Subgroups — #44543405

Order 31936.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30630343	C₂	1	1	yes	maximal, characteristic
2	#30630284	C₂	1	1	yes	maximal, characteristic
3	#30630282	C₂	1	1	yes	maximal, characteristic
4	C₄⋊(C₄₉₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
5	#30630275	C₂	1	1	yes	maximal, characteristic
6	#30630275	C₂	1	1	yes	maximal, characteristic
7	C₄×Q₈×C₄₉₉	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄₉₉⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×C₄₉₉⋊C₈	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₄₉₉	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	C₂×Q₈×C₄₉₉	C₂²	1	1	yes	characteristic
15	C₄⋊C₄×C₄₉₉	—	2	1	no	—
16	C₄⋊C₄×C₄₉₉	C₂²	1	1	yes	characteristic
17	C₄²×C₄₉₉	—	2	1	no	—
18	C₄²×C₄₉₉	C₂²	1	1	yes	characteristic
19	C₂×Dic₄₉₉	—	2	1	no	—
20	C₂×Dic₄₉₉	—	2	1	no	—
21	C₄₉₉⋊C₈	—	2	1	no	—
22	C₄₉₉⋊C₈	—	2	1	no	—
23	Q₈×C₄₉₉	—	2	1	no	—
24	Q₈×C₄₉₉	Q₈	1	2	yes	—
25	C₂×C₄×C₄₉₉	—	2	1	no	—
26	C₂×C₄×C₄₉₉	—	2	1	no	—
27	C₂×C₄×C₄₉₉	D₄	1	1	yes	characteristic
28	C₂×C₄×C₄₉₉	D₄	1	1	yes	characteristic
29	C₂×C₄×C₄₉₉	C₂³	1	1	yes	characteristic
30	C₂²×C₄₉₉	C₂×D₄	1	1	yes	characteristic
31	C₄×C₄₉₉	—	2	2	no	—
32	C₄×C₄₉₉	—	2	1	no	—
33	C₄×C₄₉₉	—	2	1	no	—
34	C₄×C₄₉₉	—	4	1	no	—
35	C₄×C₄₉₉	C₂×Q₈	1	1	yes	characteristic
36	C₄×C₄₉₉	#41	1	1	yes	characteristic
37	Dic₄₉₉	—	4	1	no	—
38	Dic₄₉₉	—	4	1	no	—
39	C₂×C₄₉₉	C₂²⋊Q₈	1	1	yes	characteristic
40	C₂×C₄₉₉	#135	1	1	yes	characteristic
41	C₂×C₄₉₉	#137	1	1	yes	characteristic
42	C₄₉₉	#479	1	1	yes	characteristic, complemented, Sylow
43	#479	—	499	1	no	maximal, Sylow
44	#125	—	499	1	no	—
45	C₄×Q₈	D₄₉₉	1	1	yes	characteristic
46	#107	—	499	1	no	—
47	#106	—	499	1	no	—
48	C₄⋊C₈	—	499	1	no	—
49	#103	—	499	1	no	—
50	#103	—	499	1	no	—
51	C₂×Q₈	C₂×D₄₉₉	1	1	yes	characteristic
52	C₂×C₈	—	499	1	no	—
53	C₂×C₈	—	499	1	no	—
54	C₄⋊C₄	—	2	1	no	—
55	C₄⋊C₄	—	998	1	no	—
56	C₄⋊C₄	—	998	1	no	—
57	C₄⋊C₄	—	499	1	no	—
58	C₄⋊C₄	—	499	1	no	—
59	C₄⋊C₄	C₂×D₄₉₉	1	1	yes	characteristic
60	C₄²	—	2	1	no	—
61	C₄²	C₂×D₄₉₉	1	1	yes	characteristic
62	Q₈	—	2	1	no	—
63	Q₈	Dic₉₉₈	1	2	yes	—
64	C₂×C₄	—	2	1	no	—
65	C₂×C₄	—	2	1	no	—
66	C₂×C₄	—	998	1	no	—
67	C₂×C₄	—	998	1	no	—
68	C₂×C₄	C₂²⋊D₄₉₉	1	1	yes	characteristic
69	C₂×C₄	C₂²⋊D₄₉₉	1	1	yes	characteristic
70	C₂×C₄	C₂²×D₄₉₉	1	1	yes	characteristic
71	C₈	—	998	1	no	—
72	C₈	—	998	1	no	—
73	C₂²	C₂×C₂²⋊D₄₉₉	1	1	yes	characteristic
74	C₄	—	2	2	no	—
75	C₄	—	2	1	no	—
76	C₄	—	2	1	no	—
77	C₄	—	4	1	no	—
78	C₄	—	1996	1	no	—
79	C₄	—	1996	1	no	—
80	C₄	C₂×Dic₉₉₈	1	1	yes	characteristic
81	C₄	#22547811	1	1	yes	characteristic
82	C₂	#30630330	1	1	yes	characteristic
83	C₂	#30630371	1	1	yes	characteristic
84	C₂	#30630382	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₄₉₉
commutator, cosocle	C₂×C₄×C₄₉₉	C₂³
Frattini	C₂×C₄	C₂²×D₄₉₉
Fitting	C₄×Q₈×C₄₉₉	C₂
socle, prime radical	C₂²×C₄₉₉	C₂×D₄
hypercenter	C₄×Q₈	D₄₉₉
hypocenter	C₄₉₉	#479
LCS	G ▷ C₂×C₄×C₄₉₉ ▷ C₂×C₄₉₉ ▷ C₄₉₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
42. C₄₉₉	#479	semidirect

Maximal subgroups

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

subgroup	index
#30630343	2
#30630284	2
#30630282	2
C₄⋊(C₄₉₉⋊C₈)	2
#30630275	2
C₄×Q₈×C₄₉₉	2
#479	499