Subgroups — C₃×C₅×#2401 #14275435

Order 1920.

864 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#401	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×C₅×#400	C₂	1	3	yes	maximal, complemented
3	C₃×C₅×#397	C₂	1	3	yes	maximal
4	C₃×C₅×#396	C₂	1	1	yes	maximal, characteristic
5	C₃×C₄×C₅×C₄⋊C₄	C₂	1	3	yes	maximal, complemented
6	C₃×C₄×C₅×C₂²⋊C₄	C₂	1	3	yes	maximal
7	C₃×C₅×#376	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#2401	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	3	yes	—
11	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	3	yes	—
12	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	3	yes	—
13	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	3	yes	—
14	C₂×C₃×C₄²×C₅	C₂²	1	3	yes	—
15	C₂×C₃×C₄²×C₅	C₂²	1	3	yes	—
16	C₃×C₅×#95	C₂²	1	1	yes	characteristic
17	C₃×C₅×#95	C₂²	1	3	yes	—
18	C₃×C₅×#95	C₂²	1	3	yes	—
19	C₃×C₅×#95	C₂²	1	3	yes	—
20	C₃×C₅×#95	C₂²	1	3	yes	—
21	C₃×C₅×#95	C₂²	1	3	yes	—
22	C₃×#2401	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
23	C₅×#401	C₂×C₃	1	1	yes	characteristic, complemented
24	C₅×#400	C₂×C₃	1	3	yes	complemented
25	C₅×#397	C₂×C₃	1	3	yes	—
26	C₅×#396	C₂×C₃	1	1	yes	characteristic
27	C₄×C₅×C₄⋊C₄	C₂×C₃	1	3	yes	complemented
28	C₄×C₅×C₂²⋊C₄	C₂×C₃	1	3	yes	—
29	C₅×#376	C₂×C₃	1	1	yes	characteristic, complemented
30	C₂⁴×C₃×C₅	C₂³	1	1	yes	characteristic
31	C₂²×C₃×C₄×C₅	—	2	6	no	—
32	C₂²×C₃×C₄×C₅	C₂³	1	3	yes	—
33	C₂²×C₃×C₄×C₅	C₂³	1	3	yes	—
34	C₂²×C₃×C₄×C₅	C₂³	1	3	yes	—
35	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
36	C₂²×C₃×C₄×C₅	C₂³	1	3	yes	—
37	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
38	C₃×C₅×C₄⋊C₄	—	2	6	no	—
39	C₃×C₅×C₄⋊C₄	—	2	6	no	—
40	C₃×C₅×C₂²⋊C₄	—	2	6	no	—
41	C₃×C₅×C₂²⋊C₄	—	2	6	no	—
42	C₃×C₄²×C₅	—	2	6	no	—
43	C₃×C₄²×C₅	—	2	6	no	—
44	C₃×#401	C₂×C₅	1	1	yes	characteristic, complemented
45	C₃×#400	C₂×C₅	1	3	yes	complemented
46	C₃×#397	C₂×C₅	1	3	yes	—
47	C₃×#396	C₂×C₅	1	1	yes	characteristic
48	C₃×C₄×C₄⋊C₄	C₂×C₅	1	3	yes	complemented
49	C₃×C₄×C₂²⋊C₄	C₂×C₅	1	3	yes	—
50	C₃×#376	C₂×C₅	1	1	yes	characteristic, complemented
51	C₂³×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
