Subgroups — #27169317

Order 12928.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20265609	C₂	1	1	yes	maximal, characteristic
2	#20265007	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₂×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#15692277	—	2	1	no	—
5	Dic₈₀₈	C₄	1	1	yes	characteristic
6	#15692200	—	2	1	no	—
7	D₁₆₁₆	C₄	1	1	yes	characteristic
8	#15692191	—	2	1	no	—
9	C₂×C₁₆×C₁₀₁	C₂²	1	1	yes	characteristic
10	C₃₂×C₁₀₁	—	2	1	no	—
11	#13108074	—	4	1	no	—
12	Dic₄₀₄	—	2	1	no	—
13	#13108063	—	4	1	no	—
14	D₈₀₈	—	2	1	no	—
15	#13108058	—	4	1	no	—
16	C₂×C₈×C₁₀₁	D₄	1	1	yes	characteristic
17	C₁₆×C₁₀₁	C₂×C₄	1	1	yes	characteristic
18	C₁₆×C₁₀₁	D₄	1	1	yes	characteristic
19	C₂×C₄×C₁₀₁	D₈	1	1	yes	characteristic
20	C₂²⋊D₁₀₁	—	8	1	no	—
21	D₄₀₄	—	4	1	no	—
22	C₄×D₁₀₁	—	8	1	no	—
23	Dic₂₀₂	—	4	1	no	—
24	C₈×C₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
25	C₈×C₁₀₁	SD₁₆	1	1	yes	characteristic
26	#11717579	—	8	1	no	—
27	C₂²×C₁₀₁	SD₃₂	1	1	yes	characteristic
28	C₂×D₁₀₁	—	8	1	no	—
29	C₄×C₁₀₁	#102	1	1	yes	characteristic
30	C₄×C₁₀₁	D₁₆	1	1	yes	characteristic
31	Dic₁₀₁	—	8	1	no	—
32	C₂×C₁₀₁	—	2	1	no	—
33	C₂×C₁₀₁	#357	1	1	yes	characteristic
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	D₁₀₁	1	1	yes	characteristic, complemented
39	#368	—	101	1	no	—
40	#135	—	202	1	no	—
41	Q₃₂	—	101	1	no	—
42	SD₃₂	—	202	1	no	—
43	D₁₆	—	101	1	no	—
44	C₂×C₁₆	C₂×D₁₀₁	1	1	yes	characteristic
45	#108	—	202	1	no	—
46	C₃₂	—	2	1	no	—
47	#41	—	404	1	no	—
48	Q₁₆	—	202	1	no	—
49	SD₁₆	—	404	1	no	—
50	D₈	—	202	1	no	—
51	M₁₆	—	404	1	no	—
52	C₂×C₈	D₄₀₄	1	1	yes	characteristic
53	C₁₆	C₄×D₁₀₁	1	1	yes	characteristic
54	C₁₆	C₂²⋊D₁₀₁	1	1	yes	characteristic
55	Q₈	—	404	1	no	—
56	D₄	—	808	1	no	—
57	D₄	—	404	1	no	—
58	C₂×C₄	—	808	1	no	—
59	C₂×C₄	D₈₀₈	1	1	yes	characteristic
60	C₈	—	808	1	no	—
61	C₈	#13108051	1	1	yes	characteristic
62	C₈	#13108063	1	1	yes	characteristic
63	C₂²	—	808	1	no	—
64	C₂²	#15692200	1	1	yes	characteristic
65	C₄	—	808	1	no	—
66	C₄	#15692177	1	1	yes	characteristic
67	C₄	D₁₆₁₆	1	1	yes	characteristic
68	C₂	—	2	1	no	—
69	C₂	—	1616	1	no	—
70	C₂	#20264979	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₁₆₁₆
commutator	C₁₆×C₁₀₁	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₁₀₁
Fitting	C₂×C₃₂×C₁₀₁	C₂
socle	C₂×C₁₀₁	#357
prime radical	#20265609	C₂
cosocle	C₂×C₁₆×C₁₀₁	C₂²
hypercenter	C₂×C₃₂	D₁₀₁
hypocenter	C₁₀₁	#1417
LCS	G ▷ C₁₆×C₁₀₁ ▷ C₈×C₁₀₁ ▷ C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂

Complements

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

N	H ≅ G/N	type
2. #20265007	C₂	semidirect
3. C₂×C₃₂×C₁₀₁	C₂	semidirect
36. C₁₀₁	#1417	semidirect
38. C₂×C₃₂	D₁₀₁	semidirect

Maximal subgroups

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

subgroup	index
#20265609	2
#20265007	2
C₂×C₃₂×C₁₀₁	2
#1417	101