Subgroups — #28385339

Order 14368.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₈₉₈	C₂	1	2	yes	maximal
2	C₄⋊Dic₄₄₉	C₂	1	1	yes	maximal, characteristic
3	#20297715	C₂	1	2	yes	maximal
4	C₄×Dic₄₄₉	C₂	1	1	yes	maximal, characteristic
5	C₄⋊C₄×C₄₄₉	C₂	1	1	yes	maximal, characteristic
6	Dic₈₉₈	—	2	4	no	—
7	C₂×Dic₄₄₉	C₂²	1	2	yes	—
8	C₂×Dic₄₄₉	C₂²	1	1	yes	characteristic
9	C₂×Dic₄₄₉	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₄₄₉	C₂²	1	2	yes	—
11	C₂×C₄×C₄₄₉	C₂²	1	1	yes	characteristic
12	C₂²×C₄₄₉	C₂³	1	1	yes	characteristic
13	C₄×C₄₄₉	—	2	2	no	—
14	C₄×C₄₄₉	Q₈	1	2	yes	complemented
15	Dic₄₄₉	—	2	2	no	—
16	Dic₄₄₉	D₄	1	2	yes	—
17	Dic₄₄₉	Q₈	1	2	yes	complemented
18	C₂×C₄₄₉	C₂×D₄	1	1	yes	characteristic
19	C₂×C₄₄₉	C₂×Q₈	1	1	yes	characteristic
20	C₂×C₄₄₉	C₂×Q₈	1	1	yes	characteristic
21	C₄₄₉	C₄⋊Q₈	1	1	yes	characteristic, complemented, Sylow
22	C₄⋊Q₈	—	449	1	no	maximal, Sylow
23	C₂×Q₈	—	449	2	no	—
24	C₄⋊C₄	—	449	2	no	—
25	C₄⋊C₄	—	449	1	no	—
26	C₄⋊C₄	D₄₄₉	1	1	yes	characteristic
27	C₄²	—	449	1	no	—
28	Q₈	—	898	4	no	—
29	C₂×C₄	—	449	2	no	—
30	C₂×C₄	—	449	1	no	—
31	C₂×C₄	—	449	1	no	—
32	C₂×C₄	C₂×D₄₄₉	1	2	yes	—
33	C₂×C₄	C₂×D₄₄₉	1	1	yes	characteristic
34	C₂²	C₂²×D₄₄₉	1	1	yes	characteristic
35	C₄	—	2	2	no	—
36	C₄	—	898	2	no	—
37	C₄	—	449	2	no	—
38	C₄	—	449	2	no	—
39	C₄	Dic₈₉₈	1	2	yes	complemented
40	C₂	D₄×D₄₄₉	1	1	yes	characteristic
41	C₂	Q₈×D₄₄₉	1	1	yes	characteristic
42	C₂	C₂×Dic₈₉₈	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₄₄₉
commutator, socle, prime radical, cosocle	C₂²×C₄₄₉	C₂³
Fitting	C₄⋊C₄×C₄₄₉	C₂
hypercenter	C₄⋊C₄	D₄₄₉
hypocenter	C₄₄₉	C₄⋊Q₈
LCS	G ▷ C₂²×C₄₄₉ ▷ C₄₄₉
UCS	C₁ ◁ C₂² ◁ C₄⋊C₄

Complements

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

N	H ≅ G/N	type
14. C₄×C₄₄₉	Q₈	semidirect
17. Dic₄₄₉	Q₈	semidirect
21. C₄₄₉	C₄⋊Q₈	semidirect
39. C₄	Dic₈₉₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₈₉₈	2
C₄⋊Dic₄₄₉	2
#20297715	2
C₄×Dic₄₄₉	2
C₄⋊C₄×C₄₄₉	2
C₄⋊Q₈	449