Subgroups — C₂₃×#2626 #15670245

Order 2944.

668 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₂²⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#399	C₂	1	2	yes	maximal, complemented
4	C₂₃×#396	C₂	1	4	yes	maximal, complemented
5	C₂₃×#393	C₂	1	2	yes	maximal, complemented
6	C₂₃×#385	C₂	1	2	yes	maximal, complemented
7	C₂₃×#380	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#379	C₂	1	2	yes	maximal
9	C₂⁵×C₂₃	C₂²	1	1	yes	characteristic
10	C₂²×Q₈×C₂₃	C₂²	1	1	yes	characteristic, complemented
11	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
12	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
13	C₂₃×C₂²⋊Q₈	—	2	4	no	—
14	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	complemented
15	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic, complemented
16	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	complemented
17	C₂×C₄⋊C₄×C₂₃	C₂²	1	4	yes	complemented
18	C₂×C₂²⋊C₄×C₂₃	—	2	4	no	—
19	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
20	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
21	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
22	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
23	C₂×C₂²⋊C₄×C₂₃	C₂²	1	4	yes	—
24	C₂×C₂²⋊C₄×C₂₃	C₂²	1	4	yes	—
25	C₂₃×#95	C₂²	1	2	yes	complemented
26	C₂₃×#95	C₂²	1	4	yes	complemented
27	C₂⁴×C₂₃	—	2	2	no	—
28	C₂⁴×C₂₃	—	2	2	no	—
29	C₂⁴×C₂₃	—	4	2	no	—
30	C₂⁴×C₂₃	—	4	2	no	—
31	C₂⁴×C₂₃	Q₈	1	4	yes	complemented
32	C₂⁴×C₂₃	C₂³	1	2	yes	—
33	C₂⁴×C₂₃	C₂³	1	1	yes	characteristic
34	C₂×Q₈×C₂₃	—	2	2	no	—
35	C₂×Q₈×C₂₃	—	4	2	no	—
36	C₂²×C₄×C₂₃	—	2	2	no	—
37	C₂²×C₄×C₂₃	—	4	2	no	—
38	C₂²×C₄×C₂₃	—	4	2	no	—
39	C₂²×C₄×C₂₃	D₄	1	4	yes	—
40	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
41	C₂²×C₄×C₂₃	C₂³	1	4	yes	—
42	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
43	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
44	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
45	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
46	C₄⋊C₄×C₂₃	—	2	2	no	—
47	C₄⋊C₄×C₂₃	—	2	4	no	—
48	C₄⋊C₄×C₂₃	—	4	2	no	—
49	C₄⋊C₄×C₂₃	—	4	4	no	—
50	C₂²⋊C₄×C₂₃	—	2	4	no	—
51	C₂²⋊C₄×C₂₃	—	2	4	no	—
52	C₂²⋊C₄×C₂₃	—	2	4	no	—
53	C₂²⋊C₄×C₂₃	—	2	4	no	—
54	C₂²⋊C₄×C₂₃	—	4	4	no	—
55	C₂²⋊C₄×C₂₃	—	4	4	no	—
56	C₂³×C₂₃	—	2	2	no	—
57	C₂³×C₂₃	—	2	2	no	—
58	C₂³×C₂₃	—	2	8	no	—
59	C₂³×C₂₃	—	2	2	no	—
60	C₂³×C₂₃	—	2	2	no	—
61	C₂³×C₂₃	—	2	4	no	—
62	C₂³×C₂₃	—	2	4	no	—
63	C₂³×C₂₃	—	4	8	no	—
64	C₂³×C₂₃	—	4	4	no	—
65	C₂³×C₂₃	—	4	4	no	—
66	C₂³×C₂₃	—	4	8	no	—
67	C₂³×C₂₃	C₂×D₄	1	2	yes	—
68	C₂³×C₂₃	C₂×Q₈	1	4	yes	—
69	C₂³×C₂₃	C₂×Q₈	1	2	yes	—
70	C₂³×C₂₃	#41	1	2	yes	—
71	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
72	Q₈×C₂₃	—	4	4	no	—
73	C₂×C₄×C₂₃	—	2	2	no	—
74	C₂×C₄×C₂₃	—	2	2	no	—
75	C₂×C₄×C₂₃	—	2	2	no	—
76	C₂×C₄×C₂₃	—	2	2	no	—
77	C₂×C₄×C₂₃	—	2	2	no	—
78	C₂×C₄×C₂₃	—	2	4	no	—
79	C₂×C₄×C₂₃	—	2	4	no	—
80	C₂×C₄×C₂₃	—	2	4	no	—
81	C₂×C₄×C₂₃	—	2	4	no	—
82	C₂×C₄×C₂₃	—	2	4	no	—
83	C₂×C₄×C₂₃	—	2	4	no	—
84	C₂×C₄×C₂₃	—	4	4	no	—
85	C₂×C₄×C₂₃	—	4	4	no	—
