Subgroups — C₂₃×#3063 #15670682

Order 2944.

472 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	C₂×C₂₃×#137	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂₃×#131	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#537	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#536	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#470	C₂	1	2	yes	maximal, complemented
6	C₂₃×#468	C₂	1	2	yes	maximal, complemented
7	C₂₃×#467	C₂	1	2	yes	maximal, complemented
8	C₂₃×#465	C₂	1	2	yes	maximal, complemented
9	C₂₃×#428	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#416	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₂₃×#103	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₂₃	C₂²	1	1	yes	characteristic
13	C₂×#41×C₂₃	C₂²	1	1	yes	characteristic
14	C₂²×Q₈×C₂₃	C₂²	1	1	yes	characteristic
15	C₂₃×#137	—	2	4	no	—
16	C₂×Q₁₆×C₂₃	C₂²	1	2	yes	—
17	C₂×SD₁₆×C₂₃	C₂²	1	2	yes	—
18	C₂₃×#131	—	2	4	no	—
19	C₂×M₁₆×C₂₃	C₂²	1	2	yes	—
20	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
21	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
22	C₂²×C₈×C₂₃	C₂²	1	2	yes	—
23	C₂₃×C₄⋊Q₈	—	2	1	no	—
24	C₂₃×#124	—	2	1	no	—
25	C₂₃×#123	—	2	2	no	—
26	C₂₃×C₂²⋊Q₈	—	2	1	no	—
27	C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	—
28	C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	—
29	C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	—
30	C₂₃×#121	—	2	1	no	—
31	C₂₃×#121	—	2	2	no	—
32	C₂₃×#121	C₂²	1	2	yes	—
33	C₂₃×#117	C₂²	1	1	yes	characteristic
34	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
35	C₂₃×#107	C₂²	1	2	yes	—
36	C₂₃×#106	C₂²	1	2	yes	—
37	C₂₃×#103	C₂²	1	2	yes	—
38	C₂₃×#103	C₂²	1	2	yes	—
39	C₂₃×#103	C₂²	1	1	yes	characteristic
40	C₂₃×#103	C₂²	1	1	yes	characteristic
41	C₂₃×#102	C₂²	1	1	yes	characteristic
42	C₂₃×#102	C₂²	1	1	yes	characteristic
43	#41×C₂₃	—	2	2	no	—
44	#41×C₂₃	—	2	2	no	—
45	#41×C₂₃	—	2	2	no	—
46	#41×C₂₃	—	2	2	no	—
47	C₂×Q₈×C₂₃	—	2	1	no	—
48	C₂×Q₈×C₂₃	—	2	2	no	—
49	C₂×Q₈×C₂₃	—	2	2	no	—
50	C₂×Q₈×C₂₃	D₄	1	1	yes	characteristic
51	C₂×Q₈×C₂₃	C₂³	1	2	yes	—
52	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
53	C₂×D₄×C₂₃	—	2	1	no	—
54	C₂×D₄×C₂₃	—	2	2	no	—
55	C₂×D₄×C₂₃	D₄	1	2	yes	—
56	C₂×D₄×C₂₃	D₄	1	1	yes	characteristic
57	C₂×D₄×C₂₃	C₂³	1	1	yes	characteristic
58	C₂²×C₄×C₂₃	—	2	1	no	—
59	C₂²×C₄×C₂₃	—	2	1	no	—
60	C₂²×C₄×C₂₃	—	2	1	no	—
61	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
62	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
63	Q₁₆×C₂₃	—	2	4	no	—
64	SD₁₆×C₂₃	—	2	4	no	—
65	M₁₆×C₂₃	—	2	4	no	—
66	M₁₆×C₂₃	D₄	1	4	yes	complemented
67	C₂×C₈×C₂₃	—	2	2	no	—
