Subgroups — D₃×Dic₃×Dic₅ #13085850

Order 1440.

462 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₃×Dic₁₅	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×D₃×Dic₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×D₃×Dic₅	C₂	1	1	yes	maximal, characteristic
4	#10623828	C₂	1	1	yes	maximal, characteristic, complemented
5	Dic₃×Dic₁₅	C₂	1	1	yes	maximal, characteristic, complemented
6	Dic₅×C₃⋊Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×Dic₃×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Dic₃×Dic₅	—	3	1	no	maximal
9	C₄×D₃×Dic₅	—	3	1	no	maximal
10	C₂²×C₃×C₅×D₃	C₂²	1	1	yes	characteristic
11	C₂×C₃⋊Dic₁₅	C₂²	1	1	yes	characteristic
12	C₂×C₅×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
13	C₂×C₃×Dic₁₅	C₂²	1	1	yes	characteristic
14	C₂×C₃×Dic₁₅	C₂²	1	1	yes	characteristic
15	C₂×C₃×C₅×Dic₃	C₂²	1	1	yes	characteristic
16	C₂×C₃²×Dic₅	C₂²	1	1	yes	characteristic
17	D₃×Dic₁₅	C₄	1	4	yes	complemented
18	C₅×D₃×Dic₃	C₄	1	4	yes	complemented
19	C₃×D₃×Dic₅	C₄	1	4	yes	complemented
20	C₄×D₃×Dic₃	—	5	1	no	maximal, Hall
21	C₂²×Dic₁₅	—	3	1	no	—
22	C₂²×C₅×Dic₃	—	3	1	no	—
23	C₂×C₄×C₅×D₃	—	3	1	no	—
24	C₂²×C₃×Dic₅	—	3	1	no	—
25	C₂×D₃×Dic₅	—	3	1	no	—
26	C₂×D₃×Dic₅	D₃	1	1	yes	characteristic
27	C₄×Dic₁₅	—	3	1	no	—
28	C₃×C₄×Dic₅	—	3	1	no	—
29	Dic₃×Dic₅	—	3	1	no	—
30	Dic₃×Dic₅	—	3	1	no	—
31	Dic₃×Dic₅	—	3	1	no	—
32	Dic₃×Dic₅	—	3	1	no	—
33	Dic₃×Dic₅	—	3	1	no	—
34	Dic₃×Dic₅	—	6	1	no	—
35	Dic₃×Dic₅	D₃	1	1	yes	characteristic, direct product factor
36	C₂²×C₃²×C₅	C₂³	1	1	yes	characteristic
37	C₂×C₃×C₅×D₃	C₂×C₄	1	2	yes	—
38	C₂×C₃×C₅×D₃	C₂×C₄	1	2	yes	—
39	C₂×C₃×C₅×D₃	C₂×C₄	1	2	yes	—
40	C₃⋊Dic₁₅	C₂×C₄	1	2	yes	complemented
41	C₅×C₃⋊Dic₃	C₂×C₄	1	2	yes	complemented
42	C₃×Dic₁₅	C₂×C₄	1	2	yes	complemented
43	C₃×Dic₁₅	C₂×C₄	1	2	yes	complemented
44	C₃×C₅×Dic₃	C₂×C₄	1	2	yes	complemented
45	C₃²×Dic₅	C₂×C₄	1	2	yes	complemented
46	C₂×C₄×Dic₅	—	9	1	no	Hall
47	C₂×C₃×C₄×D₃	—	5	1	no	—
48	C₂×D₃×Dic₃	—	5	1	no	—
49	C₂×D₃×Dic₃	D₅	1	1	yes	characteristic
50	C₄×C₃⋊Dic₃	—	5	1	no	—
51	C₃×C₄×Dic₃	—	5	1	no	—
52	Dic₃²	—	5	1	no	—
53	Dic₃²	—	5	1	no	—
54	C₂³×C₃×C₅	—	3	1	no	—
