Subgroups — #41011600

Order 27008.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27212972	C₂	1	2	yes	maximal
2	C₂×C₃₂×C₂₁₁	C₂	1	1	yes	maximal, characteristic
3	#20269911	—	2	2	no	—
4	C₂×C₁₆×C₂₁₁	C₂²	1	1	yes	characteristic
5	C₃₂×C₂₁₁	C₄	1	2	yes	complemented
6	C₄⋊Dic₂₁₁	—	4	2	no	—
7	C₂×C₈×C₂₁₁	D₄	1	1	yes	characteristic
8	C₁₆×C₂₁₁	C₂×C₄	1	1	yes	characteristic
9	C₁₆×C₂₁₁	Q₈	1	1	yes	characteristic
10	C₂×Dic₂₁₁	—	8	2	no	—
11	C₂×C₄×C₂₁₁	D₈	1	1	yes	characteristic
12	C₈×C₂₁₁	C₄⋊C₄	1	1	yes	characteristic
13	C₈×C₂₁₁	Q₁₆	1	1	yes	characteristic
14	C₂²×C₂₁₁	D₁₆	1	1	yes	characteristic
15	C₄×C₂₁₁	#107	1	1	yes	characteristic
16	C₄×C₂₁₁	Q₃₂	1	1	yes	characteristic
17	Dic₂₁₁	—	16	2	no	—
18	C₂×C₂₁₁	#366	1	1	yes	characteristic
19	C₂×C₂₁₁	SD₆₄	1	2	yes	—
20	C₂₁₁	#1424	1	1	yes	characteristic, complemented, Sylow
21	#1424	—	211	1	no	maximal, Sylow
22	C₂×C₃₂	D₂₁₁	1	1	yes	characteristic
23	#366	—	211	2	no	—
24	C₂×C₁₆	C₂×D₂₁₁	1	1	yes	characteristic
25	#107	—	422	2	no	—
26	C₃₂	Dic₂₁₁	1	2	yes	complemented
27	C₂×C₈	D₈₄₄	1	1	yes	characteristic
28	C₄⋊C₄	—	844	2	no	—
29	C₁₆	C₂×Dic₂₁₁	1	1	yes	characteristic
30	C₁₆	Dic₄₂₂	1	1	yes	characteristic
31	C₂×C₄	—	1688	2	no	—
32	C₂×C₄	D₁₆₈₈	1	1	yes	characteristic
33	C₈	C₄⋊Dic₂₁₁	1	1	yes	characteristic
34	C₈	Dic₈₄₄	1	1	yes	characteristic
35	C₂²	D₃₃₇₆	1	1	yes	characteristic
36	C₄	—	3376	2	no	—
37	C₄	#20269911	1	1	yes	characteristic
38	C₄	Dic₁₆₈₈	1	1	yes	characteristic
39	C₂	#27212972	1	1	yes	characteristic
40	C₂	#27212985	1	2	yes	—

	subgroup	G/H
center	C₂²	D₃₃₇₆
commutator	C₁₆×C₂₁₁	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₂₁₁
Fitting	C₂×C₃₂×C₂₁₁	C₂
socle, prime radical	C₂²×C₂₁₁	D₁₆
cosocle	C₂×C₁₆×C₂₁₁	C₂²
hypercenter	C₂×C₃₂	D₂₁₁
hypocenter	C₂₁₁	#1424
LCS	G ▷ C₁₆×C₂₁₁ ▷ C₈×C₂₁₁ ▷ C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂

Complements

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

N	H ≅ G/N	type
5. C₃₂×C₂₁₁	C₄	semidirect
20. C₂₁₁	#1424	semidirect
26. C₃₂	Dic₂₁₁	semidirect

Maximal subgroups

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

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