Subgroups — #30682300

Order 16832.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22554458	C₂	1	1	yes	maximal, characteristic
2	#22554393	C₂	1	2	yes	maximal, complemented
3	#22554390	C₂	1	2	yes	maximal
4	#22554381	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄⋊Q₈×C₂₆₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₅₂₆	C₂²	1	1	yes	characteristic
7	C₂×D₁₀₅₂	C₂²	1	1	yes	characteristic
8	C₂×C₂₆₃⋊C₈	C₂²	1	2	yes	—
9	(C₂×D₂₆₃)⋊C₄	—	2	1	no	—
10	(C₂×D₂₆₃)⋊C₄	—	2	1	no	—
11	C₂×Q₈×C₂₆₃	—	2	1	no	—
12	C₄⋊C₄×C₂₆₃	—	2	1	no	—
13	C₄⋊C₄×C₂₆₃	C₂²	1	2	yes	—
14	C₄²×C₂₆₃	C₂²	1	1	yes	characteristic
15	C₂²×D₂₆₃	—	2	1	no	—
16	Dic₅₂₆	—	2	2	no	—
17	D₁₀₅₂	—	2	2	no	—
18	C₂×Dic₂₆₃	—	2	1	no	—
19	C₂₆₃⋊C₈	—	2	2	no	—
20	Q₈×C₂₆₃	—	4	2	no	—
21	C₂×C₄×C₂₆₃	—	2	2	no	—
22	C₂×C₄×C₂₆₃	D₄	1	1	yes	characteristic
23	C₂×C₄×C₂₆₃	D₄	1	1	yes	characteristic
24	C₂×C₄×C₂₆₃	C₂³	1	1	yes	characteristic
25	C₂²×C₂₆₃	C₂×D₄	1	1	yes	characteristic
26	C₂×D₂₆₃	—	4	1	no	—
27	C₂×D₂₆₃	—	4	1	no	—
28	C₂×D₂₆₃	—	4	1	no	—
29	C₄×C₂₆₃	—	2	1	no	—
30	C₄×C₂₆₃	—	2	1	no	—
31	C₄×C₂₆₃	—	4	2	no	—
32	C₄×C₂₆₃	#41	1	2	yes	—
33	Dic₂₆₃	—	4	1	no	—
34	C₂×C₂₆₃	#124	1	1	yes	characteristic
35	C₂×C₂₆₃	AGL(1,ℤ/8)	1	1	yes	characteristic
36	C₂×C₂₆₃	#137	1	1	yes	characteristic
37	D₂₆₃	—	8	1	no	—
38	C₂₆₃	#489	1	1	yes	characteristic, complemented, Sylow
39	#489	—	263	1	no	maximal, Sylow
40	C₄⋊Q₈	D₂₆₃	1	1	yes	characteristic, complemented
41	#124	—	263	1	no	—
42	#103	—	263	2	no	—
43	#102	—	263	2	no	—
44	#97	—	263	1	no	—
45	C₂×Q₈	—	2	1	no	—
46	C₂×Q₈	—	263	1	no	—
47	C₂×D₄	—	263	1	no	—
48	C₂×C₈	—	263	2	no	—
49	C₄⋊C₄	—	2	1	no	—
50	C₄⋊C₄	C₂×D₂₆₃	1	2	yes	—
51	C₂²⋊C₄	—	526	1	no	—
52	C₂²⋊C₄	—	526	1	no	—
53	C₄²	C₂×D₂₆₃	1	1	yes	characteristic
54	C₂³	—	526	1	no	—
55	Q₈	—	4	2	no	—
56	Q₈	—	526	2	no	—
57	D₄	—	526	2	no	—
58	C₂×C₄	—	2	2	no	—
59	C₂×C₄	—	526	1	no	—
60	C₂×C₄	C₂²⋊D₂₆₃	1	1	yes	characteristic
61	C₂×C₄	C₂²⋊D₂₆₃	1	1	yes	characteristic
62	C₂×C₄	C₂²×D₂₆₃	1	1	yes	characteristic
63	C₈	—	526	2	no	—
64	C₂²	—	1052	1	no	—
65	C₂²	—	1052	1	no	—
66	C₂²	—	1052	1	no	—
67	C₂²	C₂×C₂²⋊D₂₆₃	1	1	yes	characteristic
68	C₄	—	2	1	no	—
69	C₄	—	2	1	no	—
70	C₄	—	4	2	no	—
71	C₄	—	1052	1	no	—
72	C₄	#16862377	1	2	yes	—
73	C₂	—	2104	1	no	—
74	C₂	#22554459	1	1	yes	characteristic
75	C₂	#22554496	1	1	yes	characteristic
76	C₂	#22554501	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₂₆₃
commutator, cosocle	C₂×C₄×C₂₆₃	C₂³
Frattini	C₂×C₄	C₂²×D₂₆₃
Fitting	C₄⋊Q₈×C₂₆₃	C₂
socle	C₂²×C₂₆₃	C₂×D₄
prime radical	C₂×D₁₀₅₂	C₂²
hypercenter	C₄⋊Q₈	D₂₆₃
hypocenter	C₂₆₃	#489
LCS	G ▷ C₂×C₄×C₂₆₃ ▷ C₂×C₂₆₃ ▷ C₂₆₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
2. #22554393	C₂	semidirect
4. #22554381	C₂	semidirect
5. C₄⋊Q₈×C₂₆₃	C₂	semidirect
38. C₂₆₃	#489	semidirect
40. C₄⋊Q₈	D₂₆₃	semidirect

Maximal subgroups

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

subgroup	index
#22554458	2
#22554393	2
#22554390	2
#22554381	2
C₄⋊Q₈×C₂₆₃	2
#489	263