Subgroups — C₃²×C₅×#117 #13088226

Order 1440.

456 conjugacy classes of subgroups.

The subgroup lattice has 120 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃²×C₄×C₅	C₂	1	1	yes	maximal, characteristic
2	C₃²×C₅×C₄⋊C₄	C₂	1	2	yes	maximal, complemented
3	C₃²×C₅×C₂²⋊C₄	C₂	1	2	yes	maximal
4	C₃²×C₄²×C₅	C₂	1	2	yes	maximal, complemented
5	C₃×C₅×#117	C₃	1	4	yes	maximal, direct product factor
6	C₂³×C₃²×C₅	C₂²	1	1	yes	characteristic
7	C₂×C₃²×C₄×C₅	C₄	1	4	yes	complemented
8	C₂×C₃²×C₄×C₅	C₂²	1	4	yes	—
9	C₂×C₃²×C₄×C₅	C₂²	1	1	yes	characteristic
10	C₂×C₃²×C₄×C₅	C₂²	1	1	yes	characteristic
11	C₃²×#117	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
12	C₂²×C₃×C₄×C₅	C₂×C₃	1	4	yes	—
13	C₃×C₅×C₄⋊C₄	C₂×C₃	1	8	yes	complemented
14	C₃×C₅×C₂²⋊C₄	C₂×C₃	1	8	yes	—
15	C₃×C₄²×C₅	C₂×C₃	1	8	yes	complemented
16	C₂²×C₃²×C₅	—	2	2	no	—
17	C₂²×C₃²×C₅	C₂×C₄	1	2	yes	—
18	C₂²×C₃²×C₅	C₂³	1	1	yes	characteristic
19	C₃²×C₄×C₅	—	2	4	no	—
20	C₃²×C₄×C₅	C₂×C₄	1	2	yes	—
21	C₃²×C₄×C₅	C₂×C₄	1	2	yes	—
22	C₅×#117	C₃²	1	1	yes	characteristic, direct product factor, Hall
23	C₂²×C₃²×C₄	C₂×C₅	1	1	yes	characteristic
24	C₃²×C₄⋊C₄	C₂×C₅	1	2	yes	complemented
25	C₃²×C₂²⋊C₄	C₂×C₅	1	2	yes	—
26	C₃²×C₄²	C₂×C₅	1	2	yes	complemented
27	C₂³×C₃×C₅	C₂²×C₃	1	4	yes	—
28	C₂×C₃×C₄×C₅	C₃×C₄	1	16	yes	complemented
29	C₂×C₃×C₄×C₅	C₂²×C₃	1	16	yes	—
30	C₂×C₃×C₄×C₅	C₂²×C₃	1	4	yes	—
31	C₂×C₃×C₄×C₅	C₂²×C₃	1	4	yes	—
32	C₃×#117	C₃×C₅	1	4	yes	direct product factor
33	C₂×C₃²×C₅	—	2	2	no	—
34	C₂×C₃²×C₅	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₃²×C₅	#41	1	2	yes	—
36	C₂²×C₄×C₅	C₂×C₃²	1	1	yes	characteristic
37	C₅×C₄⋊C₄	C₂×C₃²	1	2	yes	complemented
38	C₅×C₂²⋊C₄	C₂×C₃²	1	2	yes	—
39	C₄²×C₅	C₂×C₃²	1	2	yes	complemented
40	C₂³×C₃²	C₂²×C₅	1	1	yes	characteristic
41	C₂×C₃²×C₄	C₄×C₅	1	4	yes	complemented
42	C₂×C₃²×C₄	C₂²×C₅	1	4	yes	—
43	C₂×C₃²×C₄	C₂²×C₅	1	1	yes	characteristic
44	C₂×C₃²×C₄	C₂²×C₅	1	1	yes	characteristic
45	C₂²×C₃×C₅	—	2	8	no	—
46	C₂²×C₃×C₅	C₂×C₃×C₄	1	8	yes	—
47	C₂²×C₃×C₅	C₂³×C₃	1	4	yes	—
48	C₃×C₄×C₅	—	2	16	no	—
49	C₃×C₄×C₅	C₂×C₃×C₄	1	8	yes	—
50	C₃×C₄×C₅	C₂×C₃×C₄	1	8	yes	—
51	C₂²×C₃×C₄	C₂×C₃×C₅	1	4	yes	—
52	C₃×C₄⋊C₄	C₂×C₃×C₅	1	8	yes	complemented
53	C₃×C₂²⋊C₄	C₂×C₃×C₅	1	8	yes	—
54	C₃×C₄²	C₂×C₃×C₅	1	8	yes	complemented
