Subgroups — C₁₃₁×#7538 #45693422

Order 33536.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₆₄×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₆₄×C₁₃₁	C₂	1	1	yes	maximal, characteristic
3	#1423×C₁₃₁	C₂	1	1	yes	maximal, characteristic
4	C₂×Q₃₂×C₁₃₁	—	2	1	no	—
5	Q₆₄×C₁₃₁	—	2	1	no	—
6	Q₆₄×C₁₃₁	C₄	1	2	yes	complemented
7	C₂×C₃₂×C₁₃₁	C₂²	1	1	yes	characteristic
8	#366×C₁₃₁	—	2	1	no	—
9	C₆₄×C₁₃₁	—	2	1	no	—
10	C₂×Q₁₆×C₁₃₁	—	4	1	no	—
11	Q₃₂×C₁₃₁	—	2	2	no	—
12	Q₃₂×C₁₃₁	—	4	1	no	—
13	C₂×C₁₆×C₁₃₁	D₄	1	1	yes	characteristic
14	#107×C₁₃₁	—	4	1	no	—
15	C₃₂×C₁₃₁	C₂×C₄	1	1	yes	characteristic
16	C₃₂×C₁₃₁	D₄	1	1	yes	characteristic
17	C₂×Q₈×C₁₃₁	—	8	1	no	—
18	Q₁₆×C₁₃₁	—	4	2	no	—
19	Q₁₆×C₁₃₁	—	8	1	no	—
20	C₂×C₈×C₁₃₁	D₈	1	1	yes	characteristic
21	C₄⋊C₄×C₁₃₁	—	8	1	no	—
22	C₁₆×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
23	C₁₆×C₁₃₁	SD₁₆	1	1	yes	characteristic
24	Q₈×C₁₃₁	—	8	2	no	—
25	Q₈×C₁₃₁	—	16	1	no	—
26	C₂×C₄×C₁₃₁	—	16	1	no	—
27	C₂×C₄×C₁₃₁	—	16	1	no	—
28	C₂×C₄×C₁₃₁	D₁₆	1	1	yes	characteristic
29	C₈×C₁₃₁	#102	1	1	yes	characteristic
30	C₈×C₁₃₁	SD₃₂	1	1	yes	characteristic
31	C₂²×C₁₃₁	D₃₂	1	1	yes	characteristic
32	C₄×C₁₃₁	—	16	2	no	—
33	C₄×C₁₃₁	—	32	1	no	—
34	C₄×C₁₃₁	#357	1	1	yes	characteristic
35	C₄×C₁₃₁	SD₆₄	1	1	yes	characteristic
36	C₂×C₁₃₁	#1415	1	1	yes	characteristic
37	C₂×C₁₃₁	SD₁₂₈	1	1	yes	characteristic
38	C₂×C₁₃₁	Q₁₂₈	1	1	yes	characteristic
39	#7538	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
40	C₁₃₁	#7538	1	1	yes	characteristic, direct product factor, Sylow
41	C₂×Q₆₄	C₂×C₁₃₁	1	1	yes	characteristic
42	C₂×C₆₄	C₂×C₁₃₁	1	1	yes	characteristic
43	#1423	C₂×C₁₃₁	1	1	yes	characteristic
44	C₂×Q₃₂	—	2	1	no	—
45	Q₆₄	—	2	1	no	—
46	Q₆₄	C₄×C₁₃₁	1	2	yes	complemented
47	C₂×C₃₂	C₂²×C₁₃₁	1	1	yes	characteristic
48	#366	—	2	1	no	—
49	C₆₄	—	2	1	no	—
50	C₂×Q₁₆	—	4	1	no	—
51	Q₃₂	—	2	2	no	—
52	Q₃₂	—	4	1	no	—
53	C₂×C₁₆	D₄×C₁₃₁	1	1	yes	characteristic
54	#107	—	4	1	no	—
55	C₃₂	C₂×C₄×C₁₃₁	1	1	yes	characteristic
56	C₃₂	D₄×C₁₃₁	1	1	yes	characteristic
57	C₂×Q₈	—	8	1	no	—
58	Q₁₆	—	4	2	no	—
59	Q₁₆	—	8	1	no	—
60	C₂×C₈	D₈×C₁₃₁	1	1	yes	characteristic
61	C₄⋊C₄	—	8	1	no	—
62	C₁₆	C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
63	C₁₆	SD₁₆×C₁₃₁	1	1	yes	characteristic
64	Q₈	—	8	2	no	—
65	Q₈	—	16	1	no	—
66	C₂×C₄	—	16	1	no	—
67	C₂×C₄	—	16	1	no	—
68	C₂×C₄	D₁₆×C₁₃₁	1	1	yes	characteristic
69	C₈	#102×C₁₃₁	1	1	yes	characteristic
70	C₈	SD₃₂×C₁₃₁	1	1	yes	characteristic
71	C₂²	D₃₂×C₁₃₁	1	1	yes	characteristic
72	C₄	—	16	2	no	—
73	C₄	—	32	1	no	—
74	C₄	#357×C₁₃₁	1	1	yes	characteristic
75	C₄	SD₆₄×C₁₃₁	1	1	yes	characteristic
76	C₂	#1415×C₁₃₁	1	1	yes	characteristic
77	C₂	SD₁₂₈×C₁₃₁	1	1	yes	characteristic
78	C₂	Q₁₂₈×C₁₃₁	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₁₃₁	D₃₂
commutator	C₃₂	C₂×C₄×C₁₃₁
Frattini, cosocle	C₂×C₃₂	C₂²×C₁₃₁
LCS	G ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×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
6. Q₆₄×C₁₃₁	C₄	semidirect
39. #7538	C₁₃₁	direct (N × H)
40. C₁₃₁	#7538	direct (N × H)
46. Q₆₄	C₄×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₆₄×C₁₃₁	2
C₂×C₆₄×C₁₃₁	2
#1423×C₁₃₁	2
#7538	131

Subgroups — C131×#7538 #45693422

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#7538 #45693422