Subgroups — C₃×#10610792 #14430691

Order 1920.

768 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×#709	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#4016	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅×#423	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#65422	C₂	1	2	yes	maximal
5	C₃×#65410	C₂	1	2	yes	maximal, complemented
6	C₃×#65136	C₂	1	4	yes	maximal, complemented
7	C₃×#65118	C₂	1	4	yes	maximal, complemented
8	#10610792	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂×C₃×#709	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₅×M₁₆	C₂²	1	2	yes	—
11	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
12	C₃×C₄×C₂²⋊D₅	—	2	2	no	—
13	C₃×#4072	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₄⋊Dic₅	C₂²	1	2	yes	—
15	C₂×C₃×#683	—	2	2	no	—
16	C₃×C₄×D₂₀	C₂²	1	2	yes	—
17	C₃×#4018	—	2	1	no	—
18	C₃×C₄²×D₅	—	2	1	no	—
19	C₃×#4016	—	2	1	no	—
20	C₃×#4016	—	2	1	no	—
21	C₃×#4016	C₂²	1	4	yes	—
22	C₃×C₄×Dic₁₀	C₂²	1	2	yes	—
23	C₃×C₅×C₄⋊C₈	C₂²	1	4	yes	—
24	C₃×#3953	C₂²	1	4	yes	—
25	C₃×#3950	C₂²	1	4	yes	—
26	C₃×#3949	C₂²	1	4	yes	—
27	C₃×#3948	C₂²	1	4	yes	—
28	C₃×#3080	—	5	1	no	maximal, Hall
29	C₄×#709	C₂×C₃	1	1	yes	characteristic
30	C₂×#4016	C₂×C₃	1	1	yes	characteristic, complemented
31	C₅×#423	C₂×C₃	1	1	yes	characteristic, complemented
32	#65422	C₂×C₃	1	2	yes	—
33	#65410	C₂×C₃	1	2	yes	complemented
34	#65136	C₂×C₃	1	4	yes	complemented
35	#65118	C₂×C₃	1	4	yes	complemented
36	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
37	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
38	C₂×C₃×C₂²⋊D₅	—	2	1	no	—
39	C₂²×C₃×Dic₅	—	2	2	no	—
40	C₃×#709	—	2	2	no	—
41	C₃×#709	Q₈	1	4	yes	—
42	C₂×C₃×D₂₀	C₂³	1	1	yes	characteristic
43	C₂×C₃×C₄×D₅	—	2	1	no	—
44	C₂×C₃×C₄×D₅	—	2	2	no	—
45	C₂×C₃×Dic₁₀	C₂³	1	1	yes	characteristic
46	C₃×C₅×M₁₆	—	2	4	no	—
47	C₂×C₃×C₅×C₈	C₂³	1	4	yes	—
48	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
49	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
50	C₃×C₄²×C₅	C₂³	1	2	yes	—
51	C₃×C₂²⋊Dic₅	—	2	1	no	—
52	C₃×#685	—	2	2	no	—
53	C₃×C₄⋊Dic₅	—	2	2	no	—
54	C₃×C₄⋊Dic₅	C₂³	1	4	yes	—
55	C₃×C₄⋊Dic₅	C₂³	1	2	yes	—
56	C₃×#683	—	2	2	no	—
57	C₃×#683	—	2	4	no	—
58	C₃×#683	—	2	4	no	—
59	C₃×C₄×Dic₅	—	2	2	no	—
60	C₃×C₄×Dic₅	—	2	1	no	—
61	C₂×C₃×#125	—	5	1	no	—
62	C₃×C₄×#41	—	5	1	no	—
63	C₃×#479	—	5	4	no	—
64	C₃×#478	—	5	4	no	—
65	C₃×#428	—	5	2	no	—
66	C₃×#423	D₅	1	1	yes	characteristic, complemented
