Subgroups — #27212911

Order 13504.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16809946	C₂	1	2	yes	maximal
2	C₂²×C₈×C₂₁₁	C₂	1	1	yes	maximal, characteristic
3	#16809946	—	2	2	no	—
4	#16809946	C₄	1	4	yes	—
5	C₂×C₂₁₁⋊C₈	—	2	2	no	—
6	C₂²×C₄×C₂₁₁	C₂²	1	1	yes	characteristic
7	C₂×C₈×C₂₁₁	—	2	2	no	—
8	C₂×C₈×C₂₁₁	C₄	1	2	yes	—
9	C₂₁₁⋊C₈	—	2	4	no	—
10	C₂³×C₂₁₁	Q₈	1	1	yes	characteristic
11	C₂×C₄×C₂₁₁	C₂×C₄	1	2	yes	—
12	C₂×C₄×C₂₁₁	C₂×C₄	1	1	yes	characteristic
13	C₂×C₄×C₂₁₁	D₄	1	2	yes	—
14	C₂×C₄×C₂₁₁	D₄	1	1	yes	characteristic
15	C₈×C₂₁₁	—	2	2	no	—
16	C₂²×C₂₁₁	—	2	2	no	—
17	C₂²×C₂₁₁	C₄⋊C₄	1	2	yes	—
18	C₂²×C₂₁₁	C₄⋊C₄	1	1	yes	characteristic
19	C₄×C₂₁₁	C₄²	1	1	yes	characteristic
20	C₄×C₂₁₁	C₂²⋊C₄	1	2	yes	—
21	C₄×C₂₁₁	C₂²⋊C₄	1	1	yes	characteristic
22	C₂×C₂₁₁	—	2	2	no	—
23	C₂×C₂₁₁	#95	1	1	yes	characteristic
24	C₂×C₂₁₁	#108	1	2	yes	—
25	C₂₁₁	#341	1	1	yes	characteristic, complemented, Sylow
26	#341	—	211	1	no	maximal, Sylow
27	C₂×M₁₆	—	211	2	no	—
28	C₂²×C₈	D₂₁₁	1	1	yes	characteristic
29	C₂²×C₄	C₂×D₂₁₁	1	1	yes	characteristic
30	M₁₆	—	422	2	no	—
31	M₁₆	—	211	4	no	—
32	C₂×C₈	—	2	2	no	—
33	C₂×C₈	—	422	2	no	—
34	C₂×C₈	Dic₂₁₁	1	2	yes	—
35	C₂³	Dic₄₂₂	1	1	yes	characteristic
36	C₂×C₄	C₄×D₂₁₁	1	2	yes	—
37	C₂×C₄	C₂×Dic₂₁₁	1	1	yes	characteristic
38	C₂×C₄	C₂²⋊D₂₁₁	1	2	yes	—
39	C₂×C₄	D₈₄₄	1	1	yes	characteristic
40	C₈	—	2	2	no	—
41	C₈	—	422	4	no	—
42	C₂²	—	2	2	no	—
43	C₂²	Dic₂₁₁⋊C₄	1	2	yes	—
44	C₂²	C₄⋊Dic₂₁₁	1	1	yes	characteristic
45	C₄	C₄×Dic₂₁₁	1	1	yes	characteristic
46	C₄	(C₂×D₂₁₁)⋊C₄	1	2	yes	—
47	C₄	C₂²⋊Dic₂₁₁	1	1	yes	characteristic
48	C₂	—	2	2	no	—
49	C₂	C₂₁₁⋊#95	1	1	yes	characteristic
50	C₂	#20269913	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₈₄₄
commutator	C₄×C₂₁₁	C₄²
Frattini	C₂²×C₄	C₂×D₂₁₁
Fitting	C₂²×C₈×C₂₁₁	C₂
socle	C₂²×C₂₁₁	C₄⋊C₄
prime radical	C₂³×C₂₁₁	Q₈
cosocle	C₂²×C₄×C₂₁₁	C₂²
hypercenter	C₂²×C₈	D₂₁₁
hypocenter	C₂₁₁	#341
LCS	G ▷ C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
25. C₂₁₁	#341	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16809946	2
C₂²×C₈×C₂₁₁	2
#341	211