Subgroups — #27169197

Order 12928.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20264953	C₂	1	1	yes	maximal, characteristic
2	#20264928	C₂	1	1	yes	maximal, characteristic
3	#334×C₁₀₁	C₂	1	1	yes	maximal, characteristic
4	C₂×#13108049	C₄	1	1	yes	characteristic
5	C₂×#13108049	C₄	1	1	yes	characteristic
6	C₄⋊#11717579	—	2	1	no	—
7	C₄⋊C₈×C₁₀₁	—	2	1	no	—
8	C₄×C₈×C₁₀₁	C₂²	1	1	yes	characteristic
9	C₂×#11717579	—	4	1	no	—
10	#13108049	—	2	1	no	—
11	#13108049	C₈	1	2	yes	complemented
12	C₂×C₈×C₁₀₁	—	4	1	no	—
13	C₂×C₈×C₁₀₁	C₂×C₄	1	1	yes	characteristic
14	C₂×C₈×C₁₀₁	Q₈	1	1	yes	characteristic
15	C₄²×C₁₀₁	D₄	1	1	yes	characteristic
16	C₂×C₄×C₁₀₁	C₄⋊C₄	1	1	yes	characteristic
17	C₂×C₄×C₁₀₁	SD₁₆	1	1	yes	characteristic
18	C₂×C₄×C₁₀₁	SD₁₆	1	1	yes	characteristic
19	C₈×C₁₀₁	—	2	1	no	—
20	C₈×C₁₀₁	—	8	1	no	—
21	C₈×C₁₀₁	C₂×C₈	1	1	yes	characteristic
22	C₈×C₁₀₁	M₁₆	1	1	yes	characteristic
23	#11717579	—	8	1	no	—
24	C₂²×C₁₀₁	#106	1	1	yes	characteristic
25	C₄×C₁₀₁	—	2	1	no	—
26	C₄×C₁₀₁	—	2	1	no	—
27	C₄×C₁₀₁	C₄⋊C₈	1	1	yes	characteristic
28	C₄×C₁₀₁	#108	1	1	yes	characteristic
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	D₁₀₁	1	1	yes	characteristic
37	C₂×C₁₆	—	101	1	no	—
38	C₂×C₁₆	—	101	1	no	—
39	C₄⋊C₈	—	2	1	no	—
40	C₄⋊C₈	—	202	1	no	—
41	C₄×C₈	C₂×D₁₀₁	1	1	yes	characteristic
42	C₂×C₈	—	4	1	no	—
43	C₂×C₈	—	404	1	no	—
44	C₂×C₈	Dic₂₀₂	1	1	yes	characteristic
45	C₂×C₈	C₄×D₁₀₁	1	1	yes	characteristic
46	C₄²	C₂²⋊D₁₀₁	1	1	yes	characteristic
47	C₁₆	—	101	2	no	—
48	C₁₆	—	202	1	no	—
49	C₂×C₄	#13108053	1	1	yes	characteristic
50	C₂×C₄	#13108061	1	1	yes	characteristic
51	C₂×C₄	#13108062	1	1	yes	characteristic
52	C₈	—	2	1	no	—
53	C₈	—	8	1	no	—
54	C₈	—	808	1	no	—
55	C₈	C₈×D₁₀₁	1	1	yes	characteristic
56	C₈	#13108057	1	1	yes	characteristic
57	C₂²	#15692186	1	1	yes	characteristic
58	C₄	—	2	1	no	—
59	C₄	—	2	1	no	—
60	C₄	#15692184	1	1	yes	characteristic
61	C₄	#15692190	1	1	yes	characteristic
62	C₂	#20264927	1	1	yes	characteristic
63	C₂	#20264999	1	1	yes	characteristic
64	C₂	#20265001	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#15692186
commutator	C₈×C₁₀₁	C₂×C₈
Frattini	C₄×C₈	C₂×D₁₀₁
Fitting	#334×C₁₀₁	C₂
socle, prime radical	C₂²×C₁₀₁	#106
cosocle	C₄×C₈×C₁₀₁	C₂²
hypercenter	#334	D₁₀₁
hypocenter	C₁₀₁	#1368
LCS	G ▷ C₈×C₁₀₁ ▷ C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #334

Complements

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

N	H ≅ G/N	type
11. #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
#20264953	2
#20264928	2
#334×C₁₀₁	2
#1368	101