Subgroups — #10621528

Order 672.

92 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	#66662	C₂	1	2	yes	maximal, complemented
3	#6553	C₃	1	1	yes	maximal, characteristic, complemented, Hall
4	C₃×Dic₁₄	—	4	1	no	maximal
5	C₇×SL(2,𝔽₃)	C₂²	1	1	yes	characteristic, complemented
6	C₇×#41	C₂×C₃	1	1	yes	characteristic
7	#1161	C₂×C₃	1	2	yes	—
8	D₄×D₇	—	3	2	no	—
9	#1157	—	3	1	no	—
10	C₂×D₂₈	—	3	1	no	—
11	#994	—	7	1	no	maximal, Hall
12	C₃×C₄×C₇	—	4	1	no	—
13	C₃×Dic₇	—	4	2	no	—
14	C₂²×D₇	—	3	2	no	—
15	C₇×Q₈	C₂²×C₃	1	1	yes	characteristic
16	C₇×D₄	—	3	1	no	—
17	C₂×C₄×C₇	—	3	1	no	—
18	C₂²⋊D₇	—	3	2	no	—
19	D₂₈	—	3	2	no	—
20	D₂₈	—	3	1	no	—
21	C₄×D₇	—	3	2	no	—
22	Dic₁₄	A₄	1	1	yes	characteristic
23	#41⋊C₃	—	7	2	no	—
24	#41⋊C₃	D₇	1	1	yes	characteristic, complemented
25	C₂×C₃×C₇	—	4	1	no	—
26	H(5,𝔽₂)	—	7	1	no	Sylow
27	C₂²×C₇	—	3	1	no	—
28	C₂×D₇	—	3	2	no	—
29	C₂×D₇	—	12	2	no	—
30	C₄×C₇	—	3	1	no	—
31	C₄×C₇	C₂×A₄	1	1	yes	characteristic
32	Dic₇	C₂×A₄	1	2	yes	—
33	C₃×Q₈	—	28	1	no	—
34	SL(2,𝔽₃)	C₂×D₇	1	1	yes	characteristic, complemented
35	C₃×C₇	—	4	1	no	Hall
36	#41	—	7	2	no	—
37	#41	—	21	1	no	—
38	#41	C₃×D₇	1	1	yes	characteristic
39	C₂×D₄	—	21	1	no	—
40	C₂×D₄	—	21	2	no	—
41	C₂×C₇	—	6	1	no	—
42	C₂×C₇	C₂²×A₄	1	1	yes	characteristic
43	D₇	—	6	2	no	—
44	C₃×C₄	—	4	1	no	—
45	C₃×C₄	—	28	2	no	—
46	C₂³	—	21	2	no	—
47	Q₈	—	7	1	no	—
48	Q₈	C₂×C₃×D₇	1	1	yes	characteristic
49	D₄	—	3	1	no	—
50	D₄	—	21	2	no	—
51	D₄	—	21	2	no	—
52	D₄	—	21	1	no	—
53	C₂×C₄	—	3	1	no	—
54	C₂×C₄	—	21	2	no	—
55	C₇	#994	1	1	yes	characteristic, complemented, Sylow
56	C₂×C₃	—	4	1	no	—
57	C₂²	—	3	1	no	—
58	C₂²	—	21	2	no	—
59	C₂²	—	84	2	no	—
60	C₄	—	3	1	no	—
61	C₄	—	7	2	no	—
62	C₄	A₄×D₇	1	1	yes	characteristic
63	C₃	—	4	1	no	Sylow
64	C₂	—	6	1	no	—
65	C₂	—	42	2	no	—
66	C₂	C₂×A₄×D₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂×A₄×D₇
commutator, cosocle, hypocenter	C₇×Q₈	C₂²×C₃
Fitting	C₇×#41	C₂×C₃
socle	C₂×C₇	C₂²×A₄
hypercenter	C₄	A₄×D₇
LCS	G ▷ C₇×Q₈
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. #66662	C₂	semidirect
3. #6553	C₃	semidirect
5. C₇×SL(2,𝔽₃)	C₂²	semidirect
24. #41⋊C₃	D₇	semidirect
34. SL(2,𝔽₃)	C₂×D₇	semidirect
55. C₇	#994	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
#66662	2
#6553	3
C₃×Dic₁₄	4
#994	7