Subgroups — #22553433

Order 8384.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16862268	C₂	1	1	yes	maximal, characteristic
2	#16862254	C₂	1	2	yes	maximal, complemented
3	#16862172	C₂	1	2	yes	maximal, complemented
4	#16862156	C₂	1	1	yes	maximal, characteristic, complemented
5	#108×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#14523513	C₂²	1	2	yes	—
7	Dic₅₂₄	C₄	1	1	yes	characteristic, complemented
8	#14523502	C₄	1	2	yes	complemented
9	D₁₀₄₈	C₄	1	1	yes	characteristic, complemented
10	#14523496	—	2	2	no	—
11	C₂×C₁₃₁⋊C₈	C₂²	1	1	yes	characteristic
12	C₈×D₁₃₁	—	2	2	no	—
13	C₄×Dic₁₃₁	C₂²	1	1	yes	characteristic
14	M₁₆×C₁₃₁	C₂²	1	2	yes	—
15	C₂×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
16	Dic₂₆₂	C₂×C₄	1	2	yes	—
17	D₅₂₄	C₂×C₄	1	2	yes	—
18	C₂²⋊D₁₃₁	—	2	2	no	—
19	C₂×Dic₁₃₁	—	2	1	no	—
20	C₄×D₁₃₁	—	2	2	no	—
21	C₁₃₁⋊C₈	D₄	1	2	yes	—
22	C₂×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
23	C₈×C₁₃₁	—	2	2	no	—
24	C₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
25	C₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₁₃₁	#41	1	1	yes	characteristic
27	C₂×D₁₃₁	—	2	2	no	—
28	C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
29	C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
30	Dic₁₃₁	—	2	2	no	—
31	Dic₁₃₁	—	4	1	no	—
32	C₂×C₁₃₁	—	2	1	no	—
33	C₂×C₁₃₁	C₄×D₄	1	1	yes	characteristic
34	D₁₃₁	—	4	2	no	—
35	C₁₃₁	#444	1	1	yes	characteristic, complemented, Sylow
36	#444	—	131	1	no	maximal, Sylow
37	#135	—	131	1	no	—
38	#131	—	131	2	no	—
39	#108	D₁₃₁	1	1	yes	characteristic, complemented
40	#104	—	131	2	no	—
41	#97	—	131	1	no	—
42	#41	—	131	2	no	—
43	Q₁₆	—	131	1	no	—
44	SD₁₆	—	131	2	no	—
45	D₈	—	131	1	no	—
46	M₁₆	—	262	2	no	—
47	M₁₆	C₂×D₁₃₁	1	2	yes	—
48	C₂×C₈	—	262	2	no	—
49	C₂×C₈	—	131	1	no	—
50	C₂×C₈	C₂×D₁₃₁	1	1	yes	characteristic
51	C₄²	—	131	1	no	—
52	Q₈	—	131	2	no	—
53	D₄	—	262	2	no	—
54	D₄	—	131	2	no	—
55	C₂×C₄	—	262	2	no	—
56	C₂×C₄	—	262	1	no	—
57	C₂×C₄	C₂²×D₁₃₁	1	1	yes	characteristic
58	C₈	—	2	2	no	—
59	C₈	—	131	2	no	—
60	C₈	C₄×D₁₃₁	1	1	yes	characteristic
61	C₈	C₄×D₁₃₁	1	1	yes	characteristic
62	C₂²	—	262	2	no	—
63	C₂²	#14523514	1	1	yes	characteristic
64	C₄	—	262	2	no	—
65	C₄	—	524	1	no	—
66	C₄	C₂×C₄×D₁₃₁	1	1	yes	characteristic
67	C₄	D₄×D₁₃₁	1	1	yes	characteristic
68	C₂	—	2	1	no	—
69	C₂	—	524	2	no	—
70	C₂	#16862208	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₄×D₁₃₁
commutator	C₄×C₁₃₁	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₁₃₁
Fitting	#108×C₁₃₁	C₂
socle	C₂×C₁₃₁	C₄×D₄
hypercenter	#108	D₁₃₁
hypocenter	C₁₃₁	#444
LCS	G ▷ C₄×C₁₃₁ ▷ C₂×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #108

Complements

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

N	H ≅ G/N	type
2. #16862254	C₂	semidirect
3. #16862172	C₂	semidirect
4. #16862156	C₂	semidirect
5. #108×C₁₃₁	C₂	semidirect
7. Dic₅₂₄	C₄	semidirect
8. #14523502	C₄	semidirect
9. D₁₀₄₈	C₄	semidirect
35. C₁₃₁	#444	semidirect
39. #108	D₁₃₁	semidirect