68	C₂×C₈×C₂₃	—	2	4	no	—
69	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
70	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
71	C₂×C₈×C₂₃	C₂³	1	2	yes	—
72	C₄⋊C₄×C₂₃	—	2	1	no	—
73	C₄⋊C₄×C₂₃	—	2	2	no	—
74	C₄⋊C₄×C₂₃	—	2	2	no	—
75	C₄⋊C₄×C₂₃	—	2	2	no	—
76	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
77	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
78	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
79	C₂²⋊C₄×C₂₃	—	2	1	no	—
80	C₂²⋊C₄×C₂₃	—	2	2	no	—
81	C₂²⋊C₄×C₂₃	—	2	2	no	—
82	C₂²⋊C₄×C₂₃	—	2	2	no	—
83	C₂²⋊C₄×C₂₃	—	2	2	no	—
84	C₄²×C₂₃	—	2	1	no	—
85	C₂³×C₂₃	—	2	1	no	—
86	C₂³×C₂₃	C₂×D₄	1	2	yes	—
87	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
88	Q₈×C₂₃	—	2	2	no	—
89	Q₈×C₂₃	—	2	2	no	—
90	Q₈×C₂₃	—	2	1	no	—
91	Q₈×C₂₃	—	2	2	no	—
92	Q₈×C₂₃	—	2	1	no	—
93	Q₈×C₂₃	—	4	2	no	—
94	D₄×C₂₃	—	2	4	no	—
95	D₄×C₂₃	—	2	2	no	—
96	D₄×C₂₃	—	2	1	no	—
97	D₄×C₂₃	—	2	1	no	—
98	D₄×C₂₃	—	4	2	no	—
99	D₄×C₂₃	—	8	2	no	—
100	C₂×C₄×C₂₃	—	2	2	no	—
101	C₂×C₄×C₂₃	—	2	2	no	—
102	C₂×C₄×C₂₃	—	2	1	no	—
103	C₂×C₄×C₂₃	—	2	1	no	—
104	C₂×C₄×C₂₃	—	2	2	no	—
105	C₂×C₄×C₂₃	—	2	1	no	—
106	C₂×C₄×C₂₃	—	2	2	no	—
107	C₂×C₄×C₂₃	—	4	2	no	—
108	C₂×C₄×C₂₃	—	4	2	no	—
109	C₂×C₄×C₂₃	—	8	1	no	—
110	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
111	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
112	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
113	C₂×C₄×C₂₃	#41	1	2	yes	—
114	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
115	C₈×C₂₃	—	2	1	no	—
116	C₈×C₂₃	—	2	1	no	—
117	C₈×C₂₃	C₂×D₄	1	4	yes	—
118	#3063	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
119	C₂²×C₂₃	—	2	1	no	—
120	C₂²×C₂₃	—	2	1	no	—
121	C₂²×C₂₃	—	2	2	no	—
122	C₂²×C₂₃	—	2	2	no	—
123	C₂²×C₂₃	—	2	2	no	—
124	C₂²×C₂₃	—	4	1	no	—
125	C₂²×C₂₃	—	4	1	no	—
126	C₂²×C₂₃	—	4	1	no	—
127	C₂²×C₂₃	#121	1	2	yes	—
128	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
129	C₄×C₂₃	—	2	2	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	#121	1	2	yes	—
136	C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
137	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
138	C₂×#137	C₂×C₂₃	1	1	yes	characteristic, complemented
139	C₂×#131	C₂×C₂₃	1	1	yes	characteristic, complemented
140	#537	C₂×C₂₃	1	1	yes	characteristic
141	#536	C₂×C₂₃	1	1	yes	characteristic, complemented
142	#470	C₂×C₂₃	1	2	yes	complemented
143	#468	C₂×C₂₃	1	2	yes	complemented
144	#467	C₂×C₂₃	1	2	yes	complemented
145	#465	C₂×C₂₃	1	2	yes	complemented
146	#428	C₂×C₂₃	1	1	yes	characteristic, complemented
147	#416	C₂×C₂₃	1	1	yes	characteristic, complemented
148	C₂×#103	C₂×C₂₃	1	1	yes	characteristic, complemented