55	C₂²×C₅×D₃	C₂×D₃	1	1	yes	characteristic
56	C₂×C₃×C₄×C₅	—	3	1	no	—
57	C₂×Dic₁₅	—	3	1	no	—
58	C₂×Dic₁₅	—	3	1	no	—
59	C₂×Dic₁₅	—	3	4	no	—
60	C₂×Dic₁₅	—	6	1	no	—
61	C₂×Dic₁₅	C₂×D₃	1	1	yes	characteristic
62	C₂×Dic₁₅	C₂×D₃	1	1	yes	characteristic
63	C₂×C₅×Dic₃	—	3	1	no	—
64	C₂×C₅×Dic₃	—	3	1	no	—
65	C₂×C₅×Dic₃	—	3	4	no	—
66	C₂×C₅×Dic₃	—	6	1	no	—
67	C₂×C₅×Dic₃	C₂×D₃	1	1	yes	characteristic
68	C₄×C₅×D₃	—	3	4	no	—
69	C₂×C₃×Dic₅	—	2	1	no	—
70	C₂×C₃×Dic₅	—	3	1	no	—
71	C₂×C₃×Dic₅	—	3	1	no	—
72	C₂×C₃×Dic₅	—	3	4	no	—
73	C₂×C₃×Dic₅	C₂×D₃	1	1	yes	characteristic
74	C₂×C₃×Dic₅	C₂×D₃	1	1	yes	characteristic
75	D₃×Dic₅	—	3	4	no	—
76	D₃×Dic₅	Dic₃	1	4	yes	direct product factor
77	C₂×C₄×Dic₃	—	15	1	no	—
78	C₄²×D₃	—	15	1	no	—
79	C₂×C₃²×C₅	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₃²×C₅	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₃²×C₅	C₂²×C₄	1	1	yes	characteristic
82	C₃×C₅×D₃	C₄²	1	4	yes	complemented
83	C₂²×C₄×C₅	—	9	1	no	—
84	C₂²×Dic₅	—	3	1	no	—
85	C₂²×Dic₅	—	9	1	no	—
86	C₄×Dic₅	—	3	1	no	—
87	C₄×Dic₅	—	9	1	no	—
88	C₄×Dic₅	—	9	1	no	—
89	C₄×Dic₅	—	9	1	no	—
90	C₂²×C₃×D₃	C₂×D₅	1	1	yes	characteristic
91	C₂×C₃²×C₄	—	5	1	no	—
92	C₂×C₃⋊Dic₃	—	5	1	no	—
93	C₂×C₃⋊Dic₃	C₂×D₅	1	1	yes	characteristic
94	C₂×C₃×Dic₃	—	5	1	no	—
95	C₂×C₃×Dic₃	—	5	1	no	—
96	C₂×C₃×Dic₃	C₂×D₅	1	1	yes	characteristic
97	C₃×C₄×D₃	—	5	4	no	—
98	D₃×Dic₃	—	5	4	no	—
99	D₃×Dic₃	Dic₅	1	4	yes	direct product factor
100	C₂²×C₃×C₅	—	2	1	no	—
101	C₂²×C₃×C₅	—	3	2	no	—
102	C₂²×C₃×C₅	—	3	2	no	—
103	C₂²×C₃×C₅	—	3	2	no	—
104	C₂²×C₃×C₅	C₂²×D₃	1	1	yes	characteristic
105	C₂²×C₃×C₅	C₂²×D₃	1	1	yes	characteristic
106	C₂×C₅×D₃	C₄×D₃	1	2	yes	—
107	C₂×C₅×D₃	C₄×D₃	1	2	yes	—
108	C₂×C₅×D₃	C₂×Dic₃	1	2	yes	—
109	C₃×C₄×C₅	—	3	2	no	—
110	Dic₁₅	—	3	2	no	—
111	Dic₁₅	—	3	2	no	—
112	Dic₁₅	—	6	2	no	—
113	Dic₁₅	C₄×D₃	1	2	yes	complemented
114	Dic₁₅	C₂×Dic₃	1	2	yes	complemented
115	C₃×Dic₅	—	2	2	no	—
116	C₃×Dic₅	—	3	2	no	—
117	C₃×Dic₅	—	3	2	no	—
118	C₃×Dic₅	C₄×D₃	1	2	yes	complemented
119	C₃×Dic₅	C₂×Dic₃	1	2	yes	complemented
120	C₅×Dic₃	—	3	2	no	—
121	C₅×Dic₃	—	3	2	no	—
122	C₅×Dic₃	—	6	2	no	—
123	C₅×Dic₃	C₄×D₃	1	2	yes	complemented
124	C₂²×C₃×C₄	—	15	1	no	—
