Subgroups — C₇×#410 #90056

Order 448.

158 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#41	C₂	1	1	yes	maximal, characteristic
2	C₇×#117	C₂	1	2	yes	maximal, complemented
3	C₇×#99	C₂	1	4	yes	maximal, complemented
4	C₇×#41	—	2	2	no	—
5	C₇×#41	—	2	2	no	—
6	C₂×C₇×Q₈	C₄	1	1	yes	characteristic, complemented
7	C₂×C₇×D₄	C₄	1	1	yes	characteristic, complemented
8	C₂×C₇×D₄	C₂²	1	2	yes	—
9	C₂²×C₄×C₇	C₄	1	2	yes	complemented
10	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
11	C₇×C₄⋊C₄	—	2	2	no	—
12	C₇×C₂²⋊C₄	C₂²	1	4	yes	—
13	C₄²×C₇	—	2	2	no	—
14	#410	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
15	C₂³×C₇	C₂×C₄	1	2	yes	—
16	C₂³×C₇	C₂³	1	1	yes	characteristic
17	C₇×Q₈	—	2	2	no	—
18	C₇×D₄	—	2	4	no	—
19	C₇×D₄	—	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	2	yes	—
24	C₂×C₄×C₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×C₇	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×C₇	D₄	1	2	yes	—
27	C₂×C₄×C₇	D₄	1	2	yes	—
28	C₂×#41	C₂×C₇	1	1	yes	characteristic
29	#117	C₂×C₇	1	2	yes	complemented
30	#99	C₂×C₇	1	4	yes	complemented
31	C₂²×C₇	—	2	2	no	—
32	C₂²×C₇	—	4	1	no	—
33	C₂²×C₇	—	4	2	no	—
34	C₂²×C₇	C₂²×C₄	1	1	yes	characteristic
35	C₂²×C₇	C₂×D₄	1	2	yes	—
36	C₄×C₇	—	2	2	no	—
37	C₄×C₇	—	4	4	no	—
38	C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
39	C₄×C₇	C₂²⋊C₄	1	2	yes	—
40	C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
41	#41	—	2	2	no	—
42	#41	—	2	2	no	—
43	C₂×Q₈	C₄×C₇	1	1	yes	characteristic, complemented
44	C₂×D₄	C₄×C₇	1	1	yes	characteristic, complemented
45	C₂×D₄	C₂²×C₇	1	2	yes	—
46	C₂²×C₄	C₄×C₇	1	2	yes	complemented
47	C₂²×C₄	C₂²×C₇	1	1	yes	characteristic
48	C₄⋊C₄	—	2	2	no	—
49	C₂²⋊C₄	C₂²×C₇	1	4	yes	—
50	C₄²	—	2	2	no	—
51	C₂×C₇	—	2	2	no	—
52	C₂×C₇	—	2	1	no	—
53	C₂×C₇	—	4	2	no	—
54	C₂×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	C₂³	C₂×C₄×C₇	1	2	yes	—
56	C₂³	C₂³×C₇	1	1	yes	characteristic
57	Q₈	—	2	2	no	—
58	D₄	—	2	4	no	—
59	D₄	—	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	2	yes	—
64	C₂×C₄	C₂×C₄×C₇	1	1	yes	characteristic
65	C₂×C₄	C₂×C₄×C₇	1	1	yes	characteristic
66	C₂×C₄	C₇×D₄	1	2	yes	—
67	C₂×C₄	C₇×D₄	1	2	yes	—
68	C₇	#410	1	1	yes	characteristic, direct product factor, Sylow
69	C₂²	—	2	2	no	—
70	C₂²	—	4	1	no	—
71	C₂²	—	4	2	no	—
72	C₂²	C₂²×C₄×C₇	1	1	yes	characteristic
73	C₂²	C₂×C₇×D₄	1	2	yes	—
74	C₄	—	2	2	no	—
75	C₄	—	4	4	no	—
76	C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
77	C₄	C₇×C₂²⋊C₄	1	1	yes	characteristic
78	C₄	C₇×C₂²⋊C₄	1	2	yes	—
79	C₂	—	2	2	no	—
80	C₂	—	2	1	no	—
81	C₂	—	4	2	no	—
82	C₂	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₇	C₂²⋊C₄
commutator	C₂²	C₂²×C₄×C₇
Frattini, cosocle	C₂³	C₂³×C₇
socle	C₂×C₇	C₂×C₂²⋊C₄
prime radical	C₂×C₇×#41	C₂
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₇ ◁ C₂²×C₄×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇×#117	C₂	semidirect
3. C₇×#99	C₂	semidirect
6. C₂×C₇×Q₈	C₄	semidirect
7. C₂×C₇×D₄	C₄	semidirect
9. C₂²×C₄×C₇	C₄	semidirect
14. #410	C₇	direct (N × H)
29. #117	C₂×C₇	semidirect
30. #99	C₂×C₇	semidirect
43. C₂×Q₈	C₄×C₇	semidirect
44. C₂×D₄	C₄×C₇	semidirect
46. C₂²×C₄	C₄×C₇	semidirect
68. C₇	#410	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×#41	2
C₇×#117	2
C₇×#99	2
#410	7

Subgroups — C7×#410 #90056

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#410 #90056