Subgroups — C₂₃×#2492 #15670111

Order 2944.

508 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂₃×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#400	C₂	1	2	yes	maximal, complemented
4	C₂₃×#393	C₂	1	2	yes	maximal
5	C₂₃×#391	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#391	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#391	C₂	1	2	yes	maximal, complemented
8	C₂₃×#388	C₂	1	2	yes	maximal
9	C₂₃×#380	C₂	1	1	yes	maximal, characteristic
10	C₄×C₄⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₂²⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic
12	C₂²×Q₈×C₂₃	C₂²	1	1	yes	characteristic
13	C₂²×Q₈×C₂₃	C₂²	1	2	yes	—
14	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
15	C₂₃×C₄⋊Q₈	—	2	4	no	—
16	C₂₃×C₂²⋊Q₈	—	2	4	no	—
17	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
20	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
21	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
22	C₂×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
23	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
24	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
25	C₂×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
26	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
27	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
28	C₂×C₄²×C₂₃	C₂²	1	2	yes	—
29	C₂₃×#95	C₂²	1	1	yes	characteristic
30	C₂₃×#95	C₂²	1	1	yes	characteristic
31	C₂₃×#95	C₂²	1	2	yes	—
32	C₂₃×#95	C₂²	1	2	yes	—
33	C₂₃×#95	C₂²	1	2	yes	—
34	C₂₃×#95	C₂²	1	2	yes	—
35	C₂₃×#95	C₂²	1	2	yes	—
36	C₂₃×#95	C₂²	1	2	yes	—
37	C₂⁴×C₂₃	C₂³	1	1	yes	characteristic
38	C₂×Q₈×C₂₃	—	2	2	no	—
39	C₂×Q₈×C₂₃	—	2	4	no	—
40	C₂×Q₈×C₂₃	—	2	4	no	—
41	C₂×Q₈×C₂₃	—	2	2	no	—
42	C₂×Q₈×C₂₃	—	2	2	no	—
43	C₂×Q₈×C₂₃	—	4	2	no	—
44	C₂²×C₄×C₂₃	—	2	2	no	—
45	C₂²×C₄×C₂₃	—	2	2	no	—
46	C₂²×C₄×C₂₃	—	2	2	no	—
47	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
48	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
50	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
51	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
53	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
54	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
55	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
57	C₄⋊C₄×C₂₃	—	2	4	no	—
58	C₄⋊C₄×C₂₃	—	2	4	no	—
59	C₄⋊C₄×C₂₃	—	2	2	no	—
60	C₄⋊C₄×C₂₃	—	2	2	no	—
61	C₄⋊C₄×C₂₃	D₄	1	4	yes	—
62	C₂²⋊C₄×C₂₃	—	2	4	no	—
63	C₂²⋊C₄×C₂₃	—	2	2	no	—
64	C₂²⋊C₄×C₂₃	—	4	2	no	—
65	C₂²⋊C₄×C₂₃	Q₈	1	4	yes	complemented
66	C₄²×C₂₃	—	2	2	no	—
67	C₄²×C₂₃	—	2	2	no	—
68	C₄²×C₂₃	—	4	2	no	—
69	C₂³×C₂₃	—	2	1	no	—
70	C₂³×C₂₃	—	2	1	no	—
71	C₂³×C₂₃	—	2	1	no	—
72	C₂³×C₂₃	—	2	1	no	—
73	C₂³×C₂₃	—	2	1	no	—
74	C₂³×C₂₃	—	2	1	no	—
75	C₂³×C₂₃	C₂×Q₈	1	2	yes	—
76	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
77	Q₈×C₂₃	—	4	8	no	—
78	Q₈×C₂₃	—	4	4	no	—
79	C₂×C₄×C₂₃	—	2	1	no	—
80	C₂×C₄×C₂₃	—	2	1	no	—
81	C₂×C₄×C₂₃	—	2	1	no	—
82	C₂×C₄×C₂₃	—	2	2	no	—
83	C₂×C₄×C₂₃	—	2	1	no	—
84	C₂×C₄×C₂₃	—	2	2	no	—
85	C₂×C₄×C₂₃	—	2	1	no	—
86	C₂×C₄×C₂₃	—	2	1	no	—
87	C₂×C₄×C₂₃	—	2	2	no	—
88	C₂×C₄×C₂₃	—	2	1	no	—
89	C₂×C₄×C₂₃	—	2	2	no	—
90	C₂×C₄×C₂₃	—	2	1	no	—
91	C₂×C₄×C₂₃	—	2	1	no	—
92	C₂×C₄×C₂₃	—	2	2	no	—
93	C₂×C₄×C₂₃	—	2	1	no	—
94	C₂×C₄×C₂₃	—	2	1	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	2	no	—
101	C₂×C₄×C₂₃	—	2	1	no	—
102	C₂×C₄×C₂₃	—	4	4	no	—
