Subgroups — #14531061

Order 2176.

116 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763210	C₂	1	2	yes	maximal
2	#11762636	C₂	1	2	yes	maximal
3	#11762635	C₂	1	1	yes	maximal, characteristic
4	C₁₆×Dic₁₇	C₂	1	1	yes	maximal, characteristic
5	C₁₇×#366	C₂	1	1	yes	maximal, characteristic
6	#10587576	—	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	#10587479	—	2	2	no	—
13	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
14	C₄⋊C₄×C₁₇	—	2	2	no	—
15	C₄⋊Dic₁₇	—	2	2	no	—
16	C₄⋊Dic₁₇	—	4	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	—
21	#63214	Q₈	1	2	yes	—
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	C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
28	C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
29	#3622	Q₁₆	1	2	yes	—
30	#2253	—	17	1	no	maximal, Sylow
31	C₂²×C₁₇	C₂×D₈	1	1	yes	characteristic
32	C₄×C₁₇	—	8	2	no	—
33	C₄×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
34	C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
35	Dic₁₇	—	2	1	no	—
36	Dic₁₇	—	2	1	no	—
37	Dic₁₇	—	8	2	no	—
38	#499	—	17	2	no	—
39	#367	—	17	2	no	—
40	#366	—	17	1	no	—
41	#366	D₁₇	1	1	yes	characteristic
42	C₄×C₁₆	—	17	1	no	—
43	C₂×C₁₇	#500	1	1	yes	characteristic
44	C₂×C₁₇	#508	1	1	yes	characteristic
45	C₂×C₁₇	#508	1	1	yes	characteristic
46	#125	—	34	2	no	—
47	C₂×C₁₆	—	17	1	no	—
48	C₂×C₁₆	C₂×D₁₇	1	1	yes	characteristic
49	#107	—	17	2	no	—
50	#107	C₂×D₁₇	1	2	yes	—
51	#106	—	34	2	no	—
52	C₄×C₈	—	17	1	no	—
53	C₁₇	#2253	1	1	yes	characteristic, complemented, Sylow
54	C₂×C₈	—	17	1	no	—
55	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
56	C₄⋊C₄	—	2	2	no	—
57	C₄⋊C₄	—	34	2	no	—
58	C₄⋊C₄	—	68	2	no	—
59	C₄⋊C₄	—	68	2	no	—
60	C₄²	—	17	1	no	—
61	C₁₆	—	17	2	no	—
62	C₁₆	Dic₃₄	1	2	yes	—
63	C₂×C₄	—	4	2	no	—
64	C₂×C₄	—	17	1	no	—
65	C₂×C₄	—	17	1	no	—
66	C₂×C₄	—	68	2	no	—
67	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
68	C₈	—	17	2	no	—
69	C₈	C₂×Dic₃₄	1	1	yes	characteristic
70	C₈	Q₈×D₁₇	1	1	yes	characteristic
71	C₂²	D₈×D₁₇	1	1	yes	characteristic
72	C₄	—	8	2	no	—
73	C₄	—	34	1	no	—
74	C₄	—	34	1	no	—
75	C₄	—	136	2	no	—
76	C₄	#10587574	1	1	yes	characteristic
77	C₄	Q₁₆×D₁₇	1	1	yes	characteristic
78	C₂	#11763213	1	1	yes	characteristic
79	C₂	#11763238	1	1	yes	characteristic
80	C₂	#11763242	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	#2253
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
53. C₁₇	#2253	semidirect

Maximal subgroups

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

subgroup	index
#11763210	2
#11762636	2
#11762635	2
C₁₆×Dic₁₇	2
C₁₇×#366	2
#2253	17