Subgroups — #89852

Order 448.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	#6433	C₂	1	1	yes	maximal, characteristic
3	#6382	C₂	1	2	yes	maximal
4	#6381	C₂	1	2	yes	maximal
5	#6377	C₂	1	1	yes	maximal, characteristic
6	C₂×C₇×Q₈	—	2	1	no	—
7	C₇×C₄⋊C₄	—	2	1	no	—
8	C₇×C₄⋊C₄	C₂²	1	2	yes	—
9	C₄²×C₇	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₇	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₇	C₂²	1	1	yes	characteristic
12	Dic₇⋊C₄	—	2	1	no	—
13	Dic₇⋊C₄	—	2	1	no	—
14	C₂×C₇⋊C₈	C₂²	1	2	yes	—
15	#501	—	7	1	no	maximal, Sylow
16	C₇×Q₈	—	4	2	no	—
17	C₂×C₄×C₇	—	2	2	no	—
18	C₂×C₄×C₇	D₄	1	1	yes	characteristic
19	C₂×C₄×C₇	D₄	1	1	yes	characteristic
20	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
21	C₂×Dic₇	—	2	1	no	—
22	C₂×Dic₇	—	2	1	no	—
23	C₇⋊C₈	Q₈	1	4	yes	complemented
24	C₄⋊Q₈	D₇	1	1	yes	characteristic
25	#125	—	7	1	no	—
26	#107	—	7	2	no	—
27	#106	—	7	2	no	—
28	#97	—	7	1	no	—
29	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
30	C₄×C₇	—	2	1	no	—
31	C₄×C₇	—	2	1	no	—
32	C₄×C₇	—	4	2	no	—
33	C₄×C₇	C₂×Q₈	1	2	yes	—
34	Dic₇	—	4	1	no	—
35	Dic₇	—	4	1	no	—
36	C₂×Q₈	—	2	1	no	—
37	C₂×C₈	—	7	2	no	—
38	C₄⋊C₄	—	2	1	no	—
39	C₄⋊C₄	—	7	1	no	—
40	C₄⋊C₄	—	7	1	no	—
41	C₄⋊C₄	—	14	1	no	—
42	C₄⋊C₄	—	14	1	no	—
43	C₄⋊C₄	C₂×D₇	1	2	yes	—
44	C₄²	C₂×D₇	1	1	yes	characteristic
45	C₂×C₇	C₄⋊Q₈	1	1	yes	characteristic
46	C₂×C₇	AGL(1,ℤ/8)	1	1	yes	characteristic
47	C₂×C₇	#137	1	1	yes	characteristic
48	Q₈	—	4	2	no	—
49	C₂×C₄	—	2	2	no	—
50	C₂×C₄	—	14	1	no	—
51	C₂×C₄	—	14	1	no	—
52	C₂×C₄	C₂²⋊D₇	1	1	yes	characteristic
53	C₂×C₄	C₂²⋊D₇	1	1	yes	characteristic
54	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
55	C₈	—	7	4	no	—
56	C₇	#501	1	1	yes	characteristic, complemented, Sylow
57	C₂²	C₂×C₂²⋊D₇	1	1	yes	characteristic
58	C₄	—	2	1	no	—
59	C₄	—	2	1	no	—
60	C₄	—	4	2	no	—
61	C₄	—	28	1	no	—
62	C₄	—	28	1	no	—
63	C₄	Q₈×D₇	1	2	yes	—
64	C₂	#6495	1	1	yes	characteristic
65	C₂	#6505	1	1	yes	characteristic
66	C₂	#6507	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₇
commutator, cosocle	C₂×C₄×C₇	C₂³
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₇×C₄⋊Q₈	C₂
socle, prime radical	C₂²×C₇	C₂×D₄
hypercenter	C₄⋊Q₈	D₇
hypocenter	C₇	#501
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
23. C₇⋊C₈	Q₈	semidirect
56. C₇	#501	semidirect

Maximal subgroups

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

subgroup	index
C₇×C₄⋊Q₈	2
#6433	2
#6382	2
#6381	2
#6377	2
#501	7