Subgroups — #13062817

Order 1360.

108 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	C₂×D₅×D₁₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₇×C₂²⋊D₅	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×C₂²⋊D₁₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#10621756	C₂	1	1	yes	maximal, characteristic, complemented
6	#10621755	C₂	1	1	yes	maximal, characteristic, complemented
7	#10621753	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₅×C₁₇	C₂²	1	1	yes	characteristic, complemented
9	C₂×D₈₅	—	2	2	no	—
10	C₂×D₈₅	C₂²	1	1	yes	characteristic
11	C₂×D₈₅	C₂²	1	1	yes	characteristic, complemented
12	C₂×D₅×C₁₇	C₂²	1	1	yes	characteristic, complemented
13	C₂×C₅×D₁₇	C₂²	1	1	yes	characteristic, complemented
14	D₅×D₁₇	—	2	2	no	—
15	C₅×Dic₁₇	C₂²	1	1	yes	characteristic, complemented
16	C₁₇×Dic₅	C₂²	1	1	yes	characteristic, complemented
17	D₄×D₁₇	—	5	1	no	maximal, Hall
18	C₂×C₅×C₁₇	—	2	1	no	—
19	C₂×C₅×C₁₇	C₂³	1	1	yes	characteristic
20	D₈₅	—	2	1	no	—
21	D₈₅	D₄	1	2	yes	complemented
22	C₅×D₁₇	—	2	1	no	—
23	D₅×C₁₇	—	2	1	no	—
24	C₂²×D₁₇	—	5	1	no	—
25	C₂²×D₁₇	—	5	1	no	—
26	D₄×C₁₇	—	5	1	no	—
27	C₂²⋊D₁₇	—	5	1	no	—
28	C₂²⋊D₁₇	D₅	1	1	yes	characteristic, complemented
29	D₆₈	—	5	1	no	—
30	C₄×D₁₇	—	5	1	no	—
31	C₅×C₁₇	C₂×D₄	1	1	yes	characteristic, complemented, Hall
32	D₄×D₅	—	17	1	no	maximal, Hall
33	C₂²×C₁₇	—	5	1	no	—
34	C₂²×C₁₇	C₂×D₅	1	1	yes	characteristic, complemented
35	C₂×D₁₇	—	5	1	no	—
36	C₂×D₁₇	—	5	1	no	—
37	C₂×D₁₇	—	10	2	no	—
38	C₂×D₁₇	—	10	2	no	—
39	C₂×D₁₇	C₂×D₅	1	1	yes	characteristic, complemented
40	C₄×C₁₇	—	5	1	no	—
41	Dic₁₇	C₂×D₅	1	1	yes	characteristic, complemented
42	C₂²×D₅	—	17	1	no	—
43	C₂²×D₅	—	17	1	no	—
44	C₅×D₄	—	17	1	no	—
45	C₂²⋊D₅	—	17	1	no	—
46	C₂²⋊D₅	D₁₇	1	1	yes	characteristic, complemented
47	D₂₀	—	17	1	no	—
48	C₄×D₅	—	17	1	no	—
49	C₂×C₁₇	—	2	1	no	—
50	C₂×C₁₇	—	10	1	no	—
51	C₂×C₁₇	C₂²×D₅	1	1	yes	characteristic
52	D₁₇	—	2	1	no	—
53	D₁₇	—	5	2	no	—
54	D₁₇	—	10	1	no	—
55	C₂²×C₅	—	17	1	no	—
56	C₂²×C₅	C₂×D₁₇	1	1	yes	characteristic, complemented
57	C₂×D₅	—	17	1	no	—
58	C₂×D₅	—	17	1	no	—
59	C₂×D₅	—	34	2	no	—
60	C₂×D₅	—	34	2	no	—
61	C₂×D₅	C₂×D₁₇	1	1	yes	characteristic, complemented
62	C₄×C₅	—	17	1	no	—
63	Dic₅	C₂×D₁₇	1	1	yes	characteristic, complemented
64	C₁₇	D₄×D₅	1	1	yes	characteristic, complemented, Sylow
65	C₂×D₄	—	85	1	no	Sylow
66	C₂×C₅	—	2	1	no	—
67	C₂×C₅	—	34	1	no	—
68	C₂×C₅	C₂²×D₁₇	1	1	yes	characteristic
69	D₅	—	2	1	no	—
70	D₅	—	17	2	no	—
71	D₅	—	34	1	no	—
72	C₂³	—	85	1	no	—
73	C₂³	—	85	1	no	—
74	D₄	—	5	1	no	—
75	D₄	—	17	1	no	—
76	D₄	—	85	1	no	—
77	D₄	—	85	1	no	—
78	C₂×C₄	—	85	1	no	—
79	C₅	D₄×D₁₇	1	1	yes	characteristic, complemented, Sylow
80	C₂²	—	5	1	no	—
81	C₂²	—	17	1	no	—
82	C₂²	—	85	1	no	—
83	C₂²	—	85	1	no	—
84	C₂²	—	170	2	no	—
85	C₂²	—	170	2	no	—
86	C₂²	D₅×D₁₇	1	1	yes	characteristic, complemented
87	C₄	—	5	1	no	—
88	C₄	—	17	1	no	—
89	C₂	—	2	1	no	—
90	C₂	—	10	1	no	—
91	C₂	—	34	1	no	—
92	C₂	—	85	2	no	—
93	C₂	—	170	1	no	—
94	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	C₂×D₄
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. C₂×D₅×D₁₇	C₂	semidirect
3. C₁₇×C₂²⋊D₅	C₂	semidirect
4. C₅×C₂²⋊D₁₇	C₂	semidirect
5. #10621756	C₂	semidirect
6. #10621755	C₂	semidirect
7. #10621753	C₂	semidirect
8. C₂²×C₅×C₁₇	C₂²	semidirect
11. C₂×D₈₅	C₂²	semidirect
12. C₂×D₅×C₁₇	C₂²	semidirect
13. C₂×C₅×D₁₇	C₂²	semidirect
15. C₅×Dic₁₇	C₂²	semidirect
16. C₁₇×Dic₅	C₂²	semidirect
21. D₈₅	D₄	semidirect
28. C₂²⋊D₁₇	D₅	semidirect
31. C₅×C₁₇	C₂×D₄	semidirect
34. C₂²×C₁₇	C₂×D₅	semidirect
39. C₂×D₁₇	C₂×D₅	semidirect
41. Dic₁₇	C₂×D₅	semidirect
46. C₂²⋊D₅	D₁₇	semidirect
56. C₂²×C₅	C₂×D₁₇	semidirect
61. C₂×D₅	C₂×D₁₇	semidirect
63. Dic₅	C₂×D₁₇	semidirect
64. C₁₇	D₄×D₅	semidirect
79. C₅	D₄×D₁₇	semidirect
86. C₂²	D₅×D₁₇	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
C₂×D₅×D₁₇	2
C₁₇×C₂²⋊D₅	2
C₅×C₂²⋊D₁₇	2
#10621756	2
#10621755	2
#10621753	2
D₄×D₁₇	5
D₄×D₅	17