Subgroups — #27186043

Order 12928.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20266063	C₂	1	1	yes	maximal, characteristic
2	C₈×#11717581	C₂	1	1	yes	maximal, characteristic
3	C₁₆×Dic₁₀₁	C₂	1	1	yes	maximal, characteristic
4	C₂×#13108077	C₄	1	2	yes	—
5	C₄×#11717581	C₄	1	2	yes	—
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	C₈	1	4	yes	—
10	#13108077	C₈	1	4	yes	complemented
11	C₂×#11717579	C₂×C₄	1	1	yes	characteristic
12	C₄×Dic₁₀₁	C₂×C₄	1	1	yes	characteristic
13	#13108049	C₈	1	2	yes	complemented
14	C₂×C₈×C₁₀₁	C₂×C₄	1	1	yes	characteristic
15	C₁₆×C₁₀₁	C₈	1	2	yes	complemented
16	C₂×C₄×C₁₀₁	C₄²	1	1	yes	characteristic
17	C₂×Dic₁₀₁	C₂×C₈	1	2	yes	—
18	#11717581	—	2	4	no	—
19	C₈×C₁₀₁	C₂×C₈	1	1	yes	characteristic
20	C₈×C₁₀₁	C₂×C₈	1	1	yes	characteristic
21	#11717579	C₂×C₈	1	2	yes	—
22	C₂²×C₁₀₁	C₄×C₈	1	1	yes	characteristic
23	C₄×C₁₀₁	C₄×C₈	1	1	yes	characteristic
24	C₄×C₁₀₁	C₄×C₈	1	1	yes	characteristic
25	Dic₁₀₁	M₃₂	1	4	yes	—
26	C₂×C₁₀₁	C₈²	1	1	yes	characteristic
27	C₂×C₁₀₁	#346	1	2	yes	—
28	#1311	—	101	1	no	maximal, Sylow
29	C₁₀₁	#1311	1	1	yes	characteristic, complemented, Sylow
30	C₄×C₁₆	—	101	1	no	—
31	C₄×C₁₆	—	101	1	no	—
32	C₈²	—	101	1	no	—
33	C₂×C₁₆	—	101	2	no	—
34	C₂×C₁₆	—	101	1	no	—
35	C₂×C₁₆	#87938	1	1	yes	characteristic
36	C₄×C₈	—	101	2	no	—
37	C₄×C₈	—	101	1	no	—
38	C₂×C₈	—	101	1	no	—
39	C₂×C₈	—	101	4	no	—
40	C₂×C₈	C₂×#87938	1	1	yes	characteristic
41	C₄²	—	101	1	no	—
42	C₁₆	—	101	4	no	—
43	C₁₆	—	101	2	no	—
44	C₁₆	#11717581	1	2	yes	complemented
45	C₂×C₄	—	101	2	no	—
46	C₂×C₄	C₄×#87938	1	1	yes	characteristic
47	C₈	—	101	2	no	—
48	C₈	—	202	4	no	—
49	C₈	C₂×#11717581	1	1	yes	characteristic
50	C₈	D₁₀₁⋊C₈	1	1	yes	characteristic
51	C₂²	C₈×#87938	1	1	yes	characteristic
52	C₄	—	101	4	no	—
53	C₄	C₈×#87938	1	1	yes	characteristic
54	C₄	C₄×#11717581	1	1	yes	characteristic
55	C₂	C₈×#11717581	1	1	yes	characteristic
56	C₂	#20266064	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂×#87938
commutator	C₂×C₁₀₁	C₈²
Fitting	C₂×C₁₆×C₁₀₁	C₄
socle, prime radical	C₂²×C₁₀₁	C₄×C₈
cosocle	C₈×Dic₁₀₁	C₂²
hypercenter	C₂×C₁₆	#87938
hypocenter	C₁₀₁	#1311
LCS	G ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
10. #13108077	C₈	semidirect
13. #13108049	C₈	semidirect
15. C₁₆×C₁₀₁	C₈	semidirect
29. C₁₀₁	#1311	semidirect
44. C₁₆	#11717581	semidirect

Maximal subgroups

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

subgroup	index
#20266063	2
C₈×#11717581	2
C₁₆×Dic₁₀₁	2
#1311	101