86	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
87	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
88	#2626	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
89	C₂²×C₂₃	—	2	2	no	—
90	C₂²×C₂₃	—	2	2	no	—
91	C₂²×C₂₃	—	2	4	no	—
92	C₂²×C₂₃	—	2	4	no	—
93	C₂²×C₂₃	—	2	4	no	—
94	C₂²×C₂₃	—	2	8	no	—
95	C₂²×C₂₃	—	4	16	no	—
96	C₂²×C₂₃	—	4	2	no	—
97	C₂²×C₂₃	—	4	2	no	—
98	C₂²×C₂₃	—	4	4	no	—
99	C₂²×C₂₃	C₂²⋊Q₈	1	4	yes	—
100	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
101	C₂²×C₂₃	C₂²×Q₈	1	1	yes	characteristic
102	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
103	C₂²×C₂₃	H(5,𝔽₂)	1	2	yes	—
104	C₂²×C₂₃	H(5,𝔽₂)	1	2	yes	—
105	C₄×C₂₃	—	2	2	no	—
106	C₄×C₂₃	—	2	2	no	—
107	C₄×C₂₃	—	4	2	no	—
108	C₄×C₂₃	—	4	2	no	—
109	C₄×C₂₃	—	4	2	no	—
110	C₄×C₂₃	—	4	4	no	—
111	C₂×C₂²⋊Q₈	C₂×C₂₃	1	1	yes	characteristic, complemented
112	C₂²×C₂²⋊C₄	C₂×C₂₃	1	1	yes	characteristic
113	#399	C₂×C₂₃	1	2	yes	complemented
114	#396	C₂×C₂₃	1	4	yes	complemented
115	#393	C₂×C₂₃	1	2	yes	complemented
116	#385	C₂×C₂₃	1	2	yes	complemented
117	#380	C₂×C₂₃	1	1	yes	characteristic, complemented
118	#379	C₂×C₂₃	1	2	yes	—
119	C₂×C₂₃	—	2	4	no	—
120	C₂×C₂₃	—	4	4	no	—
121	C₂×C₂₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
122	C₂×C₂₃	#534	1	1	yes	characteristic
123	C₂×C₂₃	#535	1	1	yes	characteristic
124	C₂×C₂₃	#538	1	2	yes	—
125	C₂×C₂₃	#543	1	2	yes	—
126	C₂⁵	C₂²×C₂₃	1	1	yes	characteristic
127	C₂²×Q₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
128	C₂³×C₄	C₂²×C₂₃	1	1	yes	characteristic
129	C₂³×C₄	C₂²×C₂₃	1	1	yes	characteristic
130	C₂²⋊Q₈	—	2	4	no	—
131	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	complemented
132	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic, complemented
133	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	complemented
134	C₂×C₄⋊C₄	C₂²×C₂₃	1	4	yes	complemented
135	C₂×C₂²⋊C₄	—	2	4	no	—
136	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
137	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
138	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
139	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
140	C₂×C₂²⋊C₄	C₂²×C₂₃	1	4	yes	—
141	C₂×C₂²⋊C₄	C₂²×C₂₃	1	4	yes	—
142	#95	C₂²×C₂₃	1	2	yes	complemented
143	#95	C₂²×C₂₃	1	4	yes	complemented
144	C₂₃	#2626	1	1	yes	characteristic, direct product factor, Sylow
145	C₂⁴	—	2	2	no	—
146	C₂⁴	—	2	2	no	—
147	C₂⁴	—	4	2	no	—
148	C₂⁴	—	4	2	no	—
149	C₂⁴	Q₈×C₂₃	1	4	yes	complemented
150	C₂⁴	C₂³×C₂₃	1	2	yes	—
151	C₂⁴	C₂³×C₂₃	1	1	yes	characteristic
152	C₂×Q₈	—	2	2	no	—
153	C₂×Q₈	—	4	2	no	—
154	C₂²×C₄	—	2	2	no	—
155	C₂²×C₄	—	4	2	no	—
156	C₂²×C₄	—	4	2	no	—
157	C₂²×C₄	D₄×C₂₃	1	4	yes	—
158	C₂²×C₄	C₂³×C₂₃	1	4	yes	—
159	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
160	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
161	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
162	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
163	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
