Subgroups — C₃×C₅×D₃×C₄⋊C₄ #13087903

Order 1440.

394 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×C₅×D₃	C₂	1	2	yes	maximal
2	C₂×C₃×C₄×C₅×D₃	C₂	1	1	yes	maximal, characteristic
3	C₃²×C₅×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₅×C₄⋊Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₅×Dic₃⋊C₄	C₂	1	2	yes	maximal, complemented
6	C₂×C₃×C₅×C₄⋊C₄	—	3	1	no	maximal
7	C₅×D₃×C₄⋊C₄	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂²×C₃×C₅×D₃	C₂²	1	1	yes	characteristic
9	C₂×C₃²×C₄×C₅	C₂²	1	2	yes	—
10	C₂×C₃²×C₄×C₅	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₅×Dic₃	C₂²	1	2	yes	—
12	C₂×C₃×C₅×Dic₃	C₂²	1	1	yes	characteristic
13	C₃×C₄×C₅×D₃	—	2	4	no	—
14	C₃×C₄×C₅×D₃	C₄	1	4	yes	complemented
15	C₃×D₃×C₄⋊C₄	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
16	C₂²×C₃×C₄×C₅	—	3	1	no	—
17	C₂²×C₃×C₄×C₅	—	3	2	no	—
18	C₂×C₄×C₅×D₃	C₂×C₃	1	2	yes	—
19	C₂×C₄×C₅×D₃	C₂×C₃	1	1	yes	characteristic
20	C₃×C₅×C₄⋊C₄	—	2	1	no	—
21	C₃×C₅×C₄⋊C₄	—	3	2	no	—
22	C₃×C₅×C₄⋊C₄	—	3	1	no	—
23	C₃×C₅×C₄⋊C₄	D₃	1	1	yes	characteristic, direct product factor
24	C₃×C₅×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
25	C₅×C₄⋊Dic₃	C₂×C₃	1	1	yes	characteristic, complemented
26	C₅×Dic₃⋊C₄	C₂×C₃	1	2	yes	complemented
27	C₂²×C₃²×C₅	C₂³	1	1	yes	characteristic
28	C₂×C₃×C₅×D₃	C₂×C₄	1	2	yes	—
29	C₂×C₃×C₅×D₃	D₄	1	2	yes	—
30	C₂×C₃×C₅×D₃	Q₈	1	2	yes	—
31	C₃²×C₄×C₅	—	2	2	no	—
32	C₃²×C₄×C₅	C₂×C₄	1	2	yes	complemented
33	C₃×C₅×Dic₃	—	2	2	no	—
34	C₃×C₅×Dic₃	C₂×C₄	1	2	yes	complemented
35	C₂×C₅×C₄⋊C₄	—	3	1	no	Hall
36	C₂×C₃×C₄×D₃	C₂×C₅	1	2	yes	—
37	C₂×C₃×C₄×D₃	C₂×C₅	1	1	yes	characteristic
38	C₃²×C₄⋊C₄	C₂×C₅	1	1	yes	characteristic, complemented
39	C₃×C₄⋊Dic₃	C₂×C₅	1	1	yes	characteristic, complemented
40	C₃×Dic₃⋊C₄	C₂×C₅	1	2	yes	complemented
41	C₂³×C₃×C₅	—	3	1	no	—
42	C₂²×C₅×D₃	C₂²×C₃	1	1	yes	characteristic
43	C₂×C₃×C₄×C₅	—	2	1	no	—
44	C₂×C₃×C₄×C₅	—	2	2	no	—
45	C₂×C₃×C₄×C₅	—	3	1	no	—
46	C₂×C₃×C₄×C₅	—	3	4	no	—
47	C₂×C₃×C₄×C₅	—	3	2	no	—
48	C₂×C₃×C₄×C₅	—	6	4	no	—
