Subgroups — #50306707

Order 38336.

248 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#34062762	C₂	1	2	yes	maximal, complemented
2	#34062748	C₂	1	6	yes	maximal, complemented
3	#34062737	C₂	1	3	yes	maximal, complemented
4	C₂×Dic₂₃₉₆	C₂	1	3	yes	maximal, complemented
5	#131×C₅₉₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#23751078	C₂²	1	6	yes	—
7	#23751077	—	2	6	no	—
8	C₂×Dic₁₁₉₈	C₂²	1	6	yes	—
9	Q₈×D₅₉₉	—	2	2	no	—
10	Dic₂₃₉₆	C₂²	1	3	yes	—
11	Dic₂₃₉₆	C₂²	1	6	yes	—
12	#23751067	C₂²	1	6	yes	—
13	D₄₇₉₂	C₂²	1	1	yes	characteristic
14	#41×C₅₉₉	C₂²	1	1	yes	characteristic
15	M₁₆×C₅₉₉	C₂²	1	3	yes	—
16	C₂×C₈×C₅₉₉	C₂²	1	3	yes	—
17	Dic₁₁₉₈	—	2	6	no	—
18	Dic₁₁₉₈	C₂³	1	6	yes	—
19	D₂₃₉₆	C₂³	1	2	yes	—
20	C₂²⋊D₅₉₉	—	2	6	no	—
21	C₂×Dic₅₉₉	—	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	2	no	—
30	C₄×C₅₉₉	C₂×D₄	1	3	yes	—
31	C₄×C₅₉₉	C₂⁴	1	1	yes	characteristic
32	Dic₅₉₉	—	2	6	no	—
33	C₂×C₅₉₉	—	2	3	no	—
34	C₂×C₅₉₉	C₂²×D₄	1	1	yes	characteristic
35	D₅₉₉	—	4	2	no	—
36	C₅₉₉	#578	1	1	yes	characteristic, complemented, Sylow
37	#578	—	599	1	no	maximal, Sylow
38	2₋¹⁺⁴	—	599	2	no	—
39	#137	—	599	6	no	—
40	#135	—	599	3	no	—
41	C₂×Q₁₆	—	599	3	no	—
42	#131	D₅₉₉	1	1	yes	characteristic, complemented
43	#41	—	599	6	no	—
44	#41	—	1198	6	no	—
45	#41	C₂×D₅₉₉	1	1	yes	characteristic
46	C₂×Q₈	—	599	6	no	—
47	C₂×Q₈	—	1198	2	no	—
48	Q₁₆	—	599	3	no	—
49	Q₁₆	—	599	6	no	—
50	SD₁₆	—	599	6	no	—
51	D₈	—	599	1	no	—
52	M₁₆	C₂×D₅₉₉	1	3	yes	—
53	C₂×C₈	C₂×D₅₉₉	1	3	yes	—
54	Q₈	—	599	6	no	—
55	Q₈	—	1198	6	no	—
56	Q₈	D₂₃₉₆	1	1	yes	characteristic
57	D₄	—	599	2	no	—
58	D₄	—	1198	6	no	—
59	D₄	D₂₃₉₆	1	3	yes	—
60	C₂×C₄	—	1198	6	no	—
61	C₂×C₄	—	1198	6	no	—
62	C₂×C₄	C₂²×D₅₉₉	1	3	yes	—
63	C₈	C₂²×D₅₉₉	1	1	yes	characteristic
64	C₈	C₂²×D₅₉₉	1	3	yes	—
65	C₂²	—	1198	2	no	—
66	C₂²	C₂×D₂₃₉₆	1	3	yes	—
67	C₄	—	1198	6	no	—
68	C₄	C₂×D₂₃₉₆	1	3	yes	—
69	C₄	C₂³×D₅₉₉	1	1	yes	characteristic
70	C₂	—	2	3	no	—
71	C₂	—	2396	2	no	—
72	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	#578
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. #34062762	C₂	semidirect
2. #34062748	C₂	semidirect
3. #34062737	C₂	semidirect
4. C₂×Dic₂₃₉₆	C₂	semidirect
5. #131×C₅₉₉	C₂	semidirect
36. C₅₉₉	#578	semidirect
42. #131	D₅₉₉	semidirect

Maximal subgroups

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

subgroup	index
#34062762	2
#34062748	2
#34062737	2
C₂×Dic₂₃₉₆	2
#131×C₅₉₉	2
#578	599