Subgroups — C₇×#352 #89394

Order 448.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#123	C₂	1	1	yes	maximal, characteristic
2	C₇×#100	C₂	1	1	yes	maximal, characteristic
3	C₇×#99	C₂	1	1	yes	maximal, characteristic
4	C₂×C₇×D₄	C₂²	1	1	yes	characteristic
5	C₂²×C₄×C₇	C₄	1	1	yes	characteristic, complemented
6	C₇×M₁₆	—	2	1	no	—
7	C₇×C₄⋊C₄	—	2	1	no	—
8	C₇×C₂²⋊C₄	—	2	1	no	—
9	C₇×C₂²⋊C₄	—	2	1	no	—
10	C₇×C₂²⋊C₄	C₄	1	1	yes	characteristic, complemented
11	#352	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
12	C₂³×C₇	C₂×C₄	1	1	yes	characteristic
13	C₂³×C₇	D₄	1	1	yes	characteristic
14	C₇×D₄	—	4	1	no	—
15	C₂×C₄×C₇	—	2	1	no	—
16	C₂×C₄×C₇	—	2	1	no	—
17	C₂×C₄×C₇	—	4	1	no	—
18	C₂×C₄×C₇	—	4	1	no	—
19	C₂×C₄×C₇	D₄	1	1	yes	characteristic
20	C₇×C₈	—	4	1	no	—
21	#123	C₂×C₇	1	1	yes	characteristic
22	#100	C₂×C₇	1	1	yes	characteristic
23	#99	C₂×C₇	1	1	yes	characteristic
24	C₂²×C₇	—	2	1	no	—
25	C₂²×C₇	—	2	1	no	—
26	C₂²×C₇	—	4	1	no	—
27	C₂²×C₇	—	4	1	no	—
28	C₂²×C₇	C₂²⋊C₄	1	1	yes	characteristic
29	C₄×C₇	—	2	1	no	—
30	C₄×C₇	—	4	1	no	—
31	C₄×C₇	—	4	1	no	—
32	C₄×C₇	—	8	1	no	—
33	C₂×D₄	C₂²×C₇	1	1	yes	characteristic
34	C₂²×C₄	C₄×C₇	1	1	yes	characteristic, complemented
35	M₁₆	—	2	1	no	—
36	C₄⋊C₄	—	2	1	no	—
37	C₂²⋊C₄	—	2	1	no	—
38	C₂²⋊C₄	—	2	1	no	—
39	C₂²⋊C₄	C₄×C₇	1	1	yes	characteristic, complemented
40	C₂×C₇	—	2	1	no	—
41	C₂×C₇	—	4	1	no	—
42	C₂×C₇	—	4	1	no	—
43	C₂×C₇	#99	1	1	yes	characteristic
44	C₂³	C₂×C₄×C₇	1	1	yes	characteristic
45	C₂³	C₇×D₄	1	1	yes	characteristic
46	D₄	—	4	1	no	—
47	C₂×C₄	—	2	1	no	—
48	C₂×C₄	—	2	1	no	—
49	C₂×C₄	—	4	1	no	—
50	C₂×C₄	—	4	1	no	—
51	C₂×C₄	C₇×D₄	1	1	yes	characteristic
52	C₈	—	4	1	no	—
53	C₇	#352	1	1	yes	characteristic, direct product factor, Sylow
54	C₂²	—	2	1	no	—
55	C₂²	—	2	1	no	—
56	C₂²	—	4	1	no	—
57	C₂²	—	4	1	no	—
58	C₂²	C₇×C₂²⋊C₄	1	1	yes	characteristic
59	C₄	—	2	1	no	—
60	C₄	—	4	1	no	—
61	C₄	—	4	1	no	—
62	C₄	—	8	1	no	—
63	C₂	—	2	1	no	—
64	C₂	—	4	1	no	—
65	C₂	—	4	1	no	—
66	C₂	C₇×#99	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₇	#99
commutator	C₂³	C₂×C₄×C₇
Frattini, cosocle	C₂×D₄	C₂²×C₇
prime radical	C₂×C₇×D₄	C₂²
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂²×C₇ ◁ C₂×C₇×D₄ ◁ G

Complements

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

N	H ≅ G/N	type
5. C₂²×C₄×C₇	C₄	semidirect
10. C₇×C₂²⋊C₄	C₄	semidirect
11. #352	C₇	direct (N × H)
34. C₂²×C₄	C₄×C₇	semidirect
39. C₂²⋊C₄	C₄×C₇	semidirect
53. C₇	#352	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×#123	2
C₇×#100	2
C₇×#99	2
#352	7

Subgroups — C7×#352 #89394

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#352 #89394