Subgroups — #27133873

Order 12256.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20259458	C₂	1	1	yes	maximal, characteristic
2	Q₈×D₃₈₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#20259447	C₂	1	1	yes	maximal, characteristic, complemented
4	#20259446	C₂	1	1	yes	maximal, characteristic, complemented
5	#20259444	C₂	1	1	yes	maximal, characteristic, complemented
6	#20259440	C₂	1	1	yes	maximal, characteristic, complemented
7	Q₁₆×C₃₈₃	C₂	1	1	yes	maximal, characteristic, complemented
8	Dic₇₆₆	—	2	1	no	—
9	Dic₇₆₆	C₂²	1	1	yes	characteristic
10	D₁₅₃₂	—	2	1	no	—
11	D₁₅₃₂	C₂²	1	1	yes	characteristic
12	C₄×D₃₈₃	—	2	1	no	—
13	C₄×D₃₈₃	—	2	1	no	—
14	C₄×D₃₈₃	C₂²	1	1	yes	characteristic
15	C₃₈₃⋊C₈	C₂²	1	1	yes	characteristic
16	Q₈×C₃₈₃	C₂²	1	1	yes	characteristic
17	Q₈×C₃₈₃	C₂²	1	1	yes	characteristic
18	C₈×C₃₈₃	C₂²	1	1	yes	characteristic
19	C₂×D₃₈₃	—	2	1	no	—
20	C₂×D₃₈₃	D₄	1	1	yes	characteristic
21	C₄×C₃₈₃	—	2	1	no	—
22	C₄×C₃₈₃	—	2	1	no	—
23	C₄×C₃₈₃	C₂³	1	1	yes	characteristic
24	Dic₃₈₃	—	2	1	no	—
25	Dic₃₈₃	D₄	1	1	yes	characteristic
26	C₂×C₃₈₃	C₂×D₄	1	1	yes	characteristic
27	D₃₈₃	—	2	1	no	—
28	D₃₈₃	—	4	1	no	—
29	C₃₈₃	#137	1	1	yes	characteristic, complemented, Sylow
30	#137	—	383	1	no	maximal, Sylow
31	#41	—	383	1	no	—
32	C₂×Q₈	—	383	1	no	—
33	Q₁₆	—	383	1	no	—
34	Q₁₆	D₃₈₃	1	1	yes	characteristic, complemented
35	SD₁₆	—	383	1	no	—
36	SD₁₆	—	383	1	no	—
37	M₁₆	—	383	1	no	—
38	Q₈	—	383	1	no	—
39	Q₈	—	766	1	no	—
40	Q₈	C₂×D₃₈₃	1	1	yes	characteristic
41	Q₈	C₂×D₃₈₃	1	1	yes	characteristic
42	D₄	—	383	1	no	—
43	D₄	—	766	1	no	—
44	C₂×C₄	—	383	1	no	—
45	C₂×C₄	—	766	1	no	—
46	C₂×C₄	—	766	1	no	—
47	C₈	—	383	1	no	—
48	C₈	C₂×D₃₈₃	1	1	yes	characteristic
49	C₂²	—	383	1	no	—
50	C₂²	—	766	1	no	—
51	C₄	—	2	1	no	—
52	C₄	—	2	1	no	—
53	C₄	—	383	1	no	—
54	C₄	—	766	1	no	—
55	C₄	C₂²×D₃₈₃	1	1	yes	characteristic
56	C₂	—	1532	1	no	—
57	C₂	—	766	1	no	—
58	C₂	D₄×D₃₈₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×D₃₈₃
commutator, cosocle	C₄×C₃₈₃	C₂³
Frattini	C₄	C₂²×D₃₈₃
Fitting	Q₁₆×C₃₈₃	C₂
socle	C₂×C₃₈₃	C₂×D₄
prime radical	#20259458	C₂
hypercenter	Q₁₆	D₃₈₃
hypocenter	C₃₈₃	#137
LCS	G ▷ C₄×C₃₈₃ ▷ C₂×C₃₈₃ ▷ C₃₈₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ Q₁₆

Complements

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

N	H ≅ G/N	type
2. Q₈×D₃₈₃	C₂	semidirect
3. #20259447	C₂	semidirect
4. #20259446	C₂	semidirect
5. #20259444	C₂	semidirect
6. #20259440	C₂	semidirect
7. Q₁₆×C₃₈₃	C₂	semidirect
29. C₃₈₃	#137	semidirect
34. Q₁₆	D₃₈₃	semidirect

Maximal subgroups

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

subgroup	index
#20259458	2
Q₈×D₃₈₃	2
#20259447	2
#20259446	2
#20259444	2
#20259440	2
Q₁₆×C₃₈₃	2
#137	383