Subgroups — #89965

Order 448.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#135	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₅₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#6402	C₂	1	2	yes	maximal, complemented
4	#6400	C₂	1	2	yes	maximal, complemented
5	#6386	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#41	—	2	1	no	—
7	C₂×D₂₈	—	2	1	no	—
8	C₇×Q₁₆	C₂²	1	1	yes	characteristic, complemented
9	C₇×SD₁₆	—	2	1	no	—
10	C₇×D₈	C₂²	1	1	yes	characteristic, complemented
11	C₂×C₇×C₈	C₂²	1	1	yes	characteristic
12	D₅₆	—	2	1	no	—
13	D₅₆	C₂²	1	2	yes	—
14	C₇⋊C₁₆	C₂²	1	2	yes	complemented
15	#509	—	7	1	no	maximal, Sylow
16	C₂²×D₇	—	4	1	no	—
17	C₇×Q₈	—	2	1	no	—
18	C₇×D₄	—	2	1	no	—
19	C₇×D₄	—	4	1	no	—
20	C₂×C₄×C₇	—	4	1	no	—
21	C₂×C₄×C₇	D₄	1	1	yes	characteristic
22	D₂₈	—	2	2	no	—
23	D₂₈	—	4	1	no	—
24	C₇×C₈	D₄	1	1	yes	characteristic
25	C₇×C₈	C₂³	1	1	yes	characteristic
26	#135	D₇	1	1	yes	characteristic, complemented
27	C₂×D₈	—	7	1	no	—
28	SD₃₂	—	7	2	no	—
29	D₁₆	—	7	2	no	—
30	M₃₂	—	7	1	no	—
31	C₂²×C₇	—	4	1	no	—
32	C₂²×C₇	D₈	1	1	yes	characteristic
33	C₂×D₇	—	4	2	no	—
34	C₂×D₇	—	8	2	no	—
35	C₄×C₇	—	4	1	no	—
36	C₄×C₇	D₈	1	1	yes	characteristic
37	C₄×C₇	C₂×D₄	1	1	yes	characteristic
38	#41	—	2	1	no	—
39	C₂×D₄	—	14	1	no	—
40	Q₁₆	C₂×D₇	1	1	yes	characteristic, complemented
41	SD₁₆	—	2	1	no	—
42	D₈	—	7	2	no	—
43	D₈	—	14	1	no	—
44	D₈	C₂×D₇	1	1	yes	characteristic, complemented
45	C₂×C₈	C₂×D₇	1	1	yes	characteristic
46	C₁₆	—	7	2	no	—
47	C₂×C₇	—	2	1	no	—
48	C₂×C₇	—	8	1	no	—
49	C₂×C₇	C₂×D₈	1	1	yes	characteristic
50	D₇	—	8	2	no	—
51	C₂³	—	28	1	no	—
52	Q₈	—	2	1	no	—
53	D₄	—	2	1	no	—
54	D₄	—	4	1	no	—
55	D₄	—	14	2	no	—
56	D₄	—	28	1	no	—
57	C₂×C₄	—	4	1	no	—
58	C₂×C₄	C₂²⋊D₇	1	1	yes	characteristic
59	C₈	C₂²⋊D₇	1	1	yes	characteristic
60	C₈	C₂²×D₇	1	1	yes	characteristic
61	C₇	#509	1	1	yes	characteristic, complemented, Sylow
62	C₂²	—	4	1	no	—
63	C₂²	—	28	2	no	—
64	C₂²	—	56	2	no	—
65	C₂²	#1141	1	1	yes	characteristic
66	C₄	—	4	1	no	—
67	C₄	#1141	1	1	yes	characteristic
68	C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
69	C₂	—	2	1	no	—
70	C₂	—	8	1	no	—
71	C₂	—	56	2	no	—
72	C₂	C₂×#1141	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#1141
commutator, cosocle	C₇×C₈	C₂³
Frattini	C₈	C₂²×D₇
Fitting	C₇×#135	C₂
socle	C₂×C₇	C₂×D₈
hypercenter	#135	D₇
hypocenter	C₇	#509
LCS	G ▷ C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #135

Complements

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

N	H ≅ G/N	type
1. C₇×#135	C₂	semidirect
2. C₂×D₅₆	C₂	semidirect
3. #6402	C₂	semidirect
4. #6400	C₂	semidirect
5. #6386	C₂	semidirect
8. C₇×Q₁₆	C₂²	semidirect
10. C₇×D₈	C₂²	semidirect
14. C₇⋊C₁₆	C₂²	semidirect
26. #135	D₇	semidirect
40. Q₁₆	C₂×D₇	semidirect
44. D₈	C₂×D₇	semidirect
61. C₇	#509	semidirect

Maximal subgroups

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

subgroup	index
C₇×#135	2
C₂×D₅₆	2
#6402	2
#6400	2
#6386	2
#509	7