Subgroups — #36351765

Order 22144.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26011317	C₂	1	2	yes	maximal
2	C₂×C₃₂×C₁₇₃	C₂	1	1	yes	maximal, characteristic
3	#19149613	—	2	2	no	—
4	C₂×C₁₆×C₁₇₃	C₂²	1	1	yes	characteristic
5	C₃₂×C₁₇₃	C₄	1	2	yes	complemented
6	C₄⋊Dic₁₇₃	—	4	2	no	—
7	C₂×C₈×C₁₇₃	D₄	1	1	yes	characteristic
8	C₁₆×C₁₇₃	C₂×C₄	1	1	yes	characteristic
9	C₁₆×C₁₇₃	Q₈	1	1	yes	characteristic
10	C₂×Dic₁₇₃	—	8	2	no	—
11	C₂×C₄×C₁₇₃	D₈	1	1	yes	characteristic
12	C₈×C₁₇₃	C₄⋊C₄	1	1	yes	characteristic
13	C₈×C₁₇₃	Q₁₆	1	1	yes	characteristic
14	C₂²×C₁₇₃	D₁₆	1	1	yes	characteristic
15	C₄×C₁₇₃	#107	1	1	yes	characteristic
16	C₄×C₁₇₃	Q₃₂	1	1	yes	characteristic
17	Dic₁₇₃	—	16	2	no	—
18	C₂×C₁₇₃	#366	1	1	yes	characteristic
19	C₂×C₁₇₃	SD₆₄	1	2	yes	—
20	C₁₇₃	#1424	1	1	yes	characteristic, complemented, Sylow
21	#1424	—	173	1	no	maximal, Sylow
22	C₂×C₃₂	D₁₇₃	1	1	yes	characteristic
23	#366	—	173	2	no	—
24	C₂×C₁₆	C₂×D₁₇₃	1	1	yes	characteristic
25	#107	—	346	2	no	—
26	C₃₂	Dic₁₇₃	1	2	yes	complemented
27	C₂×C₈	D₆₉₂	1	1	yes	characteristic
28	C₄⋊C₄	—	692	2	no	—
29	C₁₆	C₂×Dic₁₇₃	1	1	yes	characteristic
30	C₁₆	Dic₃₄₆	1	1	yes	characteristic
31	C₂×C₄	—	1384	2	no	—
32	C₂×C₄	D₁₃₈₄	1	1	yes	characteristic
33	C₈	C₄⋊Dic₁₇₃	1	1	yes	characteristic
34	C₈	Dic₆₉₂	1	1	yes	characteristic
35	C₂²	D₂₇₆₈	1	1	yes	characteristic
36	C₄	—	2768	2	no	—
37	C₄	#19149613	1	1	yes	characteristic
38	C₄	Dic₁₃₈₄	1	1	yes	characteristic
39	C₂	#26011317	1	1	yes	characteristic
40	C₂	#26011330	1	2	yes	—

	subgroup	G/H
center	C₂²	D₂₇₆₈
commutator	C₁₆×C₁₇₃	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₁₇₃
Fitting	C₂×C₃₂×C₁₇₃	C₂
socle, prime radical	C₂²×C₁₇₃	D₁₆
cosocle	C₂×C₁₆×C₁₇₃	C₂²
hypercenter	C₂×C₃₂	D₁₇₃
hypocenter	C₁₇₃	#1424
LCS	G ▷ C₁₆×C₁₇₃ ▷ C₈×C₁₇₃ ▷ C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂

Complements

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

N	H ≅ G/N	type
5. C₃₂×C₁₇₃	C₄	semidirect
20. C₁₇₃	#1424	semidirect
26. C₃₂	Dic₁₇₃	semidirect

Maximal subgroups

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

subgroup	index
#26011317	2
C₂×C₃₂×C₁₇₃	2
#1424	173