125	C₂²×Dic₃	—	3	1	no	—
126	C₂²×Dic₃	—	15	1	no	—
127	C₂×C₄×D₃	—	3	1	no	—
128	C₂×C₄×D₃	—	5	1	no	—
129	C₂×C₄×D₃	—	15	1	no	—
130	C₃×C₄²	—	15	1	no	—
131	C₄×Dic₃	—	5	1	no	—
132	C₄×Dic₃	—	15	1	no	—
133	C₄×Dic₃	—	15	1	no	—
134	C₄×Dic₃	—	15	1	no	—
135	C₄×Dic₃	—	15	1	no	—
136	C₄×Dic₃	—	15	1	no	—
137	C₄×Dic₃	—	15	1	no	—
138	C₄×Dic₃	—	30	1	no	—
139	C₃²×C₅	C₂×C₄²	1	1	yes	characteristic, complemented, Hall
140	C₂³×C₅	—	3	1	no	—
141	C₂×C₄×C₅	—	3	1	no	—
142	C₂×C₄×C₅	—	9	4	no	—
143	C₂×C₄×C₅	—	9	1	no	—
144	C₂×Dic₅	—	3	1	no	—
145	C₂×Dic₅	—	3	1	no	—
146	C₂×Dic₅	—	3	4	no	—
147	C₂×Dic₅	—	9	4	no	—
148	C₂×Dic₅	—	9	1	no	—
149	C₂×Dic₅	D₃²	1	1	yes	characteristic
150	C₂²×C₃²	C₂²×D₅	1	1	yes	characteristic
151	C₂×C₃×D₃	C₄×D₅	1	2	yes	—
152	C₂×C₃×D₃	C₄×D₅	1	2	yes	—
153	C₂×C₃×D₃	C₂×Dic₅	1	2	yes	—
154	C₃²×C₄	—	5	2	no	—
155	C₃⋊Dic₃	—	5	2	no	—
156	C₃⋊Dic₃	C₂×Dic₅	1	2	yes	complemented
157	C₃×Dic₃	—	5	2	no	—
158	C₃×Dic₃	—	5	2	no	—
159	C₃×Dic₃	C₂×Dic₅	1	2	yes	complemented
160	C₂×C₄²	—	45	1	no	Sylow
161	C₂×C₃×C₅	—	2	1	no	—
162	C₂×C₃×C₅	—	2	1	no	—
163	C₂×C₃×C₅	—	2	1	no	—
164	C₂×C₃×C₅	—	3	4	no	—
165	C₂×C₃×C₅	C₂×C₄×D₃	1	1	yes	characteristic
166	C₂×C₃×C₅	C₂×C₄×D₃	1	1	yes	characteristic
167	C₂×C₃×C₅	C₂×C₄×D₃	1	1	yes	characteristic
168	C₂×C₃×C₅	C₂×C₄×D₃	1	1	yes	characteristic
169	C₂×C₃×C₅	C₂×C₄×D₃	1	1	yes	characteristic
170	C₂×C₃×C₅	C₂²×Dic₃	1	1	yes	characteristic
171	C₅×D₃	C₄×Dic₃	1	4	yes	complemented
172	C₂³×C₃	—	3	1	no	—
173	C₂²×D₃	D₃×D₅	1	1	yes	characteristic
174	C₂×C₃×C₄	—	3	1	no	—
175	C₂×C₃×C₄	—	5	1	no	—
176	C₂×C₃×C₄	—	5	1	no	—
177	C₂×C₃×C₄	—	10	1	no	—
178	C₂×C₃×C₄	—	15	4	no	—
179	C₂×C₃×C₄	—	15	1	no	—
180	C₂×C₃×C₄	—	15	1	no	—
181	C₂×Dic₃	—	3	1	no	—
182	C₂×Dic₃	—	3	1	no	—
183	C₂×Dic₃	—	3	4	no	—
184	C₂×Dic₃	—	5	1	no	—
185	C₂×Dic₃	—	5	1	no	—
186	C₂×Dic₃	—	6	1	no	—
187	C₂×Dic₃	—	15	4	no	—
188	C₂×Dic₃	—	15	1	no	—
189	C₂×Dic₃	—	15	1	no	—
190	C₂×Dic₃	—	30	1	no	—
191	C₂×Dic₃	D₃×D₅	1	1	yes	characteristic
192	C₄×D₃	—	3	4	no	—
193	C₄×D₃	—	5	4	no	—
194	C₄×D₃	—	15	4	no	—
195	C₂²×C₅	—	3	2	no	—
196	C₂²×C₅	—	3	2	no	—
197	C₂²×C₅	—	3	2	no	—
198	C₂²×C₅	C₂×D₃²	1	1	yes	characteristic
199	C₄×C₅	—	3	2	no	—
