Subgroups — #43246223

Order 28256.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#28381498	C₂	1	1	yes	maximal, characteristic
2	Q₈×D₈₈₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#28381487	C₂	1	1	yes	maximal, characteristic, complemented
4	#28381486	C₂	1	1	yes	maximal, characteristic, complemented
5	#28381484	C₂	1	1	yes	maximal, characteristic, complemented
6	#28381480	C₂	1	1	yes	maximal, characteristic, complemented
7	Q₁₆×C₈₈₃	C₂	1	1	yes	maximal, characteristic, complemented
8	Dic₁₇₆₆	—	2	1	no	—
9	Dic₁₇₆₆	C₂²	1	1	yes	characteristic
10	D₃₅₃₂	—	2	1	no	—
11	D₃₅₃₂	C₂²	1	1	yes	characteristic
12	C₄×D₈₈₃	—	2	1	no	—
13	C₄×D₈₈₃	—	2	1	no	—
14	C₄×D₈₈₃	C₂²	1	1	yes	characteristic
15	C₈₈₃⋊C₈	C₂²	1	1	yes	characteristic
16	Q₈×C₈₈₃	C₂²	1	1	yes	characteristic
17	Q₈×C₈₈₃	C₂²	1	1	yes	characteristic
18	C₈×C₈₈₃	C₂²	1	1	yes	characteristic
19	C₂×D₈₈₃	—	2	1	no	—
20	C₂×D₈₈₃	D₄	1	1	yes	characteristic
21	C₄×C₈₈₃	—	2	1	no	—
22	C₄×C₈₈₃	—	2	1	no	—
23	C₄×C₈₈₃	C₂³	1	1	yes	characteristic
24	Dic₈₈₃	—	2	1	no	—
25	Dic₈₈₃	D₄	1	1	yes	characteristic
26	C₂×C₈₈₃	C₂×D₄	1	1	yes	characteristic
27	D₈₈₃	—	2	1	no	—
28	D₈₈₃	—	4	1	no	—
29	C₈₈₃	#137	1	1	yes	characteristic, complemented, Sylow
30	#137	—	883	1	no	maximal, Sylow
31	#41	—	883	1	no	—
32	C₂×Q₈	—	883	1	no	—
33	Q₁₆	—	883	1	no	—
34	Q₁₆	D₈₈₃	1	1	yes	characteristic, complemented
35	SD₁₆	—	883	1	no	—
36	SD₁₆	—	883	1	no	—
37	M₁₆	—	883	1	no	—
38	Q₈	—	883	1	no	—
39	Q₈	—	1766	1	no	—
40	Q₈	C₂×D₈₈₃	1	1	yes	characteristic
41	Q₈	C₂×D₈₈₃	1	1	yes	characteristic
42	D₄	—	883	1	no	—
43	D₄	—	1766	1	no	—
44	C₂×C₄	—	883	1	no	—
45	C₂×C₄	—	1766	1	no	—
46	C₂×C₄	—	1766	1	no	—
47	C₈	—	883	1	no	—
48	C₈	C₂×D₈₈₃	1	1	yes	characteristic
49	C₂²	—	883	1	no	—
50	C₂²	—	1766	1	no	—
51	C₄	—	2	1	no	—
52	C₄	—	2	1	no	—
53	C₄	—	883	1	no	—
54	C₄	—	1766	1	no	—
55	C₄	C₂²×D₈₈₃	1	1	yes	characteristic
56	C₂	—	3532	1	no	—
57	C₂	—	1766	1	no	—
58	C₂	D₄×D₈₈₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×D₈₈₃
commutator, cosocle	C₄×C₈₈₃	C₂³
Frattini	C₄	C₂²×D₈₈₃
Fitting	Q₁₆×C₈₈₃	C₂
socle	C₂×C₈₈₃	C₂×D₄
prime radical	#28381498	C₂
hypercenter	Q₁₆	D₈₈₃
hypocenter	C₈₈₃	#137
LCS	G ▷ C₄×C₈₈₃ ▷ C₂×C₈₈₃ ▷ C₈₈₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ Q₁₆

Complements

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

N	H ≅ G/N	type
2. Q₈×D₈₈₃	C₂	semidirect
3. #28381487	C₂	semidirect
4. #28381486	C₂	semidirect
5. #28381484	C₂	semidirect
6. #28381480	C₂	semidirect
7. Q₁₆×C₈₈₃	C₂	semidirect
29. C₈₈₃	#137	semidirect
34. Q₁₆	D₈₈₃	semidirect

Maximal subgroups

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

subgroup	index
#28381498	2
Q₈×D₈₈₃	2
#28381487	2
#28381486	2
#28381484	2
#28381480	2
Q₁₆×C₈₈₃	2
#137	883