Subgroups — C₂₃×#3251 #15670870

Order 2944.

392 conjugacy classes of subgroups.

The subgroup lattice has 284 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	C₂₃×#529	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#478	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×#469	C₂	1	2	yes	maximal, complemented
7	C₂₃×#468	C₂	1	2	yes	maximal, complemented
8	C₂₃×#463	C₂	1	2	yes	maximal, complemented
9	C₂₃×#460	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#459	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#423	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₂₃	—	2	1	no	—
13	C₂×SD₁₆×C₂₃	C₂²	1	2	yes	—
14	C₂×D₈×C₂₃	C₂²	1	2	yes	—
15	C₂×M₁₆×C₂₃	C₂²	1	2	yes	—
16	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₂₃×#127	C₂²	1	1	yes	characteristic
18	C₂₃×#126	—	2	1	no	—
19	C₂₃×#125	C₂²	1	1	yes	characteristic
20	C₂₃×#124	—	2	1	no	—
21	C₂₃×#124	C₂²	1	1	yes	characteristic
22	C₂₃×#123	—	2	1	no	—
23	C₂₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
24	C₂₃×#121	—	2	1	no	—
25	C₂₃×#121	C₂²	1	2	yes	—
26	C₂₃×#121	C₂²	1	1	yes	characteristic
27	C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
28	C₄×D₄×C₂₃	—	2	1	no	—
29	C₄×D₄×C₂₃	—	2	1	no	—
30	C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
31	C₄×D₄×C₂₃	C₂²	1	2	yes	—
32	C₂₃×#117	—	2	1	no	—
33	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
34	C₂₃×#107	C₂²	1	2	yes	—
35	C₂₃×#106	C₂²	1	2	yes	—
36	C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
37	C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
38	C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
39	C₂₃×#103	C₂²	1	2	yes	—
40	C₂₃×#102	C₂²	1	2	yes	—
41	C₂₃×#102	C₂²	1	2	yes	—
42	C₂₃×#102	C₂²	1	2	yes	—
43	#41×C₂₃	—	4	4	no	—
44	C₂×Q₈×C₂₃	—	2	1	no	—
45	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
46	C₂×D₄×C₂₃	—	2	1	no	—
47	C₂×D₄×C₂₃	—	2	2	no	—
48	C₂×D₄×C₂₃	—	2	1	no	—
49	C₂×D₄×C₂₃	—	2	1	no	—
50	C₂×D₄×C₂₃	C₂³	1	1	yes	characteristic
51	C₂×D₄×C₂₃	C₂³	1	2	yes	—
52	C₂²×C₄×C₂₃	—	2	1	no	—
53	C₂²×C₄×C₂₃	—	2	2	no	—
54	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
55	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
56	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
57	SD₁₆×C₂₃	—	4	2	no	—
58	D₈×C₂₃	—	4	2	no	—
59	M₁₆×C₂₃	—	4	2	no	—
60	C₂×C₈×C₂₃	C₂³	1	2	yes	—
61	C₂×C₈×C₂₃	C₂³	1	2	yes	—
62	C₄⋊C₄×C₂₃	—	2	1	no	—
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	1	yes	characteristic
68	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
69	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
70	C₂²⋊C₄×C₂₃	—	2	1	no	—
71	C₂²⋊C₄×C₂₃	—	2	1	no	—
72	C₂²⋊C₄×C₂₃	—	2	1	no	—
73	C₂²⋊C₄×C₂₃	—	2	2	no	—
74	C₂²⋊C₄×C₂₃	—	2	1	no	—
75	C₂²⋊C₄×C₂₃	—	2	2	no	—
76	C₂²⋊C₄×C₂₃	—	2	1	no	—
77	C₄²×C₂₃	—	2	1	no	—
78	C₄²×C₂₃	D₄	1	1	yes	characteristic
79	C₄²×C₂₃	D₄	1	1	yes	characteristic
80	C₄²×C₂₃	C₂³	1	1	yes	characteristic
81	C₄²×C₂₃	C₂³	1	1	yes	characteristic
82	C₂³×C₂₃	—	2	1	no	—
83	C₂³×C₂₃	—	2	2	no	—
84	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
85	Q₈×C₂₃	—	2	1	no	—
86	Q₈×C₂₃	—	2	1	no	—
87	Q₈×C₂₃	—	4	2	no	—
88	D₄×C₂₃	—	2	1	no	—
89	D₄×C₂₃	—	2	2	no	—
90	D₄×C₂₃	—	2	1	no	—
91	D₄×C₂₃	—	2	2	no	—
92	D₄×C₂₃	—	4	2	no	—
93	D₄×C₂₃	—	4	2	no	—
94	D₄×C₂₃	—	4	4	no	—
95	D₄×C₂₃	—	4	2	no	—
96	C₂×C₄×C₂₃	—	2	2	no	—