67	C₃×#417	—	5	2	no	—
68	C₂×#709	C₂²×C₃	1	1	yes	characteristic
69	C₂×C₅×M₁₆	C₂²×C₃	1	2	yes	—
70	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
71	C₄×C₂²⋊D₅	—	2	2	no	—
72	#4072	C₂²×C₃	1	1	yes	characteristic
73	C₂×C₄⋊Dic₅	C₂²×C₃	1	2	yes	—
74	C₂×#683	—	2	2	no	—
75	C₄×D₂₀	C₂²×C₃	1	2	yes	—
76	#4018	—	2	1	no	—
77	C₄²×D₅	—	2	1	no	—
78	#4016	—	2	1	no	—
79	#4016	—	2	1	no	—
80	#4016	C₂²×C₃	1	4	yes	—
81	C₄×Dic₁₀	C₂²×C₃	1	2	yes	—
82	C₅×C₄⋊C₈	C₂²×C₃	1	4	yes	—
83	#3953	C₂²×C₃	1	4	yes	—
84	#3950	C₂²×C₃	1	4	yes	—
85	#3949	C₂²×C₃	1	4	yes	—
86	#3948	C₂²×C₃	1	4	yes	—
87	#3080	—	5	1	no	Sylow
88	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₃×D₅	—	2	1	no	—
90	C₂×C₃×C₄×C₅	—	4	2	no	—
91	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
92	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
93	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
94	C₂×C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
95	C₂×C₃×C₄×C₅	#41	1	2	yes	—
96	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
97	C₃×C₂²⋊D₅	—	2	4	no	—
98	C₂×C₃×Dic₅	—	2	1	no	—
99	C₂×C₃×Dic₅	—	2	2	no	—
100	C₂×C₃×Dic₅	—	2	4	no	—
101	C₂×C₃×Dic₅	—	4	4	no	—
102	C₃×D₂₀	—	2	1	no	—
103	C₃×D₂₀	C₂×Q₈	1	2	yes	—
104	C₃×C₄×D₅	—	2	4	no	—
105	C₃×C₄×D₅	—	4	4	no	—
106	C₃×Dic₁₀	—	2	1	no	—
107	C₃×Dic₁₀	C₂×Q₈	1	2	yes	—
108	C₃×C₅×C₈	—	2	4	no	—
109	C₂×C₃×#41	—	5	1	no	—
110	C₂×C₃×M₁₆	C₂×D₅	1	2	yes	—
111	C₃×#125	—	5	4	no	—
112	C₃×#125	—	10	1	no	—
113	C₃×#125	—	10	1	no	—
114	C₃×C₄×Q₈	—	5	2	no	—
115	C₃×C₄×D₄	—	5	2	no	—
116	C₃×C₄×D₄	—	10	2	no	—
117	C₃×#117	—	5	1	no	—
118	C₃×#117	—	10	1	no	—
119	C₂×C₃×C₄⋊C₄	—	5	2	no	—
120	C₂×C₃×C₄⋊C₄	—	10	2	no	—
121	C₂×C₃×C₄²	—	10	1	no	—
122	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
123	C₃×#107	—	5	4	no	—
124	C₃×#106	—	5	4	no	—
125	C₃×C₄⋊C₈	C₂×D₅	1	4	yes	—
126	C₃×#103	—	5	4	no	—
127	C₃×#102	—	5	4	no	—
128	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
129	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
130	C₂×C₂²⋊D₅	—	2	1	no	—
131	C₂²×Dic₅	—	2	2	no	—
132	#709	—	2	2	no	—
133	#709	C₃×Q₈	1	4	yes	—
134	C₂×D₂₀	C₂³×C₃	1	1	yes	characteristic
135	C₂×C₄×D₅	—	2	1	no	—
136	C₂×C₄×D₅	—	2	2	no	—
137	C₂×Dic₁₀	C₂³×C₃	1	1	yes	characteristic
138	C₅×M₁₆	—	2	4	no	—
139	C₂×C₅×C₈	C₂³×C₃	1	4	yes	—
140	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
141	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
142	C₄²×C₅	C₂³×C₃	1	2	yes	—
143	C₂²⋊Dic₅	—	2	1	no	—
