Subgroups — C₇×#484 #90130

Order 448.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₇×#107	C₂	1	2	yes	maximal, complemented
4	C₇×#103	C₂	1	2	yes	maximal, complemented
5	C₇×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic
6	C₂×C₇×Q₈	C₂²	1	1	yes	characteristic
7	C₂²×C₄×C₇	—	2	1	no	—
8	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
9	C₂×C₇×C₈	C₂²	1	2	yes	—
10	C₇×C₄⋊C₄	—	2	1	no	—
11	C₇×C₄⋊C₄	—	2	1	no	—
12	C₇×C₄⋊C₄	C₂²	1	2	yes	—
13	C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	C₇×C₂²⋊C₄	—	2	1	no	—
15	#484	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
16	C₂³×C₇	D₄	1	1	yes	characteristic
17	C₇×Q₈	—	2	2	no	—
18	C₂×C₄×C₇	—	2	2	no	—
19	C₂×C₄×C₇	—	2	1	no	—
20	C₂×C₄×C₇	—	2	1	no	—
21	C₂×C₄×C₇	—	2	2	no	—
22	C₂×C₄×C₇	—	4	2	no	—
23	C₂×C₄×C₇	D₄	1	1	yes	characteristic
24	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
25	C₇×C₈	—	2	2	no	—
26	C₂²⋊Q₈	C₂×C₇	1	1	yes	characteristic
27	C₂×C₄⋊C₄	C₂×C₇	1	1	yes	characteristic
28	#107	C₂×C₇	1	2	yes	complemented
29	#103	C₂×C₇	1	2	yes	complemented
30	C₂²⋊C₈	C₂×C₇	1	1	yes	characteristic
31	C₂²×C₇	—	2	1	no	—
32	C₂²×C₇	—	2	1	no	—
33	C₂²×C₇	Q₁₆	1	2	yes	—
34	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
35	C₄×C₇	—	2	1	no	—
36	C₄×C₇	—	4	1	no	—
37	C₄×C₇	—	4	1	no	—
38	C₄×C₇	—	4	2	no	—
39	C₄×C₇	#41	1	2	yes	—
40	C₂×Q₈	C₂²×C₇	1	1	yes	characteristic
41	C₂²×C₄	—	2	1	no	—
42	C₂²×C₄	C₂²×C₇	1	1	yes	characteristic
43	C₂×C₈	C₂²×C₇	1	2	yes	—
44	C₄⋊C₄	—	2	1	no	—
45	C₄⋊C₄	—	2	1	no	—
46	C₄⋊C₄	C₂²×C₇	1	2	yes	—
47	C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
48	C₂²⋊C₄	—	2	1	no	—
49	C₂×C₇	—	2	2	no	—
50	C₂×C₇	#123	1	1	yes	characteristic
51	C₂×C₇	C₂×Q₁₆	1	1	yes	characteristic
52	C₂×C₇	AGL(1,ℤ/8)	1	1	yes	characteristic
53	C₂³	C₇×D₄	1	1	yes	characteristic
54	Q₈	—	2	2	no	—
55	C₂×C₄	—	2	2	no	—
56	C₂×C₄	—	2	1	no	—
57	C₂×C₄	—	2	1	no	—
58	C₂×C₄	—	2	2	no	—
59	C₂×C₄	—	4	2	no	—
60	C₂×C₄	C₇×D₄	1	1	yes	characteristic
61	C₂×C₄	C₂³×C₇	1	1	yes	characteristic
62	C₈	—	2	2	no	—
63	C₇	#484	1	1	yes	characteristic, direct product factor, Sylow
64	C₂²	—	2	1	no	—
65	C₂²	—	2	1	no	—
66	C₂²	C₇×Q₁₆	1	2	yes	—
67	C₂²	C₂×C₇×D₄	1	1	yes	characteristic
68	C₄	—	2	1	no	—
69	C₄	—	4	1	no	—
70	C₄	—	4	2	no	—
71	C₄	—	4	1	no	—
72	C₄	C₇×#41	1	2	yes	—
73	C₂	—	2	2	no	—
74	C₂	C₇×#123	1	1	yes	characteristic
75	C₂	C₂×C₇×Q₁₆	1	1	yes	characteristic
76	C₂	C₇×AGL(1,ℤ/8)	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
3. C₇×#107	C₂	semidirect
4. C₇×#103	C₂	semidirect
15. #484	C₇	direct (N × H)
28. #107	C₂×C₇	semidirect
29. #103	C₂×C₇	semidirect
63. C₇	#484	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×C₂²⋊Q₈	2
C₂×C₇×C₄⋊C₄	2
C₇×#107	2
C₇×#103	2
C₇×C₂²⋊C₈	2
#484	7

Subgroups — C7×#484 #90130

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#484 #90130