Subgroups — #27212909

Order 13504.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊Dic₂₁₁	C₂	1	2	yes	maximal
2	C₂²×C₈×C₂₁₁	C₂	1	1	yes	maximal, characteristic
3	C₂²×Dic₂₁₁	—	2	2	no	—
4	C₄⋊Dic₂₁₁	—	2	2	no	—
5	C₄⋊Dic₂₁₁	C₄	1	4	yes	complemented
6	C₂²×C₄×C₂₁₁	C₂²	1	1	yes	characteristic
7	C₂×C₈×C₂₁₁	—	2	2	no	—
8	C₂×C₈×C₂₁₁	C₄	1	2	yes	complemented
9	C₂×Dic₂₁₁	—	2	4	no	—
10	C₂×Dic₂₁₁	—	4	2	no	—
11	C₂×Dic₂₁₁	—	4	2	no	—
12	C₂³×C₂₁₁	D₄	1	1	yes	characteristic
13	C₂×C₄×C₂₁₁	C₂×C₄	1	2	yes	—
14	C₂×C₄×C₂₁₁	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄×C₂₁₁	D₄	1	2	yes	—
16	C₂×C₄×C₂₁₁	Q₈	1	1	yes	characteristic
17	C₈×C₂₁₁	—	2	2	no	—
18	C₂²×C₂₁₁	C₂²⋊C₄	1	2	yes	—
19	C₂²×C₂₁₁	C₄⋊C₄	1	1	yes	characteristic
20	C₂²×C₂₁₁	D₈	1	1	yes	characteristic
21	C₂²×C₂₁₁	SD₁₆	1	2	yes	—
22	C₂²×C₂₁₁	Q₁₆	1	1	yes	characteristic
23	C₄×C₂₁₁	C₄²	1	1	yes	characteristic
24	C₄×C₂₁₁	C₂²⋊C₄	1	1	yes	characteristic
25	C₄×C₂₁₁	C₄⋊C₄	1	2	yes	—
26	Dic₂₁₁	—	4	4	no	—
27	C₂×C₂₁₁	#95	1	1	yes	characteristic
28	C₂×C₂₁₁	#102	1	2	yes	—
29	C₂×C₂₁₁	#103	1	2	yes	—
30	C₂×C₂₁₁	#106	1	1	yes	characteristic
31	C₂×C₂₁₁	#107	1	1	yes	characteristic
32	C₂₁₁	#340	1	1	yes	characteristic, complemented, Sylow
33	#340	—	211	1	no	maximal, Sylow
34	C₂²×C₈	D₂₁₁	1	1	yes	characteristic
35	C₂×C₄⋊C₄	—	211	2	no	—
36	C₂²×C₄	—	422	2	no	—
37	C₂²×C₄	C₂×D₂₁₁	1	1	yes	characteristic
38	C₂×C₈	—	2	2	no	—
39	C₂×C₈	Dic₂₁₁	1	2	yes	complemented
40	C₄⋊C₄	—	422	2	no	—
41	C₄⋊C₄	—	211	4	no	—
42	C₂³	D₈₄₄	1	1	yes	characteristic
43	C₂×C₄	—	844	2	no	—
44	C₂×C₄	—	844	2	no	—
45	C₂×C₄	—	422	4	no	—
46	C₂×C₄	C₄×D₂₁₁	1	2	yes	—
47	C₂×C₄	C₂×Dic₂₁₁	1	1	yes	characteristic
48	C₂×C₄	C₂²⋊D₂₁₁	1	2	yes	—
49	C₂×C₄	Dic₄₂₂	1	1	yes	characteristic
50	C₈	—	2	2	no	—
51	C₂²	(C₂×D₂₁₁)⋊C₄	1	2	yes	—
52	C₂²	C₄⋊Dic₂₁₁	1	1	yes	characteristic
53	C₂²	D₁₆₈₈	1	1	yes	characteristic
54	C₂²	#16809951	1	2	yes	—
55	C₂²	Dic₈₄₄	1	1	yes	characteristic
56	C₄	—	844	4	no	—
57	C₄	C₄×Dic₂₁₁	1	1	yes	characteristic
58	C₄	C₂²⋊Dic₂₁₁	1	1	yes	characteristic
59	C₄	Dic₂₁₁⋊C₄	1	2	yes	—
60	C₂	C₂₁₁⋊#95	1	1	yes	characteristic
61	C₂	#20269899	1	2	yes	—
62	C₂	#20269902	1	2	yes	—
63	C₂	#20269909	1	1	yes	characteristic
64	C₂	#20269911	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₈₄₄
commutator	C₄×C₂₁₁	C₄²
Frattini	C₂²×C₄	C₂×D₂₁₁
Fitting	C₂²×C₈×C₂₁₁	C₂
socle, prime radical	C₂³×C₂₁₁	D₄
cosocle	C₂²×C₄×C₂₁₁	C₂²
hypercenter	C₂²×C₈	D₂₁₁
hypocenter	C₂₁₁	#340
LCS	G ▷ C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
5. C₄⋊Dic₂₁₁	C₄	semidirect
8. C₂×C₈×C₂₁₁	C₄	semidirect
32. C₂₁₁	#340	semidirect
39. C₂×C₈	Dic₂₁₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊Dic₂₁₁	2
C₂²×C₈×C₂₁₁	2
#340	211