Subgroups — #10621503

Order 672.

78 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	1	yes	maximal, characteristic, complemented
3	C₇⋊#226	C₂	1	1	yes	maximal, characteristic, complemented
4	#6513	—	3	1	no	maximal, Hall
5	C₄×D₂₁	—	4	1	no	maximal
6	C₇×SL(2,𝔽₃)	C₂²	1	1	yes	characteristic
7	C₇×#41	D₃	1	1	yes	characteristic, complemented
8	#1157	—	3	1	no	—
9	#1144	—	3	1	no	—
10	#1143	—	3	1	no	—
11	#1142	—	3	1	no	—
12	#1141	—	3	1	no	—
13	C₂×C₇⋊C₈	—	3	1	no	—
14	#985	—	7	1	no	maximal, Hall
15	C₂×D₂₁	—	4	1	no	—
16	C₃×C₄×C₇	—	4	1	no	—
17	Dic₂₁	—	4	1	no	—
18	C₇×Q₈	C₂×D₃	1	1	yes	characteristic
19	C₇×D₄	—	3	1	no	—
20	C₂×C₄×C₇	—	3	1	no	—
21	C₂²⋊D₇	—	6	1	no	—
22	D₂₈	—	3	1	no	—
23	C₄×D₇	—	6	1	no	—
24	Dic₁₄	—	3	1	no	—
25	C₇⋊C₈	—	3	1	no	—
26	C₇⋊C₈	—	3	1	no	—
27	#41⋊C₃	D₇	1	1	yes	characteristic, complemented
28	GL(2,𝔽₃)	—	7	1	no	—
29	#226	—	7	1	no	—
30	C₂×C₃×C₇	—	4	1	no	—
31	D₂₁	—	4	2	no	—
32	#135	—	21	1	no	Sylow
33	C₂²×C₇	—	3	1	no	—
34	C₂×D₇	—	6	1	no	—
35	C₄×C₇	—	3	1	no	—
36	C₄×C₇	S₄	1	1	yes	characteristic
37	Dic₇	—	6	1	no	—
38	C₄×D₃	—	28	1	no	—
39	SL(2,𝔽₃)	C₂×D₇	1	1	yes	characteristic
40	C₃×C₇	—	4	1	no	Hall
41	#41	—	21	1	no	—
42	#41	D₂₁	1	1	yes	characteristic, complemented
43	Q₁₆	—	21	1	no	—
44	SD₁₆	—	21	1	no	—
45	SD₁₆	—	21	1	no	—
46	D₈	—	21	1	no	—
47	C₂×C₈	—	21	1	no	—
48	C₂×C₇	—	6	1	no	—
49	C₂×C₇	C₂×S₄	1	1	yes	characteristic
50	D₇	—	12	1	no	—
51	C₂×D₃	—	28	1	no	—
52	C₃×C₄	—	4	1	no	—
53	Dic₃	—	28	1	no	—
54	Q₈	—	21	1	no	—
55	Q₈	C₂×D₂₁	1	1	yes	characteristic
56	D₄	—	3	1	no	—
57	D₄	—	21	1	no	—
58	D₄	—	42	1	no	—
59	C₂×C₄	—	3	1	no	—
60	C₂×C₄	—	42	1	no	—
61	C₈	—	21	1	no	—
62	C₈	—	21	1	no	—
63	C₇	#985	1	1	yes	characteristic, complemented, Sylow
64	C₂×C₃	—	4	1	no	—
65	D₃	—	28	2	no	—
66	C₂²	—	3	1	no	—
67	C₂²	—	42	1	no	—
68	C₄	—	3	1	no	—
69	C₄	—	42	1	no	—
70	C₄	C₇⋊S₄	1	1	yes	characteristic
71	C₃	—	4	1	no	Sylow
72	C₂	—	6	1	no	—
73	C₂	—	84	1	no	—
74	C₂	C₂×C₇⋊S₄	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₄	C₇⋊S₄
commutator, cosocle, hypocenter	C₇×SL(2,𝔽₃)	C₂²
Frattini	C₂	C₂×C₇⋊S₄
Fitting	C₇×#41	D₃
socle	C₂×C₇	C₂×S₄
LCS	G ▷ C₇×SL(2,𝔽₃)
UCS	C₁ ◁ 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
3. C₇⋊#226	C₂	semidirect
7. C₇×#41	D₃	semidirect
27. #41⋊C₃	D₇	semidirect
42. #41	D₂₁	semidirect
63. C₇	#985	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
C₇⋊#226	2
#6513	3
C₄×D₂₁	4
#985	7