149	C₂×C₂₃	—	2	2	no	—
150	C₂×C₂₃	—	4	2	no	—
151	C₂×C₂₃	—	8	1	no	—
152	C₂×C₂₃	C₂×#121	1	1	yes	characteristic
153	C₂×C₂₃	#577	1	1	yes	characteristic
154	C₂×C₂₃	#578	1	1	yes	characteristic
155	C₂×#41	C₂²×C₂₃	1	1	yes	characteristic
156	C₂×#41	C₂²×C₂₃	1	1	yes	characteristic
157	C₂²×Q₈	C₂²×C₂₃	1	1	yes	characteristic
158	#137	—	2	4	no	—
159	C₂×Q₁₆	C₂²×C₂₃	1	2	yes	—
160	C₂×SD₁₆	C₂²×C₂₃	1	2	yes	—
161	#131	—	2	4	no	—
162	C₂×M₁₆	C₂²×C₂₃	1	2	yes	—
163	C₂×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
164	C₂²×C₈	C₂²×C₂₃	1	1	yes	characteristic
165	C₂²×C₈	C₂²×C₂₃	1	2	yes	—
166	C₄⋊Q₈	—	2	1	no	—
167	#124	—	2	1	no	—
168	#123	—	2	2	no	—
169	C₂²⋊Q₈	—	2	1	no	—
170	C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	—
171	C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	—
172	C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	—
173	#121	—	2	1	no	—
174	#121	—	2	2	no	—
175	#121	C₂²×C₂₃	1	2	yes	—
176	#117	C₂²×C₂₃	1	1	yes	characteristic
177	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
178	#107	C₂²×C₂₃	1	2	yes	—
179	#106	C₂²×C₂₃	1	2	yes	—
180	#103	C₂²×C₂₃	1	2	yes	—
181	#103	C₂²×C₂₃	1	2	yes	—
182	#103	C₂²×C₂₃	1	1	yes	characteristic
183	#103	C₂²×C₂₃	1	1	yes	characteristic
184	#102	C₂²×C₂₃	1	1	yes	characteristic
185	#102	C₂²×C₂₃	1	1	yes	characteristic
186	C₂₃	#3063	1	1	yes	characteristic, direct product factor, Sylow
187	#41	—	2	2	no	—
188	#41	—	2	2	no	—
189	#41	—	2	2	no	—
190	#41	—	2	2	no	—
191	C₂×Q₈	—	2	1	no	—
192	C₂×Q₈	—	2	2	no	—
193	C₂×Q₈	—	2	2	no	—
194	C₂×Q₈	D₄×C₂₃	1	1	yes	characteristic
195	C₂×Q₈	C₂³×C₂₃	1	2	yes	—
196	C₂×Q₈	C₂³×C₂₃	1	1	yes	characteristic
197	C₂×D₄	—	2	1	no	—
198	C₂×D₄	—	2	2	no	—
199	C₂×D₄	D₄×C₂₃	1	2	yes	—
200	C₂×D₄	D₄×C₂₃	1	1	yes	characteristic
201	C₂×D₄	C₂³×C₂₃	1	1	yes	characteristic
202	C₂²×C₄	—	2	1	no	—
203	C₂²×C₄	—	2	1	no	—
204	C₂²×C₄	—	2	1	no	—
205	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
206	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
207	Q₁₆	—	2	4	no	—
208	SD₁₆	—	2	4	no	—
209	M₁₆	—	2	4	no	—
210	M₁₆	D₄×C₂₃	1	4	yes	complemented
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	2	4	no	—
213	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
214	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
215	C₂×C₈	C₂³×C₂₃	1	2	yes	—
216	C₄⋊C₄	—	2	1	no	—
217	C₄⋊C₄	—	2	2	no	—
218	C₄⋊C₄	—	2	2	no	—
219	C₄⋊C₄	—	2	2	no	—
220	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
221	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
222	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
223	C₂²⋊C₄	—	2	1	no	—
224	C₂²⋊C₄	—	2	2	no	—
