Subgroups — #13046292

Order 1296.

50 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₈₁	C₂	1	1	yes	maximal, characteristic
2	C₈₁⋊C₈	C₂	1	2	yes	maximal, complemented
3	#88363	—	3	1	no	maximal
4	C₂²×C₈₁	C₄	1	1	yes	characteristic
5	C₄×C₈₁	C₄	1	1	yes	characteristic
6	C₄×C₈₁	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₂₇	D₃	1	1	yes	characteristic
8	C₂₇⋊C₈	—	3	2	no	—
9	C₂×C₈₁	—	2	1	no	—
10	C₂×C₈₁	C₂×C₄	1	1	yes	characteristic
11	#3667	—	9	1	no	—
12	C₂²×C₂₇	Dic₃	1	1	yes	characteristic
13	C₄×C₂₇	Dic₃	1	1	yes	characteristic
14	C₄×C₂₇	C₂×D₃	1	1	yes	characteristic
15	C₈₁	M₁₆	1	1	yes	characteristic, complemented, Sylow
16	C₂×C₄×C₉	D₉	1	1	yes	characteristic
17	C₉⋊C₈	—	9	2	no	—
18	C₂×C₂₇	—	2	1	no	—
19	C₂×C₂₇	C₂×Dic₃	1	1	yes	characteristic
20	#208	—	27	1	no	—
21	C₂²×C₉	Dic₉	1	1	yes	characteristic
22	C₄×C₉	Dic₉	1	1	yes	characteristic
23	C₄×C₉	C₂×D₉	1	1	yes	characteristic
24	C₂₇	#208	1	1	yes	characteristic
25	C₂×C₃×C₄	D₂₇	1	1	yes	characteristic
26	C₃⋊C₈	—	27	2	no	—
27	C₂×C₉	—	2	1	no	—
28	C₂×C₉	C₂×Dic₉	1	1	yes	characteristic
29	M₁₆	—	81	1	no	Sylow
30	C₂²×C₃	Dic₂₇	1	1	yes	characteristic
31	C₃×C₄	Dic₂₇	1	1	yes	characteristic
32	C₃×C₄	C₂×D₂₇	1	1	yes	characteristic
33	C₉	#3667	1	1	yes	characteristic
34	C₂×C₄	D₈₁	1	1	yes	characteristic
35	C₈	—	81	2	no	—
36	C₂×C₃	—	2	1	no	—
37	C₂×C₃	C₂×Dic₂₇	1	1	yes	characteristic
38	C₂²	Dic₈₁	1	1	yes	characteristic
39	C₄	Dic₈₁	1	1	yes	characteristic
40	C₄	C₂×D₈₁	1	1	yes	characteristic
41	C₃	#88363	1	1	yes	characteristic
42	C₂	—	2	1	no	—
43	C₂	C₂×Dic₈₁	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₈₁
commutator	C₂×C₈₁	C₂×C₄
Frattini	C₄×C₂₇	C₂×D₃
Fitting	C₂×C₄×C₈₁	C₂
socle	C₂×C₃	C₂×Dic₂₇
prime radical	C₂²×C₃	Dic₂₇
cosocle	C₄×C₈₁	C₂²
hypercenter	C₂×C₄	D₈₁
hypocenter	C₈₁	M₁₆
LCS	G ▷ C₂×C₈₁ ▷ C₈₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
2. C₈₁⋊C₈	C₂	semidirect
15. C₈₁	M₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₈₁	2
C₈₁⋊C₈	2
#88363	3