Subgroups — #27186158

Order 12928.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20266038	C₂	1	1	yes	maximal, characteristic
2	#20266038	C₂	1	1	yes	maximal, characteristic
3	#20264952	C₂	1	1	yes	maximal, characteristic
4	C₄⋊#11717581	—	2	1	no	—
5	C₄⋊#11717581	—	2	1	no	—
6	C₂×#13108049	C₄	1	1	yes	characteristic
7	C₈×Dic₁₀₁	C₂²	1	1	yes	characteristic
8	C₂×C₁₆×C₁₀₁	C₄	1	1	yes	characteristic
9	C₂×#11717581	—	4	1	no	—
10	C₂×#11717581	—	4	1	no	—
11	C₂×#11717579	Q₈	1	1	yes	characteristic
12	C₄×Dic₁₀₁	D₄	1	1	yes	characteristic
13	#13108049	C₈	1	2	yes	complemented
14	C₂×C₈×C₁₀₁	C₂×C₄	1	1	yes	characteristic
15	C₁₆×C₁₀₁	—	2	1	no	—
16	C₂×C₄×C₁₀₁	C₄⋊C₄	1	1	yes	characteristic
17	C₂×Dic₁₀₁	SD₁₆	1	1	yes	characteristic
18	C₂×Dic₁₀₁	SD₁₆	1	1	yes	characteristic
19	#11717581	—	8	1	no	—
20	#11717581	—	8	1	no	—
21	C₈×C₁₀₁	C₂×C₈	1	1	yes	characteristic
22	C₈×C₁₀₁	M₁₆	1	1	yes	characteristic
23	#11717579	—	2	1	no	—
24	C₂²×C₁₀₁	#106	1	1	yes	characteristic
25	C₄×C₁₀₁	C₄⋊C₈	1	1	yes	characteristic
26	C₄×C₁₀₁	#108	1	1	yes	characteristic
27	Dic₁₀₁	—	2	1	no	—
28	Dic₁₀₁	—	2	1	no	—
29	C₂×C₁₀₁	#334	1	1	yes	characteristic
30	C₂×C₁₀₁	#365	1	1	yes	characteristic
31	C₂×C₁₀₁	#365	1	1	yes	characteristic
32	#1368	—	101	1	no	maximal, Sylow
33	C₁₀₁	#1368	1	1	yes	characteristic, complemented, Sylow
34	#346	—	101	1	no	—
35	#334	—	101	1	no	—
36	#334	—	101	1	no	—
37	C₂×C₁₆	—	101	1	no	—
38	C₂×C₁₆	#87938	1	1	yes	characteristic
39	C₄⋊C₈	—	202	1	no	—
40	C₄⋊C₈	—	202	1	no	—
41	C₄×C₈	—	101	1	no	—
42	C₂×C₈	—	101	1	no	—
43	C₂×C₈	—	404	1	no	—
44	C₂×C₈	—	404	1	no	—
45	C₂×C₈	C₂×#87938	1	1	yes	characteristic
46	C₄²	—	101	1	no	—
47	C₁₆	—	2	1	no	—
48	C₁₆	—	101	2	no	—
49	C₂×C₄	—	101	1	no	—
50	C₂×C₄	—	101	1	no	—
51	C₂×C₄	C₄⋊#87938	1	1	yes	characteristic
52	C₈	—	808	1	no	—
53	C₈	—	808	1	no	—
54	C₈	—	202	1	no	—
55	C₈	D₁₀₁⋊C₈	1	1	yes	characteristic
56	C₈	#13108083	1	1	yes	characteristic
57	C₂²	#15692323	1	1	yes	characteristic
58	C₄	—	202	1	no	—
59	C₄	—	202	1	no	—
60	C₄	#15692322	1	1	yes	characteristic
61	C₄	#15692326	1	1	yes	characteristic
62	C₂	#20266039	1	1	yes	characteristic
63	C₂	#20266101	1	1	yes	characteristic
64	C₂	#20266102	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#15692323
commutator	C₈×C₁₀₁	C₂×C₈
Frattini	C₂×C₈	C₂×#87938
Fitting	C₂×C₁₆×C₁₀₁	C₄
socle, prime radical	C₂²×C₁₀₁	#106
cosocle	C₈×Dic₁₀₁	C₂²
hypercenter	C₂×C₁₆	#87938
hypocenter	C₁₀₁	#1368
LCS	G ▷ C₈×C₁₀₁ ▷ C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
13. #13108049	C₈	semidirect
33. C₁₀₁	#1368	semidirect

Maximal subgroups

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

subgroup	index
#20266038	2
#20264952	2
#1368	101