164	C₄⋊C₄	—	2	2	no	—
165	C₄⋊C₄	—	2	4	no	—
166	C₄⋊C₄	—	4	2	no	—
167	C₄⋊C₄	—	4	4	no	—
168	C₂²⋊C₄	—	2	4	no	—
169	C₂²⋊C₄	—	2	4	no	—
170	C₂²⋊C₄	—	2	4	no	—
171	C₂²⋊C₄	—	2	4	no	—
172	C₂²⋊C₄	—	4	4	no	—
173	C₂²⋊C₄	—	4	4	no	—
174	C₂³	—	2	2	no	—
175	C₂³	—	2	2	no	—
176	C₂³	—	2	8	no	—
177	C₂³	—	2	2	no	—
178	C₂³	—	2	2	no	—
179	C₂³	—	2	4	no	—
180	C₂³	—	2	4	no	—
181	C₂³	—	4	8	no	—
182	C₂³	—	4	4	no	—
183	C₂³	—	4	4	no	—
184	C₂³	—	4	8	no	—
185	C₂³	C₂×D₄×C₂₃	1	2	yes	—
186	C₂³	C₂×Q₈×C₂₃	1	4	yes	—
187	C₂³	C₂×Q₈×C₂₃	1	2	yes	—
188	C₂³	#41×C₂₃	1	2	yes	—
189	C₂³	C₂⁴×C₂₃	1	1	yes	characteristic
190	Q₈	—	4	4	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	4	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	2	4	no	—
199	C₂×C₄	—	2	4	no	—
200	C₂×C₄	—	2	4	no	—
201	C₂×C₄	—	2	4	no	—
202	C₂×C₄	—	4	4	no	—
203	C₂×C₄	—	4	4	no	—
204	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
205	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
206	C₂²	—	2	2	no	—
207	C₂²	—	2	2	no	—
208	C₂²	—	2	4	no	—
209	C₂²	—	2	4	no	—
210	C₂²	—	2	4	no	—
211	C₂²	—	2	8	no	—
212	C₂²	—	4	16	no	—
213	C₂²	—	4	2	no	—
214	C₂²	—	4	2	no	—
215	C₂²	—	4	4	no	—
216	C₂²	C₂₃×C₂²⋊Q₈	1	4	yes	—
217	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
218	C₂²	C₂²×Q₈×C₂₃	1	1	yes	characteristic
219	C₂²	C₂×#41×C₂₃	1	1	yes	characteristic
220	C₂²	C₂₃×H(5,𝔽₂)	1	2	yes	—
221	C₂²	C₂₃×H(5,𝔽₂)	1	2	yes	—
222	C₄	—	2	2	no	—
223	C₄	—	2	2	no	—
224	C₄	—	4	2	no	—
225	C₄	—	4	2	no	—
226	C₄	—	4	2	no	—
227	C₄	—	4	4	no	—
228	C₂	—	2	4	no	—
229	C₂	—	4	4	no	—
230	C₂	C₂×C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
231	C₂	C₂₃×#534	1	1	yes	characteristic
232	C₂	C₂₃×#535	1	1	yes	characteristic
233	C₂	C₂₃×#538	1	2	yes	—
234	C₂	C₂₃×#543	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₂₃
prime radical	C₂⁵×C₂₃	C₂²
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×C₂²⋊Q₈	C₂	semidirect
3. C₂₃×#399	C₂	semidirect
4. C₂₃×#396	C₂	semidirect
5. C₂₃×#393	C₂	semidirect
6. C₂₃×#385	C₂	semidirect
7. C₂₃×#380	C₂	semidirect
10. C₂²×Q₈×C₂₃	C₂²	semidirect
14. C₂×C₄⋊C₄×C₂₃	C₂²	semidirect
15. C₂×C₄⋊C₄×C₂₃	C₂²	semidirect
16. C₂×C₄⋊C₄×C₂₃	C₂²	semidirect
17. C₂×C₄⋊C₄×C₂₃	C₂²	semidirect
25. C₂₃×#95	C₂²	semidirect
26. C₂₃×#95	C₂²	semidirect
31. C₂⁴×C₂₃	Q₈	semidirect
88. #2626	C₂₃	direct (N × H)
111. C₂×C₂²⋊Q₈	C₂×C₂₃	semidirect
113. #399	C₂×C₂₃	semidirect
114. #396	C₂×C₂₃	semidirect
115. #393	C₂×C₂₃	semidirect
116. #385	C₂×C₂₃	semidirect
117. #380	C₂×C₂₃	semidirect
127. C₂²×Q₈	C₂²×C₂₃	semidirect
131. C₂×C₄⋊C₄	C₂²×C₂₃	semidirect
132. C₂×C₄⋊C₄	C₂²×C₂₃	semidirect
133. C₂×C₄⋊C₄	C₂²×C₂₃	semidirect
134. C₂×C₄⋊C₄	C₂²×C₂₃	semidirect
142. #95	C₂²×C₂₃	semidirect
143. #95	C₂²×C₂₃	semidirect
144. C₂₃	#2626	direct (N × H)
149. C₂⁴	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×C₂²⋊Q₈	2
C₂²×C₂²⋊C₄×C₂₃	2
C₂₃×#399	2
C₂₃×#396	2
C₂₃×#393	2
C₂₃×#385	2
C₂₃×#380	2
C₂₃×#379	2
#2626	23

Subgroups — C23×#2626 #15670245

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#2626 #15670245