Subgroups — #14528540

Order 2176.

200 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11762767	C₂	1	3	yes	maximal
2	#11762566	C₂	1	3	yes	maximal
3	C₁₇×#338	C₂	1	1	yes	maximal, characteristic
4	C₂×M₁₆×C₁₇	C₂²	1	3	yes	—
5	#10587612	C₂²	1	3	yes	—
6	C₂²×#3622	C₂²	1	1	yes	characteristic
7	#10587486	C₄	1	6	yes	complemented
8	C₈×Dic₁₇	C₄	1	6	yes	complemented
9	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
10	M₁₆×C₁₇	—	2	6	no	—
11	C₂×C₈×C₁₇	C₂×C₄	1	6	yes	—
12	C₂²⋊Dic₁₇	—	2	3	no	—
13	C₄⋊Dic₁₇	—	2	3	no	—
14	C₄×Dic₁₇	C₂×C₄	1	6	yes	—
15	C₂×#3622	C₂×C₄	1	3	yes	—
16	C₂×#3622	C₂×C₄	1	3	yes	—
17	C₂³×C₁₇	#41	1	1	yes	characteristic
18	C₂×C₄×C₁₇	C₂²×C₄	1	3	yes	—
19	C₂×C₄×C₁₇	#41	1	3	yes	—
20	C₂×Dic₁₇	—	2	6	no	—
21	C₈×C₁₇	—	2	6	no	—
22	#3622	C₄²	1	1	yes	characteristic
23	#3622	C₄²	1	3	yes	—
24	#1747	—	17	1	no	maximal, Sylow
25	C₂²×C₁₇	—	4	1	no	—
26	C₂²×C₁₇	#117	1	3	yes	—
27	C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
28	C₄×C₁₇	#117	1	3	yes	—
29	Dic₁₇	—	4	6	no	—
30	#405	—	17	3	no	—
31	#338	D₁₇	1	1	yes	characteristic
32	H(3,ℤ/4)	—	17	3	no	—
33	C₂×C₁₇	—	2	3	no	—
34	C₂×C₁₇	#376	1	1	yes	characteristic
35	C₂×M₁₆	C₂×D₁₇	1	3	yes	—
36	C₂²×C₈	—	17	1	no	—
37	#117	—	17	3	no	—
38	#97	—	17	6	no	—
39	C₄×C₈	—	17	6	no	—
40	C₁₇	#1747	1	1	yes	characteristic, complemented, Sylow
41	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
42	M₁₆	—	2	6	no	—
43	C₂×C₈	—	17	3	no	—
44	C₂×C₈	—	17	3	no	—
45	C₂×C₈	C₄×D₁₇	1	6	yes	—
46	C₄⋊C₄	—	34	3	no	—
47	C₂²⋊C₄	—	34	3	no	—
48	C₄²	—	17	6	no	—
49	C₂³	#63256	1	1	yes	characteristic
50	C₂×C₄	—	34	6	no	—
51	C₂×C₄	C₂×C₄×D₁₇	1	3	yes	—
52	C₂×C₄	#63254	1	3	yes	—
53	C₈	—	2	6	no	—
54	C₈	—	17	1	no	—
55	C₈	—	17	3	no	—
56	C₂²	—	4	1	no	—
57	C₂²	#10587578	1	3	yes	—
58	C₄	—	68	6	no	—
59	C₄	C₄²×D₁₇	1	1	yes	characteristic
60	C₄	#10587563	1	3	yes	—
61	C₂	—	2	3	no	—
62	C₂	#11762654	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₄²×D₁₇
commutator	C₄×C₁₇	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₇
Fitting	C₁₇×#338	C₂
socle	C₂×C₁₇	#376
prime radical	C₂³×C₁₇	#41
cosocle	C₂²×C₄×C₁₇	C₂³
hypercenter	#338	D₁₇
hypocenter	C₁₇	#1747
LCS	G ▷ C₄×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄ ◁ #338

Complements

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

N	H ≅ G/N	type
7. #10587486	C₄	semidirect
8. C₈×Dic₁₇	C₄	semidirect
40. C₁₇	#1747	semidirect

Maximal subgroups

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

subgroup	index
#11762767	2
#11762566	2
C₁₇×#338	2
#1747	17