Subgroups — #44543430

Order 31936.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30630331	C₂	1	1	yes	maximal, characteristic
2	C₄⋊C₄×D₄₉₉	C₂	1	1	yes	maximal, characteristic
3	#30630282	C₂	1	1	yes	maximal, characteristic, complemented
4	#30630276	C₂	1	1	yes	maximal, characteristic, complemented
5	#30630274	C₂	1	1	yes	maximal, characteristic, complemented
6	(C₂×D₄₉₉)⋊C₈	C₂	1	1	yes	maximal, characteristic
7	#106×C₄₉₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Dic₉₉₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×D₄₉₉	—	2	1	no	—
10	C₂×C₄×D₄₉₉	C₂²	1	1	yes	characteristic
11	C₂×C₄₉₉⋊C₈	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₄₉₉	—	2	1	no	—
13	C₄⋊Dic₄₉₉	C₂²	1	1	yes	characteristic
14	Dic₄₉₉⋊C₄	—	2	1	no	—
15	(C₂×D₄₉₉)⋊C₄	—	2	1	no	—
16	C₂×C₈×C₄₉₉	C₂²	1	1	yes	characteristic
17	C₄⋊C₄×C₄₉₉	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×C₄₉₉	C₂²	1	1	yes	characteristic
19	C₂²×D₄₉₉	D₄	1	1	yes	characteristic
20	Dic₉₉₈	—	2	1	no	—
21	Dic₉₉₈	—	2	1	no	—
22	C₂×Dic₄₉₉	—	2	1	no	—
23	C₂×Dic₄₉₉	—	2	1	no	—
24	C₂×Dic₄₉₉	D₄	1	1	yes	characteristic
25	C₄×D₄₉₉	—	2	1	no	—
26	C₄×D₄₉₉	—	2	1	no	—
27	C₄×D₄₉₉	—	4	2	no	—
28	C₄₉₉⋊C₈	—	2	1	no	—
29	C₂×C₄×C₄₉₉	—	2	1	no	—
30	C₂×C₄×C₄₉₉	—	2	1	no	—
31	C₂×C₄×C₄₉₉	C₂³	1	1	yes	characteristic
32	C₈×C₄₉₉	—	2	1	no	—
33	C₂²×C₄₉₉	C₂×D₄	1	1	yes	characteristic
34	C₂×D₄₉₉	—	2	1	no	—
35	C₂×D₄₉₉	—	2	1	no	—
36	C₂×D₄₉₉	SD₁₆	1	2	yes	—
37	C₄×C₄₉₉	—	4	1	no	—
38	C₄×C₄₉₉	—	4	1	no	—
39	C₄×C₄₉₉	#41	1	1	yes	characteristic
40	C₄×C₄₉₉	#41	1	1	yes	characteristic
41	Dic₄₉₉	—	2	1	no	—
42	Dic₄₉₉	—	4	1	no	—
43	Dic₄₉₉	—	4	1	no	—
44	C₂×C₄₉₉	#123	1	1	yes	characteristic
45	C₂×C₄₉₉	C₂×SD₁₆	1	1	yes	characteristic
46	C₂×C₄₉₉	#137	1	1	yes	characteristic
47	D₄₉₉	—	2	2	no	—
48	C₄₉₉	#483	1	1	yes	characteristic, complemented, Sylow
49	#483	—	499	1	no	maximal, Sylow
50	C₂²⋊Q₈	—	499	1	no	—
51	C₂×C₄⋊C₄	—	499	1	no	—
52	#106	—	499	1	no	—
53	#106	D₄₉₉	1	1	yes	characteristic, complemented
54	#103	—	499	1	no	—
55	#103	—	499	1	no	—
56	C₂²⋊C₈	—	499	1	no	—
57	C₂×Q₈	—	499	1	no	—
58	C₂²×C₄	—	998	1	no	—
59	C₂²×C₄	—	499	1	no	—
60	C₂×C₈	—	499	1	no	—
61	C₂×C₈	C₂×D₄₉₉	1	1	yes	characteristic
62	C₄⋊C₄	—	998	1	no	—
63	C₄⋊C₄	—	998	1	no	—
64	C₄⋊C₄	—	499	1	no	—
65	C₄⋊C₄	C₂×D₄₉₉	1	1	yes	characteristic
66	C₄⋊C₄	C₂×D₄₉₉	1	1	yes	characteristic
67	C₂²⋊C₄	—	998	1	no	—
68	C₂³	—	499	1	no	—
69	Q₈	—	998	1	no	—
70	Q₈	—	998	1	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	2	1	no	—
73	C₂×C₄	—	1996	2	no	—
74	C₂×C₄	—	998	1	no	—
75	C₂×C₄	—	998	1	no	—
76	C₂×C₄	—	998	1	no	—
77	C₂×C₄	—	998	1	no	—
78	C₂×C₄	—	499	1	no	—
79	C₂×C₄	C₂²×D₄₉₉	1	1	yes	characteristic
80	C₈	—	2	1	no	—
81	C₈	—	998	1	no	—
82	C₂²	—	998	1	no	—
83	C₂²	—	998	1	no	—
84	C₂²	—	499	2	no	—
85	C₂²	D₄×D₄₉₉	1	1	yes	characteristic
86	C₄	—	4	1	no	—
87	C₄	—	4	1	no	—
88	C₄	—	1996	1	no	—
89	C₄	—	1996	1	no	—
90	C₄	—	998	1	no	—
91	C₄	#22547811	1	1	yes	characteristic
92	C₄	#22547812	1	1	yes	characteristic
93	C₂	—	998	2	no	—
94	C₂	#30630336	1	1	yes	characteristic
95	C₂	SD₁₆×D₄₉₉	1	1	yes	characteristic
96	C₂	#30630380	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₄₉₉
commutator, cosocle	C₂×C₄×C₄₉₉	C₂³
Frattini	C₂×C₄	C₂²×D₄₉₉
Fitting	#106×C₄₉₉	C₂
socle	C₂²×C₄₉₉	C₂×D₄
prime radical	C₂²×D₄₉₉	D₄
hypercenter	#106	D₄₉₉
hypocenter	C₄₉₉	#483
LCS	G ▷ C₂×C₄×C₄₉₉ ▷ C₂×C₄₉₉ ▷ C₄₉₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #106

Complements

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

N	H ≅ G/N	type
3. #30630282	C₂	semidirect
4. #30630276	C₂	semidirect
5. #30630274	C₂	semidirect
7. #106×C₄₉₉	C₂	semidirect
48. C₄₉₉	#483	semidirect
53. #106	D₄₉₉	semidirect

Maximal subgroups

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

subgroup	index
#30630331	2
C₄⋊C₄×D₄₉₉	2
#30630282	2
#30630276	2
#30630274	2
(C₂×D₄₉₉)⋊C₈	2
#106×C₄₉₉	2
#483	499