Subgroups — #89644

Order 448.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₅₆	C₂	1	1	yes	maximal, characteristic
2	#6456	C₂	1	2	yes	maximal, complemented
3	C₇×#107	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈×Dic₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#6383	C₂	1	2	yes	maximal, complemented
6	C₂×D₂₈	C₂²	1	2	yes	—
7	C₂×C₄×D₇	—	2	2	no	—
8	C₂×C₇×C₈	C₂²	1	1	yes	characteristic
9	C₇×C₄⋊C₄	C₂²	1	2	yes	—
10	(C₂×D₇)⋊C₄	—	2	2	no	—
11	C₄×Dic₇	C₂²	1	1	yes	characteristic
12	C₂×C₇⋊C₈	C₂²	1	1	yes	characteristic
13	D₅₆	C₄	1	4	yes	complemented
14	C₄×D₈	—	7	1	no	maximal, Sylow
15	C₂²×D₇	—	2	2	no	—
16	C₂×C₄×C₇	—	2	2	no	—
17	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
18	C₂×Dic₇	D₄	1	1	yes	characteristic
19	C₂×Dic₇	D₄	1	1	yes	characteristic
20	D₂₈	—	2	2	no	—
21	D₂₈	C₂×C₄	1	4	yes	complemented
22	C₄×D₇	—	4	4	no	—
23	C₇×C₈	C₂×C₄	1	2	yes	complemented
24	C₇⋊C₈	—	2	1	no	—
25	C₂×D₈	—	7	1	no	—
26	C₄×D₄	—	7	2	no	—
27	#107	D₇	1	1	yes	characteristic, complemented
28	#102	—	7	2	no	—
29	C₄×C₈	—	7	1	no	—
30	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
31	C₂×D₇	—	2	4	no	—
32	C₂×D₇	—	4	2	no	—
33	C₂×D₇	—	4	2	no	—
34	C₄×C₇	—	4	2	no	—
35	C₄×C₇	C₂²×C₄	1	1	yes	characteristic
36	C₄×C₇	#41	1	1	yes	characteristic
37	Dic₇	—	2	1	no	—
38	Dic₇	D₈	1	2	yes	complemented
39	C₂×D₄	—	7	2	no	—
40	C₂²×C₄	—	14	2	no	—
41	D₈	—	7	4	no	—
42	C₂×C₈	—	7	1	no	—
43	C₂×C₈	C₂×D₇	1	1	yes	characteristic
44	C₄⋊C₄	C₂×D₇	1	2	yes	—
45	C₂²⋊C₄	—	14	2	no	—
46	C₄²	—	7	1	no	—
47	C₂×C₇	C₄×D₄	1	1	yes	characteristic
48	C₂×C₇	C₂×D₈	1	1	yes	characteristic
49	C₂×C₇	#135	1	1	yes	characteristic
50	D₇	—	4	4	no	—
51	C₂³	—	14	2	no	—
52	D₄	—	7	4	no	—
53	D₄	—	14	2	no	—
54	C₂×C₄	—	2	2	no	—
55	C₂×C₄	—	7	1	no	—
56	C₂×C₄	—	7	1	no	—
57	C₂×C₄	—	28	4	no	—
58	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
59	C₈	—	14	1	no	—
60	C₈	C₄×D₇	1	2	yes	complemented
61	C₇	C₄×D₈	1	1	yes	characteristic, complemented, Sylow
62	C₂²	—	14	4	no	—
63	C₂²	—	28	2	no	—
64	C₂²	—	28	2	no	—
65	C₂²	D₄×D₇	1	1	yes	characteristic
66	C₄	—	4	2	no	—
67	C₄	—	7	2	no	—
68	C₄	—	14	1	no	—
69	C₄	C₂×C₄×D₇	1	1	yes	characteristic
70	C₄	#1161	1	1	yes	characteristic
71	C₂	—	28	4	no	—
72	C₂	#6456	1	1	yes	characteristic
73	C₂	D₇×D₈	1	1	yes	characteristic
74	C₂	#6482	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₇
commutator	C₄×C₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₇×#107	C₂
socle	C₂²×C₇	C₂×D₄
prime radical	C₂×D₅₆	C₂
cosocle	C₂×C₄×C₇	C₂³
hypercenter	#107	D₇
hypocenter	C₇	C₄×D₈
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

Complements

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

N	H ≅ G/N	type
2. #6456	C₂	semidirect
3. C₇×#107	C₂	semidirect
4. C₈×Dic₇	C₂	semidirect
5. #6383	C₂	semidirect
13. D₅₆	C₄	semidirect
21. D₂₈	C₂×C₄	semidirect
23. C₇×C₈	C₂×C₄	semidirect
27. #107	D₇	semidirect
38. Dic₇	D₈	semidirect
60. C₈	C₄×D₇	semidirect
61. C₇	C₄×D₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₅₆	2
#6456	2
C₇×#107	2
C₈×Dic₇	2
#6383	2
C₄×D₈	7