Subgroups — C₃×C₅×#1800 #14274834

Order 1920.

672 conjugacy classes of subgroups.

The subgroup lattice has 320 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×D₄	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#423	C₂	1	2	yes	maximal, complemented
3	C₂×C₃×C₅×#100	C₂	1	2	yes	maximal
4	C₃×C₅×#343	C₂	1	2	yes	maximal, complemented
5	C₅×#1800	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂²×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
7	C₂³×C₃×C₄×C₅	C₄	1	2	yes	—
8	C₂×C₃×C₅×M₁₆	C₂²	1	4	yes	—
9	C₃×C₄×C₅×D₄	—	2	2	no	—
10	C₃×C₄×C₅×D₄	—	2	2	no	—
11	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₅×C₂²⋊C₄	C₄	1	2	yes	—
13	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
14	C₃×C₅×C₄⋊C₈	—	2	2	no	—
15	C₃×C₅×C₄⋊C₈	—	2	2	no	—
16	C₃×C₅×#100	—	2	4	no	—
17	C₃×#1800	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
18	C₂×C₄×C₅×D₄	C₂×C₃	1	1	yes	characteristic
19	C₅×#423	C₂×C₃	1	2	yes	complemented
20	C₂×C₅×#100	C₂×C₃	1	2	yes	—
21	C₅×#343	C₂×C₃	1	2	yes	complemented
22	C₂⁴×C₃×C₅	C₂×C₄	1	2	yes	—
23	C₂×C₃×C₅×D₄	—	2	2	no	—
24	C₂×C₃×C₅×D₄	—	2	2	no	—
25	C₂×C₃×C₅×D₄	D₄	1	2	yes	—
26	C₂×C₃×C₅×D₄	Q₈	1	2	yes	—
27	C₂²×C₃×C₄×C₅	—	2	2	no	—
28	C₂²×C₃×C₄×C₅	—	2	2	no	—
29	C₂²×C₃×C₄×C₅	—	4	4	no	—
30	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
32	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
34	C₃×C₅×M₁₆	—	2	4	no	—
35	C₃×C₅×M₁₆	—	2	4	no	—
36	C₂×C₃×C₅×C₈	—	2	4	no	—
37	C₃×C₅×C₄⋊C₄	—	2	2	no	—
38	C₃×C₅×C₂²⋊C₄	—	2	4	no	—
39	C₃×C₄²×C₅	D₄	1	2	yes	—
40	C₃×C₄²×C₅	D₄	1	2	yes	—
41	C₂×C₃×C₄×D₄	C₂×C₅	1	1	yes	characteristic
42	C₃×#423	C₂×C₅	1	2	yes	complemented
43	C₂×C₃×#100	C₂×C₅	1	2	yes	—
44	C₃×#343	C₂×C₅	1	2	yes	complemented
45	C₂²×C₅×D₄	C₂²×C₃	1	1	yes	characteristic
46	C₂³×C₄×C₅	C₃×C₄	1	2	yes	—
47	C₂×C₅×M₁₆	C₂²×C₃	1	4	yes	—
48	C₄×C₅×D₄	—	2	2	no	—
49	C₄×C₅×D₄	—	2	2	no	—
50	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
51	C₂×C₅×C₂²⋊C₄	C₃×C₄	1	2	yes	—
52	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
53	C₅×C₄⋊C₈	—	2	2	no	—
54	C₅×C₄⋊C₈	—	2	2	no	—
55	C₅×#100	—	2	4	no	—
56	#1800	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
57	C₂³×C₃×C₅	—	2	2	no	—
58	C₂³×C₃×C₅	—	2	2	no	—
59	C₂³×C₃×C₅	—	4	2	no	—
60	C₂³×C₃×C₅	—	4	2	no	—
61	C₂³×C₃×C₅	C₄⋊C₄	1	4	yes	—
62	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
63	C₃×C₅×D₄	—	2	4	no	—
64	C₃×C₅×D₄	—	2	4	no	—
65	C₂×C₃×C₄×C₅	—	2	2	no	—
66	C₂×C₃×C₄×C₅	—	2	2	no	—
67	C₂×C₃×C₄×C₅	—	4	1	no	—
68	C₂×C₃×C₄×C₅	—	4	4	no	—
69	C₂×C₃×C₄×C₅	—	4	4	no	—
70	C₂×C₃×C₄×C₅	—	4	2	no	—
71	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
72	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
75	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
76	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
77	C₂×C₃×C₄×C₅	#41	1	2	yes	—
78	C₃×C₅×C₈	—	4	4	no	—
79	C₂²×C₃×D₄	C₂²×C₅	1	1	yes	characteristic
80	C₂³×C₃×C₄	C₄×C₅	1	2	yes	—
81	C₂×C₃×M₁₆	C₂²×C₅	1	4	yes	—
