Subgroups — #27169338

Order 12928.

52 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20265007	C₂	1	1	yes	maximal, characteristic
2	#20265005	C₂	1	1	yes	maximal, characteristic
3	M₆₄×C₁₀₁	C₂	1	1	yes	maximal, characteristic
4	#15692191	—	2	1	no	—
5	#15692189	—	2	1	no	—
6	C₂×C₁₆×C₁₀₁	C₂²	1	1	yes	characteristic
7	C₃₂×C₁₀₁	C₄	1	2	yes	complemented
8	#13108058	—	4	1	no	—
9	C₄⋊Dic₁₀₁	—	4	1	no	—
10	C₂×C₈×C₁₀₁	D₄	1	1	yes	characteristic
11	C₁₆×C₁₀₁	C₂×C₄	1	1	yes	characteristic
12	C₁₆×C₁₀₁	Q₈	1	1	yes	characteristic
13	C₂×C₄×C₁₀₁	D₈	1	1	yes	characteristic
14	C₂×Dic₁₀₁	—	8	1	no	—
15	C₈×C₁₀₁	C₄⋊C₄	1	1	yes	characteristic
16	C₈×C₁₀₁	Q₁₆	1	1	yes	characteristic
17	#11717579	—	8	1	no	—
18	C₂²×C₁₀₁	SD₃₂	1	1	yes	characteristic
19	C₄×C₁₀₁	#107	1	1	yes	characteristic
20	C₄×C₁₀₁	SD₃₂	1	1	yes	characteristic
21	Dic₁₀₁	—	16	1	no	—
22	C₂×C₁₀₁	—	2	1	no	—
23	C₂×C₁₀₁	#367	1	1	yes	characteristic
24	#1426	—	101	1	no	maximal, Sylow
25	C₁₀₁	#1426	1	1	yes	characteristic, complemented, Sylow
26	M₆₄	D₁₀₁	1	1	yes	characteristic
27	#368	—	101	1	no	—
28	#367	—	101	1	no	—
29	C₂×C₁₆	C₂×D₁₀₁	1	1	yes	characteristic
30	#108	—	202	1	no	—
31	#107	—	202	1	no	—
32	C₃₂	Dic₁₀₁	1	2	yes	complemented
33	M₁₆	—	404	1	no	—
34	C₂×C₈	D₄₀₄	1	1	yes	characteristic
35	C₄⋊C₄	—	404	1	no	—
36	C₁₆	Dic₂₀₂	1	1	yes	characteristic
37	C₁₆	C₂×Dic₁₀₁	1	1	yes	characteristic
38	C₂×C₄	—	808	1	no	—
39	C₂×C₄	D₈₀₈	1	1	yes	characteristic
40	C₈	—	808	1	no	—
41	C₈	C₄⋊Dic₁₀₁	1	1	yes	characteristic
42	C₈	Dic₄₀₄	1	1	yes	characteristic
43	C₂²	#15692200	1	1	yes	characteristic
44	C₄	—	1616	1	no	—
45	C₄	#15692189	1	1	yes	characteristic
46	C₄	#15692200	1	1	yes	characteristic
47	C₂	—	2	1	no	—
48	C₂	#20265005	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20265005
commutator	C₁₆×C₁₀₁	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₁₀₁
Fitting	M₆₄×C₁₀₁	C₂
socle	C₂×C₁₀₁	#367
prime radical	C₂²×C₁₀₁	SD₃₂
cosocle	C₂×C₁₆×C₁₀₁	C₂²
hypercenter	M₆₄	D₁₀₁
hypocenter	C₁₀₁	#1426
LCS	G ▷ C₁₆×C₁₀₁ ▷ C₈×C₁₀₁ ▷ C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
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×C₁₀₁	C₄	semidirect
25. C₁₀₁	#1426	semidirect
32. C₃₂	Dic₁₀₁	semidirect

Maximal subgroups

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

subgroup	index
#20265007	2
#20265005	2
M₆₄×C₁₀₁	2
#1426	101