Subgroups — #27186284

Order 12928.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₁₀₁⋊C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
2	#20266105	C₂	1	1	yes	maximal, characteristic, complemented
3	#20265610	C₂	1	1	yes	maximal, characteristic
4	#15692326	—	2	1	no	—
5	#15692305	—	2	1	no	—
6	#15692278	—	2	1	no	—
7	#15692198	—	2	1	no	—
8	D₁₆₁₆	C₄	1	1	yes	characteristic
9	C₁₆×D₁₀₁	C₂²	1	1	yes	characteristic
10	Q₃₂×C₁₀₁	C₄	1	1	yes	characteristic
11	#13108084	—	4	1	no	—
12	#13108075	—	4	1	no	—
13	#13108061	—	4	1	no	—
14	D₈₀₈	—	2	1	no	—
15	C₈×D₁₀₁	D₄	1	1	yes	characteristic
16	#13108049	D₄	1	1	yes	characteristic
17	Q₁₆×C₁₀₁	—	2	1	no	—
18	C₁₆×C₁₀₁	C₂×C₄	1	1	yes	characteristic
19	Q₈×C₁₀₁	—	4	1	no	—
20	D₄₀₄	—	4	1	no	—
21	D₄₀₄	—	8	1	no	—
22	C₄×D₁₀₁	—	8	1	no	—
23	C₄×D₁₀₁	D₈	1	1	yes	characteristic
24	#11717581	—	8	1	no	—
25	C₈×C₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
26	#11717579	SD₁₆	1	1	yes	characteristic
27	C₂×D₁₀₁	—	8	1	no	—
28	C₂×D₁₀₁	SD₃₂	1	1	yes	characteristic
29	C₄×C₁₀₁	—	8	1	no	—
30	C₄×C₁₀₁	#102	1	1	yes	characteristic
31	Dic₁₀₁	D₁₆	1	1	yes	characteristic
32	C₂×C₁₀₁	#357	1	1	yes	characteristic
33	D₁₀₁	—	2	1	no	—
34	D₁₀₁	—	16	1	no	—
35	#1417	—	101	1	no	maximal, Sylow
36	C₁₀₁	#1417	1	1	yes	characteristic, complemented, Sylow
37	#508	—	101	1	no	—
38	C₂×C₃₂	—	101	1	no	—
39	#368	—	101	1	no	—
40	#135	—	202	1	no	—
41	Q₃₂	#87938	1	1	yes	characteristic
42	SD₃₂	—	202	1	no	—
43	D₁₆	—	101	1	no	—
44	C₂×C₁₆	—	101	1	no	—
45	#108	—	202	1	no	—
46	C₃₂	—	202	1	no	—
47	#41	—	404	1	no	—
48	Q₁₆	—	2	1	no	—
49	SD₁₆	—	404	1	no	—
50	D₈	—	202	1	no	—
51	M₁₆	—	404	1	no	—
52	C₂×C₈	—	101	1	no	—
53	C₁₆	—	101	1	no	—
54	C₁₆	C₂×#87938	1	1	yes	characteristic
55	Q₈	—	4	1	no	—
56	D₄	—	808	1	no	—
57	D₄	—	404	1	no	—
58	C₂×C₄	—	808	1	no	—
59	C₂×C₄	—	101	1	no	—
60	C₈	—	808	1	no	—
61	C₈	—	101	1	no	—
62	C₈	C₂²⋊#87938	1	1	yes	characteristic
63	C₂²	—	808	1	no	—
64	C₂²	—	101	1	no	—
65	C₄	—	8	1	no	—
66	C₄	—	101	1	no	—
67	C₄	#15692317	1	1	yes	characteristic
68	C₂	—	1616	1	no	—
69	C₂	—	202	1	no	—
70	C₂	#20266089	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20266089
commutator	C₁₆×C₁₀₁	C₂×C₄
Frattini	C₁₆	C₂×#87938
Fitting	Q₃₂×C₁₀₁	C₄
socle	C₂×C₁₀₁	#357
prime radical	#20265610	C₂
cosocle	C₁₆×D₁₀₁	C₂²
hypercenter	Q₃₂	#87938
hypocenter	C₁₀₁	#1417
LCS	G ▷ C₁₆×C₁₀₁ ▷ C₈×C₁₀₁ ▷ C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ Q₃₂

Complements

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

N	H ≅ G/N	type
1. D₁₀₁⋊C₃₂	C₂	semidirect
2. #20266105	C₂	semidirect
36. C₁₀₁	#1417	semidirect

Maximal subgroups

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

subgroup	index
D₁₀₁⋊C₃₂	2
#20266105	2
#20265610	2
#1417	101