Subgroups — #34062748

Order 19168.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23751078	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₁₁₉₈	C₂	1	1	yes	maximal, characteristic, complemented
3	Dic₂₃₉₆	C₂	1	2	yes	maximal, complemented
4	#23751067	C₂	1	2	yes	maximal, complemented
5	M₁₆×C₅₉₉	C₂	1	1	yes	maximal, characteristic, complemented
6	Dic₁₁₉₈	—	2	1	no	—
7	Dic₁₁₉₈	C₂²	1	2	yes	—
8	Dic₁₁₉₈	C₂²	1	1	yes	characteristic
9	D₂₃₉₆	C₂²	1	1	yes	characteristic
10	C₂²⋊D₅₉₉	—	2	1	no	—
11	C₂×Dic₅₉₉	—	2	1	no	—
12	C₄×D₅₉₉	—	2	1	no	—
13	C₂×C₄×C₅₉₉	C₂²	1	1	yes	characteristic
14	C₈×C₅₉₉	C₂²	1	2	yes	—
15	C₂²×C₅₉₉	D₄	1	1	yes	characteristic
16	C₂×D₅₉₉	—	2	1	no	—
17	C₄×C₅₉₉	D₄	1	1	yes	characteristic
18	C₄×C₅₉₉	C₂³	1	1	yes	characteristic
19	Dic₅₉₉	—	2	1	no	—
20	Dic₅₉₉	—	2	2	no	—
21	C₂×C₅₉₉	—	2	1	no	—
22	C₂×C₅₉₉	C₂×D₄	1	1	yes	characteristic
23	D₅₉₉	—	4	1	no	—
24	C₅₉₉	#137	1	1	yes	characteristic, complemented, Sylow
25	#137	—	599	1	no	maximal, Sylow
26	#41	—	599	1	no	—
27	C₂×Q₈	—	599	1	no	—
28	Q₁₆	—	599	2	no	—
29	SD₁₆	—	599	2	no	—
30	M₁₆	D₅₉₉	1	1	yes	characteristic, complemented
31	Q₈	—	599	2	no	—
32	Q₈	—	599	1	no	—
33	Q₈	—	1198	1	no	—
34	D₄	—	599	1	no	—
35	D₄	—	1198	1	no	—
36	C₂×C₄	—	1198	1	no	—
37	C₂×C₄	—	1198	1	no	—
38	C₂×C₄	C₂×D₅₉₉	1	1	yes	characteristic
39	C₈	C₂×D₅₉₉	1	2	yes	—
40	C₂²	—	1198	1	no	—
41	C₂²	D₂₃₉₆	1	1	yes	characteristic
42	C₄	—	1198	1	no	—
43	C₄	—	1198	2	no	—
44	C₄	D₂₃₉₆	1	1	yes	characteristic
45	C₄	C₂²×D₅₉₉	1	1	yes	characteristic
46	C₂	—	2	1	no	—
47	C₂	—	2396	1	no	—
48	C₂	C₂×D₂₃₉₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₂₃₉₆
commutator, cosocle	C₄×C₅₉₉	C₂³
Frattini	C₄	C₂²×D₅₉₉
Fitting	M₁₆×C₅₉₉	C₂
socle	C₂×C₅₉₉	C₂×D₄
prime radical	#23751078	C₂
hypercenter	M₁₆	D₅₉₉
hypocenter	C₅₉₉	#137
LCS	G ▷ C₄×C₅₉₉ ▷ C₂×C₅₉₉ ▷ C₅₉₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ M₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×Dic₁₁₉₈	C₂	semidirect
3. Dic₂₃₉₆	C₂	semidirect
4. #23751067	C₂	semidirect
5. M₁₆×C₅₉₉	C₂	semidirect
24. C₅₉₉	#137	semidirect
30. M₁₆	D₅₉₉	semidirect

Maximal subgroups

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

subgroup	index
#23751078	2
C₂×Dic₁₁₉₈	2
Dic₂₃₉₆	2
#23751067	2
M₁₆×C₅₉₉	2
#137	599