Subgroups — #14537037

Order 2176.

580 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11762721	C₂	1	12	yes	maximal, complemented
2	#11762665	C₂	1	2	yes	maximal
3	C₁₇×#379	C₂	1	1	yes	maximal, characteristic
4	C₂⁵×C₁₇	C₂²	1	1	yes	characteristic
5	C₂²×Dic₃₄	C₂²	1	4	yes	complemented
6	C₂×C₂²⋊C₄×C₁₇	C₂²	1	6	yes	—
7	C₂×C₂²⋊Dic₁₇	C₂²	1	12	yes	—
8	#10587502	C₂²	1	12	yes	complemented
9	C₂⁴×C₁₇	—	4	6	no	—
10	C₂⁴×C₁₇	Q₈	1	4	yes	complemented
11	C₂⁴×C₁₇	C₂³	1	3	yes	—
12	C₂²×C₄×C₁₇	C₂³	1	4	yes	—
13	C₂²×Dic₁₇	C₂³	1	8	yes	—
14	C₂×Dic₃₄	—	4	12	no	—
15	C₂²⋊C₄×C₁₇	—	4	6	no	—
16	C₂²⋊Dic₁₇	—	4	12	no	—
17	C₂³×C₁₇	—	2	12	no	—
18	C₂³×C₁₇	—	2	6	no	—
19	C₂³×C₁₇	—	4	12	no	—
20	C₂³×C₁₇	—	4	16	no	—
21	C₂³×C₁₇	C₂×Q₈	1	6	yes	—
22	C₂³×C₁₇	C₂⁴	1	1	yes	characteristic
23	C₂×C₄×C₁₇	—	2	12	no	—
24	C₂×Dic₁₇	—	2	24	no	—
25	Dic₃₄	—	4	16	no	—
26	#2840	—	17	1	no	maximal, Sylow
27	C₂²×C₁₇	—	2	12	no	—
28	C₂²×C₁₇	—	2	6	no	—
29	C₂²×C₁₇	—	4	16	no	—
30	C₂²×C₁₇	—	4	12	no	—
31	C₂²×C₁₇	C₂²×Q₈	1	1	yes	characteristic
32	C₂²×C₁₇	H(5,𝔽₂)	1	6	yes	—
33	C₄×C₁₇	—	4	4	no	—
34	Dic₁₇	—	4	8	no	—
35	#393	—	17	12	no	—
36	#379	—	17	2	no	—
37	#379	D₁₇	1	1	yes	characteristic
38	C₂×C₁₇	—	4	6	no	—
39	C₂×C₁₇	#543	1	3	yes	—
40	C₂×C₁₇	H(3,𝔽₄)	1	4	yes	—
41	C₂⁵	C₂×D₁₇	1	1	yes	characteristic
42	C₂²×Q₈	—	17	4	no	—
43	C₂×C₂²⋊C₄	—	17	12	no	—
44	C₂×C₂²⋊C₄	C₂×D₁₇	1	6	yes	—
45	#95	—	17	12	no	—
46	C₁₇	#2840	1	1	yes	characteristic, complemented, Sylow
47	C₂⁴	—	4	6	no	—
48	C₂⁴	Dic₃₄	1	4	yes	complemented
49	C₂⁴	C₂²×D₁₇	1	3	yes	—
50	C₂×Q₈	—	68	12	no	—
51	C₂²×C₄	—	17	8	no	—
52	C₂²×C₄	C₂²×D₁₇	1	4	yes	—
53	C₂²⋊C₄	—	4	6	no	—
54	C₂²⋊C₄	—	68	12	no	—
55	C₂³	—	2	6	no	—
56	C₂³	—	2	12	no	—
57	C₂³	—	4	12	no	—
58	C₂³	—	4	16	no	—
59	C₂³	C₂×Dic₃₄	1	6	yes	—
60	C₂³	C₂³×D₁₇	1	1	yes	characteristic
61	Q₈	—	68	16	no	—
62	C₂×C₄	—	2	12	no	—
63	C₂×C₄	—	34	24	no	—
64	C₂²	—	2	6	no	—
65	C₂²	—	2	12	no	—
66	C₂²	—	4	16	no	—
67	C₂²	—	4	12	no	—
68	C₂²	C₂²×Dic₃₄	1	1	yes	characteristic
69	C₂²	#10587691	1	6	yes	—
70	C₄	—	4	4	no	—
71	C₄	—	68	8	no	—
72	C₂	—	4	6	no	—
73	C₂	#11763435	1	3	yes	—
74	C₂	#11763552	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₇
commutator, socle, cosocle	C₂³×C₁₇	C₂⁴
Fitting	C₁₇×#379	C₂
prime radical	C₂⁵×C₁₇	C₂²
hypercenter	#379	D₁₇
hypocenter	C₁₇	#2840
LCS	G ▷ C₂³×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ #379

Complements

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

N	H ≅ G/N	type
1. #11762721	C₂	semidirect
5. C₂²×Dic₃₄	C₂²	semidirect
8. #10587502	C₂²	semidirect
10. C₂⁴×C₁₇	Q₈	semidirect
46. C₁₇	#2840	semidirect
48. C₂⁴	Dic₃₄	semidirect

Maximal subgroups

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

subgroup	index
#11762721	2
#11762665	2
C₁₇×#379	2
#2840	17