Subgroups — #17949615

Order 4352.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14543954	C₂	1	1	yes	maximal, characteristic
2	#14543948	C₂	1	1	yes	maximal, characteristic
3	D₁₇×Q₆₄	C₂	1	1	yes	maximal, characteristic
4	#11763737	—	2	1	no	—
5	#11763648	—	2	1	no	—
6	Q₃₂×D₁₇	—	2	1	no	—
7	Dic₂₇₂	C₄	1	1	yes	characteristic
8	#11762649	—	2	1	no	—
9	C₃₂×D₁₇	C₂²	1	1	yes	characteristic
10	C₁₇×Q₆₄	C₄	1	1	yes	characteristic
11	Q₁₆×D₁₇	—	4	1	no	—
12	#10587536	—	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	#63243	—	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	#7542	—	17	1	no	maximal, Sylow
28	Q₈×C₁₇	—	8	1	no	—
29	C₄×D₁₇	—	16	1	no	—
30	C₄×D₁₇	D₁₆	1	1	yes	characteristic
31	Dic₃₄	—	8	1	no	—
32	Dic₃₄	—	16	1	no	—
33	#3624	—	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	M₁₂₈	—	17	1	no	—
38	#1425	—	17	1	no	—
39	C₂×D₁₇	SD₆₄	1	1	yes	characteristic
40	C₄×C₁₇	—	16	1	no	—
41	C₄×C₁₇	#357	1	1	yes	characteristic
42	Dic₁₇	—	16	1	no	—
43	Dic₁₇	D₃₂	1	1	yes	characteristic
44	C₂×Q₃₂	—	34	1	no	—
45	Q₆₄	—	17	1	no	—
46	Q₆₄	—	34	1	no	—
47	Q₆₄	#595	1	1	yes	characteristic
48	C₂×C₃₂	—	17	1	no	—
49	#368	—	34	1	no	—
50	C₆₄	—	34	1	no	—
51	C₂×C₁₇	#1415	1	1	yes	characteristic
52	D₁₇	—	2	1	no	—
53	C₂×Q₁₆	—	68	1	no	—
54	Q₃₂	—	2	1	no	—
55	Q₃₂	—	34	1	no	—
56	Q₃₂	—	68	1	no	—
57	C₂×C₁₆	—	17	1	no	—
58	#108	—	68	1	no	—
59	C₃₂	—	17	1	no	—
60	C₃₂	C₂×#595	1	1	yes	characteristic
61	C₁₇	#7542	1	1	yes	characteristic, complemented, Sylow
62	C₂×Q₈	—	136	1	no	—
63	Q₁₆	—	4	1	no	—
64	Q₁₆	—	68	1	no	—
65	Q₁₆	—	136	1	no	—
66	M₁₆	—	136	1	no	—
67	C₂×C₈	—	17	1	no	—
68	C₁₆	—	17	1	no	—
69	C₁₆	C₂²⋊#595	1	1	yes	characteristic
70	Q₈	—	8	1	no	—
71	Q₈	—	272	1	no	—
72	Q₈	—	136	1	no	—
73	C₂×C₄	—	272	1	no	—
74	C₂×C₄	—	17	1	no	—
75	C₈	—	272	1	no	—
76	C₈	—	17	1	no	—
77	C₈	#10587547	1	1	yes	characteristic
78	C₂²	—	17	1	no	—
79	C₄	—	16	1	no	—
80	C₄	—	272	1	no	—
81	C₄	—	17	1	no	—
82	C₄	#11763721	1	1	yes	characteristic
83	C₂	—	34	1	no	—
84	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	SD₆₄
cosocle	C₃₂×D₁₇	C₂²
hypercenter	Q₆₄	#595
hypocenter	C₁₇	#7542
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
61. C₁₇	#7542	semidirect

Maximal subgroups

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

subgroup	index
#14543954	2
#14543948	2
D₁₇×Q₆₄	2
#7542	17