Subgroups — #13062809

Order 1360.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²⋊D₈₅	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₇×C₂²⋊D₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×Dic₁₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#10621757	C₂	1	1	yes	maximal, characteristic, complemented
5	#10621756	C₂	1	1	yes	maximal, characteristic, complemented
6	#10621753	C₂	1	1	yes	maximal, characteristic, complemented
7	D₅×Dic₁₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₅×C₁₇	C₂²	1	1	yes	characteristic
9	C₂×D₈₅	C₂²	1	1	yes	characteristic
10	C₂×D₅×C₁₇	C₂²	1	1	yes	characteristic
11	Dic₈₅	C₂²	1	1	yes	characteristic
12	C₅×Dic₁₇	C₂²	1	1	yes	characteristic
13	C₅×Dic₁₇	C₂²	1	1	yes	characteristic
14	C₁₇×Dic₅	C₂²	1	1	yes	characteristic
15	#63254	—	5	1	no	maximal, Hall
16	C₂×C₅×C₁₇	—	2	1	no	—
17	C₂×C₅×C₁₇	C₂³	1	1	yes	characteristic
18	D₈₅	—	2	1	no	—
19	D₅×C₁₇	—	2	1	no	—
20	D₄×C₁₇	—	5	1	no	—
21	C₂²⋊D₁₇	—	5	1	no	—
22	C₂²⋊D₁₇	—	5	1	no	—
23	C₂×Dic₁₇	—	5	1	no	—
24	C₂×Dic₁₇	D₅	1	1	yes	characteristic, complemented
25	C₄×D₁₇	—	5	1	no	—
26	Dic₃₄	—	5	1	no	—
27	C₅×C₁₇	#41	1	1	yes	characteristic, complemented, Hall
28	#709	—	17	1	no	maximal, Hall
29	C₂²×C₁₇	—	5	1	no	—
30	C₂²×C₁₇	C₂×D₅	1	1	yes	characteristic
31	C₂×D₁₇	—	5	1	no	—
32	C₄×C₁₇	—	5	1	no	—
33	Dic₁₇	—	5	1	no	—
34	Dic₁₇	C₂×D₅	1	1	yes	characteristic
35	Dic₁₇	C₂×D₅	1	1	yes	characteristic
36	C₂×C₄×C₅	—	17	1	no	—
37	C₂²⋊D₅	—	17	1	no	—
38	C₂²⋊D₅	D₁₇	1	1	yes	characteristic, complemented
39	D₂₀	—	17	1	no	—
40	C₄×D₅	—	17	1	no	—
41	C₄×D₅	—	17	1	no	—
42	Dic₁₀	—	17	1	no	—
43	C₂×C₁₇	—	2	1	no	—
44	C₂×C₁₇	—	10	1	no	—
45	C₂×C₁₇	C₂²×D₅	1	1	yes	characteristic
46	D₁₇	—	10	1	no	—
47	C₂²×C₅	C₂×D₁₇	1	1	yes	characteristic
48	C₂×D₅	—	17	1	no	—
49	C₂×D₅	C₂×D₁₇	1	1	yes	characteristic
50	C₄×C₅	—	17	1	no	—
51	C₄×C₅	—	17	1	no	—
52	Dic₅	—	17	1	no	—
53	Dic₅	C₂×D₁₇	1	1	yes	characteristic
54	C₁₇	#709	1	1	yes	characteristic, complemented, Sylow
55	#41	—	85	1	no	Sylow
56	C₂×C₅	—	2	1	no	—
57	C₂×C₅	C₂²×D₁₇	1	1	yes	characteristic
58	D₅	—	2	1	no	—
59	D₅	—	34	1	no	—
60	Q₈	—	85	1	no	—
61	D₄	—	5	1	no	—
62	D₄	—	85	1	no	—
63	D₄	—	85	1	no	—
64	C₂×C₄	—	17	1	no	—
65	C₂×C₄	—	85	1	no	—
66	C₂×C₄	—	85	1	no	—
67	C₅	#63254	1	1	yes	characteristic, complemented, Sylow
68	C₂²	—	5	1	no	—
69	C₂²	—	85	1	no	—
70	C₂²	D₅×D₁₇	1	1	yes	characteristic
71	C₄	—	5	1	no	—
72	C₄	—	17	1	no	—
73	C₄	—	17	1	no	—
74	C₄	—	85	1	no	—
75	C₂	—	2	1	no	—
76	C₂	—	10	1	no	—
77	C₂	—	170	1	no	—
78	C₂	C₂×D₅×D₁₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂×D₅×D₁₇
commutator, socle, cosocle	C₂×C₅×C₁₇	C₂³
Fitting	C₂²×C₅×C₁₇	C₂²
hypercenter	C₂²	D₅×D₁₇
hypocenter	C₅×C₁₇	#41
LCS	G ▷ C₂×C₅×C₁₇ ▷ C₅×C₁₇
UCS	C₁ ◁ C₂ ◁ C₂²

Complements

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

N	H ≅ G/N	type
1. C₂²⋊D₈₅	C₂	semidirect
2. C₁₇×C₂²⋊D₅	C₂	semidirect
3. C₂×C₅×Dic₁₇	C₂	semidirect
4. #10621757	C₂	semidirect
5. #10621756	C₂	semidirect
6. #10621753	C₂	semidirect
7. D₅×Dic₁₇	C₂	semidirect
24. C₂×Dic₁₇	D₅	semidirect
27. C₅×C₁₇	#41	semidirect
38. C₂²⋊D₅	D₁₇	semidirect
54. C₁₇	#709	semidirect
67. C₅	#63254	semidirect

Maximal subgroups

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

subgroup	index
C₂²⋊D₈₅	2
C₁₇×C₂²⋊D₅	2
C₂×C₅×Dic₁₇	2
#10621757	2
#10621756	2
#10621753	2
D₅×Dic₁₇	2
#63254	5
#709	17