Subgroups — C₇×#492 #90138

Order 448.

142 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×SD₁₆	C₂	1	4	yes	maximal, complemented
2	C₇×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#127	C₂	1	1	yes	maximal, characteristic
4	C₄×C₇×C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₇×Q₈	C₂²	1	2	yes	—
6	C₂×C₇×D₄	—	2	2	no	—
7	C₂×C₇×D₄	C₂²	1	2	yes	—
8	C₇×SD₁₆	—	2	8	no	—
9	C₂×C₇×C₈	C₂²	1	2	yes	—
10	C₇×C₄⋊C₄	—	2	2	no	—
11	C₄²×C₇	C₂²	1	1	yes	characteristic
12	#492	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂³×C₇	—	2	2	no	—
14	C₇×Q₈	—	2	4	no	—
15	C₇×D₄	—	2	4	no	—
16	C₇×D₄	—	4	4	no	—
17	C₂×C₄×C₇	—	2	2	no	—
18	C₂×C₄×C₇	D₄	1	2	yes	—
19	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
20	C₇×C₈	D₄	1	4	yes	complemented
21	C₂×SD₁₆	C₂×C₇	1	4	yes	complemented
22	C₄⋊Q₈	C₂×C₇	1	1	yes	characteristic, complemented
23	#127	C₂×C₇	1	1	yes	characteristic
24	C₄×C₈	C₂×C₇	1	1	yes	characteristic, complemented
25	C₂²×C₇	—	4	2	no	—
26	C₂²×C₇	—	4	4	no	—
27	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
28	C₄×C₇	—	4	2	no	—
29	C₄×C₇	SD₁₆	1	4	yes	complemented
30	C₄×C₇	C₂×D₄	1	2	yes	—
31	C₂×Q₈	C₂²×C₇	1	2	yes	—
32	C₂×D₄	—	2	2	no	—
33	C₂×D₄	C₂²×C₇	1	2	yes	—
34	SD₁₆	—	2	8	no	—
35	C₂×C₈	C₂²×C₇	1	2	yes	—
36	C₄⋊C₄	—	2	2	no	—
37	C₄²	C₂²×C₇	1	1	yes	characteristic
38	C₂×C₇	—	8	2	no	—
39	C₂×C₇	#127	1	1	yes	characteristic
40	C₂×C₇	C₂×SD₁₆	1	2	yes	—
41	C₂³	—	2	2	no	—
42	Q₈	—	2	4	no	—
43	D₄	—	2	4	no	—
44	D₄	—	4	4	no	—
45	C₂×C₄	—	2	2	no	—
46	C₂×C₄	C₇×D₄	1	2	yes	—
47	C₂×C₄	C₂³×C₇	1	1	yes	characteristic
48	C₈	C₇×D₄	1	4	yes	complemented
49	C₇	#492	1	1	yes	characteristic, direct product factor, Sylow
50	C₂²	—	4	2	no	—
51	C₂²	—	4	4	no	—
52	C₂²	C₂×C₇×D₄	1	1	yes	characteristic
53	C₄	—	4	2	no	—
54	C₄	C₇×SD₁₆	1	4	yes	complemented
55	C₄	C₂×C₇×D₄	1	2	yes	—
56	C₂	—	8	2	no	—
57	C₂	C₇×#127	1	1	yes	characteristic
58	C₂	C₂×C₇×SD₁₆	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₇	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂³×C₇
prime radical	C₇×#127	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₄²×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₇×SD₁₆	C₂	semidirect
2. C₇×C₄⋊Q₈	C₂	semidirect
4. C₄×C₇×C₈	C₂	semidirect
12. #492	C₇	direct (N × H)
20. C₇×C₈	D₄	semidirect
21. C₂×SD₁₆	C₂×C₇	semidirect
22. C₄⋊Q₈	C₂×C₇	semidirect
24. C₄×C₈	C₂×C₇	semidirect
29. C₄×C₇	SD₁₆	semidirect
48. C₈	C₇×D₄	semidirect
49. C₇	#492	direct (N × H)
54. C₄	C₇×SD₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×SD₁₆	2
C₇×C₄⋊Q₈	2
C₇×#127	2
C₄×C₇×C₈	2
#492	7

Subgroups — C7×#492 #90138

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#492 #90138