Subgroups — #17949609

Order 4352.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14543952	C₂	1	1	yes	maximal, characteristic
2	#14543945	C₂	1	1	yes	maximal, characteristic
3	D₁₇×Q₆₄	C₂	1	1	yes	maximal, characteristic
4	#11763735	—	2	1	no	—
5	#11763648	—	2	1	no	—
6	Q₃₂×D₁₇	—	2	1	no	—
7	Dic₂₇₂	C₄	1	1	yes	characteristic, complemented
8	#11762649	—	2	1	no	—
9	C₃₂×D₁₇	C₂²	1	1	yes	characteristic
10	C₁₇×Q₆₄	C₄	1	1	yes	characteristic, complemented
11	Q₁₆×D₁₇	—	4	1	no	—
12	#10587534	—	4	1	no	—
13	C₁₇×Q₃₂	—	2	1	no	—
14	#10587500	—	4	1	no	—
15	Dic₁₃₆	—	2	1	no	—
16	C₁₆×D₁₇	D₄	1	1	yes	characteristic
17	C₁₇×C₃₂	C₂×C₄	1	1	yes	characteristic
18	#10587465	D₄	1	1	yes	characteristic
19	Q₈×D₁₇	—	8	1	no	—
20	C₄⋊#595	—	8	1	no	—
21	Q₁₆×C₁₇	—	4	1	no	—
22	#63231	—	8	1	no	—
23	Dic₆₈	—	4	1	no	—
24	C₈×D₁₇	D₈	1	1	yes	characteristic
25	C₁₆×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
26	#63214	SD₁₆	1	1	yes	characteristic
27	#7538	—	17	1	no	maximal, Sylow
28	C₂×#595	—	16	1	no	—
29	Q₈×C₁₇	—	8	1	no	—
30	C₄×D₁₇	—	16	1	no	—
31	C₄×D₁₇	D₁₆	1	1	yes	characteristic
32	Dic₃₄	—	8	1	no	—
33	Dic₃₄	—	16	1	no	—
34	C₈×C₁₇	#102	1	1	yes	characteristic
35	#3622	SD₃₂	1	1	yes	characteristic
36	C₂×Q₆₄	—	17	1	no	—
37	C₂×C₆₄	—	17	1	no	—
38	#1423	—	17	1	no	—
39	C₂×D₁₇	D₃₂	1	1	yes	characteristic
40	#595	—	32	1	no	—
41	C₄×C₁₇	—	16	1	no	—
42	C₄×C₁₇	#357	1	1	yes	characteristic
43	Dic₁₇	—	16	1	no	—
44	Dic₁₇	SD₆₄	1	1	yes	characteristic
45	C₂×Q₃₂	—	34	1	no	—
46	Q₆₄	—	17	1	no	—
47	Q₆₄	—	34	1	no	—
48	Q₆₄	#595	1	1	yes	characteristic, complemented
49	C₂×C₃₂	—	17	1	no	—
50	#366	—	34	1	no	—
51	C₆₄	—	34	1	no	—
52	C₂×C₁₇	#1415	1	1	yes	characteristic
53	D₁₇	SD₁₂₈	1	1	yes	characteristic
54	D₁₇	Q₁₂₈	1	1	yes	characteristic
55	C₂×Q₁₆	—	68	1	no	—
56	Q₃₂	—	2	1	no	—
57	Q₃₂	—	34	1	no	—
58	Q₃₂	—	68	1	no	—
59	C₂×C₁₆	—	17	1	no	—
60	#107	—	68	1	no	—
61	C₃₂	—	17	1	no	—
62	C₃₂	C₂×#595	1	1	yes	characteristic
63	C₁₇	#7538	1	1	yes	characteristic, complemented, Sylow
64	C₂×Q₈	—	136	1	no	—
65	Q₁₆	—	4	1	no	—
66	Q₁₆	—	68	1	no	—
67	Q₁₆	—	136	1	no	—
68	C₂×C₈	—	17	1	no	—
69	C₄⋊C₄	—	136	1	no	—
70	C₁₆	—	17	1	no	—
71	C₁₆	C₂²⋊#595	1	1	yes	characteristic
72	Q₈	—	8	1	no	—
73	Q₈	—	272	1	no	—
74	Q₈	—	136	1	no	—
75	C₂×C₄	—	272	1	no	—
76	C₂×C₄	—	272	1	no	—
77	C₂×C₄	—	17	1	no	—
78	C₈	—	17	1	no	—
79	C₈	#10587547	1	1	yes	characteristic
80	C₂²	—	17	1	no	—
81	C₄	—	16	1	no	—
82	C₄	—	272	1	no	—
83	C₄	—	17	1	no	—
84	C₄	—	544	1	no	—
85	C₄	#11763721	1	1	yes	characteristic
86	C₂	—	17	1	no	—
87	C₂	—	17	1	no	—
88	C₂	#14543933	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#14543933
commutator	C₁₇×C₃₂	C₂×C₄
Frattini	C₃₂	C₂×#595
Fitting	C₁₇×Q₆₄	C₄
socle	C₂×C₁₇	#1415
prime radical	C₂×D₁₇	D₃₂
cosocle	C₃₂×D₁₇	C₂²
hypercenter	Q₆₄	#595
hypocenter	C₁₇	#7538
LCS	G ▷ C₁₇×C₃₂ ▷ C₁₆×C₁₇ ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ C₃₂ ◁ Q₆₄

Complements

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

N	H ≅ G/N	type
7. Dic₂₇₂	C₄	semidirect
10. C₁₇×Q₆₄	C₄	semidirect
48. Q₆₄	#595	semidirect
63. C₁₇	#7538	semidirect

Maximal subgroups

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

subgroup	index
#14543952	2
#14543945	2
D₁₇×Q₆₄	2
#7538	17