103	C₂×C₄×C₂₃	C₂×D₄	1	4	yes	—
104	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
105	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
106	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
107	C₂×C₄×C₂₃	#41	1	2	yes	—
108	C₂×C₄×C₂₃	#41	1	2	yes	—
109	C₂×C₄×C₂₃	#41	1	2	yes	—
110	#2492	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
111	C₂²×C₂₃	—	2	2	no	—
112	C₂²×C₂₃	—	2	2	no	—
113	C₂²×C₂₃	—	2	2	no	—
114	C₂²×C₂₃	—	4	1	no	—
115	C₂²×C₂₃	—	4	1	no	—
116	C₂²×C₂₃	—	4	1	no	—
117	C₂²×C₂₃	—	4	1	no	—
118	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
119	C₂²×C₂₃	C₂²×Q₈	1	1	yes	characteristic
120	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
121	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
122	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
123	C₂²×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
124	C₂²×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
125	C₄×C₂₃	—	2	4	no	—
126	C₄×C₂₃	—	2	2	no	—
127	C₄×C₂₃	—	2	2	no	—
128	C₄×C₂₃	—	2	2	no	—
129	C₄×C₂₃	—	4	1	no	—
130	C₄×C₂₃	—	4	2	no	—
131	C₄×C₂₃	—	4	2	no	—
132	C₄×C₂₃	—	4	2	no	—
133	C₄×C₂₃	—	4	1	no	—
134	C₄×C₂₃	#124	1	4	yes	—
135	C₂×C₄⋊Q₈	C₂×C₂₃	1	1	yes	characteristic, complemented
136	C₂×C₂²⋊Q₈	C₂×C₂₃	1	1	yes	characteristic
137	#400	C₂×C₂₃	1	2	yes	complemented
138	#393	C₂×C₂₃	1	2	yes	—
139	#391	C₂×C₂₃	1	2	yes	complemented
140	#391	C₂×C₂₃	1	1	yes	characteristic, complemented
141	#391	C₂×C₂₃	1	1	yes	characteristic, complemented
142	#388	C₂×C₂₃	1	2	yes	—
143	#380	C₂×C₂₃	1	1	yes	characteristic
144	C₄×C₄⋊C₄	C₂×C₂₃	1	1	yes	characteristic, complemented
145	C₄×C₂²⋊C₄	C₂×C₂₃	1	1	yes	characteristic
146	C₂×C₂₃	—	4	2	no	—
147	C₂×C₂₃	C₂×#124	1	1	yes	characteristic
148	C₂×C₂₃	#533	1	1	yes	characteristic
149	C₂×C₂₃	#542	1	1	yes	characteristic
150	C₂×C₂₃	#546	1	1	yes	characteristic
151	C₂×C₂₃	D₄×Q₈	1	1	yes	characteristic
152	C₂×C₂₃	#554	1	1	yes	characteristic
153	C₂×C₂₃	#556	1	1	yes	characteristic
154	C₂²×Q₈	C₂²×C₂₃	1	1	yes	characteristic
155	C₂²×Q₈	C₂²×C₂₃	1	2	yes	—
156	C₂³×C₄	C₂²×C₂₃	1	1	yes	characteristic
157	C₄⋊Q₈	—	2	4	no	—
158	C₂²⋊Q₈	—	2	4	no	—
159	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
160	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
161	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
162	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
163	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
164	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
165	C₂×C₂²⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
166	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
167	C₂×C₂²⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
168	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
169	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
170	C₂×C₄²	C₂²×C₂₃	1	2	yes	—
171	#95	C₂²×C₂₃	1	1	yes	characteristic
172	#95	C₂²×C₂₃	1	1	yes	characteristic
173	#95	C₂²×C₂₃	1	2	yes	—
174	#95	C₂²×C₂₃	1	2	yes	—
175	#95	C₂²×C₂₃	1	2	yes	—
176	#95	C₂²×C₂₃	1	2	yes	—
177	#95	C₂²×C₂₃	1	2	yes	—
178	#95	C₂²×C₂₃	1	2	yes	—
179	C₂₃	#2492	1	1	yes	characteristic, direct product factor, Sylow
180	C₂⁴	C₂³×C₂₃	1	1	yes	characteristic
181	C₂×Q₈	—	2	2	no	—
182	C₂×Q₈	—	2	4	no	—
183	C₂×Q₈	—	2	4	no	—
184	C₂×Q₈	—	2	2	no	—
185	C₂×Q₈	—	2	2	no	—
186	C₂×Q₈	—	4	2	no	—
