Subgroups — #11764683

Order 1120.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇⋊AGL(1,𝔽₅)	C₂	1	2	yes	maximal
2	C₂×C₄×C₇×D₅	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₇×D₅	C₂²	1	1	yes	characteristic
4	C₂×C₇⋊AGL(1,𝔽₅)	—	2	2	no	—
5	C₂×C₇⋊AGL(1,𝔽₅)	—	2	2	no	—
6	C₂×C₇⋊AGL(1,𝔽₅)	C₄	1	4	yes	complemented
7	C₂×C₄×C₅×C₇	C₄	1	1	yes	characteristic, complemented
8	C₂×C₇×Dic₅	C₄	1	1	yes	characteristic, complemented
9	C₄×C₇×D₅	—	2	2	no	—
10	C₇⋊#95	—	5	1	no	maximal, Hall
11	(C₂×AGL(1,𝔽₅))⋊C₄	—	7	1	no	maximal, Hall
12	C₂²×C₅×C₇	C₂×C₄	1	1	yes	characteristic
13	C₂×C₇×D₅	C₂×C₄	1	2	yes	—
14	C₂×C₇×D₅	D₄	1	1	yes	characteristic
15	C₂×C₇×D₅	D₄	1	2	yes	—
16	C₂×C₇×D₅	Q₈	1	1	yes	characteristic
17	C₇⋊AGL(1,𝔽₅)	—	2	4	no	—
18	C₄×C₅×C₇	—	2	1	no	—
19	C₇×Dic₅	—	2	1	no	—
20	C₂²×C₄×C₇	—	5	1	no	—
21	C₂²×Dic₇	—	5	2	no	—
22	C₂²×AGL(1,𝔽₅)	—	7	2	no	—
23	C₂×C₄×D₅	D₇	1	1	yes	characteristic
24	C₂×C₅×C₇	C₄²	1	1	yes	characteristic
25	C₂×C₅×C₇	C₂²⋊C₄	1	1	yes	characteristic
26	C₂×C₅×C₇	C₄⋊C₄	1	1	yes	characteristic
27	C₇×D₅	C₂²⋊C₄	1	2	yes	—
28	C₇×D₅	C₄⋊C₄	1	2	yes	—
29	C₂³×C₇	—	5	1	no	—
30	C₂×C₄×C₇	—	5	1	no	—
31	C₂×C₄×C₇	—	10	2	no	—
32	C₂×C₄×C₇	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
33	C₂×Dic₇	—	5	4	no	—
34	C₂×Dic₇	—	10	2	no	—
35	C₂×Dic₇	—	10	2	no	—
36	C₂²×D₅	C₂×D₇	1	1	yes	characteristic
37	C₂×AGL(1,𝔽₅)	—	7	4	no	—
38	C₂×AGL(1,𝔽₅)	—	14	2	no	—
39	C₂×AGL(1,𝔽₅)	—	14	2	no	—
40	C₂×C₄×C₅	Dic₇	1	1	yes	characteristic, complemented
41	C₂×Dic₅	Dic₇	1	1	yes	characteristic, complemented
42	C₄×D₅	—	2	2	no	—
43	C₅×C₇	#95	1	1	yes	characteristic, complemented, Hall
44	#95	—	35	1	no	Sylow
45	C₂²×C₇	—	5	2	no	—
46	C₂²×C₇	—	5	1	no	—
47	C₂²×C₇	—	5	1	no	—
48	C₂²×C₇	—	5	2	no	—
49	C₂²×C₇	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
50	C₄×C₇	—	2	1	no	—
51	C₄×C₇	—	10	1	no	—
52	Dic₇	—	10	4	no	—
53	C₂²×C₅	C₂×Dic₇	1	1	yes	characteristic
54	C₂×D₅	Dic₁₄	1	1	yes	characteristic
55	C₂×D₅	C₄×D₇	1	2	yes	—
56	C₂×D₅	D₂₈	1	1	yes	characteristic
57	C₂×D₅	C₂²⋊D₇	1	2	yes	—
58	AGL(1,𝔽₅)	—	14	4	no	—
59	C₄×C₅	—	2	1	no	—
60	Dic₅	—	2	1	no	—
61	C₂²×C₄	—	5	1	no	—
62	C₂²×C₄	—	35	2	no	—
63	C₂×C₇	—	5	2	no	—
64	C₂×C₇	—	5	2	no	—
65	C₂×C₇	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
66	C₂×C₇	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
67	C₂×C₇	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
68	C₂×C₅	C₄×Dic₇	1	1	yes	characteristic
69	C₂×C₅	C₄⋊Dic₇	1	1	yes	characteristic
70	C₂×C₅	C₂²⋊Dic₇	1	1	yes	characteristic
71	D₅	Dic₇⋊C₄	1	2	yes	—
72	D₅	(C₂×D₇)⋊C₄	1	2	yes	—
73	C₂³	—	5	1	no	—
74	C₂×C₄	—	5	1	no	—
75	C₂×C₄	—	10	2	no	—
76	C₂×C₄	—	35	4	no	—
77	C₂×C₄	—	70	2	no	—
78	C₂×C₄	—	70	2	no	—
79	C₂×C₄	C₇⋊AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
80	C₇	(C₂×AGL(1,𝔽₅))⋊C₄	1	1	yes	characteristic, complemented, Sylow
81	C₅	C₇⋊#95	1	1	yes	characteristic, complemented, Sylow
82	C₂²	—	5	2	no	—
83	C₂²	—	5	1	no	—
84	C₂²	—	5	1	no	—
85	C₂²	—	5	2	no	—
86	C₂²	C₂×C₇⋊AGL(1,𝔽₅)	1	1	yes	characteristic
87	C₄	—	2	1	no	—
88	C₄	—	10	1	no	—
89	C₄	—	70	4	no	—
90	C₂	—	5	2	no	—
91	C₂	—	5	2	no	—
92	C₂	C₄×C₇⋊AGL(1,𝔽₅)	1	1	yes	characteristic
93	C₂	C₄⋊(C₇⋊AGL(1,𝔽₅))	1	1	yes	characteristic
94	C₂	C₂²⋊(C₇⋊AGL(1,𝔽₅))	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂×C₇⋊AGL(1,𝔽₅)
commutator	C₂×C₅×C₇	C₄²
Fitting	C₂×C₄×C₅×C₇	C₄
socle	C₂²×C₅×C₇	C₂×C₄
prime radical, cosocle	C₂²×C₇×D₅	C₂²
hypercenter	C₂×C₄	C₇⋊AGL(1,𝔽₅)
hypocenter	C₅×C₇	#95
LCS	G ▷ C₂×C₅×C₇ ▷ C₅×C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
6. C₂×C₇⋊AGL(1,𝔽₅)	C₄	semidirect
7. C₂×C₄×C₅×C₇	C₄	semidirect
8. C₂×C₇×Dic₅	C₄	semidirect
32. C₂×C₄×C₇	AGL(1,𝔽₅)	semidirect
40. C₂×C₄×C₅	Dic₇	semidirect
41. C₂×Dic₅	Dic₇	semidirect
43. C₅×C₇	#95	semidirect
79. C₂×C₄	C₇⋊AGL(1,𝔽₅)	semidirect
80. C₇	(C₂×AGL(1,𝔽₅))⋊C₄	semidirect
81. C₅	C₇⋊#95	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₇⋊AGL(1,𝔽₅)	2
C₂×C₄×C₇×D₅	2
C₇⋊#95	5
(C₂×AGL(1,𝔽₅))⋊C₄	7