Subgroups — #44543008

Order 31936.

322 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₁₉₉₆	C₂	1	3	yes	maximal, complemented
2	C₂×(C₂×D₄₉₉)⋊C₄	C₂	1	3	yes	maximal, complemented
3	#95×C₄₉₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×D₄₉₉	C₂²	1	3	yes	—
5	C₂×D₁₉₉₆	—	2	6	no	—
6	C₂×D₁₉₉₆	—	2	12	no	—
7	C₂²×Dic₄₉₉	C₂²	1	1	yes	characteristic
8	(C₂×D₄₉₉)⋊C₄	—	2	6	no	—
9	C₂²×C₄×C₄₉₉	C₂²	1	3	yes	—
10	C₂²×D₄₉₉	—	2	3	no	—
11	C₂²×D₄₉₉	—	2	6	no	—
12	C₂²×D₄₉₉	—	2	6	no	—
13	C₂²×D₄₉₉	—	2	3	no	—
14	C₂²×D₄₉₉	D₄	1	6	yes	complemented
15	D₁₉₉₆	—	2	24	no	—
16	C₂×Dic₄₉₉	—	2	3	no	—
17	C₂³×C₄₉₉	C₂³	1	1	yes	characteristic
18	C₂×C₄×C₄₉₉	—	2	6	no	—
19	C₂×C₄×C₄₉₉	D₄	1	6	yes	complemented
20	C₂²×C₄₉₉	C₂×D₄	1	3	yes	—
21	C₂²×C₄₉₉	C₂×D₄	1	3	yes	—
22	C₂²×C₄₉₉	#41	1	1	yes	characteristic
23	C₂×D₄₉₉	—	2	12	no	—
24	C₂×D₄₉₉	—	2	6	no	—
25	C₂×D₄₉₉	—	4	3	no	—
26	C₂×D₄₉₉	—	4	6	no	—
27	C₂×D₄₉₉	—	4	3	no	—
28	C₄×C₄₉₉	—	2	6	no	—
29	Dic₄₉₉	—	4	1	no	—
30	C₂×C₄₉₉	#120	1	3	yes	—
31	C₂×C₄₉₉	#121	1	3	yes	—
32	C₂×C₄₉₉	#127	1	1	yes	characteristic
33	D₄₉₉	—	4	6	no	—
34	C₄₉₉	#392	1	1	yes	characteristic, complemented, Sylow
35	#392	—	499	1	no	maximal, Sylow
36	C₂²×D₄	—	499	3	no	—
37	C₂×C₂²⋊C₄	—	499	3	no	—
38	#95	D₄₉₉	1	1	yes	characteristic, complemented
39	C₂⁴	—	499	3	no	—
40	C₂×D₄	—	998	12	no	—
41	C₂×D₄	—	998	6	no	—
42	C₂²×C₄	—	499	1	no	—
43	C₂²×C₄	C₂×D₄₉₉	1	3	yes	—
44	C₂²⋊C₄	—	998	6	no	—
45	C₂³	—	998	6	no	—
46	C₂³	—	998	3	no	—
47	C₂³	—	998	6	no	—
48	C₂³	—	998	3	no	—
49	C₂³	—	499	6	no	—
50	C₂³	C₂²×D₄₉₉	1	1	yes	characteristic
51	D₄	—	998	24	no	—
52	C₂×C₄	—	2	6	no	—
53	C₂×C₄	—	998	3	no	—
54	C₂×C₄	D₁₉₉₆	1	6	yes	complemented
55	C₂²	—	1996	3	no	—
56	C₂²	—	1996	6	no	—
57	C₂²	—	1996	3	no	—
58	C₂²	—	998	12	no	—
59	C₂²	—	998	6	no	—
60	C₂²	D₄×D₄₉₉	1	3	yes	—
61	C₂²	C₂×D₁₉₉₆	1	3	yes	—
62	C₂²	#22547812	1	1	yes	characteristic
63	C₄	—	2	6	no	—
64	C₄	—	1996	1	no	—
65	C₂	—	1996	6	no	—
66	C₂	#30630321	1	3	yes	—
67	C₂	#30630326	1	3	yes	—
68	C₂	#30630348	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₄₉₉
commutator, socle, cosocle	C₂³×C₄₉₉	C₂³
Fitting	#95×C₄₉₉	C₂
hypercenter	#95	D₄₉₉
hypocenter	C₄₉₉	#392
LCS	G ▷ C₂³×C₄₉₉ ▷ C₄₉₉
UCS	C₁ ◁ C₂³ ◁ #95

Complements

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

N	H ≅ G/N	type
1. C₂²×D₁₉₉₆	C₂	semidirect
2. C₂×(C₂×D₄₉₉)⋊C₄	C₂	semidirect
3. #95×C₄₉₉	C₂	semidirect
14. C₂²×D₄₉₉	D₄	semidirect
19. C₂×C₄×C₄₉₉	D₄	semidirect
34. C₄₉₉	#392	semidirect
38. #95	D₄₉₉	semidirect
54. C₂×C₄	D₁₉₉₆	semidirect

Maximal subgroups

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

subgroup	index
C₂²×D₁₉₉₆	2
C₂×(C₂×D₄₉₉)⋊C₄	2
#95×C₄₉₉	2
#392	499