Subgroups — C₇×#533 #90554

Order 448.

222 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×C₄⋊Q₈	C₂	1	2	yes	maximal, complemented
2	C₇×#125	C₂	1	2	yes	maximal, complemented
3	C₇×C₂²⋊Q₈	C₂	1	4	yes	maximal
4	C₄×C₇×Q₈	C₂	1	4	yes	maximal, complemented
5	C₇×#117	C₂	1	2	yes	maximal
6	C₂×C₄²×C₇	C₂	1	1	yes	maximal, characteristic
7	C₂×C₇×Q₈	C₂²	1	4	yes	—
8	C₂²×C₄×C₇	C₂²	1	2	yes	—
9	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
10	C₇×C₄⋊C₄	C₂²	1	4	yes	—
11	C₇×C₄⋊C₄	C₂²	1	8	yes	—
12	C₇×C₄⋊C₄	C₂²	1	4	yes	—
13	C₇×C₂²⋊C₄	C₂²	1	4	yes	—
14	C₄²×C₇	C₂²	1	1	yes	characteristic
15	C₄²×C₇	C₂²	1	1	yes	characteristic
16	C₄²×C₇	C₂²	1	2	yes	—
17	C₄²×C₇	C₂²	1	4	yes	—
18	#533	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
19	C₂³×C₇	C₂³	1	1	yes	characteristic
20	C₇×Q₈	—	2	8	no	—
21	C₂×C₄×C₇	—	2	4	no	—
22	C₂×C₄×C₇	Q₈	1	4	yes	complemented
23	C₂×C₄×C₇	C₂³	1	2	yes	—
24	C₂×C₄×C₇	C₂³	1	8	yes	—
25	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
26	C₂×C₄×C₇	C₂³	1	2	yes	—
27	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
28	C₄⋊Q₈	C₂×C₇	1	2	yes	complemented
29	#125	C₂×C₇	1	2	yes	complemented
30	C₂²⋊Q₈	C₂×C₇	1	4	yes	—
31	C₄×Q₈	C₂×C₇	1	4	yes	complemented
32	#117	C₂×C₇	1	2	yes	—
33	C₂×C₄²	C₂×C₇	1	1	yes	characteristic
34	C₂²×C₇	—	2	2	no	—
35	C₂²×C₇	C₂×Q₈	1	2	yes	—
36	C₂²×C₇	C₂⁴	1	1	yes	characteristic
37	C₄×C₇	—	2	2	no	—
38	C₄×C₇	—	2	8	no	—
39	C₄×C₇	C₂×Q₈	1	2	yes	—
40	C₄×C₇	C₂×Q₈	1	2	yes	—
41	C₄×C₇	#41	1	4	yes	—
42	C₂×Q₈	C₂²×C₇	1	4	yes	—
43	C₂²×C₄	C₂²×C₇	1	2	yes	—
44	C₂²×C₄	C₂²×C₇	1	1	yes	characteristic
45	C₄⋊C₄	C₂²×C₇	1	4	yes	—
46	C₄⋊C₄	C₂²×C₇	1	8	yes	—
47	C₄⋊C₄	C₂²×C₇	1	4	yes	—
48	C₂²⋊C₄	C₂²×C₇	1	4	yes	—
49	C₄²	C₂²×C₇	1	2	yes	—
50	C₄²	C₂²×C₇	1	1	yes	characteristic
51	C₄²	C₂²×C₇	1	1	yes	characteristic
52	C₄²	C₂²×C₇	1	4	yes	—
53	C₂×C₇	—	2	2	no	—
54	C₂×C₇	C₂²×Q₈	1	1	yes	characteristic
55	C₂×C₇	C₂×#41	1	2	yes	—
56	C₂³	C₂³×C₇	1	1	yes	characteristic
57	Q₈	—	2	8	no	—
58	C₂×C₄	—	2	4	no	—
59	C₂×C₄	C₇×Q₈	1	4	yes	complemented
60	C₂×C₄	C₂³×C₇	1	2	yes	—
61	C₂×C₄	C₂³×C₇	1	8	yes	—
62	C₂×C₄	C₂³×C₇	1	1	yes	characteristic
63	C₂×C₄	C₂³×C₇	1	1	yes	characteristic
64	C₂×C₄	C₂³×C₇	1	2	yes	—
65	C₇	#533	1	1	yes	characteristic, direct product factor, Sylow
66	C₂²	—	2	2	no	—
67	C₂²	C₂×C₇×Q₈	1	2	yes	—
68	C₂²	C₂⁴×C₇	1	1	yes	characteristic
69	C₄	—	2	2	no	—
70	C₄	—	2	8	no	—
71	C₄	C₂×C₇×Q₈	1	2	yes	—
72	C₄	C₂×C₇×Q₈	1	2	yes	—
73	C₄	C₇×#41	1	4	yes	—
74	C₂	—	2	2	no	—
75	C₂	C₂²×C₇×Q₈	1	1	yes	characteristic
76	C₂	C₂×C₇×#41	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₇	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁴×C₇
socle	C₂²×C₇	C₂⁴
prime radical	C₂³×C₇	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇×C₄⋊Q₈	C₂	semidirect
2. C₇×#125	C₂	semidirect
4. C₄×C₇×Q₈	C₂	semidirect
18. #533	C₇	direct (N × H)
22. C₂×C₄×C₇	Q₈	semidirect
28. C₄⋊Q₈	C₂×C₇	semidirect
29. #125	C₂×C₇	semidirect
31. C₄×Q₈	C₂×C₇	semidirect
59. C₂×C₄	C₇×Q₈	semidirect
65. C₇	#533	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×C₄⋊Q₈	2
C₇×#125	2
C₇×C₂²⋊Q₈	2
C₄×C₇×Q₈	2
C₇×#117	2
C₂×C₄²×C₇	2
#533	7

Subgroups — C7×#533 #90554

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#533 #90554