Subgroups — C₂×C₃×Dic₃×Dic₅ #13086673

Order 1440.

486 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×Dic₁₅	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃×C₅×Dic₃	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₃²×Dic₅	C₂	1	1	yes	maximal, characteristic
4	C₃×Dic₃×Dic₅	C₂	1	4	yes	maximal, direct product factor
5	C₂×C₃×C₄×Dic₅	—	3	1	no	maximal
6	C₂×Dic₃×Dic₅	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₂³×C₃²×C₅	C₂²	1	1	yes	characteristic
8	C₂×C₃×Dic₁₅	C₄	1	4	yes	complemented
9	C₂×C₃×Dic₁₅	C₂²	1	2	yes	—
10	C₂×C₃×C₅×Dic₃	C₄	1	4	yes	complemented
11	C₂×C₃×C₅×Dic₃	C₂²	1	2	yes	—
12	C₂×C₃²×Dic₅	C₄	1	4	yes	complemented
13	C₂×C₃²×Dic₅	C₂²	1	2	yes	—
14	C₂×C₃×C₄×Dic₃	—	5	1	no	maximal, Hall
15	C₂²×C₃×C₄×C₅	—	3	1	no	—
16	C₂²×Dic₁₅	C₂×C₃	1	1	yes	characteristic
17	C₂²×C₅×Dic₃	C₂×C₃	1	1	yes	characteristic
18	C₂²×C₃×Dic₅	—	2	1	no	—
19	C₂²×C₃×Dic₅	—	3	1	no	—
20	C₂²×C₃×Dic₅	D₃	1	1	yes	characteristic
21	C₂²×C₃×Dic₅	C₂×C₃	1	1	yes	characteristic
22	C₃×C₄×Dic₅	—	3	4	no	—
23	Dic₃×Dic₅	C₂×C₃	1	4	yes	direct product factor
24	C₂²×C₃²×C₅	C₂×C₄	1	2	yes	—
25	C₂²×C₃²×C₅	C₂×C₄	1	2	yes	—
26	C₂²×C₃²×C₅	C₂×C₄	1	2	yes	—
27	C₂²×C₃²×C₅	C₂³	1	1	yes	characteristic
28	C₃×Dic₁₅	C₂×C₄	1	4	yes	complemented
29	C₃×C₅×Dic₃	C₂×C₄	1	4	yes	complemented
30	C₃²×Dic₅	C₂×C₄	1	4	yes	complemented
31	C₂×C₄×Dic₅	—	3	1	no	Hall
32	C₂²×C₃²×C₄	—	5	1	no	—
33	C₂²×C₃×Dic₃	—	5	1	no	—
34	C₂²×C₃×Dic₃	D₅	1	1	yes	characteristic
35	C₃×C₄×Dic₃	—	5	4	no	—
36	C₂³×C₃×C₅	—	2	1	no	—
37	C₂³×C₃×C₅	C₂×D₃	1	1	yes	characteristic
38	C₂³×C₃×C₅	C₂²×C₃	1	1	yes	characteristic
39	C₂×C₃×C₄×C₅	—	3	4	no	—
40	C₂×C₃×C₄×C₅	—	3	2	no	—
41	C₂×Dic₁₅	C₃×C₄	1	4	yes	complemented
42	C₂×Dic₁₅	C₂²×C₃	1	2	yes	—
43	C₂×C₅×Dic₃	C₃×C₄	1	4	yes	complemented
44	C₂×C₅×Dic₃	C₂²×C₃	1	2	yes	—
45	C₂×C₃×Dic₅	—	2	4	no	—
46	C₂×C₃×Dic₅	—	2	2	no	—
47	C₂×C₃×Dic₅	—	3	4	no	—
48	C₂×C₃×Dic₅	—	3	2	no	—
49	C₂×C₃×Dic₅	Dic₃	1	4	yes	direct product factor
50	C₂×C₃×Dic₅	C₃×C₄	1	4	yes	complemented
51	C₂×C₃×Dic₅	C₂×D₃	1	2	yes	—
52	C₂×C₃×Dic₅	C₂²×C₃	1	2	yes	—
53	C₂×C₃×C₄²	—	15	1	no	—
54	C₂×C₄×Dic₃	—	5	1	no	—
55	C₂×C₃²×C₅	C₄²	1	4	yes	complemented
56	C₂×C₃²×C₅	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₃²×C₅	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₃²×C₅	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₅	—	3	1	no	—
