Subgroups — #14530186

Order 2176.

196 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11762857	C₂	1	3	yes	maximal
2	C₂×#10587495	C₂	1	3	yes	maximal
3	C₁₇×#338	C₂	1	1	yes	maximal, characteristic
4	C₂²×Dic₃₄	C₂²	1	1	yes	characteristic
5	C₂×M₁₆×C₁₇	C₂²	1	3	yes	—
6	C₂×#63223	C₂²	1	3	yes	—
7	#10587495	—	2	6	no	—
8	#10587493	—	2	6	no	—
9	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
10	C₂²×Dic₁₇	Q₈	1	2	yes	—
11	C₂×Dic₃₄	—	2	3	no	—
12	C₂×Dic₃₄	—	2	3	no	—
13	M₁₆×C₁₇	—	2	3	no	—
14	M₁₆×C₁₇	—	2	3	no	—
15	C₂×C₈×C₁₇	—	2	3	no	—
16	#63223	—	2	3	no	—
17	#63223	D₄	1	6	yes	—
18	C₂×#3622	—	2	3	no	—
19	C₂³×C₁₇	C₂×Q₈	1	1	yes	characteristic
20	C₂×C₄×C₁₇	C₂×D₄	1	3	yes	—
21	C₂×C₄×C₁₇	#41	1	3	yes	—
22	C₂×Dic₁₇	—	2	6	no	—
23	Dic₃₄	—	4	3	no	—
24	Dic₃₄	—	4	1	no	—
25	C₈×C₁₇	—	4	3	no	—
26	#3622	—	2	6	no	—
27	#2070	—	17	1	no	maximal, Sylow
28	C₂²×C₁₇	—	4	1	no	—
29	C₂²×C₁₇	C₂²⋊Q₈	1	3	yes	—
30	C₄×C₁₇	#121	1	3	yes	—
31	C₄×C₁₇	#126	1	1	yes	characteristic
32	Dic₁₇	—	4	2	no	—
33	#430	—	17	3	no	—
34	C₂×#101	—	17	3	no	—
35	#338	D₁₇	1	1	yes	characteristic
36	C₂×C₁₇	—	2	3	no	—
37	C₂×C₁₇	#396	1	1	yes	characteristic
38	C₂²×Q₈	—	17	1	no	—
39	C₂×M₁₆	—	17	3	no	—
40	C₂×M₁₆	C₂×D₁₇	1	3	yes	—
41	#108	—	34	6	no	—
42	#101	—	34	6	no	—
43	C₁₇	#2070	1	1	yes	characteristic, complemented, Sylow
44	C₂×Q₈	—	34	3	no	—
45	C₂×Q₈	—	34	3	no	—
46	C₂²×C₄	—	17	2	no	—
47	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
48	M₁₆	—	2	3	no	—
49	M₁₆	—	2	3	no	—
50	M₁₆	—	17	6	no	—
51	M₁₆	—	34	3	no	—
52	C₂×C₈	—	2	3	no	—
53	C₂×C₈	—	34	3	no	—
54	C₂³	Q₈×D₁₇	1	1	yes	characteristic
55	Q₈	—	68	1	no	—
56	Q₈	—	68	3	no	—
57	C₂×C₄	—	34	6	no	—
58	C₂×C₄	#63252	1	3	yes	—
59	C₂×C₄	D₄×D₁₇	1	3	yes	—
60	C₈	—	4	3	no	—
61	C₈	—	34	6	no	—
62	C₂²	—	4	1	no	—
63	C₂²	#10587582	1	3	yes	—
64	C₄	—	68	2	no	—
65	C₄	#10587562	1	1	yes	characteristic
66	C₄	#10587570	1	3	yes	—
67	C₂	—	2	3	no	—
68	C₂	#11762733	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#11762733
commutator, cosocle	C₂²×C₄×C₁₇	C₂³
Frattini	C₂²×C₄	C₂²×D₁₇
Fitting	C₁₇×#338	C₂
socle	C₂×C₁₇	#396
prime radical	C₂³×C₁₇	C₂×Q₈
hypercenter	#338	D₁₇
hypocenter	C₁₇	#2070
LCS	G ▷ C₂²×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #338

Complements

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

N	H ≅ G/N	type
43. C₁₇	#2070	semidirect

Maximal subgroups

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

subgroup	index
#11762857	2
C₂×#10587495	2
C₁₇×#338	2
#2070	17