Subgroups — C₃×C₅×#890 #13087902

Order 1440.

250 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₅×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×C₄⋊Dic₃	C₂	1	2	yes	maximal
3	C₃×C₅×C₄⋊Dic₃	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×Dic₃⋊C₄	C₂	1	2	yes	maximal
5	C₃×C₄×C₅×Dic₃	C₂	1	1	yes	maximal, characteristic
6	C₃×C₅×#125	—	3	1	no	maximal
7	C₅×#890	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂×C₃²×C₄×C₅	C₂²	1	2	yes	—
9	C₂×C₃²×C₄×C₅	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₅×Dic₃	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₅×Dic₃	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₅×Dic₃	C₂²	1	2	yes	—
13	C₃×#890	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
14	C₃×C₅×C₄⋊C₄	—	2	1	no	—
15	C₃×C₅×C₄⋊C₄	—	3	2	no	—
16	C₃×C₅×C₄⋊C₄	—	3	2	no	—
17	C₃×C₅×C₄⋊C₄	—	3	1	no	—
18	C₃×C₅×C₄⋊C₄	D₃	1	1	yes	characteristic
19	C₃×C₅×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
20	C₃×C₄²×C₅	—	3	1	no	—
21	C₅×C₄⋊Dic₃	C₂×C₃	1	2	yes	—
22	C₅×C₄⋊Dic₃	C₂×C₃	1	1	yes	characteristic
23	C₅×Dic₃⋊C₄	C₂×C₃	1	2	yes	—
24	C₄×C₅×Dic₃	C₂×C₃	1	1	yes	characteristic
25	C₂²×C₃²×C₅	C₂³	1	1	yes	characteristic
26	C₃²×C₄×C₅	—	2	2	no	—
27	C₃²×C₄×C₅	Q₈	1	2	yes	—
28	C₃×C₅×Dic₃	—	2	1	no	—
29	C₃×C₅×Dic₃	—	2	1	no	—
30	C₃×C₅×Dic₃	—	2	2	no	—
31	C₅×#125	—	3	1	no	Hall
32	C₃²×C₄⋊C₄	C₂×C₅	1	1	yes	characteristic
33	C₃×C₄⋊Dic₃	C₂×C₅	1	2	yes	—
34	C₃×C₄⋊Dic₃	C₂×C₅	1	1	yes	characteristic
35	C₃×Dic₃⋊C₄	C₂×C₅	1	2	yes	—
36	C₃×C₄×Dic₃	C₂×C₅	1	1	yes	characteristic
37	C₂×C₃×C₄×C₅	—	2	1	no	—
38	C₂×C₃×C₄×C₅	—	2	2	no	—
39	C₂×C₃×C₄×C₅	—	3	1	no	—
40	C₂×C₃×C₄×C₅	—	3	1	no	—
41	C₂×C₃×C₄×C₅	—	3	2	no	—
42	C₂×C₃×C₄×C₅	C₂×D₃	1	2	yes	—
43	C₂×C₃×C₄×C₅	C₂×D₃	1	1	yes	characteristic
44	C₂×C₃×C₄×C₅	C₂²×C₃	1	2	yes	—
45	C₂×C₃×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
46	C₂×C₅×Dic₃	C₂²×C₃	1	1	yes	characteristic
47	C₂×C₅×Dic₃	C₂²×C₃	1	1	yes	characteristic
48	C₂×C₅×Dic₃	C₂²×C₃	1	2	yes	—
49	C₃×#125	—	3	1	no	—
50	#890	C₃×C₅	1	1	yes	characteristic, direct product factor
51	C₂×C₃²×C₅	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₃²×C₅	#41	1	1	yes	characteristic
53	C₂×C₃²×C₅	#41	1	1	yes	characteristic
54	C₅×C₄⋊C₄	—	3	2	no	—
55	C₅×C₄⋊C₄	—	3	2	no	—
56	C₅×C₄⋊C₄	—	3	1	no	—
57	C₅×C₄⋊C₄	C₃×D₃	1	1	yes	characteristic
58	C₄²×C₅	—	3	1	no	—
59	C₂×C₃²×C₄	C₂²×C₅	1	2	yes	—
60	C₂×C₃²×C₄	C₂²×C₅	1	1	yes	characteristic
61	C₂×C₃×Dic₃	C₂²×C₅	1	1	yes	characteristic
62	C₂×C₃×Dic₃	C₂²×C₅	1	1	yes	characteristic
