Subgroups — #44543382

Order 31936.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30630350	C₂	1	1	yes	maximal, characteristic
2	#30630316	C₂	1	1	yes	maximal, characteristic
3	#30630282	C₂	1	1	yes	maximal, characteristic
4	Dic₄₉₉⋊C₈	C₂	1	1	yes	maximal, characteristic
5	#30630276	C₂	1	1	yes	maximal, characteristic
6	#30630274	C₂	1	1	yes	maximal, characteristic
7	#106×C₄₉₉	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₉₉₈	—	2	1	no	—
9	C₂×Dic₉₉₈	C₂²	1	1	yes	characteristic
10	C₂×C₄₉₉⋊C₈	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₄₉₉	C₂²	1	1	yes	characteristic
12	Dic₄₉₉⋊C₄	—	2	1	no	—
13	Dic₄₉₉⋊C₄	—	2	1	no	—
14	C₄×Dic₄₉₉	—	2	1	no	—
15	C₄×Dic₄₉₉	C₂²	1	1	yes	characteristic
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	Dic₉₉₈	—	2	1	no	—
20	Dic₉₉₈	—	4	2	no	—
21	Dic₉₉₈	Q₈	1	2	yes	complemented
22	C₂×Dic₄₉₉	—	2	1	no	—
23	C₂×Dic₄₉₉	—	2	1	no	—
24	C₂×Dic₄₉₉	D₄	1	1	yes	characteristic
25	C₂×Dic₄₉₉	D₄	1	1	yes	characteristic
26	C₄₉₉⋊C₈	—	2	1	no	—
27	C₂×C₄×C₄₉₉	—	2	1	no	—
28	C₂×C₄×C₄₉₉	—	2	1	no	—
29	C₂×C₄×C₄₉₉	C₂³	1	1	yes	characteristic
30	C₈×C₄₉₉	—	2	1	no	—
31	C₂²×C₄₉₉	C₂×D₄	1	1	yes	characteristic
32	C₄×C₄₉₉	—	4	1	no	—
33	C₄×C₄₉₉	—	4	1	no	—
34	C₄×C₄₉₉	C₂×Q₈	1	1	yes	characteristic
35	C₄×C₄₉₉	#41	1	1	yes	characteristic
36	Dic₄₉₉	—	2	2	no	—
37	Dic₄₉₉	—	2	1	no	—
38	Dic₄₉₉	—	4	1	no	—
39	Dic₄₉₉	SD₁₆	1	2	yes	—
40	C₂×C₄₉₉	C₂²⋊Q₈	1	1	yes	characteristic
41	C₂×C₄₉₉	C₂×SD₁₆	1	1	yes	characteristic
42	C₂×C₄₉₉	#137	1	1	yes	characteristic
43	C₄₉₉	#475	1	1	yes	characteristic, complemented, Sylow
44	#475	—	499	1	no	maximal, Sylow
45	C₄⋊Q₈	—	499	1	no	—
46	C₄×Q₈	—	499	1	no	—
47	#106	—	499	1	no	—
48	#106	D₄₉₉	1	1	yes	characteristic
49	C₄⋊C₈	—	499	1	no	—
50	#103	—	499	1	no	—
51	#103	—	499	1	no	—
52	C₂×Q₈	—	998	1	no	—
53	C₂×Q₈	—	499	1	no	—
54	C₂×C₈	—	499	1	no	—
55	C₂×C₈	C₂×D₄₉₉	1	1	yes	characteristic
56	C₄⋊C₄	—	998	1	no	—
57	C₄⋊C₄	—	998	1	no	—
58	C₄⋊C₄	—	499	1	no	—
59	C₄⋊C₄	C₂×D₄₉₉	1	1	yes	characteristic
60	C₄⋊C₄	C₂×D₄₉₉	1	1	yes	characteristic
61	C₄²	—	998	1	no	—
62	C₄²	—	499	1	no	—
63	Q₈	—	1996	2	no	—
64	Q₈	—	998	1	no	—
65	Q₈	—	499	2	no	—
66	C₂×C₄	—	2	1	no	—
67	C₂×C₄	—	2	1	no	—
68	C₂×C₄	—	998	1	no	—
69	C₂×C₄	—	998	1	no	—
70	C₂×C₄	—	499	1	no	—
71	C₂×C₄	—	499	1	no	—
72	C₂×C₄	C₂²×D₄₉₉	1	1	yes	characteristic
73	C₈	—	2	1	no	—
74	C₈	—	998	1	no	—
75	C₂²	D₄×D₄₉₉	1	1	yes	characteristic
76	C₄	—	4	1	no	—
77	C₄	—	4	1	no	—
78	C₄	—	1996	1	no	—
79	C₄	—	998	2	no	—
80	C₄	—	998	1	no	—
81	C₄	—	499	2	no	—
82	C₄	Q₈×D₄₉₉	1	1	yes	characteristic
83	C₄	#22547811	1	1	yes	characteristic
84	C₂	#30630329	1	1	yes	characteristic
85	C₂	SD₁₆×D₄₉₉	1	1	yes	characteristic
86	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, prime radical	C₂²×C₄₉₉	C₂×D₄
hypercenter	#106	D₄₉₉
hypocenter	C₄₉₉	#475
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
21. Dic₉₉₈	Q₈	semidirect
43. C₄₉₉	#475	semidirect

Maximal subgroups

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

subgroup	index
#30630350	2
#30630316	2
#30630282	2
Dic₄₉₉⋊C₈	2
#30630276	2
#30630274	2
#106×C₄₉₉	2
#475	499