Subgroups — C₂₃×#3115 #15670734

Order 2944.

430 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#532	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#41×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#482	C₂	1	2	yes	maximal, complemented
4	C₂₃×#475	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#474	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#460	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#459	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#449	C₂	1	2	yes	maximal, complemented
9	C₂₃×#442	C₂	1	2	yes	maximal, complemented
10	C₂₃×#440	C₂	1	2	yes	maximal, complemented
11	C₂₃×#432	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₂₃	—	2	1	no	—
13	C₂×#41×C₂₃	C₂²	1	1	yes	characteristic
14	C₂×SD₁₆×C₂₃	C₂²	1	2	yes	—
15	C₂×D₈×C₂₃	C₂²	1	2	yes	—
16	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₂₃×#127	C₂²	1	1	yes	characteristic
18	C₂₃×#124	—	2	1	no	—
19	C₂₃×#121	—	2	1	no	—
20	C₂₃×#121	C₂²	1	2	yes	—
21	C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
22	C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
23	C₄×D₄×C₂₃	—	2	1	no	—
24	C₄×D₄×C₂₃	—	2	1	no	—
25	C₄×D₄×C₂₃	—	2	1	no	—
26	C₄×D₄×C₂₃	C₂²	1	2	yes	—
27	C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
28	C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
29	C₂₃×#117	—	2	1	no	—
30	C₂₃×#117	C₂²	1	1	yes	characteristic
31	C₂×C₄²×C₂₃	—	2	1	no	—
32	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
33	C₂₃×#107	C₂²	1	2	yes	—
34	C₂₃×#106	C₂²	1	2	yes	—
35	C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
36	C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
37	C₂₃×#103	C₂²	1	2	yes	—
38	C₂₃×#102	C₂²	1	2	yes	—
39	C₂₃×#102	C₂²	1	2	yes	—
40	C₂₃×#102	C₂²	1	2	yes	—
41	C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
42	C₂₃×#97	C₂²	1	2	yes	—
43	#41×C₂₃	—	2	2	no	—
44	#41×C₂₃	—	2	2	no	—
45	#41×C₂₃	—	4	4	no	—
46	C₂×Q₈×C₂₃	—	2	1	no	—
47	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
48	C₂×D₄×C₂₃	—	2	1	no	—
49	C₂×D₄×C₂₃	—	2	2	no	—
50	C₂×D₄×C₂₃	—	2	1	no	—
51	C₂×D₄×C₂₃	—	2	1	no	—
52	C₂×D₄×C₂₃	C₂³	1	2	yes	—
53	C₂×D₄×C₂₃	C₂³	1	1	yes	characteristic
54	C₂²×C₄×C₂₃	—	2	1	no	—
55	C₂²×C₄×C₂₃	—	2	1	no	—
56	C₂²×C₄×C₂₃	—	2	1	no	—
57	C₂²×C₄×C₂₃	—	2	2	no	—
58	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
59	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
60	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
61	SD₁₆×C₂₃	—	2	4	no	—
62	D₈×C₂₃	—	2	4	no	—
63	C₂×C₈×C₂₃	C₂³	1	2	yes	—
64	C₂×C₈×C₂₃	C₂³	1	2	yes	—
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	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
71	C₂²⋊C₄×C₂₃	—	2	1	no	—
72	C₂²⋊C₄×C₂₃	—	2	1	no	—
73	C₂²⋊C₄×C₂₃	—	2	1	no	—
74	C₂²⋊C₄×C₂₃	—	2	2	no	—
75	C₂²⋊C₄×C₂₃	—	2	2	no	—
76	C₄²×C₂₃	—	2	1	no	—
77	C₄²×C₂₃	—	2	1	no	—
78	C₄²×C₂₃	—	2	1	no	—
79	C₄²×C₂₃	D₄	1	1	yes	characteristic
80	C₄²×C₂₃	D₄	1	1	yes	characteristic
81	C₄²×C₂₃	C₂³	1	1	yes	characteristic
82	C₄²×C₂₃	C₂³	1	1	yes	characteristic
83	C₂³×C₂₃	—	2	1	no	—
84	C₂³×C₂₃	—	2	2	no	—
85	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
86	Q₈×C₂₃	—	2	1	no	—
87	Q₈×C₂₃	—	4	2	no	—
88	Q₈×C₂₃	#41	1	2	yes	complemented
89	D₄×C₂₃	—	2	4	no	—
90	D₄×C₂₃	—	2	1	no	—
91	D₄×C₂₃	—	2	2	no	—
92	D₄×C₂₃	—	2	2	no	—
93	D₄×C₂₃	—	4	2	no	—
94	D₄×C₂₃	—	4	4	no	—
95	D₄×C₂₃	—	4	2	no	—
