Subgroups — #50306017

Order 38336.

94 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	Dic₅₉₉⋊C₈	C₂	1	2	yes	maximal
3	#34062625	C₂	1	2	yes	maximal
4	C₄×C₅₉₉⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₄⋊C₈×C₅₉₉	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₁₁₉₈	C₄	1	1	yes	characteristic
7	C₂×C₅₉₉⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₅₉₉⋊C₈	C₂²	1	1	yes	characteristic
9	C₄⋊Dic₅₉₉	C₄	1	1	yes	characteristic
10	Dic₅₉₉⋊C₄	C₄	1	2	yes	—
11	C₄×Dic₅₉₉	C₂²	1	2	yes	—
12	C₂×C₈×C₅₉₉	C₂²	1	2	yes	—
13	C₄²×C₅₉₉	C₂²	1	1	yes	characteristic
14	Dic₁₁₉₈	—	2	2	no	—
15	C₂×Dic₅₉₉	C₂×C₄	1	2	yes	—
16	C₂×Dic₅₉₉	C₂×C₄	1	2	yes	—
17	C₅₉₉⋊C₈	—	2	1	no	—
18	C₅₉₉⋊C₈	Q₈	1	2	yes	complemented
19	C₂×C₄×C₅₉₉	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₅₉₉	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×C₅₉₉	C₂³	1	1	yes	characteristic
22	C₈×C₅₉₉	—	2	2	no	—
23	C₂²×C₅₉₉	C₂²×C₄	1	1	yes	characteristic
24	C₄×C₅₉₉	—	2	1	no	—
25	C₄×C₅₉₉	M₁₆	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	—	2	2	no	—
30	C₂×C₅₉₉	C₄×Q₈	1	1	yes	characteristic
31	C₂×C₅₉₉	C₂×M₁₆	1	1	yes	characteristic
32	C₂×C₅₉₉	#131	1	1	yes	characteristic
33	C₅₉₉	#446	1	1	yes	characteristic, complemented, Sylow
34	#446	—	599	1	no	maximal, Sylow
35	C₄×Q₈	—	599	1	no	—
36	C₄⋊C₈	—	599	2	no	—
37	C₄⋊C₈	D₅₉₉	1	1	yes	characteristic
38	#97	—	599	2	no	—
39	C₄×C₈	—	599	1	no	—
40	C₂×Q₈	—	599	1	no	—
41	C₂×C₈	—	599	1	no	—
42	C₂×C₈	—	599	1	no	—
43	C₂×C₈	C₂×D₅₉₉	1	2	yes	—
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	—	1198	2	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	2	no	—
55	C₈	—	599	2	no	—
56	C₈	—	1198	1	no	—
57	C₂²	C₂×C₄×D₅₉₉	1	1	yes	characteristic
58	C₄	—	2	1	no	—
59	C₄	—	1198	2	no	—
60	C₄	—	1198	2	no	—
61	C₄	#23751061	1	2	yes	—
62	C₄	Q₈×D₅₉₉	1	1	yes	characteristic
63	C₄	#23751077	1	1	yes	characteristic
64	C₂	#34062680	1	1	yes	characteristic
65	C₂	C₂×#23751061	1	1	yes	characteristic
66	C₂	#34062723	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₅₉₉
commutator, socle, prime radical	C₂²×C₅₉₉	C₂²×C₄
Fitting	C₄⋊C₈×C₅₉₉	C₂
cosocle	C₂×C₄×C₅₉₉	C₂³
hypercenter	C₄⋊C₈	D₅₉₉
hypocenter	C₅₉₉	#446
LCS	G ▷ C₂²×C₅₉₉ ▷ C₅₉₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
18. C₅₉₉⋊C₈	Q₈	semidirect
33. C₅₉₉	#446	semidirect

Maximal subgroups

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

subgroup	index
C₄×Dic₁₁₉₈	2
Dic₅₉₉⋊C₈	2
#34062625	2
C₄×C₅₉₉⋊C₈	2
C₄⋊C₈×C₅₉₉	2
#446	599