Subgroups — #393⋊ΓL(1,𝔽₈) #13055067

Order 1344.

114 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, complemented
2	C₇×#393	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₂²×C₂²⋊ΓL(1,𝔽₈)	—	4	1	no	maximal
4	C₂²×C₇×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
5	C₂×C₇×C₂²⋊C₄	—	3	1	no	—
6	C₇×#95	—	3	1	no	—
7	#393⋊C₃	—	7	1	no	maximal, Hall
8	C₂×C₂²⋊ΓL(1,𝔽₈)	—	4	2	no	—
9	C₂×C₂²⋊ΓL(1,𝔽₈)	—	4	1	no	—
10	C₇⋊SL(2,𝔽₃)	—	8	2	no	—
11	C₂⁴×C₇	A₄	1	1	yes	characteristic
12	C₂×C₇×Q₈	—	6	2	no	—
13	C₂²×C₄×C₇	—	3	1	no	—
14	C₂²×C₄×C₇	—	3	1	no	—
15	C₇×C₂²⋊C₄	—	6	2	no	—
16	C₂²⋊SL(2,𝔽₃)	—	7	1	no	—
17	C₂²⋊ΓL(1,𝔽₈)	—	4	1	no	—
18	C₂²×ΓL(1,𝔽₈)	—	16	1	no	—
19	#393	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
20	C₂³×C₇	—	6	1	no	—
21	C₂³×C₇	—	6	1	no	—
22	C₂³×C₇	SL(2,𝔽₃)	1	2	yes	complemented
23	C₂³×C₇	C₂×A₄	1	1	yes	characteristic
24	C₇×Q₈	—	2	2	no	—
25	C₇×Q₈	—	6	2	no	—
26	C₂×C₄×C₇	—	3	2	no	—
27	C₂×C₄×C₇	—	6	2	no	—
28	C₂×C₄×C₇	—	6	1	no	—
29	C₂×C₄×C₇	—	6	2	no	—
30	C₂²×A₄	—	28	1	no	—
31	C₂×ΓL(1,𝔽₈)	—	16	2	no	—
32	C₂×ΓL(1,𝔽₈)	—	16	1	no	—
33	C₂²×Q₈	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
34	C₂×C₂²⋊C₄	—	3	1	no	—
35	#95	—	3	1	no	—
36	C₂²×C₇	—	3	1	no	—
37	C₂²×C₇	—	3	1	no	—
38	C₂²×C₇	—	4	1	no	—
39	C₂²×C₇	—	6	2	no	—
40	C₂²×C₇	—	12	1	no	—
41	C₂²×C₇	C₂×SL(2,𝔽₃)	1	1	yes	characteristic
42	C₄×C₇	—	6	2	no	—
43	C₄×C₇	—	12	1	no	—
44	C₂×A₄	—	28	2	no	—
45	C₂×A₄	—	28	1	no	—
46	SL(2,𝔽₃)	—	56	2	no	—
47	ΓL(1,𝔽₈)	—	16	1	no	Hall
48	C₂⁴	C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic
49	C₂×Q₈	—	6	2	no	—
50	C₂²×C₄	—	3	1	no	—
51	C₂²×C₄	—	3	1	no	—
52	C₂²⋊C₄	—	6	2	no	—
53	C₂×C₇	—	3	1	no	—
54	C₂×C₇	—	3	1	no	—
55	C₂×C₇	—	4	2	no	—
56	C₂×C₇	#120⋊C₃	1	1	yes	characteristic
57	C₂²×C₃	—	112	1	no	—
58	A₄	—	28	1	no	—
59	C₂³	—	6	1	no	—
60	C₂³	—	6	1	no	—
61	C₂³	C₇⋊SL(2,𝔽₃)	1	2	yes	complemented
62	C₂³	C₂×C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic
63	Q₈	—	2	2	no	—
64	Q₈	—	6	2	no	—
65	C₂×C₄	—	3	2	no	—
66	C₂×C₄	—	6	2	no	—
67	C₂×C₄	—	6	1	no	—
68	C₂×C₄	—	6	2	no	—
69	C₇	#393⋊C₃	1	1	yes	characteristic, complemented, Sylow
70	C₂×C₃	—	112	2	no	—
71	C₂×C₃	—	112	1	no	—
72	C₂²	—	3	1	no	—
73	C₂²	—	3	1	no	—
74	C₂²	—	4	1	no	—
75	C₂²	—	6	2	no	—
76	C₂²	—	12	1	no	—
77	C₂²	C₂×C₇⋊SL(2,𝔽₃)	1	1	yes	characteristic
78	C₄	—	6	2	no	—
79	C₄	—	12	1	no	—
80	C₃	—	112	1	no	Sylow
81	C₂	—	3	1	no	—
82	C₂	—	3	1	no	—
83	C₂	—	4	2	no	—
84	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×#393	C₃
socle	C₂³×C₇	C₂×A₄
LCS	G ▷ C₂²×C₇×Q₈
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
1. C₂²⋊(C₇⋊SL(2,𝔽₃))	C₂	semidirect
2. C₇×#393	C₃	semidirect
4. C₂²×C₇×Q₈	C₂×C₃	semidirect
19. #393	ΓL(1,𝔽₈)	semidirect
22. C₂³×C₇	SL(2,𝔽₃)	semidirect
33. C₂²×Q₈	C₂×ΓL(1,𝔽₈)	semidirect
61. C₂³	C₇⋊SL(2,𝔽₃)	semidirect
69. C₇	#393⋊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×#393	3
C₂²×C₂²⋊ΓL(1,𝔽₈)	4
#393⋊C₃	7

Subgroups — #393⋊ΓL(1,𝔽8) #13055067

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #393⋊ΓL(1,𝔽₈) #13055067