Subgroups — C₂×C₅×(C₂×D₃)⋊Dic₃ #13087228

Order 1440.

582 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×C₅×D₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₅×C₃⋊Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃×C₅×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×(C₂×D₃)⋊Dic₃	C₂	1	4	yes	maximal, direct product factor
5	C₂×C₅×#217	—	3	1	no	maximal
6	C₂×C₅×(C₂×D₃)⋊C₄	—	3	1	no	maximal
7	C₂³×C₃²×C₅	C₂²	1	1	yes	characteristic
8	C₂²×C₃×C₅×D₃	—	2	2	no	—
9	C₂²×C₃×C₅×D₃	—	2	2	no	—
10	C₂²×C₃×C₅×D₃	C₄	1	4	yes	complemented
11	C₂²×C₃×C₅×D₃	C₂²	1	2	yes	—
12	C₂×C₅×C₃⋊Dic₃	—	2	2	no	—
13	C₂×C₅×C₃⋊Dic₃	C₂²	1	2	yes	complemented
14	C₂×C₃×C₅×Dic₃	—	2	2	no	—
15	C₂×C₃×C₅×Dic₃	C₂²	1	2	yes	complemented
16	C₂×(C₂×D₃)⋊Dic₃	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
17	C₂⁴×C₃×C₅	—	3	1	no	—
18	C₂³×C₅×D₃	D₃	1	1	yes	characteristic
19	C₂²×C₃×C₄×C₅	—	3	1	no	—
20	C₂²×C₅×Dic₃	—	3	1	no	—
21	C₂²×C₅×Dic₃	—	3	1	no	—
22	C₂²×C₅×Dic₃	—	6	1	no	—
23	C₂²×C₅×Dic₃	D₃	1	1	yes	characteristic, complemented
24	C₅×#217	—	3	4	no	—
25	C₅×(C₂×D₃)⋊C₄	—	3	4	no	—
26	C₂²×C₃²×C₅	C₂×C₄	1	2	yes	—
27	C₂²×C₃²×C₅	D₄	1	2	yes	—
28	C₂²×C₃²×C₅	D₄	1	2	yes	—
29	C₂²×C₃²×C₅	C₂³	1	1	yes	characteristic
30	C₂×C₃×C₅×D₃	—	2	8	no	—
31	C₂×C₃×C₅×D₃	—	2	2	no	—
32	C₂×C₃×C₅×D₃	—	2	2	no	—
33	C₂×C₃×C₅×D₃	C₂×C₄	1	4	yes	complemented
34	C₅×C₃⋊Dic₃	—	2	2	no	—
35	C₃×C₅×Dic₃	—	2	2	no	—
36	C₂×C₅×C₂²⋊C₄	—	9	1	no	Hall
37	C₂³×C₃×D₃	C₂×C₅	1	1	yes	characteristic
38	C₂²×C₃⋊Dic₃	C₂×C₅	1	1	yes	characteristic, complemented
39	C₂²×C₃×Dic₃	C₂×C₅	1	1	yes	characteristic, complemented
40	(C₂×D₃)⋊Dic₃	C₂×C₅	1	4	yes	direct product factor
41	C₂³×C₃×C₅	—	2	1	no	—
42	C₂³×C₃×C₅	—	3	4	no	—
43	C₂³×C₃×C₅	—	3	2	no	—
44	C₂³×C₃×C₅	—	6	2	no	—
45	C₂³×C₃×C₅	—	6	2	no	—
46	C₂³×C₃×C₅	C₂×D₃	1	1	yes	characteristic
47	C₂³×C₃×C₅	C₂×D₃	1	1	yes	characteristic
48	C₂²×C₅×D₃	—	2	2	no	—
49	C₂²×C₅×D₃	—	2	2	no	—
50	C₂²×C₅×D₃	Dic₃	1	4	yes	complemented
51	C₂²×C₅×D₃	C₂×D₃	1	2	yes	—
52	C₂×C₃×C₄×C₅	—	3	2	no	—
53	C₂×C₃×C₄×C₅	—	6	2	no	—
54	C₂×C₅×Dic₃	—	2	2	no	—
55	C₂×C₅×Dic₃	—	3	2	no	—
56	C₂×C₅×Dic₃	—	3	2	no	—