97	C₂×C₄×C₂₃	—	2	1	no	—
98	C₂×C₄×C₂₃	—	2	1	no	—
99	C₂×C₄×C₂₃	—	2	1	no	—
100	C₂×C₄×C₂₃	—	2	1	no	—
101	C₂×C₄×C₂₃	—	2	1	no	—
102	C₂×C₄×C₂₃	—	2	2	no	—
103	C₂×C₄×C₂₃	—	4	1	no	—
104	C₂×C₄×C₂₃	—	4	4	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	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
111	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
112	C₈×C₂₃	—	2	2	no	—
113	C₈×C₂₃	—	2	2	no	—
114	#3251	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
115	C₂²×C₂₃	—	2	1	no	—
116	C₂²×C₂₃	—	2	1	no	—
117	C₂²×C₂₃	—	2	1	no	—
118	C₂²×C₂₃	—	4	1	no	—
119	C₂²×C₂₃	—	4	2	no	—
120	C₂²×C₂₃	—	4	2	no	—
121	C₂²×C₂₃	—	4	1	no	—
122	C₂²×C₂₃	—	4	2	no	—
123	C₂²×C₂₃	—	4	1	no	—
124	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
125	C₄×C₂₃	—	2	1	no	—
126	C₄×C₂₃	—	2	1	no	—
127	C₄×C₂₃	—	2	1	no	—
128	C₄×C₂₃	—	2	1	no	—
129	C₄×C₂₃	—	4	1	no	—
130	C₄×C₂₃	—	4	1	no	—
131	C₄×C₂₃	—	4	2	no	—
132	C₄×C₂₃	—	4	1	no	—
133	C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
134	C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
135	C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
136	#532	C₂×C₂₃	1	1	yes	characteristic, complemented
137	#529	C₂×C₂₃	1	1	yes	characteristic, complemented
138	#478	C₂×C₂₃	1	2	yes	complemented
139	#475	C₂×C₂₃	1	1	yes	characteristic, complemented
140	#474	C₂×C₂₃	1	1	yes	characteristic, complemented
141	#469	C₂×C₂₃	1	2	yes	complemented
142	#468	C₂×C₂₃	1	2	yes	complemented
143	#463	C₂×C₂₃	1	2	yes	complemented
144	#460	C₂×C₂₃	1	1	yes	characteristic, complemented
145	#459	C₂×C₂₃	1	1	yes	characteristic, complemented
146	#423	C₂×C₂₃	1	1	yes	characteristic, complemented
147	C₂×C₂₃	—	4	1	no	—
148	C₂×C₂₃	—	8	2	no	—
149	C₂×C₂₃	—	8	1	no	—
150	C₂×C₂₃	#537	1	1	yes	characteristic
151	C₂×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
152	C₂×C₂₃	#575	1	1	yes	characteristic
153	C₂×#41	—	2	1	no	—
154	C₂×SD₁₆	C₂²×C₂₃	1	2	yes	—
155	C₂×D₈	C₂²×C₂₃	1	2	yes	—
156	C₂×M₁₆	C₂²×C₂₃	1	2	yes	—
157	C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
158	#127	C₂²×C₂₃	1	1	yes	characteristic
159	#126	—	2	1	no	—
160	#125	C₂²×C₂₃	1	1	yes	characteristic
161	#124	—	2	1	no	—
162	#124	C₂²×C₂₃	1	1	yes	characteristic
163	#123	—	2	1	no	—
164	C₂²⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
165	#121	—	2	1	no	—
166	#121	C₂²×C₂₃	1	1	yes	characteristic
167	#121	C₂²×C₂₃	1	2	yes	—
168	C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
169	C₄×D₄	—	2	1	no	—
170	C₄×D₄	—	2	1	no	—
171	C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
172	C₄×D₄	C₂²×C₂₃	1	2	yes	—
173	#117	—	2	1	no	—
174	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
175	#107	C₂²×C₂₃	1	2	yes	—
176	#106	C₂²×C₂₃	1	2	yes	—
177	C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
178	C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
179	C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
180	#103	C₂²×C₂₃	1	2	yes	—
181	#102	C₂²×C₂₃	1	2	yes	—
182	#102	C₂²×C₂₃	1	2	yes	—
183	#102	C₂²×C₂₃	1	2	yes	—
184	C₂₃	#3251	1	1	yes	characteristic, direct product factor, Sylow
185	#41	—	4	4	no	—
186	C₂×Q₈	—	2	1	no	—
187	C₂×Q₈	C₂³×C₂₃	1	1	yes	characteristic
188	C₂×D₄	—	2	1	no	—
189	C₂×D₄	—	2	2	no	—
190	C₂×D₄	—	2	1	no	—
191	C₂×D₄	—	2	1	no	—
192	C₂×D₄	C₂³×C₂₃	1	1	yes	characteristic
