C₈×C₂₃	—	2	2	no	—
113	#3091	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
114	C₂²×C₂₃	—	2	1	no	—
115	C₂²×C₂₃	—	2	1	no	—
116	C₂²×C₂₃	—	2	1	no	—
117	C₂²×C₂₃	—	2	1	no	—
118	C₂²×C₂₃	—	2	1	no	—
119	C₂²×C₂₃	—	2	1	no	—
120	C₂²×C₂₃	—	2	1	no	—
121	C₂²×C₂₃	—	4	4	no	—
122	C₂²×C₂₃	—	4	1	no	—
123	C₂²×C₂₃	—	4	1	no	—
124	C₂²×C₂₃	—	4	1	no	—
125	C₂²×C₂₃	#123	1	2	yes	—
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	—	4	1	no	—
131	C₄×C₂₃	—	4	2	no	—
132	C₄×C₂₃	—	4	1	no	—
133	C₄×C₂₃	—	4	2	no	—
134	C₄×C₂₃	—	4	1	no	—
135	C₄×C₂₃	#123	1	2	yes	—
136	C₄×C₂₃	C₂×#41	1	2	yes	—
137	#525	C₂×C₂₃	1	1	yes	characteristic
138	C₂×#117	C₂×C₂₃	1	1	yes	characteristic, complemented
139	#485	C₂×C₂₃	1	2	yes	complemented
140	#484	C₂×C₂₃	1	1	yes	characteristic, complemented
141	#483	C₂×C₂₃	1	1	yes	characteristic, complemented
142	#482	C₂×C₂₃	1	2	yes	complemented
143	#481	C₂×C₂₃	1	1	yes	characteristic, complemented
144	#480	C₂×C₂₃	1	1	yes	characteristic, complemented
145	#427	C₂×C₂₃	1	2	yes	complemented
146	#416	C₂×C₂₃	1	2	yes	complemented
147	C₂×C₂²⋊C₈	C₂×C₂₃	1	1	yes	characteristic, complemented
148	C₂×C₂₃	—	2	2	no	—
149	C₂×C₂₃	—	2	2	no	—
150	C₂×C₂₃	—	4	1	no	—
151	C₂×C₂₃	—	8	1	no	—
152	C₂×C₂₃	C₂×#123	1	1	yes	characteristic
153	C₂×C₂₃	C₂×#135	1	1	yes	characteristic
154	C₂×C₂₃	#575	1	1	yes	characteristic
155	C₂×#41	C₂²×C₂₃	1	1	yes	characteristic
156	C₂³×C₄	C₂²×C₂₃	1	1	yes	characteristic
157	C₂²×C₈	C₂²×C₂₃	1	2	yes	—
158	#123	—	2	1	no	—
159	C₂²⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
160	C₂²⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
161	#121	C₂²×C₂₃	1	1	yes	characteristic
162	#121	C₂²×C₂₃	1	1	yes	characteristic
163	#120	—	2	1	no	—
164	C₄×D₄	—	2	1	no	—
165	C₄×D₄	—	2	1	no	—
166	#117	—	2	1	no	—
167	#117	—	2	1	no	—
168	#117	C₂²×C₂₃	1	2	yes	—
169	#117	C₂²×C₂₃	1	2	yes	—
170	#117	C₂²×C₂₃	1	1	yes	characteristic
171	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
172	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
173	C₂×C₂²⋊C₄	—	2	1	no	—
174	C₂×C₄²	—	2	1	no	—
175	#107	C₂²×C₂₃	1	2	yes	complemented
176	#107	C₂²×C₂₃	1	2	yes	complemented
177	#106	C₂²×C₂₃	1	2	yes	complemented
178	#106	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	complemented
182	#102	C₂²×C₂₃	1	2	yes	complemented
183	C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
184	C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
185	C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
186	C₂₃	#3091	1	1	yes	characteristic, direct product factor, Sylow
187	C₂⁴	D₄×C₂₃	1	1	yes	characteristic
188	#41	—	2	4	no	—
189	C₂×Q₈	C₂³×C₂₃	1	1	yes	characteristic
190	C₂×D₄	—	2	1	no	—
191	C₂×D₄	—	2	1	no	—
192	C₂×D₄	—	2	1	no	—
193	C₂×D₄	C₂³×C₂₃	1	1	yes	characteristic
194	C₂²×C₄	—	2	4	no	—
195	C₂²×C₄	—	2	1	no	—
196	C₂²×C₄	—	2	1	no	—
197	C₂²×C₄	—	2	1	no	—
198	C₂²×C₄	D₄×C₂₃	1	1	yes	characteristic
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	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
203	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
204	C₂×C₈	—	2	4	no	—
205	C₂×C₈	C₂³×C₂₃	1	2	yes	—
206	C₂×C₈	C₂³×C₂₃	1	2	yes	—
207	C₄⋊C₄	—	2	1	no	—
208	C₄⋊C₄	—	2	1	no	—
209	C₄⋊C₄	—	2	1	no	—
210	C₄⋊C₄	—	2	1	no	—
