Subgroups — #89863

Order 448.

96 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	C₄×Dic₁₄	C₂	1	1	yes	maximal, characteristic
3	#6384	C₂	1	2	yes	maximal
4	#6381	C₂	1	2	yes	maximal
5	C₄⋊(C₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
6	C₂×C₇×Q₈	—	2	1	no	—
7	C₂×Dic₁₄	C₂²	1	1	yes	characteristic
8	C₇×C₄⋊C₄	—	2	1	no	—
9	C₇×C₄⋊C₄	C₂²	1	2	yes	—
10	C₄²×C₇	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₇	C₂²	1	1	yes	characteristic
12	Dic₇⋊C₄	—	2	1	no	—
13	C₄×Dic₇	—	2	1	no	—
14	C₂×C₇⋊C₈	C₂²	1	2	yes	—
15	#477	—	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	Dic₁₄	—	2	1	no	—
24	Dic₁₄	Q₈	1	2	yes	complemented
25	C₇⋊C₈	—	2	2	no	—
26	C₄⋊Q₈	D₇	1	1	yes	characteristic
27	C₄×Q₈	—	7	1	no	—
28	#107	—	7	2	no	—
29	C₄⋊C₈	—	7	1	no	—
30	#103	—	7	2	no	—
31	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
32	C₄×C₇	—	2	1	no	—
33	C₄×C₇	—	4	2	no	—
34	C₄×C₇	Q₁₆	1	2	yes	—
35	C₄×C₇	C₂×Q₈	1	1	yes	characteristic
36	C₄×C₇	#41	1	1	yes	characteristic
37	Dic₇	—	2	2	no	—
38	Dic₇	—	4	1	no	—
39	C₂×Q₈	—	2	1	no	—
40	C₂×Q₈	—	7	1	no	—
41	C₂×C₈	—	7	2	no	—
42	C₄⋊C₄	—	2	1	no	—
43	C₄⋊C₄	—	7	1	no	—
44	C₄⋊C₄	—	14	1	no	—
45	C₄⋊C₄	C₂×D₇	1	2	yes	—
46	C₄²	—	14	1	no	—
47	C₄²	C₂×D₇	1	1	yes	characteristic
48	C₂×C₇	C₂²⋊Q₈	1	1	yes	characteristic
49	C₂×C₇	C₂×Q₁₆	1	1	yes	characteristic
50	C₂×C₇	AGL(1,ℤ/8)	1	1	yes	characteristic
51	Q₈	—	4	2	no	—
52	Q₈	—	7	2	no	—
53	Q₈	—	14	1	no	—
54	C₂×C₄	—	2	2	no	—
55	C₂×C₄	—	14	1	no	—
56	C₂×C₄	—	14	1	no	—
57	C₂×C₄	C₂²⋊D₇	1	1	yes	characteristic
58	C₂×C₄	C₂²⋊D₇	1	1	yes	characteristic
59	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
60	C₈	—	14	2	no	—
61	C₇	#477	1	1	yes	characteristic, complemented, Sylow
62	C₂²	C₂×C₂²⋊D₇	1	1	yes	characteristic
63	C₄	—	2	1	no	—
64	C₄	—	4	2	no	—
65	C₄	—	14	2	no	—
66	C₄	—	28	1	no	—
67	C₄	#1144	1	2	yes	—
68	C₄	Q₈×D₇	1	1	yes	characteristic
69	C₄	#1161	1	1	yes	characteristic
70	C₂	#6495	1	1	yes	characteristic
71	C₂	C₂×#1144	1	1	yes	characteristic
72	C₂	#6509	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	#477
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
24. Dic₁₄	Q₈	semidirect
61. C₇	#477	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
C₄×Dic₁₄	2
#6384	2
#6381	2
C₄⋊(C₇⋊C₈)	2
#477	7