Subgroups — #27213549

Order 13504.

98 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	#20269910	C₂	1	4	yes	maximal
3	C₄×C₂₁₁⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₄⋊Q₈×C₂₁₁	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₄₂₂	—	2	1	no	—
6	C₂×C₂₁₁⋊C₈	C₂²	1	2	yes	—
7	C₄⋊Dic₂₁₁	—	2	1	no	—
8	C₄⋊Dic₂₁₁	C₂²	1	2	yes	—
9	C₂×Q₈×C₂₁₁	—	2	1	no	—
10	C₄⋊C₄×C₂₁₁	—	2	1	no	—
11	C₄⋊C₄×C₂₁₁	C₂²	1	2	yes	—
12	C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
13	Dic₄₂₂	—	4	2	no	—
14	C₂×Dic₂₁₁	—	2	2	no	—
15	C₂₁₁⋊C₈	Q₈	1	4	yes	complemented
16	Q₈×C₂₁₁	—	4	2	no	—
17	C₂×C₄×C₂₁₁	—	2	2	no	—
18	C₂×C₄×C₂₁₁	D₄	1	1	yes	characteristic
19	C₂×C₄×C₂₁₁	D₄	1	1	yes	characteristic
20	C₂×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
21	C₂²×C₂₁₁	C₂×D₄	1	1	yes	characteristic
22	C₄×C₂₁₁	—	4	2	no	—
23	C₄×C₂₁₁	D₈	1	2	yes	—
24	C₄×C₂₁₁	Q₁₆	1	2	yes	—
25	C₄×C₂₁₁	C₂×Q₈	1	2	yes	—
26	Dic₂₁₁	—	4	2	no	—
27	C₂×C₂₁₁	C₄⋊Q₈	1	1	yes	characteristic
28	C₂×C₂₁₁	C₂×D₈	1	1	yes	characteristic
29	C₂×C₂₁₁	C₂×Q₁₆	1	1	yes	characteristic
30	C₂₁₁	#500	1	1	yes	characteristic, complemented, Sylow
31	#500	—	211	1	no	maximal, Sylow
32	C₄⋊Q₈	—	211	1	no	—
33	C₄⋊Q₈	D₂₁₁	1	1	yes	characteristic
34	#107	—	211	4	no	—
35	C₄×C₈	—	211	1	no	—
36	C₂×Q₈	—	2	1	no	—
37	C₂×Q₈	—	422	1	no	—
38	C₂×C₈	—	211	2	no	—
39	C₄⋊C₄	—	2	1	no	—
40	C₄⋊C₄	—	422	1	no	—
41	C₄⋊C₄	—	211	2	no	—
42	C₄⋊C₄	C₂×D₂₁₁	1	2	yes	—
43	C₄²	C₂×D₂₁₁	1	1	yes	characteristic
44	Q₈	—	4	2	no	—
45	Q₈	—	844	2	no	—
46	C₂×C₄	—	2	2	no	—
47	C₂×C₄	—	422	2	no	—
48	C₂×C₄	C₂²⋊D₂₁₁	1	1	yes	characteristic
49	C₂×C₄	C₂²⋊D₂₁₁	1	1	yes	characteristic
50	C₂×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
51	C₈	—	211	4	no	—
52	C₂²	C₂×C₂²⋊D₂₁₁	1	1	yes	characteristic
53	C₄	—	4	2	no	—
54	C₄	—	844	2	no	—
55	C₄	#16809947	1	2	yes	—
56	C₄	#16809952	1	2	yes	—
57	C₄	Q₈×D₂₁₁	1	2	yes	—
58	C₂	#20269975	1	1	yes	characteristic
59	C₂	C₂×#16809947	1	1	yes	characteristic
60	C₂	C₂×#16809952	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₂₁₁
commutator, cosocle	C₂×C₄×C₂₁₁	C₂³
Frattini	C₂×C₄	C₂²×D₂₁₁
Fitting	C₄⋊Q₈×C₂₁₁	C₂
socle, prime radical	C₂²×C₂₁₁	C₂×D₄
hypercenter	C₄⋊Q₈	D₂₁₁
hypocenter	C₂₁₁	#500
LCS	G ▷ C₂×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
15. C₂₁₁⋊C₈	Q₈	semidirect
30. C₂₁₁	#500	semidirect

Maximal subgroups

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

subgroup	index
#20269977	2
#20269910	2
C₄×C₂₁₁⋊C₈	2
C₄⋊Q₈×C₂₁₁	2
#500	211