211	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
212	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
213	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
214	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
215	C₂²⋊C₄	—	2	2	no	—
216	C₂²⋊C₄	—	2	2	no	—
217	C₂²⋊C₄	—	2	1	no	—
218	C₂²⋊C₄	—	2	1	no	—
219	C₂²⋊C₄	—	2	1	no	—
220	C₂²⋊C₄	—	2	1	no	—
221	C₂²⋊C₄	—	2	1	no	—
222	C₄²	—	2	2	no	—
223	C₄²	—	2	2	no	—
224	C₄²	—	2	1	no	—
225	C₂³	—	2	2	no	—
226	C₂³	—	2	2	no	—
227	C₂³	—	2	1	no	—
228	C₂³	—	4	1	no	—
229	C₂³	C₂×D₄×C₂₃	1	1	yes	characteristic
230	C₂³	C₂×D₄×C₂₃	1	1	yes	characteristic
231	C₂³	C₂×D₄×C₂₃	1	1	yes	characteristic
232	Q₈	—	2	2	no	—
233	D₄	—	2	2	no	—
234	D₄	—	4	2	no	—
235	D₄	—	4	2	no	—
236	D₄	—	4	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
251	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
252	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
253	C₂×C₄	#41×C₂₃	1	4	yes	—
254	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
255	C₈	—	2	2	no	—
256	C₈	—	2	2	no	—
257	C₂²	—	2	1	no	—
258	C₂²	—	2	1	no	—
259	C₂²	—	2	1	no	—
260	C₂²	—	2	1	no	—
261	C₂²	—	2	1	no	—
262	C₂²	—	2	1	no	—
263	C₂²	—	2	1	no	—
264	C₂²	—	4	4	no	—
265	C₂²	—	4	1	no	—
266	C₂²	—	4	1	no	—
267	C₂²	—	4	1	no	—
268	C₂²	C₂₃×#123	1	2	yes	—
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	—	4	2	no	—
274	C₄	—	4	1	no	—
275	C₄	—	4	2	no	—
276	C₄	—	4	1	no	—
277	C₄	—	4	1	no	—
278	C₄	C₂₃×#123	1	2	yes	—
279	C₄	C₂×#41×C₂₃	1	2	yes	—
280	C₂	—	2	2	no	—
281	C₂	—	2	2	no	—
282	C₂	—	4	1	no	—
283	C₂	—	8	1	no	—
284	C₂	C₂×C₂₃×#123	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	C₂×C₄×C₂₃	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₂₃
socle	C₂²×C₂₃	C₂²×D₄
prime radical	C₂₃×#525	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂³×C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₂₃×#117	C₂	semidirect
3. C₂₃×#485	C₂	semidirect
4. C₂₃×#484	C₂	semidirect
5. C₂₃×#483	C₂	semidirect
6. C₂₃×#482	C₂	semidirect
7. C₂₃×#481	C₂	semidirect
8. C₂₃×#480	C₂	semidirect
9. C₂₃×#427	C₂	semidirect
10. C₂₃×#416	C₂	semidirect
11. C₂×C₂₃×C₂²⋊C₈	C₂	semidirect
32. C₂₃×#107	C₂²	semidirect
33. C₂₃×#107	C₂²	semidirect
34. C₂₃×#106	C₂²	semidirect
35. C₂₃×#106	C₂²	semidirect
36. C₂₃×#103	C₂²	semidirect
37. C₂₃×#103	C₂²	semidirect
38. C₂₃×#102	C₂²	semidirect
39. C₂₃×#102	C₂²	semidirect
113. #3091	C₂₃	direct (N × H)
138. C₂×#117	C₂×C₂₃	semidirect
139. #485	C₂×C₂₃	semidirect
140. #484	C₂×C₂₃	semidirect
141. #483	C₂×C₂₃	semidirect
142. #482	C₂×C₂₃	semidirect
143. #481	C₂×C₂₃	semidirect
144. #480	C₂×C₂₃	semidirect
145. #427	C₂×C₂₃	semidirect
146. #416	C₂×C₂₃	semidirect
147. C₂×C₂²⋊C₈	C₂×C₂₃	semidirect
175. #107	C₂²×C₂₃	semidirect
176. #107	C₂²×C₂₃	semidirect
177. #106	C₂²×C₂₃	semidirect
178. #106	C₂²×C₂₃	semidirect
179. #103	C₂²×C₂₃	semidirect
180. #103	C₂²×C₂₃	semidirect
181. #102	C₂²×C₂₃	semidirect
182. #102	C₂²×C₂₃	semidirect
186. C₂₃	#3091	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×#525	2
C₂×C₂₃×#117	2
C₂₃×#485	2
C₂₃×#484	2
C₂₃×#483	2
C₂₃×#482	2
C₂₃×#481	2
C₂₃×#480	2
C₂₃×#427	2
C₂₃×#416	2
C₂×C₂₃×C₂²⋊C₈	2
#3091	23

Subgroups — C23×#3091 #15670710

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#3091 #15670710