Subgroups — C₇×#461 #90107

Order 448.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×SD₁₆	C₂	1	2	yes	maximal
2	C₇×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₇×D₄	C₂	1	1	yes	maximal, characteristic
4	C₇×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#103	C₂	1	2	yes	maximal, complemented
6	C₂×C₇×Q₈	C₂²	1	2	yes	—
7	C₂×C₇×D₄	C₂²	1	1	yes	characteristic
8	C₂²×C₄×C₇	—	2	1	no	—
9	C₇×SD₁₆	—	2	4	no	—
10	C₂×C₇×C₈	C₂²	1	2	yes	—
11	C₇×C₄⋊C₄	—	2	1	no	—
12	C₇×C₄⋊C₄	—	2	1	no	—
13	C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	C₇×C₂²⋊C₄	—	2	1	no	—
15	C₄²×C₇	C₂²	1	1	yes	characteristic
16	#461	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
17	C₂³×C₇	—	2	1	no	—
18	C₇×Q₈	—	2	2	no	—
19	C₇×Q₈	—	2	2	no	—
20	C₇×D₄	—	2	1	no	—
21	C₇×D₄	D₄	1	2	yes	—
22	C₂×C₄×C₇	—	2	1	no	—
23	C₂×C₄×C₇	—	2	2	no	—
24	C₂×C₄×C₇	—	4	2	no	—
25	C₂×C₄×C₇	D₄	1	1	yes	characteristic
26	C₂×C₄×C₇	D₄	1	1	yes	characteristic
27	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
28	C₇×C₈	—	2	2	no	—
29	C₂×SD₁₆	C₂×C₇	1	2	yes	—
30	C₄⋊Q₈	C₂×C₇	1	1	yes	characteristic, complemented
31	C₄×D₄	C₂×C₇	1	1	yes	characteristic
32	C₄⋊C₈	C₂×C₇	1	1	yes	characteristic, complemented
33	#103	C₂×C₇	1	2	yes	complemented
34	C₂²×C₇	—	2	2	no	—
35	C₂²×C₇	—	4	1	no	—
36	C₂²×C₇	—	4	1	no	—
37	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
38	C₄×C₇	—	2	1	no	—
39	C₄×C₇	—	4	2	no	—
40	C₄×C₇	—	4	1	no	—
41	C₄×C₇	SD₁₆	1	2	yes	complemented
42	C₄×C₇	C₂×D₄	1	1	yes	characteristic
43	C₄×C₇	#41	1	1	yes	characteristic
44	C₂×Q₈	C₂²×C₇	1	2	yes	—
45	C₂×D₄	C₂²×C₇	1	1	yes	characteristic
46	C₂²×C₄	—	2	1	no	—
47	SD₁₆	—	2	4	no	—
48	C₂×C₈	C₂²×C₇	1	2	yes	—
49	C₄⋊C₄	—	2	1	no	—
50	C₄⋊C₄	—	2	1	no	—
51	C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
52	C₂²⋊C₄	—	2	1	no	—
53	C₄²	C₂²×C₇	1	1	yes	characteristic
54	C₂×C₇	—	4	2	no	—
55	C₂×C₇	#121	1	1	yes	characteristic
56	C₂×C₇	C₂×SD₁₆	1	1	yes	characteristic
57	C₂×C₇	#137	1	1	yes	characteristic
58	C₂³	—	2	1	no	—
59	Q₈	—	2	2	no	—
60	Q₈	—	2	2	no	—
61	D₄	—	2	1	no	—
62	D₄	C₇×D₄	1	2	yes	—
63	C₂×C₄	—	2	1	no	—
64	C₂×C₄	—	2	2	no	—
65	C₂×C₄	—	4	2	no	—
66	C₂×C₄	C₇×D₄	1	1	yes	characteristic
67	C₂×C₄	C₇×D₄	1	1	yes	characteristic
68	C₂×C₄	C₂³×C₇	1	1	yes	characteristic
69	C₈	—	2	2	no	—
70	C₇	#461	1	1	yes	characteristic, direct product factor, Sylow
71	C₂²	—	2	2	no	—
72	C₂²	—	4	1	no	—
73	C₂²	—	4	1	no	—
74	C₂²	C₂×C₇×D₄	1	1	yes	characteristic
75	C₄	—	2	1	no	—
76	C₄	—	4	2	no	—
77	C₄	—	4	1	no	—
78	C₄	C₇×SD₁₆	1	2	yes	complemented
79	C₄	C₂×C₇×D₄	1	1	yes	characteristic
80	C₄	C₇×#41	1	1	yes	characteristic
81	C₂	—	4	2	no	—
82	C₂	C₇×#121	1	1	yes	characteristic
83	C₂	C₂×C₇×SD₁₆	1	1	yes	characteristic
84	C₂	C₇×#137	1	1	yes	characteristic

	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
2. C₇×C₄⋊Q₈	C₂	semidirect
4. C₇×C₄⋊C₈	C₂	semidirect
5. C₇×#103	C₂	semidirect
16. #461	C₇	direct (N × H)
30. C₄⋊Q₈	C₂×C₇	semidirect
32. C₄⋊C₈	C₂×C₇	semidirect
33. #103	C₂×C₇	semidirect
41. C₄×C₇	SD₁₆	semidirect
70. C₇	#461	direct (N × H)
78. 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
C₄×C₇×D₄	2
C₇×C₄⋊C₈	2
C₇×#103	2
#461	7

Subgroups — C7×#461 #90107

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#461 #90107