Subgroups — #41011601

Order 27008.

50 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	#27212976	C₂	1	1	yes	maximal, characteristic
3	#368×C₂₁₁	C₂	1	1	yes	maximal, characteristic
4	#20269913	—	2	1	no	—
5	C₂₁₁⋊C₃₂	C₄	1	2	yes	—
6	C₂×C₁₆×C₂₁₁	C₂²	1	1	yes	characteristic
7	#108×C₂₁₁	—	2	1	no	—
8	#16809946	—	4	1	no	—
9	M₁₆×C₂₁₁	—	4	1	no	—
10	C₂×C₈×C₂₁₁	D₄	1	1	yes	characteristic
11	C₁₆×C₂₁₁	C₂×C₄	1	1	yes	characteristic
12	C₁₆×C₂₁₁	Q₈	1	1	yes	characteristic
13	C₂₁₁⋊C₈	—	8	1	no	—
14	C₂×C₄×C₂₁₁	D₈	1	1	yes	characteristic
15	C₈×C₂₁₁	—	8	1	no	—
16	C₈×C₂₁₁	C₄⋊C₄	1	1	yes	characteristic
17	C₈×C₂₁₁	Q₁₆	1	1	yes	characteristic
18	C₂²×C₂₁₁	Q₃₂	1	1	yes	characteristic
19	C₄×C₂₁₁	#107	1	1	yes	characteristic
20	C₄×C₂₁₁	D₁₆	1	1	yes	characteristic
21	C₂×C₂₁₁	—	2	1	no	—
22	C₂×C₂₁₁	#366	1	1	yes	characteristic
23	C₂₁₁	#1425	1	1	yes	characteristic, complemented, Sylow
24	#1425	—	211	1	no	maximal, Sylow
25	C₂×C₃₂	—	211	1	no	—
26	#368	—	211	1	no	—
27	#368	D₂₁₁	1	1	yes	characteristic
28	C₂×C₁₆	C₂×D₂₁₁	1	1	yes	characteristic
29	#108	—	2	1	no	—
30	#108	—	422	1	no	—
31	C₃₂	—	211	2	no	—
32	M₁₆	—	4	1	no	—
33	M₁₆	—	844	1	no	—
34	C₂×C₈	C₂²⋊D₂₁₁	1	1	yes	characteristic
35	C₁₆	C₄×D₂₁₁	1	1	yes	characteristic
36	C₁₆	Dic₄₂₂	1	1	yes	characteristic
37	C₂×C₄	#16809947	1	1	yes	characteristic
38	C₈	—	8	1	no	—
39	C₈	—	1688	1	no	—
40	C₈	Dic₂₁₁⋊C₄	1	1	yes	characteristic
41	C₈	#16809952	1	1	yes	characteristic
42	C₂²	#20269923	1	1	yes	characteristic
43	C₄	#20269910	1	1	yes	characteristic
44	C₄	#20269918	1	1	yes	characteristic
45	C₂	—	2	1	no	—
46	C₂	#27212971	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#20269918
commutator	C₁₆×C₂₁₁	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₂₁₁
Fitting	#368×C₂₁₁	C₂
socle	C₂×C₂₁₁	#366
prime radical	C₂²×C₂₁₁	Q₃₂
cosocle	C₂×C₁₆×C₂₁₁	C₂²
hypercenter	#368	D₂₁₁
hypocenter	C₂₁₁	#1425
LCS	G ▷ C₁₆×C₂₁₁ ▷ C₈×C₂₁₁ ▷ C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #368

Complements

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

N	H ≅ G/N	type
23. C₂₁₁	#1425	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
#27212976	2
#368×C₂₁₁	2
#1425	211