49	C₂×C₃×C₄×C₅	C₂×D₃	1	2	yes	—
50	C₂×C₃×C₄×C₅	C₂×D₃	1	1	yes	characteristic
51	C₂×C₃×C₄×C₅	C₂²×C₃	1	2	yes	—
52	C₂×C₃×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
53	C₂×C₅×Dic₃	C₂²×C₃	1	2	yes	—
54	C₂×C₅×Dic₃	C₂²×C₃	1	1	yes	characteristic
55	C₄×C₅×D₃	—	2	4	no	—
56	C₄×C₅×D₃	C₃×C₄	1	4	yes	complemented
57	C₂×C₃×C₄⋊C₄	—	3	1	no	—
58	D₃×C₄⋊C₄	C₃×C₅	1	1	yes	characteristic, direct product factor
59	C₂×C₃²×C₅	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₃²×C₅	C₂×D₄	1	1	yes	characteristic
61	C₂×C₃²×C₅	C₂×Q₈	1	1	yes	characteristic
62	C₃×C₅×D₃	C₄⋊C₄	1	4	yes	direct product factor
63	C₂²×C₄×C₅	—	3	1	no	—
64	C₂²×C₄×C₅	—	3	2	no	—
65	C₅×C₄⋊C₄	—	3	2	no	—
66	C₅×C₄⋊C₄	—	3	1	no	—
67	C₅×C₄⋊C₄	C₃×D₃	1	1	yes	characteristic, direct product factor
68	C₂²×C₃×D₃	C₂²×C₅	1	1	yes	characteristic
69	C₂×C₃²×C₄	C₂²×C₅	1	2	yes	—
70	C₂×C₃²×C₄	C₂²×C₅	1	1	yes	characteristic
71	C₂×C₃×Dic₃	C₂²×C₅	1	2	yes	—
72	C₂×C₃×Dic₃	C₂²×C₅	1	1	yes	characteristic
73	C₃×C₄×D₃	—	2	4	no	—
74	C₃×C₄×D₃	C₄×C₅	1	4	yes	complemented
75	C₂²×C₃×C₅	—	2	1	no	—
76	C₂²×C₃×C₅	—	3	2	no	—
77	C₂²×C₃×C₅	—	3	2	no	—
78	C₂²×C₃×C₅	—	3	2	no	—
79	C₂²×C₃×C₅	C₂²×D₃	1	1	yes	characteristic
80	C₂²×C₃×C₅	C₂³×C₃	1	1	yes	characteristic
81	C₂×C₅×D₃	C₂×C₃×C₄	1	2	yes	—
82	C₂×C₅×D₃	C₃×D₄	1	2	yes	—
83	C₂×C₅×D₃	C₃×Q₈	1	2	yes	—
84	C₃×C₄×C₅	—	2	2	no	—
85	C₃×C₄×C₅	—	2	2	no	—
86	C₃×C₄×C₅	—	2	2	no	—
87	C₃×C₄×C₅	—	3	2	no	—
88	C₃×C₄×C₅	—	4	2	no	—
89	C₃×C₄×C₅	—	6	2	no	—
90	C₃×C₄×C₅	C₄×D₃	1	2	yes	complemented
91	C₃×C₄×C₅	C₂×C₃×C₄	1	2	yes	complemented
92	C₅×Dic₃	—	2	2	no	—
93	C₅×Dic₃	C₂×C₃×C₄	1	2	yes	complemented
94	C₂²×C₃×C₄	—	3	1	no	—
95	C₂²×C₃×C₄	—	3	2	no	—
96	C₂×C₄×D₃	C₂×C₃×C₅	1	2	yes	—
97	C₂×C₄×D₃	C₂×C₃×C₅	1	1	yes	characteristic
98	C₃×C₄⋊C₄	—	2	1	no	—
99	C₃×C₄⋊C₄	—	3	2	no	—
100	C₃×C₄⋊C₄	—	3	1	no	—
101	C₃×C₄⋊C₄	C₅×D₃	1	1	yes	characteristic, direct product factor
102	C₃×C₄⋊C₄	C₂×C₃×C₅	1	1	yes	characteristic, complemented
103	C₄⋊Dic₃	C₂×C₃×C₅	1	1	yes	characteristic, complemented
104	Dic₃⋊C₄	C₂×C₃×C₅	1	2	yes	complemented