96	D₄×C₂₃	#41	1	2	yes	complemented
97	C₂×C₄×C₂₃	—	2	4	no	—
98	C₂×C₄×C₂₃	—	2	1	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	—	2	1	no	—
104	C₂×C₄×C₂₃	—	2	2	no	—
105	C₂×C₄×C₂₃	—	4	1	no	—
106	C₂×C₄×C₂₃	—	4	2	no	—
107	C₂×C₄×C₂₃	—	4	4	no	—
108	C₂×C₄×C₂₃	—	4	2	no	—
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	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
115	C₈×C₂₃	—	2	2	no	—
116	C₈×C₂₃	—	2	2	no	—
117	#3115	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
118	C₂²×C₂₃	—	2	1	no	—
119	C₂²×C₂₃	—	2	2	no	—
120	C₂²×C₂₃	—	2	1	no	—
121	C₂²×C₂₃	—	2	1	no	—
122	C₂²×C₂₃	—	4	1	no	—
123	C₂²×C₂₃	—	4	1	no	—
124	C₂²×C₂₃	—	4	2	no	—
125	C₂²×C₂₃	—	4	2	no	—
126	C₂²×C₂₃	—	4	2	no	—
127	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
128	C₄×C₂₃	—	2	1	no	—
129	C₄×C₂₃	—	2	2	no	—
130	C₄×C₂₃	—	2	1	no	—
131	C₄×C₂₃	—	2	1	no	—
132	C₄×C₂₃	—	2	1	no	—
133	C₄×C₂₃	—	4	1	no	—
134	C₄×C₂₃	—	4	2	no	—
135	C₄×C₂₃	—	4	1	no	—
136	C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
137	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
138	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
139	#532	C₂×C₂₃	1	1	yes	characteristic, complemented
140	C₄×#41	C₂×C₂₃	1	1	yes	characteristic, complemented
141	#482	C₂×C₂₃	1	2	yes	complemented
142	#475	C₂×C₂₃	1	1	yes	characteristic, complemented
143	#474	C₂×C₂₃	1	1	yes	characteristic, complemented
144	#460	C₂×C₂₃	1	1	yes	characteristic, complemented
145	#459	C₂×C₂₃	1	1	yes	characteristic, complemented
146	#449	C₂×C₂₃	1	2	yes	complemented
147	#442	C₂×C₂₃	1	2	yes	complemented
148	#440	C₂×C₂₃	1	2	yes	complemented
149	#432	C₂×C₂₃	1	1	yes	characteristic, complemented
150	C₂×C₂₃	—	4	1	no	—
151	C₂×C₂₃	—	4	2	no	—
152	C₂×C₂₃	—	8	2	no	—
153	C₂×C₂₃	#525	1	1	yes	characteristic
154	C₂×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
155	C₂×C₂₃	#575	1	1	yes	characteristic
156	C₂×#41	—	2	1	no	—
157	C₂×#41	C₂²×C₂₃	1	1	yes	characteristic
158	C₂×SD₁₆	C₂²×C₂₃	1	2	yes	—
159	C₂×D₈	C₂²×C₂₃	1	2	yes	—
160	C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
161	#127	C₂²×C₂₃	1	1	yes	characteristic
162	#124	—	2	1	no	—
163	#121	—	2	1	no	—
164	#121	C₂²×C₂₃	1	2	yes	—
165	C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
166	C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
167	C₄×D₄	—	2	1	no	—
168	C₄×D₄	—	2	1	no	—
169	C₄×D₄	—	2	1	no	—
170	C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
171	C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
172	C₄×D₄	C₂²×C₂₃	1	2	yes	—
173	#117	—	2	1	no	—
174	#117	C₂²×C₂₃	1	1	yes	characteristic
175	C₂×C₄²	—	2	1	no	—
176	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
177	#107	C₂²×C₂₃	1	2	yes	—
178	#106	C₂²×C₂₃	1	2	yes	—
179	C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
180	C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
181	#103	C₂²×C₂₃	1	2	yes	—
182	#102	C₂²×C₂₃	1	2	yes	—
183	#102	C₂²×C₂₃	1	2	yes	—
184	#102	C₂²×C₂₃	1	2	yes	—
185	C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
186	#97	C₂²×C₂₃	1	2	yes	—
187	C₂₃	#3115	1	1	yes	characteristic, direct product factor, Sylow
188	#41	—	2	2	no	—
189	#41	—	2	2	no	—
190	#41	—	4	4	no	—
191	C₂×Q₈	—	2	1	no	—
192	C₂×Q₈	C₂³×C₂₃	1	1	yes	characteristic
193	C₂×D₄	—	2	1	no	—
194	C₂×D₄	—	2	2	no	—
195	C₂×D₄	—	2	1	no	—
196	C₂×D₄	—	2	1	no	—
