Subgroups — C₇×#357 #89399

Order 448.

66 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×D₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#107	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₇×D₄	—	2	1	no	—
5	C₇×D₈	—	2	1	no	—
6	C₇×D₈	C₄	1	2	yes	complemented
7	C₂×C₇×C₈	C₂²	1	1	yes	characteristic
8	C₇×C₄⋊C₄	—	2	1	no	—
9	C₇×C₁₆	—	2	1	no	—
10	#357	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₂³×C₇	—	4	1	no	—
12	C₇×D₄	—	2	2	no	—
13	C₇×D₄	—	4	1	no	—
14	C₂×C₄×C₇	—	4	1	no	—
15	C₂×C₄×C₇	D₄	1	1	yes	characteristic
16	C₇×C₈	C₂×C₄	1	1	yes	characteristic
17	C₇×C₈	D₄	1	1	yes	characteristic
18	C₂×D₈	C₂×C₇	1	1	yes	characteristic
19	C₂×C₁₆	C₂×C₇	1	1	yes	characteristic, complemented
20	#107	C₂×C₇	1	1	yes	characteristic, complemented
21	C₂²×C₇	—	4	2	no	—
22	C₂²×C₇	—	8	1	no	—
23	C₂²×C₇	—	8	1	no	—
24	C₂²×C₇	D₈	1	1	yes	characteristic
25	C₄×C₇	—	8	1	no	—
26	C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
27	C₄×C₇	SD₁₆	1	1	yes	characteristic
28	C₂×D₄	—	2	1	no	—
29	D₈	—	2	1	no	—
30	D₈	C₄×C₇	1	2	yes	complemented
31	C₂×C₈	C₂²×C₇	1	1	yes	characteristic
32	C₄⋊C₄	—	2	1	no	—
33	C₁₆	—	2	1	no	—
34	C₂×C₇	—	8	2	no	—
35	C₂×C₇	#102	1	1	yes	characteristic
36	C₂×C₇	D₁₆	1	1	yes	characteristic
37	C₂×C₇	SD₃₂	1	1	yes	characteristic
38	C₂³	—	4	1	no	—
39	D₄	—	2	2	no	—
40	D₄	—	4	1	no	—
41	C₂×C₄	—	4	1	no	—
42	C₂×C₄	C₇×D₄	1	1	yes	characteristic
43	C₈	C₂×C₄×C₇	1	1	yes	characteristic
44	C₈	C₇×D₄	1	1	yes	characteristic
45	C₇	#357	1	1	yes	characteristic, direct product factor, Sylow
46	C₂²	—	4	2	no	—
47	C₂²	—	8	1	no	—
48	C₂²	—	8	1	no	—
49	C₂²	C₇×D₈	1	1	yes	characteristic
50	C₄	—	8	1	no	—
51	C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
52	C₄	C₇×SD₁₆	1	1	yes	characteristic
53	C₂	—	8	2	no	—
54	C₂	C₇×#102	1	1	yes	characteristic
55	C₂	C₇×D₁₆	1	1	yes	characteristic
56	C₂	C₇×SD₃₂	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	D₈
commutator	C₈	C₂×C₄×C₇
Frattini, cosocle	C₂×C₈	C₂²×C₇
prime radical	C₂×C₇×D₈	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄×C₇ ◁ C₂×C₇×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₇×C₁₆	C₂	semidirect
3. C₇×#107	C₂	semidirect
6. C₇×D₈	C₄	semidirect
10. #357	C₇	direct (N × H)
19. C₂×C₁₆	C₂×C₇	semidirect
20. #107	C₂×C₇	semidirect
30. D₈	C₄×C₇	semidirect
45. C₇	#357	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₇×D₈	2
C₂×C₇×C₁₆	2
C₇×#107	2
#357	7

Subgroups — C7×#357 #89399

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#357 #89399