200	C₄×C₅	—	9	2	no	—
201	Dic₅	—	3	2	no	—
202	Dic₅	—	3	2	no	—
203	Dic₅	—	9	2	no	—
204	Dic₅	D₃×Dic₃	1	2	yes	direct product factor
205	C₂×C₃²	C₂×C₄×D₅	1	1	yes	characteristic
206	C₂×C₃²	C₂×C₄×D₅	1	1	yes	characteristic
207	C₂×C₃²	C₂²×Dic₅	1	1	yes	characteristic
208	C₃×D₃	C₄×Dic₅	1	4	yes	complemented
209	C₂²×C₄	—	9	1	no	—
210	C₂²×C₄	—	15	1	no	—
211	C₂²×C₄	—	45	1	no	—
212	C₄²	—	15	1	no	—
213	C₄²	—	45	1	no	—
214	C₄²	—	45	1	no	—
215	C₄²	—	45	1	no	—
216	C₃×C₅	—	2	1	no	—
217	C₃×C₅	C₄²×D₃	1	1	yes	characteristic, complemented
218	C₃×C₅	C₂×C₄×Dic₃	1	1	yes	characteristic, complemented
219	C₂²×C₃	—	2	1	no	—
220	C₂²×C₃	—	3	2	no	—
221	C₂²×C₃	—	3	2	no	—
222	C₂²×C₃	—	3	2	no	—
223	C₂²×C₃	C₂×D₃×D₅	1	1	yes	characteristic
224	C₂²×C₃	C₂×D₃×D₅	1	1	yes	characteristic
225	C₂×D₃	D₅×Dic₃	1	2	yes	—
226	C₂×D₃	D₃×Dic₅	1	2	yes	—
227	C₂×D₃	#1200	1	2	yes	—
228	C₃×C₄	—	3	2	no	—
229	C₃×C₄	—	5	2	no	—
230	C₃×C₄	—	5	2	no	—
231	C₃×C₄	—	10	2	no	—
232	C₃×C₄	—	15	2	no	—
233	C₃×C₄	—	15	2	no	—
234	Dic₃	—	3	2	no	—
235	Dic₃	—	3	2	no	—
236	Dic₃	—	5	2	no	—
237	Dic₃	—	5	2	no	—
238	Dic₃	—	6	2	no	—
239	Dic₃	—	15	2	no	—
240	Dic₃	—	15	2	no	—
241	Dic₃	—	30	2	no	—
242	Dic₃	D₃×Dic₅	1	2	yes	direct product factor
243	C₂×C₅	—	3	4	no	—
244	C₂×C₅	C₄×D₃²	1	1	yes	characteristic
245	C₂×C₅	C₄×D₃²	1	1	yes	characteristic
246	C₂×C₅	C₂×D₃×Dic₃	1	1	yes	characteristic
247	C₃²	C₂×C₄×Dic₅	1	1	yes	characteristic, complemented, Sylow
248	C₂³	—	3	1	no	—
249	C₂×C₄	—	3	1	no	—
250	C₂×C₄	—	5	1	no	—
251	C₂×C₄	—	9	4	no	—
252	C₂×C₄	—	9	1	no	—
253	C₂×C₄	—	15	4	no	—
254	C₂×C₄	—	15	1	no	—
255	C₂×C₄	—	15	1	no	—
256	C₂×C₄	—	45	4	no	—
257	C₂×C₄	—	45	1	no	—
258	C₂×C₃	—	2	1	no	—
259	C₂×C₃	—	2	1	no	—
260	C₂×C₃	—	2	1	no	—
261	C₂×C₃	—	3	4	no	—
262	C₂×C₃	C₄×D₃×D₅	1	1	yes	characteristic
263	C₂×C₃	C₄×D₃×D₅	1	1	yes	characteristic
264	C₂×C₃	C₂×D₅×Dic₃	1	1	yes	characteristic
265	C₂×C₃	C₂×D₃×Dic₅	1	1	yes	characteristic
266	C₂×C₃	C₂×D₃×Dic₅	1	1	yes	characteristic
267	C₂×C₃	C₂×#1200	1	1	yes	characteristic
268	D₃	Dic₃×Dic₅	1	4	yes	direct product factor
269	C₅	C₄×D₃×Dic₃	1	1	yes	characteristic, complemented, Sylow
270	C₂²	—	3	2	no	—
271	C₂²	—	3	2	no	—
