Subgroups — #6512

Order 224.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#41	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₂₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#1143	C₂	1	2	yes	maximal, complemented
4	#1141	C₂	1	2	yes	maximal, complemented
5	#1136	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₇	—	2	1	no	—
7	C₇×Q₈	C₂²	1	1	yes	characteristic, complemented
8	C₇×D₄	—	2	1	no	—
9	C₇×D₄	C₂²	1	1	yes	characteristic, complemented
10	C₂×C₄×C₇	—	2	1	no	—
11	C₂×C₄×C₇	C₂²	1	1	yes	characteristic
12	D₂₈	—	2	1	no	—
13	D₂₈	C₂²	1	2	yes	—
14	C₇⋊C₈	C₂²	1	2	yes	complemented
15	AGL(1,ℤ/8)	—	7	1	no	maximal, Sylow
16	C₂²×C₇	—	2	1	no	—
17	C₂²×C₇	D₄	1	1	yes	characteristic
18	C₂×D₇	—	2	2	no	—
19	C₂×D₇	—	4	2	no	—
20	C₄×C₇	—	2	1	no	—
21	C₄×C₇	D₄	1	1	yes	characteristic
22	C₄×C₇	C₂³	1	1	yes	characteristic
23	#41	D₇	1	1	yes	characteristic, complemented
24	C₂×D₄	—	7	1	no	—
25	SD₁₆	—	7	2	no	—
26	D₈	—	7	2	no	—
27	M₁₆	—	7	1	no	—
28	C₂×C₇	—	2	1	no	—
29	C₂×C₇	—	4	1	no	—
30	C₂×C₇	C₂×D₄	1	1	yes	characteristic
31	D₇	—	4	2	no	—
32	C₂³	—	14	1	no	—
33	Q₈	C₂×D₇	1	1	yes	characteristic, complemented
34	D₄	—	2	1	no	—
35	D₄	—	7	2	no	—
36	D₄	—	14	1	no	—
37	D₄	C₂×D₇	1	1	yes	characteristic, complemented
38	C₂×C₄	—	2	1	no	—
39	C₂×C₄	C₂×D₇	1	1	yes	characteristic
40	C₈	—	7	2	no	—
41	C₇	AGL(1,ℤ/8)	1	1	yes	characteristic, complemented, Sylow
42	C₂²	—	2	1	no	—
43	C₂²	—	14	2	no	—
44	C₂²	—	28	2	no	—
45	C₂²	C₂²⋊D₇	1	1	yes	characteristic
46	C₄	—	2	1	no	—
47	C₄	C₂²⋊D₇	1	1	yes	characteristic
48	C₄	C₂²×D₇	1	1	yes	characteristic
49	C₂	—	2	1	no	—
50	C₂	—	4	1	no	—
51	C₂	—	28	2	no	—
52	C₂	C₂×C₂²⋊D₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×C₂²⋊D₇
commutator, cosocle	C₄×C₇	C₂³
Frattini	C₄	C₂²×D₇
Fitting	C₇×#41	C₂
socle	C₂×C₇	C₂×D₄
hypercenter	#41	D₇
hypocenter	C₇	AGL(1,ℤ/8)
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #41

Complements

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

N	H ≅ G/N	type
1. C₇×#41	C₂	semidirect
2. C₂×D₂₈	C₂	semidirect
3. #1143	C₂	semidirect
4. #1141	C₂	semidirect
5. #1136	C₂	semidirect
7. C₇×Q₈	C₂²	semidirect
9. C₇×D₄	C₂²	semidirect
14. C₇⋊C₈	C₂²	semidirect
23. #41	D₇	semidirect
33. Q₈	C₂×D₇	semidirect
37. D₄	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
C₇×#41	2
C₂×D₂₈	2
#1143	2
#1141	2
#1136	2
AGL(1,ℤ/8)	7