52	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	3	yes	—
53	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	3	yes	—
54	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	3	yes	—
55	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	3	yes	—
56	C₂×C₄²×C₅	C₂²×C₃	1	3	yes	—
57	C₂×C₄²×C₅	C₂²×C₃	1	3	yes	—
58	C₅×#95	C₂²×C₃	1	1	yes	characteristic
59	C₅×#95	C₂²×C₃	1	3	yes	—
60	C₅×#95	C₂²×C₃	1	3	yes	—
61	C₅×#95	C₂²×C₃	1	3	yes	—
62	C₅×#95	C₂²×C₃	1	3	yes	—
63	C₅×#95	C₂²×C₃	1	3	yes	—
64	#2401	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
65	C₂³×C₃×C₅	—	2	3	no	—
66	C₂³×C₃×C₅	—	2	3	no	—
67	C₂³×C₃×C₅	#41	1	2	yes	—
68	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
69	C₂×C₃×C₄×C₅	—	2	3	no	—
70	C₂×C₃×C₄×C₅	—	2	3	no	—
71	C₂×C₃×C₄×C₅	—	2	3	no	—
72	C₂×C₃×C₄×C₅	—	2	3	no	—
73	C₂×C₃×C₄×C₅	—	2	3	no	—
74	C₂×C₃×C₄×C₅	—	2	3	no	—
75	C₂×C₃×C₄×C₅	—	2	3	no	—
76	C₂×C₃×C₄×C₅	—	2	3	no	—
77	C₂×C₃×C₄×C₅	—	2	3	no	—
78	C₂×C₃×C₄×C₅	—	2	3	no	—
79	C₂×C₃×C₄×C₅	—	2	3	no	—
80	C₂×C₃×C₄×C₅	—	2	3	no	—
81	C₂×C₃×C₄×C₅	—	4	4	no	—
82	C₂×C₃×C₄×C₅	#41	1	6	yes	—
83	C₂×C₃×C₄×C₅	#41	1	6	yes	—
84	C₂³×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
85	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	3	yes	—
86	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	3	yes	—
87	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	3	yes	—
88	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	3	yes	—
89	C₂×C₃×C₄²	C₂²×C₅	1	3	yes	—
90	C₂×C₃×C₄²	C₂²×C₅	1	3	yes	—
91	C₃×#95	C₂²×C₅	1	1	yes	characteristic
92	C₃×#95	C₂²×C₅	1	3	yes	—
93	C₃×#95	C₂²×C₅	1	3	yes	—
94	C₃×#95	C₂²×C₅	1	3	yes	—
95	C₃×#95	C₂²×C₅	1	3	yes	—
96	C₃×#95	C₂²×C₅	1	3	yes	—
97	C₂⁴×C₅	C₂³×C₃	1	1	yes	characteristic
98	C₂²×C₄×C₅	—	2	6	no	—
99	C₂²×C₄×C₅	C₂³×C₃	1	3	yes	—
100	C₂²×C₄×C₅	C₂³×C₃	1	3	yes	—
101	C₂²×C₄×C₅	C₂³×C₃	1	3	yes	—
102	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
103	C₂²×C₄×C₅	C₂³×C₃	1	3	yes	—
104	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
105	C₅×C₄⋊C₄	—	2	6	no	—
106	C₅×C₄⋊C₄	—	2	6	no	—
107	C₅×C₂²⋊C₄	—	2	6	no	—
108	C₅×C₂²⋊C₄	—	2	6	no	—
