Subgroups — #89483

Order 448.

132 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	C₄×D₂₈	C₂	1	2	yes	maximal, complemented
3	C₇×#97	C₂	1	1	yes	maximal, characteristic, complemented
4	#6395	C₂	1	2	yes	maximal, complemented
5	#6391	C₂	1	1	yes	maximal, characteristic, 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	C₂×C₇×C₈	C₂²	1	1	yes	characteristic
10	C₄²×C₇	C₂²	1	1	yes	characteristic
11	(C₂×D₇)⋊C₄	—	2	2	no	—
12	C₄⋊Dic₇	C₂²	1	2	yes	—
13	D₅₆	C₄	1	4	yes	complemented
14	#442	—	7	1	no	maximal, Sylow
15	C₂²×D₇	—	2	2	no	—
16	C₂×C₄×C₇	D₄	1	2	yes	—
17	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
18	C₂×Dic₇	—	2	2	no	—
19	D₂₈	—	2	2	no	—
20	D₂₈	C₂×C₄	1	4	yes	complemented
21	C₄×D₇	—	4	4	no	—
22	C₇×C₈	—	2	1	no	—
23	C₇×C₈	C₂×C₄	1	2	yes	complemented
24	C₂×D₈	—	7	1	no	—
25	C₄×D₄	—	7	2	no	—
26	#106	—	7	1	no	—
27	#102	—	7	2	no	—
28	#97	D₇	1	1	yes	characteristic, complemented
29	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
30	C₂×D₇	—	2	4	no	—
31	C₂×D₇	—	4	2	no	—
32	C₂×D₇	—	4	2	no	—
33	C₄×C₇	—	2	2	no	—
34	C₄×C₇	C₂²×C₄	1	1	yes	characteristic
35	C₄×C₇	#41	1	1	yes	characteristic
36	Dic₇	—	4	2	no	—
37	C₂×D₄	—	7	2	no	—
38	C₂²×C₄	—	14	2	no	—
39	D₈	—	7	4	no	—
40	C₂×C₈	C₂×D₇	1	1	yes	characteristic
41	C₂×C₈	C₂×D₇	1	1	yes	characteristic
42	C₄⋊C₄	—	7	2	no	—
43	C₂²⋊C₄	—	14	2	no	—
44	C₄²	C₂×D₇	1	1	yes	characteristic
45	C₂×C₇	C₄×D₄	1	1	yes	characteristic
46	C₂×C₇	AGL(1,ℤ/8)	1	2	yes	—
47	D₇	—	4	4	no	—
48	C₂³	—	14	2	no	—
49	D₄	—	7	4	no	—
50	D₄	—	14	2	no	—
51	C₂×C₄	—	14	2	no	—
52	C₂×C₄	—	28	4	no	—
53	C₂×C₄	D₂₈	1	2	yes	—
54	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
55	C₈	—	2	1	no	—
56	C₈	C₄×D₇	1	2	yes	complemented
57	C₇	#442	1	1	yes	characteristic, complemented, Sylow
58	C₂²	—	14	4	no	—
59	C₂²	—	28	2	no	—
60	C₂²	—	28	2	no	—
61	C₂²	C₂×D₂₈	1	1	yes	characteristic
62	C₄	—	2	2	no	—
63	C₄	—	28	2	no	—
64	C₄	C₂×C₄×D₇	1	1	yes	characteristic
65	C₄	#1157	1	1	yes	characteristic
66	C₂	—	28	4	no	—
67	C₂	C₄×D₂₈	1	1	yes	characteristic
68	C₂	#6471	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₂₈
commutator	C₄×C₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₇×#97	C₂
socle	C₂²×C₇	C₂×D₄
prime radical	C₂×D₅₆	C₂
cosocle	C₂×C₄×C₇	C₂³
hypercenter	#97	D₇
hypocenter	C₇	#442
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #97

Complements

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

N	H ≅ G/N	type
2. C₄×D₂₈	C₂	semidirect
3. C₇×#97	C₂	semidirect
4. #6395	C₂	semidirect
5. #6391	C₂	semidirect
13. D₅₆	C₄	semidirect
20. D₂₈	C₂×C₄	semidirect
23. C₇×C₈	C₂×C₄	semidirect
28. #97	D₇	semidirect
56. C₈	C₄×D₇	semidirect
57. C₇	#442	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
C₄×D₂₈	2
C₇×#97	2
#6395	2
#6391	2
#442	7