Subgroups — #6513

Order 224.

62 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	#1157	C₂	1	1	yes	maximal, characteristic, complemented
3	#1144	C₂	1	1	yes	maximal, characteristic, complemented
4	#1143	C₂	1	1	yes	maximal, characteristic, complemented
5	#1142	C₂	1	1	yes	maximal, characteristic, complemented
6	#1141	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×Q₈	C₂²	1	1	yes	characteristic
9	C₇×D₄	—	2	1	no	—
10	C₇×D₄	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₇	—	2	1	no	—
12	C₂×C₄×C₇	C₂²	1	1	yes	characteristic
13	C₂²⋊D₇	—	2	1	no	—
14	D₂₈	C₂²	1	1	yes	characteristic
15	C₄×D₇	—	2	1	no	—
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	C₂²×C₇	D₄	1	1	yes	characteristic
22	C₂×D₇	—	2	1	no	—
23	C₄×C₇	—	2	1	no	—
24	C₄×C₇	D₄	1	1	yes	characteristic
25	C₄×C₇	C₂³	1	1	yes	characteristic
26	Dic₇	—	2	1	no	—
27	#41	—	7	1	no	—
28	#41	D₇	1	1	yes	characteristic, complemented
29	Q₁₆	—	7	1	no	—
30	SD₁₆	—	7	1	no	—
31	SD₁₆	—	7	1	no	—
32	D₈	—	7	1	no	—
33	C₂×C₈	—	7	1	no	—
34	C₂×C₇	—	2	1	no	—
35	C₂×C₇	—	4	1	no	—
36	C₂×C₇	C₂×D₄	1	1	yes	characteristic
37	D₇	—	4	1	no	—
38	Q₈	—	7	1	no	—
39	Q₈	C₂×D₇	1	1	yes	characteristic
40	D₄	—	2	1	no	—
41	D₄	—	7	1	no	—
42	D₄	—	14	1	no	—
43	D₄	C₂×D₇	1	1	yes	characteristic
44	C₂×C₄	—	2	1	no	—
45	C₂×C₄	—	14	1	no	—
46	C₂×C₄	C₂×D₇	1	1	yes	characteristic
47	C₈	—	7	1	no	—
48	C₈	—	7	1	no	—
49	C₇	#135	1	1	yes	characteristic, complemented, Sylow
50	C₂²	—	2	1	no	—
51	C₂²	—	14	1	no	—
52	C₂²	C₂²⋊D₇	1	1	yes	characteristic
53	C₄	—	2	1	no	—
54	C₄	—	14	1	no	—
55	C₄	C₂²⋊D₇	1	1	yes	characteristic
56	C₄	C₂²×D₇	1	1	yes	characteristic
57	C₂	—	2	1	no	—
58	C₂	—	4	1	no	—
59	C₂	—	28	1	no	—
60	C₂	C₂×C₂²⋊D₇	1	1	yes	characteristic

	subgroup	G/H
center	C₄	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	#135
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₄ ◁ 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. #1157	C₂	semidirect
3. #1144	C₂	semidirect
4. #1143	C₂	semidirect
5. #1142	C₂	semidirect
6. #1141	C₂	semidirect
7. C₂×C₇⋊C₈	C₂	semidirect
28. #41	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
C₇×#41	2
#1157	2
#1144	2
#1143	2
#1142	2
#1141	2
C₂×C₇⋊C₈	2
#135	7