Subgroups — #27186282

Order 12928.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₁₀₁⋊C₃₂	C₂	1	1	yes	maximal, characteristic
2	#20266103	C₂	1	1	yes	maximal, characteristic
3	Q₃₂×D₁₀₁	C₂	1	1	yes	maximal, characteristic
4	#15692324	—	2	1	no	—
5	#15692305	—	2	1	no	—
6	Q₁₆×D₁₀₁	—	2	1	no	—
7	Dic₈₀₈	C₄	1	1	yes	characteristic, complemented
8	#15692201	—	2	1	no	—
9	C₁₆×D₁₀₁	C₂²	1	1	yes	characteristic
10	Q₃₂×C₁₀₁	C₄	1	1	yes	characteristic, complemented
11	C₄⋊#87938	—	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	C₂×#87938	—	8	1	no	—
20	Q₈×C₁₀₁	—	4	1	no	—
21	C₄×D₁₀₁	—	8	1	no	—
22	C₄×D₁₀₁	D₈	1	1	yes	characteristic
23	Dic₂₀₂	—	4	1	no	—
24	Dic₂₀₂	—	8	1	no	—
25	C₈×C₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
26	#11717579	SD₁₆	1	1	yes	characteristic
27	C₂×D₁₀₁	D₁₆	1	1	yes	characteristic
28	#87938	—	16	1	no	—
29	C₄×C₁₀₁	—	8	1	no	—
30	C₄×C₁₀₁	#102	1	1	yes	characteristic
31	Dic₁₀₁	—	8	1	no	—
32	Dic₁₀₁	SD₃₂	1	1	yes	characteristic
33	C₂×C₁₀₁	#357	1	1	yes	characteristic
34	D₁₀₁	SD₆₄	1	1	yes	characteristic
35	D₁₀₁	Q₆₄	1	1	yes	characteristic
36	#1416	—	101	1	no	maximal, Sylow
37	C₁₀₁	#1416	1	1	yes	characteristic, complemented, Sylow
38	C₂×Q₃₂	—	101	1	no	—
39	C₂×C₃₂	—	101	1	no	—
40	#366	—	101	1	no	—
41	C₂×Q₁₆	—	202	1	no	—
42	Q₃₂	—	101	1	no	—
43	Q₃₂	—	202	1	no	—
44	Q₃₂	#87938	1	1	yes	characteristic, complemented
45	C₂×C₁₆	—	101	1	no	—
46	#107	—	202	1	no	—
47	C₃₂	—	202	1	no	—
48	C₂×Q₈	—	404	1	no	—
49	Q₁₆	—	2	1	no	—
50	Q₁₆	—	404	1	no	—
51	Q₁₆	—	202	1	no	—
52	C₂×C₈	—	101	1	no	—
53	C₄⋊C₄	—	404	1	no	—
54	C₁₆	—	101	1	no	—
55	C₁₆	C₂×#87938	1	1	yes	characteristic
56	Q₈	—	4	1	no	—
57	Q₈	—	808	1	no	—
58	Q₈	—	404	1	no	—
59	C₂×C₄	—	808	1	no	—
60	C₂×C₄	—	808	1	no	—
61	C₂×C₄	—	101	1	no	—
62	C₈	—	101	1	no	—
63	C₈	C₂²⋊#87938	1	1	yes	characteristic
64	C₂²	—	101	1	no	—
65	C₄	—	8	1	no	—
66	C₄	—	808	1	no	—
67	C₄	—	1616	1	no	—
68	C₄	—	101	1	no	—
69	C₄	#15692317	1	1	yes	characteristic
70	C₂	—	101	1	no	—
71	C₂	—	101	1	no	—
72	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	D₁₆
cosocle	C₁₆×D₁₀₁	C₂²
hypercenter	Q₃₂	#87938
hypocenter	C₁₀₁	#1416
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
7. Dic₈₀₈	C₄	semidirect
10. Q₃₂×C₁₀₁	C₄	semidirect
37. C₁₀₁	#1416	semidirect
44. Q₃₂	#87938	semidirect

Maximal subgroups

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

subgroup	index
D₁₀₁⋊C₃₂	2
#20266103	2
Q₃₂×D₁₀₁	2
#1416	101