Subgroups — C₈×Dic₁₁₉₈ #50306015

Order 38336.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Dic₁₁₉₈	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(C₅₉₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	Dic₅₉₉⋊C₈	C₂	1	2	yes	maximal
4	C₈×Dic₅₉₉	C₂	1	2	yes	maximal
5	C₄×C₈×C₅₉₉	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₁₁₉₈	C₄	1	1	yes	characteristic
7	C₂×C₅₉₉⋊C₈	C₂²	1	2	yes	—
8	C₄⋊Dic₅₉₉	C₄	1	1	yes	characteristic
9	Dic₅₉₉⋊C₄	C₄	1	2	yes	—
10	C₄×Dic₅₉₉	C₂²	1	2	yes	—
11	C₂×C₈×C₅₉₉	C₂²	1	1	yes	characteristic
12	C₂×C₈×C₅₉₉	C₂²	1	1	yes	characteristic
13	C₄²×C₅₉₉	C₂²	1	1	yes	characteristic
14	Dic₁₁₉₈	C₈	1	4	yes	direct product factor
15	C₂×Dic₅₉₉	C₂×C₄	1	2	yes	—
16	C₂×Dic₅₉₉	C₂×C₄	1	2	yes	—
17	C₅₉₉⋊C₈	—	2	2	no	—
18	C₂×C₄×C₅₉₉	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₅₉₉	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₅₉₉	C₂³	1	1	yes	characteristic
21	C₈×C₅₉₉	—	2	1	no	—
22	C₈×C₅₉₉	Q₈	1	2	yes	complemented
23	C₂²×C₅₉₉	C₂²×C₄	1	1	yes	characteristic
24	C₄×C₅₉₉	—	2	1	no	—
25	C₄×C₅₉₉	C₂×C₈	1	2	yes	—
26	C₄×C₅₉₉	C₂×Q₈	1	1	yes	characteristic
27	C₄×C₅₉₉	#41	1	1	yes	characteristic
28	Dic₅₉₉	—	2	2	no	—
29	Dic₅₉₉	C₂×C₈	1	4	yes	—
30	C₂×C₅₉₉	C₄×Q₈	1	1	yes	characteristic
31	C₂×C₅₉₉	C₂²×C₈	1	1	yes	characteristic
32	C₂×C₅₉₉	#131	1	1	yes	characteristic
33	C₅₉₉	C₈×Q₈	1	1	yes	characteristic, complemented, Sylow
34	C₈×Q₈	—	599	1	no	maximal, Sylow
35	C₄×Q₈	—	599	1	no	—
36	C₄⋊C₈	—	599	2	no	—
37	C₄⋊C₈	—	599	1	no	—
38	C₄×C₈	—	599	2	no	—
39	C₄×C₈	D₅₉₉	1	1	yes	characteristic
40	C₂×Q₈	—	599	1	no	—
41	C₂×C₈	—	599	2	no	—
42	C₂×C₈	C₂×D₅₉₉	1	1	yes	characteristic
43	C₂×C₈	C₂×D₅₉₉	1	1	yes	characteristic
44	C₄⋊C₄	—	599	2	no	—
45	C₄⋊C₄	—	599	1	no	—
46	C₄²	—	599	2	no	—
47	C₄²	C₂×D₅₉₉	1	1	yes	characteristic
48	Q₈	—	599	4	no	—
49	C₂×C₄	—	599	2	no	—
50	C₂×C₄	—	599	2	no	—
51	C₂×C₄	C₄×D₅₉₉	1	1	yes	characteristic
52	C₂×C₄	C₄×D₅₉₉	1	1	yes	characteristic
53	C₂×C₄	C₂²×D₅₉₉	1	1	yes	characteristic
54	C₈	—	2	1	no	—
55	C₈	—	1198	2	no	—
56	C₈	Dic₁₁₉₈	1	2	yes	direct product factor
57	C₂²	C₂×C₄×D₅₉₉	1	1	yes	characteristic
58	C₄	—	2	1	no	—
59	C₄	—	599	4	no	—
60	C₄	—	1198	2	no	—
61	C₄	C₈×D₅₉₉	1	2	yes	—
62	C₄	C₂×Dic₁₁₉₈	1	1	yes	characteristic
63	C₄	#23751078	1	1	yes	characteristic
64	C₂	C₄×Dic₁₁₉₈	1	1	yes	characteristic
65	C₂	C₂×C₈×D₅₉₉	1	1	yes	characteristic
66	C₂	#34062721	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₅₉₉
commutator	C₂×C₅₉₉	C₂²×C₈
Frattini	C₂×C₄	C₂²×D₅₉₉
Fitting	C₄×C₈×C₅₉₉	C₂
socle, prime radical	C₂²×C₅₉₉	C₂²×C₄
cosocle	C₂×C₄×C₅₉₉	C₂³
hypercenter	C₄×C₈	D₅₉₉
hypocenter	C₅₉₉	C₈×Q₈
LCS	G ▷ C₂×C₅₉₉ ▷ C₅₉₉
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
14. Dic₁₁₉₈	C₈	direct (N × H)
22. C₈×C₅₉₉	Q₈	semidirect
33. C₅₉₉	C₈×Q₈	semidirect
56. C₈	Dic₁₁₉₈	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×Dic₁₁₉₈	2
C₄⋊(C₅₉₉⋊C₈)	2
Dic₅₉₉⋊C₈	2
C₈×Dic₅₉₉	2
C₄×C₈×C₅₉₉	2
C₈×Q₈	599

Subgroups — C8×Dic1198 #50306015

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×Dic₁₁₉₈ #50306015