Subgroups — C₃×#10599787 #14297557

Order 1920.

672 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#4039	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#3955	C₂	1	2	yes	maximal
3	C₃×#65407	C₂	1	2	yes	maximal, complemented
4	C₃×C₅×#343	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#64750	C₂	1	1	yes	maximal, characteristic, complemented
6	#10599787	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₃×D₂₀	C₂²	1	1	yes	characteristic
8	C₂²×C₃×C₄×D₅	C₄	1	2	yes	—
9	C₂×C₃×C₅×M₁₆	C₂²	1	2	yes	—
10	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
11	C₂×C₃×#685	C₄	1	2	yes	—
12	C₂×C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
13	C₂×C₃×#681	C₂²	1	2	yes	—
14	C₃×#4039	—	2	2	no	—
15	C₃×#4039	—	2	2	no	—
16	C₃×#3955	—	2	4	no	—
17	C₃×#3946	—	2	2	no	—
18	C₃×#3946	—	2	2	no	—
19	C₃×#1800	—	5	1	no	maximal, Hall
20	C₂×#4039	C₂×C₃	1	1	yes	characteristic
21	C₂×#3955	C₂×C₃	1	2	yes	—
22	#65407	C₂×C₃	1	2	yes	complemented
23	C₅×#343	C₂×C₃	1	1	yes	characteristic, complemented
24	#64750	C₂×C₃	1	1	yes	characteristic, complemented
25	C₂³×C₃×D₅	C₂×C₄	1	2	yes	—
26	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
27	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
28	C₂²×C₃×Dic₅	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₃×Dic₅	C₂×C₄	1	1	yes	characteristic
30	C₂×C₃×D₂₀	—	2	2	no	—
31	C₂×C₃×D₂₀	—	2	2	no	—
32	C₂×C₃×D₂₀	D₄	1	2	yes	—
33	C₂×C₃×D₂₀	Q₈	1	2	yes	—
34	C₂×C₃×C₄×D₅	—	2	2	no	—
35	C₂×C₃×C₄×D₅	—	2	2	no	—
36	C₂×C₃×C₄×D₅	—	4	4	no	—
37	C₃×C₅×M₁₆	—	2	2	no	—
38	C₃×C₅×M₁₆	—	2	2	no	—
39	C₂×C₃×C₅×C₈	—	2	2	no	—
40	C₃×C₅×C₄⋊C₄	—	2	2	no	—
41	C₃×#685	—	2	4	no	—
42	C₃×C₄×Dic₅	D₄	1	2	yes	—
43	C₃×C₄×Dic₅	D₄	1	2	yes	—
44	C₃×#681	—	2	2	no	—
45	C₃×#681	—	2	2	no	—
46	C₂×C₃×#168	—	2	2	no	—
47	C₂×C₃×C₄×D₄	—	5	1	no	—
48	C₃×#423	—	5	2	no	—
49	C₂×C₃×#100	—	5	2	no	—
50	C₃×#343	—	5	1	no	—
51	C₃×#343	D₅	1	1	yes	characteristic, complemented
52	C₂²×D₂₀	C₂²×C₃	1	1	yes	characteristic
53	C₂²×C₄×D₅	C₃×C₄	1	2	yes	—
54	C₂×C₅×M₁₆	C₂²×C₃	1	2	yes	—
55	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
56	C₂×#685	C₃×C₄	1	2	yes	—
57	C₂×C₄×Dic₅	C₂²×C₃	1	1	yes	characteristic
58	C₂×#681	C₂²×C₃	1	2	yes	—
59	#4039	—	2	2	no	—
60	#4039	—	2	2	no	—
61	#3955	—	2	4	no	—
62	#3946	—	2	2	no	—
63	#3946	—	2	2	no	—
64	#1800	—	5	1	no	Sylow