60	C₂²×Dic₅	—	3	1	no	—
61	C₂²×Dic₅	C₃×D₃	1	1	yes	characteristic
62	C₄×Dic₅	—	3	4	no	—
63	C₂³×C₃²	C₂×D₅	1	1	yes	characteristic
64	C₂×C₃²×C₄	—	5	4	no	—
65	C₂×C₃²×C₄	—	5	2	no	—
66	C₂×C₃×Dic₃	—	5	4	no	—
67	C₂×C₃×Dic₃	—	5	2	no	—
68	C₂×C₃×Dic₃	Dic₅	1	4	yes	direct product factor
69	C₂×C₃×Dic₃	C₂×D₅	1	2	yes	—
70	C₂²×C₃×C₅	—	2	2	no	—
71	C₂²×C₃×C₅	—	2	2	no	—
72	C₂²×C₃×C₅	—	2	2	no	—
73	C₂²×C₃×C₅	—	2	1	no	—
74	C₂²×C₃×C₅	C₄×D₃	1	2	yes	—
75	C₂²×C₃×C₅	C₄×D₃	1	2	yes	—
76	C₂²×C₃×C₅	C₂×Dic₃	1	2	yes	—
77	C₂²×C₃×C₅	C₂×C₃×C₄	1	2	yes	—
78	C₂²×C₃×C₅	C₂×C₃×C₄	1	2	yes	—
79	C₂²×C₃×C₅	C₂×C₃×C₄	1	2	yes	—
80	C₂²×C₃×C₅	C₂²×D₃	1	1	yes	characteristic
81	C₂²×C₃×C₅	C₂³×C₃	1	1	yes	characteristic
82	C₃×C₄×C₅	—	3	4	no	—
83	Dic₁₅	C₂×C₃×C₄	1	4	yes	complemented
84	C₃×Dic₅	—	2	4	no	—
85	C₃×Dic₅	—	3	4	no	—
86	C₃×Dic₅	C₂×Dic₃	1	4	yes	direct product factor
87	C₃×Dic₅	C₂×C₃×C₄	1	4	yes	complemented
88	C₅×Dic₃	C₂×C₃×C₄	1	4	yes	complemented
89	C₂²×C₃×C₄	—	3	1	no	—
90	C₂²×C₃×C₄	—	5	1	no	—
91	C₂²×C₃×C₄	—	5	1	no	—
92	C₂²×C₃×C₄	—	10	1	no	—
93	C₂²×C₃×C₄	—	15	1	no	—
94	C₂²×Dic₃	—	5	1	no	—
95	C₂²×Dic₃	C₃×D₅	1	1	yes	characteristic
96	C₃×C₄²	—	15	4	no	—
97	C₄×Dic₃	—	5	4	no	—
98	C₃²×C₅	C₂×C₄²	1	1	yes	characteristic, complemented, Hall
99	C₂³×C₅	C₂×C₃×D₃	1	1	yes	characteristic
100	C₂×C₄×C₅	—	3	4	no	—
101	C₂×C₄×C₅	—	3	2	no	—
102	C₂×Dic₅	—	3	4	no	—
103	C₂×Dic₅	—	3	2	no	—
104	C₂×Dic₅	C₃×Dic₃	1	4	yes	direct product factor
105	C₂×Dic₅	C₂×C₃×D₃	1	2	yes	—
106	C₂²×C₃²	C₄×D₅	1	2	yes	—
107	C₂²×C₃²	C₄×D₅	1	2	yes	—
108	C₂²×C₃²	C₂×Dic₅	1	2	yes	—
109	C₂²×C₃²	C₂²×D₅	1	1	yes	characteristic
110	C₃²×C₄	—	5	4	no	—
111	C₃×Dic₃	—	5	4	no	—
112	C₃×Dic₃	C₂×Dic₅	1	4	yes	direct product factor
113	C₂×C₄²	—	15	1	no	Sylow
114	C₂×C₃×C₅	—	2	4	no	—
115	C₂×C₃×C₅	—	2	1	no	—
116	C₂×C₃×C₅	—	2	1	no	—
117	C₂×C₃×C₅	—	2	1	no	—
118	C₂×C₃×C₅	C₄×Dic₃	1	4	yes	complemented
119	C₂×C₃×C₅	C₃×C₄²	1	4	yes	complemented
120	C₂×C₃×C₅	C₂×C₄×D₃	1	1	yes	characteristic
121	C₂×C₃×C₅	C₂×C₄×D₃	1	1	yes	characteristic
122	C₂×C₃×C₅	C₂²×Dic₃	1	1	yes	characteristic
123	C₂×C₃×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
124	C₂×C₃×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
125	C₂×C₃×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
126	C₂³×C₃	—	2	1	no	—
127	C₂³×C₃	D₃×D₅	1	1	yes	characteristic
