Subgroups — #50306695

Order 38336.

268 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#34062761	C₂	1	2	yes	maximal, complemented
2	#34062743	C₂	1	6	yes	maximal, complemented
3	#34062737	C₂	1	3	yes	maximal, complemented
4	C₂×D₄₇₉₂	C₂	1	3	yes	maximal, complemented
5	#131×C₅₉₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#23751079	—	2	2	no	—
7	#23751078	C₂²	1	6	yes	complemented
8	C₂×D₂₃₉₆	C₂²	1	6	yes	complemented
9	D₄×D₅₉₉	—	2	6	no	—
10	Dic₂₃₉₆	C₂²	1	1	yes	characteristic, complemented
11	#23751067	C₂²	1	6	yes	complemented
12	D₄₇₉₂	C₂²	1	3	yes	complemented
13	D₄₇₉₂	C₂²	1	6	yes	complemented
14	#41×C₅₉₉	C₂²	1	1	yes	characteristic
15	M₁₆×C₅₉₉	C₂²	1	3	yes	complemented
16	C₂×C₈×C₅₉₉	C₂²	1	3	yes	complemented
17	C₂²×D₅₉₉	—	2	6	no	—
18	Dic₁₁₉₈	C₂³	1	2	yes	—
19	D₂₃₉₆	—	2	6	no	—
20	D₂₃₉₆	C₂³	1	6	yes	—
21	C₂²⋊D₅₉₉	—	2	6	no	—
22	C₄×D₅₉₉	—	2	6	no	—
23	Q₈×C₅₉₉	D₄	1	1	yes	characteristic
24	D₄×C₅₉₉	D₄	1	3	yes	—
25	C₂×C₄×C₅₉₉	C₂³	1	3	yes	—
26	C₈×C₅₉₉	C₂³	1	1	yes	characteristic
27	C₈×C₅₉₉	C₂³	1	3	yes	—
28	C₂²×C₅₉₉	C₂×D₄	1	3	yes	—
29	C₂×D₅₉₉	—	2	6	no	—
30	C₂×D₅₉₉	—	8	6	no	—
31	C₄×C₅₉₉	C₂×D₄	1	3	yes	—
32	C₄×C₅₉₉	C₂⁴	1	1	yes	characteristic
33	Dic₅₉₉	—	2	2	no	—
34	C₂×C₅₉₉	—	2	3	no	—
35	C₂×C₅₉₉	C₂²×D₄	1	1	yes	characteristic
36	D₅₉₉	—	4	6	no	—
37	C₅₉₉	#576	1	1	yes	characteristic, complemented, Sylow
38	#576	—	599	1	no	maximal, Sylow
39	H(5,𝔽₂)	—	599	2	no	—
40	AGL(1,ℤ/8)	—	599	6	no	—
41	#135	—	599	3	no	—
42	C₂×D₈	—	599	3	no	—
43	#131	D₅₉₉	1	1	yes	characteristic, complemented
44	#41	—	599	6	no	—
45	#41	—	1198	2	no	—
46	#41	C₂×D₅₉₉	1	1	yes	characteristic
47	C₂×D₄	—	599	6	no	—
48	C₂×D₄	—	1198	6	no	—
49	Q₁₆	—	599	1	no	—
50	SD₁₆	—	599	6	no	—
51	D₈	—	599	3	no	—
52	D₈	—	599	6	no	—
53	M₁₆	C₂×D₅₉₉	1	3	yes	complemented
54	C₂×C₈	C₂×D₅₉₉	1	3	yes	complemented
55	C₂³	—	1198	6	no	—
56	Q₈	—	599	2	no	—
57	Q₈	D₂₃₉₆	1	1	yes	characteristic
58	D₄	—	599	6	no	—
59	D₄	—	1198	6	no	—
60	D₄	—	1198	6	no	—
61	D₄	D₂₃₉₆	1	3	yes	—
62	C₂×C₄	—	1198	6	no	—
63	C₂×C₄	C₂²×D₅₉₉	1	3	yes	—
64	C₈	C₂²×D₅₉₉	1	1	yes	characteristic
65	C₈	C₂²×D₅₉₉	1	3	yes	—
66	C₂²	—	1198	6	no	—
67	C₂²	—	4792	6	no	—
68	C₂²	C₂×D₂₃₉₆	1	3	yes	—
69	C₄	—	1198	2	no	—
70	C₄	C₂×D₂₃₉₆	1	3	yes	—
71	C₄	C₂³×D₅₉₉	1	1	yes	characteristic
72	C₂	—	2	3	no	—
73	C₂	—	2396	6	no	—
74	C₂	C₂²×D₂₃₉₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²×D₂₃₉₆
commutator, cosocle	C₄×C₅₉₉	C₂⁴
Frattini	C₄	C₂³×D₅₉₉
Fitting	#131×C₅₉₉	C₂
socle	C₂×C₅₉₉	C₂²×D₄
hypercenter	#131	D₅₉₉
hypocenter	C₅₉₉	#576
LCS	G ▷ C₄×C₅₉₉ ▷ C₂×C₅₉₉ ▷ C₅₉₉
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131

Complements

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

N	H ≅ G/N	type
1. #34062761	C₂	semidirect
2. #34062743	C₂	semidirect
3. #34062737	C₂	semidirect
4. C₂×D₄₇₉₂	C₂	semidirect
5. #131×C₅₉₉	C₂	semidirect
7. #23751078	C₂²	semidirect
8. C₂×D₂₃₉₆	C₂²	semidirect
10. Dic₂₃₉₆	C₂²	semidirect
11. #23751067	C₂²	semidirect
12. D₄₇₉₂	C₂²	semidirect
13. D₄₇₉₂	C₂²	semidirect
15. M₁₆×C₅₉₉	C₂²	semidirect
16. C₂×C₈×C₅₉₉	C₂²	semidirect
37. C₅₉₉	#576	semidirect
43. #131	D₅₉₉	semidirect
53. M₁₆	C₂×D₅₉₉	semidirect
54. C₂×C₈	C₂×D₅₉₉	semidirect

Maximal subgroups

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

subgroup	index
#34062761	2
#34062743	2
#34062737	2
C₂×D₄₇₉₂	2
#131×C₅₉₉	2
#576	599