Subgroups — C₂×C₄×C₅×D₃² #13089315

Order 1440.

998 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×D₃²	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₅×C₃⋊D₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×C₄×C₅×D₃	C₂	1	2	yes	maximal, complemented
4	C₂×C₅×#624	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₅×D₃×Dic₃	C₂	1	2	yes	maximal, complemented
6	C₄×C₅×D₃²	C₂	1	8	yes	maximal, direct product factor
7	C₂²×C₄×C₅×D₃	—	3	2	no	maximal
8	C₂²×C₅×C₃⋊D₃	C₂²	1	1	yes	characteristic
9	C₂²×C₃×C₅×D₃	C₂²	1	2	yes	—
10	C₂×C₅×D₃²	C₄	1	8	yes	direct product factor
11	C₂×C₅×D₃²	C₂²	1	4	yes	—
12	C₂×C₃²×C₄×C₅	C₂²	1	1	yes	characteristic, complemented
13	C₂×C₅×C₃⋊Dic₃	C₂²	1	1	yes	characteristic, complemented
14	C₄×C₅×C₃⋊D₃	C₂²	1	4	yes	complemented
15	C₂×C₃×C₅×Dic₃	C₂²	1	2	yes	complemented
16	C₃×C₄×C₅×D₃	C₂²	1	8	yes	complemented
17	C₅×#624	C₂²	1	4	yes	complemented
18	C₅×D₃×Dic₃	C₂²	1	8	yes	complemented
19	C₂×C₄×D₃²	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
20	C₂³×C₅×D₃	—	3	2	no	—
21	C₂²×C₃×C₄×C₅	—	3	2	no	—
22	C₂²×C₅×Dic₃	—	3	2	no	—
23	C₂×C₄×C₅×D₃	—	3	2	no	—
24	C₂×C₄×C₅×D₃	—	3	2	no	—
25	C₂×C₄×C₅×D₃	—	3	2	no	—
26	C₂×C₄×C₅×D₃	—	3	16	no	—
27	C₂×C₄×C₅×D₃	—	6	1	no	—
28	C₂×C₄×C₅×D₃	D₃	1	2	yes	direct product factor
29	C₂²×C₃²×C₅	C₂³	1	1	yes	characteristic
30	C₂×C₅×C₃⋊D₃	C₂×C₄	1	4	yes	complemented
31	C₂×C₅×C₃⋊D₃	C₂³	1	2	yes	—
32	C₂×C₃×C₅×D₃	C₂×C₄	1	8	yes	complemented
33	C₂×C₃×C₅×D₃	C₂³	1	4	yes	—
34	C₅×D₃²	C₂×C₄	1	16	yes	direct product factor
35	C₃²×C₄×C₅	C₂³	1	2	yes	complemented
36	C₅×C₃⋊Dic₃	C₂³	1	2	yes	complemented
37	C₃×C₅×Dic₃	C₂³	1	4	yes	complemented
38	C₂³×C₄×C₅	—	9	1	no	Hall
39	C₂²×D₃²	C₂×C₅	1	1	yes	characteristic
40	C₂×C₄×C₃⋊D₃	C₂×C₅	1	1	yes	characteristic, complemented
41	C₂×C₃×C₄×D₃	C₂×C₅	1	2	yes	complemented
42	C₂×#624	C₂×C₅	1	1	yes	characteristic, complemented
43	C₂×D₃×Dic₃	C₂×C₅	1	2	yes	complemented
44	C₄×D₃²	C₂×C₅	1	8	yes	direct product factor
45	C₂³×C₃×C₅	—	3	2	no	—
46	C₂²×C₅×D₃	—	3	2	no	—
47	C₂²×C₅×D₃	—	3	16	no	—
48	C₂²×C₅×D₃	—	3	8	no	—
49	C₂²×C₅×D₃	—	6	1	no	—
50	C₂²×C₅×D₃	C₂×D₃	1	2	yes	—
51	C₂×C₃×C₄×C₅	—	2	1	no	—
52	C₂×C₃×C₄×C₅	—	3	2	no	—
