Subgroups — #395⋊ΓL(1,𝔽₈) #13055068

Order 1344.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²⋊(C₇⋊SL(2,𝔽₃))	C₂	1	1	yes	maximal, characteristic
2	C₇×#395	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₄×C₂²⋊ΓL(1,𝔽₈)	—	4	1	no	maximal
4	C₂²×C₇×Q₈	C₂×C₃	1	1	yes	characteristic
5	C₂×C₇×C₄⋊C₄	—	3	1	no	—
6	C₇×#95	—	3	1	no	—
7	#395⋊C₃	—	7	1	no	maximal, Hall
8	C₂×C₂²⋊ΓL(1,𝔽₈)	—	4	1	no	—
9	C₇⋊SL(2,𝔽₃)	—	8	2	no	—
10	C₂×C₇×Q₈	—	6	2	no	—
11	C₂²×C₄×C₇	—	3	1	no	—
12	C₂²×C₄×C₇	—	3	1	no	—
13	C₂²×C₄×C₇	A₄	1	1	yes	characteristic
14	C₇×C₄⋊C₄	—	6	2	no	—
15	C₂²⋊SL(2,𝔽₃)	—	7	1	no	—
16	C₂²⋊ΓL(1,𝔽₈)	—	4	1	no	—
17	C₄×ΓL(1,𝔽₈)	—	16	1	no	—
18	#395	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
19	C₂³×C₇	C₂×A₄	1	1	yes	characteristic
20	C₇×Q₈	—	2	2	no	—
21	C₇×Q₈	—	6	2	no	—
22	C₂×C₄×C₇	—	3	2	no	—
23	C₂×C₄×C₇	—	3	2	no	—
24	C₂×C₄×C₇	—	6	1	no	—
25	C₂×C₄×C₇	—	6	1	no	—
26	C₂×C₄×C₇	—	6	2	no	—
27	C₂×C₄×C₇	—	6	1	no	—
28	C₄×A₄	—	28	1	no	—
29	C₂×ΓL(1,𝔽₈)	—	16	1	no	—
30	C₂²×Q₈	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic
31	C₂×C₄⋊C₄	—	3	1	no	—
32	#95	—	3	1	no	—
33	C₂²×C₇	—	3	1	no	—
34	C₂²×C₇	—	3	1	no	—
35	C₂²×C₇	#41⋊C₃	1	1	yes	characteristic
36	C₄×C₇	—	4	1	no	—
37	C₄×C₇	—	6	2	no	—
38	C₄×C₇	—	6	2	no	—
39	C₂×A₄	—	28	1	no	—
40	SL(2,𝔽₃)	—	56	2	no	—
41	ΓL(1,𝔽₈)	—	16	1	no	Hall
42	C₂×Q₈	—	6	2	no	—
43	C₂²×C₄	—	3	1	no	—
44	C₂²×C₄	—	3	1	no	—
45	C₂²×C₄	C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic
46	C₄⋊C₄	—	6	2	no	—
47	C₂×C₇	—	3	1	no	—
48	C₂×C₇	—	3	1	no	—
49	C₂×C₇	#120⋊C₃	1	1	yes	characteristic
50	A₄	—	28	1	no	—
51	C₃×C₄	—	112	1	no	—
52	C₂³	C₂×C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic
53	Q₈	—	2	2	no	—
54	Q₈	—	6	2	no	—
55	C₂×C₄	—	3	2	no	—
56	C₂×C₄	—	3	2	no	—
57	C₂×C₄	—	6	1	no	—
58	C₂×C₄	—	6	1	no	—
59	C₂×C₄	—	6	2	no	—
60	C₂×C₄	—	6	1	no	—
61	C₇	#395⋊C₃	1	1	yes	characteristic, complemented, Sylow
62	C₂×C₃	—	112	1	no	—
63	C₂²	—	3	1	no	—
64	C₂²	—	3	1	no	—
65	C₂²	#41⋊ΓL(1,𝔽₈)	1	1	yes	characteristic
66	C₄	—	4	1	no	—
67	C₄	—	6	2	no	—
68	C₄	—	6	2	no	—
69	C₃	—	112	1	no	Sylow
70	C₂	—	3	1	no	—
71	C₂	—	3	1	no	—
72	C₂	#120⋊ΓL(1,𝔽₈)	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂	#120⋊ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₂²×C₇×Q₈	C₂×C₃
Frattini	C₂³	C₂×C₂²⋊ΓL(1,𝔽₈)
Fitting	C₇×#395	C₃
socle	C₂³×C₇	C₂×A₄
prime radical	C₂²⋊(C₇⋊SL(2,𝔽₃))	C₂
LCS	G ▷ C₂²×C₇×Q₈
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
2. C₇×#395	C₃	semidirect
18. #395	ΓL(1,𝔽₈)	semidirect
61. C₇	#395⋊C₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²⋊(C₇⋊SL(2,𝔽₃))	2
C₇×#395	3
C₄×C₂²⋊ΓL(1,𝔽₈)	4
#395⋊C₃	7

Subgroups — #395⋊ΓL(1,𝔽8) #13055068

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #395⋊ΓL(1,𝔽₈) #13055068