Subgroups — #44542909

Order 31936.

58 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30630373	C₂	1	1	yes	maximal, characteristic
2	#30630283	C₂	1	1	yes	maximal, characteristic, complemented
3	M₃₂×C₄₉₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#22547811	—	2	1	no	—
5	Dic₁₉₉₆	C₄	1	1	yes	characteristic, complemented
6	#22547800	—	2	1	no	—
7	D₃₉₉₂	C₄	1	1	yes	characteristic, complemented
8	C₄⋊Dic₄₉₉	—	2	1	no	—
9	C₂×C₈×C₄₉₉	C₂²	1	1	yes	characteristic
10	C₁₆×C₄₉₉	—	2	1	no	—
11	Dic₉₉₈	—	2	1	no	—
12	D₁₉₉₆	—	2	1	no	—
13	C₂²⋊D₄₉₉	—	4	1	no	—
14	C₂×Dic₄₉₉	—	4	1	no	—
15	C₄×D₄₉₉	—	4	1	no	—
16	C₂×C₄×C₄₉₉	D₄	1	1	yes	characteristic
17	C₈×C₄₉₉	C₂×C₄	1	1	yes	characteristic
18	C₈×C₄₉₉	D₄	1	1	yes	characteristic
19	C₂²×C₄₉₉	D₈	1	1	yes	characteristic
20	C₂×D₄₉₉	—	4	1	no	—
21	C₄×C₄₉₉	C₂²⋊C₄	1	1	yes	characteristic
22	C₄×C₄₉₉	SD₁₆	1	1	yes	characteristic
23	Dic₄₉₉	—	4	1	no	—
24	Dic₄₉₉	—	8	1	no	—
25	C₂×C₄₉₉	—	2	1	no	—
26	C₂×C₄₉₉	#102	1	1	yes	characteristic
27	D₄₉₉	—	8	1	no	—
28	C₄₉₉	#360	1	1	yes	characteristic, complemented, Sylow
29	#360	—	499	1	no	maximal, Sylow
30	#135	—	499	1	no	—
31	M₃₂	D₄₉₉	1	1	yes	characteristic, complemented
32	#106	—	499	1	no	—
33	#41	—	998	1	no	—
34	Q₁₆	—	499	1	no	—
35	SD₁₆	—	998	1	no	—
36	D₈	—	499	1	no	—
37	C₂×C₈	C₂×D₄₉₉	1	1	yes	characteristic
38	C₄⋊C₄	—	998	1	no	—
39	C₁₆	—	2	1	no	—
40	Q₈	—	998	1	no	—
41	D₄	—	1996	1	no	—
42	D₄	—	998	1	no	—
43	C₂×C₄	—	1996	1	no	—
44	C₂×C₄	—	1996	1	no	—
45	C₂×C₄	D₁₉₉₆	1	1	yes	characteristic
46	C₈	C₄×D₄₉₉	1	1	yes	characteristic
47	C₈	C₂²⋊D₄₉₉	1	1	yes	characteristic
48	C₂²	—	1996	1	no	—
49	C₂²	D₃₉₉₂	1	1	yes	characteristic
50	C₄	—	3992	1	no	—
51	C₄	—	1996	1	no	—
52	C₄	(C₂×D₄₉₉)⋊C₄	1	1	yes	characteristic
53	C₄	#22547800	1	1	yes	characteristic
54	C₂	—	2	1	no	—
55	C₂	—	3992	1	no	—
56	C₂	#30630273	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#30630273
commutator	C₈×C₄₉₉	C₂×C₄
Frattini	C₂×C₈	C₂×D₄₉₉
Fitting	M₃₂×C₄₉₉	C₂
socle	C₂×C₄₉₉	#102
prime radical	#30630373	C₂
cosocle	C₂×C₈×C₄₉₉	C₂²
hypercenter	M₃₂	D₄₉₉
hypocenter	C₄₉₉	#360
LCS	G ▷ C₈×C₄₉₉ ▷ C₄×C₄₉₉ ▷ C₂×C₄₉₉ ▷ C₄₉₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ M₃₂

Complements

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

N	H ≅ G/N	type
2. #30630283	C₂	semidirect
3. M₃₂×C₄₉₉	C₂	semidirect
5. Dic₁₉₉₆	C₄	semidirect
7. D₃₉₉₂	C₄	semidirect
28. C₄₉₉	#360	semidirect
31. M₃₂	D₄₉₉	semidirect

Maximal subgroups

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

subgroup	index
#30630373	2
#30630283	2
M₃₂×C₄₉₉	2
#360	499