Subgroups — #89602

Order 448.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Dic₁₄	C₂	1	1	yes	maximal, characteristic
2	C₇×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	Dic₇⋊C₈	C₂	1	2	yes	maximal
4	C₈×Dic₇	C₂	1	2	yes	maximal
5	C₄×C₇⋊C₈	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₁₄	C₄	1	1	yes	characteristic
7	C₂×C₇×C₈	C₂²	1	2	yes	—
8	C₄²×C₇	C₂²	1	1	yes	characteristic
9	C₄⋊Dic₇	C₄	1	1	yes	characteristic
10	Dic₇⋊C₄	C₄	1	2	yes	—
11	C₄×Dic₇	C₂²	1	2	yes	—
12	C₂×C₇⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₇⋊C₈	C₂²	1	1	yes	characteristic
14	C₈×Q₈	—	7	1	no	maximal, Sylow
15	C₂×C₄×C₇	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄×C₇	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
18	C₂×Dic₇	C₂×C₄	1	2	yes	—
19	C₂×Dic₇	C₂×C₄	1	2	yes	—
20	Dic₁₄	C₈	1	4	yes	complemented
21	C₇×C₈	—	2	2	no	—
22	C₇⋊C₈	—	2	1	no	—
23	C₇⋊C₈	Q₈	1	2	yes	complemented
24	C₄×Q₈	—	7	1	no	—
25	C₄⋊C₈	—	7	2	no	—
26	C₄⋊C₈	D₇	1	1	yes	characteristic
27	C₄×C₈	—	7	2	no	—
28	C₄×C₈	—	7	1	no	—
29	C₂²×C₇	C₂²×C₄	1	1	yes	characteristic
30	C₄×C₇	—	2	1	no	—
31	C₄×C₇	C₂×C₈	1	2	yes	—
32	C₄×C₇	C₂×Q₈	1	1	yes	characteristic
33	C₄×C₇	#41	1	1	yes	characteristic
34	Dic₇	—	2	2	no	—
35	Dic₇	C₂×C₈	1	4	yes	—
36	C₂×Q₈	—	7	1	no	—
37	C₂×C₈	—	7	1	no	—
38	C₂×C₈	—	7	1	no	—
39	C₂×C₈	C₂×D₇	1	2	yes	—
40	C₄⋊C₄	—	7	2	no	—
41	C₄⋊C₄	—	7	1	no	—
42	C₄²	—	7	2	no	—
43	C₄²	C₂×D₇	1	1	yes	characteristic
44	C₂×C₇	C₄×Q₈	1	1	yes	characteristic
45	C₂×C₇	C₂²×C₈	1	1	yes	characteristic
46	C₂×C₇	#131	1	1	yes	characteristic
47	Q₈	—	7	4	no	—
48	C₂×C₄	—	7	2	no	—
49	C₂×C₄	—	7	2	no	—
50	C₂×C₄	C₄×D₇	1	1	yes	characteristic
51	C₂×C₄	C₄×D₇	1	1	yes	characteristic
52	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
53	C₈	—	2	2	no	—
54	C₈	—	7	2	no	—
55	C₈	—	14	1	no	—
56	C₇	C₈×Q₈	1	1	yes	characteristic, complemented, Sylow
57	C₂²	C₂×C₄×D₇	1	1	yes	characteristic
58	C₄	—	2	1	no	—
59	C₄	—	7	4	no	—
60	C₄	—	14	2	no	—
61	C₄	C₈×D₇	1	2	yes	—
62	C₄	#1159	1	1	yes	characteristic
63	C₄	Q₈×D₇	1	1	yes	characteristic
64	C₂	#6450	1	1	yes	characteristic
65	C₂	C₂×C₈×D₇	1	1	yes	characteristic
66	C₂	#6470	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₇
commutator	C₂×C₇	C₂²×C₈
Fitting	C₇×C₄⋊C₈	C₂
socle, prime radical	C₂²×C₇	C₂²×C₄
cosocle	C₂×C₄×C₇	C₂³
hypercenter	C₄⋊C₈	D₇
hypocenter	C₇	C₈×Q₈
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
20. Dic₁₄	C₈	semidirect
23. C₇⋊C₈	Q₈	semidirect
56. C₇	C₈×Q₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×Dic₁₄	2
C₇×C₄⋊C₈	2
Dic₇⋊C₈	2
C₈×Dic₇	2
C₄×C₇⋊C₈	2
C₈×Q₈	7