Subgroups — #11764688

Order 1120.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×C₂²⋊D₅	C₂	1	1	yes	maximal, characteristic
2	C₂²⋊(C₇⋊AGL(1,𝔽₅))	C₂	1	2	yes	maximal, complemented
3	C₂³×C₅×C₇	C₄	1	1	yes	characteristic, complemented
4	C₂²×C₇×D₅	C₂²	1	1	yes	characteristic
5	C₂×C₇⋊AGL(1,𝔽₅)	—	2	2	no	—
6	C₇×C₂²⋊D₅	—	2	2	no	—
7	C₂×C₇×Dic₅	C₄	1	1	yes	characteristic, complemented
8	#6408	—	5	1	no	maximal, Hall
9	#4011	—	7	1	no	maximal, Hall
10	C₂²×C₅×C₇	—	2	1	no	—
11	C₂²×C₅×C₇	—	4	1	no	—
12	C₂²×C₅×C₇	C₂×C₄	1	1	yes	characteristic
13	C₂×C₇×D₅	—	4	1	no	—
14	C₂×C₇×D₅	D₄	1	2	yes	—
15	C₇⋊AGL(1,𝔽₅)	—	4	2	no	—
16	C₇×Dic₅	—	2	1	no	—
17	C₂×C₇×D₄	—	5	1	no	—
18	C₂²⋊Dic₇	—	5	2	no	—
19	C₂×C₂²⋊D₅	D₇	1	1	yes	characteristic
20	C₂²⋊AGL(1,𝔽₅)	—	7	2	no	—
21	C₂×C₅×C₇	—	2	1	no	—
22	C₂×C₅×C₇	—	4	1	no	—
23	C₂×C₅×C₇	C₂²⋊C₄	1	1	yes	characteristic
24	C₇×D₅	—	2	2	no	—
25	C₂³×C₇	—	5	1	no	—
26	C₂³×C₇	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
27	C₇×D₄	—	10	2	no	—
28	C₂×C₄×C₇	—	5	1	no	—
29	C₂×Dic₇	—	10	2	no	—
30	C₂³×C₅	Dic₇	1	1	yes	characteristic, complemented
31	C₂²×D₅	C₂×D₇	1	1	yes	characteristic
32	C₂×AGL(1,𝔽₅)	—	14	2	no	—
33	C₂²⋊D₅	—	2	2	no	—
34	C₂×Dic₅	Dic₇	1	1	yes	characteristic, complemented
35	C₅×C₇	#99	1	1	yes	characteristic, complemented, Hall
36	#99	—	35	1	no	Sylow
37	C₂²×C₇	—	2	1	no	—
38	C₂²×C₇	—	4	1	no	—
39	C₂²×C₇	—	5	2	no	—
40	C₂²×C₇	—	20	1	no	—
41	C₂²×C₇	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
42	C₄×C₇	—	10	1	no	—
43	Dic₇	—	20	2	no	—
44	C₂²×C₅	—	2	1	no	—
45	C₂²×C₅	—	4	1	no	—
46	C₂²×C₅	C₂×Dic₇	1	1	yes	characteristic
47	C₂×D₅	—	4	1	no	—
48	C₂×D₅	C₂²⋊D₇	1	2	yes	—
49	AGL(1,𝔽₅)	—	28	2	no	—
50	Dic₅	—	2	1	no	—
51	C₂×D₄	—	5	1	no	—
52	C₂²⋊C₄	—	35	2	no	—
53	C₂×C₇	—	2	1	no	—
54	C₂×C₇	—	4	1	no	—
55	C₂×C₇	—	10	2	no	—
56	C₂×C₇	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
57	C₂×C₅	—	2	1	no	—
58	C₂×C₅	—	4	1	no	—
59	C₂×C₅	C₂²⋊Dic₇	1	1	yes	characteristic
60	D₅	—	2	2	no	—
61	C₂³	—	5	1	no	—
62	C₂³	C₇⋊AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
63	D₄	—	10	2	no	—
64	C₂×C₄	—	5	1	no	—
65	C₂×C₄	—	70	2	no	—
66	C₇	#4011	1	1	yes	characteristic, complemented, Sylow
67	C₅	#6408	1	1	yes	characteristic, complemented, Sylow
68	C₂²	—	2	1	no	—
69	C₂²	—	4	1	no	—
70	C₂²	—	5	2	no	—
71	C₂²	—	20	1	no	—
72	C₂²	C₂×C₇⋊AGL(1,𝔽₅)	1	1	yes	characteristic
73	C₄	—	10	1	no	—
74	C₄	—	140	2	no	—
75	C₂	—	2	1	no	—
76	C₂	—	4	1	no	—
77	C₂	—	10	2	no	—
78	C₂	C₂²⋊(C₇⋊AGL(1,𝔽₅))	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²⋊(C₇⋊AGL(1,𝔽₅))
commutator	C₂²×C₅×C₇	C₂×C₄
Frattini	C₂²	C₂×C₇⋊AGL(1,𝔽₅)
Fitting	C₂³×C₅×C₇	C₄
socle	C₂×C₅×C₇	C₂²⋊C₄
prime radical	C₂×C₇×C₂²⋊D₅	C₂
cosocle	C₂²×C₇×D₅	C₂²
hypercenter	C₂³	C₇⋊AGL(1,𝔽₅)
hypocenter	C₅×C₇	#99
LCS	G ▷ C₂²×C₅×C₇ ▷ C₂×C₅×C₇ ▷ C₅×C₇
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂³

Complements

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

N	H ≅ G/N	type
2. C₂²⋊(C₇⋊AGL(1,𝔽₅))	C₂	semidirect
3. C₂³×C₅×C₇	C₄	semidirect
7. C₂×C₇×Dic₅	C₄	semidirect
26. C₂³×C₇	AGL(1,𝔽₅)	semidirect
30. C₂³×C₅	Dic₇	semidirect
34. C₂×Dic₅	Dic₇	semidirect
35. C₅×C₇	#99	semidirect
62. C₂³	C₇⋊AGL(1,𝔽₅)	semidirect
66. C₇	#4011	semidirect
67. C₅	#6408	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×C₂²⋊D₅	2
C₂²⋊(C₇⋊AGL(1,𝔽₅))	2
#6408	5
#4011	7