Subgroups — C₃×#10602436 #14295631

Order 1920.

640 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×C₅×#41	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#3969	C₂	1	2	yes	maximal
3	C₃×#65221	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₄×#681	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#64752	C₂	1	2	yes	maximal, complemented
6	#10602436	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₅×#41	C₂²	1	1	yes	characteristic
8	C₃×C₄×C₅×Q₈	C₄	1	2	yes	complemented
9	C₃×C₄×C₅×D₄	—	2	2	no	—
10	C₃×C₄×C₅×D₄	C₄	1	2	yes	complemented
11	C₃×C₅×#117	—	2	1	no	—
12	C₃×C₅×#117	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₄²×C₅	—	2	1	no	—
14	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
15	C₂×C₃×C₄×Dic₅	C₂²	1	2	yes	—
16	C₂×C₃×#681	C₂²	1	2	yes	—
17	C₃×#3969	C₄	1	8	yes	complemented
18	C₃×#3963	—	2	2	no	—
19	C₃×#3935	—	2	1	no	—
20	C₃×C₄×#168	—	2	1	no	—
21	C₃×#1759	—	5	1	no	maximal, Hall
22	C₄×C₅×#41	C₂×C₃	1	1	yes	characteristic
23	C₂×#3969	C₂×C₃	1	2	yes	—
24	#65221	C₂×C₃	1	1	yes	characteristic, complemented
25	C₄×#681	C₂×C₃	1	1	yes	characteristic, complemented
26	#64752	C₂×C₃	1	2	yes	complemented
27	C₃×C₅×#41	—	2	2	no	—
28	C₃×C₅×#41	C₂×C₄	1	4	yes	—
29	C₂×C₃×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
30	C₂×C₃×C₅×D₄	—	2	1	no	—
31	C₂×C₃×C₅×D₄	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₃×C₄×C₅	—	2	1	no	—
33	C₂²×C₃×C₄×C₅	—	2	2	no	—
34	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
35	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
36	C₂²×C₃×Dic₅	—	2	2	no	—
37	C₃×C₅×C₄⋊C₄	—	2	2	no	—
38	C₃×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
39	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
40	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
41	C₃×C₄²×C₅	—	2	2	no	—
42	C₃×C₄²×C₅	—	2	1	no	—
43	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
44	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
45	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
46	C₃×C₄×Dic₅	—	2	2	no	—
47	C₃×C₄×Dic₅	C₂×C₄	1	4	yes	complemented
48	C₃×#681	—	2	2	no	—
49	C₃×#681	C₂×C₄	1	4	yes	complemented
50	C₂×C₃×#168	—	2	2	no	—
51	C₃×C₄×#41	D₅	1	1	yes	characteristic
52	C₂×C₃×#104	—	5	2	no	—
53	C₃×C₄×M₁₆	—	5	1	no	—
54	C₃×#376	—	5	1	no	—
55	C₃×#339	—	5	2	no	—
56	C₂×C₅×#41	C₂²×C₃	1	1	yes	characteristic
57	C₄×C₅×Q₈	C₃×C₄	1	2	yes	complemented
58	C₄×C₅×D₄	—	2	2	no	—
59	C₄×C₅×D₄	C₃×C₄	1	2	yes	complemented
60	C₅×#117	—	2	1	no	—
61	C₅×#117	C₂²×C₃	1	1	yes	characteristic
62	C₂×C₄²×C₅	—	2	1	no	—
63	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
64	C₂×C₄×Dic₅	C₂²×C₃	1	2	yes	—
65	C₂×#681	C₂²×C₃	1	2	yes	—
