Subgroups — #6474

Order 224.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#1159	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×D₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×D₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#1142	C₂	1	1	yes	maximal, characteristic, complemented
5	#1141	C₂	1	1	yes	maximal, characteristic, complemented
6	#1132	C₂	1	1	yes	maximal, characteristic, complemented
7	#1131	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₇	—	2	1	no	—
9	C₇×D₄	C₂²	1	1	yes	characteristic
10	C₇×D₄	C₂²	1	1	yes	characteristic, complemented
11	C₂²⋊D₇	—	2	1	no	—
12	C₂²⋊D₇	—	2	1	no	—
13	C₂×Dic₇	—	2	1	no	—
14	D₂₈	C₂²	1	1	yes	characteristic
15	C₄×D₇	C₂²	1	1	yes	characteristic
16	Dic₁₄	C₂²	1	1	yes	characteristic, complemented
17	C₇×C₈	C₂²	1	1	yes	characteristic, complemented
18	C₇⋊C₈	C₂²	1	1	yes	characteristic, complemented
19	AGL(1,ℤ/8)	—	7	1	no	maximal, Sylow
20	C₂²×C₇	—	2	1	no	—
21	C₂²×C₇	—	2	1	no	—
22	C₂×D₇	—	2	1	no	—
23	C₂×D₇	—	4	2	no	—
24	C₂×D₇	D₄	1	1	yes	characteristic
25	C₄×C₇	C₂³	1	1	yes	characteristic
26	Dic₇	—	2	1	no	—
27	Dic₇	D₄	1	1	yes	characteristic
28	#41	—	7	1	no	—
29	C₂×D₄	—	7	1	no	—
30	SD₁₆	—	7	1	no	—
31	SD₁₆	—	7	1	no	—
32	D₈	—	7	1	no	—
33	D₈	D₇	1	1	yes	characteristic, complemented
34	M₁₆	—	7	1	no	—
35	C₂×C₇	—	4	1	no	—
36	C₂×C₇	—	4	1	no	—
37	C₂×C₇	C₂×D₄	1	1	yes	characteristic
38	D₇	—	2	1	no	—
39	D₇	—	4	1	no	—
40	C₂³	—	14	1	no	—
41	Q₈	—	7	1	no	—
42	D₄	—	7	1	no	—
43	D₄	—	14	1	no	—
44	D₄	—	14	1	no	—
45	D₄	C₂×D₇	1	1	yes	characteristic
46	D₄	C₂×D₇	1	1	yes	characteristic, complemented
47	C₂×C₄	—	7	1	no	—
48	C₂×C₄	—	14	1	no	—
49	C₈	—	7	1	no	—
50	C₈	C₂×D₇	1	1	yes	characteristic, complemented
51	C₇	AGL(1,ℤ/8)	1	1	yes	characteristic, complemented, Sylow
52	C₂²	—	2	1	no	—
53	C₂²	—	2	1	no	—
54	C₂²	—	7	1	no	—
55	C₂²	—	14	1	no	—
56	C₂²	—	28	2	no	—
57	C₄	—	7	1	no	—
58	C₄	—	14	1	no	—
59	C₄	C₂²×D₇	1	1	yes	characteristic
60	C₂	—	4	1	no	—
61	C₂	—	4	1	no	—
62	C₂	—	14	1	no	—
63	C₂	—	28	1	no	—
64	C₂	D₄×D₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×D₇
commutator, cosocle	C₄×C₇	C₂³
Frattini	C₄	C₂²×D₇
Fitting	C₇×D₈	C₂
socle	C₂×C₇	C₂×D₄
hypercenter	D₈	D₇
hypocenter	C₇	AGL(1,ℤ/8)
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ D₈

Complements

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

N	H ≅ G/N	type
1. #1159	C₂	semidirect
2. D₄×D₇	C₂	semidirect
3. C₇×D₈	C₂	semidirect
4. #1142	C₂	semidirect
5. #1141	C₂	semidirect
6. #1132	C₂	semidirect
7. #1131	C₂	semidirect
10. C₇×D₄	C₂²	semidirect
16. Dic₁₄	C₂²	semidirect
17. C₇×C₈	C₂²	semidirect
18. C₇⋊C₈	C₂²	semidirect
33. D₈	D₇	semidirect
46. D₄	C₂×D₇	semidirect
50. C₈	C₂×D₇	semidirect
51. 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
#1159	2
D₄×D₇	2
C₇×D₈	2
#1142	2
#1141	2
#1132	2
#1131	2
AGL(1,ℤ/8)	7