Subgroups — #50305943

Order 38336.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊(C₅₉₉⋊C₈)	C₂	1	2	yes	maximal, complemented
2	C₂²⋊(C₅₉₉⋊C₈)	C₂	1	2	yes	maximal
3	C₄×C₅₉₉⋊C₈	C₂	1	2	yes	maximal, complemented
4	C₂×C₄²×C₅₉₉	C₂	1	1	yes	maximal, characteristic
5	C₂×C₅₉₉⋊C₈	C₂²	1	4	yes	—
6	C₂²×C₄×C₅₉₉	C₄	1	1	yes	characteristic
7	C₂²×C₄×C₅₉₉	C₄	1	1	yes	characteristic
8	C₂²×C₄×C₅₉₉	C₂²	1	1	yes	characteristic
9	C₄²×C₅₉₉	C₄	1	1	yes	characteristic
10	C₄²×C₅₉₉	C₄	1	1	yes	characteristic
11	C₄²×C₅₉₉	C₂²	1	1	yes	characteristic
12	C₄²×C₅₉₉	C₂²	1	1	yes	characteristic
13	C₅₉₉⋊C₈	—	2	4	no	—
14	C₂³×C₅₉₉	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄×C₅₉₉	—	2	2	no	—
16	C₂×C₄×C₅₉₉	—	2	2	no	—
17	C₂×C₄×C₅₉₉	C₈	1	4	yes	complemented
18	C₂×C₄×C₅₉₉	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₅₉₉	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₅₉₉	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×C₅₉₉	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×C₅₉₉	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×C₅₉₉	C₂³	1	1	yes	characteristic
24	C₂²×C₅₉₉	—	2	1	no	—
25	C₂²×C₅₉₉	—	2	1	no	—
26	C₂²×C₅₉₉	C₂×C₈	1	2	yes	—
27	C₂²×C₅₉₉	C₂²×C₄	1	1	yes	characteristic
28	C₄×C₅₉₉	—	2	1	no	—
29	C₄×C₅₉₉	—	2	1	no	—
30	C₄×C₅₉₉	C₂×C₈	1	2	yes	—
31	C₄×C₅₉₉	C₂×C₈	1	2	yes	—
32	C₄×C₅₉₉	M₁₆	1	2	yes	—
33	C₄×C₅₉₉	#41	1	2	yes	—
34	C₂×C₅₉₉	—	2	2	no	—
35	C₂×C₅₉₉	#117	1	1	yes	characteristic
36	C₂×C₅₉₉	C₂²×C₈	1	1	yes	characteristic
37	C₂×C₅₉₉	C₂×M₁₆	1	1	yes	characteristic
38	C₅₉₉	#431	1	1	yes	characteristic, complemented, Sylow
39	#431	—	599	1	no	maximal, Sylow
40	C₂×C₄²	D₅₉₉	1	1	yes	characteristic
41	C₄⋊C₈	—	599	2	no	—
42	C₂²⋊C₈	—	599	2	no	—
43	C₄×C₈	—	599	2	no	—
44	C₂²×C₄	Dic₅₉₉	1	1	yes	characteristic
45	C₂²×C₄	Dic₅₉₉	1	1	yes	characteristic
46	C₂²×C₄	C₂×D₅₉₉	1	1	yes	characteristic
47	C₂×C₈	—	599	4	no	—
48	C₄²	Dic₅₉₉	1	1	yes	characteristic
49	C₄²	Dic₅₉₉	1	1	yes	characteristic
50	C₄²	C₂×D₅₉₉	1	1	yes	characteristic
51	C₄²	C₂×D₅₉₉	1	1	yes	characteristic
52	C₂³	C₂×Dic₅₉₉	1	1	yes	characteristic
53	C₂×C₄	—	2	2	no	—
54	C₂×C₄	—	2	2	no	—
55	C₂×C₄	C₅₉₉⋊C₈	1	4	yes	complemented
56	C₂×C₄	C₂×Dic₅₉₉	1	1	yes	characteristic
57	C₂×C₄	C₂×Dic₅₉₉	1	1	yes	characteristic
58	C₂×C₄	C₂×Dic₅₉₉	1	1	yes	characteristic
59	C₂×C₄	C₂×Dic₅₉₉	1	1	yes	characteristic
60	C₂×C₄	C₂×Dic₅₉₉	1	1	yes	characteristic
61	C₂×C₄	C₂²×D₅₉₉	1	1	yes	characteristic
62	C₈	—	1198	4	no	—
63	C₂²	—	2	1	no	—
64	C₂²	—	2	1	no	—
65	C₂²	C₂×C₅₉₉⋊C₈	1	2	yes	—
66	C₂²	C₂²×Dic₅₉₉	1	1	yes	characteristic
67	C₄	—	2	1	no	—
68	C₄	—	2	1	no	—
69	C₄	C₂×C₅₉₉⋊C₈	1	2	yes	—
70	C₄	C₂×C₅₉₉⋊C₈	1	2	yes	—
71	C₄	#23751062	1	2	yes	—
72	C₄	#23751078	1	2	yes	—
73	C₂	—	2	2	no	—
74	C₂	#34062674	1	1	yes	characteristic
75	C₂	C₂²×C₅₉₉⋊C₈	1	1	yes	characteristic
76	C₂	C₂×#23751062	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₂×D₅₉₉
commutator	C₂×C₅₉₉	C₂²×C₈
Frattini	C₂×C₄	C₂²×D₅₉₉
Fitting	C₂×C₄²×C₅₉₉	C₂
socle	C₂²×C₅₉₉	C₂²×C₄
prime radical	C₂³×C₅₉₉	C₂×C₄
cosocle	C₂×C₄×C₅₉₉	C₂³
hypercenter	C₂×C₄²	D₅₉₉
hypocenter	C₅₉₉	#431
LCS	G ▷ C₂×C₅₉₉ ▷ C₅₉₉
UCS	C₁ ◁ C₄² ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
1. C₄⋊(C₅₉₉⋊C₈)	C₂	semidirect
3. C₄×C₅₉₉⋊C₈	C₂	semidirect
17. C₂×C₄×C₅₉₉	C₈	semidirect
38. C₅₉₉	#431	semidirect
55. C₂×C₄	C₅₉₉⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊(C₅₉₉⋊C₈)	2
C₂²⋊(C₅₉₉⋊C₈)	2
C₄×C₅₉₉⋊C₈	2
C₂×C₄²×C₅₉₉	2
#431	599