Subgroups — #36351769

Order 22144.

52 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26011321	C₂	1	1	yes	maximal, characteristic
2	#26011319	C₂	1	1	yes	maximal, characteristic
3	M₆₄×C₁₇₃	C₂	1	1	yes	maximal, characteristic
4	#19149615	—	2	1	no	—
5	#19149613	—	2	1	no	—
6	C₂×C₁₆×C₁₇₃	C₂²	1	1	yes	characteristic
7	C₃₂×C₁₇₃	C₄	1	2	yes	complemented
8	#14583810	—	4	1	no	—
9	C₄⋊Dic₁₇₃	—	4	1	no	—
10	C₂×C₈×C₁₇₃	D₄	1	1	yes	characteristic
11	C₁₆×C₁₇₃	C₂×C₄	1	1	yes	characteristic
12	C₁₆×C₁₇₃	Q₈	1	1	yes	characteristic
13	C₂×Dic₁₇₃	—	8	1	no	—
14	#13063436	—	8	1	no	—
15	C₂×C₄×C₁₇₃	D₈	1	1	yes	characteristic
16	C₈×C₁₇₃	C₄⋊C₄	1	1	yes	characteristic
17	C₈×C₁₇₃	Q₁₆	1	1	yes	characteristic
18	C₂²×C₁₇₃	SD₃₂	1	1	yes	characteristic
19	C₄×C₁₇₃	#107	1	1	yes	characteristic
20	C₄×C₁₇₃	SD₃₂	1	1	yes	characteristic
21	Dic₁₇₃	—	16	1	no	—
22	C₂×C₁₇₃	—	2	1	no	—
23	C₂×C₁₇₃	#367	1	1	yes	characteristic
24	C₁₇₃	#1426	1	1	yes	characteristic, complemented, Sylow
25	#1426	—	173	1	no	maximal, Sylow
26	M₆₄	D₁₇₃	1	1	yes	characteristic
27	#368	—	173	1	no	—
28	#367	—	173	1	no	—
29	C₂×C₁₆	C₂×D₁₇₃	1	1	yes	characteristic
30	#108	—	346	1	no	—
31	#107	—	346	1	no	—
32	C₃₂	Dic₁₇₃	1	2	yes	complemented
33	M₁₆	—	692	1	no	—
34	C₂×C₈	D₆₉₂	1	1	yes	characteristic
35	C₄⋊C₄	—	692	1	no	—
36	C₁₆	C₂×Dic₁₇₃	1	1	yes	characteristic
37	C₁₆	Dic₃₄₆	1	1	yes	characteristic
38	C₂×C₄	—	1384	1	no	—
39	C₂×C₄	D₁₃₈₄	1	1	yes	characteristic
40	C₈	—	1384	1	no	—
41	C₈	C₄⋊Dic₁₇₃	1	1	yes	characteristic
42	C₈	Dic₆₉₂	1	1	yes	characteristic
43	C₂²	#19149624	1	1	yes	characteristic
44	C₄	—	2768	1	no	—
45	C₄	#19149613	1	1	yes	characteristic
46	C₄	#19149624	1	1	yes	characteristic
47	C₂	—	2	1	no	—
48	C₂	#26011319	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#26011319
commutator	C₁₆×C₁₇₃	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₁₇₃
Fitting	M₆₄×C₁₇₃	C₂
socle	C₂×C₁₇₃	#367
prime radical	C₂²×C₁₇₃	SD₃₂
cosocle	C₂×C₁₆×C₁₇₃	C₂²
hypercenter	M₆₄	D₁₇₃
hypocenter	C₁₇₃	#1426
LCS	G ▷ C₁₆×C₁₇₃ ▷ C₈×C₁₇₃ ▷ C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
7. C₃₂×C₁₇₃	C₄	semidirect
24. C₁₇₃	#1426	semidirect
32. C₃₂	Dic₁₇₃	semidirect

Maximal subgroups

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

subgroup	index
#26011321	2
#26011319	2
M₆₄×C₁₇₃	2
#1426	173