Subgroups — #10621504

Order 672.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#41⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇⋊GL(2,𝔽₃)	C₂	1	2	yes	maximal, complemented
3	#6512	—	3	1	no	maximal, Hall
4	D₈₄	—	4	1	no	maximal
5	C₇×SL(2,𝔽₃)	C₂²	1	1	yes	characteristic, complemented
6	C₇×#41	D₃	1	1	yes	characteristic, complemented
7	C₂×D₂₈	—	3	1	no	—
8	#1143	—	3	2	no	—
9	#1141	—	3	2	no	—
10	#1136	—	3	1	no	—
11	#986	—	7	1	no	maximal, Hall
12	C₂×D₂₁	—	4	2	no	—
13	C₃×C₄×C₇	—	4	1	no	—
14	C₂²×D₇	—	6	1	no	—
15	C₇×Q₈	C₂×D₃	1	1	yes	characteristic
16	C₇×D₄	—	3	1	no	—
17	C₂×C₄×C₇	—	3	1	no	—
18	D₂₈	—	3	2	no	—
19	D₂₈	—	6	1	no	—
20	C₇⋊C₈	—	3	2	no	—
21	#41⋊C₃	D₇	1	1	yes	characteristic, complemented
22	GL(2,𝔽₃)	—	7	2	no	—
23	C₂×C₃×C₇	—	4	1	no	—
24	D₂₁	—	8	2	no	—
25	AGL(1,ℤ/8)	—	21	1	no	Sylow
26	C₂²×C₇	—	3	1	no	—
27	C₂×D₇	—	6	2	no	—
28	C₂×D₇	—	12	2	no	—
29	C₄×C₇	—	3	1	no	—
30	C₄×C₇	S₄	1	1	yes	characteristic
31	D₁₂	—	28	1	no	—
32	SL(2,𝔽₃)	C₂×D₇	1	1	yes	characteristic, complemented
33	C₃×C₇	—	4	1	no	Hall
34	#41	D₂₁	1	1	yes	characteristic, complemented
35	C₂×D₄	—	21	1	no	—
36	SD₁₆	—	21	2	no	—
37	D₈	—	21	2	no	—
38	M₁₆	—	21	1	no	—
39	C₂×C₇	—	6	1	no	—
40	C₂×C₇	C₂×S₄	1	1	yes	characteristic
41	D₇	—	12	2	no	—
42	C₂×D₃	—	28	2	no	—
43	C₃×C₄	—	4	1	no	—
44	C₂³	—	42	1	no	—
45	Q₈	C₂×D₂₁	1	1	yes	characteristic
46	D₄	—	3	1	no	—
47	D₄	—	21	2	no	—
48	D₄	—	42	1	no	—
49	C₂×C₄	—	3	1	no	—
50	C₈	—	21	2	no	—
51	C₇	#986	1	1	yes	characteristic, complemented, Sylow
52	C₂×C₃	—	4	1	no	—
53	D₃	—	56	2	no	—
54	C₂²	—	3	1	no	—
55	C₂²	—	42	2	no	—
56	C₂²	—	84	2	no	—
57	C₄	—	3	1	no	—
58	C₄	C₇⋊S₄	1	1	yes	characteristic
59	C₃	—	4	1	no	Sylow
60	C₂	—	6	1	no	—
61	C₂	—	84	2	no	—
62	C₂	C₂×C₇⋊S₄	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂×C₇⋊S₄
commutator, cosocle, hypocenter	C₇×SL(2,𝔽₃)	C₂²
Fitting	C₇×#41	D₃
socle	C₂×C₇	C₂×S₄
hypercenter	C₄	C₇⋊S₄
LCS	G ▷ C₇×SL(2,𝔽₃)
UCS	C₁ ◁ C₂ ◁ C₄

Complements

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

N	H ≅ G/N	type
1. C₇×#41⋊C₃	C₂	semidirect
2. C₇⋊GL(2,𝔽₃)	C₂	semidirect
5. C₇×SL(2,𝔽₃)	C₂²	semidirect
6. C₇×#41	D₃	semidirect
21. #41⋊C₃	D₇	semidirect
32. SL(2,𝔽₃)	C₂×D₇	semidirect
34. #41	D₂₁	semidirect
51. C₇	#986	semidirect

Maximal subgroups

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

subgroup	index
C₇×#41⋊C₃	2
C₇⋊GL(2,𝔽₃)	2
#6512	3
D₈₄	4
#986	7