Subgroups — #89691

Order 448.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#6482	C₂	1	2	yes	maximal, complemented
2	C₇×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#6402	C₂	1	2	yes	maximal, complemented
4	D₁₁₂	C₂	1	1	yes	maximal, characteristic, complemented
5	D₇×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	#1161	—	2	2	no	—
7	C₇×Q₁₆	C₂²	1	2	yes	—
8	#1143	—	2	2	no	—
9	D₅₆	C₂²	1	2	yes	—
10	C₈×D₇	C₂²	1	1	yes	characteristic
11	C₇×C₁₆	C₂²	1	1	yes	characteristic
12	C₇⋊C₁₆	C₂²	1	1	yes	characteristic
13	#508	—	7	1	no	maximal, Sylow
14	C₇×Q₈	—	2	2	no	—
15	D₂₈	—	2	2	no	—
16	D₂₈	—	4	2	no	—
17	C₄×D₇	—	4	2	no	—
18	C₄×D₇	D₄	1	1	yes	characteristic
19	C₇×C₈	C₂³	1	1	yes	characteristic
20	C₇⋊C₈	D₄	1	1	yes	characteristic
21	#135	—	7	2	no	—
22	Q₃₂	D₇	1	1	yes	characteristic, complemented
23	SD₃₂	—	7	2	no	—
24	D₁₆	—	7	1	no	—
25	C₂×C₁₆	—	7	1	no	—
26	C₂×D₇	—	4	2	no	—
27	C₂×D₇	D₈	1	1	yes	characteristic
28	C₄×C₇	—	4	2	no	—
29	C₄×C₇	C₂×D₄	1	1	yes	characteristic
30	Dic₇	D₈	1	1	yes	characteristic
31	#41	—	14	2	no	—
32	Q₁₆	C₂×D₇	1	2	yes	—
33	SD₁₆	—	14	2	no	—
34	D₈	—	7	2	no	—
35	C₂×C₈	—	7	1	no	—
36	C₁₆	—	7	1	no	—
37	C₁₆	C₂×D₇	1	1	yes	characteristic
38	C₂×C₇	C₂×D₈	1	1	yes	characteristic
39	D₇	—	2	1	no	—
40	D₇	—	8	2	no	—
41	Q₈	—	2	2	no	—
42	D₄	—	14	2	no	—
43	D₄	—	28	2	no	—
44	C₂×C₄	—	7	1	no	—
45	C₂×C₄	—	28	2	no	—
46	C₈	—	7	1	no	—
47	C₈	C₂²×D₇	1	1	yes	characteristic
48	C₇	#508	1	1	yes	characteristic, complemented, Sylow
49	C₂²	—	7	1	no	—
50	C₂²	—	28	2	no	—
51	C₄	—	4	2	no	—
52	C₄	—	7	1	no	—
53	C₄	D₄×D₇	1	1	yes	characteristic
54	C₂	—	14	1	no	—
55	C₂	—	56	2	no	—
56	C₂	D₇×D₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₇×D₈
commutator, cosocle	C₇×C₈	C₂³
Frattini	C₈	C₂²×D₇
Fitting	C₇×Q₃₂	C₂
socle	C₂×C₇	C₂×D₈
hypercenter	Q₃₂	D₇
hypocenter	C₇	#508
LCS	G ▷ C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ Q₃₂

Complements

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

N	H ≅ G/N	type
1. #6482	C₂	semidirect
2. C₇×Q₃₂	C₂	semidirect
3. #6402	C₂	semidirect
4. D₁₁₂	C₂	semidirect
5. D₇×C₁₆	C₂	semidirect
22. Q₃₂	D₇	semidirect
48. C₇	#508	semidirect

Maximal subgroups

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

subgroup	index
#6482	2
C₇×Q₃₂	2
#6402	2
D₁₁₂	2
D₇×C₁₆	2
#508	7