Subgroups — C₃×C₄×D₅×Dic₃ #13086522

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

	subgroup	G/H
center, hypercenter	C₂×C₃×C₄	D₃×D₅
commutator, hypocenter	C₃×C₅	C₂×C₃×C₄²
Frattini	C₂²	C₂×C₃×D₃×D₅
Fitting	C₂×C₃²×C₄×C₅	C₂²
socle	C₂²×C₃²×C₅	C₂³
prime radical	C₂²×C₃²×D₅	C₂²
cosocle	C₂²×C₃×C₅	C₂³×C₃
LCS	G ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₃×C₄

Complements

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

N	H ≅ G/N	type
3. C₃×C₄×Dic₁₅	C₂	semidirect
4. C₃×C₄×C₅×Dic₃	C₂	semidirect
5. C₃×Dic₃×Dic₅	C₂	semidirect
7. C₄×D₅×Dic₃	C₃	direct (N × H)
13. C₃²×C₄×D₅	C₄	semidirect
14. C₃×D₅×Dic₃	C₄	direct (N × H)
22. C₄×Dic₁₅	C₂×C₃	semidirect
23. C₄×C₅×Dic₃	C₂×C₃	semidirect
26. Dic₃×Dic₅	C₂×C₃	semidirect
30. C₃²×C₄×C₅	C₂×C₄	semidirect
31. C₃×Dic₁₅	C₂×C₄	semidirect
32. C₃×C₅×Dic₃	C₂×C₄	semidirect
33. C₃²×Dic₅	C₂×C₄	semidirect
39. C₃×C₄×Dic₃	D₅	direct (N × H)
55. C₃×C₄×D₅	Dic₃	direct (N × H)
56. C₃×C₄×D₅	C₃×C₄	semidirect
57. D₅×Dic₃	C₃×C₄	direct (N × H)
62. C₃²×D₅	C₄²	semidirect
86. C₃×C₄×C₅	C₂×Dic₃	semidirect
87. C₃×C₄×C₅	C₂×C₃×C₄	semidirect
88. Dic₁₅	C₂×C₃×C₄	semidirect
91. C₃×Dic₅	C₂×Dic₃	semidirect
92. C₃×Dic₅	C₂×C₃×C₄	semidirect
93. C₅×Dic₃	C₂×C₃×C₄	semidirect
104. C₄×Dic₃	C₃×D₅	direct (N × H)
105. C₃²×C₅	C₂×C₄²	semidirect
112. C₄×D₅	C₃×Dic₃	direct (N × H)
117. C₃²×C₄	C₄×D₅	semidirect
119. C₃×Dic₃	C₄×D₅	direct (N × H)
128. C₃×D₅	C₄×Dic₃	direct (N × H)
129. C₃×D₅	C₃×C₄²	semidirect
152. C₄×C₅	C₂×C₃×Dic₃	semidirect
154. Dic₅	C₂×C₃×Dic₃	semidirect
164. C₃×C₅	C₂×C₄×Dic₃	semidirect
165. C₃×C₅	C₂×C₃×C₄²	semidirect
181. C₃×C₄	D₅×Dic₃	direct (N × H)
182. C₃×C₄	C₃×C₄×D₅	semidirect
184. Dic₃	C₃×C₄×D₅	direct (N × H)
187. D₅	C₃×C₄×Dic₃	direct (N × H)
188. C₃²	C₄²×D₅	semidirect
205. C₅	C₂×C₃×C₄×Dic₃	semidirect
212. C₄	C₃×D₅×Dic₃	direct (N × H)
214. C₃	C₄×D₅×Dic₃	direct (N × H)
215. C₃	C₃×C₄²×D₅	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃²×C₄×D₅	2
C₂×C₃×D₅×Dic₃	2
C₃×C₄×Dic₁₅	2
C₃×C₄×C₅×Dic₃	2
C₃×Dic₃×Dic₅	2
C₃×C₄²×D₅	3
C₄×D₅×Dic₃	3
C₂×C₃×C₄×Dic₃	5

Subgroups — C3×C4×D5×Dic3 #13086522

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₄×D₅×Dic₃ #13086522