Subgroups — #14531058

Order 2176.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763213	C₂	1	2	yes	maximal
2	#11762635	C₂	1	1	yes	maximal, characteristic
3	#11762634	C₂	1	2	yes	maximal
4	C₁₆×Dic₁₇	C₂	1	1	yes	maximal, characteristic
5	C₁₇×#366	C₂	1	1	yes	maximal, characteristic
6	#10587574	—	2	2	no	—
7	C₂×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
8	C₁₇×#107	C₂²	1	2	yes	—
9	#10587489	C₂²	1	2	yes	—
10	C₈×Dic₁₇	C₂²	1	1	yes	characteristic
11	C₂×#63214	C₂²	1	1	yes	characteristic
12	#10587478	—	2	2	no	—
13	C₂×Dic₃₄	—	4	2	no	—
14	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
15	C₄⋊C₄×C₁₇	—	2	2	no	—
16	C₄⋊Dic₁₇	—	2	2	no	—
17	#63225	—	4	2	no	—
18	C₄×Dic₁₇	D₄	1	1	yes	characteristic
19	C₂×#3622	D₄	1	1	yes	characteristic
20	C₁₆×C₁₇	Q₈	1	2	yes	complemented
21	#63214	Q₈	1	2	yes	complemented
22	C₂×C₄×C₁₇	—	4	2	no	—
23	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
24	C₂×Dic₁₇	—	4	2	no	—
25	C₂×Dic₁₇	D₈	1	1	yes	characteristic
26	C₂×Dic₁₇	D₈	1	1	yes	characteristic
27	Dic₃₄	—	8	4	no	—
28	C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
29	C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
30	#3622	Q₁₆	1	2	yes	—
31	#2252	—	17	1	no	maximal, Sylow
32	C₂²×C₁₇	C₂×D₈	1	1	yes	characteristic
33	C₄×C₁₇	—	8	2	no	—
34	C₄×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
35	C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
36	Dic₁₇	—	8	2	no	—
37	Dic₁₇	D₁₆	1	2	yes	—
38	Dic₁₇	Q₃₂	1	2	yes	—
39	#500	—	17	2	no	—
40	#366	—	17	1	no	—
41	#366	—	17	2	no	—
42	#366	D₁₇	1	1	yes	characteristic
43	C₄×C₁₆	—	17	1	no	—
44	C₂×C₁₇	#500	1	1	yes	characteristic
45	C₂×C₁₇	C₂×D₁₆	1	1	yes	characteristic
46	C₂×C₁₇	C₂×Q₃₂	1	1	yes	characteristic
47	C₄⋊Q₈	—	34	2	no	—
48	C₂×C₁₆	—	17	1	no	—
49	C₂×C₁₆	C₂×D₁₇	1	1	yes	characteristic
50	#107	—	17	2	no	—
51	#107	—	34	2	no	—
52	#107	C₂×D₁₇	1	2	yes	—
53	C₄×C₈	—	17	1	no	—
54	C₁₇	#2252	1	1	yes	characteristic, complemented, Sylow
55	C₂×Q₈	—	68	2	no	—
56	C₂×C₈	—	17	1	no	—
57	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
58	C₄⋊C₄	—	2	2	no	—
59	C₄⋊C₄	—	34	2	no	—
60	C₄⋊C₄	—	68	2	no	—
61	C₄²	—	17	1	no	—
62	C₁₆	—	17	2	no	—
63	C₁₆	Dic₃₄	1	2	yes	complemented
64	Q₈	—	136	4	no	—
65	C₂×C₄	—	4	2	no	—
66	C₂×C₄	—	17	1	no	—
67	C₂×C₄	—	17	1	no	—
68	C₂×C₄	—	68	2	no	—
69	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
70	C₈	—	17	2	no	—
71	C₈	C₂×Dic₃₄	1	1	yes	characteristic
72	C₈	Q₈×D₁₇	1	1	yes	characteristic
73	C₂²	D₈×D₁₇	1	1	yes	characteristic
74	C₄	—	8	2	no	—
75	C₄	—	17	2	no	—
76	C₄	—	17	2	no	—
77	C₄	—	136	2	no	—
78	C₄	#10587574	1	1	yes	characteristic
79	C₄	Q₁₆×D₁₇	1	1	yes	characteristic
80	C₂	#11763213	1	1	yes	characteristic
81	C₂	D₁₆×D₁₇	1	1	yes	characteristic
82	C₂	Q₃₂×D₁₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₈×D₁₇
commutator, cosocle	C₂×C₈×C₁₇	C₂³
Frattini	C₂×C₈	C₂²×D₁₇
Fitting	C₁₇×#366	C₂
socle, prime radical	C₂²×C₁₇	C₂×D₈
hypercenter	#366	D₁₇
hypocenter	C₁₇	#2252
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #366

Complements

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

N	H ≅ G/N	type
20. C₁₆×C₁₇	Q₈	semidirect
21. #63214	Q₈	semidirect
54. C₁₇	#2252	semidirect
63. C₁₆	Dic₃₄	semidirect

Maximal subgroups

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

subgroup	index
#11763213	2
#11762635	2
#11762634	2
C₁₆×Dic₁₇	2
C₁₇×#366	2
#2252	17