57	C₂×C₅×Dic₃	—	6	2	no	—
58	C₂×C₅×Dic₃	—	6	4	no	—
59	C₂×C₅×Dic₃	—	6	2	no	—
60	C₂×C₅×Dic₃	—	6	2	no	—
61	C₂×C₅×Dic₃	C₂×D₃	1	2	yes	complemented
62	C₂×#217	—	3	1	no	—
63	C₂×(C₂×D₃)⋊C₄	—	3	1	no	—
64	C₂×C₃²×C₅	C₂²⋊C₄	1	4	yes	complemented
65	C₂×C₃²×C₅	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₃²×C₅	C₂×D₄	1	1	yes	characteristic
67	C₂×C₃²×C₅	C₂×D₄	1	1	yes	characteristic
68	C₃×C₅×D₃	—	2	4	no	—
69	C₂⁴×C₅	—	3	1	no	—
70	C₂²×C₄×C₅	—	3	1	no	—
71	C₂²×C₄×C₅	—	9	1	no	—
72	C₅×C₂²⋊C₄	—	9	4	no	—
73	C₂³×C₃²	C₂²×C₅	1	1	yes	characteristic
74	C₂²×C₃×D₃	—	2	2	no	—
75	C₂²×C₃×D₃	—	2	2	no	—
76	C₂²×C₃×D₃	C₄×C₅	1	4	yes	complemented
77	C₂²×C₃×D₃	C₂²×C₅	1	2	yes	—
78	C₂×C₃⋊Dic₃	—	2	2	no	—
79	C₂×C₃⋊Dic₃	C₂²×C₅	1	2	yes	complemented
80	C₂×C₃×Dic₃	—	2	2	no	—
81	C₂×C₃×Dic₃	C₂²×C₅	1	2	yes	complemented
82	C₂²×C₃×C₅	—	2	2	no	—
83	C₂²×C₃×C₅	—	2	2	no	—
84	C₂²×C₃×C₅	—	2	2	no	—
85	C₂²×C₃×C₅	—	2	1	no	—
86	C₂²×C₃×C₅	—	3	4	no	—
87	C₂²×C₃×C₅	—	6	8	no	—
88	C₂²×C₃×C₅	—	6	2	no	—
89	C₂²×C₃×C₅	—	6	2	no	—
90	C₂²×C₃×C₅	C₄×D₃	1	2	yes	—
91	C₂²×C₃×C₅	D₁₂	1	2	yes	—
92	C₂²×C₃×C₅	C₂×Dic₃	1	2	yes	—
93	C₂²×C₃×C₅	#67	1	2	yes	—
94	C₂²×C₃×C₅	#67	1	2	yes	—
95	C₂²×C₃×C₅	#67	1	2	yes	—
96	C₂²×C₃×C₅	C₂²×D₃	1	1	yes	characteristic
97	C₂²×C₃×C₅	C₂²×D₃	1	1	yes	characteristic
98	C₂×C₅×D₃	—	2	8	no	—
99	C₂×C₅×D₃	—	2	2	no	—
100	C₂×C₅×D₃	—	2	2	no	—
101	C₂×C₅×D₃	C₂×Dic₃	1	4	yes	complemented
102	C₃×C₄×C₅	—	6	2	no	—
103	C₅×Dic₃	—	2	2	no	—
104	C₅×Dic₃	—	6	4	no	—
105	C₅×Dic₃	—	6	2	no	—
106	C₅×Dic₃	—	6	2	no	—
107	C₂⁴×C₃	—	3	1	no	—
108	C₂³×D₃	C₅×D₃	1	1	yes	characteristic
109	C₂²×C₃×C₄	—	3	1	no	—
110	C₂²×Dic₃	—	3	1	no	—
111	C₂²×Dic₃	—	3	1	no	—
112	C₂²×Dic₃	—	6	1	no	—
113	C₂²×Dic₃	C₅×D₃	1	1	yes	characteristic, complemented
114	#217	—	3	4	no	—
115	(C₂×D₃)⋊C₄	—	3	4	no	—
116	C₃²×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic, complemented, Hall
117	C₂³×C₅	—	3	4	no	—
118	C₂³×C₅	—	3	2	no	—
119	C₂³×C₅	—	6	2	no	—
120	C₂³×C₅	—	6	2	no	—
121	C₂³×C₅	D₃²	1	1	yes	characteristic
122	C₂×C₄×C₅	—	3	2	no	—
123	C₂×C₄×C₅	—	6	2	no	—
124	C₂×C₄×C₅	—	9	2	no	—
125	C₂×C₄×C₅	—	18	2	no	—
126	C₂²×C₃²	C₂×C₄×C₅	1	2	yes	—
