Subgroups — #89279

Order 448.

70 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	C₄×C₇⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₁₄	C₄	1	1	yes	characteristic
5	C₂×C₇×C₈	—	2	1	no	—
6	C₄²×C₇	C₂²	1	1	yes	characteristic
7	C₄⋊Dic₇	C₄	1	1	yes	characteristic
8	Dic₇⋊C₄	—	2	1	no	—
9	C₄×Dic₇	—	2	1	no	—
10	C₂×C₇⋊C₈	—	2	1	no	—
11	Q₈⋊C₈	—	7	1	no	maximal, Sylow
12	C₂×C₄×C₇	C₂×C₄	1	1	yes	characteristic
13	C₂×C₄×C₇	D₄	1	1	yes	characteristic
14	C₂×C₄×C₇	D₄	1	1	yes	characteristic
15	C₂×Dic₇	—	2	1	no	—
16	C₂×Dic₇	—	2	1	no	—
17	Dic₁₄	—	2	1	no	—
18	Dic₁₄	C₈	1	2	yes	complemented
19	C₇×C₈	—	4	1	no	—
20	C₇⋊C₈	—	2	2	no	—
21	C₄×Q₈	—	7	1	no	—
22	C₄⋊C₈	D₇	1	1	yes	characteristic
23	C₄×C₈	—	7	1	no	—
24	C₂²×C₇	C₂²⋊C₄	1	1	yes	characteristic
25	C₄×C₇	—	2	1	no	—
26	C₄×C₇	C₂×C₈	1	1	yes	characteristic
27	C₄×C₇	M₁₆	1	1	yes	characteristic
28	C₄×C₇	SD₁₆	1	1	yes	characteristic
29	C₄×C₇	Q₁₆	1	1	yes	characteristic
30	Dic₇	—	2	2	no	—
31	Dic₇	—	4	1	no	—
32	C₂×Q₈	—	7	1	no	—
33	C₂×C₈	—	2	1	no	—
34	C₂×C₈	—	14	1	no	—
35	C₄⋊C₄	—	7	1	no	—
36	C₄⋊C₄	—	14	1	no	—
37	C₄²	—	14	1	no	—
38	C₄²	C₂×D₇	1	1	yes	characteristic
39	C₂×C₇	C₂²⋊C₈	1	1	yes	characteristic
40	C₂×C₇	#103	1	1	yes	characteristic
41	C₂×C₇	#104	1	1	yes	characteristic
42	Q₈	—	7	2	no	—
43	Q₈	—	14	1	no	—
44	C₂×C₄	—	14	1	no	—
45	C₂×C₄	—	14	1	no	—
46	C₂×C₄	C₄×D₇	1	1	yes	characteristic
47	C₂×C₄	D₂₈	1	1	yes	characteristic
48	C₂×C₄	C₂²⋊D₇	1	1	yes	characteristic
49	C₈	—	4	1	no	—
50	C₈	—	14	2	no	—
51	C₇	Q₈⋊C₈	1	1	yes	characteristic, complemented, Sylow
52	C₂²	(C₂×D₇)⋊C₄	1	1	yes	characteristic
53	C₄	—	2	1	no	—
54	C₄	—	14	2	no	—
55	C₄	—	28	1	no	—
56	C₄	C₈×D₇	1	1	yes	characteristic
57	C₄	#1131	1	1	yes	characteristic
58	C₄	#1142	1	1	yes	characteristic
59	C₄	#1144	1	1	yes	characteristic
60	C₂	#6384	1	1	yes	characteristic
61	C₂	(C₂×D₇)⋊C₈	1	1	yes	characteristic
62	C₂	#6399	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊D₇
commutator	C₄×C₇	C₂×C₈
Frattini	C₄²	C₂×D₇
Fitting	C₇×C₄⋊C₈	C₂
socle, prime radical	C₂²×C₇	C₂²⋊C₄
cosocle	C₄²×C₇	C₂²
hypercenter	C₄⋊C₈	D₇
hypocenter	C₇	Q₈⋊C₈
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
18. Dic₁₄	C₈	semidirect
51. C₇	Q₈⋊C₈	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
C₄×C₇⋊C₈	2
Q₈⋊C₈	7