82	C₃×C₄×D₄	—	2	2	no	—
83	C₃×C₄×D₄	—	2	2	no	—
84	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
85	C₂×C₃×C₂²⋊C₄	C₄×C₅	1	2	yes	—
86	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
87	C₃×C₄⋊C₈	—	2	2	no	—
88	C₃×C₄⋊C₈	—	2	2	no	—
89	C₃×#100	—	2	4	no	—
90	C₂⁴×C₅	C₂×C₃×C₄	1	2	yes	—
91	C₂×C₅×D₄	—	2	2	no	—
92	C₂×C₅×D₄	—	2	2	no	—
93	C₂×C₅×D₄	C₃×D₄	1	2	yes	—
94	C₂×C₅×D₄	C₃×Q₈	1	2	yes	—
95	C₂²×C₄×C₅	—	2	2	no	—
96	C₂²×C₄×C₅	—	2	2	no	—
97	C₂²×C₄×C₅	—	4	4	no	—
98	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
99	C₂²×C₄×C₅	C₂×C₃×C₄	1	2	yes	—
100	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
101	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
102	C₅×M₁₆	—	2	4	no	—
103	C₅×M₁₆	—	2	4	no	—
104	C₂×C₅×C₈	—	2	4	no	—
105	C₅×C₄⋊C₄	—	2	2	no	—
106	C₅×C₂²⋊C₄	—	2	4	no	—
107	C₄²×C₅	C₃×D₄	1	2	yes	—
108	C₄²×C₅	C₃×D₄	1	2	yes	—
109	C₂×C₄×D₄	C₂×C₃×C₅	1	1	yes	characteristic
110	#423	C₂×C₃×C₅	1	2	yes	complemented
111	C₂×#100	C₂×C₃×C₅	1	2	yes	—
112	#343	C₂×C₃×C₅	1	2	yes	complemented
113	C₂²×C₃×C₅	—	2	1	no	—
114	C₂²×C₃×C₅	—	2	1	no	—
115	C₂²×C₃×C₅	—	2	4	no	—
116	C₂²×C₃×C₅	—	4	4	no	—
117	C₂²×C₃×C₅	—	4	4	no	—
118	C₂²×C₃×C₅	—	4	2	no	—
119	C₂²×C₃×C₅	—	4	2	no	—
120	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂²×C₃×C₅	#117	1	1	yes	characteristic
123	C₃×C₄×C₅	—	2	1	no	—
124	C₃×C₄×C₅	—	2	2	no	—
125	C₃×C₄×C₅	—	4	2	no	—
126	C₃×C₄×C₅	#100	1	2	yes	complemented
127	C₃×C₄×C₅	#121	1	2	yes	—
128	C₃×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
129	C₂⁴×C₃	C₂×C₄×C₅	1	2	yes	—
130	C₂×C₃×D₄	—	2	2	no	—
131	C₂×C₃×D₄	—	2	2	no	—
132	C₂×C₃×D₄	C₅×D₄	1	2	yes	—
133	C₂×C₃×D₄	C₅×Q₈	1	2	yes	—
134	C₂²×C₃×C₄	—	2	2	no	—
135	C₂²×C₃×C₄	—	2	2	no	—
136	C₂²×C₃×C₄	—	4	4	no	—
137	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
138	C₂²×C₃×C₄	C₂×C₄×C₅	1	2	yes	—
139	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
140	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
141	C₃×M₁₆	—	2	4	no	—
142	C₃×M₁₆	—	2	4	no	—
143	C₂×C₃×C₈	—	2	4	no	—
144	C₃×C₄⋊C₄	—	2	2	no	—
145	C₃×C₂²⋊C₄	—	2	4	no	—
146	C₃×C₄²	C₅×D₄	1	2	yes	—
147	C₃×C₄²	C₅×D₄	1	2	yes	—
148	C₂³×C₅	—	2	2	no	—
149	C₂³×C₅	—	2	2	no	—
150	C₂³×C₅	—	4	2	no	—
151	C₂³×C₅	—	4	2	no	—
152	C₂³×C₅	C₃×C₄⋊C₄	1	4	yes	—
153	C₂³×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
154	C₅×D₄	—	2	4	no	—
155	C₅×D₄	—	2	4	no	—
156	C₂×C₄×C₅	—	2	2	no	—
157	C₂×C₄×C₅	—	2	2	no	—
158	C₂×C₄×C₅	—	4	1	no	—
159	C₂×C₄×C₅	—	4	4	no	—
160	C₂×C₄×C₅	—	4	4	no	—
161	C₂×C₄×C₅	—	4	2	no	—
162	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	2	yes	—
163	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
164	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
165	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
166	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
167	C₂×C₄×C₅	C₂×C₃×Q₈	1	1	yes	characteristic
168	C₂×C₄×C₅	C₃×#41	1	2	yes	—
169	C₅×C₈	—	4	4	no	—
170	C₂²×D₄	C₂²×C₃×C₅	1	1	yes	characteristic