127	C₂²×C₃²	C₅×D₄	1	2	yes	—
128	C₂²×C₃²	C₅×D₄	1	2	yes	—
129	C₂²×C₃²	C₂³×C₅	1	1	yes	characteristic
130	C₂×C₃×D₃	—	2	8	no	—
131	C₂×C₃×D₃	—	2	2	no	—
132	C₂×C₃×D₃	—	2	2	no	—
133	C₂×C₃×D₃	C₂×C₄×C₅	1	4	yes	complemented
134	C₃⋊Dic₃	—	2	2	no	—
135	C₃×Dic₃	—	2	2	no	—
136	C₂×C₂²⋊C₄	—	9	1	no	Sylow
137	C₂×C₃×C₅	—	2	4	no	—
138	C₂×C₃×C₅	—	2	1	no	—
139	C₂×C₃×C₅	—	2	1	no	—
140	C₂×C₃×C₅	—	2	1	no	—
141	C₂×C₃×C₅	—	6	4	no	—
142	C₂×C₃×C₅	(C₂×D₃)⋊C₄	1	4	yes	complemented
143	C₂×C₃×C₅	#217	1	4	yes	complemented
144	C₂×C₃×C₅	C₂×C₄×D₃	1	1	yes	characteristic
145	C₂×C₃×C₅	C₂×D₁₂	1	1	yes	characteristic
146	C₂×C₃×C₅	C₂²×Dic₃	1	1	yes	characteristic
147	C₂×C₃×C₅	C₂×#67	1	1	yes	characteristic
148	C₂×C₃×C₅	C₂×#67	1	1	yes	characteristic
149	C₂×C₃×C₅	C₂×#67	1	1	yes	characteristic
150	C₅×D₃	—	2	4	no	—
151	C₂³×C₃	—	2	1	no	—
152	C₂³×C₃	—	3	4	no	—
153	C₂³×C₃	—	3	2	no	—
154	C₂³×C₃	—	6	2	no	—
155	C₂³×C₃	—	6	2	no	—
156	C₂³×C₃	C₂×C₅×D₃	1	1	yes	characteristic
157	C₂³×C₃	C₂×C₅×D₃	1	1	yes	characteristic
158	C₂²×D₃	—	2	2	no	—
159	C₂²×D₃	—	2	2	no	—
160	C₂²×D₃	C₅×Dic₃	1	4	yes	complemented
161	C₂²×D₃	C₂×C₅×D₃	1	2	yes	—
162	C₂×C₃×C₄	—	3	2	no	—
163	C₂×C₃×C₄	—	6	2	no	—
164	C₂×Dic₃	—	2	2	no	—
165	C₂×Dic₃	—	3	2	no	—
166	C₂×Dic₃	—	3	2	no	—
167	C₂×Dic₃	—	6	2	no	—
168	C₂×Dic₃	—	6	4	no	—
169	C₂×Dic₃	—	6	2	no	—
170	C₂×Dic₃	—	6	2	no	—
171	C₂×Dic₃	C₂×C₅×D₃	1	2	yes	complemented
172	C₂²×C₅	—	3	4	no	—
173	C₂²×C₅	—	6	8	no	—
174	C₂²×C₅	—	6	2	no	—
175	C₂²×C₅	—	6	2	no	—
176	C₂²×C₅	D₃×Dic₃	1	2	yes	—
177	C₂²×C₅	#625	1	2	yes	—
178	C₂²×C₅	#626	1	2	yes	—
179	C₂²×C₅	C₂×D₃²	1	1	yes	characteristic
180	C₄×C₅	—	6	2	no	—
181	C₄×C₅	—	18	2	no	—
182	C₂×C₃²	C₅×C₂²⋊C₄	1	4	yes	complemented
183	C₂×C₃²	C₂²×C₄×C₅	1	1	yes	characteristic
184	C₂×C₃²	C₂×C₅×D₄	1	1	yes	characteristic
185	C₂×C₃²	C₂×C₅×D₄	1	1	yes	characteristic
186	C₃×D₃	—	2	4	no	—
187	C₂⁴	—	3	1	no	—
188	C₂²×C₄	—	3	1	no	—
189	C₂²×C₄	—	9	1	no	—
190	C₂²⋊C₄	—	9	4	no	—
191	C₃×C₅	—	2	1	no	—
192	C₃×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic, complemented
193	C₃×C₅	C₂×#217	1	1	yes	characteristic, complemented
194	C₂²×C₃	—	2	2	no	—