187	C₂²×C₄	—	2	2	no	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
191	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
192	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
193	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
194	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
195	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
196	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
197	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
198	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
199	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
200	C₄⋊C₄	—	2	4	no	—
201	C₄⋊C₄	—	2	4	no	—
202	C₄⋊C₄	—	2	2	no	—
203	C₄⋊C₄	—	2	2	no	—
204	C₄⋊C₄	D₄×C₂₃	1	4	yes	—
205	C₂²⋊C₄	—	2	4	no	—
206	C₂²⋊C₄	—	2	2	no	—
207	C₂²⋊C₄	—	4	2	no	—
208	C₂²⋊C₄	Q₈×C₂₃	1	4	yes	complemented
209	C₄²	—	2	2	no	—
210	C₄²	—	2	2	no	—
211	C₄²	—	4	2	no	—
212	C₂³	—	2	1	no	—
213	C₂³	—	2	1	no	—
214	C₂³	—	2	1	no	—
215	C₂³	—	2	1	no	—
216	C₂³	—	2	1	no	—
217	C₂³	—	2	1	no	—
218	C₂³	C₂×Q₈×C₂₃	1	2	yes	—
219	C₂³	C₂⁴×C₂₃	1	1	yes	characteristic
220	Q₈	—	4	8	no	—
221	Q₈	—	4	4	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	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	1	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	4	4	no	—
246	C₂×C₄	C₂×D₄×C₂₃	1	4	yes	—
247	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
248	C₂×C₄	C₂×Q₈×C₂₃	1	2	yes	—
249	C₂×C₄	C₂×Q₈×C₂₃	1	2	yes	—
250	C₂×C₄	#41×C₂₃	1	2	yes	—
251	C₂×C₄	#41×C₂₃	1	2	yes	—
252	C₂×C₄	#41×C₂₃	1	2	yes	—
253	C₂²	—	2	2	no	—
254	C₂²	—	2	2	no	—
255	C₂²	—	2	2	no	—
256	C₂²	—	4	1	no	—
257	C₂²	—	4	1	no	—
258	C₂²	—	4	1	no	—
259	C₂²	—	4	1	no	—
260	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
261	C₂²	C₂²×Q₈×C₂₃	1	1	yes	characteristic
262	C₂²	C₂×#41×C₂₃	1	1	yes	characteristic
263	C₂²	C₂×#41×C₂₃	1	1	yes	characteristic
264	C₂²	C₂×#41×C₂₃	1	1	yes	characteristic
265	C₂²	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
266	C₂²	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
267	C₄	—	2	2	no	—
268	C₄	—	2	2	no	—
269	C₄	—	2	4	no	—
270	C₄	—	2	2	no	—
271	C₄	—	4	1	no	—
272	C₄	—	4	1	no	—
273	C₄	—	4	2	no	—
274	C₄	—	4	2	no	—
275	C₄	—	4	2	no	—
276	C₄	C₂₃×#124	1	4	yes	—
277	C₂	—	4	2	no	—
278	C₂	C₂×C₂₃×#124	1	1	yes	characteristic
279	C₂	C₂₃×#533	1	1	yes	characteristic
280	C₂	C₂₃×#542	1	1	yes	characteristic
281	C₂	C₂₃×#546	1	1	yes	characteristic
282	C₂	D₄×Q₈×C₂₃	1	1	yes	characteristic
283	C₂	C₂₃×#554	1	1	yes	characteristic
284	C₂	C₂₃×#556	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₂₃
prime radical	C₂⁴×C₂₃	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₃×C₄⋊Q₈	C₂	semidirect
3. C₂₃×#400	C₂	semidirect
5. C₂₃×#391	C₂	semidirect
6. C₂₃×#391	C₂	semidirect
7. C₂₃×#391	C₂	semidirect
10. C₄×C₄⋊C₄×C₂₃	C₂	semidirect
65. C₂²⋊C₄×C₂₃	Q₈	semidirect
110. #2492	C₂₃	direct (N × H)
135. C₂×C₄⋊Q₈	C₂×C₂₃	semidirect
137. #400	C₂×C₂₃	semidirect
139. #391	C₂×C₂₃	semidirect
140. #391	C₂×C₂₃	semidirect
141. #391	C₂×C₂₃	semidirect
144. C₄×C₄⋊C₄	C₂×C₂₃	semidirect
179. C₂₃	#2492	direct (N × H)
208. C₂²⋊C₄	Q₈×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×C₄⋊Q₈	2
C₂×C₂₃×C₂²⋊Q₈	2
C₂₃×#400	2
C₂₃×#393	2
C₂₃×#391	2
C₂₃×#388	2
C₂₃×#380	2
C₄×C₄⋊C₄×C₂₃	2
C₄×C₂²⋊C₄×C₂₃	2
#2492	23

Subgroups — C23×#2492 #15670111

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#2492 #15670111