Subgroups — #30682305

Order 16832.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22554468	C₂	1	1	yes	maximal, characteristic
2	#22554461	C₂	1	1	yes	maximal, characteristic
3	#22554395	C₂	1	1	yes	maximal, characteristic
4	#22554394	C₂	1	1	yes	maximal, characteristic
5	#22554393	C₂	1	1	yes	maximal, characteristic
6	#22554380	C₂	1	1	yes	maximal, characteristic
7	#103×C₂₆₃	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₅₂₆	C₂²	1	1	yes	characteristic
9	C₂×C₂₆₃⋊C₈	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₂₆₃	—	2	1	no	—
11	C₄⋊Dic₂₆₃	C₂²	1	1	yes	characteristic
12	Dic₂₆₃⋊C₄	—	2	1	no	—
13	Dic₂₆₃⋊C₄	—	2	1	no	—
14	Dic₂₆₃⋊C₄	—	2	1	no	—
15	C₄×Dic₂₆₃	C₂²	1	1	yes	characteristic
16	C₂×Q₈×C₂₆₃	C₂²	1	1	yes	characteristic
17	C₂×C₈×C₂₆₃	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×C₂₆₃	C₂²	1	1	yes	characteristic
19	Dic₅₂₆	—	2	1	no	—
20	Dic₅₂₆	—	2	1	no	—
21	C₂×Dic₂₆₃	—	2	1	no	—
22	C₂×Dic₂₆₃	—	2	1	no	—
23	C₂×Dic₂₆₃	D₄	1	1	yes	characteristic
24	C₂×Dic₂₆₃	D₄	1	1	yes	characteristic
25	C₂₆₃⋊C₈	—	2	1	no	—
26	Q₈×C₂₆₃	—	2	1	no	—
27	Q₈×C₂₆₃	—	2	1	no	—
28	C₂×C₄×C₂₆₃	—	2	1	no	—
29	C₂×C₄×C₂₆₃	—	2	1	no	—
30	C₂×C₄×C₂₆₃	C₂³	1	1	yes	characteristic
31	C₈×C₂₆₃	—	2	1	no	—
32	C₂²×C₂₆₃	C₂×D₄	1	1	yes	characteristic
33	C₄×C₂₆₃	—	4	1	no	—
34	C₄×C₂₆₃	—	4	1	no	—
35	C₄×C₂₆₃	#41	1	1	yes	characteristic
36	C₄×C₂₆₃	#41	1	1	yes	characteristic
37	Dic₂₆₃	—	2	1	no	—
38	Dic₂₆₃	—	2	1	no	—
39	Dic₂₆₃	—	4	1	no	—
40	Dic₂₆₃	—	4	1	no	—
41	C₂×C₂₆₃	#124	1	1	yes	characteristic
42	C₂×C₂₆₃	#137	1	1	yes	characteristic
43	C₂×C₂₆₃	#137	1	1	yes	characteristic
44	C₂₆₃	#491	1	1	yes	characteristic, complemented, Sylow
45	#491	—	263	1	no	maximal, Sylow
46	C₄⋊Q₈	—	263	1	no	—
47	#125	—	263	1	no	—
48	#103	—	263	1	no	—
49	#103	—	263	1	no	—
50	#103	—	263	1	no	—
51	#103	D₂₆₃	1	1	yes	characteristic
52	#97	—	263	1	no	—
53	C₂×Q₈	—	263	1	no	—
54	C₂×Q₈	C₂×D₂₆₃	1	1	yes	characteristic
55	C₂×C₈	—	263	1	no	—
56	C₂×C₈	C₂×D₂₆₃	1	1	yes	characteristic
57	C₄⋊C₄	—	263	1	no	—
58	C₄⋊C₄	—	526	1	no	—
59	C₄⋊C₄	—	526	1	no	—
60	C₄⋊C₄	—	526	1	no	—
61	C₄⋊C₄	—	526	1	no	—
62	C₄⋊C₄	C₂×D₂₆₃	1	1	yes	characteristic
63	C₄²	—	263	1	no	—
64	Q₈	—	2	1	no	—
65	Q₈	—	2	1	no	—
66	Q₈	—	526	1	no	—
67	Q₈	—	526	1	no	—
68	C₂×C₄	—	2	1	no	—
69	C₂×C₄	—	2	1	no	—
70	C₂×C₄	—	263	1	no	—
71	C₂×C₄	—	263	1	no	—
72	C₂×C₄	—	526	1	no	—
73	C₂×C₄	—	526	1	no	—
74	C₂×C₄	C₂²×D₂₆₃	1	1	yes	characteristic
75	C₈	—	2	1	no	—
76	C₈	—	526	1	no	—
77	C₂²	D₄×D₂₆₃	1	1	yes	characteristic
78	C₄	—	4	1	no	—
79	C₄	—	4	1	no	—
80	C₄	—	526	1	no	—
81	C₄	—	526	1	no	—
82	C₄	—	1052	1	no	—
83	C₄	—	1052	1	no	—
84	C₄	#16862375	1	1	yes	characteristic
85	C₄	#16862376	1	1	yes	characteristic
86	C₂	#22554456	1	1	yes	characteristic
87	C₂	#22554499	1	1	yes	characteristic
88	C₂	#22554502	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₂₆₃
commutator, cosocle	C₂×C₄×C₂₆₃	C₂³
Frattini	C₂×C₄	C₂²×D₂₆₃
Fitting	#103×C₂₆₃	C₂
socle, prime radical	C₂²×C₂₆₃	C₂×D₄
hypercenter	#103	D₂₆₃
hypocenter	C₂₆₃	#491
LCS	G ▷ C₂×C₄×C₂₆₃ ▷ C₂×C₂₆₃ ▷ C₂₆₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #103

Complements

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

N	H ≅ G/N	type
44. C₂₆₃	#491	semidirect

Maximal subgroups

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

subgroup	index
#22554468	2
#22554461	2
#22554395	2
#22554394	2
#22554393	2
#22554380	2
#103×C₂₆₃	2
#491	263