Subgroups — C₁₃₁×#7540 #45693424

Order 33536.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2262×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	M₁₂₈×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#1424×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#508×C₁₃₁	—	2	1	no	—
5	Q₆₄×C₁₃₁	C₄	1	1	yes	characteristic, complemented
6	SD₆₄×C₁₃₁	—	2	1	no	—
7	D₃₂×C₁₃₁	C₄	1	1	yes	characteristic, complemented
8	C₂×C₃₂×C₁₃₁	C₂²	1	1	yes	characteristic
9	#366×C₁₃₁	—	2	1	no	—
10	C₆₄×C₁₃₁	—	2	1	no	—
11	#135×C₁₃₁	—	4	1	no	—
12	Q₃₂×C₁₃₁	—	2	1	no	—
13	SD₃₂×C₁₃₁	—	4	1	no	—
14	D₁₆×C₁₃₁	—	2	1	no	—
15	C₂×C₁₆×C₁₃₁	D₄	1	1	yes	characteristic
16	#107×C₁₃₁	—	4	1	no	—
17	C₃₂×C₁₃₁	C₂×C₄	1	1	yes	characteristic
18	C₃₂×C₁₃₁	D₄	1	1	yes	characteristic
19	#41×C₁₃₁	—	8	1	no	—
20	Q₁₆×C₁₃₁	—	4	1	no	—
21	SD₁₆×C₁₃₁	—	8	1	no	—
22	D₈×C₁₃₁	—	4	1	no	—
23	C₂×C₈×C₁₃₁	D₈	1	1	yes	characteristic
24	C₄⋊C₄×C₁₃₁	—	8	1	no	—
25	C₁₆×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
26	C₁₆×C₁₃₁	SD₁₆	1	1	yes	characteristic
27	Q₈×C₁₃₁	—	8	1	no	—
28	D₄×C₁₃₁	—	8	1	no	—
29	D₄×C₁₃₁	—	16	1	no	—
30	C₂×C₄×C₁₃₁	—	16	1	no	—
31	C₂×C₄×C₁₃₁	—	16	1	no	—
32	C₂×C₄×C₁₃₁	D₁₆	1	1	yes	characteristic
33	C₈×C₁₃₁	#102	1	1	yes	characteristic
34	C₈×C₁₃₁	SD₃₂	1	1	yes	characteristic
35	C₂²×C₁₃₁	—	16	1	no	—
36	C₂²×C₁₃₁	D₃₂	1	1	yes	characteristic
37	C₄×C₁₃₁	—	16	1	no	—
38	C₄×C₁₃₁	—	32	1	no	—
39	C₄×C₁₃₁	#357	1	1	yes	characteristic
40	C₄×C₁₃₁	SD₆₄	1	1	yes	characteristic
41	C₂×C₁₃₁	—	2	1	no	—
42	C₂×C₁₃₁	—	32	1	no	—
43	C₂×C₁₃₁	#1415	1	1	yes	characteristic
44	#7540	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
45	C₁₃₁	#7540	1	1	yes	characteristic, direct product factor, Sylow
46	#2262	C₂×C₁₃₁	1	1	yes	characteristic
47	M₁₂₈	C₂×C₁₃₁	1	1	yes	characteristic, complemented
48	#1424	C₂×C₁₃₁	1	1	yes	characteristic, complemented
49	#508	—	2	1	no	—
50	Q₆₄	C₄×C₁₃₁	1	1	yes	characteristic, complemented
51	SD₆₄	—	2	1	no	—
52	D₃₂	C₄×C₁₃₁	1	1	yes	characteristic, complemented
53	C₂×C₃₂	C₂²×C₁₃₁	1	1	yes	characteristic
54	#366	—	2	1	no	—
55	C₆₄	—	2	1	no	—
56	#135	—	4	1	no	—
57	Q₃₂	—	2	1	no	—
58	SD₃₂	—	4	1	no	—
59	D₁₆	—	2	1	no	—
60	C₂×C₁₆	D₄×C₁₃₁	1	1	yes	characteristic
61	#107	—	4	1	no	—
62	C₃₂	C₂×C₄×C₁₃₁	1	1	yes	characteristic
63	C₃₂	D₄×C₁₃₁	1	1	yes	characteristic
64	#41	—	8	1	no	—
65	Q₁₆	—	4	1	no	—
66	SD₁₆	—	8	1	no	—
67	D₈	—	4	1	no	—
68	C₂×C₈	D₈×C₁₃₁	1	1	yes	characteristic
69	C₄⋊C₄	—	8	1	no	—
70	C₁₆	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
71	C₁₆	SD₁₆×C₁₃₁	1	1	yes	characteristic
72	Q₈	—	8	1	no	—
73	D₄	—	8	1	no	—
74	D₄	—	16	1	no	—
75	C₂×C₄	—	16	1	no	—
76	C₂×C₄	—	16	1	no	—
77	C₂×C₄	D₁₆×C₁₃₁	1	1	yes	characteristic
78	C₈	#102×C₁₃₁	1	1	yes	characteristic
79	C₈	SD₃₂×C₁₃₁	1	1	yes	characteristic
80	C₂²	—	16	1	no	—
81	C₂²	D₃₂×C₁₃₁	1	1	yes	characteristic
82	C₄	—	16	1	no	—
83	C₄	—	32	1	no	—
84	C₄	#357×C₁₃₁	1	1	yes	characteristic
85	C₄	SD₆₄×C₁₃₁	1	1	yes	characteristic
86	C₂	—	2	1	no	—
87	C₂	—	32	1	no	—
88	C₂	#1415×C₁₃₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₁₃₁	#1415
commutator	C₃₂	C₂×C₄×C₁₃₁
Frattini, cosocle	C₂×C₃₂	C₂²×C₁₃₁
prime radical	#2262×C₁₃₁	C₂
LCS	G ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₃₁ ◁ C₂×C₄×C₁₃₁ ◁ C₂×C₈×C₁₃₁ ◁ C₂×C₁₆×C₁₃₁ ◁ C₂×C₃₂×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. M₁₂₈×C₁₃₁	C₂	semidirect
3. #1424×C₁₃₁	C₂	semidirect
5. Q₆₄×C₁₃₁	C₄	semidirect
7. D₃₂×C₁₃₁	C₄	semidirect
44. #7540	C₁₃₁	direct (N × H)
45. C₁₃₁	#7540	direct (N × H)
47. M₁₂₈	C₂×C₁₃₁	semidirect
48. #1424	C₂×C₁₃₁	semidirect
50. Q₆₄	C₄×C₁₃₁	semidirect
52. D₃₂	C₄×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
#2262×C₁₃₁	2
M₁₂₈×C₁₃₁	2
#1424×C₁₃₁	2
#7540	131

Subgroups — C131×#7540 #45693424

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#7540 #45693424