Subgroups — #27169324

Order 12928.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₈₀₈	C₂	1	1	yes	maximal, characteristic
2	#20265011	C₂	1	1	yes	maximal, characteristic
3	#368×C₁₀₁	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₄₀₄	—	2	1	no	—
5	Dic₈₀₈	—	2	1	no	—
6	Dic₈₀₈	C₄	1	2	yes	—
7	#15692161	—	2	1	no	—
8	C₂×C₁₆×C₁₀₁	C₂²	1	1	yes	characteristic
9	#108×C₁₀₁	—	2	1	no	—
10	C₂×Dic₂₀₂	—	4	1	no	—
11	Dic₄₀₄	—	2	2	no	—
12	Dic₄₀₄	—	4	1	no	—
13	M₁₆×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	C₂×C₄×C₁₀₁	D₈	1	1	yes	characteristic
18	C₂×Dic₁₀₁	—	8	1	no	—
19	Dic₂₀₂	—	4	2	no	—
20	Dic₂₀₂	—	8	1	no	—
21	C₈×C₁₀₁	—	8	1	no	—
22	C₈×C₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
23	C₈×C₁₀₁	SD₁₆	1	1	yes	characteristic
24	C₂²×C₁₀₁	SD₃₂	1	1	yes	characteristic
25	C₄×C₁₀₁	#102	1	1	yes	characteristic
26	C₄×C₁₀₁	D₁₆	1	1	yes	characteristic
27	Dic₁₀₁	—	8	2	no	—
28	C₂×C₁₀₁	—	2	1	no	—
29	C₂×C₁₀₁	#357	1	1	yes	characteristic
30	#1420	—	101	1	no	maximal, Sylow
31	C₁₀₁	#1420	1	1	yes	characteristic, complemented, Sylow
32	C₂×Q₃₂	—	101	1	no	—
33	M₆₄	—	101	1	no	—
34	#368	D₁₀₁	1	1	yes	characteristic
35	C₂×Q₁₆	—	202	1	no	—
36	Q₃₂	—	101	2	no	—
37	Q₃₂	—	202	1	no	—
38	C₂×C₁₆	C₂×D₁₀₁	1	1	yes	characteristic
39	#108	—	2	1	no	—
40	C₃₂	—	202	1	no	—
41	C₂×Q₈	—	404	1	no	—
42	Q₁₆	—	404	1	no	—
43	Q₁₆	—	202	2	no	—
44	M₁₆	—	4	1	no	—
45	C₂×C₈	C₂²⋊D₁₀₁	1	1	yes	characteristic
46	C₁₆	C₄×D₁₀₁	1	1	yes	characteristic
47	C₁₆	D₄₀₄	1	1	yes	characteristic
48	Q₈	—	808	1	no	—
49	Q₈	—	404	2	no	—
50	C₂×C₄	—	808	1	no	—
51	C₂×C₄	#13108059	1	1	yes	characteristic
52	C₈	—	8	1	no	—
53	C₈	#13108051	1	1	yes	characteristic
54	C₈	#13108061	1	1	yes	characteristic
55	C₂²	#15692198	1	1	yes	characteristic
56	C₄	—	808	2	no	—
57	C₄	#15692175	1	1	yes	characteristic
58	C₄	#15692196	1	1	yes	characteristic
59	C₂	—	2	1	no	—
60	C₂	#20264977	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20264977
commutator	C₁₆×C₁₀₁	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₁₀₁
Fitting	#368×C₁₀₁	C₂
socle	C₂×C₁₀₁	#357
prime radical	C₂²×C₁₀₁	SD₃₂
cosocle	C₂×C₁₆×C₁₀₁	C₂²
hypercenter	#368	D₁₀₁
hypocenter	C₁₀₁	#1420
LCS	G ▷ C₁₆×C₁₀₁ ▷ C₈×C₁₀₁ ▷ C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #368

Complements

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

N	H ≅ G/N	type
31. C₁₀₁	#1420	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₈₀₈	2
#20265011	2
#368×C₁₀₁	2
#1420	101