Subgroups — #89260

Order 448.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#6433	C₂	1	1	yes	maximal, characteristic
2	C₇×#97	C₂	1	1	yes	maximal, characteristic
3	#6377	C₂	1	1	yes	maximal, characteristic
4	C₂×C₇×C₈	—	2	1	no	—
5	C₄²×C₇	C₂²	1	1	yes	characteristic
6	C₄⋊Dic₇	C₄	1	1	yes	characteristic
7	C₄⋊Dic₇	C₄	1	1	yes	characteristic
8	Dic₇⋊C₄	—	2	1	no	—
9	Dic₇⋊C₄	—	2	1	no	—
10	C₂×C₇⋊C₈	—	2	1	no	—
11	#330	—	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	C₇×C₈	—	2	2	no	—
18	C₇⋊C₈	—	2	2	no	—
19	#125	—	7	1	no	—
20	#97	—	7	1	no	—
21	#97	D₇	1	1	yes	characteristic
22	C₂²×C₇	C₂²⋊C₄	1	1	yes	characteristic
23	C₄×C₇	—	2	1	no	—
24	C₄×C₇	—	2	1	no	—
25	C₄×C₇	—	2	1	no	—
26	Dic₇	—	4	1	no	—
27	Dic₇	—	4	1	no	—
28	C₂×C₈	—	2	1	no	—
29	C₂×C₈	—	14	1	no	—
30	C₄⋊C₄	—	7	1	no	—
31	C₄⋊C₄	—	7	1	no	—
32	C₄⋊C₄	—	14	1	no	—
33	C₄⋊C₄	—	14	1	no	—
34	C₄²	C₂×D₇	1	1	yes	characteristic
35	C₂×C₇	#101	1	1	yes	characteristic
36	C₂×C₇	#104	1	1	yes	characteristic
37	C₂×C₇	#104	1	1	yes	characteristic
38	C₂×C₄	—	14	1	no	—
39	C₂×C₄	—	14	1	no	—
40	C₂×C₄	C₄×D₇	1	1	yes	characteristic
41	C₂×C₄	D₂₈	1	1	yes	characteristic
42	C₂×C₄	C₂²⋊D₇	1	1	yes	characteristic
43	C₈	—	2	2	no	—
44	C₈	—	14	2	no	—
45	C₇	#330	1	1	yes	characteristic, complemented, Sylow
46	C₂²	(C₂×D₇)⋊C₄	1	1	yes	characteristic
47	C₄	—	2	1	no	—
48	C₄	—	2	1	no	—
49	C₄	—	2	1	no	—
50	C₄	—	28	1	no	—
51	C₄	—	28	1	no	—
52	C₂	#6379	1	1	yes	characteristic
53	C₂	#6398	1	1	yes	characteristic
54	C₂	#6399	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	(C₂×D₇)⋊C₄
commutator	C₂×C₄×C₇	C₂×C₄
Frattini	C₄²	C₂×D₇
Fitting	C₇×#97	C₂
socle, prime radical	C₂²×C₇	C₂²⋊C₄
cosocle	C₄²×C₇	C₂²
hypercenter	#97	D₇
hypocenter	C₇	#330
LCS	G ▷ C₂×C₄×C₇ ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #97

Complements

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

N	H ≅ G/N	type
45. C₇	#330	semidirect

Maximal subgroups

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

subgroup	index
#6433	2
C₇×#97	2
#6377	2
#330	7