Subgroups — #16950746

Order 4352.

278 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14530550	C₂	1	3	yes	maximal
2	#14530543	C₂	1	3	yes	maximal
3	C₁₇×#1747	C₂	1	1	yes	maximal, characteristic
4	C₂×#10587483	C₂²	1	3	yes	—
5	C₂²×#63214	C₂²	1	1	yes	characteristic
6	#11762630	—	2	6	no	—
7	C₄⋊#63214	—	2	6	no	—
8	C₁₇×#405	C₂²	1	3	yes	—
9	C₁₇×#338	C₄	1	1	yes	characteristic
10	C₁₇×H(3,ℤ/4)	C₄	1	3	yes	—
11	C₂×M₁₆×C₁₇	C₂×C₄	1	3	yes	—
12	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
13	C₁₇×#117	C₂×C₄	1	3	yes	—
14	C₁₇×#97	Q₈	1	6	yes	—
15	C₄×C₈×C₁₇	D₄	1	6	yes	—
16	#10587483	—	2	6	no	—
17	C₂×#63214	—	2	3	no	—
18	C₂×#63214	—	2	3	no	—
19	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
20	M₁₆×C₁₇	—	2	6	no	—
21	C₂×C₈×C₁₇	C₄⋊C₄	1	6	yes	—
22	C₂×C₈×C₁₇	C₂×D₄	1	3	yes	—
23	C₂×C₈×C₁₇	C₂×Q₈	1	3	yes	—
24	C₄⋊C₄×C₁₇	—	2	3	no	—
25	C₂²⋊C₄×C₁₇	—	2	3	no	—
26	C₄²×C₁₇	C₄⋊C₄	1	6	yes	—
27	#63214	—	4	1	no	—
28	#63214	—	4	3	no	—
29	#12902	—	17	1	no	maximal, Sylow
30	C₂³×C₁₇	#131	1	1	yes	characteristic
31	C₂×C₄×C₁₇	—	2	6	no	—
32	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	3	yes	—
33	C₂×C₄×C₁₇	#131	1	3	yes	—
34	C₈×C₁₇	—	2	6	no	—
35	C₈×C₁₇	#127	1	1	yes	characteristic
36	C₈×C₁₇	C₄⋊Q₈	1	3	yes	—
37	#2153	—	17	3	no	—
38	#2151	—	17	3	no	—
39	#1747	D₁₇	1	1	yes	characteristic
40	C₂²×C₁₇	—	4	1	no	—
41	C₂²×C₁₇	#424	1	3	yes	—
42	C₄×C₁₇	—	4	6	no	—
43	C₄×C₁₇	#384	1	1	yes	characteristic
44	C₄×C₁₇	#424	1	3	yes	—
45	C₂×M₃₂	—	17	3	no	—
46	C₂²×C₁₆	—	17	1	no	—
47	#405	C₂×D₁₇	1	3	yes	—
48	#364	—	34	6	no	—
49	C₄⋊C₁₆	—	34	6	no	—
50	#338	Dic₁₇	1	1	yes	characteristic
51	H(3,ℤ/4)	Dic₁₇	1	3	yes	—
52	C₂×C₁₇	—	2	3	no	—
53	C₂×C₁₇	#1843	1	1	yes	characteristic
54	C₂×M₁₆	C₂×Dic₁₇	1	3	yes	—
55	C₂²×C₈	C₂²×D₁₇	1	1	yes	characteristic
56	#117	C₂×Dic₁₇	1	3	yes	—
57	M₃₂	—	34	6	no	—
58	C₂×C₁₆	—	34	3	no	—
59	C₂×C₁₆	—	34	3	no	—
60	#97	Dic₃₄	1	6	yes	—
61	C₄×C₈	D₆₈	1	6	yes	—
62	C₁₇	#12902	1	1	yes	characteristic, complemented, Sylow
63	C₂²×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
64	M₁₆	—	2	6	no	—
65	C₂×C₈	C₄⋊Dic₁₇	1	6	yes	—
66	C₂×C₈	C₂×Dic₃₄	1	3	yes	—
67	C₂×C₈	C₂×D₆₈	1	3	yes	—
68	C₄⋊C₄	—	2	3	no	—
69	C₂²⋊C₄	—	2	3	no	—
70	C₄²	C₄⋊Dic₁₇	1	6	yes	—
71	C₁₆	—	68	1	no	—
72	C₁₆	—	68	3	no	—
73	C₂³	#10587634	1	1	yes	characteristic
74	C₂×C₄	—	2	6	no	—
75	C₂×C₄	C₂×C₄⋊Dic₁₇	1	3	yes	—
76	C₂×C₄	#10587634	1	3	yes	—
77	C₈	—	2	6	no	—
78	C₈	#10587555	1	3	yes	—
79	C₈	#10587560	1	1	yes	characteristic
80	C₂²	—	4	1	no	—
81	C₂²	#11762836	1	3	yes	—
82	C₄	—	4	6	no	—
83	C₄	#11762679	1	1	yes	characteristic
84	C₄	#11762836	1	3	yes	—
85	C₂	—	2	3	no	—
86	C₂	#14528938	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#10587560
commutator	C₂²×C₄×C₁₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₇
Fitting	C₁₇×#1747	C₂
socle	C₂×C₁₇	#1843
prime radical	C₂³×C₁₇	#131
cosocle	C₂²×C₈×C₁₇	C₂³
hypercenter	#1747	D₁₇
hypocenter	C₁₇	#12902
LCS	G ▷ C₂²×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #1747

Complements

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

N	H ≅ G/N	type
62. C₁₇	#12902	semidirect

Maximal subgroups

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

subgroup	index
#14530550	2
#14530543	2
C₁₇×#1747	2
#12902	17