Subgroups — #13092494

Order 1472.

122 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#67262	C₂	1	2	yes	maximal, complemented
2	C₄×D₉₂	C₂	1	1	yes	maximal, characteristic
3	C₂₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	(C₂×D₂₃)⋊C₈	C₂	1	2	yes	maximal, complemented
5	C₄×C₂₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₉₂	C₄	1	1	yes	characteristic
7	C₂×C₄×D₂₃	C₂²	1	2	yes	—
8	C₂×C₈×C₂₃	C₂²	1	2	yes	—
9	C₄²×C₂₃	C₂²	1	1	yes	characteristic
10	(C₂×D₂₃)⋊C₄	C₄	1	2	yes	—
11	C₄⋊Dic₂₃	C₄	1	1	yes	characteristic
12	C₂×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
14	#67262	—	2	4	no	—
15	C₂²×D₂₃	C₂×C₄	1	2	yes	—
16	C₂×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
19	C₂×Dic₂₃	C₂×C₄	1	2	yes	—
20	D₉₂	—	2	2	no	—
21	C₄×D₂₃	—	2	2	no	—
22	C₄×D₂₃	—	2	2	no	—
23	C₈×C₂₃	—	2	2	no	—
24	C₂₃⋊C₈	—	2	1	no	—
25	C₂₃⋊C₈	D₄	1	2	yes	complemented
26	C₂²×C₂₃	C₂²×C₄	1	1	yes	characteristic
27	C₂×D₂₃	—	2	2	no	—
28	C₂×D₂₃	—	2	2	no	—
29	C₂×D₂₃	—	2	2	no	—
30	C₄×C₂₃	—	2	1	no	—
31	C₄×C₂₃	M₁₆	1	2	yes	complemented
32	C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
33	C₄×C₂₃	#41	1	1	yes	characteristic
34	Dic₂₃	—	2	2	no	—
35	#436	—	23	1	no	maximal, Sylow
36	C₂×C₂₃	C₄×D₄	1	1	yes	characteristic
37	C₂×C₂₃	C₂×M₁₆	1	1	yes	characteristic
38	C₂×C₂₃	#131	1	1	yes	characteristic
39	D₂₃	—	4	2	no	—
40	C₂×M₁₆	—	23	2	no	—
41	C₄×D₄	—	23	1	no	—
42	C₄⋊C₈	D₂₃	1	1	yes	characteristic, complemented
43	C₂²⋊C₈	—	23	2	no	—
44	C₄×C₈	—	23	1	no	—
45	C₂₃	#436	1	1	yes	characteristic, complemented, Sylow
46	C₂×D₄	—	23	1	no	—
47	C₂²×C₄	—	23	2	no	—
48	M₁₆	—	46	4	no	—
49	C₂×C₈	—	23	1	no	—
50	C₂×C₈	—	23	1	no	—
51	C₂×C₈	C₂×D₂₃	1	2	yes	—
52	C₄⋊C₄	—	23	1	no	—
53	C₂²⋊C₄	—	23	2	no	—
54	C₄²	C₂×D₂₃	1	1	yes	characteristic
55	C₂³	—	23	2	no	—
56	D₄	—	46	2	no	—
57	C₂×C₄	—	23	2	no	—
58	C₂×C₄	—	46	2	no	—
59	C₂×C₄	—	46	2	no	—
60	C₂×C₄	C₄×D₂₃	1	1	yes	characteristic
61	C₂×C₄	C₄×D₂₃	1	1	yes	characteristic
62	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
63	C₈	—	2	2	no	—
64	C₈	—	23	2	no	—
65	C₈	—	46	1	no	—
66	C₂²	—	46	2	no	—
67	C₂²	—	46	2	no	—
68	C₂²	—	46	2	no	—
69	C₂²	C₂×C₄×D₂₃	1	1	yes	characteristic
70	C₄	—	2	1	no	—
71	C₄	—	46	2	no	—
72	C₄	#67262	1	2	yes	complemented
73	C₄	D₄×D₂₃	1	1	yes	characteristic
74	C₄	#67292	1	1	yes	characteristic
75	C₂	—	92	2	no	—
76	C₂	#10625064	1	1	yes	characteristic
77	C₂	C₂×#67262	1	1	yes	characteristic
78	C₂	#10625078	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₂₃
commutator, socle	C₂²×C₂₃	C₂²×C₄
Fitting	C₂₃×C₄⋊C₈	C₂
prime radical	C₂×D₉₂	C₄
cosocle	C₂×C₄×C₂₃	C₂³
hypercenter	C₄⋊C₈	D₂₃
hypocenter	C₂₃	#436
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
1. C₂×#67262	C₂	semidirect
3. C₂₃×C₄⋊C₈	C₂	semidirect
4. (C₂×D₂₃)⋊C₈	C₂	semidirect
5. C₄×C₂₃⋊C₈	C₂	semidirect
25. C₂₃⋊C₈	D₄	semidirect
31. C₄×C₂₃	M₁₆	semidirect
42. C₄⋊C₈	D₂₃	semidirect
45. C₂₃	#436	semidirect
72. C₄	#67262	semidirect

Maximal subgroups

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

subgroup	index
C₂×#67262	2
C₄×D₉₂	2
C₂₃×C₄⋊C₈	2
(C₂×D₂₃)⋊C₈	2
C₄×C₂₃⋊C₈	2
#436	23