Subgroups — C₂₃×#3081 #15670700

Order 2944.

398 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#544	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#518	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#478	C₂	1	2	yes	maximal, complemented
4	C₂₃×#478	C₂	1	2	yes	maximal, complemented
5	C₂₃×#476	C₂	1	2	yes	maximal, complemented
6	C₂₃×#474	C₂	1	2	yes	maximal, complemented
7	C₂₃×#428	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#427	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#424	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#418	C₂	1	1	yes	maximal, characteristic
11	C₂×C₂₃×#102	C₂	1	1	yes	maximal, characteristic
12	C₂²×D₄×C₂₃	C₂²	1	1	yes	characteristic
13	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
15	C₂₃×C₄⋊Q₈	—	2	1	no	—
16	C₂₃×C₄⋊Q₈	C₂²	1	2	yes	complemented
17	C₂₃×#125	—	2	1	no	—
18	C₂₃×#125	C₂²	1	2	yes	complemented
19	C₂₃×C₂²⋊Q₈	—	2	2	no	—
20	C₄×D₄×C₂₃	—	2	1	no	—
21	C₄×D₄×C₂₃	—	2	1	no	—
22	C₄×D₄×C₂₃	C₂²	1	2	yes	—
23	C₄×D₄×C₂₃	C₂²	1	2	yes	—
24	C₂₃×#117	C₂²	1	1	yes	characteristic
25	C₂₃×#117	C₂²	1	1	yes	characteristic
26	C₂₃×#117	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
28	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
29	C₂₃×#107	C₂²	1	1	yes	characteristic, complemented
30	C₂₃×#107	C₂²	1	1	yes	characteristic, complemented
31	C₂₃×#107	C₂²	1	2	yes	complemented
32	C₂₃×#106	C₂²	1	1	yes	characteristic, complemented
33	C₂₃×#106	C₂²	1	1	yes	characteristic, complemented
34	C₂₃×#106	C₂²	1	2	yes	complemented
35	C₂₃×C₄⋊C₈	C₂²	1	2	yes	complemented
36	C₂₃×C₄⋊C₈	C₂²	1	2	yes	complemented
37	C₂₃×#102	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	C₂⁴×C₂₃	—	2	1	no	—
42	C₂×Q₈×C₂₃	—	2	2	no	—
43	C₂×D₄×C₂₃	—	2	1	no	—
44	C₂×D₄×C₂₃	—	2	2	no	—
45	C₂×D₄×C₂₃	Q₈	1	4	yes	complemented
46	C₂×D₄×C₂₃	C₂³	1	2	yes	—
47	C₂²×C₄×C₂₃	—	2	1	no	—
48	C₂²×C₄×C₂₃	—	2	2	no	—
49	C₂²×C₄×C₂₃	—	2	2	no	—
50	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
51	M₁₆×C₂₃	—	2	2	no	—
52	C₂×C₈×C₂₃	—	2	2	no	—
53	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
54	C₂×C₈×C₂₃	C₂³	1	2	yes	—
55	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
56	C₄⋊C₄×C₂₃	—	2	1	no	—
57	C₄⋊C₄×C₂₃	—	2	2	no	—
58	C₄⋊C₄×C₂₃	—	2	2	no	—
59	C₄⋊C₄×C₂₃	—	2	2	no	—
60	C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
61	C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
62	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
63	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
64	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
65	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
66	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
67	C₂²⋊C₄×C₂₃	—	2	1	no	—
68	C₂²⋊C₄×C₂₃	—	2	1	no	—
69	C₂²⋊C₄×C₂₃	—	2	2	no	—
70	C₂²⋊C₄×C₂₃	—	2	2	no	—
71	C₂²⋊C₄×C₂₃	D₄	1	2	yes	—
72	C₄²×C₂₃	—	2	1	no	—
73	C₄²×C₂₃	—	2	1	no	—
74	C₄²×C₂₃	C₂³	1	2	yes	—
75	C₂³×C₂₃	—	2	2	no	—
76	C₂³×C₂₃	—	2	4	no	—
77	C₂³×C₂₃	—	8	1	no	—
78	C₂³×C₂₃	—	8	1	no	—
79	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
80	Q₈×C₂₃	—	8	2	no	—
81	D₄×C₂₃	—	2	4	no	—
82	D₄×C₂₃	—	2	2	no	—
83	D₄×C₂₃	C₂×Q₈	1	4	yes	—
84	C₂×C₄×C₂₃	—	2	1	no	—
85	C₂×C₄×C₂₃	—	2	1	no	—
86	C₂×C₄×C₂₃	—	2	1	no	—
87	C₂×C₄×C₂₃	—	2	1	no	—
88	C₂×C₄×C₂₃	—	2	2	no	—
89	C₂×C₄×C₂₃	—	8	2	no	—
90	C₂×C₄×C₂₃	—	8	2	no	—
91	C₂×C₄×C₂₃	—	8	1	no	—
92	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
93	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