65	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₃×D₅	—	2	2	no	—
67	C₂²×C₃×D₅	—	2	2	no	—
68	C₂²×C₃×D₅	—	4	2	no	—
69	C₂²×C₃×D₅	—	4	2	no	—
70	C₂²×C₃×D₅	C₄⋊C₄	1	4	yes	—
71	C₂×C₃×C₄×C₅	—	2	2	no	—
72	C₂×C₃×C₄×C₅	—	4	2	no	—
73	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
74	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
75	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
76	C₂×C₃×C₄×C₅	#41	1	2	yes	—
77	C₂×C₃×Dic₅	—	2	2	no	—
78	C₂×C₃×Dic₅	—	4	1	no	—
79	C₂×C₃×Dic₅	C₂²⋊C₄	1	2	yes	—
80	C₂×C₃×Dic₅	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₃×Dic₅	C₂²⋊C₄	1	1	yes	characteristic
82	C₃×D₂₀	—	2	4	no	—
83	C₃×D₂₀	—	2	4	no	—
84	C₃×C₄×D₅	—	4	4	no	—
85	C₃×C₄×D₅	—	4	4	no	—
86	C₃×C₅×C₈	—	4	2	no	—
87	C₃×#168	—	4	2	no	—
88	C₂²×C₃×D₄	—	5	1	no	—
89	C₂³×C₃×C₄	—	5	2	no	—
90	C₂×C₃×M₁₆	—	5	2	no	—
91	C₂×C₃×M₁₆	C₂×D₅	1	2	yes	—
92	C₃×C₄×D₄	—	10	2	no	—
93	C₃×C₄×D₄	—	10	2	no	—
94	C₂×C₃×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
95	C₂×C₃×C₂²⋊C₄	—	5	2	no	—
96	C₂×C₃×C₄²	—	5	1	no	—
97	C₃×C₄⋊C₈	—	10	2	no	—
98	C₃×C₄⋊C₈	—	10	2	no	—
99	C₃×#100	—	10	4	no	—
100	C₂³×D₅	C₂×C₃×C₄	1	2	yes	—
101	C₂²×C₄×C₅	C₂×C₃×C₄	1	2	yes	—
102	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
103	C₂²×Dic₅	C₂×C₃×C₄	1	1	yes	characteristic
104	C₂²×Dic₅	C₂×C₃×C₄	1	1	yes	characteristic
105	C₂×D₂₀	—	2	2	no	—
106	C₂×D₂₀	—	2	2	no	—
107	C₂×D₂₀	C₃×D₄	1	2	yes	—
108	C₂×D₂₀	C₃×Q₈	1	2	yes	—
109	C₂×C₄×D₅	—	2	2	no	—
110	C₂×C₄×D₅	—	2	2	no	—
111	C₂×C₄×D₅	—	4	4	no	—
112	C₅×M₁₆	—	2	2	no	—
113	C₅×M₁₆	—	2	2	no	—
114	C₂×C₅×C₈	—	2	2	no	—
115	C₅×C₄⋊C₄	—	2	2	no	—
116	#685	—	2	4	no	—
117	C₄×Dic₅	C₃×D₄	1	2	yes	—
118	C₄×Dic₅	C₃×D₄	1	2	yes	—
119	#681	—	2	2	no	—
120	#681	—	2	2	no	—
121	C₂×#168	—	2	2	no	—
122	C₂×C₄×D₄	—	5	1	no	—
123	#423	—	5	2	no	—
124	C₂×#100	—	5	2	no	—
125	#343	—	5	1	no	—
126	#343	C₃×D₅	1	1	yes	characteristic, complemented
127	C₂²×C₃×C₅	—	2	1	no	—
128	C₂²×C₃×C₅	—	2	1	no	—
129	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
130	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
131	C₂²×C₃×C₅	#117	1	1	yes	characteristic
132	C₂×C₃×D₅	—	2	4	no	—
133	C₂×C₃×D₅	—	4	4	no	—
134	C₂×C₃×D₅	—	4	2	no	—
135	C₂×C₃×D₅	—	4	2	no	—
136	C₂×C₃×D₅	—	4	4	no	—
137	C₃×C₄×C₅	—	4	2	no	—
138	C₃×C₄×C₅	#121	1	2	yes	—
139	C₃×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
