Subgroups — #72244992

Order 2352.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×ΓL(1,𝔽₈)⋊D₇	C₂	1	1	yes	maximal, characteristic
2	ΓL(1,𝔽₈)×C₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×ΓL(1,𝔽₈)⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#11716031	C₃	1	1	yes	maximal, characteristic, complemented, Hall
5	C₂×ΓL(1,𝔽₈)⋊D₇	C₄	1	1	yes	characteristic
6	C₄×C₇×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
7	ΓL(1,𝔽₈)⋊Dic₇	C₄	1	1	yes	characteristic
8	C₄×C₇⋊D₇	C₂×C₃	1	1	yes	characteristic
9	C₇×C₇⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
10	C₇×C₇⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
11	C₃×#1131	—	7	1	no	maximal
12	#1131⋊C₃	—	7	1	no	maximal
13	C₂×C₇×ΓL(1,𝔽₈)	C₂×C₄	1	1	yes	characteristic
14	ΓL(1,𝔽₈)⋊D₇	—	2	1	no	—
15	C₂×C₇⋊D₇	C₃×C₄	1	1	yes	characteristic
16	C₄×C₇²	C₂²×C₃	1	1	yes	characteristic
17	C₇⋊Dic₇	C₃×C₄	1	1	yes	characteristic
18	C₃×C₄×D₇	—	7	1	no	—
19	C₄×AGL(1,𝔽₇)	—	7	1	no	—
20	C₃×C₇×C₈	—	7	1	no	—
21	C₃×C₇⋊C₈	—	14	1	no	—
22	C₈×ΓL(1,𝔽₈)	—	7	1	no	—
23	ΓL(1,𝔽₈)⋊C₈	D₇	1	1	yes	characteristic, complemented
24	C₇×ΓL(1,𝔽₈)	M₁₆	1	1	yes	characteristic, complemented, Hall
25	#1131	—	7	1	no	—
26	#1131	—	7	1	no	—
27	C₂×C₇²	C₂×C₃×C₄	1	1	yes	characteristic
28	C₇⋊D₇	—	2	1	no	—
29	C₂×C₃×D₇	—	7	1	no	—
30	C₂×AGL(1,𝔽₇)	—	7	1	no	—
31	C₃×C₄×C₇	—	14	1	no	—
32	C₃×Dic₇	—	7	1	no	—
33	C₄×ΓL(1,𝔽₈)	C₂×D₇	1	1	yes	characteristic
34	ΓL(1,𝔽₈)⋊C₄	—	7	1	no	—
35	C₄×D₇	—	7	1	no	—
36	C₄×D₇	—	7	1	no	—
37	C₄×D₇	—	42	1	no	—
38	C₇×C₈	—	7	1	no	—
39	C₇×C₈	—	7	1	no	—
40	C₇⋊C₈	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
41	C₇⋊C₈	C₃×D₇	1	1	yes	characteristic, complemented
42	C₇²	C₃×M₁₆	1	1	yes	characteristic, complemented, Sylow
43	C₃×M₁₆	—	49	1	no	Hall
44	C₂×C₃×C₇	—	14	1	no	—
45	C₃×D₇	—	14	1	no	—
46	C₂×ΓL(1,𝔽₈)	C₄×D₇	1	1	yes	characteristic
47	AGL(1,𝔽₇)	—	14	1	no	—
48	C₂×D₇	—	7	1	no	—
49	C₂×D₇	—	7	1	no	—
50	C₂×D₇	—	42	1	no	—
51	C₄×C₇	—	6	1	no	—
52	C₄×C₇	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
53	C₄×C₇	C₂×C₃×D₇	1	1	yes	characteristic
54	Dic₇	—	7	1	no	—
55	Dic₇	—	7	1	no	—
56	Dic₇	—	42	1	no	—
57	C₂×C₃×C₄	—	49	1	no	—
58	C₃×C₈	—	7	1	no	—
59	C₃×C₈	—	98	1	no	—
60	C₃×C₇	—	14	1	no	—
61	ΓL(1,𝔽₈)	#1131	1	1	yes	characteristic, complemented
62	M₁₆	—	49	1	no	Sylow
63	C₂×C₇	—	6	1	no	—
64	C₂×C₇	C₄×AGL(1,𝔽₇)	1	1	yes	characteristic
65	C₂×C₇	C₃×C₄×D₇	1	1	yes	characteristic
66	D₇	—	14	1	no	—
67	D₇	—	14	1	no	—
68	D₇	—	42	1	no	—
69	D₇	—	42	1	no	—
70	C₂²×C₃	—	49	1	no	—
71	C₃×C₄	—	14	1	no	—
72	C₃×C₄	—	49	1	no	—
73	C₂×C₄	—	49	1	no	—
74	C₈	—	7	1	no	—
75	C₈	—	7	1	no	—
76	C₇	—	6	1	no	—
77	C₇	#1131⋊C₃	1	1	yes	characteristic, complemented
78	C₇	C₃×#1131	1	1	yes	characteristic, complemented
79	C₂×C₃	—	14	1	no	—
80	C₂×C₃	—	98	1	no	—
81	C₂²	—	49	1	no	—
82	C₄	—	49	1	no	—
83	C₄	D₇×AGL(1,𝔽₇)	1	1	yes	characteristic
84	C₃	—	14	1	no	Sylow
85	C₂	—	98	1	no	—
86	C₂	(ΓL(1,𝔽₈)⋊D₇)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₄	D₇×AGL(1,𝔽₇)
commutator, socle	C₂×C₇²	C₂×C₃×C₄
Fitting, cosocle	C₄×C₇²	C₂²×C₃
prime radical	C₂×ΓL(1,𝔽₈)⋊D₇	C₄
hypocenter	C₇²	C₃×M₁₆
LCS	G ▷ C₂×C₇² ▷ C₇²
UCS	C₁ ◁ C₄

Complements

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

N	H ≅ G/N	type
2. ΓL(1,𝔽₈)×C₇⋊C₈	C₂	semidirect
3. C₇×ΓL(1,𝔽₈)⋊C₈	C₂	semidirect
4. #11716031	C₃	semidirect
9. C₇×C₇⋊C₈	C₂×C₃	semidirect
10. C₇×C₇⋊C₈	C₂×C₃	semidirect
23. ΓL(1,𝔽₈)⋊C₈	D₇	semidirect
24. C₇×ΓL(1,𝔽₈)	M₁₆	semidirect
40. C₇⋊C₈	AGL(1,𝔽₇)	semidirect
41. C₇⋊C₈	C₃×D₇	semidirect
42. C₇²	C₃×M₁₆	semidirect
61. ΓL(1,𝔽₈)	#1131	semidirect
77. C₇	#1131⋊C₃	semidirect
78. C₇	C₃×#1131	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
ΓL(1,𝔽₈)×C₇⋊C₈	2
C₇×ΓL(1,𝔽₈)⋊C₈	2
#11716031	3
C₃×#1131	7
#1131⋊C₃	7