Subgroups — #27213875

Order 13504.

200 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20269977	C₂	1	3	yes	maximal
2	#20269968	C₂	1	6	yes	maximal
3	Q₈×Dic₂₁₁	C₂	1	2	yes	maximal
4	C₄×Dic₄₂₂	C₂	1	3	yes	maximal
5	C₄×Q₈×C₂₁₁	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₄₂₂	C₂²	1	3	yes	—
7	C₄⋊Dic₂₁₁	C₂²	1	1	yes	characteristic
8	C₄⋊Dic₂₁₁	C₂²	1	6	yes	—
9	C₄⋊Dic₂₁₁	C₂²	1	6	yes	—
10	Dic₂₁₁⋊C₄	C₂²	1	6	yes	—
11	C₄×Dic₂₁₁	C₂²	1	6	yes	—
12	C₂×Q₈×C₂₁₁	C₂²	1	1	yes	characteristic
13	C₄⋊C₄×C₂₁₁	C₂²	1	3	yes	—
14	C₄²×C₂₁₁	C₂²	1	3	yes	—
15	Dic₄₂₂	—	2	6	no	—
16	C₂×Dic₂₁₁	C₂³	1	2	yes	—
17	C₂×Dic₂₁₁	C₂³	1	6	yes	—
18	Q₈×C₂₁₁	Q₈	1	4	yes	complemented
19	C₂×C₄×C₂₁₁	C₂³	1	3	yes	—
20	C₂×C₄×C₂₁₁	C₂³	1	3	yes	—
21	C₂×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
22	C₂²×C₂₁₁	C₂⁴	1	1	yes	characteristic
23	C₄×C₂₁₁	—	2	3	no	—
24	C₄×C₂₁₁	C₂×Q₈	1	6	yes	—
25	C₄×C₂₁₁	#41	1	2	yes	—
26	Dic₂₁₁	—	2	6	no	—
27	Dic₂₁₁	—	2	2	no	—
28	C₂×C₂₁₁	C₂²×Q₈	1	1	yes	characteristic
29	C₂×C₂₁₁	C₂×#41	1	1	yes	characteristic
30	C₂×C₂₁₁	2₋¹⁺⁴	1	1	yes	characteristic
31	C₂₁₁	#557	1	1	yes	characteristic, complemented, Sylow
32	#557	—	211	1	no	maximal, Sylow
33	C₄⋊Q₈	—	211	3	no	—
34	#125	—	211	6	no	—
35	C₄×Q₈	—	211	2	no	—
36	C₄×Q₈	—	211	3	no	—
37	C₄×Q₈	D₂₁₁	1	1	yes	characteristic
38	C₂×Q₈	—	211	3	no	—
39	C₂×Q₈	C₂×D₂₁₁	1	1	yes	characteristic
40	C₄⋊C₄	—	211	1	no	—
41	C₄⋊C₄	—	211	6	no	—
42	C₄⋊C₄	—	211	6	no	—
43	C₄⋊C₄	—	211	6	no	—
44	C₄⋊C₄	C₂×D₂₁₁	1	3	yes	—
45	C₄²	—	211	6	no	—
46	C₄²	C₂×D₂₁₁	1	3	yes	—
47	Q₈	—	422	6	no	—
48	Q₈	Dic₄₂₂	1	4	yes	complemented
49	C₂×C₄	—	211	2	no	—
50	C₂×C₄	—	211	6	no	—
51	C₂×C₄	C₂²×D₂₁₁	1	3	yes	—
52	C₂×C₄	C₂²×D₂₁₁	1	3	yes	—
53	C₂×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
54	C₂²	C₂³×D₂₁₁	1	1	yes	characteristic
55	C₄	—	2	3	no	—
56	C₄	—	422	6	no	—
57	C₄	—	422	2	no	—
58	C₄	C₂×Dic₄₂₂	1	6	yes	—
59	C₄	#16809963	1	2	yes	—
60	C₂	C₂²×Dic₄₂₂	1	1	yes	characteristic
61	C₂	C₂×#16809963	1	1	yes	characteristic
62	C₂	#20270024	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₁₁
commutator, socle, prime radical, cosocle	C₂²×C₂₁₁	C₂⁴
Fitting	C₄×Q₈×C₂₁₁	C₂
hypercenter	C₄×Q₈	D₂₁₁
hypocenter	C₂₁₁	#557
LCS	G ▷ C₂²×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
18. Q₈×C₂₁₁	Q₈	semidirect
31. C₂₁₁	#557	semidirect
48. Q₈	Dic₄₂₂	semidirect

Maximal subgroups

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

subgroup	index
#20269977	2
#20269968	2
Q₈×Dic₂₁₁	2
C₄×Dic₄₂₂	2
C₄×Q₈×C₂₁₁	2
#557	211