Subgroups — C₇×#497 #90143

Order 448.

124 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×Q₁₆	C₂	1	2	yes	maximal, complemented
2	C₂×C₇×SD₁₆	C₂	1	2	yes	maximal
3	C₇×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#124	C₂	1	1	yes	maximal, characteristic
5	C₇×#97	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₇×Q₈	C₂²	1	1	yes	characteristic
7	C₂×C₇×Q₈	C₂²	1	2	yes	—
8	C₂×C₇×D₄	C₂²	1	1	yes	characteristic
9	C₇×Q₁₆	—	2	4	no	—
10	C₇×SD₁₆	—	2	4	no	—
11	C₂×C₇×C₈	C₂²	1	2	yes	—
12	C₇×C₄⋊C₄	—	2	2	no	—
13	C₇×C₂²⋊C₄	—	2	2	no	—
14	C₄²×C₇	C₂²	1	1	yes	characteristic
15	#497	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
16	C₂³×C₇	—	2	1	no	—
17	C₇×Q₈	—	2	2	no	—
18	C₇×Q₈	—	2	2	no	—
19	C₇×Q₈	—	2	2	no	—
20	C₇×D₄	—	2	2	no	—
21	C₂×C₄×C₇	—	2	1	no	—
22	C₂×C₄×C₇	—	2	2	no	—
23	C₂×C₄×C₇	D₄	1	2	yes	—
24	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
25	C₇×C₈	D₄	1	4	yes	—
26	C₂×Q₁₆	C₂×C₇	1	2	yes	complemented
27	C₂×SD₁₆	C₂×C₇	1	2	yes	—
28	C₄⋊Q₈	C₂×C₇	1	1	yes	characteristic, complemented
29	#124	C₂×C₇	1	1	yes	characteristic
30	#97	C₂×C₇	1	1	yes	characteristic, complemented
31	C₂²×C₇	—	4	1	no	—
32	C₂²×C₇	—	4	2	no	—
33	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
34	C₄×C₇	—	2	2	no	—
35	C₄×C₇	—	4	1	no	—
36	C₄×C₇	—	4	2	no	—
37	C₄×C₇	C₂×D₄	1	2	yes	—
38	C₂×Q₈	C₂²×C₇	1	1	yes	characteristic
39	C₂×Q₈	C₂²×C₇	1	2	yes	—
40	C₂×D₄	C₂²×C₇	1	1	yes	characteristic
41	Q₁₆	—	2	4	no	—
42	SD₁₆	—	2	4	no	—
43	C₂×C₈	C₂²×C₇	1	2	yes	—
44	C₄⋊C₄	—	2	2	no	—
45	C₂²⋊C₄	—	2	2	no	—
46	C₄²	C₂²×C₇	1	1	yes	characteristic
47	C₂×C₇	—	8	1	no	—
48	C₂×C₇	#127	1	1	yes	characteristic
49	C₂×C₇	#137	1	2	yes	—
50	C₂³	—	2	1	no	—
51	Q₈	—	2	2	no	—
52	Q₈	—	2	2	no	—
53	Q₈	—	2	2	no	—
54	D₄	—	2	2	no	—
55	C₂×C₄	—	2	1	no	—
56	C₂×C₄	—	2	2	no	—
57	C₂×C₄	C₇×D₄	1	2	yes	—
58	C₂×C₄	C₂³×C₇	1	1	yes	characteristic
59	C₈	C₇×D₄	1	4	yes	—
60	C₇	#497	1	1	yes	characteristic, direct product factor, Sylow
61	C₂²	—	4	1	no	—
62	C₂²	—	4	2	no	—
63	C₂²	C₂×C₇×D₄	1	1	yes	characteristic
64	C₄	—	2	2	no	—
65	C₄	—	4	1	no	—
66	C₄	—	4	2	no	—
67	C₄	C₂×C₇×D₄	1	2	yes	—
68	C₂	—	8	1	no	—
69	C₂	C₇×#127	1	1	yes	characteristic
70	C₂	C₇×#137	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₇	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂³×C₇
prime radical	C₂×C₇×D₄	C₂²
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×Q₁₆	C₂	semidirect
3. C₇×C₄⋊Q₈	C₂	semidirect
5. C₇×#97	C₂	semidirect
15. #497	C₇	direct (N × H)
26. C₂×Q₁₆	C₂×C₇	semidirect
28. C₄⋊Q₈	C₂×C₇	semidirect
30. #97	C₂×C₇	semidirect
60. C₇	#497	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₇×Q₁₆	2
C₂×C₇×SD₁₆	2
C₇×C₄⋊Q₈	2
C₇×#124	2
C₇×#97	2
#497	7

Subgroups — C7×#497 #90143

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#497 #90143