Subgroups — #6481

Order 224.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#1161	C₂	1	1	yes	maximal, characteristic
2	Q₈×D₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#1144	C₂	1	1	yes	maximal, characteristic, complemented
5	#1143	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	C₇×Q₈	C₂²	1	1	yes	characteristic
9	C₇×Q₈	C₂²	1	1	yes	characteristic
10	D₂₈	—	2	1	no	—
11	D₂₈	C₂²	1	1	yes	characteristic
12	C₄×D₇	—	2	1	no	—
13	C₄×D₇	—	2	1	no	—
14	C₄×D₇	C₂²	1	1	yes	characteristic
15	Dic₁₄	—	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	#137	—	7	1	no	maximal, Sylow
20	C₂×D₇	—	2	1	no	—
21	C₂×D₇	D₄	1	1	yes	characteristic
22	C₄×C₇	—	2	1	no	—
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	C₂×Q₈	—	7	1	no	—
29	Q₁₆	—	7	1	no	—
30	Q₁₆	D₇	1	1	yes	characteristic, complemented
31	SD₁₆	—	7	1	no	—
32	SD₁₆	—	7	1	no	—
33	M₁₆	—	7	1	no	—
34	C₂×C₇	C₂×D₄	1	1	yes	characteristic
35	D₇	—	2	1	no	—
36	D₇	—	4	1	no	—
37	Q₈	—	7	1	no	—
38	Q₈	—	14	1	no	—
39	Q₈	C₂×D₇	1	1	yes	characteristic
40	Q₈	C₂×D₇	1	1	yes	characteristic
41	D₄	—	7	1	no	—
42	D₄	—	14	1	no	—
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	—	7	1	no	—
50	C₂²	—	14	1	no	—
51	C₄	—	2	1	no	—
52	C₄	—	2	1	no	—
53	C₄	—	7	1	no	—
54	C₄	—	14	1	no	—
55	C₄	C₂²×D₇	1	1	yes	characteristic
56	C₂	—	14	1	no	—
57	C₂	—	28	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×Q₁₆	C₂
socle	C₂×C₇	C₂×D₄
prime radical	#1161	C₂
hypercenter	Q₁₆	D₇
hypocenter	C₇	#137
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ Q₁₆

Complements

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

N	H ≅ G/N	type
2. Q₈×D₇	C₂	semidirect
3. C₇×Q₁₆	C₂	semidirect
4. #1144	C₂	semidirect
5. #1143	C₂	semidirect
6. #1132	C₂	semidirect
7. #1131	C₂	semidirect
30. Q₁₆	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
#1161	2
Q₈×D₇	2
C₇×Q₁₆	2
#1144	2
#1143	2
#1132	2
#1131	2
#137	7