Subgroups — C₇×#984 #10621491

Order 672.

72 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
2	C₇×#226	C₂	1	2	yes	maximal, complemented
3	C₇×#137	—	3	1	no	maximal, Hall
4	C₇×Dic₆	—	4	1	no	maximal
5	C₇×SL(2,𝔽₃)	C₂²	1	1	yes	characteristic
6	C₇×#41	D₃	1	1	yes	characteristic
7	C₂×C₇×Q₈	—	3	1	no	—
8	C₇×Q₁₆	—	3	2	no	—
9	C₇×SD₁₆	—	3	2	no	—
10	C₇×M₁₆	—	3	1	no	—
11	#984	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
12	C₃×C₄×C₇	—	4	1	no	—
13	C₇×Dic₃	—	4	2	no	—
14	C₇×Q₈	—	3	2	no	—
15	C₇×Q₈	—	6	1	no	—
16	C₇×Q₈	C₂×D₃	1	1	yes	characteristic
17	C₇×D₄	—	3	1	no	—
18	C₂×C₄×C₇	—	3	1	no	—
19	C₂×C₄×C₇	—	6	1	no	—
20	C₇×C₈	—	3	2	no	—
21	#41⋊C₃	C₂×C₇	1	1	yes	characteristic
22	#226	C₂×C₇	1	2	yes	complemented
23	C₂×C₃×C₇	—	4	1	no	—
24	#137	—	3	1	no	Sylow
25	C₂²×C₇	—	3	1	no	—
26	C₄×C₇	—	3	1	no	—
27	C₄×C₇	—	6	2	no	—
28	C₄×C₇	S₄	1	1	yes	characteristic
29	Dic₆	—	4	1	no	—
30	SL(2,𝔽₃)	C₂²×C₇	1	1	yes	characteristic
31	C₃×C₇	—	4	1	no	Hall
32	#41	D₃×C₇	1	1	yes	characteristic
33	C₂×Q₈	—	3	1	no	—
34	Q₁₆	—	3	2	no	—
35	SD₁₆	—	3	2	no	—
36	M₁₆	—	3	1	no	—
37	C₂×C₇	—	6	1	no	—
38	C₂×C₇	C₂×S₄	1	1	yes	characteristic
39	C₃×C₄	—	4	1	no	—
40	Dic₃	—	4	2	no	—
41	Q₈	—	3	2	no	—
42	Q₈	—	6	1	no	—
43	Q₈	C₂×D₃×C₇	1	1	yes	characteristic
44	D₄	—	3	1	no	—
45	C₂×C₄	—	3	1	no	—
46	C₂×C₄	—	6	1	no	—
47	C₈	—	3	2	no	—
48	C₇	#984	1	1	yes	characteristic, direct product factor, Sylow
49	C₂×C₃	—	4	1	no	—
50	C₂²	—	3	1	no	—
51	C₄	—	3	1	no	—
52	C₄	—	6	2	no	—
53	C₄	C₇×S₄	1	1	yes	characteristic
54	C₃	—	4	1	no	Sylow
55	C₂	—	6	1	no	—
56	C₂	C₂×C₇×S₄	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₇	C₂×S₄
commutator, cosocle, hypocenter	SL(2,𝔽₃)	C₂²×C₇
Frattini	C₂	C₂×C₇×S₄
Fitting	C₇×#41	D₃
prime radical	C₇×#41⋊C₃	C₂
hypercenter	C₄×C₇	S₄
LCS	G ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂×C₇ ◁ C₄×C₇

Complements

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

N	H ≅ G/N	type
2. C₇×#226	C₂	semidirect
11. #984	C₇	direct (N × H)
22. #226	C₂×C₇	semidirect
48. C₇	#984	direct (N × H)

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×#226	2
C₇×#137	3
C₇×Dic₆	4
#984	7

Subgroups — C7×#984 #10621491

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#984 #10621491