Subgroups — #27186288

Order 12928.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20266110	C₂	1	1	yes	maximal, characteristic
2	#20266106	C₂	1	1	yes	maximal, characteristic
3	Q₃₂×D₁₀₁	C₂	1	1	yes	maximal, characteristic
4	#15692326	—	2	1	no	—
5	#15692305	—	2	1	no	—
6	Q₁₆×D₁₀₁	—	2	1	no	—
7	Dic₈₀₈	C₄	1	1	yes	characteristic
8	#15692201	—	2	1	no	—
9	C₁₆×D₁₀₁	C₂²	1	1	yes	characteristic
10	Q₃₂×C₁₀₁	C₄	1	1	yes	characteristic
11	#13108084	—	4	1	no	—
12	Q₈×D₁₀₁	—	4	1	no	—
13	Dic₄₀₄	—	2	1	no	—
14	#13108064	—	4	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	C₄×D₁₀₁	—	8	1	no	—
21	C₄×D₁₀₁	D₈	1	1	yes	characteristic
22	Dic₂₀₂	—	4	1	no	—
23	Dic₂₀₂	—	8	1	no	—
24	#11717581	—	8	1	no	—
25	C₈×C₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
26	#11717579	SD₁₆	1	1	yes	characteristic
27	C₂×D₁₀₁	SD₃₂	1	1	yes	characteristic
28	C₄×C₁₀₁	—	8	1	no	—
29	C₄×C₁₀₁	#102	1	1	yes	characteristic
30	Dic₁₀₁	—	8	1	no	—
31	Dic₁₀₁	D₁₆	1	1	yes	characteristic
32	C₂×C₁₀₁	#357	1	1	yes	characteristic
33	D₁₀₁	—	2	1	no	—
34	#1420	—	101	1	no	maximal, Sylow
35	C₁₀₁	#1420	1	1	yes	characteristic, complemented, Sylow
36	C₂×Q₃₂	—	101	1	no	—
37	M₆₄	—	101	1	no	—
38	#368	—	101	1	no	—
39	C₂×Q₁₆	—	202	1	no	—
40	Q₃₂	—	101	1	no	—
41	Q₃₂	—	202	1	no	—
42	Q₃₂	#87938	1	1	yes	characteristic
43	C₂×C₁₆	—	101	1	no	—
44	#108	—	202	1	no	—
45	C₃₂	—	202	1	no	—
46	C₂×Q₈	—	404	1	no	—
47	Q₁₆	—	2	1	no	—
48	Q₁₆	—	404	1	no	—
49	Q₁₆	—	202	1	no	—
50	M₁₆	—	404	1	no	—
51	C₂×C₈	—	101	1	no	—
52	C₁₆	—	101	1	no	—
53	C₁₆	C₂×#87938	1	1	yes	characteristic
54	Q₈	—	4	1	no	—
55	Q₈	—	808	1	no	—
56	Q₈	—	404	1	no	—
57	C₂×C₄	—	808	1	no	—
58	C₂×C₄	—	101	1	no	—
59	C₈	—	808	1	no	—
60	C₈	—	101	1	no	—
61	C₈	C₂²⋊#87938	1	1	yes	characteristic
62	C₂²	—	101	1	no	—
63	C₄	—	8	1	no	—
64	C₄	—	808	1	no	—
65	C₄	—	101	1	no	—
66	C₄	#15692317	1	1	yes	characteristic
67	C₂	—	202	1	no	—
68	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	C₂×D₁₀₁	SD₃₂
cosocle	C₁₆×D₁₀₁	C₂²
hypercenter	Q₃₂	#87938
hypocenter	C₁₀₁	#1420
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
35. C₁₀₁	#1420	semidirect

Maximal subgroups

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

subgroup	index
#20266110	2
#20266106	2
Q₃₂×D₁₀₁	2
#1420	101