Subgroups — #72245197

Order 2352.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊(ΓL(1,𝔽₈)⋊D₇)	C₂	1	2	yes	maximal, complemented
2	C₇×C₈×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented
3	#11716063	C₃	1	1	yes	maximal, characteristic, complemented, Hall
4	C₂×ΓL(1,𝔽₈)⋊D₇	—	2	2	no	—
5	C₄×C₇×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
6	C₄⋊(C₇⋊D₇)	C₂×C₃	1	2	yes	complemented
7	C₇²×C₈	C₂×C₃	1	1	yes	characteristic, complemented
8	C₃×D₅₆	—	7	1	no	maximal
9	D₅₆⋊C₃	—	7	1	no	maximal
10	C₂×C₇×ΓL(1,𝔽₈)	D₄	1	1	yes	characteristic
11	ΓL(1,𝔽₈)⋊D₇	—	4	2	no	—
12	C₂×C₇⋊D₇	—	2	2	no	—
13	C₄×C₇²	C₂²×C₃	1	1	yes	characteristic
14	C₃×D₂₈	—	7	2	no	—
15	C₄⋊AGL(1,𝔽₇)	—	7	2	no	—
16	C₃×C₇×C₈	—	14	1	no	—
17	C₈×ΓL(1,𝔽₈)	D₇	1	1	yes	characteristic, complemented
18	C₇×ΓL(1,𝔽₈)	D₈	1	1	yes	characteristic, complemented, Hall
19	D₅₆	—	7	1	no	—
20	D₅₆	—	7	1	no	—
21	D₅₆	—	21	2	no	—
22	C₂×C₇²	C₃×D₄	1	1	yes	characteristic
23	C₇⋊D₇	—	4	2	no	—
24	C₂×C₃×D₇	—	14	2	no	—
25	C₂×AGL(1,𝔽₇)	—	14	2	no	—
26	C₃×C₄×C₇	—	14	1	no	—
27	C₄×ΓL(1,𝔽₈)	C₂×D₇	1	1	yes	characteristic
28	D₂₈	—	7	2	no	—
29	D₂₈	—	7	2	no	—
30	D₂₈	—	21	4	no	—
31	C₇×C₈	—	3	2	no	—
32	C₇×C₈	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
33	C₇×C₈	C₃×D₇	1	1	yes	characteristic, complemented
34	C₇²	C₃×D₈	1	1	yes	characteristic, complemented, Sylow
35	C₃×D₈	—	49	1	no	Hall
36	C₂×C₃×C₇	—	14	1	no	—
37	C₃×D₇	—	28	2	no	—
38	C₂×ΓL(1,𝔽₈)	D₂₈	1	1	yes	characteristic
39	AGL(1,𝔽₇)	—	28	2	no	—
40	C₂×D₇	—	14	2	no	—
41	C₂×D₇	—	14	2	no	—
42	C₂×D₇	—	42	4	no	—
43	C₄×C₇	—	3	2	no	—
44	C₄×C₇	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
45	C₄×C₇	C₂×C₃×D₇	1	1	yes	characteristic
46	C₃×D₄	—	49	2	no	—
47	C₃×C₈	—	14	1	no	—
48	C₃×C₇	—	14	1	no	—
49	ΓL(1,𝔽₈)	D₅₆	1	1	yes	characteristic, complemented
50	D₈	—	49	1	no	Sylow
51	C₂×C₇	—	3	2	no	—
52	C₂×C₇	C₄⋊AGL(1,𝔽₇)	1	1	yes	characteristic
53	C₂×C₇	C₃×D₂₈	1	1	yes	characteristic
54	D₇	—	28	2	no	—
55	D₇	—	28	2	no	—
56	D₇	—	84	4	no	—
57	C₂²×C₃	—	98	2	no	—
58	C₃×C₄	—	14	1	no	—
59	D₄	—	49	2	no	—
60	C₈	ΓL(1,𝔽₈)⋊D₇	1	1	yes	characteristic, complemented
61	C₇	—	3	2	no	—
62	C₇	D₅₆⋊C₃	1	1	yes	characteristic, complemented
63	C₇	C₃×D₅₆	1	1	yes	characteristic, complemented
64	C₂×C₃	—	14	1	no	—
65	C₂×C₃	—	196	2	no	—
66	C₂²	—	98	2	no	—
67	C₄	C₂×ΓL(1,𝔽₈)⋊D₇	1	1	yes	characteristic
68	C₃	—	14	1	no	Sylow
69	C₂	—	196	2	no	—
70	C₂	C₄⋊(ΓL(1,𝔽₈)⋊D₇)	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₄⋊(ΓL(1,𝔽₈)⋊D₇)
commutator, cosocle	C₄×C₇²	C₂²×C₃
Frattini	C₄	C₂×ΓL(1,𝔽₈)⋊D₇
Fitting	C₇²×C₈	C₂×C₃
socle	C₂×C₇²	C₃×D₄
hypercenter	C₈	ΓL(1,𝔽₈)⋊D₇
hypocenter	C₇²	C₃×D₈
LCS	G ▷ C₄×C₇² ▷ C₂×C₇² ▷ C₇²
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

Complements

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

N	H ≅ G/N	type
1. C₄⋊(ΓL(1,𝔽₈)⋊D₇)	C₂	semidirect
2. C₇×C₈×ΓL(1,𝔽₈)	C₂	semidirect
3. #11716063	C₃	semidirect
6. C₄⋊(C₇⋊D₇)	C₂×C₃	semidirect
7. C₇²×C₈	C₂×C₃	semidirect
17. C₈×ΓL(1,𝔽₈)	D₇	semidirect
18. C₇×ΓL(1,𝔽₈)	D₈	semidirect
32. C₇×C₈	AGL(1,𝔽₇)	semidirect
33. C₇×C₈	C₃×D₇	semidirect
34. C₇²	C₃×D₈	semidirect
49. ΓL(1,𝔽₈)	D₅₆	semidirect
60. C₈	ΓL(1,𝔽₈)⋊D₇	semidirect
62. C₇	D₅₆⋊C₃	semidirect
63. C₇	C₃×D₅₆	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊(ΓL(1,𝔽₈)⋊D₇)	2
C₇×C₈×ΓL(1,𝔽₈)	2
#11716063	3
C₃×D₅₆	7
D₅₆⋊C₃	7