63	C₂×C₃×Dic₃	C₂²×C₅	1	2	yes	—
64	C₂²×C₃×C₅	—	2	1	no	—
65	C₂²×C₃×C₅	C₂²×D₃	1	1	yes	characteristic
66	C₂²×C₃×C₅	C₂³×C₃	1	1	yes	characteristic
67	C₃×C₄×C₅	—	2	2	no	—
68	C₃×C₄×C₅	—	2	2	no	—
69	C₃×C₄×C₅	—	2	2	no	—
70	C₃×C₄×C₅	—	4	2	no	—
71	C₃×C₄×C₅	—	6	1	no	—
72	C₃×C₄×C₅	—	6	1	no	—
73	C₃×C₄×C₅	—	6	2	no	—
74	C₃×C₄×C₅	Dic₆	1	2	yes	—
75	C₃×C₄×C₅	C₃×Q₈	1	2	yes	—
76	C₅×Dic₃	—	2	1	no	—
77	C₅×Dic₃	—	2	1	no	—
78	C₅×Dic₃	—	2	2	no	—
79	C₃×C₄⋊C₄	—	2	1	no	—
80	C₃×C₄⋊C₄	—	3	2	no	—
81	C₃×C₄⋊C₄	—	3	2	no	—
82	C₃×C₄⋊C₄	—	3	1	no	—
83	C₃×C₄⋊C₄	C₅×D₃	1	1	yes	characteristic
84	C₃×C₄⋊C₄	C₂×C₃×C₅	1	1	yes	characteristic
85	C₃×C₄²	—	3	1	no	—
86	C₄⋊Dic₃	C₂×C₃×C₅	1	2	yes	—
87	C₄⋊Dic₃	C₂×C₃×C₅	1	1	yes	characteristic
88	Dic₃⋊C₄	C₂×C₃×C₅	1	2	yes	—
89	C₄×Dic₃	C₂×C₃×C₅	1	1	yes	characteristic
90	C₃²×C₅	#125	1	1	yes	characteristic, complemented, Hall
91	C₂×C₄×C₅	—	3	1	no	—
92	C₂×C₄×C₅	—	3	1	no	—
93	C₂×C₄×C₅	—	3	2	no	—
94	C₂×C₄×C₅	C₂×C₃×D₃	1	2	yes	—
95	C₂×C₄×C₅	C₂×C₃×D₃	1	1	yes	characteristic
96	C₂²×C₃²	C₂³×C₅	1	1	yes	characteristic
97	C₃²×C₄	—	2	2	no	—
98	C₃²×C₄	C₅×Q₈	1	2	yes	—
99	C₃×Dic₃	—	2	1	no	—
100	C₃×Dic₃	—	2	1	no	—
101	C₃×Dic₃	—	2	2	no	—
102	#125	—	3	1	no	Sylow
103	C₂×C₃×C₅	—	2	1	no	—
104	C₂×C₃×C₅	—	2	1	no	—
105	C₂×C₃×C₅	—	2	1	no	—
106	C₂×C₃×C₅	C₂×Dic₆	1	1	yes	characteristic
107	C₂×C₃×C₅	#237	1	1	yes	characteristic
108	C₂×C₃×C₅	#239	1	1	yes	characteristic
109	C₂×C₃×C₅	C₂×C₃×Q₈	1	1	yes	characteristic
110	C₂×C₃×C₅	C₃×#41	1	1	yes	characteristic
111	C₂×C₃×C₅	C₃×#41	1	1	yes	characteristic
112	C₂×C₃×C₄	—	2	1	no	—
113	C₂×C₃×C₄	—	2	2	no	—
114	C₂×C₃×C₄	—	3	1	no	—
115	C₂×C₃×C₄	—	3	1	no	—
116	C₂×C₃×C₄	—	3	2	no	—
117	C₂×C₃×C₄	C₂×C₅×D₃	1	2	yes	—
118	C₂×C₃×C₄	C₂×C₅×D₃	1	1	yes	characteristic
119	C₂×C₃×C₄	C₂²×C₃×C₅	1	2	yes	—
120	C₂×C₃×C₄	C₂²×C₃×C₅	1	1	yes	characteristic
121	C₂×Dic₃	C₂²×C₃×C₅	1	1	yes	characteristic
122	C₂×Dic₃	C₂²×C₃×C₅	1	1	yes	characteristic
123	C₂×Dic₃	C₂²×C₃×C₅	1	2	yes	—
124	C₂²×C₅	C₂²×C₃×D₃	1	1	yes	characteristic
125	C₄×C₅	—	2	2	no	—
126	C₄×C₅	—	6	1	no	—
127	C₄×C₅	—	6	1	no	—
128	C₄×C₅	—	6	2	no	—
129	C₄×C₅	C₃×Dic₆	1	2	yes	—
130	C₂×C₃²	C₂×C₅×Q₈	1	1	yes	characteristic
131	C₂×C₃²	C₅×#41	1	1	yes	characteristic
132	C₂×C₃²	C₅×#41	1	1	yes	characteristic
133	C₄⋊C₄	—	3	2	no	—
134	C₄⋊C₄	—	3	2	no	—
135	C₄⋊C₄	—	3	1	no	—
136	C₄⋊C₄	C₃×C₅×D₃	1	1	yes	characteristic
137	C₄²	—	3	1	no	—
138	C₃×C₅	—	2	1	no	—
139	C₃×C₅	#890	1	1	yes	characteristic, direct product factor