53	C₂×C₃×C₄×C₅	—	3	8	no	—
54	C₂×C₃×C₄×C₅	C₂×D₃	1	2	yes	complemented
55	C₂×C₅×Dic₃	—	3	2	no	—
56	C₂×C₅×Dic₃	—	3	8	no	—
57	C₂×C₅×Dic₃	—	6	1	no	—
58	C₂×C₅×Dic₃	C₂×D₃	1	2	yes	complemented
59	C₄×C₅×D₃	—	3	8	no	—
60	C₄×C₅×D₃	—	3	8	no	—
61	C₄×C₅×D₃	—	3	8	no	—
62	C₄×C₅×D₃	—	6	4	no	—
63	C₄×C₅×D₃	C₂×D₃	1	8	yes	direct product factor
64	C₂²×C₄×D₃	—	3	2	no	—
65	C₂×C₃²×C₅	C₂²×C₄	1	2	yes	complemented
66	C₂×C₃²×C₅	C₂⁴	1	1	yes	characteristic
67	C₅×C₃⋊D₃	C₂²×C₄	1	4	yes	complemented
68	C₃×C₅×D₃	C₂²×C₄	1	8	yes	complemented
69	C₂⁴×C₅	—	9	1	no	—
70	C₂²×C₄×C₅	—	3	2	no	—
71	C₂²×C₄×C₅	—	9	1	no	—
72	C₂²×C₄×C₅	—	9	8	no	—
73	C₂²×C₄×C₅	—	9	2	no	—
74	C₂²×C₄×C₅	—	9	1	no	—
75	C₂²×C₃⋊D₃	C₂²×C₅	1	1	yes	characteristic
76	C₂²×C₃×D₃	C₂²×C₅	1	2	yes	—
77	C₂×D₃²	C₄×C₅	1	8	yes	direct product factor
78	C₂×D₃²	C₂²×C₅	1	4	yes	—
79	C₂×C₃²×C₄	C₂²×C₅	1	1	yes	characteristic, complemented
80	C₂×C₃⋊Dic₃	C₂²×C₅	1	1	yes	characteristic, complemented
81	C₄×C₃⋊D₃	C₂²×C₅	1	4	yes	complemented
82	C₂×C₃×Dic₃	C₂²×C₅	1	2	yes	complemented
83	C₃×C₄×D₃	C₂²×C₅	1	8	yes	complemented
84	#624	C₂²×C₅	1	4	yes	complemented
85	D₃×Dic₃	C₂²×C₅	1	8	yes	complemented
86	C₂²×C₃×C₅	—	2	1	no	—
87	C₂²×C₃×C₅	—	3	8	no	—
88	C₂²×C₃×C₅	—	3	4	no	—
89	C₂²×C₃×C₅	C₂²×D₃	1	2	yes	—
90	C₂×C₅×D₃	—	3	8	no	—
91	C₂×C₅×D₃	—	3	4	no	—
92	C₂×C₅×D₃	—	3	32	no	—
93	C₂×C₅×D₃	—	6	4	no	—
94	C₂×C₅×D₃	—	6	2	no	—
95	C₂×C₅×D₃	C₄×D₃	1	8	yes	direct product factor
96	C₂×C₅×D₃	C₂²×D₃	1	4	yes	—
97	C₃×C₄×C₅	—	2	2	no	—
98	C₃×C₄×C₅	—	3	4	no	—
99	C₃×C₄×C₅	C₂²×D₃	1	4	yes	complemented
100	C₅×Dic₃	—	3	4	no	—
101	C₅×Dic₃	—	6	2	no	—
102	C₅×Dic₃	C₂²×D₃	1	4	yes	complemented
103	C₂³×D₃	—	3	2	no	—
104	C₂²×C₃×C₄	—	3	2	no	—
105	C₂²×Dic₃	—	3	2	no	—
106	C₂×C₄×D₃	—	3	2	no	—
107	C₂×C₄×D₃	—	3	2	no	—
108	C₂×C₄×D₃	—	3	2	no	—
109	C₂×C₄×D₃	—	3	16	no	—
110	C₂×C₄×D₃	—	6	1	no	—
111	C₂×C₄×D₃	C₅×D₃	1	2	yes	direct product factor
112	C₃²×C₅	C₂³×C₄	1	1	yes	characteristic, complemented, Hall
113	C₂³×C₅	—	3	2	no	—
114	C₂³×C₅	—	9	4	no	—
115	C₂³×C₅	—	9	8	no	—
116	C₂³×C₅	—	9	1	no	—
117	C₂×C₄×C₅	—	3	2	no	—
118	C₂×C₄×C₅	—	3	8	no	—
119	C₂×C₄×C₅	—	9	4	no	—
120	C₂×C₄×C₅	—	9	8	no	—
121	C₂×C₄×C₅	—	9	1	no	—