55	C₃²×C₅	#117	1	1	yes	characteristic, direct product factor, Hall
56	C₂³×C₅	C₂²×C₃²	1	1	yes	characteristic
57	C₂×C₄×C₅	C₃²×C₄	1	4	yes	complemented
58	C₂×C₄×C₅	C₂²×C₃²	1	4	yes	—
59	C₂×C₄×C₅	C₂²×C₃²	1	1	yes	characteristic
60	C₂×C₄×C₅	C₂²×C₃²	1	1	yes	characteristic
61	C₂²×C₃²	—	2	2	no	—
62	C₂²×C₃²	C₂×C₄×C₅	1	2	yes	—
63	C₂²×C₃²	C₂³×C₅	1	1	yes	characteristic
64	C₃²×C₄	—	2	4	no	—
65	C₃²×C₄	C₂×C₄×C₅	1	2	yes	—
66	C₃²×C₄	C₂×C₄×C₅	1	2	yes	—
67	#117	C₃²×C₅	1	1	yes	characteristic, direct product factor, Sylow
68	C₂×C₃×C₅	—	2	8	no	—
69	C₂×C₃×C₅	C₂²×C₃×C₄	1	4	yes	—
70	C₂×C₃×C₅	C₃×#41	1	8	yes	—
71	C₂³×C₃	C₂²×C₃×C₅	1	4	yes	—
72	C₂×C₃×C₄	C₃×C₄×C₅	1	16	yes	complemented
73	C₂×C₃×C₄	C₂²×C₃×C₅	1	16	yes	—
74	C₂×C₃×C₄	C₂²×C₃×C₅	1	4	yes	—
75	C₂×C₃×C₄	C₂²×C₃×C₅	1	4	yes	—
76	C₂²×C₅	—	2	2	no	—
77	C₂²×C₅	C₂×C₃²×C₄	1	2	yes	—
78	C₂²×C₅	C₂³×C₃²	1	1	yes	characteristic
79	C₄×C₅	—	2	4	no	—
80	C₄×C₅	C₂×C₃²×C₄	1	2	yes	—
81	C₄×C₅	C₂×C₃²×C₄	1	2	yes	—
82	C₂×C₃²	—	2	2	no	—
83	C₂×C₃²	C₂²×C₄×C₅	1	1	yes	characteristic
84	C₂×C₃²	C₅×#41	1	2	yes	—
85	C₂²×C₄	C₂×C₃²×C₅	1	1	yes	characteristic
86	C₄⋊C₄	C₂×C₃²×C₅	1	2	yes	complemented
87	C₂²⋊C₄	C₂×C₃²×C₅	1	2	yes	—
88	C₄²	C₂×C₃²×C₅	1	2	yes	complemented
89	C₃×C₅	C₃×#117	1	4	yes	direct product factor
90	C₂²×C₃	—	2	8	no	—
91	C₂²×C₃	C₂×C₃×C₄×C₅	1	8	yes	—
92	C₂²×C₃	C₂³×C₃×C₅	1	4	yes	—
93	C₃×C₄	—	2	16	no	—
94	C₃×C₄	C₂×C₃×C₄×C₅	1	8	yes	—
95	C₃×C₄	C₂×C₃×C₄×C₅	1	8	yes	—
96	C₂×C₅	—	2	2	no	—
97	C₂×C₅	C₂²×C₃²×C₄	1	1	yes	characteristic
98	C₂×C₅	C₃²×#41	1	2	yes	—
99	C₃²	C₅×#117	1	1	yes	characteristic, direct product factor, Sylow
100	C₂³	C₂²×C₃²×C₅	1	1	yes	characteristic
101	C₂×C₄	C₃²×C₄×C₅	1	4	yes	complemented
102	C₂×C₄	C₂²×C₃²×C₅	1	4	yes	—
103	C₂×C₄	C₂²×C₃²×C₅	1	1	yes	characteristic
104	C₂×C₄	C₂²×C₃²×C₅	1	1	yes	characteristic
105	C₂×C₃	—	2	8	no	—
106	C₂×C₃	C₂²×C₃×C₄×C₅	1	4	yes	—
107	C₂×C₃	C₃×C₅×#41	1	8	yes	—
108	C₅	C₃²×#117	1	1	yes	characteristic, direct product factor, Sylow
109	C₂²	—	2	2	no	—
110	C₂²	C₂×C₃²×C₄×C₅	1	2	yes	—
111	C₂²	C₂³×C₃²×C₅	1	1	yes	characteristic
112	C₄	—	2	4	no	—
113	C₄	C₂×C₃²×C₄×C₅	1	2	yes	—
114	C₄	C₂×C₃²×C₄×C₅	1	2	yes	—
115	C₃	C₃×C₅×#117	1	4	yes	direct product factor
116	C₂	—	2	2	no	—
117	C₂	C₂²×C₃²×C₄×C₅	1	1	yes	characteristic
118	C₂	C₃²×C₅×#41	1	2	yes	—

