Subgroups — #14531029

Order 2176.

132 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763211	C₂	1	1	yes	maximal, characteristic
2	#11762613	C₂	1	4	yes	maximal
3	#11762585	C₂	1	1	yes	maximal, characteristic
4	C₁₇×#494	C₂	1	1	yes	maximal, characteristic
5	C₂×Q₁₆×C₁₇	—	2	1	no	—
6	C₂×Q₁₆×C₁₇	C₂²	1	2	yes	—
7	C₁₇×C₄⋊Q₈	—	2	1	no	—
8	#10587556	—	2	1	no	—
9	C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
10	#10587489	C₂²	1	2	yes	—
11	#10587488	—	2	1	no	—
12	C₂×#63214	C₂²	1	2	yes	—
13	C₂×Q₈×C₁₇	—	2	2	no	—
14	Q₁₆×C₁₇	—	2	4	no	—
15	Q₁₆×C₁₇	—	4	2	no	—
16	C₂×C₈×C₁₇	D₄	1	1	yes	characteristic
17	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
18	C₄⋊C₄×C₁₇	—	4	2	no	—
19	C₄²×C₁₇	D₄	1	1	yes	characteristic
20	C₄⋊Dic₁₇	—	2	2	no	—
21	#63225	—	4	2	no	—
22	#63214	—	2	2	no	—
23	Q₈×C₁₇	—	4	2	no	—
24	Q₈×C₁₇	—	4	2	no	—
25	C₂×C₄×C₁₇	—	4	2	no	—
26	C₂×C₄×C₁₇	D₈	1	2	yes	—
27	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
28	C₂×Dic₁₇	—	4	2	no	—
29	C₈×C₁₇	SD₁₆	1	2	yes	—
30	C₈×C₁₇	#41	1	2	yes	—
31	#2245	—	17	1	no	maximal, Sylow
32	C₂²×C₁₇	C₂×D₈	1	1	yes	characteristic
33	C₄×C₁₇	—	2	2	no	—
34	C₄×C₁₇	—	8	2	no	—
35	C₄×C₁₇	#124	1	1	yes	characteristic
36	C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
37	Dic₁₇	—	8	2	no	—
38	#499	—	17	1	no	—
39	#494	D₁₇	1	1	yes	characteristic
40	#358	—	17	4	no	—
41	#346	—	17	1	no	—
42	C₂×C₁₇	#486	1	1	yes	characteristic
43	C₂×C₁₇	#510	1	2	yes	—
44	C₂×Q₁₆	—	2	1	no	—
45	C₂×Q₁₆	C₂×D₁₇	1	2	yes	—
46	C₄⋊Q₈	—	2	1	no	—
47	#125	—	34	1	no	—
48	C₂×C₁₆	—	17	2	no	—
49	#107	—	17	2	no	—
50	#106	—	34	1	no	—
51	C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
52	C₁₇	#2245	1	1	yes	characteristic, complemented, Sylow
53	C₂×Q₈	—	2	2	no	—
54	Q₁₆	—	2	4	no	—
55	Q₁₆	—	4	2	no	—
56	C₂×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
57	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
58	C₄⋊C₄	—	4	2	no	—
59	C₄⋊C₄	—	34	2	no	—
60	C₄⋊C₄	—	68	2	no	—
61	C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
62	C₁₆	—	34	2	no	—
63	Q₈	—	4	2	no	—
64	Q₈	—	4	2	no	—
65	C₂×C₄	—	4	2	no	—
66	C₂×C₄	—	68	2	no	—
67	C₂×C₄	#63228	1	2	yes	—
68	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
69	C₈	#63229	1	2	yes	—
70	C₈	#63254	1	2	yes	—
71	C₂²	C₂×#63228	1	1	yes	characteristic
72	C₄	—	2	2	no	—
73	C₄	—	8	2	no	—
74	C₄	—	136	2	no	—
75	C₄	C₂×#63229	1	1	yes	characteristic
76	C₄	#10587622	1	1	yes	characteristic
77	C₂	#11763155	1	1	yes	characteristic
78	C₂	#11763250	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#63228
commutator, cosocle	C₂×C₈×C₁₇	C₂³
Frattini	C₂×C₈	C₂²×D₁₇
Fitting	C₁₇×#494	C₂
socle, prime radical	C₂²×C₁₇	C₂×D₈
hypercenter	#494	D₁₇
hypocenter	C₁₇	#2245
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #494

Complements

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

N	H ≅ G/N	type
52. C₁₇	#2245	semidirect

Maximal subgroups

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

subgroup	index
#11763211	2
#11762613	2
#11762585	2
C₁₇×#494	2
#2245	17