144	#685	—	2	2	no	—
145	C₄⋊Dic₅	—	2	2	no	—
146	C₄⋊Dic₅	C₂³×C₃	1	4	yes	—
147	C₄⋊Dic₅	C₂³×C₃	1	2	yes	—
148	#683	—	2	2	no	—
149	#683	—	2	4	no	—
150	#683	—	2	4	no	—
151	C₄×Dic₅	—	2	2	no	—
152	C₄×Dic₅	—	2	1	no	—
153	C₂×#125	—	5	1	no	—
154	C₄×#41	—	5	1	no	—
155	#479	—	5	4	no	—
156	#478	—	5	4	no	—
157	#428	—	5	2	no	—
158	#423	C₃×D₅	1	1	yes	characteristic, complemented
159	#417	—	5	2	no	—
160	C₂²×C₃×C₅	—	2	2	no	—
161	C₂²×C₃×C₅	C₂²⋊Q₈	1	2	yes	—
162	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
163	C₂×C₃×D₅	—	2	2	no	—
164	C₂×C₃×D₅	—	4	2	no	—
165	C₃×C₄×C₅	—	2	2	no	—
166	C₃×C₄×C₅	—	2	2	no	—
167	C₃×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
168	C₃×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
169	C₃×C₄×C₅	C₂×#41	1	1	yes	characteristic
170	C₃×Dic₅	—	2	2	no	—
171	C₃×Dic₅	—	4	2	no	—
172	C₃×Dic₅	—	4	4	no	—
173	C₃×#41	—	5	4	no	—
174	C₃×#41	—	10	2	no	—
175	C₂×C₃×Q₈	—	5	1	no	—
176	C₂×C₃×D₄	—	5	1	no	—
177	C₂×C₃×D₄	—	10	1	no	—
178	C₂²×C₃×C₄	—	10	1	no	—
179	C₂²×C₃×C₄	—	10	2	no	—
180	C₂²×C₃×C₄	—	10	2	no	—
181	C₂²×C₃×C₄	D₂₀	1	2	yes	—
182	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
183	C₃×M₁₆	—	2	4	no	—
184	C₂×C₃×C₈	C₂²×D₅	1	4	yes	—
185	C₃×C₄⋊C₄	—	5	4	no	—
186	C₃×C₄⋊C₄	—	5	2	no	—
187	C₃×C₄⋊C₄	—	10	4	no	—
188	C₃×C₄⋊C₄	—	10	4	no	—
189	C₃×C₄⋊C₄	—	10	2	no	—
190	C₃×C₄⋊C₄	—	10	2	no	—
191	C₃×C₂²⋊C₄	—	10	1	no	—
192	C₃×C₂²⋊C₄	—	10	2	no	—
193	C₃×C₄²	—	10	1	no	—
194	C₃×C₄²	—	10	2	no	—
195	C₃×C₄²	D₂₀	1	1	yes	characteristic
196	C₃×C₄²	D₂₀	1	1	yes	characteristic
197	C₃×C₄²	C₂²×D₅	1	2	yes	—
198	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
199	C₂²×D₅	—	2	1	no	—
200	C₂×C₄×C₅	—	4	2	no	—
201	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
202	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
203	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
204	C₂×C₄×C₅	C₂×C₃×Q₈	1	2	yes	—
205	C₂×C₄×C₅	C₃×#41	1	2	yes	—
206	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
207	C₂²⋊D₅	—	2	4	no	—
208	C₂×Dic₅	—	2	1	no	—
209	C₂×Dic₅	—	2	2	no	—
210	C₂×Dic₅	—	2	4	no	—
211	C₂×Dic₅	—	4	4	no	—
212	D₂₀	—	2	1	no	—
213	D₂₀	C₂×C₃×Q₈	1	2	yes	—
214	C₄×D₅	—	2	4	no	—
215	C₄×D₅	—	4	4	no	—
216	Dic₁₀	—	2	1	no	—
217	Dic₁₀	C₂×C₃×Q₈	1	2	yes	—
218	C₅×C₈	—	2	4	no	—
219	C₂×#41	—	5	1	no	—
220	C₂×M₁₆	C₂×C₃×D₅	1	2	yes	—
221	#125	—	5	4	no	—
222	#125	—	10	1	no	—
223	#125	—	10	1	no	—
224	C₄×Q₈	—	5	2	no	—
225	C₄×D₄	—	5	2	no	—
226	C₄×D₄	—	10	2	no	—
227	#117	—	5	1	no	—
