Subgroups — C₇×#948 #10621345

Order 672.

124 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₇×#41	C₂	1	1	yes	maximal, characteristic
2	C₇×#208	C₂	1	3	yes	maximal, complemented
3	C₂×C₇×C₃⋊C₈	C₂	1	3	yes	maximal, complemented
4	C₇×#131	—	3	1	no	maximal, Hall
5	C₃×C₇×Q₈	C₄	1	1	yes	characteristic
6	C₃×C₇×D₄	C₄	1	3	yes	—
7	C₂×C₃×C₄×C₇	C₂²	1	3	yes	—
8	C₇×C₃⋊C₈	C₂²	1	1	yes	characteristic
9	C₇×C₃⋊C₈	C₂²	1	3	yes	—
10	C₇×#41	D₃	1	1	yes	characteristic
11	C₇×M₁₆	—	3	3	no	—
12	C₂×C₇×C₈	—	3	3	no	—
13	#948	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
14	C₂²×C₃×C₇	C₂×C₄	1	3	yes	—
15	C₃×C₄×C₇	C₂×C₄	1	3	yes	—
16	C₃×C₄×C₇	C₂³	1	1	yes	characteristic
17	C₇×Q₈	Dic₃	1	1	yes	characteristic
18	C₇×D₄	Dic₃	1	3	yes	—
19	C₂×C₄×C₇	C₂×D₃	1	3	yes	—
20	C₇×C₈	—	3	1	no	—
21	C₇×C₈	—	3	3	no	—
22	C₃×#41	C₂×C₇	1	1	yes	characteristic
23	#208	C₂×C₇	1	3	yes	complemented
24	C₂×C₃⋊C₈	C₂×C₇	1	3	yes	complemented
25	C₂×C₃×C₇	—	2	3	no	—
26	C₂×C₃×C₇	C₂²×C₄	1	1	yes	characteristic
27	#131	—	3	1	no	Sylow
28	C₂²×C₇	C₂×Dic₃	1	3	yes	—
29	C₄×C₇	C₂×Dic₃	1	3	yes	—
30	C₄×C₇	C₂²×D₃	1	1	yes	characteristic
31	C₃×Q₈	C₄×C₇	1	1	yes	characteristic
32	C₃×D₄	C₄×C₇	1	3	yes	—
33	C₂×C₃×C₄	C₂²×C₇	1	3	yes	—
34	C₃⋊C₈	C₂²×C₇	1	1	yes	characteristic
35	C₃⋊C₈	C₂²×C₇	1	3	yes	—
36	C₃×C₇	#131	1	1	yes	characteristic, complemented, Hall
37	#41	D₃×C₇	1	1	yes	characteristic
38	M₁₆	—	3	3	no	—
39	C₂×C₈	—	3	3	no	—
40	C₂×C₇	—	2	3	no	—
41	C₂×C₇	C₂²×Dic₃	1	1	yes	characteristic
42	C₂²×C₃	C₂×C₄×C₇	1	3	yes	—
43	C₃×C₄	C₂×C₄×C₇	1	3	yes	—
44	C₃×C₄	C₂³×C₇	1	1	yes	characteristic
45	Q₈	C₇×Dic₃	1	1	yes	characteristic
46	D₄	C₇×Dic₃	1	3	yes	—
47	C₂×C₄	C₂×D₃×C₇	1	3	yes	—
48	C₈	—	3	1	no	—
49	C₈	—	3	3	no	—
50	C₇	#948	1	1	yes	characteristic, direct product factor, Sylow
51	C₂×C₃	—	2	3	no	—
52	C₂×C₃	C₂²×C₄×C₇	1	1	yes	characteristic
53	C₂²	C₂×C₇×Dic₃	1	3	yes	—
54	C₄	C₂×C₇×Dic₃	1	3	yes	—
55	C₄	C₂²×D₃×C₇	1	1	yes	characteristic
56	C₃	C₇×#131	1	1	yes	characteristic, complemented, Sylow
57	C₂	—	2	3	no	—
58	C₂	C₂²×C₇×Dic₃	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₇	C₂²×D₃
commutator	C₂×C₃	C₂²×C₄×C₇
Frattini	C₄	C₂²×D₃×C₇
Fitting, prime radical	C₃×C₇×#41	C₂
socle	C₂×C₃×C₇	C₂²×C₄
cosocle	C₃×C₄	C₂³×C₇
hypercenter	C₇×#41	D₃
hypocenter	C₃	C₇×#131
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄×C₇ ◁ C₇×#41

Complements

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

N	H ≅ G/N	type
2. C₇×#208	C₂	semidirect
3. C₂×C₇×C₃⋊C₈	C₂	semidirect
13. #948	C₇	direct (N × H)
23. #208	C₂×C₇	semidirect
24. C₂×C₃⋊C₈	C₂×C₇	semidirect
36. C₃×C₇	#131	semidirect
50. C₇	#948	direct (N × H)
56. C₃	C₇×#131	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₇×#41	2
C₇×#208	2
C₂×C₇×C₃⋊C₈	2
C₇×#131	3
#948	7

Subgroups — C7×#948 #10621345

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#948 #10621345