Subgroups — #27213575

Order 13504.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20269996	C₂	1	2	yes	maximal, complemented
2	Dic₁₆₈₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#20269921	C₂	1	2	yes	maximal, complemented
4	C₁₆×D₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
5	D₁₆×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#16809961	—	2	2	no	—
7	Dic₈₄₄	C₂²	1	2	yes	—
8	#16809950	—	2	2	no	—
9	C₈×D₂₁₁	C₂²	1	1	yes	characteristic
10	C₂₁₁⋊C₁₆	C₂²	1	1	yes	characteristic
11	D₈×C₂₁₁	C₂²	1	2	yes	—
12	C₁₆×C₂₁₁	C₂²	1	1	yes	characteristic
13	Dic₄₂₂	—	2	2	no	—
14	C₂²⋊D₂₁₁	—	4	2	no	—
15	C₂×Dic₂₁₁	—	4	2	no	—
16	C₄×D₂₁₁	D₄	1	1	yes	characteristic
17	C₂₁₁⋊C₈	D₄	1	1	yes	characteristic
18	D₄×C₂₁₁	—	2	2	no	—
19	C₈×C₂₁₁	C₂³	1	1	yes	characteristic
20	C₂²×C₂₁₁	—	4	2	no	—
21	C₂×D₂₁₁	D₈	1	1	yes	characteristic
22	C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
23	Dic₂₁₁	—	4	2	no	—
24	Dic₂₁₁	D₈	1	1	yes	characteristic
25	C₂×C₂₁₁	—	8	2	no	—
26	C₂×C₂₁₁	C₂×D₈	1	1	yes	characteristic
27	D₂₁₁	—	2	1	no	—
28	C₂₁₁	#508	1	1	yes	characteristic, complemented, Sylow
29	#508	—	211	1	no	maximal, Sylow
30	#135	—	211	2	no	—
31	Q₃₂	—	211	1	no	—
32	SD₃₂	—	211	2	no	—
33	D₁₆	D₂₁₁	1	1	yes	characteristic, complemented
34	C₂×C₁₆	—	211	1	no	—
35	#41	—	422	2	no	—
36	Q₁₆	—	211	2	no	—
37	SD₁₆	—	422	2	no	—
38	D₈	C₂×D₂₁₁	1	2	yes	—
39	C₂×C₈	—	211	1	no	—
40	C₁₆	—	211	1	no	—
41	C₁₆	C₂×D₂₁₁	1	1	yes	characteristic
42	Q₈	—	422	2	no	—
43	D₄	—	2	2	no	—
44	D₄	—	844	2	no	—
45	C₂×C₄	—	844	2	no	—
46	C₂×C₄	—	211	1	no	—
47	C₈	—	211	1	no	—
48	C₈	C₂²×D₂₁₁	1	1	yes	characteristic
49	C₂²	—	4	2	no	—
50	C₂²	—	211	1	no	—
51	C₄	—	844	2	no	—
52	C₄	—	211	1	no	—
53	C₄	D₄×D₂₁₁	1	1	yes	characteristic
54	C₂	—	8	2	no	—
55	C₂	—	422	1	no	—
56	C₂	D₈×D₂₁₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₈×D₂₁₁
commutator, cosocle	C₈×C₂₁₁	C₂³
Frattini	C₈	C₂²×D₂₁₁
Fitting	D₁₆×C₂₁₁	C₂
socle	C₂×C₂₁₁	C₂×D₈
hypercenter	D₁₆	D₂₁₁
hypocenter	C₂₁₁	#508
LCS	G ▷ C₈×C₂₁₁ ▷ C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ D₁₆

Complements

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

N	H ≅ G/N	type
1. #20269996	C₂	semidirect
2. Dic₁₆₈₈	C₂	semidirect
3. #20269921	C₂	semidirect
4. C₁₆×D₂₁₁	C₂	semidirect
5. D₁₆×C₂₁₁	C₂	semidirect
28. C₂₁₁	#508	semidirect
33. D₁₆	D₂₁₁	semidirect

Maximal subgroups

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

subgroup	index
#20269996	2
Dic₁₆₈₈	2
#20269921	2
C₁₆×D₂₁₁	2
D₁₆×C₂₁₁	2
#508	211