Subgroups — #46782907

Order 33664.

280 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#22554404	C₂	1	4	yes	maximal
2	#30681808	C₂	1	2	yes	maximal
3	C₂×C₄×C₈×C₂₆₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊Dic₂₆₃	—	2	4	no	—
5	C₂×C₄⋊Dic₂₆₃	C₂²	1	4	yes	—
6	#22554404	—	2	8	no	—
7	C₂²×C₈×C₂₆₃	C₂²	1	2	yes	—
8	C₂×C₄²×C₂₆₃	C₂²	1	1	yes	characteristic
9	C₄×C₈×C₂₆₃	C₄	1	4	yes	complemented
10	C₂²×Dic₂₆₃	—	2	4	no	—
11	C₄⋊Dic₂₆₃	—	2	8	no	—
12	C₄⋊Dic₂₆₃	—	4	8	no	—
13	C₂²×C₄×C₂₆₃	D₄	1	2	yes	—
14	C₂²×C₄×C₂₆₃	C₂³	1	1	yes	characteristic
15	C₂×C₈×C₂₆₃	C₂×C₄	1	4	yes	—
16	C₂×C₈×C₂₆₃	D₄	1	4	yes	—
17	C₂×C₈×C₂₆₃	Q₈	1	4	yes	—
18	C₄²×C₂₆₃	C₂×C₄	1	2	yes	—
19	C₄²×C₂₆₃	Q₈	1	2	yes	—
20	C₂×Dic₂₆₃	—	4	4	no	—
21	C₂×Dic₂₆₃	—	4	8	no	—
22	C₂³×C₂₆₃	C₂×D₄	1	1	yes	characteristic
23	C₂×C₄×C₂₆₃	C₄⋊C₄	1	4	yes	—
24	C₂×C₄×C₂₆₃	D₈	1	4	yes	—
25	C₂×C₄×C₂₆₃	Q₁₆	1	4	yes	—
26	C₂×C₄×C₂₆₃	C₂²×C₄	1	1	yes	characteristic
27	C₂×C₄×C₂₆₃	C₂×D₄	1	2	yes	—
28	C₂×C₄×C₂₆₃	C₂×Q₈	1	2	yes	—
29	C₂×C₄×C₂₆₃	C₂×Q₈	1	1	yes	characteristic
30	C₈×C₂₆₃	C₄⋊C₄	1	8	yes	complemented
31	C₂²×C₂₆₃	C₂×C₄⋊C₄	1	1	yes	characteristic
32	C₂²×C₂₆₃	#127	1	1	yes	characteristic
33	C₂²×C₂₆₃	C₄⋊Q₈	1	1	yes	characteristic
34	C₂²×C₂₆₃	C₂×D₈	1	2	yes	—
35	C₂²×C₂₆₃	C₂×Q₁₆	1	2	yes	—
36	C₄×C₂₆₃	#107	1	8	yes	complemented
37	C₄×C₂₆₃	C₂×C₄⋊C₄	1	2	yes	—
38	C₄×C₂₆₃	C₄⋊Q₈	1	2	yes	—
39	Dic₂₆₃	—	8	4	no	—
40	C₂×C₂₆₃	#384	1	1	yes	characteristic
41	C₂×C₂₆₃	C₂×#107	1	2	yes	—
42	C₂×C₂₆₃	#493	1	1	yes	characteristic
43	C₂×C₂₆₃	#494	1	1	yes	characteristic
44	C₂×C₂₆₃	#500	1	2	yes	—
45	C₂₆₃	#1845	1	1	yes	characteristic, complemented, Sylow
46	#1845	—	263	1	no	maximal, Sylow
47	C₂×#107	—	263	4	no	—
48	C₂×C₄×C₈	D₂₆₃	1	1	yes	characteristic
49	#384	—	263	2	no	—
50	C₂²×C₈	C₂×D₂₆₃	1	2	yes	—
51	C₂×C₄⋊C₄	—	263	4	no	—
52	C₂×C₄⋊C₄	—	526	4	no	—
53	C₂×C₄²	C₂×D₂₆₃	1	1	yes	characteristic
