Subgroups — #89923

Order 448.

162 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×D₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#6503	C₂	1	2	yes	maximal, complemented
3	C₂×#1141	C₂	1	2	yes	maximal, complemented
4	C₂×D₅₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈×Dic₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₇×D₄	C₂²	1	2	yes	—
7	C₂×C₂²⋊D₇	—	2	4	no	—
8	C₂×D₂₈	C₂²	1	2	yes	—
9	C₇×D₈	—	2	2	no	—
10	C₂×C₇×C₈	C₂²	1	1	yes	characteristic
11	#1141	—	2	4	no	—
12	C₄×Dic₇	C₂²	1	1	yes	characteristic
13	C₂×C₇⋊C₈	C₂²	1	1	yes	characteristic
14	D₅₆	—	2	2	no	—
15	#493	—	7	1	no	maximal, Sylow
16	C₂³×C₇	—	2	2	no	—
17	C₂²×D₇	—	2	2	no	—
18	C₇×D₄	—	2	2	no	—
19	C₇×D₄	—	2	2	no	—
20	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
21	C₂²⋊D₇	—	4	8	no	—
22	C₂×Dic₇	D₄	1	2	yes	—
23	D₂₈	—	2	2	no	—
24	D₂₈	—	2	2	no	—
25	C₇×C₈	D₄	1	2	yes	complemented
26	C₇⋊C₈	D₄	1	2	yes	complemented
27	C₂×D₈	—	7	2	no	—
28	C₂×D₈	—	7	1	no	—
29	C₂×D₈	D₇	1	1	yes	characteristic, complemented
30	#127	—	7	2	no	—
31	C₄×C₈	—	7	1	no	—
32	C₂²×C₇	—	4	2	no	—
33	C₂²×C₇	—	4	4	no	—
34	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
35	C₂×D₇	—	4	2	no	—
36	C₂×D₇	—	4	4	no	—
37	C₄×C₇	C₂×D₄	1	1	yes	characteristic
38	C₄×C₇	C₂×D₄	1	1	yes	characteristic
39	Dic₇	D₈	1	4	yes	complemented
40	C₂×D₄	—	7	2	no	—
41	C₂×D₄	—	14	4	no	—
42	C₂×D₄	C₂×D₇	1	2	yes	—
43	D₈	—	2	2	no	—
44	D₈	—	14	4	no	—
45	D₈	—	14	2	no	—
46	C₂×C₈	—	7	1	no	—
47	C₂×C₈	C₂×D₇	1	1	yes	characteristic
48	C₄²	—	7	1	no	—
49	C₂×C₇	—	8	2	no	—
50	C₂×C₇	#127	1	1	yes	characteristic
51	C₂×C₇	C₂×D₈	1	2	yes	—
52	D₇	—	8	2	no	—
53	C₂³	—	2	2	no	—
54	C₂³	—	14	2	no	—
55	D₄	—	2	2	no	—
56	D₄	—	2	2	no	—
57	D₄	—	14	2	no	—
58	D₄	—	14	2	no	—
59	D₄	—	28	8	no	—
60	C₂×C₄	—	7	2	no	—
61	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
62	C₈	—	7	2	no	—
63	C₈	C₂²⋊D₇	1	2	yes	complemented
64	C₇	#493	1	1	yes	characteristic, complemented, Sylow
65	C₂²	—	4	4	no	—
66	C₂²	—	4	2	no	—
67	C₂²	—	28	4	no	—
68	C₂²	—	28	2	no	—
69	C₂²	D₄×D₇	1	1	yes	characteristic
70	C₄	—	7	4	no	—
71	C₄	D₄×D₇	1	1	yes	characteristic
72	C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
73	C₂	—	8	2	no	—
74	C₂	—	56	2	no	—
75	C₂	D₇×D₈	1	2	yes	—
76	C₂	#6503	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₇
commutator, cosocle	C₂×C₄×C₇	C₂³
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₂×C₇×D₈	C₂
socle	C₂²×C₇	C₂×D₄
hypercenter	C₂×D₈	D₇
hypocenter	C₇	#493
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×D₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₇×D₈	C₂	semidirect
2. #6503	C₂	semidirect
3. C₂×#1141	C₂	semidirect
4. C₂×D₅₆	C₂	semidirect
5. C₈×Dic₇	C₂	semidirect
25. C₇×C₈	D₄	semidirect
26. C₇⋊C₈	D₄	semidirect
29. C₂×D₈	D₇	semidirect
39. Dic₇	D₈	semidirect
63. C₈	C₂²⋊D₇	semidirect
64. C₇	#493	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×D₈	2
#6503	2
C₂×#1141	2
C₂×D₅₆	2
C₈×Dic₇	2
#493	7