Subgroups — #46780713

Order 33664.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×D₂₁₀₄	C₂	1	1	yes	maximal, characteristic
2	#30681714	C₂	1	1	yes	maximal, characteristic, complemented
3	#334×C₂₆₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₂₁₀₄	C₄	1	1	yes	characteristic
5	C₄×D₁₀₅₂	—	2	1	no	—
6	C₂×C₂₆₃⋊C₁₆	—	2	1	no	—
7	#22554404	C₄	1	1	yes	characteristic
8	#22554392	—	2	1	no	—
9	C₄⋊C₈×C₂₆₃	—	2	1	no	—
10	C₄×C₈×C₂₆₃	C₂²	1	1	yes	characteristic
11	C₂×D₁₀₅₂	—	2	1	no	—
12	C₂×C₄×D₂₆₃	—	4	1	no	—
13	D₂₁₀₄	—	2	1	no	—
14	D₂₁₀₄	C₈	1	2	yes	complemented
15	C₄⋊Dic₂₆₃	—	2	1	no	—
16	(C₂×D₂₆₃)⋊C₄	—	4	1	no	—
17	C₂₆₃⋊C₁₆	—	2	2	no	—
18	C₂×C₈×C₂₆₃	—	4	1	no	—
19	C₂×C₈×C₂₆₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₈×C₂₆₃	D₄	1	1	yes	characteristic
21	C₄²×C₂₆₃	D₄	1	1	yes	characteristic
22	C₂²×D₂₆₃	—	4	1	no	—
23	D₁₀₅₂	—	2	2	no	—
24	D₁₀₅₂	—	4	1	no	—
25	C₂×Dic₂₆₃	—	4	1	no	—
26	C₄×D₂₆₃	—	8	2	no	—
27	C₂×C₄×C₂₆₃	C₂²⋊C₄	1	1	yes	characteristic
28	C₂×C₄×C₂₆₃	D₈	1	1	yes	characteristic
29	C₂×C₄×C₂₆₃	SD₁₆	1	1	yes	characteristic
30	C₈×C₂₆₃	—	2	1	no	—
31	C₈×C₂₆₃	—	8	1	no	—
32	C₈×C₂₆₃	C₂×C₈	1	1	yes	characteristic
33	C₈×C₂₆₃	M₁₆	1	1	yes	characteristic
34	C₂²×C₂₆₃	#102	1	1	yes	characteristic
35	C₂×D₂₆₃	—	4	2	no	—
36	C₂×D₂₆₃	—	8	1	no	—
37	C₂×D₂₆₃	—	8	1	no	—
38	C₄×C₂₆₃	—	2	1	no	—
39	C₄×C₂₆₃	—	2	1	no	—
40	C₄×C₂₆₃	C₂²⋊C₈	1	1	yes	characteristic
41	C₄×C₂₆₃	#104	1	1	yes	characteristic
42	Dic₂₆₃	—	8	1	no	—
43	C₂×C₂₆₃	D₄⋊C₈	1	1	yes	characteristic
44	C₂×C₂₆₃	#360	1	1	yes	characteristic
45	C₂×C₂₆₃	#361	1	1	yes	characteristic
46	D₂₆₃	—	8	2	no	—
47	C₂₆₃	#1333	1	1	yes	characteristic, complemented, Sylow
48	#1333	—	263	1	no	maximal, Sylow
49	C₄×D₈	—	263	1	no	—
50	#346	—	263	1	no	—
51	#334	D₂₆₃	1	1	yes	characteristic, complemented
52	C₂×D₈	—	263	1	no	—
53	C₄×D₄	—	526	1	no	—
54	C₂×C₁₆	—	526	1	no	—
55	#107	—	263	1	no	—
56	C₄⋊C₈	—	2	1	no	—
57	#102	—	526	1	no	—
58	C₄×C₈	C₂×D₂₆₃	1	1	yes	characteristic
59	C₂×D₄	—	526	1	no	—
60	C₂²×C₄	—	1052	1	no	—
61	D₈	—	263	2	no	—
62	D₈	—	526	1	no	—
63	C₂×C₈	—	4	1	no	—
64	C₂×C₈	C₄×D₂₆₃	1	1	yes	characteristic
65	C₂×C₈	D₁₀₅₂	1	1	yes	characteristic
66	C₄⋊C₄	—	526	1	no	—
67	C₂²⋊C₄	—	1052	1	no	—
68	C₄²	C₂²⋊D₂₆₃	1	1	yes	characteristic
69	C₁₆	—	526	2	no	—
70	C₂³	—	1052	1	no	—
71	D₄	—	526	2	no	—
72	D₄	—	1052	1	no	—
73	C₂×C₄	—	1052	1	no	—
74	C₂×C₄	—	2104	2	no	—
75	C₂×C₄	(C₂×D₂₆₃)⋊C₄	1	1	yes	characteristic
76	C₂×C₄	#16862361	1	1	yes	characteristic
77	C₂×C₄	#16862363	1	1	yes	characteristic
78	C₈	—	2	1	no	—
79	C₈	—	8	1	no	—
80	C₈	C₈×D₂₆₃	1	1	yes	characteristic
81	C₈	#16862359	1	1	yes	characteristic
82	C₂²	—	1052	2	no	—
83	C₂²	—	2104	1	no	—
84	C₂²	—	2104	1	no	—
85	C₂²	#22554390	1	1	yes	characteristic
86	C₄	—	2	1	no	—
87	C₄	—	2	1	no	—
88	C₄	—	2104	1	no	—
89	C₄	(C₂×D₂₆₃)⋊C₈	1	1	yes	characteristic
90	C₄	#22554396	1	1	yes	characteristic
91	C₂	—	2104	2	no	—
92	C₂	#30681665	1	1	yes	characteristic
93	C₂	#30681747	1	1	yes	characteristic
94	C₂	#30681750	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#22554390
commutator	C₈×C₂₆₃	C₂×C₈
Frattini	C₄×C₈	C₂×D₂₆₃
Fitting	#334×C₂₆₃	C₂
socle	C₂²×C₂₆₃	#102
prime radical	C₂×D₂₁₀₄	C₄
cosocle	C₄×C₈×C₂₆₃	C₂²
hypercenter	#334	D₂₆₃
hypocenter	C₂₆₃	#1333
LCS	G ▷ C₈×C₂₆₃ ▷ C₄×C₂₆₃ ▷ C₂×C₂₆₃ ▷ C₂₆₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #334

Complements

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

N	H ≅ G/N	type
2. #30681714	C₂	semidirect
3. #334×C₂₆₃	C₂	semidirect
14. D₂₁₀₄	C₈	semidirect
47. C₂₆₃	#1333	semidirect
51. #334	D₂₆₃	semidirect

Maximal subgroups

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

subgroup	index
C₄×D₂₁₀₄	2
#30681714	2
#334×C₂₆₃	2
#1333	263