Subgroups — #28380775

Order 14048.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20296330	C₂	1	1	yes	maximal, characteristic
2	Q₈×D₄₃₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#20296319	C₂	1	1	yes	maximal, characteristic, complemented
4	#20296318	C₂	1	1	yes	maximal, characteristic, complemented
5	#20296316	C₂	1	1	yes	maximal, characteristic, complemented
6	#20296312	C₂	1	1	yes	maximal, characteristic, complemented
7	Q₁₆×C₄₃₉	C₂	1	1	yes	maximal, characteristic, complemented
8	Dic₈₇₈	—	2	1	no	—
9	Dic₈₇₈	C₂²	1	1	yes	characteristic
10	D₁₇₅₆	—	2	1	no	—
11	D₁₇₅₆	C₂²	1	1	yes	characteristic
12	C₄×D₄₃₉	—	2	1	no	—
13	C₄×D₄₃₉	—	2	1	no	—
14	C₄×D₄₃₉	C₂²	1	1	yes	characteristic
15	C₄₃₉⋊C₈	C₂²	1	1	yes	characteristic
16	Q₈×C₄₃₉	C₂²	1	1	yes	characteristic
17	Q₈×C₄₃₉	C₂²	1	1	yes	characteristic
18	C₈×C₄₃₉	C₂²	1	1	yes	characteristic
19	C₂×D₄₃₉	—	2	1	no	—
20	C₂×D₄₃₉	D₄	1	1	yes	characteristic
21	C₄×C₄₃₉	—	2	1	no	—
22	C₄×C₄₃₉	—	2	1	no	—
23	C₄×C₄₃₉	C₂³	1	1	yes	characteristic
24	Dic₄₃₉	—	2	1	no	—
25	Dic₄₃₉	D₄	1	1	yes	characteristic
26	C₂×C₄₃₉	C₂×D₄	1	1	yes	characteristic
27	D₄₃₉	—	2	1	no	—
28	D₄₃₉	—	4	1	no	—
29	C₄₃₉	#137	1	1	yes	characteristic, complemented, Sylow
30	#137	—	439	1	no	maximal, Sylow
31	#41	—	439	1	no	—
32	C₂×Q₈	—	439	1	no	—
33	Q₁₆	—	439	1	no	—
34	Q₁₆	D₄₃₉	1	1	yes	characteristic, complemented
35	SD₁₆	—	439	1	no	—
36	SD₁₆	—	439	1	no	—
37	M₁₆	—	439	1	no	—
38	Q₈	—	878	1	no	—
39	Q₈	—	439	1	no	—
40	Q₈	C₂×D₄₃₉	1	1	yes	characteristic
41	Q₈	C₂×D₄₃₉	1	1	yes	characteristic
42	D₄	—	878	1	no	—
43	D₄	—	439	1	no	—
44	C₂×C₄	—	878	1	no	—
45	C₂×C₄	—	878	1	no	—
46	C₂×C₄	—	439	1	no	—
47	C₈	—	439	1	no	—
48	C₈	C₂×D₄₃₉	1	1	yes	characteristic
49	C₂²	—	878	1	no	—
50	C₂²	—	439	1	no	—
51	C₄	—	2	1	no	—
52	C₄	—	2	1	no	—
53	C₄	—	878	1	no	—
54	C₄	—	439	1	no	—
55	C₄	C₂²×D₄₃₉	1	1	yes	characteristic
56	C₂	—	878	1	no	—
57	C₂	—	1756	1	no	—
58	C₂	D₄×D₄₃₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×D₄₃₉
commutator, cosocle	C₄×C₄₃₉	C₂³
Frattini	C₄	C₂²×D₄₃₉
Fitting	Q₁₆×C₄₃₉	C₂
socle	C₂×C₄₃₉	C₂×D₄
prime radical	#20296330	C₂
hypercenter	Q₁₆	D₄₃₉
hypocenter	C₄₃₉	#137
LCS	G ▷ C₄×C₄₃₉ ▷ C₂×C₄₃₉ ▷ C₄₃₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ Q₁₆

Complements

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

N	H ≅ G/N	type
2. Q₈×D₄₃₉	C₂	semidirect
3. #20296319	C₂	semidirect
4. #20296318	C₂	semidirect
5. #20296316	C₂	semidirect
6. #20296312	C₂	semidirect
7. Q₁₆×C₄₃₉	C₂	semidirect
29. C₄₃₉	#137	semidirect
34. Q₁₆	D₄₃₉	semidirect

Maximal subgroups

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

subgroup	index
#20296330	2
Q₈×D₄₃₉	2
#20296319	2
#20296318	2
#20296316	2
#20296312	2
Q₁₆×C₄₃₉	2
#137	439