122	C₂×C₄×C₅	—	9	4	no	—
123	C₂×C₄×C₅	D₃²	1	1	yes	characteristic, direct product factor
124	C₂²×C₃²	C₂³×C₅	1	1	yes	characteristic
125	C₂×C₃⋊D₃	C₂×C₄×C₅	1	4	yes	complemented
126	C₂×C₃⋊D₃	C₂³×C₅	1	2	yes	—
127	C₂×C₃×D₃	C₂×C₄×C₅	1	8	yes	complemented
128	C₂×C₃×D₃	C₂³×C₅	1	4	yes	—
129	D₃²	C₂×C₄×C₅	1	16	yes	direct product factor
130	C₃²×C₄	C₂³×C₅	1	2	yes	complemented
131	C₃⋊Dic₃	C₂³×C₅	1	2	yes	complemented
132	C₃×Dic₃	C₂³×C₅	1	4	yes	complemented
133	C₂³×C₄	—	9	1	no	Sylow
134	C₂×C₃×C₅	—	2	1	no	—
135	C₂×C₃×C₅	—	2	2	no	—
136	C₂×C₃×C₅	—	3	8	no	—
137	C₂×C₃×C₅	C₂×C₄×D₃	1	4	yes	complemented
138	C₂×C₃×C₅	C₂³×D₃	1	2	yes	—
139	C₅×D₃	—	3	8	no	—
140	C₅×D₃	—	6	4	no	—
141	C₅×D₃	C₂×C₄×D₃	1	8	yes	direct product factor
142	C₂³×C₃	—	3	2	no	—
143	C₂²×D₃	—	3	2	no	—
144	C₂²×D₃	—	3	16	no	—
145	C₂²×D₃	—	3	8	no	—
146	C₂²×D₃	—	6	1	no	—
147	C₂²×D₃	C₂×C₅×D₃	1	2	yes	—
148	C₂×C₃×C₄	—	2	1	no	—
149	C₂×C₃×C₄	—	3	2	no	—
150	C₂×C₃×C₄	—	3	8	no	—
151	C₂×C₃×C₄	C₂×C₅×D₃	1	2	yes	complemented
152	C₂×Dic₃	—	3	2	no	—
153	C₂×Dic₃	—	3	8	no	—
154	C₂×Dic₃	—	6	1	no	—
155	C₂×Dic₃	C₂×C₅×D₃	1	2	yes	complemented
156	C₄×D₃	—	3	8	no	—
157	C₄×D₃	—	3	8	no	—
158	C₄×D₃	—	3	8	no	—
159	C₄×D₃	—	6	4	no	—
160	C₄×D₃	C₂×C₅×D₃	1	8	yes	direct product factor
161	C₂²×C₅	—	3	4	no	—
162	C₂²×C₅	—	3	8	no	—
163	C₂²×C₅	—	9	16	no	—
164	C₂²×C₅	—	9	2	no	—
165	C₂²×C₅	—	9	4	no	—
166	C₂²×C₅	C₂×D₃²	1	1	yes	characteristic
167	C₄×C₅	—	3	4	no	—
168	C₄×C₅	—	9	2	no	—
169	C₄×C₅	C₂×D₃²	1	2	yes	direct product factor
170	C₂×C₃²	C₂²×C₄×C₅	1	2	yes	complemented
171	C₂×C₃²	C₂⁴×C₅	1	1	yes	characteristic
172	C₃⋊D₃	C₂²×C₄×C₅	1	4	yes	complemented
173	C₃×D₃	C₂²×C₄×C₅	1	8	yes	complemented
174	C₂⁴	—	9	1	no	—
175	C₂²×C₄	—	3	2	no	—
176	C₂²×C₄	—	9	1	no	—
177	C₂²×C₄	—	9	8	no	—
178	C₂²×C₄	—	9	2	no	—
179	C₂²×C₄	—	9	1	no	—
180	C₃×C₅	—	2	1	no	—
181	C₃×C₅	C₂²×C₄×D₃	1	2	yes	complemented
182	C₂²×C₃	—	2	1	no	—
183	C₂²×C₃	—	3	8	no	—
184	C₂²×C₃	—	3	4	no	—
185	C₂²×C₃	C₂²×C₅×D₃	1	2	yes	—
186	C₂×D₃	—	3	8	no	—
187	C₂×D₃	—	3	4	no	—
188	C₂×D₃	—	3	32	no	—
189	C₂×D₃	—	6	2	no	—
190	C₂×D₃	—	6	4	no	—
191	C₂×D₃	C₄×C₅×D₃	1	8	yes	direct product factor
192	C₂×D₃	C₂²×C₅×D₃	1	4	yes	—
193	C₃×C₄	—	2	2	no	—