	subgroup	G/H
center	C₂×C₃²×C₄×C₅	C₂²
commutator	C₂	C₂²×C₃²×C₄×C₅
Frattini, cosocle	C₂²	C₂³×C₃²×C₅
socle	C₂²×C₃²×C₅	C₂³
prime radical	C₂³×C₃²×C₅	C₂²
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃²×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃²×C₅×C₄⋊C₄	C₂	semidirect
4. C₃²×C₄²×C₅	C₂	semidirect
5. C₃×C₅×#117	C₃	direct (N × H)
7. C₂×C₃²×C₄×C₅	C₄	semidirect
11. C₃²×#117	C₅	direct (N × H)
13. C₃×C₅×C₄⋊C₄	C₂×C₃	semidirect
15. C₃×C₄²×C₅	C₂×C₃	semidirect
22. C₅×#117	C₃²	direct (N × H)
24. C₃²×C₄⋊C₄	C₂×C₅	semidirect
26. C₃²×C₄²	C₂×C₅	semidirect
28. C₂×C₃×C₄×C₅	C₃×C₄	semidirect
32. C₃×#117	C₃×C₅	direct (N × H)
37. C₅×C₄⋊C₄	C₂×C₃²	semidirect
39. C₄²×C₅	C₂×C₃²	semidirect
41. C₂×C₃²×C₄	C₄×C₅	semidirect
52. C₃×C₄⋊C₄	C₂×C₃×C₅	semidirect
54. C₃×C₄²	C₂×C₃×C₅	semidirect
55. C₃²×C₅	#117	direct (N × H)
57. C₂×C₄×C₅	C₃²×C₄	semidirect
67. #117	C₃²×C₅	direct (N × H)
72. C₂×C₃×C₄	C₃×C₄×C₅	semidirect
86. C₄⋊C₄	C₂×C₃²×C₅	semidirect
88. C₄²	C₂×C₃²×C₅	semidirect
89. C₃×C₅	C₃×#117	direct (N × H)
99. C₃²	C₅×#117	direct (N × H)
101. C₂×C₄	C₃²×C₄×C₅	semidirect
108. C₅	C₃²×#117	direct (N × H)
115. C₃	C₃×C₅×#117	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃²×C₄×C₅	2
C₃²×C₅×C₄⋊C₄	2
C₃²×C₅×C₂²⋊C₄	2
C₃²×C₄²×C₅	2
C₃×C₅×#117	3
C₃²×#117	5

Subgroups — C32×C5×#117 #13088226

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₅×#117 #13088226