140	C₃×C₅	C₃×#125	1	1	yes	characteristic, complemented
141	C₂²×C₃	—	2	1	no	—
142	C₂²×C₃	C₂²×C₅×D₃	1	1	yes	characteristic
143	C₂²×C₃	C₂³×C₃×C₅	1	1	yes	characteristic
144	C₃×C₄	—	2	2	no	—
145	C₃×C₄	—	2	2	no	—
146	C₃×C₄	—	2	2	no	—
147	C₃×C₄	—	4	2	no	—
148	C₃×C₄	—	6	1	no	—
149	C₃×C₄	—	6	1	no	—
150	C₃×C₄	—	6	2	no	—
151	C₃×C₄	C₅×Dic₆	1	2	yes	—
152	C₃×C₄	C₃×C₅×Q₈	1	2	yes	—
153	Dic₃	—	2	1	no	—
154	Dic₃	—	2	2	no	—
155	Dic₃	—	2	1	no	—
156	C₂×C₅	C₂×C₃×Dic₆	1	1	yes	characteristic
157	C₂×C₅	C₃×#237	1	1	yes	characteristic
158	C₂×C₅	C₃×#239	1	1	yes	characteristic
159	C₃²	C₅×#125	1	1	yes	characteristic, complemented, Sylow
160	C₂×C₄	—	3	1	no	—
161	C₂×C₄	—	3	1	no	—
162	C₂×C₄	—	3	2	no	—
163	C₂×C₄	C₂×C₃×C₅×D₃	1	2	yes	—
164	C₂×C₄	C₂×C₃×C₅×D₃	1	1	yes	characteristic
165	C₂×C₃	—	2	1	no	—
166	C₂×C₃	—	2	1	no	—
167	C₂×C₃	—	2	1	no	—
168	C₂×C₃	C₂×C₅×Dic₆	1	1	yes	characteristic
169	C₂×C₃	C₅×#237	1	1	yes	characteristic
170	C₂×C₃	C₅×#239	1	1	yes	characteristic
171	C₂×C₃	C₂×C₃×C₅×Q₈	1	1	yes	characteristic
172	C₂×C₃	C₃×C₅×#41	1	1	yes	characteristic
173	C₂×C₃	C₃×C₅×#41	1	1	yes	characteristic
174	C₅	C₃×#890	1	1	yes	characteristic, direct product factor, Sylow
175	C₂²	C₂²×C₃×C₅×D₃	1	1	yes	characteristic
176	C₄	—	2	2	no	—
177	C₄	—	6	1	no	—
178	C₄	—	6	2	no	—
179	C₄	—	6	1	no	—
180	C₄	C₃×C₅×Dic₆	1	2	yes	—
181	C₃	—	2	1	no	—
182	C₃	C₅×#890	1	1	yes	characteristic, direct product factor
183	C₃	C₃×C₅×#125	1	1	yes	characteristic, complemented
184	C₂	C₂×C₃×C₅×Dic₆	1	1	yes	characteristic
185	C₂	C₃×C₅×#237	1	1	yes	characteristic
186	C₂	C₃×C₅×#239	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃×C₅	C₂²×D₃
commutator, cosocle	C₂²×C₃	C₂³×C₃×C₅
Frattini	C₂²	C₂²×C₃×C₅×D₃
Fitting	C₃²×C₅×C₄⋊C₄	C₂
socle, prime radical	C₂²×C₃²×C₅	C₂³
hypercenter	C₃×C₅×C₄⋊C₄	D₃
hypocenter	C₃	C₃×C₅×#125
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃×C₅ ◁ C₃×C₅×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
7. C₅×#890	C₃	direct (N × H)
13. C₃×#890	C₅	direct (N × H)
50. #890	C₃×C₅	direct (N × H)
90. C₃²×C₅	#125	semidirect
139. C₃×C₅	#890	direct (N × H)
140. C₃×C₅	C₃×#125	semidirect
159. C₃²	C₅×#125	semidirect
174. C₅	C₃×#890	direct (N × H)
182. C₃	C₅×#890	direct (N × H)
183. C₃	C₃×C₅×#125	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₅×C₄⋊C₄	2
C₃×C₅×C₄⋊Dic₃	2
C₃×C₅×Dic₃⋊C₄	2
C₃×C₄×C₅×Dic₃	2
C₃×C₅×#125	3
C₅×#890	3
C₃×#890	5

Subgroups — C3×C5×#890 #13087902

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#890 #13087902