128	C₂³×C₃	C₂×C₃×D₅	1	1	yes	characteristic
129	C₂×C₃×C₄	—	3	4	no	—
130	C₂×C₃×C₄	—	3	2	no	—
131	C₂×C₃×C₄	—	5	4	no	—
132	C₂×C₃×C₄	—	5	4	no	—
133	C₂×C₃×C₄	—	5	2	no	—
134	C₂×C₃×C₄	—	5	2	no	—
135	C₂×C₃×C₄	—	10	4	no	—
136	C₂×C₃×C₄	—	10	2	no	—
137	C₂×C₃×C₄	—	15	4	no	—
138	C₂×C₃×C₄	—	15	2	no	—
139	C₂×Dic₃	—	5	4	no	—
140	C₂×Dic₃	—	5	2	no	—
141	C₂×Dic₃	C₃×Dic₅	1	4	yes	direct product factor
142	C₂×Dic₃	C₂×C₃×D₅	1	2	yes	—
143	C₂²×C₅	C₃×C₄×D₃	1	2	yes	—
144	C₂²×C₅	C₃×C₄×D₃	1	2	yes	—
145	C₂²×C₅	C₂×C₃×Dic₃	1	2	yes	—
146	C₂²×C₅	C₂²×C₃×D₃	1	1	yes	characteristic
147	C₄×C₅	—	3	4	no	—
148	Dic₅	—	3	4	no	—
149	Dic₅	C₂×C₃×Dic₃	1	4	yes	direct product factor
150	C₂×C₃²	C₄×Dic₅	1	4	yes	complemented
151	C₂×C₃²	C₂×C₄×D₅	1	1	yes	characteristic
152	C₂×C₃²	C₂×C₄×D₅	1	1	yes	characteristic
153	C₂×C₃²	C₂²×Dic₅	1	1	yes	characteristic
154	C₂²×C₄	—	3	1	no	—
155	C₂²×C₄	—	5	1	no	—
156	C₂²×C₄	—	15	1	no	—
157	C₄²	—	15	4	no	—
158	C₃×C₅	—	2	1	no	—
159	C₃×C₅	C₂×C₄×Dic₃	1	1	yes	characteristic, complemented
160	C₃×C₅	C₂×C₃×C₄²	1	1	yes	characteristic, complemented
161	C₂²×C₃	—	2	2	no	—
162	C₂²×C₃	—	2	2	no	—
163	C₂²×C₃	—	2	2	no	—
164	C₂²×C₃	—	2	1	no	—
165	C₂²×C₃	D₅×Dic₃	1	2	yes	—
166	C₂²×C₃	D₃×Dic₅	1	2	yes	—
167	C₂²×C₃	#1200	1	2	yes	—
168	C₂²×C₃	C₃×C₄×D₅	1	2	yes	—
169	C₂²×C₃	C₃×C₄×D₅	1	2	yes	—
170	C₂²×C₃	C₂×C₃×Dic₅	1	2	yes	—
171	C₂²×C₃	C₂×D₃×D₅	1	1	yes	characteristic
172	C₂²×C₃	C₂²×C₃×D₅	1	1	yes	characteristic
173	C₃×C₄	—	3	4	no	—
174	C₃×C₄	—	5	4	no	—
175	C₃×C₄	—	5	4	no	—
176	C₃×C₄	—	10	4	no	—
177	C₃×C₄	—	15	4	no	—
178	Dic₃	—	5	4	no	—
179	Dic₃	C₂×C₃×Dic₅	1	4	yes	direct product factor
180	C₂×C₅	C₃×C₄×Dic₃	1	4	yes	complemented
181	C₂×C₅	C₂×C₃×C₄×D₃	1	1	yes	characteristic
182	C₂×C₅	C₂×C₃×C₄×D₃	1	1	yes	characteristic
183	C₂×C₅	C₂²×C₃×Dic₃	1	1	yes	characteristic
184	C₃²	C₂×C₄×Dic₅	1	1	yes	characteristic, complemented, Sylow
185	C₂³	C₃×D₃×D₅	1	1	yes	characteristic
186	C₂×C₄	—	3	4	no	—
187	C₂×C₄	—	3	2	no	—
188	C₂×C₄	—	5	4	no	—
189	C₂×C₄	—	5	2	no	—
190	C₂×C₄	—	15	4	no	—
191	C₂×C₄	—	15	2	no	—
192	C₂×C₃	—	2	4	no	—
193	C₂×C₃	—	2	1	no	—
194	C₂×C₃	—	2	1	no	—
195	C₂×C₃	—	2	1	no	—
196	C₂×C₃	Dic₃×Dic₅	1	4	yes	direct product factor
197	C₂×C₃	C₃×C₄×Dic₅	1	4	yes	complemented
198	C₂×C₃	C₂×D₅×Dic₃	1	1	yes	characteristic
199	C₂×C₃	C₂×D₃×Dic₅	1	1	yes	characteristic
200	C₂×C₃	C₂×#1200	1	1	yes	characteristic
