Subgroups — #90444

Order 448.

248 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#6554	C₂	1	2	yes	maximal, complemented
2	C₇×#131	C₂	1	1	yes	maximal, characteristic, complemented
3	#6472	C₂	1	6	yes	maximal, complemented
4	C₂×Dic₂₈	C₂	1	3	yes	maximal, complemented
5	#6467	C₂	1	3	yes	maximal, complemented
6	C₇×#41	C₂²	1	1	yes	characteristic
7	Q₈×D₇	—	2	2	no	—
8	#1159	—	2	6	no	—
9	#1157	C₂²	1	6	yes	—
10	C₂×Dic₁₄	C₂²	1	6	yes	—
11	C₇×M₁₆	C₂²	1	3	yes	—
12	C₂×C₇×C₈	C₂²	1	3	yes	—
13	Dic₂₈	C₂²	1	3	yes	—
14	Dic₂₈	C₂²	1	6	yes	—
15	D₅₆	C₂²	1	1	yes	characteristic
16	#1132	C₂²	1	6	yes	—
17	#578	—	7	1	no	maximal, Sylow
18	C₇×Q₈	D₄	1	1	yes	characteristic
19	C₇×D₄	D₄	1	3	yes	—
20	C₂×C₄×C₇	C₂³	1	3	yes	—
21	C₂²⋊D₇	—	2	6	no	—
22	C₂×Dic₇	—	2	6	no	—
23	D₂₈	C₂³	1	2	yes	—
24	C₄×D₇	—	2	6	no	—
25	Dic₁₄	—	2	6	no	—
26	Dic₁₄	C₂³	1	6	yes	—
27	C₇×C₈	C₂³	1	3	yes	—
28	C₇×C₈	C₂³	1	1	yes	characteristic
29	2₋¹⁺⁴	—	7	2	no	—
30	#137	—	7	6	no	—
31	#135	—	7	3	no	—
32	C₂×Q₁₆	—	7	3	no	—
33	#131	D₇	1	1	yes	characteristic, complemented
34	C₂²×C₇	C₂×D₄	1	3	yes	—
35	C₂×D₇	—	2	2	no	—
36	C₄×C₇	C₂×D₄	1	3	yes	—
37	C₄×C₇	C₂⁴	1	1	yes	characteristic
38	Dic₇	—	2	6	no	—
39	#41	—	7	6	no	—
40	#41	—	14	6	no	—
41	#41	C₂×D₇	1	1	yes	characteristic
42	C₂×Q₈	—	7	6	no	—
43	C₂×Q₈	—	14	2	no	—
44	Q₁₆	—	7	3	no	—
45	Q₁₆	—	7	6	no	—
46	SD₁₆	—	7	6	no	—
47	D₈	—	7	1	no	—
48	M₁₆	C₂×D₇	1	3	yes	—
49	C₂×C₈	C₂×D₇	1	3	yes	—
50	C₂×C₇	—	2	3	no	—
51	C₂×C₇	C₂²×D₄	1	1	yes	characteristic
52	D₇	—	4	2	no	—
53	Q₈	—	7	6	no	—
54	Q₈	—	14	6	no	—
55	Q₈	D₂₈	1	1	yes	characteristic
56	D₄	—	7	2	no	—
57	D₄	—	14	6	no	—
58	D₄	D₂₈	1	3	yes	—
59	C₂×C₄	—	14	6	no	—
60	C₂×C₄	—	14	6	no	—
61	C₂×C₄	C₂²×D₇	1	3	yes	—
62	C₈	C₂²×D₇	1	1	yes	characteristic
63	C₈	C₂²×D₇	1	3	yes	—
64	C₇	#578	1	1	yes	characteristic, complemented, Sylow
65	C₂²	—	14	2	no	—
66	C₂²	C₂×D₂₈	1	3	yes	—
67	C₄	—	14	6	no	—
68	C₄	C₂×D₂₈	1	3	yes	—
69	C₄	C₂³×D₇	1	1	yes	characteristic
70	C₂	—	2	3	no	—
71	C₂	—	28	2	no	—
72	C₂	C₂²×D₂₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²×D₂₈
commutator, cosocle	C₄×C₇	C₂⁴
Frattini	C₄	C₂³×D₇
Fitting	C₇×#131	C₂
socle	C₂×C₇	C₂²×D₄
hypercenter	#131	D₇
hypocenter	C₇	#578
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131

Complements

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

N	H ≅ G/N	type
1. #6554	C₂	semidirect
2. C₇×#131	C₂	semidirect
3. #6472	C₂	semidirect
4. C₂×Dic₂₈	C₂	semidirect
5. #6467	C₂	semidirect
33. #131	D₇	semidirect
64. C₇	#578	semidirect

Maximal subgroups

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

subgroup	index
#6554	2
C₇×#131	2
#6472	2
C₂×Dic₂₈	2
#6467	2
#578	7