Subgroups — C₇×#798 #10620596

Order 672.

52 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₃×C₇×C₈	C₂	1	1	yes	maximal, characteristic
2	C₃×C₇×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×C₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×M₃₂	—	3	1	no	maximal, Hall
5	C₄×D₃×C₇	C₄	1	1	yes	characteristic
6	C₃×C₇×C₈	C₂²	1	1	yes	characteristic
7	C₇×C₃⋊C₈	C₄	1	1	yes	characteristic
8	C₂×C₇×C₈	—	3	1	no	—
9	C₇×C₁₆	—	3	1	no	—
10	C₇×C₁₆	D₃	1	1	yes	characteristic, complemented
11	#798	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
12	C₂×D₃×C₇	C₈	1	1	yes	characteristic
13	C₃×C₄×C₇	C₂×C₄	1	1	yes	characteristic
14	C₇×Dic₃	C₈	1	1	yes	characteristic
15	C₂×C₄×C₇	—	3	1	no	—
16	C₇×C₈	—	3	1	no	—
17	C₇×C₈	C₂×D₃	1	1	yes	characteristic
18	D₃×C₈	C₂×C₇	1	1	yes	characteristic
19	C₃×C₁₆	C₂×C₇	1	1	yes	characteristic, complemented
20	C₃⋊C₁₆	C₂×C₇	1	1	yes	characteristic, complemented
21	C₂×C₃×C₇	C₂×C₈	1	1	yes	characteristic
22	D₃×C₇	—	2	1	no	—
23	M₃₂	—	3	1	no	Sylow
24	C₂²×C₇	—	3	1	no	—
25	C₄×C₇	—	3	1	no	—
26	C₄×C₇	C₄×D₃	1	1	yes	characteristic
27	C₄×D₃	C₄×C₇	1	1	yes	characteristic
28	C₃×C₈	C₂²×C₇	1	1	yes	characteristic
29	C₃⋊C₈	C₄×C₇	1	1	yes	characteristic
30	C₃×C₇	M₃₂	1	1	yes	characteristic, complemented, Hall
31	C₂×C₈	—	3	1	no	—
32	C₁₆	—	3	1	no	—
33	C₁₆	D₃×C₇	1	1	yes	characteristic, complemented
34	C₂×C₇	—	6	1	no	—
35	C₂×C₇	D₃×C₈	1	1	yes	characteristic
36	C₂×D₃	C₇×C₈	1	1	yes	characteristic
37	C₃×C₄	C₂×C₄×C₇	1	1	yes	characteristic
38	Dic₃	C₇×C₈	1	1	yes	characteristic
39	C₂×C₄	—	3	1	no	—
40	C₈	—	3	1	no	—
41	C₈	C₂×D₃×C₇	1	1	yes	characteristic
42	C₇	#798	1	1	yes	characteristic, direct product factor, Sylow
43	C₂×C₃	C₂×C₇×C₈	1	1	yes	characteristic
44	D₃	—	2	1	no	—
45	C₂²	—	3	1	no	—
46	C₄	—	3	1	no	—
47	C₄	C₄×D₃×C₇	1	1	yes	characteristic
48	C₃	C₇×M₃₂	1	1	yes	characteristic, complemented, Sylow
49	C₂	—	6	1	no	—
50	C₂	D₃×C₇×C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₇×C₈	C₂×D₃
commutator	C₂×C₃	C₂×C₇×C₈
Frattini	C₈	C₂×D₃×C₇
Fitting	C₃×C₇×C₁₆	C₂
socle	C₂×C₃×C₇	C₂×C₈
prime radical	C₂×D₃×C₇	C₈
cosocle	C₃×C₈	C₂²×C₇
hypercenter	C₇×C₁₆	D₃
hypocenter	C₃	C₇×M₃₂
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₇×C₈ ◁ C₇×C₁₆

Complements

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

N	H ≅ G/N	type
2. C₃×C₇×C₁₆	C₂	semidirect
3. C₇×C₃⋊C₁₆	C₂	semidirect
10. C₇×C₁₆	D₃	semidirect
11. #798	C₇	direct (N × H)
19. C₃×C₁₆	C₂×C₇	semidirect
20. C₃⋊C₁₆	C₂×C₇	semidirect
30. C₃×C₇	M₃₂	semidirect
33. C₁₆	D₃×C₇	semidirect
42. C₇	#798	direct (N × H)
48. C₃	C₇×M₃₂	semidirect

Maximal subgroups

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

subgroup	index
D₃×C₇×C₈	2
C₃×C₇×C₁₆	2
C₇×C₃⋊C₁₆	2
C₇×M₃₂	3
#798	7

Subgroups — C7×#798 #10620596

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#798 #10620596