54	#107	—	526	8	no	—
55	C₄×C₈	Dic₂₆₃	1	4	yes	complemented
56	C₂²×C₄	—	526	4	no	—
57	C₂²×C₄	D₁₀₅₂	1	2	yes	—
58	C₂²×C₄	C₂²×D₂₆₃	1	1	yes	characteristic
59	C₂×C₈	C₂×Dic₂₆₃	1	4	yes	—
60	C₂×C₈	D₁₀₅₂	1	4	yes	—
61	C₂×C₈	Dic₅₂₆	1	4	yes	—
62	C₄⋊C₄	—	526	8	no	—
63	C₄⋊C₄	—	1052	8	no	—
64	C₄²	C₂×Dic₂₆₃	1	2	yes	—
65	C₄²	Dic₅₂₆	1	2	yes	—
66	C₂³	C₂×D₁₀₅₂	1	1	yes	characteristic
67	C₂×C₄	—	1052	4	no	—
68	C₂×C₄	—	1052	8	no	—
69	C₂×C₄	C₄⋊Dic₂₆₃	1	4	yes	—
70	C₂×C₄	D₂₁₀₄	1	4	yes	—
71	C₂×C₄	Dic₁₀₅₂	1	4	yes	—
72	C₂×C₄	C₂²×Dic₂₆₃	1	1	yes	characteristic
73	C₂×C₄	C₂×D₁₀₅₂	1	2	yes	—
74	C₂×C₄	C₂×Dic₅₂₆	1	2	yes	—
75	C₂×C₄	C₂×Dic₅₂₆	1	1	yes	characteristic
76	C₈	C₄⋊Dic₂₆₃	1	8	yes	complemented
77	C₂²	C₂×C₄⋊Dic₂₆₃	1	1	yes	characteristic
78	C₂²	#22554467	1	1	yes	characteristic
79	C₂²	#22554470	1	1	yes	characteristic
80	C₂²	C₂×D₂₁₀₄	1	2	yes	—
81	C₂²	C₂×Dic₁₀₅₂	1	2	yes	—
82	C₄	—	2104	4	no	—
83	C₄	#22554404	1	8	yes	complemented
84	C₄	C₂×C₄⋊Dic₂₆₃	1	2	yes	—
85	C₄	#22554470	1	2	yes	—
86	C₂	#30681808	1	1	yes	characteristic
87	C₂	C₂×#22554404	1	2	yes	—
88	C₂	#30682315	1	1	yes	characteristic
89	C₂	#30682318	1	1	yes	characteristic
90	C₂	#30682343	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₁₀₅₂
commutator	C₂×C₄×C₂₆₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₆₃
Fitting	C₂×C₄×C₈×C₂₆₃	C₂
socle, prime radical	C₂³×C₂₆₃	C₂×D₄
cosocle	C₂²×C₄×C₂₆₃	C₂³
hypercenter	C₂×C₄×C₈	D₂₆₃
hypocenter	C₂₆₃	#1845
LCS	G ▷ C₂×C₄×C₂₆₃ ▷ C₂×C₂₆₃ ▷ C₂₆₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
9. C₄×C₈×C₂₆₃	C₄	semidirect
30. C₈×C₂₆₃	C₄⋊C₄	semidirect
36. C₄×C₂₆₃	#107	semidirect
45. C₂₆₃	#1845	semidirect
55. C₄×C₈	Dic₂₆₃	semidirect
76. C₈	C₄⋊Dic₂₆₃	semidirect
83. C₄	#22554404	semidirect

Maximal subgroups

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

subgroup	index
C₂×#22554404	2
#30681808	2
C₂×C₄×C₈×C₂₆₃	2
#1845	263