Subgroups — #27212894

Order 13504.

64 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20269977	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(C₂₁₁⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₈×C₂₁₁	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₄₂₂	C₄	1	2	yes	—
5	C₂×C₂₁₁⋊C₈	—	2	1	no	—
6	C₄⋊Dic₂₁₁	—	2	1	no	—
7	C₄⋊Dic₂₁₁	—	2	1	no	—
8	C₂×C₈×C₂₁₁	—	2	1	no	—
9	C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
10	Dic₄₂₂	—	4	2	no	—
11	C₂×Dic₂₁₁	—	2	2	no	—
12	C₂₁₁⋊C₈	—	4	1	no	—
13	C₂×C₄×C₂₁₁	C₂×C₄	1	1	yes	characteristic
14	C₂×C₄×C₂₁₁	D₄	1	1	yes	characteristic
15	C₂×C₄×C₂₁₁	D₄	1	1	yes	characteristic
16	C₈×C₂₁₁	—	4	1	no	—
17	C₂²×C₂₁₁	C₂²⋊C₄	1	1	yes	characteristic
18	C₄×C₂₁₁	—	2	1	no	—
19	C₄×C₂₁₁	SD₁₆	1	1	yes	characteristic
20	C₄×C₂₁₁	SD₁₆	1	1	yes	characteristic
21	C₄×C₂₁₁	Q₁₆	1	1	yes	characteristic
22	C₄×C₂₁₁	Q₁₆	1	1	yes	characteristic
23	Dic₂₁₁	—	4	2	no	—
24	C₂×C₂₁₁	#100	1	1	yes	characteristic
25	C₂×C₂₁₁	#103	1	1	yes	characteristic
26	C₂×C₂₁₁	#103	1	1	yes	characteristic
27	C₂₁₁	#333	1	1	yes	characteristic, complemented, Sylow
28	#333	—	211	1	no	maximal, Sylow
29	C₄⋊Q₈	—	211	1	no	—
30	C₄⋊C₈	—	211	1	no	—
31	C₄⋊C₈	D₂₁₁	1	1	yes	characteristic
32	C₂×Q₈	—	211	2	no	—
33	C₂×C₈	—	2	1	no	—
34	C₂×C₈	—	422	1	no	—
35	C₄⋊C₄	—	422	1	no	—
36	C₄⋊C₄	—	422	1	no	—
37	C₄²	C₂×D₂₁₁	1	1	yes	characteristic
38	Q₈	—	844	2	no	—
39	C₂×C₄	—	422	2	no	—
40	C₂×C₄	C₄×D₂₁₁	1	1	yes	characteristic
41	C₂×C₄	C₂²⋊D₂₁₁	1	1	yes	characteristic
42	C₂×C₄	D₈₄₄	1	1	yes	characteristic
43	C₈	—	4	1	no	—
44	C₈	—	844	1	no	—
45	C₂²	(C₂×D₂₁₁)⋊C₄	1	1	yes	characteristic
46	C₄	—	2	1	no	—
47	C₄	—	844	2	no	—
48	C₄	#16809950	1	1	yes	characteristic
49	C₄	#16809951	1	1	yes	characteristic
50	C₄	#16809952	1	1	yes	characteristic
51	C₄	Dic₈₄₄	1	1	yes	characteristic
52	C₂	#20269893	1	1	yes	characteristic
53	C₂	#20269900	1	1	yes	characteristic
54	C₂	#20269902	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	(C₂×D₂₁₁)⋊C₄
commutator	C₂×C₄×C₂₁₁	C₂×C₄
Frattini	C₄²	C₂×D₂₁₁
Fitting	C₄⋊C₈×C₂₁₁	C₂
socle, prime radical	C₂²×C₂₁₁	C₂²⋊C₄
cosocle	C₄²×C₂₁₁	C₂²
hypercenter	C₄⋊C₈	D₂₁₁
hypocenter	C₂₁₁	#333
LCS	G ▷ C₂×C₄×C₂₁₁ ▷ C₂²×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
27. C₂₁₁	#333	semidirect

Maximal subgroups

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

subgroup	index
#20269977	2
C₄⋊(C₂₁₁⋊C₈)	2
C₄⋊C₈×C₂₁₁	2
#333	211