272	C₂²	—	3	2	no	—
273	C₂²	D₃²×D₅	1	1	yes	characteristic
274	C₄	—	3	2	no	—
275	C₄	—	5	2	no	—
276	C₄	—	9	2	no	—
277	C₄	—	15	2	no	—
278	C₄	—	15	2	no	—
279	C₄	—	45	2	no	—
280	C₃	—	2	1	no	—
281	C₃	C₄×D₃×Dic₅	1	1	yes	characteristic, complemented
282	C₃	C₂×Dic₃×Dic₅	1	1	yes	characteristic, complemented
283	C₂	—	3	4	no	—
284	C₂	D₃×D₅×Dic₃	1	1	yes	characteristic
285	C₂	D₃²×Dic₅	1	1	yes	characteristic
286	C₂	D₃×#1200	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂²	D₃²×D₅
commutator, hypocenter	C₃²×C₅	C₂×C₄²
Fitting, socle, cosocle	C₂²×C₃²×C₅	C₂³
prime radical	C₂²×C₃×C₅×D₃	C₂²
LCS	G ▷ C₃²×C₅
UCS	C₁ ◁ C₂²

Complements

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

N	H ≅ G/N	type
4. #10623828	C₂	semidirect
5. Dic₃×Dic₁₅	C₂	semidirect
6. Dic₅×C₃⋊Dic₃	C₂	semidirect
7. C₃×Dic₃×Dic₅	C₂	semidirect
17. D₃×Dic₁₅	C₄	semidirect
18. C₅×D₃×Dic₃	C₄	semidirect
19. C₃×D₃×Dic₅	C₄	semidirect
35. Dic₃×Dic₅	D₃	direct (N × H)
40. C₃⋊Dic₁₅	C₂×C₄	semidirect
41. C₅×C₃⋊Dic₃	C₂×C₄	semidirect
42. C₃×Dic₁₅	C₂×C₄	semidirect
43. C₃×Dic₁₅	C₂×C₄	semidirect
44. C₃×C₅×Dic₃	C₂×C₄	semidirect
45. C₃²×Dic₅	C₂×C₄	semidirect
76. D₃×Dic₅	Dic₃	direct (N × H)
82. C₃×C₅×D₃	C₄²	semidirect
99. D₃×Dic₃	Dic₅	direct (N × H)
113. Dic₁₅	C₄×D₃	semidirect
114. Dic₁₅	C₂×Dic₃	semidirect
118. C₃×Dic₅	C₄×D₃	semidirect
119. C₃×Dic₅	C₂×Dic₃	semidirect
123. C₅×Dic₃	C₄×D₃	semidirect
139. C₃²×C₅	C₂×C₄²	semidirect
156. C₃⋊Dic₃	C₂×Dic₅	semidirect
159. C₃×Dic₃	C₂×Dic₅	semidirect
171. C₅×D₃	C₄×Dic₃	semidirect
204. Dic₅	D₃×Dic₃	direct (N × H)
208. C₃×D₃	C₄×Dic₅	semidirect
217. C₃×C₅	C₄²×D₃	semidirect
218. C₃×C₅	C₂×C₄×Dic₃	semidirect
242. Dic₃	D₃×Dic₅	direct (N × H)
247. C₃²	C₂×C₄×Dic₅	semidirect
268. D₃	Dic₃×Dic₅	direct (N × H)
269. C₅	C₄×D₃×Dic₃	semidirect
281. C₃	C₄×D₃×Dic₅	semidirect
282. C₃	C₂×Dic₃×Dic₅	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₃×Dic₁₅	2
C₂×C₅×D₃×Dic₃	2
C₂×C₃×D₃×Dic₅	2
#10623828	2
Dic₃×Dic₁₅	2
Dic₅×C₃⋊Dic₃	2
C₃×Dic₃×Dic₅	2
C₂×Dic₃×Dic₅	3
C₄×D₃×Dic₅	3
C₄×D₃×Dic₃	5

Subgroups — D3×Dic3×Dic5 #13085850

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₃×Dic₃×Dic₅ #13085850