Subgroups — #41011473

Order 27008.

62 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	C₄×C₂₁₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	#364×C₂₁₁	C₂	1	1	yes	maximal, characteristic
4	#20269917	—	2	1	no	—
5	C₂×C₂₁₁⋊C₁₆	C₄	1	1	yes	characteristic
6	C₂×C₂₁₁⋊C₁₆	C₄	1	1	yes	characteristic
7	M₃₂×C₂₁₁	—	2	1	no	—
8	C₄×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
9	C₂₁₁⋊C₁₆	—	2	1	no	—
10	C₂₁₁⋊C₁₆	—	4	1	no	—
11	C₂₁₁⋊C₁₆	C₈	1	2	yes	—
12	C₂×C₈×C₂₁₁	C₂×C₄	1	1	yes	characteristic
13	C₂×C₈×C₂₁₁	D₄	1	1	yes	characteristic
14	C₄²×C₂₁₁	Q₈	1	1	yes	characteristic
15	C₁₆×C₂₁₁	—	4	1	no	—
16	C₂×C₄×C₂₁₁	—	2	1	no	—
17	C₂×C₄×C₂₁₁	C₄⋊C₄	1	1	yes	characteristic
18	C₈×C₂₁₁	C₂×C₈	1	1	yes	characteristic
19	C₈×C₂₁₁	M₁₆	1	1	yes	characteristic
20	C₈×C₂₁₁	D₈	1	1	yes	characteristic
21	C₈×C₂₁₁	Q₁₆	1	1	yes	characteristic
22	C₂²×C₂₁₁	#108	1	1	yes	characteristic
23	C₄×C₂₁₁	—	2	2	no	—
24	C₄×C₂₁₁	C₄⋊C₈	1	1	yes	characteristic
25	C₄×C₂₁₁	#107	1	1	yes	characteristic
26	C₂×C₂₁₁	—	2	1	no	—
27	C₂×C₂₁₁	#335	1	1	yes	characteristic
28	C₂₁₁	#1373	1	1	yes	characteristic, complemented, Sylow
29	#1373	—	211	1	no	maximal, Sylow
30	#364	—	211	1	no	—
31	#364	D₂₁₁	1	1	yes	characteristic
32	C₄×C₁₆	—	211	1	no	—
33	M₃₂	—	2	1	no	—
34	M₃₂	—	422	1	no	—
35	C₂×C₁₆	—	211	1	no	—
36	C₂×C₁₆	—	211	1	no	—
37	C₄×C₈	C₂×D₂₁₁	1	1	yes	characteristic
38	C₂×C₈	C₄×D₂₁₁	1	1	yes	characteristic
39	C₂×C₈	C₂²⋊D₂₁₁	1	1	yes	characteristic
40	C₄²	Dic₄₂₂	1	1	yes	characteristic
41	C₁₆	—	4	1	no	—
42	C₁₆	—	844	1	no	—
43	C₁₆	—	422	1	no	—
44	C₁₆	—	211	2	no	—
45	C₂×C₄	—	2	1	no	—
46	C₂×C₄	Dic₂₁₁⋊C₄	1	1	yes	characteristic
47	C₈	C₈×D₂₁₁	1	1	yes	characteristic
48	C₈	#16809945	1	1	yes	characteristic
49	C₈	#16809947	1	1	yes	characteristic
50	C₈	#16809952	1	1	yes	characteristic
51	C₂²	#20269912	1	1	yes	characteristic
52	C₄	—	2	2	no	—
53	C₄	Dic₂₁₁⋊C₈	1	1	yes	characteristic
54	C₄	#20269910	1	1	yes	characteristic
55	C₂	—	2	1	no	—
56	C₂	#27212898	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#16809947
commutator	C₈×C₂₁₁	C₂×C₈
Frattini	C₄×C₈	C₂×D₂₁₁
Fitting	#364×C₂₁₁	C₂
socle	C₂×C₂₁₁	#335
prime radical	C₂²×C₂₁₁	#108
cosocle	C₄×C₈×C₂₁₁	C₂²
hypercenter	#364	D₂₁₁
hypocenter	C₂₁₁	#1373
LCS	G ▷ C₈×C₂₁₁ ▷ C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #364

Complements

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

N	H ≅ G/N	type
28. C₂₁₁	#1373	semidirect

Maximal subgroups

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

subgroup	index
#27212967	2
C₄×C₂₁₁⋊C₁₆	2
#364×C₂₁₁	2
#1373	211