140	C₃×Dic₅	—	2	1	no	—
141	C₃×Dic₅	—	2	2	no	—
142	C₃×Dic₅	#100	1	2	yes	complemented
143	C₂⁴×C₃	—	5	2	no	—
144	C₂×C₃×D₄	—	5	2	no	—
145	C₂×C₃×D₄	—	5	2	no	—
146	C₂×C₃×D₄	—	10	2	no	—
147	C₂×C₃×D₄	—	10	2	no	—
148	C₂²×C₃×C₄	—	5	1	no	—
149	C₂²×C₃×C₄	—	5	1	no	—
150	C₂²×C₃×C₄	—	10	2	no	—
151	C₂²×C₃×C₄	—	10	2	no	—
152	C₂²×C₃×C₄	—	20	4	no	—
153	C₂²×C₃×C₄	C₄×D₅	1	2	yes	—
154	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
155	C₃×M₁₆	—	2	2	no	—
156	C₃×M₁₆	—	2	2	no	—
157	C₃×M₁₆	—	10	2	no	—
158	C₃×M₁₆	—	10	2	no	—
159	C₂×C₃×C₈	—	2	2	no	—
160	C₂×C₃×C₈	—	10	2	no	—
161	C₃×C₄⋊C₄	—	2	2	no	—
162	C₃×C₂²⋊C₄	—	10	4	no	—
163	C₃×C₄²	—	5	2	no	—
164	C₃×C₄²	—	5	2	no	—
165	C₂³×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
166	C₂²×D₅	—	2	2	no	—
167	C₂²×D₅	—	2	2	no	—
168	C₂²×D₅	—	4	2	no	—
169	C₂²×D₅	—	4	2	no	—
170	C₂²×D₅	C₃×C₄⋊C₄	1	4	yes	—
171	C₂×C₄×C₅	—	2	2	no	—
172	C₂×C₄×C₅	—	4	2	no	—
173	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
174	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
175	C₂×C₄×C₅	C₂×C₃×Q₈	1	1	yes	characteristic
176	C₂×C₄×C₅	C₃×#41	1	2	yes	—
177	C₂×Dic₅	—	2	2	no	—
178	C₂×Dic₅	—	4	1	no	—
179	C₂×Dic₅	C₃×C₂²⋊C₄	1	2	yes	—
180	C₂×Dic₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
181	C₂×Dic₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
182	D₂₀	—	2	4	no	—
183	D₂₀	—	2	4	no	—
184	C₄×D₅	—	4	4	no	—
185	C₄×D₅	—	4	4	no	—
186	C₅×C₈	—	4	2	no	—
187	#168	—	4	2	no	—
188	C₂²×D₄	—	5	1	no	—
189	C₂³×C₄	—	5	2	no	—
190	C₂×M₁₆	—	5	2	no	—
191	C₂×M₁₆	C₂×C₃×D₅	1	2	yes	—
192	C₄×D₄	—	10	2	no	—
193	C₄×D₄	—	10	2	no	—
194	C₂×C₄⋊C₄	C₂×C₃×D₅	1	1	yes	characteristic
195	C₂×C₂²⋊C₄	—	5	2	no	—
196	C₂×C₄²	—	5	1	no	—
197	C₄⋊C₈	—	10	2	no	—
198	C₄⋊C₈	—	10	2	no	—
199	#100	—	10	4	no	—
200	C₂×C₃×C₅	—	2	1	no	—
201	C₂×C₃×C₅	—	2	1	no	—
202	C₂×C₃×C₅	#380	1	1	yes	characteristic
203	C₂×C₃×C₅	C₂×#100	1	1	yes	characteristic
204	C₂×C₃×C₅	#413	1	1	yes	characteristic
205	C₃×D₅	—	4	4	no	—
206	C₂³×C₃	—	5	4	no	—
207	C₂³×C₃	—	10	2	no	—
208	C₂³×C₃	—	10	2	no	—
209	C₂³×C₃	—	20	2	no	—
210	C₂³×C₃	—	20	2	no	—
211	C₂³×C₃	C₂×C₄×D₅	1	1	yes	characteristic
212	C₃×D₄	—	10	4	no	—
213	C₃×D₄	—	10	4	no	—
214	C₂×C₃×C₄	—	2	2	no	—
215	C₂×C₃×C₄	—	4	2	no	—
216	C₂×C₃×C₄	—	5	1	no	—
217	C₂×C₃×C₄	—	5	2	no	—
218	C₂×C₃×C₄	—	5	1	no	—
