Subgroups — #27169322

Order 12928.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20265011	C₂	1	1	yes	maximal, characteristic, complemented
2	#20265007	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×D₁₆×C₁₀₁	C₂	1	1	yes	maximal, characteristic
4	#15692191	—	2	1	no	—
5	#15692161	—	2	1	no	—
6	C₂×D₈×C₁₀₁	—	2	1	no	—
7	D₁₆×C₁₀₁	—	2	1	no	—
8	D₁₆×C₁₀₁	C₄	1	2	yes	—
9	C₂×C₁₆×C₁₀₁	C₂²	1	1	yes	characteristic
10	#13108058	—	4	1	no	—
11	C₂×D₄×C₁₀₁	—	4	1	no	—
12	D₈×C₁₀₁	—	2	2	no	—
13	D₈×C₁₀₁	—	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	C₂³×C₁₀₁	—	8	1	no	—
18	D₄×C₁₀₁	—	4	2	no	—
19	D₄×C₁₀₁	—	8	1	no	—
20	C₂×C₄×C₁₀₁	D₈	1	1	yes	characteristic
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	—	8	2	no	—
25	C₂²×C₁₀₁	—	16	2	no	—
26	C₂²×C₁₀₁	SD₃₂	1	1	yes	characteristic
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	—	16	2	no	—
31	C₂×C₁₀₁	#357	1	1	yes	characteristic
32	#1419	—	101	1	no	maximal, Sylow
33	C₁₀₁	#1419	1	1	yes	characteristic, complemented, Sylow
34	C₂×D₁₆	D₁₀₁	1	1	yes	characteristic
35	M₆₄	—	101	1	no	—
36	#368	—	101	1	no	—
37	C₂×D₈	—	2	1	no	—
38	D₁₆	—	2	1	no	—
39	D₁₆	Dic₁₀₁	1	2	yes	—
40	C₂×C₁₆	C₂×D₁₀₁	1	1	yes	characteristic
41	#108	—	202	1	no	—
42	C₃₂	—	202	1	no	—
43	C₂×D₄	—	4	1	no	—
44	D₈	—	2	2	no	—
45	D₈	—	4	1	no	—
46	M₁₆	—	404	1	no	—
47	C₂×C₈	C₂²⋊D₁₀₁	1	1	yes	characteristic
48	C₁₆	C₂×Dic₁₀₁	1	1	yes	characteristic
49	C₁₆	C₂²⋊D₁₀₁	1	1	yes	characteristic
50	C₂³	—	8	1	no	—
51	D₄	—	4	2	no	—
52	D₄	—	8	1	no	—
53	C₂×C₄	#13108059	1	1	yes	characteristic
54	C₈	—	808	1	no	—
55	C₈	C₂²⋊Dic₁₀₁	1	1	yes	characteristic
56	C₈	#13108062	1	1	yes	characteristic
57	C₂²	—	8	2	no	—
58	C₂²	—	16	2	no	—
59	C₂²	#15692199	1	1	yes	characteristic
60	C₄	#15692176	1	1	yes	characteristic
61	C₄	#15692196	1	1	yes	characteristic
62	C₂	—	2	1	no	—
63	C₂	—	16	2	no	—
64	C₂	#20264978	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20264978
commutator	C₁₆×C₁₀₁	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₁₀₁
Fitting, prime radical	C₂×D₁₆×C₁₀₁	C₂
socle	C₂×C₁₀₁	#357
cosocle	C₂×C₁₆×C₁₀₁	C₂²
hypercenter	C₂×D₁₆	D₁₀₁
hypocenter	C₁₀₁	#1419
LCS	G ▷ C₁₆×C₁₀₁ ▷ C₈×C₁₀₁ ▷ C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×D₁₆

Complements

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

N	H ≅ G/N	type
1. #20265011	C₂	semidirect
2. #20265007	C₂	semidirect
33. C₁₀₁	#1419	semidirect

Maximal subgroups

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

subgroup	index
#20265011	2
#20265007	2
C₂×D₁₆×C₁₀₁	2
#1419	101