Subgroups — #13055073

Order 1344.

66 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×#401	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₄×C₂²⋊ΓL(1,𝔽₈)	—	4	1	no	maximal
4	C₇×#95	—	3	1	no	—
5	C₇×#95	—	3	1	no	—
6	C₇×#95	C₂×C₃	1	1	yes	characteristic
7	#401⋊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	#95⋊C₃	—	7	1	no	—
13	C₂²⋊ΓL(1,𝔽₈)	—	4	1	no	—
14	C₄×ΓL(1,𝔽₈)	—	16	1	no	—
15	#401	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
16	C₂³×C₇	C₂×A₄	1	1	yes	characteristic
17	C₂×C₄×C₇	—	6	1	no	—
18	C₂×C₄×C₇	—	6	1	no	—
19	C₂×C₄×C₇	—	6	2	no	—
20	C₂×C₄×C₇	—	6	1	no	—
21	C₂×C₄×C₇	—	6	1	no	—
22	C₂×C₄×C₇	—	6	1	no	—
23	C₄×A₄	—	28	1	no	—
24	C₂×ΓL(1,𝔽₈)	—	16	1	no	—
25	#95	—	3	1	no	—
26	#95	—	3	1	no	—
27	#95	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic
28	C₂²×C₇	—	3	1	no	—
29	C₂²×C₇	—	3	1	no	—
30	C₂²×C₇	#41⋊C₃	1	1	yes	characteristic
31	C₄×C₇	—	4	1	no	—
32	C₄×C₇	—	12	1	no	—
33	C₄×C₇	—	12	1	no	—
34	C₂×A₄	—	28	1	no	—
35	ΓL(1,𝔽₈)	—	16	1	no	Hall
36	C₂²×C₄	—	3	1	no	—
37	C₂²×C₄	—	3	1	no	—
38	C₂²×C₄	C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic
39	C₂×C₇	—	3	1	no	—
40	C₂×C₇	—	3	1	no	—
41	C₂×C₇	#126⋊C₃	1	1	yes	characteristic
42	A₄	—	28	1	no	—
43	C₃×C₄	—	112	1	no	—
44	C₂³	C₂×C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic
45	C₂×C₄	—	6	1	no	—
46	C₂×C₄	—	6	2	no	—
47	C₂×C₄	—	6	1	no	—
48	C₂×C₄	—	6	1	no	—
49	C₂×C₄	—	6	1	no	—
50	C₂×C₄	—	6	1	no	—
51	C₇	#401⋊C₃	1	1	yes	characteristic, complemented, Sylow
52	C₂×C₃	—	112	1	no	—
53	C₂²	—	3	1	no	—
54	C₂²	—	3	1	no	—
55	C₂²	#41⋊ΓL(1,𝔽₈)	1	1	yes	characteristic
56	C₄	—	4	1	no	—
57	C₄	—	12	1	no	—
58	C₄	—	12	1	no	—
59	C₃	—	112	1	no	Sylow
60	C₂	—	3	1	no	—
61	C₂	—	3	1	no	—
62	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×#401	C₃
socle	C₂³×C₇	C₂×A₄
prime radical	C₇⋊(#95⋊C₃)	C₂
LCS	G ▷ C₇×#95
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
2. C₇×#401	C₃	semidirect
15. #401	ΓL(1,𝔽₈)	semidirect
51. C₇	#401⋊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×#401	3
C₄×C₂²⋊ΓL(1,𝔽₈)	4
#401⋊C₃	7