94	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
96	C₂×C₄×C₂₃	#41	1	2	yes	—
97	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
98	C₈×C₂₃	—	2	1	no	—
99	C₈×C₂₃	—	2	1	no	—
100	C₈×C₂₃	—	2	2	no	—
101	#3081	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
102	C₂²×C₂₃	—	2	1	no	—
103	C₂²×C₂₃	—	2	1	no	—
104	C₂²×C₂₃	—	2	4	no	—
105	C₂²×C₂₃	—	4	2	no	—
106	C₂²×C₂₃	—	4	2	no	—
107	C₂²×C₂₃	—	8	4	no	—
108	C₂²×C₂₃	C₂²⋊Q₈	1	2	yes	—
109	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
110	C₄×C₂₃	—	2	2	no	—
111	C₄×C₂₃	—	2	2	no	—
112	C₄×C₂₃	—	4	1	no	—
113	C₄×C₂₃	—	4	2	no	—
114	C₄×C₂₃	—	4	1	no	—
115	C₄×C₂₃	—	4	1	no	—
116	C₄×C₂₃	—	4	1	no	—
117	C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
118	C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
119	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
120	#544	C₂×C₂₃	1	1	yes	characteristic, complemented
121	#518	C₂×C₂₃	1	1	yes	characteristic
122	#478	C₂×C₂₃	1	2	yes	complemented
123	#478	C₂×C₂₃	1	2	yes	complemented
124	#476	C₂×C₂₃	1	2	yes	complemented
125	#474	C₂×C₂₃	1	2	yes	complemented
126	#428	C₂×C₂₃	1	1	yes	characteristic, complemented
127	#427	C₂×C₂₃	1	1	yes	characteristic, complemented
128	#424	C₂×C₂₃	1	1	yes	characteristic, complemented
129	#418	C₂×C₂₃	1	1	yes	characteristic
130	C₂×#102	C₂×C₂₃	1	1	yes	characteristic
131	C₂×C₂₃	—	2	2	no	—
132	C₂×C₂₃	—	4	4	no	—
133	C₂×C₂₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
134	C₂×C₂₃	#576	1	1	yes	characteristic
135	C₂×C₂₃	#577	1	1	yes	characteristic
136	C₂²×D₄	C₂²×C₂₃	1	1	yes	characteristic
137	C₂×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
138	C₂²×C₈	C₂²×C₂₃	1	1	yes	characteristic
139	C₄⋊Q₈	—	2	1	no	—
140	C₄⋊Q₈	C₂²×C₂₃	1	2	yes	complemented
141	#125	—	2	1	no	—
142	#125	C₂²×C₂₃	1	2	yes	complemented
143	C₂²⋊Q₈	—	2	2	no	—
144	C₄×D₄	—	2	1	no	—
145	C₄×D₄	—	2	1	no	—
146	C₄×D₄	C₂²×C₂₃	1	2	yes	—
147	C₄×D₄	C₂²×C₂₃	1	2	yes	—
148	#117	C₂²×C₂₃	1	1	yes	characteristic
149	#117	C₂²×C₂₃	1	1	yes	characteristic
150	#117	C₂²×C₂₃	1	1	yes	characteristic
151	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
152	C₂×C₂²⋊C₄	—	2	2	no	—
153	#107	C₂²×C₂₃	1	1	yes	characteristic, complemented
154	#107	C₂²×C₂₃	1	1	yes	characteristic, complemented
155	#107	C₂²×C₂₃	1	2	yes	complemented
156	#106	C₂²×C₂₃	1	1	yes	characteristic, complemented
157	#106	C₂²×C₂₃	1	1	yes	characteristic, complemented
158	#106	C₂²×C₂₃	1	2	yes	complemented
159	C₄⋊C₈	C₂²×C₂₃	1	2	yes	complemented
160	C₄⋊C₈	C₂²×C₂₃	1	2	yes	complemented
161	#102	C₂²×C₂₃	1	2	yes	—
162	#102	C₂²×C₂₃	1	2	yes	—
163	#102	C₂²×C₂₃	1	2	yes	—
164	#102	C₂²×C₂₃	1	2	yes	—
165	C₂₃	#3081	1	1	yes	characteristic, direct product factor, Sylow
166	C₂⁴	—	2	1	no	—
167	C₂×Q₈	—	2	2	no	—
168	C₂×D₄	—	2	1	no	—
169	C₂×D₄	—	2	2	no	—
170	C₂×D₄	Q₈×C₂₃	1	4	yes	complemented
171	C₂×D₄	C₂³×C₂₃	1	2	yes	—
172	C₂²×C₄	—	2	1	no	—
173	C₂²×C₄	—	2	2	no	—
174	C₂²×C₄	—	2	2	no	—
175	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
176	M₁₆	—	2	2	no	—
177	C₂×C₈	—	2	2	no	—
178	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
179	C₂×C₈	C₂³×C₂₃	1	2	yes	—
180	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
181	C₄⋊C₄	—	2	1	no	—
182	C₄⋊C₄	—	2	2	no	—
183	C₄⋊C₄	—	2	2	no	—
184	C₄⋊C₄	—	2	2	no	—
185	C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
186	C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