171	C₂³×C₄	C₃×C₄×C₅	1	2	yes	—
172	C₂×M₁₆	C₂²×C₃×C₅	1	4	yes	—
173	C₄×D₄	—	2	2	no	—
174	C₄×D₄	—	2	2	no	—
175	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	1	yes	characteristic
176	C₂×C₂²⋊C₄	C₃×C₄×C₅	1	2	yes	—
177	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
178	C₄⋊C₈	—	2	2	no	—
179	C₄⋊C₈	—	2	2	no	—
180	#100	—	2	4	no	—
181	C₂×C₃×C₅	—	2	1	no	—
182	C₂×C₃×C₅	—	2	1	no	—
183	C₂×C₃×C₅	—	4	4	no	—
184	C₂×C₃×C₅	#380	1	1	yes	characteristic
185	C₂×C₃×C₅	C₂×#100	1	1	yes	characteristic
186	C₂×C₃×C₅	#413	1	1	yes	characteristic
187	C₂³×C₃	—	2	2	no	—
188	C₂³×C₃	—	2	2	no	—
189	C₂³×C₃	—	4	2	no	—
190	C₂³×C₃	—	4	2	no	—
191	C₂³×C₃	C₅×C₄⋊C₄	1	4	yes	—
192	C₂³×C₃	C₂²×C₄×C₅	1	1	yes	characteristic
193	C₃×D₄	—	2	4	no	—
194	C₃×D₄	—	2	4	no	—
195	C₂×C₃×C₄	—	2	2	no	—
196	C₂×C₃×C₄	—	2	2	no	—
197	C₂×C₃×C₄	—	4	1	no	—
198	C₂×C₃×C₄	—	4	4	no	—
199	C₂×C₃×C₄	—	4	4	no	—
200	C₂×C₃×C₄	—	4	2	no	—
201	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	yes	—
202	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
203	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
204	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
205	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
206	C₂×C₃×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
207	C₂×C₃×C₄	C₅×#41	1	2	yes	—
208	C₃×C₈	—	4	4	no	—
209	C₂²×C₅	—	2	1	no	—
210	C₂²×C₅	—	2	1	no	—
211	C₂²×C₅	—	2	4	no	—
212	C₂²×C₅	—	4	4	no	—
213	C₂²×C₅	—	4	4	no	—
214	C₂²×C₅	—	4	2	no	—
215	C₂²×C₅	—	4	2	no	—
216	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
217	C₂²×C₅	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
218	C₂²×C₅	C₃×#117	1	1	yes	characteristic
219	C₄×C₅	—	2	1	no	—
220	C₄×C₅	—	2	2	no	—
221	C₄×C₅	—	4	2	no	—
222	C₄×C₅	C₃×#100	1	2	yes	complemented
223	C₄×C₅	C₃×#121	1	2	yes	—
224	C₄×C₅	C₃×C₂²⋊Q₈	1	2	yes	—
225	C₂⁴	C₂×C₃×C₄×C₅	1	2	yes	—
226	C₂×D₄	—	2	2	no	—
227	C₂×D₄	—	2	2	no	—
228	C₂×D₄	C₃×C₅×D₄	1	2	yes	—
229	C₂×D₄	C₃×C₅×Q₈	1	2	yes	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	4	4	no	—
233	C₂²×C₄	C₂×C₃×C₄×C₅	1	1	yes	characteristic
234	C₂²×C₄	C₂×C₃×C₄×C₅	1	2	yes	—
235	C₂²×C₄	C₂×C₃×C₄×C₅	1	1	yes	characteristic
236	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
237	M₁₆	—	2	4	no	—
238	M₁₆	—	2	4	no	—
239	C₂×C₈	—	2	4	no	—
240	C₄⋊C₄	—	2	2	no	—
241	C₂²⋊C₄	—	2	4	no	—
242	C₄²	C₃×C₅×D₄	1	2	yes	—
243	C₄²	C₃×C₅×D₄	1	2	yes	—
244	C₃×C₅	#1800	1	1	yes	characteristic, direct product factor, Hall
245	C₂²×C₃	—	2	1	no	—
246	C₂²×C₃	—	2	1	no	—
247	C₂²×C₃	—	2	4	no	—
248	C₂²×C₃	—	4	4	no	—
249	C₂²×C₃	—	4	4	no	—
250	C₂²×C₃	—	4	2	no	—
251	C₂²×C₃	—	4	2	no	—
252	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
253	C₂²×C₃	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
254	C₂²×C₃	C₅×#117	1	1	yes	characteristic
255	C₃×C₄	—	2	1	no	—
256	C₃×C₄	—	2	2	no	—
257	C₃×C₄	—	4	2	no	—
258	C₃×C₄	C₅×#100	1	2	yes	complemented
259	C₃×C₄	C₅×#121	1	2	yes	—
260	C₃×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