105	C₃²×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic, complemented, Hall
106	C₂³×C₅	—	3	1	no	—
107	C₂×C₄×C₅	—	3	1	no	—
108	C₂×C₄×C₅	—	3	4	no	—
109	C₂×C₄×C₅	—	3	2	no	—
110	C₂×C₄×C₅	—	6	4	no	—
111	C₂×C₄×C₅	C₂×C₃×D₃	1	2	yes	—
112	C₂×C₄×C₅	C₂×C₃×D₃	1	1	yes	characteristic
113	C₂²×C₃²	C₂³×C₅	1	1	yes	characteristic
114	C₂×C₃×D₃	C₂×C₄×C₅	1	2	yes	—
115	C₂×C₃×D₃	C₅×D₄	1	2	yes	—
116	C₂×C₃×D₃	C₅×Q₈	1	2	yes	—
117	C₃²×C₄	—	2	2	no	—
118	C₃²×C₄	C₂×C₄×C₅	1	2	yes	complemented
119	C₃×Dic₃	—	2	2	no	—
120	C₃×Dic₃	C₂×C₄×C₅	1	2	yes	complemented
121	C₂×C₄⋊C₄	—	3	1	no	Sylow
122	C₂×C₃×C₅	—	2	1	no	—
123	C₂×C₃×C₅	—	2	1	no	—
124	C₂×C₃×C₅	—	2	1	no	—
125	C₂×C₃×C₅	—	3	4	no	—
126	C₂×C₃×C₅	C₂×C₄×D₃	1	1	yes	characteristic
127	C₂×C₃×C₅	D₃×D₄	1	1	yes	characteristic
128	C₂×C₃×C₅	D₃×Q₈	1	1	yes	characteristic
129	C₂×C₃×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
130	C₂×C₃×C₅	C₂×C₃×D₄	1	1	yes	characteristic
131	C₂×C₃×C₅	C₂×C₃×Q₈	1	1	yes	characteristic
132	C₅×D₃	C₃×C₄⋊C₄	1	4	yes	direct product factor
133	C₂³×C₃	—	3	1	no	—
134	C₂²×D₃	C₂²×C₃×C₅	1	1	yes	characteristic
135	C₂×C₃×C₄	—	2	1	no	—
136	C₂×C₃×C₄	—	2	2	no	—
137	C₂×C₃×C₄	—	3	1	no	—
138	C₂×C₃×C₄	—	3	4	no	—
139	C₂×C₃×C₄	—	3	2	no	—
140	C₂×C₃×C₄	—	6	4	no	—
141	C₂×C₃×C₄	C₂×C₅×D₃	1	2	yes	—
142	C₂×C₃×C₄	C₂×C₅×D₃	1	1	yes	characteristic
143	C₂×C₃×C₄	C₂²×C₃×C₅	1	2	yes	—
144	C₂×C₃×C₄	C₂²×C₃×C₅	1	1	yes	characteristic
145	C₂×Dic₃	C₂²×C₃×C₅	1	2	yes	—
146	C₂×Dic₃	C₂²×C₃×C₅	1	1	yes	characteristic
147	C₄×D₃	—	2	4	no	—
148	C₄×D₃	C₃×C₄×C₅	1	4	yes	complemented
149	C₂²×C₅	—	3	2	no	—
150	C₂²×C₅	—	3	2	no	—
151	C₂²×C₅	—	3	2	no	—
152	C₂²×C₅	C₂²×C₃×D₃	1	1	yes	characteristic
153	C₄×C₅	—	2	2	no	—
154	C₄×C₅	—	3	2	no	—
155	C₄×C₅	—	6	2	no	—
156	C₄×C₅	C₃×C₄×D₃	1	2	yes	complemented
157	C₂×C₃²	C₂²×C₄×C₅	1	1	yes	characteristic
158	C₂×C₃²	C₂×C₅×D₄	1	1	yes	characteristic
159	C₂×C₃²	C₂×C₅×Q₈	1	1	yes	characteristic
160	C₃×D₃	C₅×C₄⋊C₄	1	4	yes	direct product factor
161	C₂²×C₄	—	3	1	no	—
162	C₂²×C₄	—	3	2	no	—
163	C₄⋊C₄	—	3	2	no	—
164	C₄⋊C₄	—	3	1	no	—