109	C₄²×C₅	—	2	6	no	—
110	C₄²×C₅	—	2	6	no	—
111	#401	C₂×C₃×C₅	1	1	yes	characteristic, complemented
112	#400	C₂×C₃×C₅	1	3	yes	complemented
113	#397	C₂×C₃×C₅	1	3	yes	—
114	#396	C₂×C₃×C₅	1	1	yes	characteristic
115	C₄×C₄⋊C₄	C₂×C₃×C₅	1	3	yes	complemented
116	C₄×C₂²⋊C₄	C₂×C₃×C₅	1	3	yes	—
117	#376	C₂×C₃×C₅	1	1	yes	characteristic, complemented
118	C₂²×C₃×C₅	—	2	6	no	—
119	C₂²×C₃×C₅	—	4	1	no	—
120	C₂²×C₃×C₅	—	4	3	no	—
121	C₂²×C₃×C₅	C₂×#41	1	3	yes	—
122	C₂²×C₃×C₅	C₂×#41	1	1	yes	characteristic
123	C₂²×C₃×C₅	C₂×#41	1	3	yes	—
124	C₃×C₄×C₅	—	2	6	no	—
125	C₃×C₄×C₅	—	4	1	no	—
126	C₃×C₄×C₅	—	4	3	no	—
127	C₃×C₄×C₅	—	4	3	no	—
128	C₃×C₄×C₅	—	4	3	no	—
129	C₃×C₄×C₅	#126	1	4	yes	—
130	C₂⁴×C₃	C₂³×C₅	1	1	yes	characteristic
131	C₂²×C₃×C₄	—	2	6	no	—
132	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
133	C₂²×C₃×C₄	C₂³×C₅	1	3	yes	—
134	C₂²×C₃×C₄	C₂³×C₅	1	3	yes	—
135	C₂²×C₃×C₄	C₂³×C₅	1	3	yes	—
136	C₂²×C₃×C₄	C₂³×C₅	1	3	yes	—
137	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
138	C₃×C₄⋊C₄	—	2	6	no	—
139	C₃×C₄⋊C₄	—	2	6	no	—
140	C₃×C₂²⋊C₄	—	2	6	no	—
141	C₃×C₂²⋊C₄	—	2	6	no	—
142	C₃×C₄²	—	2	6	no	—
143	C₃×C₄²	—	2	6	no	—
144	C₂³×C₅	—	2	3	no	—
145	C₂³×C₅	—	2	3	no	—
146	C₂³×C₅	C₃×#41	1	2	yes	—
147	C₂³×C₅	C₂⁴×C₃	1	1	yes	characteristic
148	C₂×C₄×C₅	—	2	3	no	—
149	C₂×C₄×C₅	—	2	3	no	—
150	C₂×C₄×C₅	—	2	3	no	—
151	C₂×C₄×C₅	—	2	3	no	—
152	C₂×C₄×C₅	—	2	3	no	—
153	C₂×C₄×C₅	—	2	3	no	—
154	C₂×C₄×C₅	—	2	3	no	—
155	C₂×C₄×C₅	—	2	3	no	—
156	C₂×C₄×C₅	—	2	3	no	—
157	C₂×C₄×C₅	—	2	3	no	—
158	C₂×C₄×C₅	—	2	3	no	—
159	C₂×C₄×C₅	—	2	3	no	—
160	C₂×C₄×C₅	—	4	4	no	—
161	C₂×C₄×C₅	C₃×#41	1	6	yes	—
162	C₂×C₄×C₅	C₃×#41	1	6	yes	—
163	C₂³×C₄	C₂²×C₃×C₅	1	1	yes	characteristic
164	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	3	yes	—
165	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	3	yes	—
166	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	3	yes	—
167	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	3	yes	—
168	C₂×C₄²	C₂²×C₃×C₅	1	3	yes	—
169	C₂×C₄²	C₂²×C₃×C₅	1	3	yes	—
170	#95	C₂²×C₃×C₅	1	1	yes	characteristic
171	#95	C₂²×C₃×C₅	1	3	yes	—
172	#95	C₂²×C₃×C₅	1	3	yes	—
