Subgroups — #50306258

Order 38336.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#34062707	C₂	1	1	yes	maximal, characteristic
2	#34062639	C₂	1	4	yes	maximal
3	#34062626	C₂	1	1	yes	maximal, characteristic
4	C₄⋊Q₈×C₅₉₉	C₂	1	1	yes	maximal, characteristic
5	C₂×C₅₉₉⋊C₈	C₂²	1	2	yes	—
6	C₄⋊Dic₅₉₉	C₂²	1	2	yes	—
7	Dic₅₉₉⋊C₄	—	2	2	no	—
8	C₂×Q₈×C₅₉₉	C₂²	1	2	yes	—
9	C₄⋊C₄×C₅₉₉	—	2	2	no	—
10	C₄²×C₅₉₉	C₂²	1	1	yes	characteristic
11	C₂×Dic₅₉₉	—	2	2	no	—
12	C₅₉₉⋊C₈	—	2	2	no	—
13	Q₈×C₅₉₉	—	2	4	no	—
14	C₂×C₄×C₅₉₉	—	2	2	no	—
15	C₂×C₄×C₅₉₉	D₄	1	2	yes	—
16	C₂×C₄×C₅₉₉	C₂³	1	1	yes	characteristic
17	C₂²×C₅₉₉	C₂×D₄	1	1	yes	characteristic
18	C₄×C₅₉₉	—	2	2	no	—
19	C₄×C₅₉₉	—	4	2	no	—
20	C₄×C₅₉₉	#41	1	2	yes	—
21	Dic₅₉₉	—	4	2	no	—
22	C₂×C₅₉₉	#124	1	1	yes	characteristic
23	C₂×C₅₉₉	#137	1	2	yes	—
24	C₅₉₉	#491	1	1	yes	characteristic, complemented, Sylow
25	#491	—	599	1	no	maximal, Sylow
26	C₄⋊Q₈	D₅₉₉	1	1	yes	characteristic
27	#125	—	599	1	no	—
28	#103	—	599	4	no	—
29	#97	—	599	1	no	—
30	C₂×Q₈	C₂×D₅₉₉	1	2	yes	—
31	C₂×C₈	—	599	2	no	—
32	C₄⋊C₄	—	2	2	no	—
33	C₄⋊C₄	—	599	2	no	—
34	C₄⋊C₄	—	1198	2	no	—
35	C₄²	C₂×D₅₉₉	1	1	yes	characteristic
36	Q₈	—	2	4	no	—
37	C₂×C₄	—	2	2	no	—
38	C₂×C₄	—	1198	2	no	—
39	C₂×C₄	C₂²⋊D₅₉₉	1	2	yes	—
40	C₂×C₄	C₂²×D₅₉₉	1	1	yes	characteristic
41	C₈	—	1198	2	no	—
42	C₂²	C₂×C₂²⋊D₅₉₉	1	1	yes	characteristic
43	C₄	—	2	2	no	—
44	C₄	—	4	2	no	—
45	C₄	—	2396	2	no	—
46	C₄	#23751077	1	2	yes	—
47	C₂	#34062702	1	1	yes	characteristic
48	C₂	#34062746	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₅₉₉
commutator, cosocle	C₂×C₄×C₅₉₉	C₂³
Frattini	C₂×C₄	C₂²×D₅₉₉
Fitting	C₄⋊Q₈×C₅₉₉	C₂
socle, prime radical	C₂²×C₅₉₉	C₂×D₄
hypercenter	C₄⋊Q₈	D₅₉₉
hypocenter	C₅₉₉	#491
LCS	G ▷ C₂×C₄×C₅₉₉ ▷ C₂×C₅₉₉ ▷ C₅₉₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
24. C₅₉₉	#491	semidirect

Maximal subgroups

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

subgroup	index
#34062707	2
#34062639	2
#34062626	2
C₄⋊Q₈×C₅₉₉	2
#491	599