194	C₃×C₄	—	3	4	no	—
195	C₃×C₄	C₂²×C₅×D₃	1	4	yes	complemented
196	Dic₃	—	3	4	no	—
197	Dic₃	—	6	2	no	—
198	Dic₃	C₂²×C₅×D₃	1	4	yes	complemented
199	C₂×C₅	—	3	8	no	—
200	C₂×C₅	—	9	4	no	—
201	C₂×C₅	C₄×D₃²	1	2	yes	direct product factor
202	C₂×C₅	C₂²×D₃²	1	1	yes	characteristic
203	C₃²	C₂³×C₄×C₅	1	1	yes	characteristic, complemented, Sylow
204	C₂³	—	3	2	no	—
205	C₂³	—	9	4	no	—
206	C₂³	—	9	8	no	—
207	C₂³	—	9	1	no	—
208	C₂×C₄	—	3	2	no	—
209	C₂×C₄	—	3	8	no	—
210	C₂×C₄	—	9	4	no	—
211	C₂×C₄	—	9	8	no	—
212	C₂×C₄	—	9	1	no	—
213	C₂×C₄	—	9	4	no	—
214	C₂×C₄	C₅×D₃²	1	1	yes	characteristic, direct product factor
215	C₂×C₃	—	2	1	no	—
216	C₂×C₃	—	2	2	no	—
217	C₂×C₃	—	3	8	no	—
218	C₂×C₃	C₂×C₄×C₅×D₃	1	4	yes	complemented
219	C₂×C₃	C₂³×C₅×D₃	1	2	yes	—
220	D₃	—	3	8	no	—
221	D₃	—	6	4	no	—
222	D₃	C₂×C₄×C₅×D₃	1	8	yes	direct product factor
223	C₅	C₂×C₄×D₃²	1	1	yes	characteristic, direct product factor, Sylow
224	C₂²	—	3	4	no	—
225	C₂²	—	3	8	no	—
226	C₂²	—	9	16	no	—
227	C₂²	—	9	2	no	—
228	C₂²	—	9	4	no	—
229	C₂²	C₂×C₅×D₃²	1	1	yes	characteristic
230	C₄	—	3	4	no	—
231	C₄	—	9	2	no	—
232	C₄	C₂×C₅×D₃²	1	2	yes	direct product factor
233	C₃	—	2	1	no	—
234	C₃	C₂²×C₄×C₅×D₃	1	2	yes	complemented
235	C₂	—	3	8	no	—
236	C₂	—	9	4	no	—
237	C₂	C₄×C₅×D₃²	1	2	yes	direct product factor
238	C₂	C₂²×C₅×D₃²	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₄×C₅	D₃²
commutator, hypocenter	C₃²	C₂³×C₄×C₅
Frattini	C₂	C₂²×C₅×D₃²
Fitting	C₂×C₃²×C₄×C₅	C₂²
socle	C₂²×C₃²×C₅	C₂³
prime radical	C₂²×C₅×D₃²	C₂
cosocle	C₂×C₃²	C₂⁴×C₅
LCS	G ▷ C₃²
UCS	C₁ ◁ C₂×C₄×C₅

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×C₅×C₃⋊D₃	C₂	semidirect
3. C₂×C₃×C₄×C₅×D₃	C₂	semidirect
4. C₂×C₅×#624	C₂	semidirect
5. C₂×C₅×D₃×Dic₃	C₂	semidirect
6. C₄×C₅×D₃²	C₂	direct (N × H)
10. C₂×C₅×D₃²	C₄	direct (N × H)
12. C₂×C₃²×C₄×C₅	C₂²	semidirect
13. C₂×C₅×C₃⋊Dic₃	C₂²	semidirect
14. C₄×C₅×C₃⋊D₃	C₂²	semidirect
15. C₂×C₃×C₅×Dic₃	C₂²	semidirect
16. C₃×C₄×C₅×D₃	C₂²	semidirect
17. C₅×#624	C₂²	semidirect
18. C₅×D₃×Dic₃	C₂²	semidirect
19. C₂×C₄×D₃²	C₅	direct (N × H)
28. C₂×C₄×C₅×D₃	D₃	direct (N × H)
30. C₂×C₅×C₃⋊D₃	C₂×C₄	semidirect
32. C₂×C₃×C₅×D₃	C₂×C₄	semidirect
