Subgroups — #89648

Order 448.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#6453	C₂	1	1	yes	maximal, characteristic
2	#6451	C₂	1	1	yes	maximal, characteristic
3	C₇×#107	C₂	1	1	yes	maximal, characteristic
4	#6391	C₂	1	1	yes	maximal, characteristic
5	#6389	C₂	1	1	yes	maximal, characteristic
6	#6382	C₂	1	1	yes	maximal, characteristic
7	#6381	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₁₄	—	2	1	no	—
9	C₂×C₇×C₈	C₂²	1	1	yes	characteristic
10	C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
11	C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₇	—	2	1	no	—
13	C₄⋊Dic₇	C₂²	1	1	yes	characteristic
14	C₄⋊Dic₇	C₂²	1	1	yes	characteristic
15	Dic₇⋊C₄	—	2	1	no	—
16	Dic₇⋊C₄	—	2	1	no	—
17	C₄×Dic₇	C₂²	1	1	yes	characteristic
18	C₂×C₇⋊C₈	C₂²	1	1	yes	characteristic
19	#501	—	7	1	no	maximal, Sylow
20	C₂×C₄×C₇	—	2	1	no	—
21	C₂×C₄×C₇	—	2	1	no	—
22	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
23	C₂×Dic₇	—	2	1	no	—
24	C₂×Dic₇	—	2	1	no	—
25	C₂×Dic₇	D₄	1	1	yes	characteristic
26	C₂×Dic₇	D₄	1	1	yes	characteristic
27	Dic₁₄	—	4	2	no	—
28	C₇×C₈	Q₈	1	2	yes	complemented
29	C₇⋊C₈	Q₈	1	2	yes	complemented
30	C₄⋊Q₈	—	7	1	no	—
31	#125	—	7	1	no	—
32	#107	—	7	1	no	—
33	#107	D₇	1	1	yes	characteristic
34	#106	—	7	1	no	—
35	#106	—	7	1	no	—
36	#97	—	7	1	no	—
37	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
38	C₄×C₇	—	4	1	no	—
39	C₄×C₇	—	4	1	no	—
40	C₄×C₇	C₂×Q₈	1	1	yes	characteristic
41	C₄×C₇	C₂×Q₈	1	1	yes	characteristic
42	Dic₇	—	2	1	no	—
43	Dic₇	—	2	1	no	—
44	Dic₇	—	4	1	no	—
45	Dic₇	—	4	1	no	—
46	C₂×Q₈	—	14	1	no	—
47	C₂×C₈	—	7	1	no	—
48	C₂×C₈	C₂×D₇	1	1	yes	characteristic
49	C₄⋊C₄	—	7	1	no	—
50	C₄⋊C₄	—	7	1	no	—
51	C₄⋊C₄	—	14	1	no	—
52	C₄⋊C₄	—	14	1	no	—
53	C₄⋊C₄	—	14	1	no	—
54	C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
55	C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
56	C₄²	—	7	1	no	—
57	C₂×C₇	C₄⋊Q₈	1	1	yes	characteristic
58	C₂×C₇	AGL(1,ℤ/8)	1	1	yes	characteristic
59	C₂×C₇	#137	1	1	yes	characteristic
60	Q₈	—	28	2	no	—
61	C₂×C₄	—	2	1	no	—
62	C₂×C₄	—	2	1	no	—
63	C₂×C₄	—	7	1	no	—
64	C₂×C₄	—	7	1	no	—
65	C₂×C₄	—	14	1	no	—
66	C₂×C₄	—	14	1	no	—
67	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
68	C₈	—	7	2	no	—
69	C₈	Dic₁₄	1	2	yes	complemented
70	C₇	#501	1	1	yes	characteristic, complemented, Sylow
71	C₂²	D₄×D₇	1	1	yes	characteristic
72	C₄	—	4	1	no	—
73	C₄	—	4	1	no	—
74	C₄	—	14	1	no	—
75	C₄	—	14	1	no	—
76	C₄	—	28	1	no	—
77	C₄	—	28	1	no	—
78	C₄	C₂×Dic₁₄	1	1	yes	characteristic
79	C₄	Q₈×D₇	1	1	yes	characteristic
80	C₂	#6451	1	1	yes	characteristic
81	C₂	#6474	1	1	yes	characteristic
82	C₂	#6481	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₇
commutator, cosocle	C₂×C₄×C₇	C₂³
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₇×#107	C₂
socle, prime radical	C₂²×C₇	C₂×D₄
hypercenter	#107	D₇
hypocenter	C₇	#501
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

Complements

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

N	H ≅ G/N	type
28. C₇×C₈	Q₈	semidirect
29. C₇⋊C₈	Q₈	semidirect
69. C₈	Dic₁₄	semidirect
70. C₇	#501	semidirect

Maximal subgroups

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

subgroup	index
#6453	2
#6451	2
C₇×#107	2
#6391	2
#6389	2
#6382	2
#6381	2
#501	7