261	C₂×C₅	—	2	1	no	—
262	C₂×C₅	—	2	1	no	—
263	C₂×C₅	—	4	4	no	—
264	C₂×C₅	C₃×#380	1	1	yes	characteristic
265	C₂×C₅	C₂×C₃×#100	1	1	yes	characteristic
266	C₂×C₅	C₃×#413	1	1	yes	characteristic
267	C₂³	—	2	2	no	—
268	C₂³	—	2	2	no	—
269	C₂³	—	4	2	no	—
270	C₂³	—	4	2	no	—
271	C₂³	C₃×C₅×C₄⋊C₄	1	4	yes	—
272	C₂³	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
273	D₄	—	2	4	no	—
274	D₄	—	2	4	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	4	1	no	—
278	C₂×C₄	—	4	4	no	—
279	C₂×C₄	—	4	4	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	2	yes	—
282	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
283	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
284	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
285	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
286	C₂×C₄	C₂×C₃×C₅×Q₈	1	1	yes	characteristic
287	C₂×C₄	C₃×C₅×#41	1	2	yes	—
288	C₈	—	4	4	no	—
289	C₂×C₃	—	2	1	no	—
290	C₂×C₃	—	2	1	no	—
291	C₂×C₃	—	4	4	no	—
292	C₂×C₃	C₅×#380	1	1	yes	characteristic
293	C₂×C₃	C₂×C₅×#100	1	1	yes	characteristic
294	C₂×C₃	C₅×#413	1	1	yes	characteristic
295	C₅	C₃×#1800	1	1	yes	characteristic, direct product factor, Sylow
296	C₂²	—	2	1	no	—
297	C₂²	—	2	1	no	—
298	C₂²	—	2	4	no	—
299	C₂²	—	4	4	no	—
300	C₂²	—	4	4	no	—
301	C₂²	—	4	2	no	—
302	C₂²	—	4	2	no	—
303	C₂²	C₂×C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
304	C₂²	C₂×C₃×C₅×C₄⋊C₄	1	1	yes	characteristic
305	C₂²	C₃×C₅×#117	1	1	yes	characteristic
306	C₄	—	2	1	no	—
307	C₄	—	2	2	no	—
308	C₄	—	4	2	no	—
309	C₄	C₃×C₅×#100	1	2	yes	complemented
310	C₄	C₃×C₅×#121	1	2	yes	—
311	C₄	C₃×C₅×C₂²⋊Q₈	1	2	yes	—
312	C₃	C₅×#1800	1	1	yes	characteristic, direct product factor, Sylow
313	C₂	—	2	1	no	—
314	C₂	—	2	1	no	—
315	C₂	—	4	4	no	—
316	C₂	C₃×C₅×#380	1	1	yes	characteristic
317	C₂	C₂×C₃×C₅×#100	1	1	yes	characteristic
318	C₂	C₃×C₅×#413	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
commutator	C₂³	C₂²×C₃×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂³×C₃×C₅
prime radical	C₂²×C₃×C₅×D₄	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₅ ◁ C₂×C₃×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×#423	C₂	semidirect
4. C₃×C₅×#343	C₂	semidirect
5. C₅×#1800	C₃	direct (N × H)
17. C₃×#1800	C₅	direct (N × H)
19. C₅×#423	C₂×C₃	semidirect
21. C₅×#343	C₂×C₃	semidirect
42. C₃×#423	C₂×C₅	semidirect
44. C₃×#343	C₂×C₅	semidirect
56. #1800	C₃×C₅	direct (N × H)
110. #423	C₂×C₃×C₅	semidirect
112. #343	C₂×C₃×C₅	semidirect
126. C₃×C₄×C₅	#100	semidirect
222. C₄×C₅	C₃×#100	semidirect
244. C₃×C₅	#1800	direct (N × H)
258. C₃×C₄	C₅×#100	semidirect
295. C₅	C₃×#1800	direct (N × H)
309. C₄	C₃×C₅×#100	semidirect
312. C₃	C₅×#1800	direct (N × H)

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×D₄	2
C₃×C₅×#423	2
C₂×C₃×C₅×#100	2
C₃×C₅×#343	2
C₅×#1800	3
C₃×#1800	5

Subgroups — C3×C5×#1800 #14274834

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1800 #14274834