Subgroups — #11765202

Order 1120.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₃₅	C₂	1	1	yes	maximal, characteristic
2	C₄²×C₅×C₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#10587902	C₂	1	2	yes	maximal
4	#10587900	C₂	1	2	yes	maximal, complemented
5	C₄×Dic₃₅	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₃₅	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₅×C₇	C₂²	1	2	yes	—
8	C₂×C₄×C₅×C₇	C₂²	1	1	yes	characteristic
9	C₂×Dic₃₅	C₂²	1	2	yes	—
10	C₂×Dic₃₅	C₂²	1	1	yes	characteristic
11	C₄×D₃₅	C₄	1	4	yes	complemented
12	#6435	—	5	1	no	maximal, Hall
13	#4018	—	7	1	no	maximal, Hall
14	C₂²×C₅×C₇	C₂³	1	1	yes	characteristic
15	C₂×D₃₅	—	2	2	no	—
16	C₂×D₃₅	C₂×C₄	1	2	yes	—
17	C₄×C₅×C₇	—	2	2	no	—
18	C₄×C₅×C₇	C₂×C₄	1	2	yes	—
19	Dic₃₅	—	2	2	no	—
20	Dic₃₅	C₂×C₄	1	2	yes	—
21	C₂×C₄×D₇	—	5	1	no	—
22	C₄²×C₇	D₅	1	1	yes	characteristic, complemented
23	(C₂×D₇)⋊C₄	—	5	2	no	—
24	Dic₇⋊C₄	—	5	2	no	—
25	C₄×Dic₇	—	5	1	no	—
26	C₂×C₄×D₅	—	7	1	no	—
27	C₄²×C₅	D₇	1	1	yes	characteristic, complemented
28	#685	—	7	2	no	—
29	#683	—	7	2	no	—
30	C₄×Dic₅	—	7	1	no	—
31	C₂×C₅×C₇	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₅×C₇	#41	1	2	yes	—
33	D₃₅	—	2	2	no	—
34	C₂²×D₇	—	5	1	no	—
35	C₂×C₄×C₇	C₂×D₅	1	2	yes	—
36	C₂×C₄×C₇	C₂×D₅	1	1	yes	characteristic
37	C₂×Dic₇	—	5	1	no	—
38	C₂×Dic₇	—	5	2	no	—
39	C₄×D₇	—	5	4	no	—
40	C₂²×D₅	—	7	1	no	—
41	C₂×C₄×C₅	C₂×D₇	1	2	yes	—
42	C₂×C₄×C₅	C₂×D₇	1	1	yes	characteristic
43	C₂×Dic₅	—	7	1	no	—
44	C₂×Dic₅	—	7	2	no	—
45	C₄×D₅	—	7	4	no	—
46	C₅×C₇	#117	1	1	yes	characteristic, complemented, Hall
47	#117	—	35	1	no	Sylow
48	C₂²×C₇	C₂²×D₅	1	1	yes	characteristic
49	C₂×D₇	—	5	2	no	—
50	C₂×D₇	—	10	2	no	—
51	C₄×C₇	—	2	2	no	—
52	C₄×C₇	C₄×D₅	1	2	yes	—
53	Dic₇	—	5	2	no	—
54	Dic₇	—	10	2	no	—
55	C₂²×C₅	C₂²×D₇	1	1	yes	characteristic
56	C₂×D₅	—	7	2	no	—
57	C₂×D₅	—	14	2	no	—
58	C₄×C₅	—	2	2	no	—
59	C₄×C₅	C₄×D₇	1	2	yes	—
60	Dic₅	—	7	2	no	—
61	Dic₅	—	14	2	no	—
62	C₂²×C₄	—	35	1	no	—
63	C₄⋊C₄	—	35	2	no	—
64	C₂²⋊C₄	—	35	2	no	—
65	C₄²	—	35	1	no	—
66	C₄²	D₃₅	1	1	yes	characteristic, complemented
67	C₂×C₇	C₂×C₄×D₅	1	1	yes	characteristic
68	C₂×C₇	#709	1	2	yes	—
69	D₇	—	10	2	no	—
70	C₂×C₅	C₂×C₄×D₇	1	1	yes	characteristic
71	C₂×C₅	#1157	1	2	yes	—
72	D₅	—	14	2	no	—
73	C₂³	—	35	1	no	—
74	C₂×C₄	—	35	1	no	—
75	C₂×C₄	—	35	4	no	—
76	C₂×C₄	—	35	2	no	—
77	C₂×C₄	C₂×D₃₅	1	2	yes	—
78	C₂×C₄	C₂×D₃₅	1	1	yes	characteristic
79	C₇	#4018	1	1	yes	characteristic, complemented, Sylow
80	C₅	#6435	1	1	yes	characteristic, complemented, Sylow
81	C₂²	—	35	2	no	—
82	C₂²	—	70	2	no	—
83	C₂²	C₂²×D₃₅	1	1	yes	characteristic
84	C₄	—	2	2	no	—
85	C₄	—	35	2	no	—
86	C₄	—	70	2	no	—
87	C₄	C₄×D₃₅	1	2	yes	—
88	C₂	—	70	2	no	—
89	C₂	C₂×C₄×D₃₅	1	1	yes	characteristic
90	C₂	#10587987	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₃₅
commutator	C₂×C₅×C₇	C₂²×C₄
Frattini	C₂²	C₂²×D₃₅
Fitting	C₄²×C₅×C₇	C₂
socle, cosocle	C₂²×C₅×C₇	C₂³
prime radical	C₂²×D₃₅	C₂²
hypercenter	C₄²	D₃₅
hypocenter	C₅×C₇	#117
LCS	G ▷ C₂×C₅×C₇ ▷ C₅×C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄²

Complements

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

N	H ≅ G/N	type
2. C₄²×C₅×C₇	C₂	semidirect
4. #10587900	C₂	semidirect
5. C₄×Dic₃₅	C₂	semidirect
11. C₄×D₃₅	C₄	semidirect
22. C₄²×C₇	D₅	semidirect
27. C₄²×C₅	D₇	semidirect
46. C₅×C₇	#117	semidirect
66. C₄²	D₃₅	semidirect
79. C₇	#4018	semidirect
80. C₅	#6435	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₃₅	2
C₄²×C₅×C₇	2
#10587902	2
#10587900	2
C₄×Dic₃₅	2
#6435	5
#4018	7