Subgroups — #16922218

Order 4352.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₂₇₂	C₂	1	1	yes	maximal, characteristic
2	#14526988	C₂	1	1	yes	maximal, characteristic
3	C₁₇×M₁₂₈	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₁₃₆	—	2	1	no	—
5	Dic₂₇₂	—	2	1	no	—
6	Dic₂₇₂	C₄	1	2	yes	—
7	#11762639	—	2	1	no	—
8	C₂×C₁₇×C₃₂	C₂²	1	1	yes	characteristic
9	C₁₇×C₆₄	—	2	1	no	—
10	C₂×Dic₆₈	—	4	1	no	—
11	C₂×C₁₆×C₁₇	D₄	1	1	yes	characteristic
12	#10587490	—	4	1	no	—
13	Dic₁₃₆	—	2	2	no	—
14	Dic₁₃₆	—	4	1	no	—
15	C₁₇×C₃₂	C₂×C₄	1	1	yes	characteristic
16	C₁₇×C₃₂	D₄	1	1	yes	characteristic
17	C₂×Dic₃₄	—	8	1	no	—
18	C₂×C₈×C₁₇	D₈	1	1	yes	characteristic
19	#63223	—	8	1	no	—
20	Dic₆₈	—	4	2	no	—
21	Dic₆₈	—	8	1	no	—
22	C₁₆×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
23	C₁₆×C₁₇	SD₁₆	1	1	yes	characteristic
24	#7542	—	17	1	no	maximal, Sylow
25	C₂×C₄×C₁₇	D₁₆	1	1	yes	characteristic
26	C₂×Dic₁₇	—	16	1	no	—
27	Dic₃₄	—	8	2	no	—
28	Dic₃₄	—	16	1	no	—
29	C₈×C₁₇	#102	1	1	yes	characteristic
30	C₈×C₁₇	SD₃₂	1	1	yes	characteristic
31	#3622	—	16	1	no	—
32	C₂×Q₆₄	—	17	1	no	—
33	M₁₂₈	D₁₇	1	1	yes	characteristic
34	#1425	—	17	1	no	—
35	C₂²×C₁₇	SD₆₄	1	1	yes	characteristic
36	C₄×C₁₇	#357	1	1	yes	characteristic
37	C₄×C₁₇	D₃₂	1	1	yes	characteristic
38	Dic₁₇	—	16	2	no	—
39	C₂×Q₃₂	—	34	1	no	—
40	Q₆₄	—	17	2	no	—
41	Q₆₄	—	34	1	no	—
42	C₂×C₃₂	C₂×D₁₇	1	1	yes	characteristic
43	#368	—	34	1	no	—
44	C₆₄	—	2	1	no	—
45	C₂×C₁₇	—	2	1	no	—
46	C₂×C₁₇	#1415	1	1	yes	characteristic
47	C₂×Q₁₆	—	68	1	no	—
48	Q₃₂	—	34	2	no	—
49	Q₃₂	—	68	1	no	—
50	C₂×C₁₆	D₆₈	1	1	yes	characteristic
51	#108	—	68	1	no	—
52	C₃₂	C₄×D₁₇	1	1	yes	characteristic
53	C₃₂	C₂²⋊D₁₇	1	1	yes	characteristic
54	C₁₇	#7542	1	1	yes	characteristic, complemented, Sylow
55	C₂×Q₈	—	136	1	no	—
56	Q₁₆	—	68	2	no	—
57	Q₁₆	—	136	1	no	—
58	M₁₆	—	136	1	no	—
59	C₂×C₈	D₁₃₆	1	1	yes	characteristic
60	C₁₆	#63219	1	1	yes	characteristic
61	C₁₆	#63227	1	1	yes	characteristic
62	Q₈	—	272	1	no	—
63	Q₈	—	136	2	no	—
64	C₂×C₄	—	272	1	no	—
65	C₂×C₄	D₂₇₂	1	1	yes	characteristic
66	C₈	—	272	1	no	—
67	C₈	#10587471	1	1	yes	characteristic
68	C₈	#10587492	1	1	yes	characteristic
69	C₂²	#11762648	1	1	yes	characteristic
70	C₄	—	272	2	no	—
71	C₄	#11762611	1	1	yes	characteristic
72	C₄	D₅₄₄	1	1	yes	characteristic
73	C₂	—	2	1	no	—
74	C₂	#14526963	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#14526963
commutator	C₁₇×C₃₂	C₂×C₄
Frattini	C₂×C₃₂	C₂×D₁₇
Fitting	C₁₇×M₁₂₈	C₂
socle	C₂×C₁₇	#1415
prime radical	C₂²×C₁₇	SD₆₄
cosocle	C₂×C₁₇×C₃₂	C₂²
hypercenter	M₁₂₈	D₁₇
hypocenter	C₁₇	#7542
LCS	G ▷ C₁₇×C₃₂ ▷ C₁₆×C₁₇ ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂ ◁ M₁₂₈

Complements

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

N	H ≅ G/N	type
54. C₁₇	#7542	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₂₇₂	2
#14526988	2
C₁₇×M₁₂₈	2
#7542	17