Subgroups — #16922005

Order 4352.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14526860	C₂	1	2	yes	maximal
2	C₄×C₁₇×C₃₂	C₂	1	1	yes	maximal, characteristic
3	#11762630	—	2	2	no	—
4	C₂×C₁₇×C₃₂	C₄	1	1	yes	characteristic
5	C₂×C₁₇×C₃₂	C₄	1	1	yes	characteristic
6	C₄×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
7	C₂×C₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
8	C₂×C₁₆×C₁₇	Q₈	1	1	yes	characteristic
9	C₄×C₈×C₁₇	D₄	1	1	yes	characteristic
10	#10587483	—	4	2	no	—
11	C₁₇×C₃₂	—	2	1	no	—
12	C₁₇×C₃₂	C₈	1	2	yes	—
13	C₂×C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
14	C₂×C₈×C₁₇	D₈	1	1	yes	characteristic
15	C₄²×C₁₇	Q₁₆	1	1	yes	characteristic
16	C₁₆×C₁₇	—	2	1	no	—
17	C₁₆×C₁₇	C₂×C₈	1	1	yes	characteristic
18	C₁₆×C₁₇	M₁₆	1	1	yes	characteristic
19	#63214	—	8	2	no	—
20	#7462	—	17	1	no	maximal, Sylow
21	C₂×C₄×C₁₇	—	2	1	no	—
22	C₂×C₄×C₁₇	#107	1	1	yes	characteristic
23	C₈×C₁₇	C₄⋊C₈	1	1	yes	characteristic
24	C₈×C₁₇	#108	1	1	yes	characteristic
25	C₈×C₁₇	D₁₆	1	1	yes	characteristic
26	C₈×C₁₇	Q₃₂	1	1	yes	characteristic
27	C₄×C₃₂	D₁₇	1	1	yes	characteristic
28	#1373	—	17	2	no	—
29	C₂²×C₁₇	#368	1	1	yes	characteristic
30	C₄×C₁₇	—	2	2	no	—
31	C₄×C₁₇	#335	1	1	yes	characteristic
32	C₄×C₁₇	#366	1	1	yes	characteristic
33	C₂×C₃₂	Dic₁₇	1	1	yes	characteristic
34	C₂×C₃₂	Dic₁₇	1	1	yes	characteristic
35	#364	—	34	2	no	—
36	C₄×C₁₆	C₂×D₁₇	1	1	yes	characteristic
37	C₂×C₁₇	—	2	1	no	—
38	C₂×C₁₇	#1371	1	1	yes	characteristic
39	M₃₂	—	68	2	no	—
40	C₂×C₁₆	Dic₃₄	1	1	yes	characteristic
41	C₂×C₁₆	C₂×Dic₁₇	1	1	yes	characteristic
42	C₄×C₈	D₆₈	1	1	yes	characteristic
43	C₃₂	—	2	1	no	—
44	C₃₂	#3622	1	2	yes	—
45	C₁₇	#7462	1	1	yes	characteristic, complemented, Sylow
46	C₂×C₈	D₁₃₆	1	1	yes	characteristic
47	C₂×C₈	C₄⋊Dic₁₇	1	1	yes	characteristic
48	C₄²	Dic₆₈	1	1	yes	characteristic
49	C₁₆	—	2	1	no	—
50	C₁₆	—	136	2	no	—
51	C₁₆	C₂×#3622	1	1	yes	characteristic
52	C₁₆	#63223	1	1	yes	characteristic
53	C₂×C₄	—	2	1	no	—
54	C₂×C₄	#10587489	1	1	yes	characteristic
55	C₈	D₂₇₂	1	1	yes	characteristic
56	C₈	Dic₁₃₆	1	1	yes	characteristic
57	C₈	C₄⋊#3622	1	1	yes	characteristic
58	C₈	#10587490	1	1	yes	characteristic
59	C₂²	#11762639	1	1	yes	characteristic
60	C₄	—	2	2	no	—
61	C₄	#11762562	1	1	yes	characteristic
62	C₄	#11762635	1	1	yes	characteristic
63	C₂	—	2	1	no	—
64	C₂	#14526856	1	1	yes	characteristic

	subgroup	G/H
center	C₈	D₂₇₂
commutator	C₁₆×C₁₇	C₂×C₈
Frattini	C₄×C₁₆	C₂×D₁₇
Fitting	C₄×C₁₇×C₃₂	C₂
socle	C₂×C₁₇	#1371
prime radical	C₂²×C₁₇	#368
cosocle	C₄×C₁₆×C₁₇	C₂²
hypercenter	C₄×C₃₂	D₁₇
hypocenter	C₁₇	#7462
LCS	G ▷ C₁₆×C₁₇ ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄×C₃₂

Complements

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

N	H ≅ G/N	type
45. C₁₇	#7462	semidirect

Maximal subgroups

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

subgroup	index
#14526860	2
C₄×C₁₇×C₃₂	2
#7462	17