197	C₂×D₄	C₂³×C₂₃	1	2	yes	—
198	C₂×D₄	C₂³×C₂₃	1	1	yes	characteristic
199	C₂²×C₄	—	2	1	no	—
200	C₂²×C₄	—	2	1	no	—
201	C₂²×C₄	—	2	1	no	—
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	D₄×C₂₃	1	1	yes	characteristic
204	C₂²×C₄	D₄×C₂₃	1	1	yes	characteristic
205	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
206	SD₁₆	—	2	4	no	—
207	D₈	—	2	4	no	—
208	C₂×C₈	C₂³×C₂₃	1	2	yes	—
209	C₂×C₈	C₂³×C₂₃	1	2	yes	—
210	C₄⋊C₄	—	2	1	no	—
211	C₄⋊C₄	—	2	1	no	—
212	C₄⋊C₄	—	2	1	no	—
213	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
214	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
215	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
216	C₂²⋊C₄	—	2	1	no	—
217	C₂²⋊C₄	—	2	1	no	—
218	C₂²⋊C₄	—	2	1	no	—
219	C₂²⋊C₄	—	2	2	no	—
220	C₂²⋊C₄	—	2	2	no	—
221	C₄²	—	2	1	no	—
222	C₄²	—	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	C₂×D₄×C₂₃	1	1	yes	characteristic
231	Q₈	—	2	1	no	—
232	Q₈	—	4	2	no	—
233	Q₈	#41×C₂₃	1	2	yes	complemented
234	D₄	—	2	4	no	—
235	D₄	—	2	1	no	—
236	D₄	—	2	2	no	—
237	D₄	—	2	2	no	—
238	D₄	—	4	2	no	—
239	D₄	—	4	4	no	—
240	D₄	—	4	2	no	—
241	D₄	#41×C₂₃	1	2	yes	complemented
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	4	1	no	—
251	C₂×C₄	—	4	4	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	4	2	no	—
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	C₂×D₄×C₂₃	1	1	yes	characteristic
258	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
259	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
260	C₈	—	2	2	no	—
261	C₈	—	2	2	no	—
262	C₂²	—	2	1	no	—
263	C₂²	—	2	1	no	—
264	C₂²	—	2	1	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	4	1	no	—
267	C₂²	—	4	1	no	—
268	C₂²	—	4	2	no	—
269	C₂²	—	4	2	no	—
270	C₂²	—	4	2	no	—
271	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	2	2	no	—
277	C₄	—	4	2	no	—
278	C₄	—	4	1	no	—
279	C₄	—	4	1	no	—
280	C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
281	C₄	C₂×#41×C₂₃	1	1	yes	characteristic
282	C₄	C₂×#41×C₂₃	1	1	yes	characteristic
283	C₂	—	4	1	no	—
284	C₂	—	4	2	no	—
285	C₂	—	8	2	no	—
286	C₂	C₂₃×#525	1	1	yes	characteristic
287	C₂	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
288	C₂	C₂₃×#575	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₂₃
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×#532	C₂	semidirect
2. C₄×#41×C₂₃	C₂	semidirect
3. C₂₃×#482	C₂	semidirect
4. C₂₃×#475	C₂	semidirect
5. C₂₃×#474	C₂	semidirect
6. C₂₃×#460	C₂	semidirect
7. C₂₃×#459	C₂	semidirect
8. C₂₃×#449	C₂	semidirect
9. C₂₃×#442	C₂	semidirect
10. C₂₃×#440	C₂	semidirect
11. C₂₃×#432	C₂	semidirect
88. Q₈×C₂₃	#41	semidirect
96. D₄×C₂₃	#41	semidirect
117. #3115	C₂₃	direct (N × H)
139. #532	C₂×C₂₃	semidirect
140. C₄×#41	C₂×C₂₃	semidirect
141. #482	C₂×C₂₃	semidirect
142. #475	C₂×C₂₃	semidirect
143. #474	C₂×C₂₃	semidirect
144. #460	C₂×C₂₃	semidirect
145. #459	C₂×C₂₃	semidirect
146. #449	C₂×C₂₃	semidirect
147. #442	C₂×C₂₃	semidirect
148. #440	C₂×C₂₃	semidirect
149. #432	C₂×C₂₃	semidirect
187. C₂₃	#3115	direct (N × H)
233. Q₈	#41×C₂₃	semidirect
241. D₄	#41×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#532	2
C₄×#41×C₂₃	2
C₂₃×#482	2
C₂₃×#475	2
C₂₃×#474	2
C₂₃×#460	2
C₂₃×#459	2
C₂₃×#449	2
C₂₃×#442	2
C₂₃×#440	2
C₂₃×#432	2
#3115	23

Subgroups — C23×#3115 #15670734

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#3115 #15670734