Subgroups — #11764757

Order 1120.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₇×D₅	C₂	1	1	yes	maximal, characteristic
2	C₄×Dic₃₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₅×Dic₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#10587859	C₂	1	2	yes	maximal, complemented
5	#10587855	C₂	1	2	yes	maximal
6	C₂²×C₇×D₅	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₅×C₇	C₂²	1	1	yes	characteristic
8	C₂×Dic₃₅	C₂²	1	2	yes	—
9	C₂×C₇×Dic₅	C₂²	1	1	yes	characteristic
10	C₄×C₇×D₅	C₄	1	4	yes	complemented
11	C₂×C₅×Dic₇	C₂²	1	2	yes	—
12	#6489	—	5	1	no	maximal, Hall
13	#4018	—	7	1	no	maximal, Hall
14	C₂²×C₅×C₇	C₂³	1	1	yes	characteristic
15	C₂×C₇×D₅	—	2	2	no	—
16	C₂×C₇×D₅	C₂×C₄	1	2	yes	—
17	C₄×C₅×C₇	C₂×C₄	1	2	yes	—
18	Dic₃₅	—	2	2	no	—
19	C₅×Dic₇	—	2	2	no	—
20	C₇×Dic₅	C₂×C₄	1	2	yes	—
21	C₂²×C₄×C₇	—	5	1	no	—
22	C₂²⋊Dic₇	—	5	2	no	—
23	C₄⋊Dic₇	—	5	2	no	—
24	C₄×Dic₇	—	5	1	no	—
25	C₄×Dic₇	D₅	1	1	yes	characteristic, complemented
26	C₂×C₄×D₅	D₇	1	1	yes	characteristic
27	C₄²×C₅	—	7	1	no	—
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	C₇×D₅	—	2	2	no	—
34	C₂³×C₇	—	5	1	no	—
35	C₂×C₄×C₇	—	5	1	no	—
36	C₂×C₄×C₇	—	5	4	no	—
37	C₂×C₄×C₇	C₂×D₅	1	1	yes	characteristic
38	C₂×Dic₇	—	5	2	no	—
39	C₂×Dic₇	C₂×D₅	1	2	yes	—
40	C₂²×D₅	C₂×D₇	1	1	yes	characteristic
41	C₂×C₄×C₅	—	7	2	no	—
42	C₂×C₄×C₅	C₂×D₇	1	1	yes	characteristic
43	C₂×Dic₅	—	7	2	no	—
44	C₂×Dic₅	C₂×D₇	1	1	yes	characteristic
45	C₄×D₅	Dic₇	1	4	yes	complemented
46	C₅×C₇	#117	1	1	yes	characteristic, complemented, Hall
47	#117	—	35	1	no	Sylow
48	C₂²×C₇	—	5	2	no	—
49	C₂²×C₇	—	10	2	no	—
50	C₂²×C₇	C₂²×D₅	1	1	yes	characteristic
51	C₄×C₇	—	5	2	no	—
52	C₄×C₇	C₄×D₅	1	2	yes	—
53	Dic₇	—	2	2	no	—
54	Dic₇	—	10	2	no	—
55	C₂²×C₅	C₂²×D₇	1	1	yes	characteristic
56	C₂×D₅	—	2	2	no	—
57	C₂×D₅	C₂×Dic₇	1	2	yes	—
58	C₄×C₅	—	14	2	no	—
59	C₄×C₅	C₂×Dic₇	1	2	yes	—
60	Dic₅	—	14	2	no	—
61	Dic₅	C₂×Dic₇	1	2	yes	—
62	C₂²×C₄	—	5	1	no	—
63	C₄⋊C₄	—	35	2	no	—
64	C₂²⋊C₄	—	35	2	no	—
65	C₄²	—	7	1	no	—
66	C₄²	—	35	1	no	—
67	C₂×C₇	—	10	2	no	—
68	C₂×C₇	C₂×C₄×D₅	1	1	yes	characteristic
69	C₂×C₇	#709	1	2	yes	—
70	C₂×C₅	#1157	1	2	yes	—
71	C₂×C₅	C₂²×Dic₇	1	1	yes	characteristic
72	D₅	—	2	2	no	—
73	C₂³	—	5	1	no	—
74	C₂×C₄	—	5	1	no	—
75	C₂×C₄	—	5	4	no	—
76	C₂×C₄	—	7	2	no	—
77	C₂×C₄	—	35	2	no	—
78	C₂×C₄	D₅×D₇	1	1	yes	characteristic
79	C₇	#4018	1	1	yes	characteristic, complemented, Sylow
80	C₅	#6489	1	1	yes	characteristic, complemented, Sylow
81	C₂²	—	5	2	no	—
82	C₂²	—	10	2	no	—
83	C₂²	C₂×D₅×D₇	1	1	yes	characteristic
84	C₄	—	5	2	no	—
85	C₄	—	14	2	no	—
86	C₄	—	70	2	no	—
87	C₄	D₅×Dic₇	1	2	yes	—
88	C₂	—	10	2	no	—
89	C₂	#10587944	1	2	yes	—
90	C₂	C₂×D₅×Dic₇	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₄	D₅×D₇
commutator	C₂×C₅×C₇	C₂²×C₄
Frattini	C₂²	C₂×D₅×D₇
Fitting	C₂×C₄×C₅×C₇	C₂²
socle, cosocle	C₂²×C₅×C₇	C₂³
prime radical	C₂²×C₇×D₅	C₂²
hypocenter	C₅×C₇	#117
LCS	G ▷ C₂×C₅×C₇ ▷ C₅×C₇
UCS	C₁ ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
2. C₄×Dic₃₅	C₂	semidirect
3. C₄×C₅×Dic₇	C₂	semidirect
4. #10587859	C₂	semidirect
10. C₄×C₇×D₅	C₄	semidirect
25. C₄×Dic₇	D₅	semidirect
45. C₄×D₅	Dic₇	semidirect
46. C₅×C₇	#117	semidirect
79. C₇	#4018	semidirect
80. C₅	#6489	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₇×D₅	2
C₄×Dic₃₅	2
C₄×C₅×Dic₇	2
#10587859	2
#10587855	2
#6489	5
#4018	7