Subgroups — C₇×#557 #90578

Order 448.

200 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×C₄⋊Q₈	C₂	1	3	yes	maximal
2	C₇×#125	C₂	1	6	yes	maximal
3	C₄×C₇×Q₈	C₂	1	2	yes	maximal
4	C₄×C₇×Q₈	C₂	1	3	yes	maximal
5	C₄×C₇×Q₈	C₂	1	1	yes	maximal, characteristic
6	C₂×C₇×Q₈	C₂²	1	3	yes	—
7	C₂×C₇×Q₈	C₂²	1	1	yes	characteristic
8	C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
9	C₇×C₄⋊C₄	C₂²	1	6	yes	—
10	C₇×C₄⋊C₄	C₂²	1	3	yes	—
11	C₇×C₄⋊C₄	C₂²	1	6	yes	—
12	C₇×C₄⋊C₄	C₂²	1	6	yes	—
13	C₄²×C₇	C₂²	1	6	yes	—
14	C₄²×C₇	C₂²	1	3	yes	—
15	#557	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
16	C₇×Q₈	—	2	6	no	—
17	C₇×Q₈	Q₈	1	4	yes	complemented
18	C₂×C₄×C₇	C₂³	1	2	yes	—
19	C₂×C₄×C₇	C₂³	1	3	yes	—
20	C₂×C₄×C₇	C₂³	1	6	yes	—
21	C₂×C₄×C₇	C₂³	1	3	yes	—
22	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
23	C₄⋊Q₈	C₂×C₇	1	3	yes	—
24	#125	C₂×C₇	1	6	yes	—
25	C₄×Q₈	C₂×C₇	1	2	yes	—
26	C₄×Q₈	C₂×C₇	1	3	yes	—
27	C₄×Q₈	C₂×C₇	1	1	yes	characteristic
28	C₂²×C₇	C₂⁴	1	1	yes	characteristic
29	C₄×C₇	—	2	6	no	—
30	C₄×C₇	—	2	2	no	—
31	C₄×C₇	—	2	3	no	—
32	C₄×C₇	C₂×Q₈	1	6	yes	—
33	C₄×C₇	#41	1	2	yes	—
34	C₂×Q₈	C₂²×C₇	1	3	yes	—
35	C₂×Q₈	C₂²×C₇	1	1	yes	characteristic
36	C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
37	C₄⋊C₄	C₂²×C₇	1	6	yes	—
38	C₄⋊C₄	C₂²×C₇	1	3	yes	—
39	C₄⋊C₄	C₂²×C₇	1	6	yes	—
40	C₄⋊C₄	C₂²×C₇	1	6	yes	—
41	C₄²	C₂²×C₇	1	6	yes	—
42	C₄²	C₂²×C₇	1	3	yes	—
43	C₂×C₇	C₂²×Q₈	1	1	yes	characteristic
44	C₂×C₇	C₂×#41	1	1	yes	characteristic
45	C₂×C₇	2₋¹⁺⁴	1	1	yes	characteristic
46	Q₈	—	2	6	no	—
47	Q₈	C₇×Q₈	1	4	yes	complemented
48	C₂×C₄	C₂³×C₇	1	2	yes	—
49	C₂×C₄	C₂³×C₇	1	3	yes	—
50	C₂×C₄	C₂³×C₇	1	6	yes	—
51	C₂×C₄	C₂³×C₇	1	3	yes	—
52	C₂×C₄	C₂³×C₇	1	1	yes	characteristic
53	C₇	#557	1	1	yes	characteristic, direct product factor, Sylow
54	C₂²	C₂⁴×C₇	1	1	yes	characteristic
55	C₄	—	2	6	no	—
56	C₄	—	2	2	no	—
57	C₄	—	2	3	no	—
58	C₄	C₂×C₇×Q₈	1	6	yes	—
59	C₄	C₇×#41	1	2	yes	—
60	C₂	C₂²×C₇×Q₈	1	1	yes	characteristic
61	C₂	C₂×C₇×#41	1	1	yes	characteristic
62	C₂	C₇×2₋¹⁺⁴	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
15. #557	C₇	direct (N × H)
17. C₇×Q₈	Q₈	semidirect
47. Q₈	C₇×Q₈	semidirect
53. C₇	#557	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
C₄×C₇×Q₈	2
#557	7

Subgroups — C7×#557 #90578

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#557 #90578