225	C₂²⋊C₄	—	2	2	no	—
226	C₂²⋊C₄	—	2	2	no	—
227	C₂²⋊C₄	—	2	2	no	—
228	C₄²	—	2	1	no	—
229	C₂³	—	2	1	no	—
230	C₂³	C₂×D₄×C₂₃	1	2	yes	—
231	C₂³	C₂×D₄×C₂₃	1	1	yes	characteristic
232	Q₈	—	2	2	no	—
233	Q₈	—	2	2	no	—
234	Q₈	—	2	1	no	—
235	Q₈	—	2	2	no	—
236	Q₈	—	2	1	no	—
237	Q₈	—	4	2	no	—
238	D₄	—	2	4	no	—
239	D₄	—	2	2	no	—
240	D₄	—	2	1	no	—
241	D₄	—	2	1	no	—
242	D₄	—	4	2	no	—
243	D₄	—	8	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	8	1	no	—
254	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
255	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
256	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
257	C₂×C₄	#41×C₂₃	1	2	yes	—
258	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
259	C₈	—	2	1	no	—
260	C₈	—	2	1	no	—
261	C₈	C₂×D₄×C₂₃	1	4	yes	—
262	C₂²	—	2	1	no	—
263	C₂²	—	2	1	no	—
264	C₂²	—	2	2	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	2	2	no	—
267	C₂²	—	4	1	no	—
268	C₂²	—	4	1	no	—
269	C₂²	—	4	1	no	—
270	C₂²	C₂₃×#121	1	2	yes	—
271	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
272	C₄	—	2	2	no	—
273	C₄	—	4	2	no	—
274	C₄	—	4	1	no	—
275	C₄	—	4	1	no	—
276	C₄	—	4	1	no	—
277	C₄	—	4	2	no	—
278	C₄	C₂₃×#121	1	2	yes	—
279	C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
280	C₄	C₂×#41×C₂₃	1	1	yes	characteristic
281	C₂	—	2	2	no	—
282	C₂	—	4	2	no	—
283	C₂	—	8	1	no	—
284	C₂	C₂×C₂₃×#121	1	1	yes	characteristic
285	C₂	C₂₃×#577	1	1	yes	characteristic
286	C₂	C₂₃×#578	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₂₃
prime radical	C₂₃×#537	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×#41×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₃×#137	C₂	semidirect
2. C₂×C₂₃×#131	C₂	semidirect
4. C₂₃×#536	C₂	semidirect
5. C₂₃×#470	C₂	semidirect
6. C₂₃×#468	C₂	semidirect
7. C₂₃×#467	C₂	semidirect
8. C₂₃×#465	C₂	semidirect
9. C₂₃×#428	C₂	semidirect
10. C₂₃×#416	C₂	semidirect
11. C₂×C₂₃×#103	C₂	semidirect
66. M₁₆×C₂₃	D₄	semidirect
118. #3063	C₂₃	direct (N × H)
138. C₂×#137	C₂×C₂₃	semidirect
139. C₂×#131	C₂×C₂₃	semidirect
141. #536	C₂×C₂₃	semidirect
142. #470	C₂×C₂₃	semidirect
143. #468	C₂×C₂₃	semidirect
144. #467	C₂×C₂₃	semidirect
145. #465	C₂×C₂₃	semidirect
146. #428	C₂×C₂₃	semidirect
147. #416	C₂×C₂₃	semidirect
148. C₂×#103	C₂×C₂₃	semidirect
186. C₂₃	#3063	direct (N × H)
210. M₁₆	D₄×C₂₃	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₂₃×#137	2
C₂×C₂₃×#131	2
C₂₃×#537	2
C₂₃×#536	2
C₂₃×#470	2
C₂₃×#468	2
C₂₃×#467	2
C₂₃×#465	2
C₂₃×#428	2
C₂₃×#416	2
C₂×C₂₃×#103	2
#3063	23

Subgroups — C23×#3063 #15670682

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#3063 #15670682