Subgroups — #14528396

Order 2176.

178 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763160	C₂	1	3	yes	maximal, complemented
2	#11763156	C₂	1	3	yes	maximal
3	C₈²×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#10587560	C₂²	1	1	yes	characteristic
5	#10587555	C₂²	1	3	yes	—
6	C₄×C₈×C₁₇	C₂²	1	3	yes	—
7	#10587492	—	2	6	no	—
8	#10587487	—	2	6	no	—
9	C₂×D₆₈	—	2	3	no	—
10	C₂×Dic₃₄	—	2	3	no	—
11	C₂×C₈×C₁₇	D₄	1	6	yes	—
12	C₄²×C₁₇	C₂³	1	1	yes	characteristic
13	C₄⋊Dic₁₇	—	2	3	no	—
14	C₄⋊Dic₁₇	—	2	3	no	—
15	C₂²×D₁₇	—	4	1	no	—
16	C₂×C₄×C₁₇	C₂×D₄	1	3	yes	—
17	C₂×Dic₁₇	—	4	3	no	—
18	D₆₈	—	4	6	no	—
19	Dic₃₄	—	4	6	no	—
20	C₈×C₁₇	—	2	6	no	—
21	#1711	—	17	1	no	maximal, Sylow
22	C₂²×C₁₇	#127	1	1	yes	characteristic
23	C₂×D₁₇	—	8	3	no	—
24	C₄×C₁₇	#135	1	6	yes	—
25	Dic₁₇	—	8	3	no	—
26	#487	—	17	3	no	—
27	#486	—	17	3	no	—
28	C₈²	D₁₇	1	1	yes	characteristic, complemented
29	C₂×C₁₇	#495	1	3	yes	—
30	D₁₇	—	16	1	no	—
31	C₄⋊Q₈	—	17	3	no	—
32	#127	—	17	1	no	—
33	#103	—	34	6	no	—
34	#102	—	34	6	no	—
35	C₄×C₈	C₂×D₁₇	1	3	yes	—
36	C₁₇	#1711	1	1	yes	characteristic, complemented, Sylow
37	C₂×Q₈	—	34	3	no	—
38	C₂×D₄	—	34	3	no	—
39	C₂×C₈	D₆₈	1	6	yes	—
40	C₄⋊C₄	—	34	3	no	—
41	C₄⋊C₄	—	34	3	no	—
42	C₄²	C₂²×D₁₇	1	1	yes	characteristic
43	C₂³	—	68	1	no	—
44	Q₈	—	68	6	no	—
45	D₄	—	68	6	no	—
46	C₂×C₄	—	68	3	no	—
47	C₂×C₄	C₂×D₆₈	1	3	yes	—
48	C₈	—	2	6	no	—
49	C₂²	—	136	3	no	—
50	C₂²	#10587560	1	1	yes	characteristic
51	C₄	—	136	3	no	—
52	C₄	#10587590	1	6	yes	—
53	C₂	—	272	1	no	—
54	C₂	#11763193	1	3	yes	—

	subgroup	G/H
center	C₂²	#10587560
commutator, cosocle	C₄²×C₁₇	C₂³
Frattini	C₄²	C₂²×D₁₇
Fitting	C₈²×C₁₇	C₂
socle	C₂²×C₁₇	#127
prime radical	#10587560	C₂²
hypercenter	C₈²	D₁₇
hypocenter	C₁₇	#1711
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈²

Complements

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

N	H ≅ G/N	type
1. #11763160	C₂	semidirect
3. C₈²×C₁₇	C₂	semidirect
28. C₈²	D₁₇	semidirect
36. C₁₇	#1711	semidirect

Maximal subgroups

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

subgroup	index
#11763160	2
#11763156	2
C₈²×C₁₇	2
#1711	17