165	C₄⋊C₄	C₃×C₅×D₃	1	1	yes	characteristic, direct product factor
166	C₃×C₅	—	2	1	no	—
167	C₃×C₅	D₃×C₄⋊C₄	1	1	yes	characteristic, direct product factor
168	C₃×C₅	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic, complemented
169	C₂²×C₃	—	2	1	no	—
170	C₂²×C₃	—	3	2	no	—
171	C₂²×C₃	—	3	2	no	—
172	C₂²×C₃	—	3	2	no	—
173	C₂²×C₃	C₂²×C₅×D₃	1	1	yes	characteristic
174	C₂²×C₃	C₂³×C₃×C₅	1	1	yes	characteristic
175	C₂×D₃	C₂×C₃×C₄×C₅	1	2	yes	—
176	C₂×D₃	C₃×C₅×D₄	1	2	yes	—
177	C₂×D₃	C₃×C₅×Q₈	1	2	yes	—
178	C₃×C₄	—	2	2	no	—
179	C₃×C₄	—	2	2	no	—
180	C₃×C₄	—	2	2	no	—
181	C₃×C₄	—	3	2	no	—
182	C₃×C₄	—	4	2	no	—
183	C₃×C₄	—	6	2	no	—
184	C₃×C₄	C₄×C₅×D₃	1	2	yes	complemented
185	C₃×C₄	C₂×C₃×C₄×C₅	1	2	yes	complemented
186	Dic₃	—	2	2	no	—
187	Dic₃	C₂×C₃×C₄×C₅	1	2	yes	complemented
188	C₂×C₅	—	3	4	no	—
189	C₂×C₅	C₂×C₃×C₄×D₃	1	1	yes	characteristic
190	C₂×C₅	C₃×D₃×D₄	1	1	yes	characteristic
191	C₂×C₅	C₃×D₃×Q₈	1	1	yes	characteristic
192	C₃²	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
193	C₂³	—	3	1	no	—
194	C₂×C₄	—	3	1	no	—
195	C₂×C₄	—	3	4	no	—
196	C₂×C₄	—	3	2	no	—
197	C₂×C₄	—	6	4	no	—
198	C₂×C₄	C₂×C₃×C₅×D₃	1	2	yes	—
199	C₂×C₄	C₂×C₃×C₅×D₃	1	1	yes	characteristic
200	C₂×C₃	—	2	1	no	—
201	C₂×C₃	—	2	1	no	—
202	C₂×C₃	—	2	1	no	—
203	C₂×C₃	—	3	4	no	—
204	C₂×C₃	C₂×C₄×C₅×D₃	1	1	yes	characteristic
205	C₂×C₃	C₅×D₃×D₄	1	1	yes	characteristic
206	C₂×C₃	C₅×D₃×Q₈	1	1	yes	characteristic
207	C₂×C₃	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
208	C₂×C₃	C₂×C₃×C₅×D₄	1	1	yes	characteristic
209	C₂×C₃	C₂×C₃×C₅×Q₈	1	1	yes	characteristic
210	D₃	C₃×C₅×C₄⋊C₄	1	4	yes	direct product factor
211	C₅	C₃×D₃×C₄⋊C₄	1	1	yes	characteristic, direct product factor, Sylow
212	C₂²	—	3	2	no	—
213	C₂²	—	3	2	no	—
214	C₂²	—	3	2	no	—
215	C₂²	C₂²×C₃×C₅×D₃	1	1	yes	characteristic
216	C₄	—	2	2	no	—
217	C₄	—	3	2	no	—
218	C₄	—	6	2	no	—
219	C₄	C₃×C₄×C₅×D₃	1	2	yes	complemented
220	C₃	—	2	1	no	—
221	C₃	C₅×D₃×C₄⋊C₄	1	1	yes	characteristic, direct product factor