195	C₂²×C₃	—	2	2	no	—
196	C₂²×C₃	—	2	2	no	—
197	C₂²×C₃	—	2	1	no	—
198	C₂²×C₃	—	3	4	no	—
199	C₂²×C₃	—	6	8	no	—
200	C₂²×C₃	—	6	2	no	—
201	C₂²×C₃	—	6	2	no	—
202	C₂²×C₃	C₄×C₅×D₃	1	2	yes	—
203	C₂²×C₃	C₅×D₁₂	1	2	yes	—
204	C₂²×C₃	C₂×C₅×Dic₃	1	2	yes	—
205	C₂²×C₃	C₅×#67	1	2	yes	—
206	C₂²×C₃	C₅×#67	1	2	yes	—
207	C₂²×C₃	C₅×#67	1	2	yes	—
208	C₂²×C₃	C₂²×C₅×D₃	1	1	yes	characteristic
209	C₂²×C₃	C₂²×C₅×D₃	1	1	yes	characteristic
210	C₂×D₃	—	2	8	no	—
211	C₂×D₃	—	2	2	no	—
212	C₂×D₃	—	2	2	no	—
213	C₂×D₃	C₂×C₅×Dic₃	1	4	yes	complemented
214	C₃×C₄	—	6	2	no	—
215	Dic₃	—	2	2	no	—
216	Dic₃	—	6	4	no	—
217	Dic₃	—	6	2	no	—
218	Dic₃	—	6	2	no	—
219	C₂×C₅	—	6	4	no	—
220	C₂×C₅	(C₂×D₃)⋊Dic₃	1	4	yes	direct product factor
221	C₂×C₅	C₂×D₃×Dic₃	1	1	yes	characteristic
222	C₂×C₅	C₂×#625	1	1	yes	characteristic
223	C₂×C₅	C₂×#626	1	1	yes	characteristic
224	C₃²	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
225	C₂³	—	3	4	no	—
226	C₂³	—	3	2	no	—
227	C₂³	—	6	2	no	—
228	C₂³	—	6	2	no	—
229	C₂³	C₅×D₃²	1	1	yes	characteristic
230	C₂×C₄	—	3	2	no	—
231	C₂×C₄	—	6	2	no	—
232	C₂×C₄	—	9	2	no	—
233	C₂×C₄	—	18	2	no	—
234	C₂×C₃	—	2	4	no	—
235	C₂×C₃	—	2	1	no	—
236	C₂×C₃	—	2	1	no	—
237	C₂×C₃	—	2	1	no	—
238	C₂×C₃	—	6	4	no	—
239	C₂×C₃	C₅×(C₂×D₃)⋊C₄	1	4	yes	complemented
240	C₂×C₃	C₅×#217	1	4	yes	complemented
241	C₂×C₃	C₂×C₄×C₅×D₃	1	1	yes	characteristic
242	C₂×C₃	C₂×C₅×D₁₂	1	1	yes	characteristic
243	C₂×C₃	C₂²×C₅×Dic₃	1	1	yes	characteristic
244	C₂×C₃	C₂×C₅×#67	1	1	yes	characteristic
245	C₂×C₃	C₂×C₅×#67	1	1	yes	characteristic
246	C₂×C₃	C₂×C₅×#67	1	1	yes	characteristic
247	D₃	—	2	4	no	—
248	C₅	C₂×(C₂×D₃)⋊Dic₃	1	1	yes	characteristic, direct product factor, Sylow
249	C₂²	—	3	4	no	—
250	C₂²	—	6	8	no	—
251	C₂²	—	6	2	no	—
252	C₂²	—	6	2	no	—
253	C₂²	C₅×D₃×Dic₃	1	2	yes	—
254	C₂²	C₅×#625	1	2	yes	—
255	C₂²	C₅×#626	1	2	yes	—
256	C₂²	C₂×C₅×D₃²	1	1	yes	characteristic
257	C₄	—	6	2	no	—
258	C₄	—	18	2	no	—
259	C₃	—	2	1	no	—
260	C₃	C₂×C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic, complemented
261	C₃	C₂×C₅×#217	1	1	yes	characteristic, complemented
262	C₂	—	6	4	no	—
