Subgroups — #30682061

Order 16832.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22554492	C₂	1	1	yes	maximal, characteristic
2	#22554476	C₂	1	2	yes	maximal, complemented
3	#22554406	C₂	1	1	yes	maximal, characteristic, complemented
4	#22554398	C₂	1	2	yes	maximal, complemented
5	#97×C₂₆₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#16862376	C₂²	1	2	yes	—
7	Dic₁₀₅₂	C₄	1	1	yes	characteristic, complemented
8	#16862365	C₄	1	2	yes	complemented
9	D₂₁₀₄	C₄	1	1	yes	characteristic, complemented
10	#16862360	C₂²	1	2	yes	—
11	#16862359	—	2	2	no	—
12	C₈×D₂₆₃	—	2	2	no	—
13	C₂×C₈×C₂₆₃	C₂²	1	1	yes	characteristic
14	C₂×C₈×C₂₆₃	C₂²	1	1	yes	characteristic
15	C₄²×C₂₆₃	C₂²	1	1	yes	characteristic
16	Dic₅₂₆	C₂×C₄	1	2	yes	—
17	D₁₀₅₂	C₂×C₄	1	2	yes	—
18	C₂²⋊D₂₆₃	—	2	2	no	—
19	C₄×D₂₆₃	—	2	2	no	—
20	C₂₆₃⋊C₈	—	2	2	no	—
21	C₂×C₄×C₂₆₃	—	2	1	no	—
22	C₂×C₄×C₂₆₃	C₂³	1	1	yes	characteristic
23	C₈×C₂₆₃	C₂×C₄	1	1	yes	characteristic
24	C₈×C₂₆₃	C₂×C₄	1	1	yes	characteristic
25	C₈×C₂₆₃	D₄	1	2	yes	—
26	C₂²×C₂₆₃	#41	1	1	yes	characteristic
27	C₂×D₂₆₃	—	2	2	no	—
28	C₄×C₂₆₃	—	4	1	no	—
29	C₄×C₂₆₃	C₂²×C₄	1	1	yes	characteristic
30	C₄×C₂₆₃	C₂×D₄	1	1	yes	characteristic
31	Dic₂₆₃	—	2	2	no	—
32	C₂×C₂₆₃	—	2	1	no	—
33	C₂×C₂₆₃	C₄×D₄	1	1	yes	characteristic
34	D₂₆₃	—	4	2	no	—
35	C₂₆₃	#444	1	1	yes	characteristic, complemented, Sylow
36	#444	—	263	1	no	maximal, Sylow
37	#135	—	263	1	no	—
38	#131	—	263	2	no	—
39	#108	—	263	1	no	—
40	#104	—	263	2	no	—
41	#97	D₂₆₃	1	1	yes	characteristic, complemented
42	#41	—	263	2	no	—
43	Q₁₆	—	263	1	no	—
44	SD₁₆	—	263	2	no	—
45	D₈	—	263	1	no	—
46	M₁₆	—	263	2	no	—
47	M₁₆	—	526	2	no	—
48	C₂×C₈	—	526	2	no	—
49	C₂×C₈	C₂×D₂₆₃	1	1	yes	characteristic
50	C₂×C₈	C₂×D₂₆₃	1	1	yes	characteristic
51	C₄²	C₂×D₂₆₃	1	1	yes	characteristic
52	Q₈	—	263	2	no	—
53	D₄	—	263	2	no	—
54	D₄	—	526	2	no	—
55	C₂×C₄	—	2	1	no	—
56	C₂×C₄	—	526	2	no	—
57	C₂×C₄	C₂²×D₂₆₃	1	1	yes	characteristic
58	C₈	—	526	2	no	—
59	C₈	C₄×D₂₆₃	1	1	yes	characteristic
60	C₈	C₄×D₂₆₃	1	1	yes	characteristic
61	C₈	D₁₀₅₂	1	2	yes	—
62	C₂²	—	526	2	no	—
63	C₂²	#16862376	1	1	yes	characteristic
64	C₄	—	4	1	no	—
65	C₄	—	526	2	no	—
66	C₄	C₂×C₄×D₂₆₃	1	1	yes	characteristic
67	C₄	C₂×D₁₀₅₂	1	1	yes	characteristic
68	C₂	—	2	1	no	—
69	C₂	—	1052	2	no	—
70	C₂	C₄×D₁₀₅₂	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₁₀₅₂
commutator	C₄×C₂₆₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₂₆₃
Fitting	#97×C₂₆₃	C₂
socle	C₂×C₂₆₃	C₄×D₄
prime radical	#22554492	C₂
cosocle	C₂×C₄×C₂₆₃	C₂³
hypercenter	#97	D₂₆₃
hypocenter	C₂₆₃	#444
LCS	G ▷ C₄×C₂₆₃ ▷ C₂×C₂₆₃ ▷ C₂₆₃
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ #97

Complements

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

N	H ≅ G/N	type
2. #22554476	C₂	semidirect
3. #22554406	C₂	semidirect
4. #22554398	C₂	semidirect
5. #97×C₂₆₃	C₂	semidirect
7. Dic₁₀₅₂	C₄	semidirect
8. #16862365	C₄	semidirect
9. D₂₁₀₄	C₄	semidirect
35. C₂₆₃	#444	semidirect
41. #97	D₂₆₃	semidirect

Maximal subgroups

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

subgroup	index
#22554492	2
#22554476	2
#22554406	2
#22554398	2
#97×C₂₆₃	2
#444	263