Subgroups — C₇×#407 #90053

Order 448.

190 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄×C₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×M₁₆	C₂	1	2	yes	maximal, complemented
3	C₇×C₂²⋊C₈	C₂	1	4	yes	maximal, complemented
4	C₂⁴×C₇	C₄	1	1	yes	characteristic
5	C₂²×C₄×C₇	—	2	4	no	—
6	C₂²×C₄×C₇	C₄	1	1	yes	characteristic
7	C₂²×C₄×C₇	C₄	1	2	yes	—
8	C₂²×C₄×C₇	C₂²	1	2	yes	—
9	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
10	C₇×M₁₆	—	2	4	no	—
11	C₂×C₇×C₈	C₂²	1	4	yes	complemented
12	#407	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂³×C₇	—	2	2	no	—
14	C₂³×C₇	—	2	4	no	—
15	C₂³×C₇	C₂×C₄	1	1	yes	characteristic
16	C₂³×C₇	C₂×C₄	1	2	yes	—
17	C₂×C₄×C₇	—	2	2	no	—
18	C₂×C₄×C₇	—	2	4	no	—
19	C₂×C₄×C₇	—	2	4	no	—
20	C₂×C₄×C₇	C₂×C₄	1	2	yes	—
21	C₂×C₄×C₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×C₇	D₄	1	4	yes	—
23	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
24	C₇×C₈	—	2	4	no	—
25	C₂³×C₄	C₂×C₇	1	1	yes	characteristic
26	C₂×M₁₆	C₂×C₇	1	2	yes	complemented
27	C₂²⋊C₈	C₂×C₇	1	4	yes	complemented
28	C₂²×C₇	—	2	8	no	—
29	C₂²×C₇	—	2	2	no	—
30	C₂²×C₇	—	2	2	no	—
31	C₂²×C₇	—	2	2	no	—
32	C₂²×C₇	C₂²⋊C₄	1	2	yes	—
33	C₂²×C₇	M₁₆	1	4	yes	complemented
34	C₂²×C₇	C₂²×C₄	1	1	yes	characteristic
35	C₄×C₇	—	2	2	no	—
36	C₄×C₇	C₂²⋊C₄	1	2	yes	—
37	C₄×C₇	C₂×D₄	1	2	yes	—
38	C₂⁴	C₄×C₇	1	1	yes	characteristic
39	C₂²×C₄	—	2	4	no	—
40	C₂²×C₄	C₄×C₇	1	1	yes	characteristic
41	C₂²×C₄	C₄×C₇	1	2	yes	—
42	C₂²×C₄	C₂²×C₇	1	2	yes	—
43	C₂²×C₄	C₂²×C₇	1	1	yes	characteristic
44	M₁₆	—	2	4	no	—
45	C₂×C₈	C₂²×C₇	1	4	yes	complemented
46	C₂×C₇	—	2	2	no	—
47	C₂×C₇	—	2	4	no	—
48	C₂×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₇	C₂×M₁₆	1	2	yes	—
50	C₂³	—	2	2	no	—
51	C₂³	—	2	4	no	—
52	C₂³	C₂×C₄×C₇	1	1	yes	characteristic
53	C₂³	C₂×C₄×C₇	1	2	yes	—
54	C₂×C₄	—	2	2	no	—
55	C₂×C₄	—	2	4	no	—
56	C₂×C₄	—	2	4	no	—
57	C₂×C₄	C₂×C₄×C₇	1	2	yes	—
58	C₂×C₄	C₂×C₄×C₇	1	1	yes	characteristic
59	C₂×C₄	C₇×D₄	1	4	yes	—
60	C₂×C₄	C₂³×C₇	1	1	yes	characteristic
61	C₈	—	2	4	no	—
62	C₇	#407	1	1	yes	characteristic, direct product factor, Sylow
63	C₂²	—	2	8	no	—
64	C₂²	—	2	2	no	—
65	C₂²	—	2	2	no	—
66	C₂²	—	2	2	no	—
67	C₂²	C₇×C₂²⋊C₄	1	2	yes	—
68	C₂²	C₇×M₁₆	1	4	yes	complemented
69	C₂²	C₂²×C₄×C₇	1	1	yes	characteristic
70	C₄	—	2	2	no	—
71	C₄	C₇×C₂²⋊C₄	1	2	yes	—
72	C₄	C₂×C₇×D₄	1	2	yes	—
73	C₂	—	2	2	no	—
74	C₂	—	2	4	no	—
75	C₂	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
76	C₂	C₂×C₇×M₁₆	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₇	C₂³
commutator	C₂²	C₂²×C₄×C₇
Frattini, cosocle	C₂×C₄	C₂³×C₇
socle	C₂²×C₇	C₂²×C₄
prime radical	C₂⁴×C₇	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₇×M₁₆	C₂	semidirect
3. C₇×C₂²⋊C₈	C₂	semidirect
11. C₂×C₇×C₈	C₂²	semidirect
12. #407	C₇	direct (N × H)
26. C₂×M₁₆	C₂×C₇	semidirect
27. C₂²⋊C₈	C₂×C₇	semidirect
33. C₂²×C₇	M₁₆	semidirect
45. C₂×C₈	C₂²×C₇	semidirect
62. C₇	#407	direct (N × H)
68. C₂²	C₇×M₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₄×C₇	2
C₂×C₇×M₁₆	2
C₇×C₂²⋊C₈	2
#407	7

Subgroups — C7×#407 #90053

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#407 #90053