219	C₂×C₃×C₄	—	10	2	no	—
220	C₂×C₃×C₄	—	20	4	no	—
221	C₂×C₃×C₄	—	20	4	no	—
222	C₂×C₃×C₄	—	20	1	no	—
223	C₂×C₃×C₄	#709	1	2	yes	—
224	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
225	C₂×C₃×C₄	D₄×D₅	1	2	yes	—
226	C₂×C₃×C₄	Q₈×D₅	1	1	yes	characteristic
227	C₃×C₈	—	4	2	no	—
228	C₃×C₈	—	20	2	no	—
229	C₂²×C₅	—	2	1	no	—
230	C₂²×C₅	—	2	1	no	—
231	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
232	C₂²×C₅	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
233	C₂²×C₅	C₃×#117	1	1	yes	characteristic
234	C₂×D₅	—	2	4	no	—
235	C₂×D₅	—	4	4	no	—
236	C₂×D₅	—	4	2	no	—
237	C₂×D₅	—	4	2	no	—
238	C₂×D₅	—	4	4	no	—
239	C₄×C₅	—	4	2	no	—
240	C₄×C₅	C₃×#121	1	2	yes	—
241	C₄×C₅	C₃×C₂²⋊Q₈	1	2	yes	—
242	Dic₅	—	2	1	no	—
243	Dic₅	—	2	2	no	—
244	Dic₅	C₃×#100	1	2	yes	complemented
245	C₂⁴	—	5	2	no	—
246	C₂×D₄	—	5	2	no	—
247	C₂×D₄	—	5	2	no	—
248	C₂×D₄	—	10	2	no	—
249	C₂×D₄	—	10	2	no	—
250	C₂²×C₄	—	5	1	no	—
251	C₂²×C₄	—	5	1	no	—
252	C₂²×C₄	—	10	2	no	—
253	C₂²×C₄	—	10	2	no	—
254	C₂²×C₄	—	20	4	no	—
255	C₂²×C₄	C₃×C₄×D₅	1	2	yes	—
256	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
257	M₁₆	—	2	2	no	—
258	M₁₆	—	2	2	no	—
259	M₁₆	—	10	2	no	—
260	M₁₆	—	10	2	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	10	2	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₂²⋊C₄	—	10	4	no	—
265	C₄²	—	5	2	no	—
266	C₄²	—	5	2	no	—
267	C₃×C₅	#1800	1	1	yes	characteristic, complemented, Hall
268	C₂²×C₃	—	2	1	no	—
269	C₂²×C₃	—	2	1	no	—
270	C₂²×C₃	—	10	4	no	—
271	C₂²×C₃	—	20	4	no	—
272	C₂²×C₃	—	20	4	no	—
273	C₂²×C₃	—	20	2	no	—
274	C₂²×C₃	—	20	2	no	—
275	C₂²×C₃	#4018	1	1	yes	characteristic
276	C₂²×C₃	D₅×C₂²⋊C₄	1	1	yes	characteristic
277	C₂²×C₃	D₅×C₄⋊C₄	1	1	yes	characteristic
278	C₃×C₄	—	4	2	no	—
279	C₃×C₄	—	5	2	no	—
280	C₃×C₄	—	10	1	no	—
281	C₃×C₄	—	10	2	no	—
282	C₃×C₄	#4030	1	2	yes	—
283	C₃×C₄	#4042	1	2	yes	—
284	C₂×C₅	—	2	1	no	—
285	C₂×C₅	—	2	1	no	—
286	C₂×C₅	C₃×#380	1	1	yes	characteristic
287	C₂×C₅	C₂×C₃×#100	1	1	yes	characteristic
288	C₂×C₅	C₃×#413	1	1	yes	characteristic
289	D₅	—	4	4	no	—
290	C₂³	—	5	4	no	—
291	C₂³	—	10	2	no	—
292	C₂³	—	10	2	no	—
293	C₂³	—	20	2	no	—
294	C₂³	—	20	2	no	—
295	C₂³	C₂×C₃×C₄×D₅	1	1	yes	characteristic
296	D₄	—	10	4	no	—
297	D₄	—	10	4	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	5	1	no	—
301	C₂×C₄	—	5	2	no	—
