Subgroups — #6471

Order 224.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#1157	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₂₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	D₅₆	C₂	1	2	yes	maximal, complemented
5	#1132	C₂	1	2	yes	maximal, complemented
6	C₂²×D₇	—	2	1	no	—
7	C₂×C₄×C₇	C₂²	1	1	yes	characteristic
8	C₂²⋊D₇	—	2	1	no	—
9	D₂₈	—	2	1	no	—
10	D₂₈	C₂²	1	2	yes	—
11	D₂₈	C₂²	1	1	yes	characteristic, complemented
12	C₄×D₇	—	2	1	no	—
13	Dic₁₄	C₂²	1	1	yes	characteristic, complemented
14	C₇×C₈	C₂²	1	2	yes	complemented
15	AGL(1,ℤ/8)	—	7	1	no	maximal, Sylow
16	C₂²×C₇	D₄	1	1	yes	characteristic
17	C₂×D₇	—	2	1	no	—
18	C₂×D₇	—	2	2	no	—
19	C₂×D₇	—	4	2	no	—
20	C₄×C₇	D₄	1	1	yes	characteristic
21	C₄×C₇	C₂³	1	1	yes	characteristic
22	Dic₇	—	2	1	no	—
23	#41	—	7	1	no	—
24	C₂×D₄	—	7	1	no	—
25	SD₁₆	—	7	2	no	—
26	D₈	—	7	2	no	—
27	M₁₆	D₇	1	1	yes	characteristic, complemented
28	C₂×C₇	—	2	1	no	—
29	C₂×C₇	C₂×D₄	1	1	yes	characteristic
30	D₇	—	4	1	no	—
31	D₇	—	4	2	no	—
32	C₂³	—	14	1	no	—
33	Q₈	—	7	1	no	—
34	D₄	—	7	2	no	—
35	D₄	—	7	1	no	—
36	D₄	—	14	1	no	—
37	D₄	—	14	1	no	—
38	C₂×C₄	—	14	1	no	—
39	C₂×C₄	C₂×D₇	1	1	yes	characteristic
40	C₈	C₂×D₇	1	2	yes	complemented
41	C₇	AGL(1,ℤ/8)	1	1	yes	characteristic, complemented, Sylow
42	C₂²	—	14	1	no	—
43	C₂²	—	14	2	no	—
44	C₂²	—	28	2	no	—
45	C₂²	D₂₈	1	1	yes	characteristic
46	C₄	—	14	1	no	—
47	C₄	D₂₈	1	1	yes	characteristic
48	C₄	C₂²×D₇	1	1	yes	characteristic
49	C₂	—	2	1	no	—
50	C₂	—	28	2	no	—
51	C₂	—	28	1	no	—
52	C₂	C₂×D₂₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₂₈
commutator, cosocle	C₄×C₇	C₂³
Frattini	C₄	C₂²×D₇
Fitting	C₇×M₁₆	C₂
socle	C₂×C₇	C₂×D₄
hypercenter	M₁₆	D₇
hypocenter	C₇	AGL(1,ℤ/8)
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ M₁₆

Complements

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

N	H ≅ G/N	type
1. #1157	C₂	semidirect
2. C₂×D₂₈	C₂	semidirect
3. C₇×M₁₆	C₂	semidirect
4. D₅₆	C₂	semidirect
5. #1132	C₂	semidirect
11. D₂₈	C₂²	semidirect
13. Dic₁₄	C₂²	semidirect
14. C₇×C₈	C₂²	semidirect
27. M₁₆	D₇	semidirect
40. C₈	C₂×D₇	semidirect
41. C₇	AGL(1,ℤ/8)	semidirect

Maximal subgroups

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

subgroup	index
#1157	2
C₂×D₂₈	2
C₇×M₁₆	2
D₅₆	2
#1132	2
AGL(1,ℤ/8)	7