Subgroups — #89839

Order 448.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	#6437	C₂	1	1	yes	maximal, characteristic
3	#6383	C₂	1	4	yes	maximal, complemented
4	#6377	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₂₈	—	2	2	no	—
6	C₂×D₂₈	C₂²	1	2	yes	—
7	C₇×C₄⋊C₄	—	2	2	no	—
8	C₇×C₄⋊C₄	C₂²	1	2	yes	—
9	C₄²×C₇	C₂²	1	1	yes	characteristic
10	C₂×C₇⋊C₈	C₂²	1	2	yes	—
11	#490	—	7	1	no	maximal, Sylow
12	C₂²×D₇	—	2	2	no	—
13	C₂×C₄×C₇	—	2	2	no	—
14	C₂×C₄×C₇	D₄	1	2	yes	—
15	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
16	D₂₈	—	2	4	no	—
17	D₂₈	—	4	4	no	—
18	C₇⋊C₈	—	2	2	no	—
19	#127	—	7	1	no	—
20	#125	D₇	1	1	yes	characteristic, complemented
21	#102	—	7	4	no	—
22	#97	—	7	1	no	—
23	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
24	C₂×D₇	—	4	2	no	—
25	C₂×D₇	—	4	4	no	—
26	C₄×C₇	—	2	2	no	—
27	C₄×C₇	—	4	2	no	—
28	C₄×C₇	#41	1	2	yes	—
29	C₂×D₄	—	7	2	no	—
30	C₂×D₄	—	14	2	no	—
31	C₂×C₈	—	7	2	no	—
32	C₄⋊C₄	—	2	2	no	—
33	C₄⋊C₄	C₂×D₇	1	2	yes	—
34	C₄²	C₂×D₇	1	1	yes	characteristic
35	C₂×C₇	#124	1	1	yes	characteristic
36	C₂×C₇	AGL(1,ℤ/8)	1	2	yes	—
37	D₇	—	8	2	no	—
38	C₂³	—	14	2	no	—
39	D₄	—	14	4	no	—
40	D₄	—	28	4	no	—
41	C₂×C₄	—	2	2	no	—
42	C₂×C₄	C₂²⋊D₇	1	2	yes	—
43	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
44	C₈	—	14	2	no	—
45	C₇	#490	1	1	yes	characteristic, complemented, Sylow
46	C₂²	—	28	2	no	—
47	C₂²	—	28	4	no	—
48	C₂²	C₂×C₂²⋊D₇	1	1	yes	characteristic
49	C₄	—	2	2	no	—
50	C₄	—	4	2	no	—
51	C₄	#1161	1	2	yes	—
52	C₂	—	56	2	no	—
53	C₂	#6510	1	1	yes	characteristic
54	C₂	#6512	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₇
commutator, cosocle	C₂×C₄×C₇	C₂³
Frattini	C₂×C₄	C₂²×D₇
Fitting	C₇×#125	C₂
socle	C₂²×C₇	C₂×D₄
prime radical	#6437	C₂
hypercenter	#125	D₇
hypocenter	C₇	#490
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #125

Complements

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

N	H ≅ G/N	type
1. C₇×#125	C₂	semidirect
3. #6383	C₂	semidirect
4. #6377	C₂	semidirect
20. #125	D₇	semidirect
45. C₇	#490	semidirect

Maximal subgroups

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

subgroup	index
C₇×#125	2
#6437	2
#6383	2
#6377	2
#490	7