66	#3969	C₃×C₄	1	8	yes	complemented
67	#3963	—	2	2	no	—
68	#3935	—	2	1	no	—
69	C₄×#168	—	2	1	no	—
70	#1759	—	5	1	no	Sylow
71	C₂³×C₃×C₅	—	2	1	no	—
72	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
73	C₃×C₅×Q₈	—	2	1	no	—
74	C₃×C₅×Q₈	C₄²	1	2	yes	—
75	C₃×C₅×D₄	—	2	4	no	—
76	C₃×C₅×D₄	—	2	1	no	—
77	C₃×C₅×D₄	C₄²	1	2	yes	—
78	C₂×C₃×C₄×C₅	—	2	1	no	—
79	C₂×C₃×C₄×C₅	—	2	4	no	—
80	C₂×C₃×C₄×C₅	—	2	2	no	—
81	C₂×C₃×C₄×C₅	—	4	2	no	—
82	C₂×C₃×C₄×C₅	—	4	4	no	—
83	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
84	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
85	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
87	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
88	C₂×C₃×C₄×C₅	#41	1	2	yes	—
89	C₂×C₃×Dic₅	—	2	4	no	—
90	C₂×C₃×Dic₅	—	4	4	no	—
91	C₃×#168	—	2	4	no	—
92	C₂×C₃×#41	C₂×D₅	1	1	yes	characteristic
93	C₂×C₃×M₁₆	—	5	2	no	—
94	C₃×C₄×Q₈	Dic₅	1	2	yes	complemented
95	C₃×C₄×D₄	—	2	2	no	—
96	C₃×C₄×D₄	Dic₅	1	2	yes	complemented
97	C₃×#117	—	2	1	no	—
98	C₃×#117	C₂×D₅	1	1	yes	characteristic
99	C₂×C₃×C₄²	—	2	1	no	—
100	C₂×C₃×C₄²	—	5	2	no	—
101	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
102	C₃×#104	—	5	8	no	—
103	C₃×#97	—	10	1	no	—
104	C₃×C₄×C₈	—	10	1	no	—
105	C₃×#95	—	10	2	no	—
106	C₅×#41	—	2	2	no	—
107	C₅×#41	C₂×C₃×C₄	1	4	yes	—
108	C₂×C₅×Q₈	C₂×C₃×C₄	1	1	yes	characteristic
109	C₂×C₅×D₄	—	2	1	no	—
110	C₂×C₅×D₄	C₂×C₃×C₄	1	1	yes	characteristic
111	C₂²×C₄×C₅	—	2	1	no	—
112	C₂²×C₄×C₅	—	2	2	no	—
113	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
114	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
115	C₂²×Dic₅	—	2	2	no	—
116	C₅×C₄⋊C₄	—	2	2	no	—
117	C₅×C₄⋊C₄	C₂×C₃×C₄	1	2	yes	—
118	C₅×C₂²⋊C₄	—	2	1	no	—
119	C₅×C₂²⋊C₄	—	2	2	no	—
120	C₄²×C₅	—	2	2	no	—
121	C₄²×C₅	—	2	1	no	—
122	C₄²×C₅	C₂×C₃×C₄	1	2	yes	—
123	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
124	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
125	C₄×Dic₅	—	2	2	no	—
126	C₄×Dic₅	C₂×C₃×C₄	1	4	yes	complemented
127	#681	—	2	2	no	—
128	#681	C₂×C₃×C₄	1	4	yes	complemented
129	C₂×#168	—	2	2	no	—
130	C₄×#41	C₃×D₅	1	1	yes	characteristic
131	C₂×#104	—	5	2	no	—
132	C₄×M₁₆	—	5	1	no	—
133	#376	—	5	1	no	—
134	#339	—	5	2	no	—
135	C₂²×C₃×C₅	—	2	2	no	—
136	C₂²×C₃×C₅	—	2	2	no	—
137	C₂²×C₃×C₅	—	4	2	no	—
138	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
139	C₂²×C₃×C₅	C₄×D₄	1	2	yes	—
140	C₃×C₄×C₅	—	2	2	no	—
141	C₃×C₄×C₅	—	2	2	no	—
142	C₃×C₄×C₅	—	2	2	no	—
143	C₃×C₄×C₅	—	4	2	no	—
144	C₃×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
145	C₃×C₄×C₅	#117	1	1	yes	characteristic
146	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
147	C₃×Dic₅	—	4	4	no	—
