Subgroups — #11763552

Order 1088.

260 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₇×#120	C₂	1	1	yes	maximal, characteristic, complemented
2	#10587639	C₂	1	2	yes	maximal, complemented
3	#10587623	C₂	1	3	yes	maximal, complemented
4	#10587622	C₂	1	3	yes	maximal, complemented
5	#10587571	C₂	1	6	yes	maximal, complemented
6	C₂⁴×C₁₇	C₂²	1	1	yes	characteristic, complemented
7	C₂³×D₁₇	C₂²	1	1	yes	characteristic, complemented
8	C₂×D₄×C₁₇	C₂²	1	3	yes	—
9	C₂×C₂²⋊D₁₇	C₂²	1	6	yes	—
10	C₂×Dic₃₄	C₂²	1	3	yes	complemented
11	C₂²⋊C₄×C₁₇	C₂²	1	3	yes	complemented
12	C₂²⋊Dic₁₇	C₂²	1	6	yes	complemented
13	C₂²⋊Dic₁₇	C₂²	1	3	yes	complemented
14	#63227	C₂²	1	6	yes	complemented
15	C₄×Dic₁₇	C₂²	1	3	yes	complemented
16	C₂³×C₁₇	—	4	3	no	—
17	C₂³×C₁₇	C₂³	1	1	yes	characteristic
18	C₂³×C₁₇	C₂³	1	3	yes	—
19	C₂²×D₁₇	—	4	3	no	—
20	C₂²×D₁₇	C₂³	1	2	yes	—
21	D₄×C₁₇	—	4	3	no	—
22	C₂×C₄×C₁₇	C₂³	1	3	yes	—
23	C₂²⋊D₁₇	—	4	6	no	—
24	C₂×Dic₁₇	C₂³	1	6	yes	—
25	Dic₃₄	—	4	3	no	—
26	C₂²×C₁₇	—	2	6	no	—
27	C₂²×C₁₇	—	2	3	no	—
28	C₂²×C₁₇	—	2	3	no	—
29	C₂²×C₁₇	—	4	4	no	—
30	C₂²×C₁₇	C₂⁴	1	1	yes	characteristic
31	C₂×D₁₇	—	2	6	no	—
32	C₂×D₁₇	—	4	4	no	—
33	C₄×C₁₇	—	2	3	no	—
34	Dic₁₇	—	2	6	no	—
35	H(3,𝔽₄)	—	17	1	no	maximal, Sylow
36	C₂×C₁₇	—	4	1	no	—
37	C₂×C₁₇	—	4	3	no	—
38	C₂×C₁₇	H(5,𝔽₂)	1	3	yes	—
39	D₁₇	—	4	2	no	—
40	#124	—	17	3	no	—
41	#124	—	17	6	no	—
42	#120	—	17	2	no	—
43	#120	—	17	3	no	—
44	#120	D₁₇	1	1	yes	characteristic, complemented
45	C₁₇	H(3,𝔽₄)	1	1	yes	characteristic, complemented, Sylow
46	C₂⁴	—	17	1	no	—
47	C₂⁴	C₂×D₁₇	1	1	yes	characteristic, complemented
48	C₂×Q₈	—	17	3	no	—
49	C₂×D₄	—	17	6	no	—
50	C₂×D₄	C₂×D₁₇	1	3	yes	—
51	C₂²⋊C₄	—	17	3	no	—
52	C₂²⋊C₄	—	17	6	no	—
53	C₂²⋊C₄	—	17	6	no	—
54	C₂²⋊C₄	C₂×D₁₇	1	3	yes	complemented
55	C₄²	—	17	3	no	—
56	C₂³	—	4	3	no	—
57	C₂³	—	17	2	no	—
58	C₂³	—	68	3	no	—
59	C₂³	C₂²×D₁₇	1	1	yes	characteristic
60	C₂³	C₂²×D₁₇	1	3	yes	—
61	Q₈	—	68	3	no	—
62	D₄	—	4	3	no	—
63	D₄	—	68	6	no	—
64	C₂×C₄	—	17	6	no	—
65	C₂×C₄	C₂²×D₁₇	1	3	yes	—
66	C₂²	—	2	3	no	—
67	C₂²	—	2	3	no	—
68	C₂²	—	2	6	no	—
69	C₂²	—	4	4	no	—
70	C₂²	—	34	6	no	—
71	C₂²	—	68	4	no	—
72	C₂²	C₂³×D₁₇	1	1	yes	characteristic
73	C₄	—	2	3	no	—
74	C₄	—	34	6	no	—
75	C₂	—	4	3	no	—
76	C₂	—	4	1	no	—
77	C₂	—	68	2	no	—
78	C₂	#10587691	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₇
commutator, socle, cosocle	C₂²×C₁₇	C₂⁴
Fitting	C₁₇×#120	C₂
hypercenter	#120	D₁₇
hypocenter	C₁₇	H(3,𝔽₄)
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ #120

Complements

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

N	H ≅ G/N	type
1. C₁₇×#120	C₂	semidirect
2. #10587639	C₂	semidirect
3. #10587623	C₂	semidirect
4. #10587622	C₂	semidirect
5. #10587571	C₂	semidirect
6. C₂⁴×C₁₇	C₂²	semidirect
7. C₂³×D₁₇	C₂²	semidirect
10. C₂×Dic₃₄	C₂²	semidirect
11. C₂²⋊C₄×C₁₇	C₂²	semidirect
12. C₂²⋊Dic₁₇	C₂²	semidirect
13. C₂²⋊Dic₁₇	C₂²	semidirect
14. #63227	C₂²	semidirect
15. C₄×Dic₁₇	C₂²	semidirect
44. #120	D₁₇	semidirect
45. C₁₇	H(3,𝔽₄)	semidirect
47. C₂⁴	C₂×D₁₇	semidirect
54. C₂²⋊C₄	C₂×D₁₇	semidirect

Maximal subgroups

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

subgroup	index
C₁₇×#120	2
#10587639	2
#10587623	2
#10587622	2
#10587571	2
H(3,𝔽₄)	17