193	C₂×D₄	C₂³×C₂₃	1	2	yes	—
194	C₂²×C₄	—	2	1	no	—
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	D₄×C₂₃	1	1	yes	characteristic
197	C₂²×C₄	D₄×C₂₃	1	1	yes	characteristic
198	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
199	SD₁₆	—	4	2	no	—
200	D₈	—	4	2	no	—
201	M₁₆	—	4	2	no	—
202	C₂×C₈	C₂³×C₂₃	1	2	yes	—
203	C₂×C₈	C₂³×C₂₃	1	2	yes	—
204	C₄⋊C₄	—	2	1	no	—
205	C₄⋊C₄	—	2	1	no	—
206	C₄⋊C₄	—	2	1	no	—
207	C₄⋊C₄	—	2	1	no	—
208	C₄⋊C₄	—	2	1	no	—
209	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
210	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
211	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
212	C₂²⋊C₄	—	2	1	no	—
213	C₂²⋊C₄	—	2	1	no	—
214	C₂²⋊C₄	—	2	1	no	—
215	C₂²⋊C₄	—	2	2	no	—
216	C₂²⋊C₄	—	2	1	no	—
217	C₂²⋊C₄	—	2	2	no	—
218	C₂²⋊C₄	—	2	1	no	—
219	C₄²	—	2	1	no	—
220	C₄²	D₄×C₂₃	1	1	yes	characteristic
221	C₄²	D₄×C₂₃	1	1	yes	characteristic
222	C₄²	C₂³×C₂₃	1	1	yes	characteristic
223	C₄²	C₂³×C₂₃	1	1	yes	characteristic
224	C₂³	—	2	1	no	—
225	C₂³	—	2	2	no	—
226	C₂³	C₂×D₄×C₂₃	1	1	yes	characteristic
227	Q₈	—	2	1	no	—
228	Q₈	—	2	1	no	—
229	Q₈	—	4	2	no	—
230	D₄	—	2	1	no	—
231	D₄	—	2	2	no	—
232	D₄	—	2	1	no	—
233	D₄	—	2	2	no	—
234	D₄	—	4	2	no	—
235	D₄	—	4	4	no	—
236	D₄	—	4	2	no	—
237	D₄	—	4	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	4	1	no	—
246	C₂×C₄	—	4	4	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
249	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
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	C₂⁴×C₂₃	1	1	yes	characteristic
254	C₈	—	2	2	no	—
255	C₈	—	2	2	no	—
256	C₂²	—	2	1	no	—
257	C₂²	—	2	1	no	—
258	C₂²	—	2	1	no	—
259	C₂²	—	4	2	no	—
260	C₂²	—	4	1	no	—
261	C₂²	—	4	2	no	—
262	C₂²	—	4	1	no	—
263	C₂²	—	4	1	no	—
264	C₂²	—	4	2	no	—
265	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
266	C₄	—	2	1	no	—
267	C₄	—	2	1	no	—
268	C₄	—	2	1	no	—
269	C₄	—	2	1	no	—
270	C₄	—	4	2	no	—
271	C₄	—	4	1	no	—
272	C₄	—	4	1	no	—
273	C₄	—	4	1	no	—
274	C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
275	C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
276	C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
277	C₂	—	4	1	no	—
278	C₂	—	8	2	no	—
279	C₂	—	8	1	no	—
280	C₂	C₂₃×#537	1	1	yes	characteristic
281	C₂	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
282	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. C₂₃×#529	C₂	semidirect
3. C₂₃×#478	C₂	semidirect
4. C₂₃×#475	C₂	semidirect
5. C₂₃×#474	C₂	semidirect
6. C₂₃×#469	C₂	semidirect
7. C₂₃×#468	C₂	semidirect
8. C₂₃×#463	C₂	semidirect
9. C₂₃×#460	C₂	semidirect
10. C₂₃×#459	C₂	semidirect
11. C₂₃×#423	C₂	semidirect
114. #3251	C₂₃	direct (N × H)
136. #532	C₂×C₂₃	semidirect
137. #529	C₂×C₂₃	semidirect
138. #478	C₂×C₂₃	semidirect
139. #475	C₂×C₂₃	semidirect
140. #474	C₂×C₂₃	semidirect
141. #469	C₂×C₂₃	semidirect
142. #468	C₂×C₂₃	semidirect
143. #463	C₂×C₂₃	semidirect
144. #460	C₂×C₂₃	semidirect
145. #459	C₂×C₂₃	semidirect
146. #423	C₂×C₂₃	semidirect
184. C₂₃	#3251	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×#532	2
C₂₃×#529	2
C₂₃×#478	2
C₂₃×#475	2
C₂₃×#474	2
C₂₃×#469	2
C₂₃×#468	2
C₂₃×#463	2
C₂₃×#460	2
C₂₃×#459	2
C₂₃×#423	2
#3251	23

Subgroups — C23×#3251 #15670870

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#3251 #15670870