222	C₃	C₂×C₃×C₅×C₄⋊C₄	1	1	yes	characteristic, complemented
223	C₂	—	3	4	no	—
224	C₂	C₂×C₃×C₄×C₅×D₃	1	1	yes	characteristic
225	C₂	C₃×C₅×D₃×D₄	1	1	yes	characteristic
226	C₂	C₃×C₅×D₃×Q₈	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
3. C₃²×C₅×C₄⋊C₄	C₂	semidirect
4. C₃×C₅×C₄⋊Dic₃	C₂	semidirect
5. C₃×C₅×Dic₃⋊C₄	C₂	semidirect
7. C₅×D₃×C₄⋊C₄	C₃	direct (N × H)
14. C₃×C₄×C₅×D₃	C₄	semidirect
15. C₃×D₃×C₄⋊C₄	C₅	direct (N × H)
23. C₃×C₅×C₄⋊C₄	D₃	direct (N × H)
24. C₃×C₅×C₄⋊C₄	C₂×C₃	semidirect
25. C₅×C₄⋊Dic₃	C₂×C₃	semidirect
26. C₅×Dic₃⋊C₄	C₂×C₃	semidirect
32. C₃²×C₄×C₅	C₂×C₄	semidirect
34. C₃×C₅×Dic₃	C₂×C₄	semidirect
38. C₃²×C₄⋊C₄	C₂×C₅	semidirect
39. C₃×C₄⋊Dic₃	C₂×C₅	semidirect
40. C₃×Dic₃⋊C₄	C₂×C₅	semidirect
56. C₄×C₅×D₃	C₃×C₄	semidirect
58. D₃×C₄⋊C₄	C₃×C₅	direct (N × H)
62. C₃×C₅×D₃	C₄⋊C₄	direct (N × H)
67. C₅×C₄⋊C₄	C₃×D₃	direct (N × H)
74. C₃×C₄×D₃	C₄×C₅	semidirect
90. C₃×C₄×C₅	C₄×D₃	semidirect
91. C₃×C₄×C₅	C₂×C₃×C₄	semidirect
93. C₅×Dic₃	C₂×C₃×C₄	semidirect
101. C₃×C₄⋊C₄	C₅×D₃	direct (N × H)
102. C₃×C₄⋊C₄	C₂×C₃×C₅	semidirect
103. C₄⋊Dic₃	C₂×C₃×C₅	semidirect
104. Dic₃⋊C₄	C₂×C₃×C₅	semidirect
105. C₃²×C₅	C₂×C₄⋊C₄	semidirect
118. C₃²×C₄	C₂×C₄×C₅	semidirect
120. C₃×Dic₃	C₂×C₄×C₅	semidirect
132. C₅×D₃	C₃×C₄⋊C₄	direct (N × H)
148. C₄×D₃	C₃×C₄×C₅	semidirect
156. C₄×C₅	C₃×C₄×D₃	semidirect
160. C₃×D₃	C₅×C₄⋊C₄	direct (N × H)
165. C₄⋊C₄	C₃×C₅×D₃	direct (N × H)
167. C₃×C₅	D₃×C₄⋊C₄	direct (N × H)
168. C₃×C₅	C₂×C₃×C₄⋊C₄	semidirect
184. C₃×C₄	C₄×C₅×D₃	semidirect
185. C₃×C₄	C₂×C₃×C₄×C₅	semidirect
187. Dic₃	C₂×C₃×C₄×C₅	semidirect
192. C₃²	C₂×C₅×C₄⋊C₄	semidirect
210. D₃	C₃×C₅×C₄⋊C₄	direct (N × H)
211. C₅	C₃×D₃×C₄⋊C₄	direct (N × H)
219. C₄	C₃×C₄×C₅×D₃	semidirect
221. C₃	C₅×D₃×C₄⋊C₄	direct (N × H)
222. C₃	C₂×C₃×C₅×C₄⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄×C₅×D₃	2
C₃²×C₅×C₄⋊C₄	2
C₃×C₅×C₄⋊Dic₃	2
C₃×C₅×Dic₃⋊C₄	2
C₂×C₃×C₅×C₄⋊C₄	3
C₅×D₃×C₄⋊C₄	3
C₃×D₃×C₄⋊C₄	5

Subgroups — C3×C5×D3×C4⋊C4 #13087903

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×D₃×C₄⋊C₄ #13087903