Subgroups — #14526990

Order 2176.

52 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11762639	C₂	1	1	yes	maximal, characteristic
2	#11762637	C₂	1	1	yes	maximal, characteristic
3	C₁₇×M₆₄	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
5	#10587490	—	2	1	no	—
6	#10587489	—	2	1	no	—
7	C₁₇×C₃₂	C₄	1	2	yes	complemented
8	C₂×C₈×C₁₇	D₄	1	1	yes	characteristic
9	C₄⋊Dic₁₇	—	4	1	no	—
10	#63223	—	4	1	no	—
11	C₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
12	C₁₆×C₁₇	Q₈	1	1	yes	characteristic
13	C₂×C₄×C₁₇	D₈	1	1	yes	characteristic
14	C₂×Dic₁₇	—	8	1	no	—
15	C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
16	C₈×C₁₇	Q₁₆	1	1	yes	characteristic
17	#3622	—	8	1	no	—
18	#1426	—	17	1	no	maximal, Sylow
19	C₂²×C₁₇	SD₃₂	1	1	yes	characteristic
20	C₄×C₁₇	#107	1	1	yes	characteristic
21	C₄×C₁₇	SD₃₂	1	1	yes	characteristic
22	Dic₁₇	—	16	1	no	—
23	M₆₄	D₁₇	1	1	yes	characteristic
24	#368	—	17	1	no	—
25	#367	—	17	1	no	—
26	C₂×C₁₇	—	2	1	no	—
27	C₂×C₁₇	#367	1	1	yes	characteristic
28	C₂×C₁₆	C₂×D₁₇	1	1	yes	characteristic
29	#108	—	34	1	no	—
30	#107	—	34	1	no	—
31	C₃₂	Dic₁₇	1	2	yes	complemented
32	C₁₇	#1426	1	1	yes	characteristic, complemented, Sylow
33	M₁₆	—	68	1	no	—
34	C₂×C₈	D₆₈	1	1	yes	characteristic
35	C₄⋊C₄	—	68	1	no	—
36	C₁₆	Dic₃₄	1	1	yes	characteristic
37	C₁₆	C₂×Dic₁₇	1	1	yes	characteristic
38	C₂×C₄	—	136	1	no	—
39	C₂×C₄	D₁₃₆	1	1	yes	characteristic
40	C₈	—	136	1	no	—
41	C₈	Dic₆₈	1	1	yes	characteristic
42	C₈	C₄⋊Dic₁₇	1	1	yes	characteristic
43	C₂²	#10587471	1	1	yes	characteristic
44	C₄	—	272	1	no	—
45	C₄	#10587471	1	1	yes	characteristic
46	C₄	#10587489	1	1	yes	characteristic
47	C₂	—	2	1	no	—
48	C₂	#11762637	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#11762637
commutator	C₁₆×C₁₇	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₁₇
Fitting	C₁₇×M₆₄	C₂
socle	C₂×C₁₇	#367
prime radical	C₂²×C₁₇	SD₃₂
cosocle	C₂×C₁₆×C₁₇	C₂²
hypercenter	M₆₄	D₁₇
hypocenter	C₁₇	#1426
LCS	G ▷ C₁₆×C₁₇ ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
7. C₁₇×C₃₂	C₄	semidirect
31. C₃₂	Dic₁₇	semidirect
32. C₁₇	#1426	semidirect

Maximal subgroups

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

subgroup	index
#11762639	2
#11762637	2
C₁₇×M₆₄	2
#1426	17