Subgroups — #13092793

Order 1472.

162 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#127	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#67272	C₂	1	4	yes	maximal, complemented
3	#10625045	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×C₂₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₄×C₂₃	—	2	2	no	—
6	C₂×D₄×C₂₃	C₂²	1	2	yes	—
7	C₂×D₉₂	—	2	2	no	—
8	C₂×D₉₂	C₂²	1	2	yes	—
9	C₄²×C₂₃	C₂²	1	1	yes	characteristic
10	#67272	—	2	8	no	—
11	C₂×C₂₃⋊C₈	C₂²	1	2	yes	—
12	C₂³×C₂₃	—	2	2	no	—
13	C₂²×D₂₃	—	2	2	no	—
14	D₄×C₂₃	—	2	4	no	—
15	D₄×C₂₃	—	4	4	no	—
16	C₂×C₄×C₂₃	D₄	1	2	yes	—
17	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
18	D₉₂	—	2	4	no	—
19	D₉₂	—	4	4	no	—
20	C₂₃⋊C₈	D₄	1	4	yes	complemented
21	C₂²×C₂₃	—	4	2	no	—
22	C₂²×C₂₃	—	4	4	no	—
23	C₂²×C₂₃	C₂×D₄	1	1	yes	characteristic
24	C₂×D₂₃	—	4	2	no	—
25	C₂×D₂₃	—	4	4	no	—
26	C₄×C₂₃	D₈	1	4	yes	complemented
27	C₄×C₂₃	C₂×D₄	1	2	yes	—
28	#493	—	23	1	no	maximal, Sylow
29	C₂×C₂₃	—	8	2	no	—
30	C₂×C₂₃	#127	1	1	yes	characteristic
31	C₂×C₂₃	C₂×D₈	1	2	yes	—
32	D₂₃	—	8	2	no	—
33	C₂×D₈	—	23	4	no	—
34	#127	—	23	1	no	—
35	#127	D₂₃	1	1	yes	characteristic, complemented
36	C₄×C₈	—	23	1	no	—
37	C₂₃	#493	1	1	yes	characteristic, complemented, Sylow
38	C₂×D₄	—	2	2	no	—
39	C₂×D₄	—	23	2	no	—
40	C₂×D₄	—	46	2	no	—
41	C₂×D₄	C₂×D₂₃	1	2	yes	—
42	D₈	—	46	8	no	—
43	C₂×C₈	—	23	2	no	—
44	C₄²	C₂×D₂₃	1	1	yes	characteristic
45	C₂³	—	2	2	no	—
46	C₂³	—	46	2	no	—
47	D₄	—	2	4	no	—
48	D₄	—	4	4	no	—
49	D₄	—	46	4	no	—
50	D₄	—	92	4	no	—
51	C₂×C₄	C₂²⋊D₂₃	1	2	yes	—
52	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
53	C₈	—	23	4	no	—
54	C₂²	—	4	4	no	—
55	C₂²	—	4	2	no	—
56	C₂²	—	92	4	no	—
57	C₂²	—	92	2	no	—
58	C₂²	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
59	C₄	#67272	1	4	yes	complemented
60	C₄	D₄×D₂₃	1	2	yes	—
61	C₂	—	8	2	no	—
62	C₂	—	184	2	no	—
63	C₂	C₂×#67272	1	2	yes	—
64	C₂	#10625111	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₂₃
commutator, cosocle	C₂×C₄×C₂₃	C₂³
Frattini	C₂×C₄	C₂²×D₂₃
Fitting	C₂₃×#127	C₂
socle	C₂²×C₂₃	C₂×D₄
hypercenter	#127	D₂₃
hypocenter	C₂₃	#493
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #127

Complements

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

N	H ≅ G/N	type
1. C₂₃×#127	C₂	semidirect
2. C₂×#67272	C₂	semidirect
3. #10625045	C₂	semidirect
4. C₄×C₂₃⋊C₈	C₂	semidirect
20. C₂₃⋊C₈	D₄	semidirect
26. C₄×C₂₃	D₈	semidirect
35. #127	D₂₃	semidirect
37. C₂₃	#493	semidirect
59. C₄	#67272	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#127	2
C₂×#67272	2
#10625045	2
C₄×C₂₃⋊C₈	2
#493	23