228	#117	—	10	1	no	—
229	C₂×C₄⋊C₄	—	5	2	no	—
230	C₂×C₄⋊C₄	—	10	2	no	—
231	C₂×C₄²	—	10	1	no	—
232	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
233	#107	—	5	4	no	—
234	#106	—	5	4	no	—
235	C₄⋊C₈	C₂×C₃×D₅	1	4	yes	—
236	#103	—	5	4	no	—
237	#102	—	5	4	no	—
238	C₂×C₃×C₅	—	2	2	no	—
239	C₂×C₃×C₅	C₂×C₂²⋊Q₈	1	1	yes	characteristic
240	C₂×C₃×C₅	#575	1	2	yes	—
241	C₃×D₅	—	4	2	no	—
242	C₂³×C₃	—	10	1	no	—
243	C₂³×C₃	C₂×D₂₀	1	1	yes	characteristic
244	C₃×Q₈	—	5	2	no	—
245	C₃×Q₈	—	10	1	no	—
246	C₃×D₄	—	5	2	no	—
247	C₃×D₄	—	10	4	no	—
248	C₃×D₄	—	10	1	no	—
249	C₂×C₃×C₄	—	4	2	no	—
250	C₂×C₃×C₄	—	10	1	no	—
251	C₂×C₃×C₄	—	10	4	no	—
252	C₂×C₃×C₄	—	10	2	no	—
253	C₂×C₃×C₄	—	10	4	no	—
254	C₂×C₃×C₄	—	20	4	no	—
255	C₂×C₃×C₄	—	20	4	no	—
256	C₂×C₃×C₄	C₂×D₂₀	1	2	yes	—
257	C₂×C₃×C₄	C₂×D₂₀	1	2	yes	—
258	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
259	C₂×C₃×C₄	#711	1	2	yes	—
260	C₂×C₃×C₄	Q₈×D₅	1	2	yes	—
261	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
262	C₃×C₈	—	2	4	no	—
263	C₂²×C₅	—	2	2	no	—
264	C₂²×C₅	C₃×C₂²⋊Q₈	1	2	yes	—
265	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
266	C₂×D₅	—	2	2	no	—
267	C₂×D₅	—	4	2	no	—
268	C₄×C₅	—	2	2	no	—
269	C₄×C₅	—	2	2	no	—
270	C₄×C₅	C₃×C₂²⋊Q₈	1	2	yes	—
271	C₄×C₅	C₂²×C₃×Q₈	1	1	yes	characteristic
272	C₄×C₅	C₂×C₃×#41	1	1	yes	characteristic
273	Dic₅	—	2	2	no	—
274	Dic₅	—	4	2	no	—
275	Dic₅	—	4	4	no	—
276	#41	—	5	4	no	—
277	#41	—	10	2	no	—
278	C₂×Q₈	—	5	1	no	—
279	C₂×D₄	—	5	1	no	—
280	C₂×D₄	—	10	1	no	—
281	C₂²×C₄	—	10	1	no	—
282	C₂²×C₄	—	10	2	no	—
283	C₂²×C₄	—	10	2	no	—
284	C₂²×C₄	C₃×D₂₀	1	2	yes	—
285	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
286	M₁₆	—	2	4	no	—
287	C₂×C₈	C₂²×C₃×D₅	1	4	yes	—
288	C₄⋊C₄	—	5	4	no	—
289	C₄⋊C₄	—	5	2	no	—
290	C₄⋊C₄	—	10	2	no	—
291	C₄⋊C₄	—	10	4	no	—
292	C₄⋊C₄	—	10	4	no	—
293	C₄⋊C₄	—	10	2	no	—
294	C₂²⋊C₄	—	10	1	no	—
295	C₂²⋊C₄	—	10	2	no	—
296	C₄²	—	10	1	no	—
297	C₄²	—	10	2	no	—
298	C₄²	C₃×D₂₀	1	1	yes	characteristic
299	C₄²	C₃×D₂₀	1	1	yes	characteristic
300	C₄²	C₂²×C₃×D₅	1	2	yes	—
301	C₃×C₅	#3080	1	1	yes	characteristic, complemented, Hall
302	C₂²×C₃	—	2	2	no	—
303	C₂²×C₃	—	10	2	no	—
304	C₂²×C₃	—	20	2	no	—
305	C₂²×C₃	#4043	1	2	yes	—
306	C₂²×C₃	C₂²×D₂₀	1	1	yes	characteristic
307	C₃×C₄	—	2	2	no	—
308	C₃×C₄	—	2	2	no	—
309	C₃×C₄	—	10	2	no	—
310	C₃×C₄	—	20	2	no	—
311	C₃×C₄	—	20	4	no	—
312	C₃×C₄	#4043	1	2	yes	—
313	C₃×C₄	C₂×#711	1	1	yes	characteristic
314	C₃×C₄	C₂×Q₈×D₅	1	1	yes	characteristic
