Subgroups — #16955351

Order 4352.

158 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14531059	C₂	1	2	yes	maximal
2	#14526986	C₂	1	4	yes	maximal
3	C₄×C₁₇×C₃₂	C₂	1	1	yes	maximal, characteristic
4	#11763214	—	2	2	no	—
5	#11762635	—	2	2	no	—
6	#11762635	C₂²	1	4	yes	—
7	C₂×C₁₇×C₃₂	C₂²	1	2	yes	—
8	C₄×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
9	#10587555	—	4	2	no	—
10	C₂×C₁₆×C₁₇	D₄	1	1	yes	characteristic
11	C₂×C₁₆×C₁₇	C₂³	1	1	yes	characteristic
12	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
13	#10587489	—	2	4	no	—
14	#10587489	—	4	2	no	—
15	C₁₇×C₃₂	Q₈	1	4	yes	complemented
16	C₂×Dic₃₄	—	8	2	no	—
17	C₂×C₈×C₁₇	D₈	1	1	yes	characteristic
18	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
19	C₄²×C₁₇	D₈	1	1	yes	characteristic
20	C₄⋊Dic₁₇	—	4	4	no	—
21	C₄⋊Dic₁₇	—	8	2	no	—
22	C₁₆×C₁₇	Q₁₆	1	2	yes	—
23	C₁₆×C₁₇	C₂×Q₈	1	2	yes	—
24	#13733	—	17	1	no	maximal, Sylow
25	C₂×C₄×C₁₇	D₁₆	1	2	yes	—
26	C₂×C₄×C₁₇	C₂×D₈	1	1	yes	characteristic
27	C₂×Dic₁₇	—	8	4	no	—
28	Dic₃₄	—	16	4	no	—
29	C₈×C₁₇	Q₃₂	1	2	yes	—
30	C₈×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
31	C₈×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
32	#2252	—	17	2	no	—
33	#1424	—	17	4	no	—
34	C₄×C₃₂	D₁₇	1	1	yes	characteristic
35	C₂²×C₁₇	C₂×D₁₆	1	1	yes	characteristic
36	C₄×C₁₇	SD₆₄	1	4	yes	—
37	C₄×C₁₇	#500	1	1	yes	characteristic
38	C₄×C₁₇	C₂×Q₃₂	1	1	yes	characteristic
39	Dic₁₇	—	16	4	no	—
40	#500	—	34	2	no	—
41	C₂×C₃₂	C₂×D₁₇	1	2	yes	—
42	#366	—	17	4	no	—
43	#366	—	34	2	no	—
44	C₄×C₁₆	C₂×D₁₇	1	1	yes	characteristic
45	C₂×C₁₇	#2252	1	1	yes	characteristic
46	C₂×C₁₇	C₂×SD₆₄	1	2	yes	—
47	C₄⋊Q₈	—	68	2	no	—
48	C₂×C₁₆	D₆₈	1	1	yes	characteristic
49	C₂×C₁₆	C₂²×D₁₇	1	1	yes	characteristic
50	#107	—	34	4	no	—
51	#107	—	68	2	no	—
52	C₄×C₈	D₆₈	1	1	yes	characteristic
53	C₃₂	Dic₃₄	1	4	yes	complemented
54	C₁₇	#13733	1	1	yes	characteristic, complemented, Sylow
55	C₂×Q₈	—	136	2	no	—
56	C₂×C₈	D₁₃₆	1	1	yes	characteristic
57	C₂×C₈	C₂×D₆₈	1	1	yes	characteristic
58	C₄⋊C₄	—	68	4	no	—
59	C₄⋊C₄	—	136	2	no	—
60	C₄²	D₁₃₆	1	1	yes	characteristic
61	C₁₆	Dic₆₈	1	2	yes	—
62	C₁₆	C₂×Dic₃₄	1	2	yes	—
63	Q₈	—	272	4	no	—
64	C₂×C₄	—	136	4	no	—
65	C₂×C₄	D₂₇₂	1	2	yes	—
66	C₂×C₄	C₂×D₁₃₆	1	1	yes	characteristic
67	C₈	Dic₁₃₆	1	2	yes	—
68	C₈	#10587555	1	1	yes	characteristic
69	C₈	C₂×Dic₆₈	1	1	yes	characteristic
70	C₂²	C₂×D₂₇₂	1	1	yes	characteristic
71	C₄	—	272	4	no	—
72	C₄	#11762648	1	4	yes	—
73	C₄	#11763214	1	1	yes	characteristic
74	C₄	C₂×Dic₁₃₆	1	1	yes	characteristic
75	C₂	#14531059	1	1	yes	characteristic
76	C₂	C₂×#11762648	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₂₇₂
commutator, cosocle	C₂×C₁₆×C₁₇	C₂³
Frattini	C₂×C₁₆	C₂²×D₁₇
Fitting	C₄×C₁₇×C₃₂	C₂
socle, prime radical	C₂²×C₁₇	C₂×D₁₆
hypercenter	C₄×C₃₂	D₁₇
hypocenter	C₁₇	#13733
LCS	G ▷ C₂×C₁₆×C₁₇ ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄×C₃₂

Complements

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

N	H ≅ G/N	type
15. C₁₇×C₃₂	Q₈	semidirect
53. C₃₂	Dic₃₄	semidirect
54. C₁₇	#13733	semidirect

Maximal subgroups

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

subgroup	index
#14531059	2
#14526986	2
C₄×C₁₇×C₃₂	2
#13733	17