Subgroups — #6478

Order 224.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×D₇	C₂	1	1	yes	maximal, characteristic, complemented
2	#1159	C₂	1	1	yes	maximal, characteristic
3	C₇×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#1144	C₂	1	1	yes	maximal, characteristic, complemented
5	#1142	C₂	1	1	yes	maximal, characteristic, complemented
6	Dic₂₈	C₂	1	1	yes	maximal, characteristic, complemented
7	#1131	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	C₄×D₇	—	2	1	no	—
13	C₄×D₇	C₂²	1	1	yes	characteristic
14	Dic₁₄	—	2	1	no	—
15	Dic₁₄	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	#137	—	7	1	no	maximal, Sylow
20	C₂²×C₇	—	2	1	no	—
21	C₂×D₇	D₄	1	1	yes	characteristic
22	C₄×C₇	—	2	1	no	—
23	C₄×C₇	C₂³	1	1	yes	characteristic
24	Dic₇	—	2	1	no	—
25	Dic₇	—	2	1	no	—
26	Dic₇	D₄	1	1	yes	characteristic
27	#41	—	7	1	no	—
28	C₂×Q₈	—	7	1	no	—
29	Q₁₆	—	7	1	no	—
30	Q₁₆	—	7	1	no	—
31	SD₁₆	—	7	1	no	—
32	SD₁₆	D₇	1	1	yes	characteristic, complemented
33	M₁₆	—	7	1	no	—
34	C₂×C₇	—	4	1	no	—
35	C₂×C₇	C₂×D₄	1	1	yes	characteristic
36	D₇	—	2	1	no	—
37	Q₈	—	7	1	no	—
38	Q₈	—	7	1	no	—
39	Q₈	—	14	1	no	—
40	Q₈	C₂×D₇	1	1	yes	characteristic
41	D₄	—	14	1	no	—
42	D₄	C₂×D₇	1	1	yes	characteristic
43	C₂×C₄	—	7	1	no	—
44	C₂×C₄	—	14	1	no	—
45	C₂×C₄	—	14	1	no	—
46	C₈	—	7	1	no	—
47	C₈	C₂×D₇	1	1	yes	characteristic
48	C₇	#137	1	1	yes	characteristic, complemented, Sylow
49	C₂²	—	2	1	no	—
50	C₂²	—	7	1	no	—
51	C₄	—	2	1	no	—
52	C₄	—	7	1	no	—
53	C₄	—	14	1	no	—
54	C₄	—	14	1	no	—
55	C₄	C₂²×D₇	1	1	yes	characteristic
56	C₂	—	4	1	no	—
57	C₂	—	14	1	no	—
58	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
prime radical	#1159	C₂
hypercenter	SD₁₆	D₇
hypocenter	C₇	#137
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. Q₈×D₇	C₂	semidirect
3. C₇×SD₁₆	C₂	semidirect
4. #1144	C₂	semidirect
5. #1142	C₂	semidirect
6. Dic₂₈	C₂	semidirect
7. #1131	C₂	semidirect
32. SD₁₆	D₇	semidirect
48. C₇	#137	semidirect

Maximal subgroups

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

subgroup	index
Q₈×D₇	2
#1159	2
C₇×SD₁₆	2
#1144	2
#1142	2
Dic₂₈	2
#1131	2
#137	7