Subgroups — #46782333

Order 33664.

214 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30682336	C₂	1	1	yes	maximal, characteristic, complemented
2	#30682334	C₂	1	2	yes	maximal, complemented
3	#30682310	C₂	1	2	yes	maximal
4	C₈×C₂₆₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	#492×C₂₆₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#16862364	—	2	4	no	—
7	#22554470	C₂²	1	2	yes	—
8	#22554402	—	2	2	no	—
9	#22554401	—	2	4	no	—
10	C₄×C₂₆₃⋊C₈	C₂²	1	2	yes	—
11	C₂×SD₁₆×C₂₆₃	—	2	2	no	—
12	C₄⋊Q₈×C₂₆₃	C₂²	1	1	yes	characteristic
13	#127×C₂₆₃	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₂₆₃	C₂²	1	1	yes	characteristic
15	C₂×Dic₅₂₆	—	2	2	no	—
16	#16862364	—	4	8	no	—
17	C₂×C₂₆₃⋊C₈	D₄	1	4	yes	—
18	C₄⋊Dic₂₆₃	—	2	2	no	—
19	C₄⋊Dic₂₆₃	—	2	2	no	—
20	C₂×Q₈×C₂₆₃	—	2	1	no	—
21	C₂×D₄×C₂₆₃	—	2	2	no	—
22	C₂×D₄×C₂₆₃	—	2	1	no	—
23	SD₁₆×C₂₆₃	—	4	4	no	—
24	C₂×C₈×C₂₆₃	D₄	1	2	yes	—
25	C₄⋊C₄×C₂₆₃	—	2	2	no	—
26	C₄²×C₂₆₃	C₂³	1	1	yes	characteristic
27	Dic₅₂₆	—	4	4	no	—
28	C₂×Dic₂₆₃	—	4	2	no	—
29	C₂₆₃⋊C₈	SD₁₆	1	8	yes	complemented
30	C₂³×C₂₆₃	—	4	1	no	—
31	Q₈×C₂₆₃	—	4	2	no	—
32	D₄×C₂₆₃	—	4	2	no	—
33	D₄×C₂₆₃	—	4	4	no	—
34	C₂×C₄×C₂₆₃	—	4	1	no	—
35	C₂×C₄×C₂₆₃	C₂×D₄	1	2	yes	—
36	C₂×C₄×C₂₆₃	C₂×D₄	1	1	yes	characteristic
37	C₈×C₂₆₃	SD₁₆	1	4	yes	complemented
38	C₂²×C₂₆₃	—	8	2	no	—
39	C₂²×C₂₆₃	—	8	1	no	—
40	C₂²×C₂₆₃	#127	1	1	yes	characteristic
41	C₄×C₂₆₃	—	8	1	no	—
42	C₄×C₂₆₃	C₂×SD₁₆	1	4	yes	—
43	C₄×C₂₆₃	C₂×SD₁₆	1	2	yes	—
44	Dic₂₆₃	—	8	2	no	—
45	C₂×C₂₆₃	—	16	1	no	—
46	C₂×C₂₆₃	#492	1	2	yes	—
47	C₂×C₂₆₃	#492	1	1	yes	characteristic
48	C₂₆₃	#1705	1	1	yes	characteristic, complemented, Sylow
49	#1705	—	263	1	no	maximal, Sylow
50	#498	—	263	2	no	—
51	#498	—	263	1	no	—
52	#492	—	263	2	no	—
53	#492	D₂₆₃	1	1	yes	characteristic
54	C₈²	—	263	1	no	—
55	C₂×SD₁₆	—	2	2	no	—
56	C₂×SD₁₆	—	526	4	no	—
57	C₄⋊Q₈	—	263	2	no	—
58	C₄⋊Q₈	C₂×D₂₆₃	1	1	yes	characteristic
59	#127	C₂×D₂₆₃	1	1	yes	characteristic
60	#106	—	526	4	no	—
61	#106	—	526	2	no	—
62	C₄×C₈	—	263	2	no	—
63	C₄×C₈	C₂×D₂₆₃	1	1	yes	characteristic
64	C₂×Q₈	—	2	1	no	—
65	C₂×Q₈	—	526	2	no	—
66	C₂×D₄	—	2	2	no	—
67	C₂×D₄	—	2	1	no	—
68	SD₁₆	—	4	4	no	—
69	SD₁₆	—	1052	8	no	—
70	C₂×C₈	—	263	4	no	—
71	C₂×C₈	C₂²⋊D₂₆₃	1	2	yes	—
72	C₄⋊C₄	—	2	2	no	—
73	C₄⋊C₄	—	526	2	no	—
74	C₄⋊C₄	—	526	2	no	—
75	C₄²	C₂²×D₂₆₃	1	1	yes	characteristic
76	C₂³	—	4	1	no	—
77	Q₈	—	4	2	no	—
78	Q₈	—	1052	4	no	—
79	D₄	—	4	4	no	—
80	D₄	—	4	2	no	—
81	C₂×C₄	—	4	1	no	—
82	C₂×C₄	—	1052	2	no	—
83	C₂×C₄	D₄×D₂₆₃	1	2	yes	—
84	C₂×C₄	C₂×C₂²⋊D₂₆₃	1	1	yes	characteristic
85	C₈	—	263	8	no	—
86	C₈	#16862364	1	4	yes	complemented
87	C₂²	—	8	2	no	—
88	C₂²	—	8	1	no	—
89	C₂²	#22554466	1	1	yes	characteristic
90	C₄	—	8	1	no	—
91	C₄	—	2104	2	no	—
92	C₄	SD₁₆×D₂₆₃	1	4	yes	—
93	C₄	C₂×#16862364	1	2	yes	—
94	C₂	—	16	1	no	—
95	C₂	#30682309	1	2	yes	—
96	C₂	#30682310	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#22554466
commutator, cosocle	C₄²×C₂₆₃	C₂³
Frattini	C₄²	C₂²×D₂₆₃
Fitting	#492×C₂₆₃	C₂
socle	C₂²×C₂₆₃	#127
prime radical	#127×C₂₆₃	C₂²
hypercenter	#492	D₂₆₃
hypocenter	C₂₆₃	#1705
LCS	G ▷ C₄²×C₂₆₃ ▷ C₂²×C₂₆₃ ▷ C₂₆₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

Complements

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

N	H ≅ G/N	type
1. #30682336	C₂	semidirect
2. #30682334	C₂	semidirect
4. C₈×C₂₆₃⋊C₈	C₂	semidirect
29. C₂₆₃⋊C₈	SD₁₆	semidirect
37. C₈×C₂₆₃	SD₁₆	semidirect
48. C₂₆₃	#1705	semidirect
86. C₈	#16862364	semidirect

Maximal subgroups

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

subgroup	index
#30682336	2
#30682334	2
#30682310	2
C₈×C₂₆₃⋊C₈	2
#492×C₂₆₃	2
#1705	263