173	#95	C₂²×C₃×C₅	1	3	yes	—
174	#95	C₂²×C₃×C₅	1	3	yes	—
175	#95	C₂²×C₃×C₅	1	3	yes	—
176	C₂×C₃×C₅	—	4	2	no	—
177	C₂×C₃×C₅	C₂×#126	1	1	yes	characteristic
178	C₂×C₃×C₅	#529	1	3	yes	—
179	C₂×C₃×C₅	#529	1	3	yes	—
180	C₂³×C₃	—	2	3	no	—
181	C₂³×C₃	—	2	3	no	—
182	C₂³×C₃	C₅×#41	1	2	yes	—
183	C₂³×C₃	C₂⁴×C₅	1	1	yes	characteristic
184	C₂×C₃×C₄	—	2	3	no	—
185	C₂×C₃×C₄	—	2	3	no	—
186	C₂×C₃×C₄	—	2	3	no	—
187	C₂×C₃×C₄	—	2	3	no	—
188	C₂×C₃×C₄	—	2	3	no	—
189	C₂×C₃×C₄	—	2	3	no	—
190	C₂×C₃×C₄	—	2	3	no	—
191	C₂×C₃×C₄	—	2	3	no	—
192	C₂×C₃×C₄	—	2	3	no	—
193	C₂×C₃×C₄	—	2	3	no	—
194	C₂×C₃×C₄	—	2	3	no	—
195	C₂×C₃×C₄	—	2	3	no	—
196	C₂×C₃×C₄	—	4	4	no	—
197	C₂×C₃×C₄	C₅×#41	1	6	yes	—
198	C₂×C₃×C₄	C₅×#41	1	6	yes	—
199	C₂²×C₅	—	2	6	no	—
200	C₂²×C₅	—	4	1	no	—
201	C₂²×C₅	—	4	3	no	—
202	C₂²×C₅	C₂×C₃×#41	1	3	yes	—
203	C₂²×C₅	C₂×C₃×#41	1	1	yes	characteristic
204	C₂²×C₅	C₂×C₃×#41	1	3	yes	—
205	C₄×C₅	—	2	6	no	—
206	C₄×C₅	—	4	1	no	—
207	C₄×C₅	—	4	3	no	—
208	C₄×C₅	—	4	3	no	—
209	C₄×C₅	—	4	3	no	—
210	C₄×C₅	C₃×#126	1	4	yes	—
211	C₂⁴	C₂³×C₃×C₅	1	1	yes	characteristic
212	C₂²×C₄	—	2	6	no	—
213	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
214	C₂²×C₄	C₂³×C₃×C₅	1	3	yes	—
215	C₂²×C₄	C₂³×C₃×C₅	1	3	yes	—
216	C₂²×C₄	C₂³×C₃×C₅	1	3	yes	—
217	C₂²×C₄	C₂³×C₃×C₅	1	3	yes	—
218	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
219	C₄⋊C₄	—	2	6	no	—
220	C₄⋊C₄	—	2	6	no	—
221	C₂²⋊C₄	—	2	6	no	—
222	C₂²⋊C₄	—	2	6	no	—
223	C₄²	—	2	6	no	—
224	C₄²	—	2	6	no	—
225	C₃×C₅	#2401	1	1	yes	characteristic, direct product factor, Hall
226	C₂²×C₃	—	2	6	no	—
227	C₂²×C₃	—	4	1	no	—
228	C₂²×C₃	—	4	3	no	—
229	C₂²×C₃	C₂×C₅×#41	1	3	yes	—
230	C₂²×C₃	C₂×C₅×#41	1	1	yes	characteristic
231	C₂²×C₃	C₂×C₅×#41	1	3	yes	—
232	C₃×C₄	—	2	6	no	—
233	C₃×C₄	—	4	1	no	—
234	C₃×C₄	—	4	3	no	—
235	C₃×C₄	—	4	3	no	—
236	C₃×C₄	—	4	3	no	—
237	C₃×C₄	C₅×#126	1	4	yes	—
238	C₂×C₅	—	4	2	no	—
239	C₂×C₅	C₂×C₃×#126	1	1	yes	characteristic
240	C₂×C₅	C₃×#529	1	3	yes	—
241	C₂×C₅	C₃×#529	1	3	yes	—
242	C₂³	—	2	3	no	—
243	C₂³	—	2	3	no	—
244	C₂³	C₃×C₅×#41	1	2	yes	—
245	C₂³	C₂⁴×C₃×C₅	1	1	yes	characteristic
246	C₂×C₄	—	2	3	no	—
247	C₂×C₄	—	2	3	no	—
