Subgroups — #89850

Order 448.

98 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	#6432	C₂	1	1	yes	maximal, characteristic
3	#6381	C₂	1	4	yes	maximal
4	C₄×C₇⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₂×C₇×Q₈	—	2	1	no	—
6	C₂×Dic₁₄	—	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	—	2	1	no	—
11	C₄⋊Dic₇	C₂²	1	2	yes	—
12	C₂×C₇⋊C₈	C₂²	1	2	yes	—
13	#500	—	7	1	no	maximal, Sylow
14	C₇×Q₈	—	4	2	no	—
15	C₂×C₄×C₇	—	2	2	no	—
16	C₂×C₄×C₇	D₄	1	1	yes	characteristic
17	C₂×C₄×C₇	D₄	1	1	yes	characteristic
18	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
19	C₂×Dic₇	—	2	2	no	—
20	Dic₁₄	—	4	2	no	—
21	C₇⋊C₈	Q₈	1	4	yes	complemented
22	C₄⋊Q₈	—	7	1	no	—
23	C₄⋊Q₈	D₇	1	1	yes	characteristic
24	#107	—	7	4	no	—
25	C₄×C₈	—	7	1	no	—
26	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
27	C₄×C₇	—	4	2	no	—
28	C₄×C₇	D₈	1	2	yes	—
29	C₄×C₇	Q₁₆	1	2	yes	—
30	C₄×C₇	C₂×Q₈	1	2	yes	—
31	Dic₇	—	4	2	no	—
32	C₂×Q₈	—	2	1	no	—
33	C₂×Q₈	—	14	1	no	—
34	C₂×C₈	—	7	2	no	—
35	C₄⋊C₄	—	2	1	no	—
36	C₄⋊C₄	—	7	2	no	—
37	C₄⋊C₄	—	14	1	no	—
38	C₄⋊C₄	C₂×D₇	1	2	yes	—
39	C₄²	C₂×D₇	1	1	yes	characteristic
40	C₂×C₇	C₄⋊Q₈	1	1	yes	characteristic
41	C₂×C₇	C₂×D₈	1	1	yes	characteristic
42	C₂×C₇	C₂×Q₁₆	1	1	yes	characteristic
43	Q₈	—	4	2	no	—
44	Q₈	—	28	2	no	—
45	C₂×C₄	—	2	2	no	—
46	C₂×C₄	—	14	2	no	—
47	C₂×C₄	C₂²⋊D₇	1	1	yes	characteristic
48	C₂×C₄	C₂²⋊D₇	1	1	yes	characteristic
49	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
50	C₈	—	7	4	no	—
51	C₇	#500	1	1	yes	characteristic, complemented, Sylow
52	C₂²	C₂×C₂²⋊D₇	1	1	yes	characteristic
53	C₄	—	4	2	no	—
54	C₄	—	28	2	no	—
55	C₄	#1141	1	2	yes	—
56	C₄	#1144	1	2	yes	—
57	C₄	Q₈×D₇	1	2	yes	—
58	C₂	C₂×#1141	1	1	yes	characteristic
59	C₂	C₂×#1144	1	1	yes	characteristic
60	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	#500
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
21. C₇⋊C₈	Q₈	semidirect
51. C₇	#500	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
#6432	2
#6381	2
C₄×C₇⋊C₈	2
#500	7