Subgroups — #27213951

Order 13504.

268 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20270023	C₂	1	2	yes	maximal, complemented
2	#20270005	C₂	1	6	yes	maximal, complemented
3	#20269999	C₂	1	3	yes	maximal, complemented
4	C₂×D₁₆₈₈	C₂	1	3	yes	maximal, complemented
5	#131×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#16809963	—	2	2	no	—
7	#16809962	C₂²	1	6	yes	complemented
8	C₂×D₈₄₄	C₂²	1	6	yes	complemented
9	D₄×D₂₁₁	—	2	6	no	—
10	Dic₈₄₄	C₂²	1	1	yes	characteristic, complemented
11	#16809951	C₂²	1	6	yes	complemented
12	D₁₆₈₈	C₂²	1	3	yes	complemented
13	D₁₆₈₈	C₂²	1	6	yes	complemented
14	#41×C₂₁₁	C₂²	1	1	yes	characteristic
15	M₁₆×C₂₁₁	C₂²	1	3	yes	complemented
16	C₂×C₈×C₂₁₁	C₂²	1	3	yes	complemented
17	C₂²×D₂₁₁	—	2	6	no	—
18	Dic₄₂₂	C₂³	1	2	yes	—
19	D₈₄₄	—	2	6	no	—
20	D₈₄₄	C₂³	1	6	yes	—
21	C₂²⋊D₂₁₁	—	2	6	no	—
22	C₄×D₂₁₁	—	2	6	no	—
23	Q₈×C₂₁₁	D₄	1	1	yes	characteristic
24	D₄×C₂₁₁	D₄	1	3	yes	—
25	C₂×C₄×C₂₁₁	C₂³	1	3	yes	—
26	C₈×C₂₁₁	C₂³	1	1	yes	characteristic
27	C₈×C₂₁₁	C₂³	1	3	yes	—
28	C₂²×C₂₁₁	C₂×D₄	1	3	yes	—
29	C₂×D₂₁₁	—	2	6	no	—
30	C₂×D₂₁₁	—	8	6	no	—
31	C₄×C₂₁₁	C₂×D₄	1	3	yes	—
32	C₄×C₂₁₁	C₂⁴	1	1	yes	characteristic
33	Dic₂₁₁	—	2	2	no	—
34	C₂×C₂₁₁	—	2	3	no	—
35	C₂×C₂₁₁	C₂²×D₄	1	1	yes	characteristic
36	D₂₁₁	—	4	6	no	—
37	C₂₁₁	#576	1	1	yes	characteristic, complemented, Sylow
38	#576	—	211	1	no	maximal, Sylow
39	H(5,𝔽₂)	—	211	2	no	—
40	AGL(1,ℤ/8)	—	211	6	no	—
41	#135	—	211	3	no	—
42	C₂×D₈	—	211	3	no	—
43	#131	D₂₁₁	1	1	yes	characteristic, complemented
44	#41	—	422	2	no	—
45	#41	—	211	6	no	—
46	#41	C₂×D₂₁₁	1	1	yes	characteristic
47	C₂×D₄	—	422	6	no	—
48	C₂×D₄	—	211	6	no	—
49	Q₁₆	—	211	1	no	—
50	SD₁₆	—	211	6	no	—
51	D₈	—	211	3	no	—
52	D₈	—	211	6	no	—
53	M₁₆	C₂×D₂₁₁	1	3	yes	complemented
54	C₂×C₈	C₂×D₂₁₁	1	3	yes	complemented
55	C₂³	—	422	6	no	—
56	Q₈	—	211	2	no	—
57	Q₈	D₈₄₄	1	1	yes	characteristic
58	D₄	—	422	6	no	—
59	D₄	—	422	6	no	—
60	D₄	—	211	6	no	—
61	D₄	D₈₄₄	1	3	yes	—
62	C₂×C₄	—	422	6	no	—
63	C₂×C₄	C₂²×D₂₁₁	1	3	yes	—
64	C₈	C₂²×D₂₁₁	1	1	yes	characteristic
65	C₈	C₂²×D₂₁₁	1	3	yes	—
66	C₂²	—	1688	6	no	—
67	C₂²	—	422	6	no	—
68	C₂²	C₂×D₈₄₄	1	3	yes	—
69	C₄	—	422	2	no	—
70	C₄	C₂×D₈₄₄	1	3	yes	—
71	C₄	C₂³×D₂₁₁	1	1	yes	characteristic
72	C₂	—	2	3	no	—
73	C₂	—	844	6	no	—
74	C₂	C₂²×D₈₄₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²×D₈₄₄
commutator, cosocle	C₄×C₂₁₁	C₂⁴
Frattini	C₄	C₂³×D₂₁₁
Fitting	#131×C₂₁₁	C₂
socle	C₂×C₂₁₁	C₂²×D₄
hypercenter	#131	D₂₁₁
hypocenter	C₂₁₁	#576
LCS	G ▷ C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131

Complements

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

N	H ≅ G/N	type
1. #20270023	C₂	semidirect
2. #20270005	C₂	semidirect
3. #20269999	C₂	semidirect
4. C₂×D₁₆₈₈	C₂	semidirect
5. #131×C₂₁₁	C₂	semidirect
7. #16809962	C₂²	semidirect
8. C₂×D₈₄₄	C₂²	semidirect
10. Dic₈₄₄	C₂²	semidirect
11. #16809951	C₂²	semidirect
12. D₁₆₈₈	C₂²	semidirect
13. D₁₆₈₈	C₂²	semidirect
15. M₁₆×C₂₁₁	C₂²	semidirect
16. C₂×C₈×C₂₁₁	C₂²	semidirect
37. C₂₁₁	#576	semidirect
43. #131	D₂₁₁	semidirect
53. M₁₆	C₂×D₂₁₁	semidirect
54. C₂×C₈	C₂×D₂₁₁	semidirect

Maximal subgroups

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

subgroup	index
#20270023	2
#20270005	2
#20269999	2
C₂×D₁₆₈₈	2
#131×C₂₁₁	2
#576	211