148	C₃×#41	—	2	2	no	—
149	C₃×#41	C₄×D₅	1	4	yes	—
150	C₂×C₃×Q₈	C₂×Dic₅	1	1	yes	characteristic
151	C₂×C₃×D₄	—	2	1	no	—
152	C₂×C₃×D₄	C₂×Dic₅	1	1	yes	characteristic
153	C₂²×C₃×C₄	—	2	1	no	—
154	C₂²×C₃×C₄	—	2	2	no	—
155	C₂²×C₃×C₄	—	10	2	no	—
156	C₂²×C₃×C₄	C₂²⋊D₅	1	2	yes	—
157	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
158	C₃×M₁₆	—	5	4	no	—
159	C₃×M₁₆	—	10	2	no	—
160	C₂×C₃×C₈	—	10	2	no	—
161	C₃×C₄⋊C₄	—	2	2	no	—
162	C₃×C₄⋊C₄	C₂×Dic₅	1	2	yes	—
163	C₃×C₂²⋊C₄	—	2	1	no	—
164	C₃×C₂²⋊C₄	—	2	2	no	—
165	C₃×C₄²	—	2	2	no	—
166	C₃×C₄²	—	2	1	no	—
167	C₃×C₄²	—	5	4	no	—
168	C₃×C₄²	—	10	2	no	—
169	C₃×C₄²	C₂×Dic₅	1	2	yes	—
170	C₃×C₄²	C₂²⋊D₅	1	1	yes	characteristic
171	C₃×C₄²	C₂²⋊D₅	1	1	yes	characteristic
172	C₂³×C₅	—	2	1	no	—
173	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
174	C₅×Q₈	—	2	1	no	—
175	C₅×Q₈	C₃×C₄²	1	2	yes	—
176	C₅×D₄	—	2	4	no	—
177	C₅×D₄	—	2	1	no	—
178	C₅×D₄	C₃×C₄²	1	2	yes	—
179	C₂×C₄×C₅	—	2	1	no	—
180	C₂×C₄×C₅	—	2	4	no	—
181	C₂×C₄×C₅	—	2	2	no	—
182	C₂×C₄×C₅	—	4	2	no	—
183	C₂×C₄×C₅	—	4	4	no	—
184	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	2	yes	—
185	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	2	yes	—
186	C₂×C₄×C₅	C₂²×C₃×C₄	1	2	yes	—
187	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
188	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
189	C₂×C₄×C₅	C₃×#41	1	2	yes	—
190	C₂×Dic₅	—	2	4	no	—
191	C₂×Dic₅	—	4	4	no	—
192	#168	—	2	4	no	—
193	C₂×#41	C₂×C₃×D₅	1	1	yes	characteristic
194	C₂×M₁₆	—	5	2	no	—
195	C₄×Q₈	C₃×Dic₅	1	2	yes	complemented
196	C₄×D₄	—	2	2	no	—
197	C₄×D₄	C₃×Dic₅	1	2	yes	complemented
198	#117	—	2	1	no	—
199	#117	C₂×C₃×D₅	1	1	yes	characteristic
200	C₂×C₄²	—	2	1	no	—
201	C₂×C₄²	—	5	2	no	—
202	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
203	#104	—	5	8	no	—
204	#97	—	10	1	no	—
205	C₄×C₈	—	10	1	no	—
206	#95	—	10	2	no	—
207	C₂×C₃×C₅	—	2	2	no	—
208	C₂×C₃×C₅	—	4	2	no	—
209	C₂×C₃×C₅	C₄×C₂²⋊C₄	1	1	yes	characteristic
210	C₂×C₃×C₅	#421	1	2	yes	—
211	C₂³×C₃	—	2	1	no	—
212	C₂³×C₃	C₂×C₂²⋊D₅	1	1	yes	characteristic
213	C₃×Q₈	—	2	1	no	—
214	C₃×Q₈	C₄×Dic₅	1	2	yes	—
215	C₃×D₄	—	2	4	no	—
216	C₃×D₄	—	2	1	no	—
217	C₃×D₄	C₄×Dic₅	1	2	yes	—
218	C₂×C₃×C₄	—	2	1	no	—
219	C₂×C₃×C₄	—	2	4	no	—
220	C₂×C₃×C₄	—	2	2	no	—
221	C₂×C₃×C₄	—	4	2	no	—
222	C₂×C₃×C₄	—	4	4	no	—
223	C₂×C₃×C₄	—	10	4	no	—
224	C₂×C₃×C₄	—	20	4	no	—
225	C₂×C₃×C₄	C₂²⋊Dic₅	1	2	yes	—
226	C₂×C₃×C₄	C₂²⋊Dic₅	1	2	yes	—
227	C₂×C₃×C₄	C₂×C₄×D₅	1	2	yes	—
228	C₂×C₃×C₄	#709	1	2	yes	—
229	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
230	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
231	C₃×C₈	—	10	4	no	—
232	C₂²×C₅	—	2	2	no	—
233	C₂²×C₅	—	2	2	no	—
