Subgroups — #28385372

Order 14368.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20297737	C₂	1	1	yes	maximal, characteristic
2	Q₈×D₄₄₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#20297726	C₂	1	1	yes	maximal, characteristic, complemented
4	#20297725	C₂	1	1	yes	maximal, characteristic, complemented
5	#20297723	C₂	1	1	yes	maximal, characteristic, complemented
6	#20297719	C₂	1	1	yes	maximal, characteristic, complemented
7	Q₁₆×C₄₄₉	C₂	1	1	yes	maximal, characteristic, complemented
8	Dic₈₉₈	—	2	1	no	—
9	Dic₈₉₈	C₂²	1	1	yes	characteristic
10	D₁₇₉₆	—	2	1	no	—
11	D₁₇₉₆	C₂²	1	1	yes	characteristic
12	C₄×D₄₄₉	—	2	1	no	—
13	C₄×D₄₄₉	—	2	1	no	—
14	C₄×D₄₄₉	C₂²	1	1	yes	characteristic
15	#16813311	C₂²	1	1	yes	characteristic
16	Q₈×C₄₄₉	C₂²	1	1	yes	characteristic
17	Q₈×C₄₄₉	C₂²	1	1	yes	characteristic
18	C₈×C₄₄₉	C₂²	1	1	yes	characteristic
19	C₂×D₄₄₉	—	2	1	no	—
20	C₂×D₄₄₉	D₄	1	1	yes	characteristic
21	C₄×C₄₄₉	—	2	1	no	—
22	C₄×C₄₄₉	—	2	1	no	—
23	C₄×C₄₄₉	C₂³	1	1	yes	characteristic
24	Dic₄₄₉	—	2	1	no	—
25	Dic₄₄₉	D₄	1	1	yes	characteristic
26	C₂×C₄₄₉	C₂×D₄	1	1	yes	characteristic
27	D₄₄₉	—	2	1	no	—
28	D₄₄₉	—	4	1	no	—
29	C₄₄₉	#137	1	1	yes	characteristic, complemented, Sylow
30	#137	—	449	1	no	maximal, Sylow
31	#41	—	449	1	no	—
32	C₂×Q₈	—	449	1	no	—
33	Q₁₆	—	449	1	no	—
34	Q₁₆	D₄₄₉	1	1	yes	characteristic, complemented
35	SD₁₆	—	449	1	no	—
36	SD₁₆	—	449	1	no	—
37	M₁₆	—	449	1	no	—
38	Q₈	—	898	1	no	—
39	Q₈	—	449	1	no	—
40	Q₈	C₂×D₄₄₉	1	1	yes	characteristic
41	Q₈	C₂×D₄₄₉	1	1	yes	characteristic
42	D₄	—	898	1	no	—
43	D₄	—	449	1	no	—
44	C₂×C₄	—	898	1	no	—
45	C₂×C₄	—	898	1	no	—
46	C₂×C₄	—	449	1	no	—
47	C₈	—	449	1	no	—
48	C₈	C₂×D₄₄₉	1	1	yes	characteristic
49	C₂²	—	898	1	no	—
50	C₂²	—	449	1	no	—
51	C₄	—	2	1	no	—
52	C₄	—	2	1	no	—
53	C₄	—	898	1	no	—
54	C₄	—	449	1	no	—
55	C₄	C₂²×D₄₄₉	1	1	yes	characteristic
56	C₂	—	1796	1	no	—
57	C₂	—	898	1	no	—
58	C₂	D₄×D₄₄₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×D₄₄₉
commutator, cosocle	C₄×C₄₄₉	C₂³
Frattini	C₄	C₂²×D₄₄₉
Fitting	Q₁₆×C₄₄₉	C₂
socle	C₂×C₄₄₉	C₂×D₄
prime radical	#20297737	C₂
hypercenter	Q₁₆	D₄₄₉
hypocenter	C₄₄₉	#137
LCS	G ▷ C₄×C₄₄₉ ▷ C₂×C₄₄₉ ▷ C₄₄₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ Q₁₆

Complements

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

N	H ≅ G/N	type
2. Q₈×D₄₄₉	C₂	semidirect
3. #20297726	C₂	semidirect
4. #20297725	C₂	semidirect
5. #20297723	C₂	semidirect
6. #20297719	C₂	semidirect
7. Q₁₆×C₄₄₉	C₂	semidirect
29. C₄₄₉	#137	semidirect
34. Q₁₆	D₄₄₉	semidirect

Maximal subgroups

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

subgroup	index
#20297737	2
Q₈×D₄₄₉	2
#20297726	2
#20297725	2
#20297723	2
#20297719	2
Q₁₆×C₄₄₉	2
#137	449