Subgroups — #22553097

Order 8384.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16862252	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	M₃₂×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#14523513	C₄	1	1	yes	characteristic
5	#14523497	C₄	1	1	yes	characteristic
6	#14523496	—	2	1	no	—
7	C₈×D₁₃₁	—	2	1	no	—
8	C₁₃₁⋊C₁₆	—	2	1	no	—
9	C₂×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
10	C₁₆×C₁₃₁	—	2	1	no	—
11	Dic₂₆₂	C₈	1	1	yes	characteristic
12	D₅₂₄	C₈	1	1	yes	characteristic
13	C₂²⋊D₁₃₁	—	2	1	no	—
14	C₄×D₁₃₁	—	2	1	no	—
15	C₁₃₁⋊C₈	—	2	1	no	—
16	C₂×C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
17	C₈×C₁₃₁	D₄	1	1	yes	characteristic
18	C₈×C₁₃₁	D₄	1	1	yes	characteristic
19	C₂²×C₁₃₁	M₁₆	1	1	yes	characteristic
20	C₂×D₁₃₁	—	2	1	no	—
21	C₄×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
22	C₄×C₁₃₁	C₂×C₈	1	1	yes	characteristic
23	Dic₁₃₁	—	2	1	no	—
24	C₂×C₁₃₁	—	2	1	no	—
25	C₂×C₁₃₁	C₂²⋊C₈	1	1	yes	characteristic
26	D₁₃₁	—	4	1	no	—
27	C₁₃₁	#350	1	1	yes	characteristic, complemented, Sylow
28	#350	—	131	1	no	maximal, Sylow
29	#131	—	131	1	no	—
30	M₃₂	D₁₃₁	1	1	yes	characteristic, complemented
31	C₂×C₁₆	—	131	1	no	—
32	#41	—	131	1	no	—
33	M₁₆	—	262	1	no	—
34	M₁₆	—	131	1	no	—
35	C₂×C₈	—	262	1	no	—
36	C₂×C₈	C₂×D₁₃₁	1	1	yes	characteristic
37	C₁₆	—	2	1	no	—
38	C₁₆	—	262	1	no	—
39	Q₈	—	131	1	no	—
40	D₄	—	262	1	no	—
41	D₄	—	131	1	no	—
42	C₂×C₄	—	262	1	no	—
43	C₂×C₄	C₄×D₁₃₁	1	1	yes	characteristic
44	C₈	—	262	1	no	—
45	C₈	C₂²⋊D₁₃₁	1	1	yes	characteristic
46	C₈	D₅₂₄	1	1	yes	characteristic
47	C₂²	—	262	1	no	—
48	C₂²	#14523496	1	1	yes	characteristic
49	C₄	—	262	1	no	—
50	C₄	(C₂×D₁₃₁)⋊C₄	1	1	yes	characteristic
51	C₄	C₈×D₁₃₁	1	1	yes	characteristic
52	C₂	—	2	1	no	—
53	C₂	—	524	1	no	—
54	C₂	(C₂×D₁₃₁)⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂²⋊D₁₃₁
commutator	C₄×C₁₃₁	C₂×C₈
Frattini	C₂×C₈	C₂×D₁₃₁
Fitting	M₃₂×C₁₃₁	C₂
socle	C₂×C₁₃₁	C₂²⋊C₈
hypercenter	M₃₂	D₁₃₁
hypocenter	C₁₃₁	#350
LCS	G ▷ C₄×C₁₃₁ ▷ C₂×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ M₃₂

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₃₁⋊C₁₆	C₂	semidirect
3. M₃₂×C₁₃₁	C₂	semidirect
27. C₁₃₁	#350	semidirect
30. M₃₂	D₁₃₁	semidirect