Subgroups — #27133874

Order 12256.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20259457	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₇₆₆	C₂	1	1	yes	maximal, characteristic, complemented
3	Dic₁₅₃₂	C₂	1	2	yes	maximal, complemented
4	#20259446	C₂	1	2	yes	maximal, complemented
5	M₁₆×C₃₈₃	C₂	1	1	yes	maximal, characteristic, complemented
6	Dic₇₆₆	—	2	1	no	—
7	Dic₇₆₆	C₂²	1	2	yes	—
8	Dic₇₆₆	C₂²	1	1	yes	characteristic
9	D₁₅₃₂	C₂²	1	1	yes	characteristic
10	C₂²⋊D₃₈₃	—	2	1	no	—
11	C₂×Dic₃₈₃	—	2	1	no	—
12	C₄×D₃₈₃	—	2	1	no	—
13	C₂×C₄×C₃₈₃	C₂²	1	1	yes	characteristic
14	C₈×C₃₈₃	C₂²	1	2	yes	—
15	C₂²×C₃₈₃	D₄	1	1	yes	characteristic
16	C₂×D₃₈₃	—	2	1	no	—
17	C₄×C₃₈₃	D₄	1	1	yes	characteristic
18	C₄×C₃₈₃	C₂³	1	1	yes	characteristic
19	Dic₃₈₃	—	2	1	no	—
20	Dic₃₈₃	—	2	2	no	—
21	C₂×C₃₈₃	—	2	1	no	—
22	C₂×C₃₈₃	C₂×D₄	1	1	yes	characteristic
23	D₃₈₃	—	4	1	no	—
24	C₃₈₃	#137	1	1	yes	characteristic, complemented, Sylow
25	#137	—	383	1	no	maximal, Sylow
26	#41	—	383	1	no	—
27	C₂×Q₈	—	383	1	no	—
28	Q₁₆	—	383	2	no	—
29	SD₁₆	—	383	2	no	—
30	M₁₆	D₃₈₃	1	1	yes	characteristic, complemented
31	Q₈	—	383	2	no	—
32	Q₈	—	383	1	no	—
33	Q₈	—	766	1	no	—
34	D₄	—	383	1	no	—
35	D₄	—	766	1	no	—
36	C₂×C₄	—	766	1	no	—
37	C₂×C₄	—	766	1	no	—
38	C₂×C₄	C₂×D₃₈₃	1	1	yes	characteristic
39	C₈	C₂×D₃₈₃	1	2	yes	—
40	C₂²	—	766	1	no	—
41	C₂²	D₁₅₃₂	1	1	yes	characteristic
42	C₄	—	766	1	no	—
43	C₄	—	766	2	no	—
44	C₄	D₁₅₃₂	1	1	yes	characteristic
45	C₄	C₂²×D₃₈₃	1	1	yes	characteristic
46	C₂	—	2	1	no	—
47	C₂	—	1532	1	no	—
48	C₂	C₂×D₁₅₃₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₁₅₃₂
commutator, cosocle	C₄×C₃₈₃	C₂³
Frattini	C₄	C₂²×D₃₈₃
Fitting	M₁₆×C₃₈₃	C₂
socle	C₂×C₃₈₃	C₂×D₄
prime radical	#20259457	C₂
hypercenter	M₁₆	D₃₈₃
hypocenter	C₃₈₃	#137
LCS	G ▷ C₄×C₃₈₃ ▷ C₂×C₃₈₃ ▷ C₃₈₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ M₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×Dic₇₆₆	C₂	semidirect
3. Dic₁₅₃₂	C₂	semidirect
4. #20259446	C₂	semidirect
5. M₁₆×C₃₈₃	C₂	semidirect
24. C₃₈₃	#137	semidirect
30. M₁₆	D₃₈₃	semidirect

Maximal subgroups

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

subgroup	index
#20259457	2
C₂×Dic₇₆₆	2
Dic₁₅₃₂	2
#20259446	2
M₁₆×C₃₈₃	2
#137	383