Subgroups — #27169199

Order 12928.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20264998	C₂	1	1	yes	maximal, characteristic
2	#20264953	C₂	1	1	yes	maximal, characteristic
3	#364×C₁₀₁	C₂	1	1	yes	maximal, characteristic
4	#15692195	—	2	1	no	—
5	C₂×#13108049	C₄	1	1	yes	characteristic
6	C₂×#13108049	C₄	1	1	yes	characteristic
7	M₃₂×C₁₀₁	—	2	1	no	—
8	C₄×C₈×C₁₀₁	C₂²	1	1	yes	characteristic
9	#13108049	—	2	1	no	—
10	#13108049	—	4	1	no	—
11	#13108049	C₈	1	2	yes	—
12	C₂×C₈×C₁₀₁	C₂×C₄	1	1	yes	characteristic
13	C₂×C₈×C₁₀₁	D₄	1	1	yes	characteristic
14	C₄²×C₁₀₁	Q₈	1	1	yes	characteristic
15	C₁₆×C₁₀₁	—	4	1	no	—
16	C₂×C₄×C₁₀₁	—	2	1	no	—
17	C₂×C₄×C₁₀₁	C₄⋊C₄	1	1	yes	characteristic
18	C₈×C₁₀₁	C₂×C₈	1	1	yes	characteristic
19	C₈×C₁₀₁	M₁₆	1	1	yes	characteristic
20	C₈×C₁₀₁	SD₁₆	1	1	yes	characteristic
21	C₈×C₁₀₁	SD₁₆	1	1	yes	characteristic
22	C₂²×C₁₀₁	#108	1	1	yes	characteristic
23	C₄×C₁₀₁	—	4	1	no	—
24	C₄×C₁₀₁	C₄⋊C₈	1	1	yes	characteristic
25	C₄×C₁₀₁	#106	1	1	yes	characteristic
26	C₂×C₁₀₁	—	2	1	no	—
27	C₂×C₁₀₁	#334	1	1	yes	characteristic
28	#1369	—	101	1	no	maximal, Sylow
29	C₁₀₁	#1369	1	1	yes	characteristic, complemented, Sylow
30	#364	—	101	1	no	—
31	#364	D₁₀₁	1	1	yes	characteristic
32	#346	—	101	1	no	—
33	M₃₂	—	2	1	no	—
34	M₃₂	—	202	1	no	—
35	C₂×C₁₆	—	101	1	no	—
36	C₂×C₁₆	—	101	1	no	—
37	C₄×C₈	C₂×D₁₀₁	1	1	yes	characteristic
38	C₂×C₈	C₄×D₁₀₁	1	1	yes	characteristic
39	C₂×C₈	C₂²⋊D₁₀₁	1	1	yes	characteristic
40	C₄²	Dic₂₀₂	1	1	yes	characteristic
41	C₁₆	—	4	1	no	—
42	C₁₆	—	101	2	no	—
43	C₁₆	—	404	1	no	—
44	C₁₆	—	202	1	no	—
45	C₂×C₄	—	2	1	no	—
46	C₂×C₄	#13108053	1	1	yes	characteristic
47	C₈	C₈×D₁₀₁	1	1	yes	characteristic
48	C₈	#13108057	1	1	yes	characteristic
49	C₈	#13108061	1	1	yes	characteristic
50	C₈	#13108062	1	1	yes	characteristic
51	C₂²	#15692190	1	1	yes	characteristic
52	C₄	—	4	1	no	—
53	C₄	#15692184	1	1	yes	characteristic
54	C₄	#15692186	1	1	yes	characteristic
55	C₂	—	2	1	no	—
56	C₂	#20264927	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#15692186
commutator	C₈×C₁₀₁	C₂×C₈
Frattini	C₄×C₈	C₂×D₁₀₁
Fitting	#364×C₁₀₁	C₂
socle	C₂×C₁₀₁	#334
prime radical	C₂²×C₁₀₁	#108
cosocle	C₄×C₈×C₁₀₁	C₂²
hypercenter	#364	D₁₀₁
hypocenter	C₁₀₁	#1369
LCS	G ▷ C₈×C₁₀₁ ▷ C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #364

Complements

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

N	H ≅ G/N	type
29. C₁₀₁	#1369	semidirect

Maximal subgroups

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

subgroup	index
#20264998	2
#20264953	2
#364×C₁₀₁	2
#1369	101