Subgroups — #41011599

Order 27008.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₁₁⋊C₃₂	C₂	1	1	yes	maximal, characteristic
2	#27212972	C₂	1	1	yes	maximal, characteristic
3	#366×C₂₁₁	C₂	1	1	yes	maximal, characteristic
4	#20269911	—	2	1	no	—
5	C₂₁₁⋊C₃₂	C₄	1	2	yes	complemented
6	C₂×C₁₆×C₂₁₁	C₂²	1	1	yes	characteristic
7	#107×C₂₁₁	—	2	1	no	—
8	C₄⋊Dic₂₁₁	—	4	1	no	—
9	C₂×C₈×C₂₁₁	D₄	1	1	yes	characteristic
10	C₄⋊C₄×C₂₁₁	—	4	1	no	—
11	C₁₆×C₂₁₁	C₂×C₄	1	1	yes	characteristic
12	C₁₆×C₂₁₁	Q₈	1	1	yes	characteristic
13	C₂×Dic₂₁₁	—	8	1	no	—
14	C₂×C₄×C₂₁₁	—	8	1	no	—
15	C₂×C₄×C₂₁₁	D₈	1	1	yes	characteristic
16	C₈×C₂₁₁	C₄⋊C₄	1	1	yes	characteristic
17	C₈×C₂₁₁	Q₁₆	1	1	yes	characteristic
18	C₂²×C₂₁₁	D₁₆	1	1	yes	characteristic
19	C₄×C₂₁₁	—	16	1	no	—
20	C₄×C₂₁₁	#107	1	1	yes	characteristic
21	C₄×C₂₁₁	Q₃₂	1	1	yes	characteristic
22	Dic₂₁₁	—	16	1	no	—
23	C₂×C₂₁₁	#366	1	1	yes	characteristic
24	C₂×C₂₁₁	SD₆₄	1	1	yes	characteristic
25	C₂×C₂₁₁	SD₆₄	1	1	yes	characteristic
26	C₂₁₁	#1424	1	1	yes	characteristic, complemented, Sylow
27	#1424	—	211	1	no	maximal, Sylow
28	C₂×C₃₂	—	211	1	no	—
29	#366	—	211	1	no	—
30	#366	D₂₁₁	1	1	yes	characteristic
31	C₂×C₁₆	C₂×D₂₁₁	1	1	yes	characteristic
32	#107	—	2	1	no	—
33	#107	—	422	1	no	—
34	C₃₂	—	211	2	no	—
35	C₂×C₈	C₂²⋊D₂₁₁	1	1	yes	characteristic
36	C₄⋊C₄	—	4	1	no	—
37	C₄⋊C₄	—	844	1	no	—
38	C₁₆	C₄×D₂₁₁	1	1	yes	characteristic
39	C₁₆	Dic₄₂₂	1	1	yes	characteristic
40	C₂×C₄	—	8	1	no	—
41	C₂×C₄	—	1688	1	no	—
42	C₂×C₄	#16809947	1	1	yes	characteristic
43	C₈	Dic₂₁₁⋊C₄	1	1	yes	characteristic
44	C₈	#16809952	1	1	yes	characteristic
45	C₂²	#20269918	1	1	yes	characteristic
46	C₄	—	16	1	no	—
47	C₄	—	3376	1	no	—
48	C₄	#20269910	1	1	yes	characteristic
49	C₄	#20269923	1	1	yes	characteristic
50	C₂	#27212971	1	1	yes	characteristic
51	C₂	#27212983	1	1	yes	characteristic
52	C₂	#27212984	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#20269918
commutator	C₁₆×C₂₁₁	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₂₁₁
Fitting	#366×C₂₁₁	C₂
socle, prime radical	C₂²×C₂₁₁	D₁₆
cosocle	C₂×C₁₆×C₂₁₁	C₂²
hypercenter	#366	D₂₁₁
hypocenter	C₂₁₁	#1424
LCS	G ▷ C₁₆×C₂₁₁ ▷ C₈×C₂₁₁ ▷ C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #366

Complements

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

N	H ≅ G/N	type
5. C₂₁₁⋊C₃₂	C₄	semidirect
26. C₂₁₁	#1424	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₁₁⋊C₃₂	2
#27212972	2
#366×C₂₁₁	2
#1424	211