Subgroups — C₇×#405 #90051

Order 448.

130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₇×C₈	C₂	1	1	yes	maximal, characteristic
3	C₇×#117	C₂	1	1	yes	maximal, characteristic
4	C₇×#97	C₂	1	2	yes	maximal, complemented
5	C₄×C₇×C₈	C₂	1	2	yes	maximal, complemented
6	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
7	C₇×M₁₆	C₄	1	4	yes	complemented
8	C₂×C₇×C₈	C₄	1	4	yes	complemented
9	C₂×C₇×C₈	C₂²	1	2	yes	—
10	C₂×C₇×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₇×C₈	C₂²	1	1	yes	characteristic
12	C₇×C₄⋊C₄	C₄	1	2	yes	—
13	C₇×C₂²⋊C₄	C₄	1	2	yes	—
14	C₄²×C₇	C₂²	1	2	yes	—
15	#405	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
16	C₂³×C₇	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×C₇	C₂×C₄	1	4	yes	—
18	C₂×C₄×C₇	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₇	C₂×C₄	1	2	yes	—
20	C₂×C₄×C₇	C₂×C₄	1	2	yes	—
21	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
22	C₇×C₈	C₂×C₄	1	2	yes	—
23	C₇×C₈	C₂×C₄	1	2	yes	—
24	C₇×C₈	C₂×C₄	1	4	yes	—
25	C₂×M₁₆	C₂×C₇	1	1	yes	characteristic
26	C₂²×C₈	C₂×C₇	1	1	yes	characteristic
27	#117	C₂×C₇	1	1	yes	characteristic
28	#97	C₂×C₇	1	2	yes	complemented
29	C₄×C₈	C₂×C₇	1	2	yes	complemented
30	C₂²×C₇	—	2	2	no	—
31	C₂²×C₇	C₄²	1	2	yes	—
32	C₂²×C₇	C₂²×C₄	1	1	yes	characteristic
33	C₄×C₇	—	2	4	no	—
34	C₄×C₇	C₄²	1	2	yes	—
35	C₄×C₇	C₂²×C₄	1	1	yes	characteristic
36	C₄×C₇	C₂²×C₄	1	1	yes	characteristic
37	C₂²×C₄	C₂²×C₇	1	1	yes	characteristic
38	M₁₆	C₄×C₇	1	4	yes	complemented
39	C₂×C₈	C₄×C₇	1	4	yes	complemented
40	C₂×C₈	C₂²×C₇	1	1	yes	characteristic
41	C₂×C₈	C₂²×C₇	1	2	yes	—
42	C₂×C₈	C₂²×C₇	1	1	yes	characteristic
43	C₄⋊C₄	C₄×C₇	1	2	yes	—
44	C₂²⋊C₄	C₄×C₇	1	2	yes	—
45	C₄²	C₂²×C₇	1	2	yes	—
46	C₂×C₇	—	2	2	no	—
47	C₂×C₇	C₂×C₄²	1	1	yes	characteristic
48	C₂×C₇	#131	1	2	yes	—
49	C₂³	C₂×C₄×C₇	1	1	yes	characteristic
50	C₂×C₄	C₂×C₄×C₇	1	4	yes	—
51	C₂×C₄	C₂×C₄×C₇	1	1	yes	characteristic
52	C₂×C₄	C₂×C₄×C₇	1	2	yes	—
53	C₂×C₄	C₂×C₄×C₇	1	2	yes	—
54	C₂×C₄	C₂³×C₇	1	1	yes	characteristic
55	C₈	C₂×C₄×C₇	1	2	yes	—
56	C₈	C₂×C₄×C₇	1	2	yes	—
57	C₈	C₂×C₄×C₇	1	4	yes	—
58	C₇	#405	1	1	yes	characteristic, direct product factor, Sylow
59	C₂²	—	2	2	no	—
60	C₂²	C₄²×C₇	1	2	yes	—
61	C₂²	C₂²×C₄×C₇	1	1	yes	characteristic
62	C₄	—	2	4	no	—
63	C₄	C₄²×C₇	1	2	yes	—
64	C₄	C₂²×C₄×C₇	1	1	yes	characteristic
65	C₄	C₂²×C₄×C₇	1	1	yes	characteristic
66	C₂	—	2	2	no	—
67	C₂	C₂×C₄²×C₇	1	1	yes	characteristic
68	C₂	C₇×#131	1	2	yes	—

	subgroup	G/H
center	C₂×C₇×C₈	C₂²
commutator	C₂	C₂×C₄²×C₇
Frattini, cosocle	C₂×C₄	C₂³×C₇
socle	C₂²×C₇	C₂²×C₄
prime radical	C₂³×C₇	C₂×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₇×#97	C₂	semidirect
5. C₄×C₇×C₈	C₂	semidirect
7. C₇×M₁₆	C₄	semidirect
8. C₂×C₇×C₈	C₄	semidirect
15. #405	C₇	direct (N × H)
28. #97	C₂×C₇	semidirect
29. C₄×C₈	C₂×C₇	semidirect
38. M₁₆	C₄×C₇	semidirect
39. C₂×C₈	C₄×C₇	semidirect
58. C₇	#405	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×M₁₆	2
C₂²×C₇×C₈	2
C₇×#117	2
C₇×#97	2
C₄×C₇×C₈	2
#405	7

Subgroups — C7×#405 #90051

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#405 #90051