Subgroups — #27169194

Order 12928.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20265579	C₂	1	1	yes	maximal, characteristic
2	#20264998	C₂	1	1	yes	maximal, characteristic
3	#364×C₁₀₁	C₂	1	1	yes	maximal, characteristic
4	#15692247	C₄	1	2	yes	—
5	#15692195	—	2	1	no	—
6	#15692189	—	2	1	no	—
7	#15692187	—	2	1	no	—
8	M₃₂×C₁₀₁	—	2	1	no	—
9	C₄×C₈×C₁₀₁	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₁₀₁	—	2	2	no	—
11	#13108053	—	4	2	no	—
12	#13108049	—	4	1	no	—
13	C₂×C₈×C₁₀₁	D₄	1	1	yes	characteristic
14	C₂×C₈×C₁₀₁	D₄	1	1	yes	characteristic
15	C₄²×C₁₀₁	C₂×C₄	1	1	yes	characteristic
16	C₁₆×C₁₀₁	—	4	1	no	—
17	C₂×C₄×C₁₀₁	—	2	1	no	—
18	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
19	C₂×Dic₁₀₁	—	4	2	no	—
20	C₈×C₁₀₁	SD₁₆	1	1	yes	characteristic
21	C₈×C₁₀₁	SD₁₆	1	1	yes	characteristic
22	C₈×C₁₀₁	Q₁₆	1	1	yes	characteristic
23	C₈×C₁₀₁	Q₁₆	1	1	yes	characteristic
24	C₂²×C₁₀₁	#100	1	1	yes	characteristic
25	C₄×C₁₀₁	—	4	1	no	—
26	C₄×C₁₀₁	#103	1	1	yes	characteristic
27	C₄×C₁₀₁	#103	1	1	yes	characteristic
28	Dic₁₀₁	—	8	2	no	—
29	C₂×C₁₀₁	—	2	1	no	—
30	C₂×C₁₀₁	#333	1	1	yes	characteristic
31	#1366	—	101	1	no	maximal, Sylow
32	C₁₀₁	#1366	1	1	yes	characteristic, complemented, Sylow
33	#499	—	101	1	no	—
34	#364	—	101	1	no	—
35	#364	D₁₀₁	1	1	yes	characteristic
36	#125	—	101	2	no	—
37	M₃₂	—	2	1	no	—
38	M₃₂	—	202	1	no	—
39	#107	—	202	1	no	—
40	#106	—	202	1	no	—
41	C₄×C₈	C₂×D₁₀₁	1	1	yes	characteristic
42	C₂×C₈	D₄₀₄	1	1	yes	characteristic
43	C₂×C₈	C₂²⋊D₁₀₁	1	1	yes	characteristic
44	C₄⋊C₄	—	404	2	no	—
45	C₄⋊C₄	—	202	2	no	—
46	C₄²	C₄×D₁₀₁	1	1	yes	characteristic
47	C₁₆	—	4	1	no	—
48	C₁₆	—	404	1	no	—
49	C₂×C₄	—	2	1	no	—
50	C₂×C₄	—	404	2	no	—
51	C₂×C₄	#13108051	1	1	yes	characteristic
52	C₈	#13108062	1	1	yes	characteristic
53	C₈	#13108063	1	1	yes	characteristic
54	C₈	#13108064	1	1	yes	characteristic
55	C₈	Dic₄₀₄	1	1	yes	characteristic
56	C₂²	#15692171	1	1	yes	characteristic
57	C₄	—	4	1	no	—
58	C₄	—	808	2	no	—
59	C₄	#15692178	1	1	yes	characteristic
60	C₄	#15692180	1	1	yes	characteristic
61	C₂	—	2	1	no	—
62	C₂	#20264925	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20264925
commutator	C₄²×C₁₀₁	C₂×C₄
Frattini	C₄×C₈	C₂×D₁₀₁
Fitting	#364×C₁₀₁	C₂
socle	C₂×C₁₀₁	#333
prime radical	C₂²×C₁₀₁	#100
cosocle	C₄×C₈×C₁₀₁	C₂²
hypercenter	#364	D₁₀₁
hypocenter	C₁₀₁	#1366
LCS	G ▷ C₄²×C₁₀₁ ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #364

Complements

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

N	H ≅ G/N	type
32. C₁₀₁	#1366	semidirect

Maximal subgroups

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

subgroup	index
#20265579	2
#20264998	2
#364×C₁₀₁	2
#1366	101