Subgroups — #14531065

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	#11762637	C₂	1	1	yes	maximal, characteristic
3	#11762636	C₂	1	2	yes	maximal
4	C₁₆×Dic₁₇	C₂	1	1	yes	maximal, characteristic
5	C₁₇×#367	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	2	yes	—
26	Dic₃₄	—	8	4	no	—
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	#2254	—	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	—	8	2	no	—
36	Dic₁₇	SD₃₂	1	4	yes	—
37	#500	—	17	2	no	—
38	#367	—	17	1	no	—
39	#367	—	17	2	no	—
40	#367	D₁₇	1	1	yes	characteristic
41	C₄×C₁₆	—	17	1	no	—
42	C₂×C₁₇	#500	1	1	yes	characteristic
43	C₂×C₁₇	C₂×SD₃₂	1	2	yes	—
44	C₄⋊Q₈	—	34	2	no	—
45	C₂×C₁₆	—	17	1	no	—
46	C₂×C₁₆	C₂×D₁₇	1	1	yes	characteristic
47	#107	—	17	2	no	—
48	#107	—	34	2	no	—
49	#107	C₂×D₁₇	1	2	yes	—
50	C₄×C₈	—	17	1	no	—
51	C₁₇	#2254	1	1	yes	characteristic, complemented, Sylow
52	C₂×Q₈	—	68	2	no	—
53	C₂×C₈	—	17	1	no	—
54	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
55	C₄⋊C₄	—	2	2	no	—
56	C₄⋊C₄	—	34	2	no	—
57	C₄⋊C₄	—	68	2	no	—
58	C₄²	—	17	1	no	—
59	C₁₆	—	17	2	no	—
60	C₁₆	Dic₃₄	1	2	yes	complemented
61	Q₈	—	136	4	no	—
62	C₂×C₄	—	4	2	no	—
63	C₂×C₄	—	17	2	no	—
64	C₂×C₄	—	68	2	no	—
65	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
66	C₈	—	17	2	no	—
67	C₈	C₂×Dic₃₄	1	1	yes	characteristic
68	C₈	Q₈×D₁₇	1	1	yes	characteristic
69	C₂²	D₈×D₁₇	1	1	yes	characteristic
70	C₄	—	8	2	no	—
71	C₄	—	17	4	no	—
72	C₄	—	136	2	no	—
73	C₄	#10587574	1	1	yes	characteristic
74	C₄	Q₁₆×D₁₇	1	1	yes	characteristic
75	C₂	#11763213	1	1	yes	characteristic
76	C₂	SD₃₂×D₁₇	1	2	yes	—

	subgroup	G/H
center	C₂²	D₈×D₁₇
commutator, cosocle	C₂×C₈×C₁₇	C₂³
Frattini	C₂×C₈	C₂²×D₁₇
Fitting	C₁₇×#367	C₂
socle, prime radical	C₂²×C₁₇	C₂×D₈
hypercenter	#367	D₁₇
hypocenter	C₁₇	#2254
LCS	G ▷ C₂×C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #367

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
51. C₁₇	#2254	semidirect
60. 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
#11762637	2
#11762636	2
C₁₆×Dic₁₇	2
C₁₇×#367	2
#2254	17