Subgroups — #27169320

Order 12928.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20265609	C₂	1	1	yes	maximal, characteristic
2	#20265005	C₂	1	1	yes	maximal, characteristic, complemented
3	M₆₄×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#15692277	—	2	1	no	—
5	Dic₈₀₈	C₄	1	1	yes	characteristic, complemented
6	#15692200	—	2	1	no	—
7	D₁₆₁₆	C₄	1	1	yes	characteristic, complemented
8	#15692189	—	2	1	no	—
9	C₂×C₁₆×C₁₀₁	C₂²	1	1	yes	characteristic
10	C₃₂×C₁₀₁	—	2	1	no	—
11	#13108074	—	4	1	no	—
12	Dic₄₀₄	—	2	1	no	—
13	#13108063	—	4	1	no	—
14	D₈₀₈	—	2	1	no	—
15	C₄⋊Dic₁₀₁	—	4	1	no	—
16	C₂×C₈×C₁₀₁	D₄	1	1	yes	characteristic
17	C₁₆×C₁₀₁	C₂×C₄	1	1	yes	characteristic
18	C₁₆×C₁₀₁	D₄	1	1	yes	characteristic
19	C₂×C₄×C₁₀₁	D₈	1	1	yes	characteristic
20	C₂²⋊D₁₀₁	—	8	1	no	—
21	C₂×Dic₁₀₁	—	8	1	no	—
22	D₄₀₄	—	4	1	no	—
23	C₄×D₁₀₁	—	8	1	no	—
24	Dic₂₀₂	—	4	1	no	—
25	C₈×C₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
26	C₈×C₁₀₁	SD₁₆	1	1	yes	characteristic
27	C₂²×C₁₀₁	D₁₆	1	1	yes	characteristic
28	C₂×D₁₀₁	—	8	1	no	—
29	C₄×C₁₀₁	#102	1	1	yes	characteristic
30	C₄×C₁₀₁	SD₃₂	1	1	yes	characteristic
31	Dic₁₀₁	—	8	1	no	—
32	Dic₁₀₁	—	16	1	no	—
33	C₂×C₁₀₁	—	2	1	no	—
34	C₂×C₁₀₁	#357	1	1	yes	characteristic
35	D₁₀₁	—	16	1	no	—
36	#1418	—	101	1	no	maximal, Sylow
37	C₁₀₁	#1418	1	1	yes	characteristic, complemented, Sylow
38	#508	—	101	1	no	—
39	M₆₄	D₁₀₁	1	1	yes	characteristic, complemented
40	#367	—	101	1	no	—
41	#135	—	202	1	no	—
42	Q₃₂	—	101	1	no	—
43	SD₃₂	—	202	1	no	—
44	D₁₆	—	101	1	no	—
45	C₂×C₁₆	C₂×D₁₀₁	1	1	yes	characteristic
46	#107	—	202	1	no	—
47	C₃₂	—	2	1	no	—
48	#41	—	404	1	no	—
49	Q₁₆	—	202	1	no	—
50	SD₁₆	—	404	1	no	—
51	D₈	—	202	1	no	—
52	C₂×C₈	D₄₀₄	1	1	yes	characteristic
53	C₄⋊C₄	—	404	1	no	—
54	C₁₆	C₄×D₁₀₁	1	1	yes	characteristic
55	C₁₆	C₂²⋊D₁₀₁	1	1	yes	characteristic
56	Q₈	—	404	1	no	—
57	D₄	—	808	1	no	—
58	D₄	—	404	1	no	—
59	C₂×C₄	—	808	1	no	—
60	C₂×C₄	—	808	1	no	—
61	C₂×C₄	D₈₀₈	1	1	yes	characteristic
62	C₈	#13108051	1	1	yes	characteristic
63	C₈	#13108063	1	1	yes	characteristic
64	C₂²	—	808	1	no	—
65	C₂²	D₁₆₁₆	1	1	yes	characteristic
66	C₄	—	808	1	no	—
67	C₄	—	1616	1	no	—
68	C₄	#15692177	1	1	yes	characteristic
69	C₄	#15692200	1	1	yes	characteristic
70	C₂	—	2	1	no	—
71	C₂	—	1616	1	no	—
72	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	#20265609	C₂
cosocle	C₂×C₁₆×C₁₀₁	C₂²
hypercenter	M₆₄	D₁₀₁
hypocenter	C₁₀₁	#1418
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. #20265005	C₂	semidirect
3. M₆₄×C₁₀₁	C₂	semidirect
5. Dic₈₀₈	C₄	semidirect
7. D₁₆₁₆	C₄	semidirect
37. C₁₀₁	#1418	semidirect
39. M₆₄	D₁₀₁	semidirect

Maximal subgroups

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

subgroup	index
#20265609	2
#20265005	2
M₆₄×C₁₀₁	2
#1418	101