201	C₂×C₃	C₂×C₃×C₄×D₅	1	1	yes	characteristic
202	C₂×C₃	C₂×C₃×C₄×D₅	1	1	yes	characteristic
203	C₂×C₃	C₂²×C₃×Dic₅	1	1	yes	characteristic
204	C₅	C₂×C₃×C₄×Dic₃	1	1	yes	characteristic, complemented, Sylow
205	C₂²	C₃×D₅×Dic₃	1	2	yes	—
206	C₂²	C₃×D₃×Dic₅	1	2	yes	—
207	C₂²	C₃×#1200	1	2	yes	—
208	C₂²	C₂×C₃×D₃×D₅	1	1	yes	characteristic
209	C₄	—	3	4	no	—
210	C₄	—	5	4	no	—
211	C₄	—	15	4	no	—
212	C₃	—	2	1	no	—
213	C₃	C₂×Dic₃×Dic₅	1	1	yes	characteristic, direct product factor
214	C₃	C₂×C₃×C₄×Dic₅	1	1	yes	characteristic, complemented
215	C₂	C₃×Dic₃×Dic₅	1	4	yes	direct product factor
216	C₂	C₂×C₃×D₅×Dic₃	1	1	yes	characteristic
217	C₂	C₂×C₃×D₃×Dic₅	1	1	yes	characteristic
218	C₂	C₂×C₃×#1200	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
4. C₃×Dic₃×Dic₅	C₂	direct (N × H)
6. C₂×Dic₃×Dic₅	C₃	direct (N × H)
8. C₂×C₃×Dic₁₅	C₄	semidirect
10. C₂×C₃×C₅×Dic₃	C₄	semidirect
12. C₂×C₃²×Dic₅	C₄	semidirect
23. Dic₃×Dic₅	C₂×C₃	direct (N × H)
28. C₃×Dic₁₅	C₂×C₄	semidirect
29. C₃×C₅×Dic₃	C₂×C₄	semidirect
30. C₃²×Dic₅	C₂×C₄	semidirect
41. C₂×Dic₁₅	C₃×C₄	semidirect
43. C₂×C₅×Dic₃	C₃×C₄	semidirect
49. C₂×C₃×Dic₅	Dic₃	direct (N × H)
50. C₂×C₃×Dic₅	C₃×C₄	semidirect
55. C₂×C₃²×C₅	C₄²	semidirect
68. C₂×C₃×Dic₃	Dic₅	direct (N × H)
83. Dic₁₅	C₂×C₃×C₄	semidirect
86. C₃×Dic₅	C₂×Dic₃	direct (N × H)
87. C₃×Dic₅	C₂×C₃×C₄	semidirect
88. C₅×Dic₃	C₂×C₃×C₄	semidirect
98. C₃²×C₅	C₂×C₄²	semidirect
104. C₂×Dic₅	C₃×Dic₃	direct (N × H)
112. C₃×Dic₃	C₂×Dic₅	direct (N × H)
118. C₂×C₃×C₅	C₄×Dic₃	semidirect
119. C₂×C₃×C₅	C₃×C₄²	semidirect
141. C₂×Dic₃	C₃×Dic₅	direct (N × H)
149. Dic₅	C₂×C₃×Dic₃	direct (N × H)
150. C₂×C₃²	C₄×Dic₅	semidirect
159. C₃×C₅	C₂×C₄×Dic₃	semidirect
160. C₃×C₅	C₂×C₃×C₄²	semidirect
179. Dic₃	C₂×C₃×Dic₅	direct (N × H)
180. C₂×C₅	C₃×C₄×Dic₃	semidirect
184. C₃²	C₂×C₄×Dic₅	semidirect
196. C₂×C₃	Dic₃×Dic₅	direct (N × H)
197. C₂×C₃	C₃×C₄×Dic₅	semidirect
204. C₅	C₂×C₃×C₄×Dic₃	semidirect
213. C₃	C₂×Dic₃×Dic₅	direct (N × H)
214. C₃	C₂×C₃×C₄×Dic₅	semidirect
215. C₂	C₃×Dic₃×Dic₅	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×C₃×Dic₁₅	2
C₂²×C₃×C₅×Dic₃	2
C₂²×C₃²×Dic₅	2
C₃×Dic₃×Dic₅	2
C₂×C₃×C₄×Dic₅	3
C₂×Dic₃×Dic₅	3
C₂×C₃×C₄×Dic₃	5

Subgroups — C2×C3×Dic3×Dic5 #13086673

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×Dic₃×Dic₅ #13086673