Subgroups — #14537061

Order 2176.

3422 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10587639	C₂	1	7	yes	maximal, complemented
2	#11762665	C₂	1	7	yes	maximal, complemented
3	C₂⁶×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂⁵×C₁₇	—	2	28	no	—
5	C₂⁵×C₁₇	C₂²	1	7	yes	—
6	C₂²×C₂²⋊D₁₇	C₂²	1	7	yes	complemented
7	#10587639	—	2	28	no	—
8	C₂×C₂²⋊Dic₁₇	C₂²	1	21	yes	complemented
9	C₂⁴×C₁₇	—	2	224	no	—
10	C₂⁴×C₁₇	—	2	84	no	—
11	C₂⁴×C₁₇	D₄	1	28	yes	complemented
12	C₂⁴×C₁₇	C₂³	1	7	yes	—
13	C₂³×D₁₇	C₂³	1	1	yes	characteristic, complemented
14	C₂×C₂²⋊D₁₇	—	2	42	no	—
15	C₂²×Dic₁₇	C₂³	1	7	yes	complemented
16	C₂²⋊Dic₁₇	—	2	42	no	—
17	C₂³×C₁₇	—	2	256	no	—
18	C₂³×C₁₇	—	2	224	no	—
19	C₂³×C₁₇	—	2	168	no	—
20	C₂³×C₁₇	—	2	28	no	—
21	C₂³×C₁₇	C₂×D₄	1	42	yes	—
22	C₂³×C₁₇	C₂⁴	1	1	yes	characteristic
23	C₂²×D₁₇	—	2	7	no	—
24	C₂²⋊D₁₇	—	4	28	no	—
25	C₂×Dic₁₇	—	2	21	no	—
26	#2846	—	17	1	no	maximal, Sylow
27	C₂²×C₁₇	—	2	224	no	—
28	C₂²×C₁₇	—	2	84	no	—
29	C₂²×C₁₇	#120	1	28	yes	complemented
30	C₂²×C₁₇	C₂²×D₄	1	7	yes	—
31	C₂×D₁₇	—	4	7	no	—
32	Dic₁₇	—	4	7	no	—
33	C₂⁶	D₁₇	1	1	yes	characteristic, complemented
34	C₂×#120	—	17	7	no	—
35	#379	—	17	7	no	—
36	C₂×C₁₇	—	2	28	no	—
37	C₂×C₁₇	C₂×#120	1	7	yes	—
38	D₁₇	—	8	1	no	—
39	C₂⁵	—	2	28	no	—
40	C₂⁵	C₂×D₁₇	1	7	yes	—
41	C₂²×D₄	—	17	7	no	—
42	#120	—	34	28	no	—
43	C₂×C₂²⋊C₄	—	17	21	no	—
44	C₁₇	#2846	1	1	yes	characteristic, complemented, Sylow
45	C₂⁴	—	2	224	no	—
46	C₂⁴	—	2	84	no	—
47	C₂⁴	—	17	1	no	—
48	C₂⁴	C₂²⋊D₁₇	1	28	yes	complemented
49	C₂⁴	C₂²×D₁₇	1	7	yes	—
50	C₂×D₄	—	34	42	no	—
51	C₂²×C₄	—	17	7	no	—
52	C₂²⋊C₄	—	34	42	no	—
53	C₂³	—	2	256	no	—
54	C₂³	—	2	224	no	—
55	C₂³	—	2	28	no	—
56	C₂³	—	2	168	no	—
57	C₂³	—	34	7	no	—
58	C₂³	C₂×C₂²⋊D₁₇	1	42	yes	—
59	C₂³	C₂³×D₁₇	1	1	yes	characteristic
60	D₄	—	68	28	no	—
61	C₂×C₄	—	34	21	no	—
62	C₂²	—	2	224	no	—
63	C₂²	—	2	84	no	—
64	C₂²	—	68	7	no	—
65	C₂²	#10587639	1	28	yes	complemented
66	C₂²	C₂²×C₂²⋊D₁₇	1	7	yes	—
67	C₄	—	68	7	no	—
68	C₂	—	2	28	no	—
69	C₂	—	136	1	no	—
70	C₂	C₂×#10587639	1	7	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₇
commutator, socle, cosocle	C₂³×C₁₇	C₂⁴
Fitting	C₂⁶×C₁₇	C₂
hypercenter	C₂⁶	D₁₇
hypocenter	C₁₇	#2846
LCS	G ▷ C₂³×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂⁶

Complements

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

N	H ≅ G/N	type
1. C₂×#10587639	C₂	semidirect
2. #11762665	C₂	semidirect
3. C₂⁶×C₁₇	C₂	semidirect
6. C₂²×C₂²⋊D₁₇	C₂²	semidirect
8. C₂×C₂²⋊Dic₁₇	C₂²	semidirect
11. C₂⁴×C₁₇	D₄	semidirect
13. C₂³×D₁₇	C₂³	semidirect
15. C₂²×Dic₁₇	C₂³	semidirect
29. C₂²×C₁₇	#120	semidirect
33. C₂⁶	D₁₇	semidirect
44. C₁₇	#2846	semidirect
48. C₂⁴	C₂²⋊D₁₇	semidirect
65. C₂²	#10587639	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10587639	2
#11762665	2
C₂⁶×C₁₇	2
#2846	17