Subgroups — #36368729

Order 22144.

52 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26012420	C₂	1	1	yes	maximal, characteristic
2	#26012418	C₂	1	1	yes	maximal, characteristic
3	#26011324	C₂	1	1	yes	maximal, characteristic
4	#19149750	—	2	1	no	—
5	#19149748	—	2	1	no	—
6	C₁₆×D₁₇₃	C₂²	1	1	yes	characteristic
7	#19149585	C₄	1	1	yes	characteristic, complemented
8	C₃₂×C₁₇₃	C₄	1	1	yes	characteristic, complemented
9	#14583836	—	4	1	no	—
10	C₄⋊#10621924	—	4	1	no	—
11	C₈×D₁₇₃	D₄	1	1	yes	characteristic
12	#14583801	Q₈	1	1	yes	characteristic
13	C₁₆×C₁₇₃	C₂×C₄	1	1	yes	characteristic
14	C₂×#10621924	—	8	1	no	—
15	#13063443	—	8	1	no	—
16	C₄×D₁₇₃	D₈	1	1	yes	characteristic
17	#13063436	Q₁₆	1	1	yes	characteristic
18	C₈×C₁₇₃	C₄⋊C₄	1	1	yes	characteristic
19	C₂×D₁₇₃	SD₃₂	1	1	yes	characteristic
20	#10621924	—	16	1	no	—
21	C₄×C₁₇₃	#107	1	1	yes	characteristic
22	Dic₁₇₃	SD₃₂	1	1	yes	characteristic
23	C₂×C₁₇₃	#367	1	1	yes	characteristic
24	D₁₇₃	—	2	1	no	—
25	C₁₇₃	#1426	1	1	yes	characteristic, complemented, Sylow
26	#1426	—	173	1	no	maximal, Sylow
27	M₆₄	—	173	1	no	—
28	#368	—	173	1	no	—
29	#367	—	173	1	no	—
30	C₂×C₁₆	—	173	1	no	—
31	#108	—	346	1	no	—
32	#107	—	346	1	no	—
33	C₃₂	—	173	1	no	—
34	C₃₂	#10621924	1	1	yes	characteristic, complemented
35	M₁₆	—	692	1	no	—
36	C₂×C₈	—	173	1	no	—
37	C₄⋊C₄	—	692	1	no	—
38	C₁₆	—	173	1	no	—
39	C₁₆	C₂×#10621924	1	1	yes	characteristic
40	C₂×C₄	—	1384	1	no	—
41	C₂×C₄	—	173	1	no	—
42	C₈	—	1384	1	no	—
43	C₈	—	173	1	no	—
44	C₈	C₄⋊#10621924	1	1	yes	characteristic
45	C₂²	—	173	1	no	—
46	C₄	—	173	1	no	—
47	C₄	—	2768	1	no	—
48	C₄	#19149748	1	1	yes	characteristic
49	C₂	—	346	1	no	—
50	C₂	#26012418	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#26012418
commutator	C₁₆×C₁₇₃	C₂×C₄
Frattini	C₁₆	C₂×#10621924
Fitting	C₃₂×C₁₇₃	C₄
socle	C₂×C₁₇₃	#367
prime radical	C₂×D₁₇₃	SD₃₂
cosocle	C₁₆×D₁₇₃	C₂²
hypercenter	C₃₂	#10621924
hypocenter	C₁₇₃	#1426
LCS	G ▷ C₁₆×C₁₇₃ ▷ C₈×C₁₇₃ ▷ C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ C₃₂

Complements

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

N	H ≅ G/N	type
7. #19149585	C₄	semidirect
8. C₃₂×C₁₇₃	C₄	semidirect
25. C₁₇₃	#1426	semidirect
34. C₃₂	#10621924	semidirect

Maximal subgroups

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

subgroup	index
#26012420	2
#26012418	2
#26011324	2
#1426	173