Subgroups — #399⋊ΓL(1,𝔽₈) #13055071

Order 1344.

102 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, complemented
2	C₇×#399	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₂²×C₂²⋊ΓL(1,𝔽₈)	—	4	1	no	maximal
4	C₂×C₇×C₄⋊C₄	—	3	1	no	—
5	C₂×C₇×C₂²⋊C₄	—	3	1	no	—
6	C₇×#95	C₂×C₃	1	1	yes	characteristic, complemented
7	#399⋊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	C₂⁴×C₇	A₄	1	1	yes	characteristic
11	C₂²×C₄×C₇	—	3	1	no	—
12	C₂²×C₄×C₇	—	3	1	no	—
13	C₇×C₄⋊C₄	—	6	2	no	—
14	C₇×C₂²⋊C₄	—	6	2	no	—
15	#95⋊C₃	—	7	1	no	—
16	C₂²⋊ΓL(1,𝔽₈)	—	4	1	no	—
17	C₂²×ΓL(1,𝔽₈)	—	16	1	no	—
18	#399	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
19	C₂³×C₇	—	6	1	no	—
20	C₂³×C₇	—	6	1	no	—
21	C₂³×C₇	SL(2,𝔽₃)	1	2	yes	—
22	C₂³×C₇	C₂×A₄	1	1	yes	characteristic
23	C₂×C₄×C₇	—	3	2	no	—
24	C₂×C₄×C₇	—	6	2	no	—
25	C₂×C₄×C₇	—	6	1	no	—
26	C₂×C₄×C₇	—	6	2	no	—
27	C₂²×A₄	—	28	1	no	—
28	C₂×ΓL(1,𝔽₈)	—	16	2	no	—
29	C₂×ΓL(1,𝔽₈)	—	16	1	no	—
30	C₂×C₄⋊C₄	—	3	1	no	—
31	C₂×C₂²⋊C₄	—	3	1	no	—
32	#95	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
33	C₂²×C₇	—	3	1	no	—
34	C₂²×C₇	—	3	1	no	—
35	C₂²×C₇	—	4	1	no	—
36	C₂²×C₇	—	6	2	no	—
37	C₂²×C₇	—	12	1	no	—
38	C₂²×C₇	C₂×SL(2,𝔽₃)	1	1	yes	characteristic
39	C₄×C₇	—	6	2	no	—
40	C₄×C₇	—	12	1	no	—
41	C₂×A₄	—	28	2	no	—
42	C₂×A₄	—	28	1	no	—
43	ΓL(1,𝔽₈)	—	16	1	no	Hall
44	C₂⁴	C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic
45	C₂²×C₄	—	3	1	no	—
46	C₂²×C₄	—	3	1	no	—
47	C₄⋊C₄	—	6	2	no	—
48	C₂²⋊C₄	—	6	2	no	—
49	C₂×C₇	—	3	1	no	—
50	C₂×C₇	—	3	1	no	—
51	C₂×C₇	—	4	2	no	—
52	C₂×C₇	#127⋊C₃	1	1	yes	characteristic
53	C₂²×C₃	—	112	1	no	—
54	A₄	—	28	1	no	—
55	C₂³	—	6	1	no	—
56	C₂³	—	6	1	no	—
57	C₂³	C₇⋊SL(2,𝔽₃)	1	2	yes	—
58	C₂³	C₂×C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic
59	C₂×C₄	—	3	2	no	—
60	C₂×C₄	—	6	2	no	—
61	C₂×C₄	—	6	1	no	—
62	C₂×C₄	—	6	2	no	—
63	C₇	#399⋊C₃	1	1	yes	characteristic, complemented, Sylow
64	C₂×C₃	—	112	2	no	—
65	C₂×C₃	—	112	1	no	—
66	C₂²	—	3	1	no	—
67	C₂²	—	3	1	no	—
68	C₂²	—	4	1	no	—
69	C₂²	—	6	2	no	—
70	C₂²	—	12	1	no	—
71	C₂²	C₂×C₇⋊SL(2,𝔽₃)	1	1	yes	characteristic
72	C₄	—	6	2	no	—
73	C₄	—	12	1	no	—
74	C₃	—	112	1	no	Sylow
75	C₂	—	3	1	no	—
76	C₂	—	3	1	no	—
77	C₂	—	4	2	no	—
78	C₂	#127⋊ΓL(1,𝔽₈)	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂	#127⋊ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₇×#95	C₂×C₃
Frattini	C₂³	C₂×C₂²⋊ΓL(1,𝔽₈)
Fitting	C₇×#399	C₃
socle	C₂³×C₇	C₂×A₄
LCS	G ▷ C₇×#95
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
1. C₇⋊(#95⋊C₃)	C₂	semidirect
2. C₇×#399	C₃	semidirect
6. C₇×#95	C₂×C₃	semidirect
18. #399	ΓL(1,𝔽₈)	semidirect
32. #95	C₂×ΓL(1,𝔽₈)	semidirect
63. C₇	#399⋊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×#399	3
C₂²×C₂²⋊ΓL(1,𝔽₈)	4
#399⋊C₃	7

Subgroups — #399⋊ΓL(1,𝔽8) #13055071

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #399⋊ΓL(1,𝔽₈) #13055071