263	C₂	C₅×(C₂×D₃)⋊Dic₃	1	4	yes	direct product factor
264	C₂	C₂×C₅×D₃×Dic₃	1	1	yes	characteristic
265	C₂	C₂×C₅×#625	1	1	yes	characteristic
266	C₂	C₂×C₅×#626	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂³×C₅	D₃²
commutator	C₂×C₃²	C₂²×C₄×C₅
Frattini	C₂²	C₂×C₅×D₃²
Fitting, socle	C₂³×C₃²×C₅	C₂²
prime radical	C₂³×C₃×C₅×D₃	C₂
cosocle	C₂²×C₃²	C₂³×C₅
hypocenter	C₃²	C₂×C₅×C₂²⋊C₄
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂³×C₅

Complements

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

N	H ≅ G/N	type
2. C₂²×C₅×C₃⋊Dic₃	C₂	semidirect
3. C₂²×C₃×C₅×Dic₃	C₂	semidirect
4. C₅×(C₂×D₃)⋊Dic₃	C₂	direct (N × H)
10. C₂²×C₃×C₅×D₃	C₄	semidirect
13. C₂×C₅×C₃⋊Dic₃	C₂²	semidirect
15. C₂×C₃×C₅×Dic₃	C₂²	semidirect
16. C₂×(C₂×D₃)⋊Dic₃	C₅	direct (N × H)
23. C₂²×C₅×Dic₃	D₃	semidirect
33. C₂×C₃×C₅×D₃	C₂×C₄	semidirect
38. C₂²×C₃⋊Dic₃	C₂×C₅	semidirect
39. C₂²×C₃×Dic₃	C₂×C₅	semidirect
40. (C₂×D₃)⋊Dic₃	C₂×C₅	direct (N × H)
50. C₂²×C₅×D₃	Dic₃	semidirect
61. C₂×C₅×Dic₃	C₂×D₃	semidirect
64. C₂×C₃²×C₅	C₂²⋊C₄	semidirect
76. C₂²×C₃×D₃	C₄×C₅	semidirect
79. C₂×C₃⋊Dic₃	C₂²×C₅	semidirect
81. C₂×C₃×Dic₃	C₂²×C₅	semidirect
101. C₂×C₅×D₃	C₂×Dic₃	semidirect
113. C₂²×Dic₃	C₅×D₃	semidirect
116. C₃²×C₅	C₂×C₂²⋊C₄	semidirect
133. C₂×C₃×D₃	C₂×C₄×C₅	semidirect
142. C₂×C₃×C₅	(C₂×D₃)⋊C₄	semidirect
143. C₂×C₃×C₅	#217	semidirect
160. C₂²×D₃	C₅×Dic₃	semidirect
171. C₂×Dic₃	C₂×C₅×D₃	semidirect
182. C₂×C₃²	C₅×C₂²⋊C₄	semidirect
192. C₃×C₅	C₂×(C₂×D₃)⋊C₄	semidirect
193. C₃×C₅	C₂×#217	semidirect
213. C₂×D₃	C₂×C₅×Dic₃	semidirect
220. C₂×C₅	(C₂×D₃)⋊Dic₃	direct (N × H)
224. C₃²	C₂×C₅×C₂²⋊C₄	semidirect
239. C₂×C₃	C₅×(C₂×D₃)⋊C₄	semidirect
240. C₂×C₃	C₅×#217	semidirect
248. C₅	C₂×(C₂×D₃)⋊Dic₃	direct (N × H)
260. C₃	C₂×C₅×(C₂×D₃)⋊C₄	semidirect
261. C₃	C₂×C₅×#217	semidirect
263. C₂	C₅×(C₂×D₃)⋊Dic₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₃×C₅×D₃	2
C₂²×C₅×C₃⋊Dic₃	2
C₂²×C₃×C₅×Dic₃	2
C₅×(C₂×D₃)⋊Dic₃	2
C₂×C₅×#217	3
C₂×C₅×(C₂×D₃)⋊C₄	3
C₂×(C₂×D₃)⋊Dic₃	5

Subgroups — C2×C5×(C2×D3)⋊Dic3 #13087228

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×(C₂×D₃)⋊Dic₃ #13087228