Subgroups — #14528402

Order 2176.

192 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763214	C₂	1	4	yes	maximal
2	#11763189	C₂	1	2	yes	maximal
3	C₈²×C₁₇	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₆₈	—	2	4	no	—
5	#10587555	C₂²	1	4	yes	—
6	C₄×C₈×C₁₇	C₂²	1	2	yes	—
7	C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
8	#10587489	—	2	8	no	—
9	C₂×Dic₃₄	—	2	4	no	—
10	C₂×C₈×C₁₇	D₄	1	2	yes	—
11	C₂×C₈×C₁₇	D₄	1	4	yes	—
12	C₄²×C₁₇	C₂³	1	1	yes	characteristic
13	C₄⋊Dic₁₇	—	2	8	no	—
14	Dic₆₈	—	4	8	no	—
15	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
16	C₂×C₄×C₁₇	C₂×D₄	1	2	yes	—
17	C₂×Dic₁₇	—	4	4	no	—
18	Dic₃₄	—	4	8	no	—
19	C₈×C₁₇	D₈	1	4	yes	—
20	C₈×C₁₇	Q₁₆	1	8	yes	complemented
21	#1713	—	17	1	no	maximal, Sylow
22	C₂²×C₁₇	#127	1	1	yes	characteristic
23	C₄×C₁₇	C₂×D₈	1	2	yes	—
24	C₄×C₁₇	C₂×Q₁₆	1	4	yes	—
25	Dic₁₇	—	8	4	no	—
26	#500	—	17	4	no	—
27	#494	—	17	2	no	—
28	C₈²	D₁₇	1	1	yes	characteristic
29	C₂×C₁₇	#493	1	1	yes	characteristic
30	C₂×C₁₇	#494	1	2	yes	—
31	C₂×Q₁₆	—	34	4	no	—
32	C₄⋊Q₈	—	17	4	no	—
33	#107	—	34	8	no	—
34	C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
35	C₄×C₈	C₂×D₁₇	1	2	yes	—
36	C₁₇	#1713	1	1	yes	characteristic, complemented, Sylow
37	C₂×Q₈	—	34	4	no	—
38	Q₁₆	—	68	8	no	—
39	C₂×C₈	D₆₈	1	2	yes	—
40	C₂×C₈	D₆₈	1	4	yes	—
41	C₄⋊C₄	—	34	8	no	—
42	C₄²	C₂²×D₁₇	1	1	yes	characteristic
43	Q₈	—	68	8	no	—
44	C₂×C₄	—	68	4	no	—
45	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
46	C₂×C₄	C₂×D₆₈	1	2	yes	—
47	C₈	D₁₃₆	1	4	yes	—
48	C₈	Dic₆₈	1	8	yes	complemented
49	C₂²	#10587560	1	1	yes	characteristic
50	C₄	—	136	4	no	—
51	C₄	C₂×D₁₃₆	1	2	yes	—
52	C₄	C₂×Dic₆₈	1	4	yes	—
53	C₂	#11763186	1	1	yes	characteristic
54	C₂	#11763189	1	2	yes	—

	subgroup	G/H
center	C₂²	#10587560
commutator, cosocle	C₄²×C₁₇	C₂³
Frattini	C₄²	C₂²×D₁₇
Fitting	C₈²×C₁₇	C₂
socle, prime radical	C₂²×C₁₇	#127
hypercenter	C₈²	D₁₇
hypocenter	C₁₇	#1713
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
20. C₈×C₁₇	Q₁₆	semidirect
36. C₁₇	#1713	semidirect
48. C₈	Dic₆₈	semidirect

Maximal subgroups

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

subgroup	index
#11763214	2
#11763189	2
C₈²×C₁₇	2
#1713	17