Subgroups — #27169323

Order 12928.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₆₁₆	C₂	1	1	yes	maximal, characteristic
2	#20265007	C₂	1	1	yes	maximal, characteristic, complemented
3	M₆₄×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₈₀₈	—	2	1	no	—
5	D₁₆₁₆	—	2	1	no	—
6	D₁₆₁₆	C₄	1	2	yes	—
7	#15692191	—	2	1	no	—
8	C₂×C₁₆×C₁₀₁	C₂²	1	1	yes	characteristic
9	C₃₂×C₁₀₁	—	2	1	no	—
10	C₂×D₄₀₄	—	4	1	no	—
11	D₈₀₈	—	2	2	no	—
12	D₈₀₈	—	4	1	no	—
13	#13108058	—	4	1	no	—
14	C₂×C₈×C₁₀₁	D₄	1	1	yes	characteristic
15	C₁₆×C₁₀₁	C₂×C₄	1	1	yes	characteristic
16	C₁₆×C₁₀₁	D₄	1	1	yes	characteristic
17	C₂²×D₁₀₁	—	8	1	no	—
18	C₂×C₄×C₁₀₁	D₈	1	1	yes	characteristic
19	D₄₀₄	—	4	2	no	—
20	D₄₀₄	—	8	1	no	—
21	C₈×C₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
22	C₈×C₁₀₁	SD₁₆	1	1	yes	characteristic
23	#11717579	—	8	1	no	—
24	C₂²×C₁₀₁	SD₃₂	1	1	yes	characteristic
25	C₂×D₁₀₁	—	8	2	no	—
26	C₂×D₁₀₁	—	16	2	no	—
27	C₄×C₁₀₁	#102	1	1	yes	characteristic
28	C₄×C₁₀₁	D₁₆	1	1	yes	characteristic
29	C₂×C₁₀₁	—	2	1	no	—
30	C₂×C₁₀₁	#357	1	1	yes	characteristic
31	D₁₀₁	—	16	2	no	—
32	#1419	—	101	1	no	maximal, Sylow
33	C₁₀₁	#1419	1	1	yes	characteristic, complemented, Sylow
34	C₂×D₁₆	—	101	1	no	—
35	M₆₄	D₁₀₁	1	1	yes	characteristic, complemented
36	#368	—	101	1	no	—
37	C₂×D₈	—	202	1	no	—
38	D₁₆	—	101	2	no	—
39	D₁₆	—	202	1	no	—
40	C₂×C₁₆	C₂×D₁₀₁	1	1	yes	characteristic
41	#108	—	202	1	no	—
42	C₃₂	—	2	1	no	—
43	C₂×D₄	—	404	1	no	—
44	D₈	—	404	1	no	—
45	D₈	—	202	2	no	—
46	M₁₆	—	404	1	no	—
47	C₂×C₈	D₄₀₄	1	1	yes	characteristic
48	C₁₆	C₄×D₁₀₁	1	1	yes	characteristic
49	C₁₆	C₂²⋊D₁₀₁	1	1	yes	characteristic
50	C₂³	—	808	1	no	—
51	D₄	—	808	1	no	—
52	D₄	—	404	2	no	—
53	C₂×C₄	D₈₀₈	1	1	yes	characteristic
54	C₈	—	808	1	no	—
55	C₈	#13108051	1	1	yes	characteristic
56	C₈	#13108063	1	1	yes	characteristic
57	C₂²	—	808	2	no	—
58	C₂²	—	1616	2	no	—
59	C₂²	#15692200	1	1	yes	characteristic
60	C₄	#15692177	1	1	yes	characteristic
61	C₄	D₁₆₁₆	1	1	yes	characteristic
62	C₂	—	2	1	no	—
63	C₂	—	1616	2	no	—
64	C₂	#20264979	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20264979
commutator	C₁₆×C₁₀₁	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₁₀₁
Fitting	M₆₄×C₁₀₁	C₂
socle	C₂×C₁₀₁	#357
prime radical	C₂×D₁₆₁₆	C₂
cosocle	C₂×C₁₆×C₁₀₁	C₂²
hypercenter	M₆₄	D₁₀₁
hypocenter	C₁₀₁	#1419
LCS	G ▷ C₁₆×C₁₀₁ ▷ C₈×C₁₀₁ ▷ C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
2. #20265007	C₂	semidirect
3. M₆₄×C₁₀₁	C₂	semidirect
33. C₁₀₁	#1419	semidirect
35. M₆₄	D₁₀₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₁₆₁₆	2
#20265007	2
M₆₄×C₁₀₁	2
#1419	101