34. C₅×D₃²	C₂×C₄	direct (N × H)
35. C₃²×C₄×C₅	C₂³	semidirect
36. C₅×C₃⋊Dic₃	C₂³	semidirect
37. C₃×C₅×Dic₃	C₂³	semidirect
40. C₂×C₄×C₃⋊D₃	C₂×C₅	semidirect
41. C₂×C₃×C₄×D₃	C₂×C₅	semidirect
42. C₂×#624	C₂×C₅	semidirect
43. C₂×D₃×Dic₃	C₂×C₅	semidirect
44. C₄×D₃²	C₂×C₅	direct (N × H)
54. C₂×C₃×C₄×C₅	C₂×D₃	semidirect
58. C₂×C₅×Dic₃	C₂×D₃	semidirect
63. C₄×C₅×D₃	C₂×D₃	direct (N × H)
65. C₂×C₃²×C₅	C₂²×C₄	semidirect
67. C₅×C₃⋊D₃	C₂²×C₄	semidirect
68. C₃×C₅×D₃	C₂²×C₄	semidirect
77. C₂×D₃²	C₄×C₅	direct (N × H)
79. C₂×C₃²×C₄	C₂²×C₅	semidirect
80. C₂×C₃⋊Dic₃	C₂²×C₅	semidirect
81. C₄×C₃⋊D₃	C₂²×C₅	semidirect
82. C₂×C₃×Dic₃	C₂²×C₅	semidirect
83. C₃×C₄×D₃	C₂²×C₅	semidirect
84. #624	C₂²×C₅	semidirect
85. D₃×Dic₃	C₂²×C₅	semidirect
95. C₂×C₅×D₃	C₄×D₃	direct (N × H)
99. C₃×C₄×C₅	C₂²×D₃	semidirect
102. C₅×Dic₃	C₂²×D₃	semidirect
111. C₂×C₄×D₃	C₅×D₃	direct (N × H)
112. C₃²×C₅	C₂³×C₄	semidirect
123. C₂×C₄×C₅	D₃²	direct (N × H)
125. C₂×C₃⋊D₃	C₂×C₄×C₅	semidirect
127. C₂×C₃×D₃	C₂×C₄×C₅	semidirect
129. D₃²	C₂×C₄×C₅	direct (N × H)
130. C₃²×C₄	C₂³×C₅	semidirect
131. C₃⋊Dic₃	C₂³×C₅	semidirect
132. C₃×Dic₃	C₂³×C₅	semidirect
137. C₂×C₃×C₅	C₂×C₄×D₃	semidirect
141. C₅×D₃	C₂×C₄×D₃	direct (N × H)
151. C₂×C₃×C₄	C₂×C₅×D₃	semidirect
155. C₂×Dic₃	C₂×C₅×D₃	semidirect
160. C₄×D₃	C₂×C₅×D₃	direct (N × H)
169. C₄×C₅	C₂×D₃²	direct (N × H)
170. C₂×C₃²	C₂²×C₄×C₅	semidirect
172. C₃⋊D₃	C₂²×C₄×C₅	semidirect
173. C₃×D₃	C₂²×C₄×C₅	semidirect
181. C₃×C₅	C₂²×C₄×D₃	semidirect
191. C₂×D₃	C₄×C₅×D₃	direct (N × H)
195. C₃×C₄	C₂²×C₅×D₃	semidirect
198. Dic₃	C₂²×C₅×D₃	semidirect
201. C₂×C₅	C₄×D₃²	direct (N × H)
203. C₃²	C₂³×C₄×C₅	semidirect
214. C₂×C₄	C₅×D₃²	direct (N × H)
218. C₂×C₃	C₂×C₄×C₅×D₃	semidirect
222. D₃	C₂×C₄×C₅×D₃	direct (N × H)
223. C₅	C₂×C₄×D₃²	direct (N × H)
232. C₄	C₂×C₅×D₃²	direct (N × H)
234. C₃	C₂²×C₄×C₅×D₃	semidirect
237. C₂	C₄×C₅×D₃²	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×C₅×D₃²	2
C₂×C₄×C₅×C₃⋊D₃	2
C₂×C₃×C₄×C₅×D₃	2
C₂×C₅×#624	2
C₂×C₅×D₃×Dic₃	2
C₄×C₅×D₃²	2
C₂²×C₄×C₅×D₃	3
C₂×C₄×D₃²	5

Subgroups — C2×C4×C5×D32 #13089315

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×C₅×D₃² #13089315