187	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
188	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
189	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
190	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
191	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
192	C₂²⋊C₄	—	2	1	no	—
193	C₂²⋊C₄	—	2	1	no	—
194	C₂²⋊C₄	—	2	2	no	—
195	C₂²⋊C₄	—	2	2	no	—
196	C₂²⋊C₄	D₄×C₂₃	1	2	yes	—
197	C₄²	—	2	1	no	—
198	C₄²	—	2	1	no	—
199	C₄²	C₂³×C₂₃	1	2	yes	—
200	C₂³	—	2	2	no	—
201	C₂³	—	2	4	no	—
202	C₂³	—	8	1	no	—
203	C₂³	—	8	1	no	—
204	C₂³	C₂×D₄×C₂₃	1	1	yes	characteristic
205	Q₈	—	8	2	no	—
206	D₄	—	2	4	no	—
207	D₄	—	2	2	no	—
208	D₄	C₂×Q₈×C₂₃	1	4	yes	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	8	2	no	—
215	C₂×C₄	—	8	2	no	—
216	C₂×C₄	—	8	1	no	—
217	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
218	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
219	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
220	C₂×C₄	C₂×Q₈×C₂₃	1	2	yes	—
221	C₂×C₄	#41×C₂₃	1	2	yes	—
222	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
223	C₈	—	2	1	no	—
224	C₈	—	2	1	no	—
225	C₈	—	2	2	no	—
226	C₂²	—	2	1	no	—
227	C₂²	—	2	1	no	—
228	C₂²	—	2	4	no	—
229	C₂²	—	4	2	no	—
230	C₂²	—	4	2	no	—
231	C₂²	—	8	4	no	—
232	C₂²	C₂₃×C₂²⋊Q₈	1	2	yes	—
233	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
234	C₄	—	2	2	no	—
235	C₄	—	2	2	no	—
236	C₄	—	4	1	no	—
237	C₄	—	4	2	no	—
238	C₄	—	4	1	no	—
239	C₄	—	4	1	no	—
240	C₄	—	4	1	no	—
241	C₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
242	C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
243	C₄	C₂×#41×C₂₃	1	1	yes	characteristic
244	C₂	—	2	2	no	—
245	C₂	—	4	4	no	—
246	C₂	C₂×C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
247	C₂	C₂₃×#576	1	1	yes	characteristic
248	C₂	C₂₃×#577	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₂₃
prime radical	C₂²×D₄×C₂₃	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂₃×#117 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#544	C₂	semidirect
3. C₂₃×#478	C₂	semidirect
4. C₂₃×#478	C₂	semidirect
5. C₂₃×#476	C₂	semidirect
6. C₂₃×#474	C₂	semidirect
7. C₂₃×#428	C₂	semidirect
8. C₂₃×#427	C₂	semidirect
9. C₂₃×#424	C₂	semidirect
16. C₂₃×C₄⋊Q₈	C₂²	semidirect
18. C₂₃×#125	C₂²	semidirect
29. C₂₃×#107	C₂²	semidirect
30. C₂₃×#107	C₂²	semidirect
31. C₂₃×#107	C₂²	semidirect
32. C₂₃×#106	C₂²	semidirect
33. C₂₃×#106	C₂²	semidirect
34. C₂₃×#106	C₂²	semidirect
35. C₂₃×C₄⋊C₈	C₂²	semidirect
36. C₂₃×C₄⋊C₈	C₂²	semidirect
45. C₂×D₄×C₂₃	Q₈	semidirect
101. #3081	C₂₃	direct (N × H)
120. #544	C₂×C₂₃	semidirect
122. #478	C₂×C₂₃	semidirect
123. #478	C₂×C₂₃	semidirect
124. #476	C₂×C₂₃	semidirect
125. #474	C₂×C₂₃	semidirect
126. #428	C₂×C₂₃	semidirect
127. #427	C₂×C₂₃	semidirect
128. #424	C₂×C₂₃	semidirect
140. C₄⋊Q₈	C₂²×C₂₃	semidirect
142. #125	C₂²×C₂₃	semidirect
153. #107	C₂²×C₂₃	semidirect
154. #107	C₂²×C₂₃	semidirect
155. #107	C₂²×C₂₃	semidirect
156. #106	C₂²×C₂₃	semidirect
157. #106	C₂²×C₂₃	semidirect
158. #106	C₂²×C₂₃	semidirect
159. C₄⋊C₈	C₂²×C₂₃	semidirect
160. C₄⋊C₈	C₂²×C₂₃	semidirect
165. C₂₃	#3081	direct (N × H)
170. C₂×D₄	Q₈×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#544	2
C₂₃×#518	2
C₂₃×#478	2
C₂₃×#476	2
C₂₃×#474	2
C₂₃×#428	2
C₂₃×#427	2
C₂₃×#424	2
C₂₃×#418	2
C₂×C₂₃×#102	2
#3081	23

Subgroups — C23×#3081 #15670700

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#3081 #15670700