Subgroups — #13055070

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×#396	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	#396⋊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	#396	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	1	no	—
25	C₂×C₄×C₇	—	6	1	no	—
26	C₂×C₄×C₇	—	6	1	no	—
27	C₂×C₄×C₇	—	6	1	no	—
28	C₂×C₄×C₇	—	6	1	no	—
29	C₂²×A₄	—	28	1	no	—
30	C₂×ΓL(1,𝔽₈)	—	16	2	no	—
31	C₂×ΓL(1,𝔽₈)	—	16	1	no	—
32	C₂×C₄⋊C₄	—	3	1	no	—
33	C₂×C₂²⋊C₄	—	3	1	no	—
34	#95	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
35	C₂²×C₇	—	3	1	no	—
36	C₂²×C₇	—	3	1	no	—
37	C₂²×C₇	—	4	1	no	—
38	C₂²×C₇	—	6	2	no	—
39	C₂²×C₇	—	12	1	no	—
40	C₂²×C₇	C₂×SL(2,𝔽₃)	1	1	yes	characteristic
41	C₄×C₇	—	6	2	no	—
42	C₄×C₇	—	12	1	no	—
43	C₂×A₄	—	28	2	no	—
44	C₂×A₄	—	28	1	no	—
45	ΓL(1,𝔽₈)	—	16	1	no	Hall
46	C₂⁴	C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic
47	C₂²×C₄	—	3	1	no	—
48	C₂²×C₄	—	3	1	no	—
49	C₄⋊C₄	—	6	2	no	—
50	C₂²⋊C₄	—	6	2	no	—
51	C₂×C₇	—	3	1	no	—
52	C₂×C₇	—	3	1	no	—
53	C₂×C₇	—	4	2	no	—
54	C₂×C₇	#126⋊C₃	1	1	yes	characteristic
55	C₂²×C₃	—	112	1	no	—
56	A₄	—	28	1	no	—
57	C₂³	—	6	1	no	—
58	C₂³	—	6	1	no	—
59	C₂³	C₇⋊SL(2,𝔽₃)	1	2	yes	—
60	C₂³	C₂×C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic
61	C₂×C₄	—	3	2	no	—
62	C₂×C₄	—	6	1	no	—
63	C₂×C₄	—	6	1	no	—
64	C₂×C₄	—	6	1	no	—
65	C₂×C₄	—	6	1	no	—
66	C₂×C₄	—	6	1	no	—
67	C₇	#396⋊C₃	1	1	yes	characteristic, complemented, Sylow
68	C₂×C₃	—	112	2	no	—
69	C₂×C₃	—	112	1	no	—
70	C₂²	—	3	1	no	—
71	C₂²	—	3	1	no	—
72	C₂²	—	4	1	no	—
73	C₂²	—	6	2	no	—
74	C₂²	—	12	1	no	—
75	C₂²	C₂×C₇⋊SL(2,𝔽₃)	1	1	yes	characteristic
76	C₄	—	6	2	no	—
77	C₄	—	12	1	no	—
78	C₃	—	112	1	no	Sylow
79	C₂	—	3	1	no	—
80	C₂	—	3	1	no	—
81	C₂	—	4	2	no	—
82	C₂	#10621174	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂	#10621174
commutator, cosocle, hypocenter	C₇×#95	C₂×C₃
Frattini	C₂³	C₂×C₂²⋊ΓL(1,𝔽₈)
Fitting	C₇×#396	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×#396	C₃	semidirect
6. C₇×#95	C₂×C₃	semidirect
18. #396	ΓL(1,𝔽₈)	semidirect
34. #95	C₂×ΓL(1,𝔽₈)	semidirect
67. C₇	#396⋊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×#396	3
C₂²×C₂²⋊ΓL(1,𝔽₈)	4
#396⋊C₃	7