Subgroups — ΓL(1,𝔽₈)×#1136 #72244994

Order 2352.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₇×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic
2	ΓL(1,𝔽₈)×C₇⋊C₈	C₂	1	2	yes	maximal, complemented
3	C₇×#1136	C₃	1	1	yes	maximal, characteristic, complemented, Hall
4	C₂²×C₇×ΓL(1,𝔽₈)	C₄	1	1	yes	characteristic
5	C₄×C₇×ΓL(1,𝔽₈)	C₄	1	1	yes	characteristic
6	C₄×C₇×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₇²	C₂×C₃	1	1	yes	characteristic
8	C₇×C₇⋊C₈	C₂×C₃	1	2	yes	complemented
9	C₃×#1136	—	7	1	no	maximal
10	M₁₆×ΓL(1,𝔽₈)	—	7	1	no	maximal
11	C₂×C₇×ΓL(1,𝔽₈)	—	2	1	no	—
12	C₂×C₇×ΓL(1,𝔽₈)	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₇²	C₃×C₄	1	1	yes	characteristic
14	C₄×C₇²	C₃×C₄	1	1	yes	characteristic
15	C₄×C₇²	C₂²×C₃	1	1	yes	characteristic
16	C₂×C₃×C₄×C₇	—	7	1	no	—
17	C₂×C₄×ΓL(1,𝔽₈)	D₇	1	1	yes	characteristic
18	C₃×C₇⋊C₈	—	7	2	no	—
19	C₈×ΓL(1,𝔽₈)	—	7	2	no	—
20	C₇×ΓL(1,𝔽₈)	M₁₆	1	1	yes	characteristic, complemented, Hall
21	C₇×M₁₆	—	7	1	no	—
22	#1136	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
23	C₂×C₇²	—	2	1	no	—
24	C₂×C₇²	C₂×C₃×C₄	1	1	yes	characteristic
25	C₂²×C₃×C₇	—	7	1	no	—
26	C₂²×ΓL(1,𝔽₈)	Dic₇	1	1	yes	characteristic
27	C₃×C₄×C₇	—	7	1	no	—
28	C₃×C₄×C₇	—	7	1	no	—
29	C₄×ΓL(1,𝔽₈)	Dic₇	1	1	yes	characteristic
30	C₄×ΓL(1,𝔽₈)	C₂×D₇	1	1	yes	characteristic
31	C₂×C₄×C₇	—	6	1	no	—
32	C₂×C₄×C₇	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic
33	C₂×C₄×C₇	C₃×D₇	1	1	yes	characteristic
34	C₇×C₈	—	7	2	no	—
35	C₇⋊C₈	C₂×ΓL(1,𝔽₈)	1	2	yes	complemented
36	C₇²	C₃×M₁₆	1	1	yes	characteristic, complemented, Sylow
37	C₃×M₁₆	—	49	1	no	Hall
38	C₂×C₃×C₇	—	7	1	no	—
39	C₂×C₃×C₇	—	14	1	no	—
40	C₂×ΓL(1,𝔽₈)	—	2	1	no	—
41	C₂×ΓL(1,𝔽₈)	C₂×Dic₇	1	1	yes	characteristic
42	C₂²×C₇	—	6	1	no	—
43	C₂²×C₇	C₄×ΓL(1,𝔽₈)	1	1	yes	characteristic
44	C₂²×C₇	C₃×Dic₇	1	1	yes	characteristic
45	C₄×C₇	—	6	1	no	—
46	C₄×C₇	—	6	1	no	—
47	C₄×C₇	C₄×ΓL(1,𝔽₈)	1	1	yes	characteristic
48	C₄×C₇	C₃×Dic₇	1	1	yes	characteristic
49	C₄×C₇	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
50	C₄×C₇	C₂×C₃×D₇	1	1	yes	characteristic
51	C₂×C₃×C₄	—	7	1	no	—
52	C₃×C₈	—	49	2	no	—
53	C₃×C₇	—	7	1	no	—
54	ΓL(1,𝔽₈)	#1136	1	1	yes	characteristic, direct product factor
55	M₁₆	—	7	1	no	Sylow
56	C₂×C₇	—	2	1	no	—
57	C₂×C₇	—	2	1	no	—
58	C₂×C₇	—	6	1	no	—
59	C₂×C₇	—	6	1	no	—
60	C₂×C₇	—	6	1	no	—
61	C₂×C₇	C₂×C₄×ΓL(1,𝔽₈)	1	1	yes	characteristic
62	C₂×C₇	C₂×C₃×Dic₇	1	1	yes	characteristic
63	C₂²×C₃	—	7	1	no	—
64	C₃×C₄	—	7	1	no	—
65	C₃×C₄	—	7	1	no	—
66	C₂×C₄	D₇×ΓL(1,𝔽₈)	1	1	yes	characteristic
67	C₈	—	7	2	no	—
68	C₇	—	6	1	no	—
69	C₇	M₁₆×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
70	C₇	C₃×#1136	1	1	yes	characteristic, complemented
71	C₂×C₃	—	7	1	no	—
72	C₂×C₃	—	14	1	no	—
73	C₂²	ΓL(1,𝔽₈)×Dic₇	1	1	yes	characteristic
74	C₄	ΓL(1,𝔽₈)×Dic₇	1	1	yes	characteristic
75	C₄	C₂×D₇×ΓL(1,𝔽₈)	1	1	yes	characteristic
76	C₃	—	7	1	no	Sylow
77	C₂	—	2	1	no	—
78	C₂	C₂×ΓL(1,𝔽₈)×Dic₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄	C₂×D₇×ΓL(1,𝔽₈)
commutator, socle	C₂×C₇²	C₂×C₃×C₄
Fitting	C₂×C₄×C₇²	C₂×C₃
prime radical	C₂²×C₇×ΓL(1,𝔽₈)	C₄
cosocle	C₄×C₇²	C₂²×C₃
hypercenter	C₂×C₄	D₇×ΓL(1,𝔽₈)
hypocenter	C₇²	C₃×M₁₆
LCS	G ▷ C₂×C₇² ▷ C₇²
UCS	C₁ ◁ C₄ ◁ C₂×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×#1136	C₃	semidirect
8. C₇×C₇⋊C₈	C₂×C₃	semidirect
20. C₇×ΓL(1,𝔽₈)	M₁₆	semidirect
22. #1136	ΓL(1,𝔽₈)	direct (N × H)
35. C₇⋊C₈	C₂×ΓL(1,𝔽₈)	semidirect
36. C₇²	C₃×M₁₆	semidirect
54. ΓL(1,𝔽₈)	#1136	direct (N × H)
69. C₇	M₁₆×ΓL(1,𝔽₈)	semidirect
70. C₇	C₃×#1136	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₇×ΓL(1,𝔽₈)	2
ΓL(1,𝔽₈)×C₇⋊C₈	2
C₇×#1136	3
C₃×#1136	7
M₁₆×ΓL(1,𝔽₈)	7

Subgroups — ΓL(1,𝔽8)×#1136 #72244994

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — ΓL(1,𝔽₈)×#1136 #72244994