248	C₂×C₄	—	2	3	no	—
249	C₂×C₄	—	2	3	no	—
250	C₂×C₄	—	2	3	no	—
251	C₂×C₄	—	2	3	no	—
252	C₂×C₄	—	2	3	no	—
253	C₂×C₄	—	2	3	no	—
254	C₂×C₄	—	2	3	no	—
255	C₂×C₄	—	2	3	no	—
256	C₂×C₄	—	2	3	no	—
257	C₂×C₄	—	2	3	no	—
258	C₂×C₄	—	4	4	no	—
259	C₂×C₄	C₃×C₅×#41	1	6	yes	—
260	C₂×C₄	C₃×C₅×#41	1	6	yes	—
261	C₂×C₃	—	4	2	no	—
262	C₂×C₃	C₂×C₅×#126	1	1	yes	characteristic
263	C₂×C₃	C₅×#529	1	3	yes	—
264	C₂×C₃	C₅×#529	1	3	yes	—
265	C₅	C₃×#2401	1	1	yes	characteristic, direct product factor, Sylow
266	C₂²	—	2	6	no	—
267	C₂²	—	4	1	no	—
268	C₂²	—	4	3	no	—
269	C₂²	C₂×C₃×C₅×#41	1	3	yes	—
270	C₂²	C₂×C₃×C₅×#41	1	1	yes	characteristic
271	C₂²	C₂×C₃×C₅×#41	1	3	yes	—
272	C₄	—	2	6	no	—
273	C₄	—	4	1	no	—
274	C₄	—	4	3	no	—
275	C₄	—	4	3	no	—
276	C₄	—	4	3	no	—
277	C₄	C₃×C₅×#126	1	4	yes	—
278	C₃	C₅×#2401	1	1	yes	characteristic, direct product factor, Sylow
279	C₂	—	4	2	no	—
280	C₂	C₂×C₃×C₅×#126	1	1	yes	characteristic
281	C₂	C₃×C₅×#529	1	3	yes	—
282	C₂	C₃×C₅×#529	1	3	yes	—

	subgroup	G/H
center	C₂²×C₃×C₄×C₅	C₂³
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃×C₅
socle	C₂³×C₃×C₅	C₂⁴
prime radical	C₂⁴×C₃×C₅	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×#401	C₂	semidirect
2. C₃×C₅×#400	C₂	semidirect
5. C₃×C₄×C₅×C₄⋊C₄	C₂	semidirect
7. C₃×C₅×#376	C₂	semidirect
8. C₅×#2401	C₃	direct (N × H)
22. C₃×#2401	C₅	direct (N × H)
23. C₅×#401	C₂×C₃	semidirect
24. C₅×#400	C₂×C₃	semidirect
27. C₄×C₅×C₄⋊C₄	C₂×C₃	semidirect
29. C₅×#376	C₂×C₃	semidirect
44. C₃×#401	C₂×C₅	semidirect
45. C₃×#400	C₂×C₅	semidirect
48. C₃×C₄×C₄⋊C₄	C₂×C₅	semidirect
50. C₃×#376	C₂×C₅	semidirect
64. #2401	C₃×C₅	direct (N × H)
111. #401	C₂×C₃×C₅	semidirect
112. #400	C₂×C₃×C₅	semidirect
115. C₄×C₄⋊C₄	C₂×C₃×C₅	semidirect
117. #376	C₂×C₃×C₅	semidirect
225. C₃×C₅	#2401	direct (N × H)
265. C₅	C₃×#2401	direct (N × H)
278. C₃	C₅×#2401	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₅×#401	2
C₃×C₅×#400	2
C₃×C₅×#397	2
C₃×C₅×#396	2
C₃×C₄×C₅×C₄⋊C₄	2
C₃×C₄×C₅×C₂²⋊C₄	2
C₃×C₅×#376	2
C₅×#2401	3
C₃×#2401	5

Subgroups — C3×C5×#2401 #14275435

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2401 #14275435