Subgroups — C₇×#391 #90037

Order 448.

186 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄²×C₇	C₂	1	1	yes	maximal, characteristic
4	C₇×#95	C₂	1	4	yes	maximal
5	C₂×C₇×Q₈	—	2	2	no	—
6	C₂×C₇×Q₈	—	2	2	no	—
7	C₂×C₇×Q₈	C₄	1	4	yes	complemented
8	C₂²×C₄×C₇	C₂²	1	2	yes	—
9	C₂²×C₄×C₇	C₂²	1	2	yes	—
10	C₂²×C₄×C₇	C₂²	1	2	yes	—
11	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
12	C₇×C₄⋊C₄	—	2	2	no	—
13	C₄²×C₇	—	2	2	no	—
14	#391	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
15	C₂³×C₇	C₂³	1	1	yes	characteristic
16	C₇×Q₈	—	2	8	no	—
17	C₂×C₄×C₇	—	2	2	no	—
18	C₂×C₄×C₇	—	2	2	no	—
19	C₂×C₄×C₇	—	2	2	no	—
20	C₂×C₄×C₇	—	2	2	no	—
21	C₂×C₄×C₇	—	2	2	no	—
22	C₂×C₄×C₇	—	2	4	no	—
23	C₂×C₄×C₇	C₂×C₄	1	4	yes	—
24	C₂×C₄×C₇	C₂×C₄	1	2	yes	—
25	C₂×C₄×C₇	D₄	1	2	yes	—
26	C₂×C₄×C₇	D₄	1	2	yes	—
27	C₂×C₄×C₇	Q₈	1	4	yes	complemented
28	C₂²×Q₈	C₂×C₇	1	1	yes	characteristic
29	C₂×C₄⋊C₄	C₂×C₇	1	1	yes	characteristic
30	C₂×C₄²	C₂×C₇	1	1	yes	characteristic
31	#95	C₂×C₇	1	4	yes	—
32	C₂²×C₇	C₂²×C₄	1	1	yes	characteristic
33	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
34	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₇	C₂×Q₈	1	2	yes	—
36	C₂²×C₇	#41	1	2	yes	—
37	C₄×C₇	—	2	4	no	—
38	C₄×C₇	—	2	4	no	—
39	C₄×C₇	—	4	2	no	—
40	C₄×C₇	C₂²⋊C₄	1	4	yes	—
41	C₂×Q₈	—	2	2	no	—
42	C₂×Q₈	—	2	2	no	—
43	C₂×Q₈	C₄×C₇	1	4	yes	complemented
44	C₂²×C₄	C₂²×C₇	1	2	yes	—
45	C₂²×C₄	C₂²×C₇	1	2	yes	—
46	C₂²×C₄	C₂²×C₇	1	2	yes	—
47	C₂²×C₄	C₂²×C₇	1	1	yes	characteristic
48	C₄⋊C₄	—	2	2	no	—
49	C₄²	—	2	2	no	—
50	C₂×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₇	C₄×Q₈	1	2	yes	—
52	C₂×C₇	C₂²⋊Q₈	1	2	yes	—
53	C₂×C₇	#124	1	1	yes	characteristic
54	C₂×C₇	C₄⋊Q₈	1	1	yes	characteristic
55	C₂³	C₂³×C₇	1	1	yes	characteristic
56	Q₈	—	2	8	no	—
57	C₂×C₄	—	2	2	no	—
58	C₂×C₄	—	2	2	no	—
59	C₂×C₄	—	2	2	no	—
60	C₂×C₄	—	2	2	no	—
61	C₂×C₄	—	2	2	no	—
62	C₂×C₄	—	2	4	no	—
63	C₂×C₄	C₂×C₄×C₇	1	4	yes	—
64	C₂×C₄	C₂×C₄×C₇	1	2	yes	—
65	C₂×C₄	C₇×D₄	1	2	yes	—
66	C₂×C₄	C₇×D₄	1	2	yes	—
67	C₂×C₄	C₇×Q₈	1	4	yes	complemented
68	C₇	#391	1	1	yes	characteristic, direct product factor, Sylow
69	C₂²	C₂²×C₄×C₇	1	1	yes	characteristic
70	C₂²	C₂×C₇×D₄	1	1	yes	characteristic
71	C₂²	C₂×C₇×D₄	1	1	yes	characteristic
72	C₂²	C₂×C₇×Q₈	1	2	yes	—
73	C₂²	C₇×#41	1	2	yes	—
74	C₄	—	2	4	no	—
75	C₄	—	2	4	no	—
76	C₄	—	4	2	no	—
77	C₄	C₇×C₂²⋊C₄	1	4	yes	—
78	C₂	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
79	C₂	C₄×C₇×Q₈	1	2	yes	—
80	C₂	C₇×C₂²⋊Q₈	1	2	yes	—
81	C₂	C₇×#124	1	1	yes	characteristic
82	C₂	C₇×C₄⋊Q₈	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
7. C₂×C₇×Q₈	C₄	semidirect
14. #391	C₇	direct (N × H)
27. C₂×C₄×C₇	Q₈	semidirect
43. C₂×Q₈	C₄×C₇	semidirect
67. C₂×C₄	C₇×Q₈	semidirect
68. C₇	#391	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₇×Q₈	2
C₂×C₇×C₄⋊C₄	2
C₂×C₄²×C₇	2
C₇×#95	2
#391	7

Subgroups — C7×#391 #90037

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#391 #90037