302	C₂×C₄	—	5	1	no	—
303	C₂×C₄	—	10	2	no	—
304	C₂×C₄	—	20	4	no	—
305	C₂×C₄	—	20	4	no	—
306	C₂×C₄	—	20	1	no	—
307	C₂×C₄	C₃×#709	1	2	yes	—
308	C₂×C₄	C₃×D₄×D₅	1	1	yes	characteristic
309	C₂×C₄	C₃×D₄×D₅	1	2	yes	—
310	C₂×C₄	C₃×Q₈×D₅	1	1	yes	characteristic
311	C₈	—	4	2	no	—
312	C₈	—	20	2	no	—
313	C₂×C₃	—	2	1	no	—
314	C₂×C₃	—	2	1	no	—
315	C₂×C₃	—	20	4	no	—
316	C₂×C₃	#64956	1	1	yes	characteristic
317	C₂×C₃	D₅×#100	1	1	yes	characteristic
318	C₂×C₃	#65040	1	1	yes	characteristic
319	C₅	C₃×#1800	1	1	yes	characteristic, complemented, Sylow
320	C₂²	—	2	1	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	10	4	no	—
323	C₂²	—	20	4	no	—
324	C₂²	—	20	4	no	—
325	C₂²	—	20	2	no	—
326	C₂²	—	20	2	no	—
327	C₂²	C₃×#4018	1	1	yes	characteristic
328	C₂²	C₃×D₅×C₂²⋊C₄	1	1	yes	characteristic
329	C₂²	C₃×D₅×C₄⋊C₄	1	1	yes	characteristic
330	C₄	—	4	2	no	—
331	C₄	—	5	2	no	—
332	C₄	—	10	1	no	—
333	C₄	—	10	2	no	—
334	C₄	C₃×#4030	1	2	yes	—
335	C₄	C₃×#4042	1	2	yes	—
336	C₃	#10599787	1	1	yes	characteristic, direct product factor, Sylow
337	C₂	—	2	1	no	—
338	C₂	—	2	1	no	—
339	C₂	—	20	4	no	—
340	C₂	C₃×#64956	1	1	yes	characteristic
341	C₂	C₃×D₅×#100	1	1	yes	characteristic
342	C₂	C₃×#65040	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	D₅×C₂²⋊C₄
commutator	C₂³×C₅	C₂²×C₃×C₄
Frattini	C₂²×C₄	C₂²×C₃×D₅
Fitting	C₃×C₅×#343	C₂
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₂²×C₃×D₂₀	C₂²
cosocle	C₂²×C₄×C₅	C₂³×C₃
hypercenter	C₃×#343	D₅
hypocenter	C₅	C₃×#1800
LCS	G ▷ C₂³×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×C₄ ◁ C₃×#343

Complements

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

N	H ≅ G/N	type
3. C₃×#65407	C₂	semidirect
4. C₃×C₅×#343	C₂	semidirect
5. C₃×#64750	C₂	semidirect
6. #10599787	C₃	direct (N × H)
22. #65407	C₂×C₃	semidirect
23. C₅×#343	C₂×C₃	semidirect
24. #64750	C₂×C₃	semidirect
51. C₃×#343	D₅	semidirect
126. #343	C₃×D₅	semidirect
142. C₃×Dic₅	#100	semidirect
244. Dic₅	C₃×#100	semidirect
267. C₃×C₅	#1800	semidirect
319. C₅	C₃×#1800	semidirect
336. C₃	#10599787	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×#4039	2
C₂×C₃×#3955	2
C₃×#65407	2
C₃×C₅×#343	2
C₃×#64750	2
#10599787	3
C₃×#1800	5

Subgroups — C3×#10599787 #14297557

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10599787 #14297557