315	C₂×C₅	—	2	2	no	—
316	C₂×C₅	C₂×C₃×C₂²⋊Q₈	1	1	yes	characteristic
317	C₂×C₅	C₃×#575	1	2	yes	—
318	D₅	—	4	2	no	—
319	C₂³	—	10	1	no	—
320	C₂³	C₂×C₃×D₂₀	1	1	yes	characteristic
321	Q₈	—	5	2	no	—
322	Q₈	—	10	1	no	—
323	D₄	—	5	2	no	—
324	D₄	—	10	4	no	—
325	D₄	—	10	1	no	—
326	C₂×C₄	—	4	2	no	—
327	C₂×C₄	—	10	1	no	—
328	C₂×C₄	—	10	4	no	—
329	C₂×C₄	—	10	2	no	—
330	C₂×C₄	—	10	4	no	—
331	C₂×C₄	—	20	4	no	—
332	C₂×C₄	—	20	4	no	—
333	C₂×C₄	C₂×C₃×D₂₀	1	2	yes	—
334	C₂×C₄	C₂×C₃×D₂₀	1	2	yes	—
335	C₂×C₄	C₂×C₃×D₂₀	1	1	yes	characteristic
336	C₂×C₄	C₃×#711	1	2	yes	—
337	C₂×C₄	C₃×Q₈×D₅	1	2	yes	—
338	C₂×C₄	C₂³×C₃×D₅	1	1	yes	characteristic
339	C₈	—	2	4	no	—
340	C₂×C₃	—	2	2	no	—
341	C₂×C₃	—	20	2	no	—
342	C₂×C₃	C₂×#4043	1	1	yes	characteristic
343	C₂×C₃	#66082	1	2	yes	—
344	C₅	C₃×#3080	1	1	yes	characteristic, complemented, Sylow
345	C₂²	—	2	2	no	—
346	C₂²	—	10	2	no	—
347	C₂²	—	20	2	no	—
348	C₂²	C₃×#4043	1	2	yes	—
349	C₂²	C₂²×C₃×D₂₀	1	1	yes	characteristic
350	C₄	—	2	2	no	—
351	C₄	—	2	2	no	—
352	C₄	—	10	2	no	—
353	C₄	—	20	2	no	—
354	C₄	—	20	4	no	—
355	C₄	C₃×#4043	1	2	yes	—
356	C₄	C₂×C₃×#711	1	1	yes	characteristic
357	C₄	C₂×C₃×Q₈×D₅	1	1	yes	characteristic
358	C₃	#10610792	1	1	yes	characteristic, direct product factor, Sylow
359	C₂	—	2	2	no	—
360	C₂	—	20	2	no	—
361	C₂	C₂×C₃×#4043	1	1	yes	characteristic
362	C₂	C₃×#66082	1	2	yes	—

	subgroup	G/H
center	C₂²×C₃	C₂²×D₂₀
commutator, cosocle	C₂×C₄×C₅	C₂⁴×C₃
Frattini	C₂×C₄	C₂³×C₃×D₅
Fitting	C₃×C₅×#423	C₂
socle	C₂²×C₃×C₅	C₂²×D₄
prime radical	C₂×C₃×#709	C₂²
hypercenter	C₃×#423	D₅
hypocenter	C₅	C₃×#3080
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄² ◁ C₃×#423

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×#4016	C₂	semidirect
3. C₃×C₅×#423	C₂	semidirect
5. C₃×#65410	C₂	semidirect
6. C₃×#65136	C₂	semidirect
7. C₃×#65118	C₂	semidirect
8. #10610792	C₃	direct (N × H)
30. C₂×#4016	C₂×C₃	semidirect
31. C₅×#423	C₂×C₃	semidirect
33. #65410	C₂×C₃	semidirect
34. #65136	C₂×C₃	semidirect
35. #65118	C₂×C₃	semidirect
66. C₃×#423	D₅	semidirect
158. #423	C₃×D₅	semidirect
301. C₃×C₅	#3080	semidirect
344. C₅	C₃×#3080	semidirect
358. C₃	#10610792	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×C₄×#709	2
C₂×C₃×#4016	2
C₃×C₅×#423	2
C₃×#65422	2
C₃×#65410	2
C₃×#65136	2
C₃×#65118	2
#10610792	3
C₃×#3080	5

Subgroups — C3×#10610792 #14430691

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10610792 #14430691