Subgroups — #41011459

Order 27008.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27212967	C₂	1	1	yes	maximal, characteristic
2	#27212967	C₂	1	1	yes	maximal, characteristic
3	#499×C₂₁₁	C₂	1	1	yes	maximal, characteristic
4	#20269917	—	2	1	no	—
5	#20269917	—	2	1	no	—
6	#125×C₂₁₁	C₄	1	2	yes	—
7	#107×C₂₁₁	—	2	1	no	—
8	#106×C₂₁₁	—	2	1	no	—
9	C₄×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
10	C₂₁₁⋊C₁₆	—	4	1	no	—
11	C₂₁₁⋊C₁₆	—	4	1	no	—
12	C₂×C₈×C₂₁₁	D₄	1	1	yes	characteristic
13	C₂×C₈×C₂₁₁	D₄	1	1	yes	characteristic
14	C₄⋊C₄×C₂₁₁	—	2	2	no	—
15	C₄⋊C₄×C₂₁₁	—	4	2	no	—
16	C₄²×C₂₁₁	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×C₂₁₁	—	2	1	no	—
18	C₂×C₄×C₂₁₁	—	4	2	no	—
19	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	1	yes	characteristic
20	C₈×C₂₁₁	SD₁₆	1	1	yes	characteristic
21	C₈×C₂₁₁	SD₁₆	1	1	yes	characteristic
22	C₈×C₂₁₁	Q₁₆	1	1	yes	characteristic
23	C₈×C₂₁₁	Q₁₆	1	1	yes	characteristic
24	C₂²×C₂₁₁	#100	1	1	yes	characteristic
25	C₄×C₂₁₁	—	4	1	no	—
26	C₄×C₂₁₁	—	8	2	no	—
27	C₄×C₂₁₁	#103	1	1	yes	characteristic
28	C₄×C₂₁₁	#103	1	1	yes	characteristic
29	C₂×C₂₁₁	—	2	1	no	—
30	C₂×C₂₁₁	#333	1	1	yes	characteristic
31	C₂₁₁	#1366	1	1	yes	characteristic, complemented, Sylow
32	#1366	—	211	1	no	maximal, Sylow
33	#499	D₂₁₁	1	1	yes	characteristic
34	#364	—	211	1	no	—
35	#364	—	211	1	no	—
36	#125	Dic₂₁₁	1	2	yes	—
37	M₃₂	—	422	1	no	—
38	M₃₂	—	422	1	no	—
39	#107	—	2	1	no	—
40	#106	—	2	1	no	—
41	C₄×C₈	C₂×D₂₁₁	1	1	yes	characteristic
42	C₂×C₈	C₂²⋊D₂₁₁	1	1	yes	characteristic
43	C₂×C₈	C₂²⋊D₂₁₁	1	1	yes	characteristic
44	C₄⋊C₄	—	2	2	no	—
45	C₄⋊C₄	—	4	2	no	—
46	C₄²	C₂×Dic₂₁₁	1	1	yes	characteristic
47	C₁₆	—	844	1	no	—
48	C₁₆	—	844	1	no	—
49	C₂×C₄	—	2	1	no	—
50	C₂×C₄	—	4	2	no	—
51	C₂×C₄	C₂²⋊Dic₂₁₁	1	1	yes	characteristic
52	C₈	#16809949	1	1	yes	characteristic
53	C₈	#16809949	1	1	yes	characteristic
54	C₈	#16809952	1	1	yes	characteristic
55	C₈	#16809952	1	1	yes	characteristic
56	C₂²	#20269894	1	1	yes	characteristic
57	C₄	—	4	1	no	—
58	C₄	—	8	2	no	—
59	C₄	#20269901	1	1	yes	characteristic
60	C₄	#20269901	1	1	yes	characteristic
61	C₂	—	2	1	no	—
62	C₂	#27212895	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#27212895
commutator	C₄²×C₂₁₁	C₂×C₄
Frattini	C₄×C₈	C₂×D₂₁₁
Fitting	#499×C₂₁₁	C₂
socle	C₂×C₂₁₁	#333
prime radical	C₂²×C₂₁₁	#100
cosocle	C₄×C₈×C₂₁₁	C₂²
hypercenter	#499	D₂₁₁
hypocenter	C₂₁₁	#1366
LCS	G ▷ C₄²×C₂₁₁ ▷ C₂×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #499

Complements

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

N	H ≅ G/N	type
31. C₂₁₁	#1366	semidirect

Maximal subgroups

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

subgroup	index
#27212967	2
#499×C₂₁₁	2
#1366	211