Subgroups — #376⋊ΓL(1,𝔽₈) #13055061

Order 1344.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇⋊(#95⋊C₃)	C₂	1	1	yes	maximal, characteristic
2	C₇×#376	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₄×C₂²⋊ΓL(1,𝔽₈)	—	4	1	no	maximal
4	C₂×C₄²×C₇	—	3	1	no	—
5	C₇×#95	—	3	1	no	—
6	C₇×#95	C₂×C₃	1	1	yes	characteristic
7	#376⋊C₃	—	7	1	no	maximal, Hall
8	C₂×C₂²⋊ΓL(1,𝔽₈)	—	4	1	no	—
9	C₂²×C₄×C₇	—	3	1	no	—
10	C₂²×C₄×C₇	—	3	1	no	—
11	C₂²×C₄×C₇	A₄	1	1	yes	characteristic
12	C₄²×C₇	—	3	2	no	—
13	C₄²×C₇	—	3	2	no	—
14	#95⋊C₃	—	7	1	no	—
15	C₂²⋊ΓL(1,𝔽₈)	—	4	1	no	—
16	C₄×ΓL(1,𝔽₈)	—	16	1	no	—
17	#376	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
18	C₂³×C₇	C₂×A₄	1	1	yes	characteristic
19	C₂×C₄×C₇	—	3	1	no	—
20	C₂×C₄×C₇	—	3	1	no	—
21	C₂×C₄×C₇	—	3	2	no	—
22	C₂×C₄×C₇	—	3	2	no	—
23	C₂×C₄×C₇	—	6	2	no	—
24	C₂×C₄×C₇	—	6	1	no	—
25	C₂×C₄×C₇	—	6	1	no	—
26	C₄×A₄	—	28	1	no	—
27	C₂×ΓL(1,𝔽₈)	—	16	1	no	—
28	C₂×C₄²	—	3	1	no	—
29	#95	—	3	1	no	—
30	#95	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic
31	C₂²×C₇	—	3	1	no	—
32	C₂²×C₇	—	3	1	no	—
33	C₂²×C₇	#41⋊C₃	1	1	yes	characteristic
34	C₄×C₇	—	3	1	no	—
35	C₄×C₇	—	6	2	no	—
36	C₄×C₇	—	6	2	no	—
37	C₄×C₇	C₄²⋊C₃	1	1	yes	characteristic
38	C₂×A₄	—	28	1	no	—
39	ΓL(1,𝔽₈)	—	16	1	no	Hall
40	C₂²×C₄	—	3	1	no	—
41	C₂²×C₄	—	3	1	no	—
42	C₂²×C₄	C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic
43	C₄²	—	3	2	no	—
44	C₄²	—	3	2	no	—
45	C₂×C₇	—	3	1	no	—
46	C₂×C₇	—	3	1	no	—
47	C₂×C₇	C₂×C₄²⋊C₃	1	1	yes	characteristic
48	A₄	—	28	1	no	—
49	C₃×C₄	—	112	1	no	—
50	C₂³	C₂×C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic
51	C₂×C₄	—	3	1	no	—
52	C₂×C₄	—	3	1	no	—
53	C₂×C₄	—	3	2	no	—
54	C₂×C₄	—	3	2	no	—
55	C₂×C₄	—	6	2	no	—
56	C₂×C₄	—	6	1	no	—
57	C₂×C₄	—	6	1	no	—
58	C₇	#376⋊C₃	1	1	yes	characteristic, complemented, Sylow
59	C₂×C₃	—	112	1	no	—
60	C₂²	—	3	1	no	—
61	C₂²	—	3	1	no	—
62	C₂²	#41⋊ΓL(1,𝔽₈)	1	1	yes	characteristic
63	C₄	—	3	1	no	—
64	C₄	—	6	2	no	—
65	C₄	—	6	2	no	—
66	C₄	C₇⋊(C₄²⋊C₃)	1	1	yes	characteristic
67	C₃	—	112	1	no	Sylow
68	C₂	—	3	1	no	—
69	C₂	—	3	1	no	—
70	C₂	C₂×C₇⋊(C₄²⋊C₃)	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₄	C₇⋊(C₄²⋊C₃)
commutator, cosocle, hypocenter	C₇×#95	C₂×C₃
Frattini	C₂³	C₂×C₂²⋊ΓL(1,𝔽₈)
Fitting	C₇×#376	C₃
socle	C₂³×C₇	C₂×A₄
prime radical	C₇⋊(#95⋊C₃)	C₂
LCS	G ▷ C₇×#95
UCS	C₁ ◁ C₄

Complements

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

N	H ≅ G/N	type
2. C₇×#376	C₃	semidirect
17. #376	ΓL(1,𝔽₈)	semidirect
58. C₇	#376⋊C₃	semidirect

Maximal subgroups

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

subgroup	index
C₇⋊(#95⋊C₃)	2
C₇×#376	3
C₄×C₂²⋊ΓL(1,𝔽₈)	4
#376⋊C₃	7

Subgroups — #376⋊ΓL(1,𝔽8) #13055061

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #376⋊ΓL(1,𝔽₈) #13055061