Subgroups — #34062746

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	#23751068	C₂	1	2	yes	maximal, complemented
3	#23751065	C₂	1	2	yes	maximal, complemented
4	#23751062	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×Q₈×C₅₉₉	C₂	1	1	yes	maximal, characteristic, complemented
6	Dic₁₁₉₈	C₂²	1	1	yes	characteristic
7	D₂₃₉₆	C₂²	1	1	yes	characteristic
8	C₂²⋊D₅₉₉	—	2	1	no	—
9	C₄×D₅₉₉	—	2	1	no	—
10	C₅₉₉⋊C₈	C₂²	1	2	yes	—
11	Q₈×C₅₉₉	—	2	1	no	—
12	Q₈×C₅₉₉	C₂²	1	2	yes	—
13	C₂×C₄×C₅₉₉	—	2	1	no	—
14	C₂×C₄×C₅₉₉	C₂²	1	1	yes	characteristic
15	C₂²×C₅₉₉	D₄	1	1	yes	characteristic
16	C₂×D₅₉₉	—	2	1	no	—
17	C₄×C₅₉₉	—	2	2	no	—
18	C₄×C₅₉₉	D₄	1	1	yes	characteristic
19	C₄×C₅₉₉	C₂³	1	1	yes	characteristic
20	Dic₅₉₉	—	2	1	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	D₅₉₉	1	1	yes	characteristic, complemented
28	Q₁₆	—	599	2	no	—
29	SD₁₆	—	599	2	no	—
30	M₁₆	—	599	1	no	—
31	Q₈	—	2	1	no	—
32	Q₈	—	599	1	no	—
33	Q₈	C₂×D₅₉₉	1	2	yes	—
34	D₄	—	599	1	no	—
35	D₄	—	1198	1	no	—
36	C₂×C₄	—	2	1	no	—
37	C₂×C₄	—	1198	1	no	—
38	C₂×C₄	C₂×D₅₉₉	1	1	yes	characteristic
39	C₈	—	599	2	no	—
40	C₂²	—	1198	1	no	—
41	C₂²	C₂²⋊D₅₉₉	1	1	yes	characteristic
42	C₄	—	2	2	no	—
43	C₄	—	1198	1	no	—
44	C₄	C₂²⋊D₅₉₉	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×C₂²⋊D₅₉₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×C₂²⋊D₅₉₉
commutator, cosocle	C₄×C₅₉₉	C₂³
Frattini	C₄	C₂²×D₅₉₉
Fitting	C₂×Q₈×C₅₉₉	C₂
socle	C₂×C₅₉₉	C₂×D₄
prime radical	#23751078	C₂
hypercenter	C₂×Q₈	D₅₉₉
hypocenter	C₅₉₉	#137
LCS	G ▷ C₄×C₅₉₉ ▷ C₂×C₅₉₉ ▷ C₅₉₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
2. #23751068	C₂	semidirect
3. #23751065	C₂	semidirect
4. #23751062	C₂	semidirect
5. C₂×Q₈×C₅₉₉	C₂	semidirect
24. C₅₉₉	#137	semidirect
27. C₂×Q₈	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
#23751068	2
#23751065	2
#23751062	2
C₂×Q₈×C₅₉₉	2
#137	599