Subgroups — #27169275

Order 12928.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20265593	C₂	1	1	yes	maximal, characteristic
2	C₂×#15692161	C₂	1	1	yes	maximal, characteristic, complemented
3	M₆₄×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#15692261	C₄	1	1	yes	characteristic
5	#15692195	C₄	1	1	yes	characteristic
6	#15692194	—	2	1	no	—
7	C₁₆×D₁₀₁	—	2	1	no	—
8	#15692161	—	2	1	no	—
9	C₂×C₁₆×C₁₀₁	C₂²	1	1	yes	characteristic
10	C₃₂×C₁₀₁	—	2	1	no	—
11	#13108074	C₈	1	1	yes	characteristic
12	#13108058	C₈	1	1	yes	characteristic
13	#13108057	—	2	1	no	—
14	C₈×D₁₀₁	—	2	1	no	—
15	#13108049	—	2	1	no	—
16	C₂×C₈×C₁₀₁	C₂×C₄	1	1	yes	characteristic
17	C₁₆×C₁₀₁	D₄	1	1	yes	characteristic
18	C₁₆×C₁₀₁	D₄	1	1	yes	characteristic
19	C₂×C₄×C₁₀₁	C₂×C₈	1	1	yes	characteristic
20	C₂²⋊D₁₀₁	—	2	1	no	—
21	D₄₀₄	C₁₆	1	1	yes	characteristic
22	C₄×D₁₀₁	—	2	1	no	—
23	Dic₂₀₂	C₁₆	1	1	yes	characteristic
24	C₈×C₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
25	C₈×C₁₀₁	M₁₆	1	1	yes	characteristic
26	#11717579	—	2	1	no	—
27	C₂²×C₁₀₁	M₃₂	1	1	yes	characteristic
28	C₂×D₁₀₁	—	2	1	no	—
29	C₄×C₁₀₁	C₂²⋊C₈	1	1	yes	characteristic
30	C₄×C₁₀₁	C₂×C₁₆	1	1	yes	characteristic
31	Dic₁₀₁	—	2	1	no	—
32	C₂×C₁₀₁	—	2	1	no	—
33	C₂×C₁₀₁	C₂²⋊C₁₆	1	1	yes	characteristic
34	D₁₀₁	—	4	1	no	—
35	#1401	—	101	1	no	maximal, Sylow
36	C₁₀₁	#1401	1	1	yes	characteristic, complemented, Sylow
37	#504	—	101	1	no	—
38	M₆₄	D₁₀₁	1	1	yes	characteristic, complemented
39	C₂×C₃₂	—	101	1	no	—
40	#131	—	101	1	no	—
41	M₃₂	—	101	1	no	—
42	M₃₂	—	202	1	no	—
43	C₂×C₁₆	—	202	1	no	—
44	C₂×C₁₆	C₂×D₁₀₁	1	1	yes	characteristic
45	C₃₂	—	2	1	no	—
46	C₃₂	—	202	1	no	—
47	#41	—	101	1	no	—
48	M₁₆	—	101	1	no	—
49	M₁₆	—	202	1	no	—
50	C₂×C₈	—	202	1	no	—
51	C₂×C₈	C₄×D₁₀₁	1	1	yes	characteristic
52	C₁₆	—	202	1	no	—
53	C₁₆	D₄₀₄	1	1	yes	characteristic
54	C₁₆	C₂²⋊D₁₀₁	1	1	yes	characteristic
55	Q₈	—	101	1	no	—
56	D₄	—	101	1	no	—
57	D₄	—	202	1	no	—
58	C₂×C₄	—	202	1	no	—
59	C₂×C₄	C₈×D₁₀₁	1	1	yes	characteristic
60	C₈	—	202	1	no	—
61	C₈	#13108051	1	1	yes	characteristic
62	C₈	#13108057	1	1	yes	characteristic
63	C₂²	—	202	1	no	—
64	C₂²	#15692194	1	1	yes	characteristic
65	C₄	—	202	1	no	—
66	C₄	#15692167	1	1	yes	characteristic
67	C₄	C₁₆×D₁₀₁	1	1	yes	characteristic
68	C₂	—	2	1	no	—
69	C₂	—	404	1	no	—
70	C₂	#20264956	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	C₂²⋊D₁₀₁
commutator	C₄×C₁₀₁	C₂×C₁₆
Frattini	C₂×C₁₆	C₂×D₁₀₁
Fitting	M₆₄×C₁₀₁	C₂
socle	C₂×C₁₀₁	C₂²⋊C₁₆
prime radical	#13108074	C₈
cosocle	C₂×C₁₆×C₁₀₁	C₂²
hypercenter	M₆₄	D₁₀₁
hypocenter	C₁₀₁	#1401
LCS	G ▷ C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₁₆ ◁ C₂×C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
2. C₂×#15692161	C₂	semidirect
3. M₆₄×C₁₀₁	C₂	semidirect
36. C₁₀₁	#1401	semidirect
38. M₆₄	D₁₀₁	semidirect

Maximal subgroups

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

subgroup	index
#20265593	2
C₂×#15692161	2
M₆₄×C₁₀₁	2
#1401	101