Subgroups — #6479

Order 224.

62 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#1161	C₂	1	1	yes	maximal, characteristic, complemented
2	#1159	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#1144	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	C₈×D₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×Q₈	C₂²	1	1	yes	characteristic
9	C₇×D₄	C₂²	1	1	yes	characteristic
10	C₂²⋊D₇	—	2	1	no	—
11	C₂×Dic₇	—	2	1	no	—
12	D₂₈	—	2	1	no	—
13	D₂₈	C₂²	1	1	yes	characteristic
14	C₄×D₇	—	2	1	no	—
15	C₄×D₇	C₂²	1	1	yes	characteristic
16	Dic₁₄	C₂²	1	1	yes	characteristic
17	C₇×C₈	C₂²	1	1	yes	characteristic
18	C₇⋊C₈	C₂²	1	1	yes	characteristic
19	#135	—	7	1	no	maximal, Sylow
20	C₂²×C₇	—	2	1	no	—
21	C₂×D₇	—	2	1	no	—
22	C₂×D₇	D₄	1	1	yes	characteristic
23	C₄×C₇	—	2	1	no	—
24	C₄×C₇	C₂³	1	1	yes	characteristic
25	Dic₇	—	2	1	no	—
26	Dic₇	D₄	1	1	yes	characteristic
27	#41	—	7	1	no	—
28	#41	—	7	1	no	—
29	Q₁₆	—	7	1	no	—
30	SD₁₆	—	7	1	no	—
31	SD₁₆	D₇	1	1	yes	characteristic, complemented
32	D₈	—	7	1	no	—
33	C₂×C₈	—	7	1	no	—
34	C₂×C₇	—	4	1	no	—
35	C₂×C₇	C₂×D₄	1	1	yes	characteristic
36	D₇	—	2	1	no	—
37	D₇	—	4	1	no	—
38	Q₈	—	7	1	no	—
39	Q₈	C₂×D₇	1	1	yes	characteristic
40	D₄	—	7	1	no	—
41	D₄	—	14	1	no	—
42	D₄	—	14	1	no	—
43	D₄	C₂×D₇	1	1	yes	characteristic
44	C₂×C₄	—	7	1	no	—
45	C₂×C₄	—	14	1	no	—
46	C₂×C₄	—	14	1	no	—
47	C₈	—	7	1	no	—
48	C₈	C₂×D₇	1	1	yes	characteristic
49	C₇	#135	1	1	yes	characteristic, complemented, Sylow
50	C₂²	—	2	1	no	—
51	C₂²	—	7	1	no	—
52	C₂²	—	14	1	no	—
53	C₄	—	2	1	no	—
54	C₄	—	7	1	no	—
55	C₄	—	14	1	no	—
56	C₄	C₂²×D₇	1	1	yes	characteristic
57	C₂	—	4	1	no	—
58	C₂	—	14	1	no	—
59	C₂	—	28	1	no	—
60	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×SD₁₆	C₂
socle	C₂×C₇	C₂×D₄
hypercenter	SD₁₆	D₇
hypocenter	C₇	#135
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ SD₁₆

Complements

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

N	H ≅ G/N	type
1. #1161	C₂	semidirect
2. #1159	C₂	semidirect
3. C₇×SD₁₆	C₂	semidirect
4. #1144	C₂	semidirect
5. #1141	C₂	semidirect
6. #1132	C₂	semidirect
7. C₈×D₇	C₂	semidirect
31. SD₁₆	D₇	semidirect
49. C₇	#135	semidirect

Maximal subgroups

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

subgroup	index
#1161	2
#1159	2
C₇×SD₁₆	2
#1144	2
#1141	2
#1132	2
C₈×D₇	2
#135	7