234	C₂²×C₅	—	4	2	no	—
235	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
236	C₂²×C₅	C₃×C₄×D₄	1	2	yes	—
237	C₄×C₅	—	2	2	no	—
238	C₄×C₅	—	2	2	no	—
239	C₄×C₅	—	2	2	no	—
240	C₄×C₅	—	4	2	no	—
241	C₄×C₅	C₂×C₃×C₄²	1	1	yes	characteristic
242	C₄×C₅	C₃×#117	1	1	yes	characteristic
243	C₄×C₅	C₃×C₄×D₄	1	2	yes	—
244	Dic₅	—	4	4	no	—
245	#41	—	2	2	no	—
246	#41	C₃×C₄×D₅	1	4	yes	—
247	C₂×Q₈	C₂×C₃×Dic₅	1	1	yes	characteristic
248	C₂×D₄	—	2	1	no	—
249	C₂×D₄	C₂×C₃×Dic₅	1	1	yes	characteristic
250	C₂²×C₄	—	2	1	no	—
251	C₂²×C₄	—	2	2	no	—
252	C₂²×C₄	—	10	2	no	—
253	C₂²×C₄	C₃×C₂²⋊D₅	1	2	yes	—
254	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
255	M₁₆	—	5	4	no	—
256	M₁₆	—	10	2	no	—
257	C₂×C₈	—	10	2	no	—
258	C₄⋊C₄	—	2	2	no	—
259	C₄⋊C₄	C₂×C₃×Dic₅	1	2	yes	—
260	C₂²⋊C₄	—	2	1	no	—
261	C₂²⋊C₄	—	2	2	no	—
262	C₄²	—	2	2	no	—
263	C₄²	—	2	1	no	—
264	C₄²	—	5	4	no	—
265	C₄²	—	10	2	no	—
266	C₄²	C₂×C₃×Dic₅	1	2	yes	—
267	C₄²	C₃×C₂²⋊D₅	1	1	yes	characteristic
268	C₄²	C₃×C₂²⋊D₅	1	1	yes	characteristic
269	C₃×C₅	#1759	1	1	yes	characteristic, complemented, Hall
270	C₂²×C₃	—	2	2	no	—
271	C₂²×C₃	—	2	2	no	—
272	C₂²×C₃	—	4	2	no	—
273	C₂²×C₃	C₄×C₂²⋊D₅	1	2	yes	—
274	C₂²×C₃	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
275	C₃×C₄	—	2	2	no	—
276	C₃×C₄	—	2	2	no	—
277	C₃×C₄	—	2	2	no	—
278	C₃×C₄	—	4	2	no	—
279	C₃×C₄	—	20	4	no	—
280	C₃×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
281	C₃×C₄	#4072	1	1	yes	characteristic
282	C₃×C₄	C₄×C₂²⋊D₅	1	2	yes	—
283	C₂×C₅	—	2	2	no	—
284	C₂×C₅	—	4	2	no	—
285	C₂×C₅	C₃×C₄×C₂²⋊C₄	1	1	yes	characteristic
286	C₂×C₅	C₃×#421	1	2	yes	—
287	C₂³	—	2	1	no	—
288	C₂³	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
289	Q₈	—	2	1	no	—
290	Q₈	C₃×C₄×Dic₅	1	2	yes	—
291	D₄	—	2	4	no	—
292	D₄	—	2	1	no	—
293	D₄	C₃×C₄×Dic₅	1	2	yes	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	4	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	4	no	—
299	C₂×C₄	—	10	4	no	—
300	C₂×C₄	—	20	4	no	—
301	C₂×C₄	C₃×C₂²⋊Dic₅	1	2	yes	—
302	C₂×C₄	C₃×C₂²⋊Dic₅	1	2	yes	—
303	C₂×C₄	C₂×C₃×C₄×D₅	1	2	yes	—
304	C₂×C₄	C₃×#709	1	2	yes	—
305	C₂×C₄	C₂²×C₃×Dic₅	1	1	yes	characteristic
306	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
307	C₈	—	10	4	no	—
308	C₂×C₃	—	2	2	no	—
309	C₂×C₃	—	4	2	no	—
310	C₂×C₃	C₄×C₂²⋊Dic₅	1	1	yes	characteristic
311	C₂×C₃	#65525	1	2	yes	—
312	C₅	C₃×#1759	1	1	yes	characteristic, complemented, Sylow
313	C₂²	—	2	2	no	—
314	C₂²	—	2	2	no	—
315	C₂²	—	4	2	no	—
316	C₂²	C₃×C₄×C₂²⋊D₅	1	2	yes	—
317	C₂²	C₂×C₃×C₂²⋊Dic₅	1	1	yes	characteristic
318	C₄	—	2	2	no	—
319	C₄	—	2	2	no	—
320	C₄	—	2	2	no	—
321	C₄	—	4	2	no	—
322	C₄	—	20	4	no	—
323	C₄	C₂×C₃×C₄×Dic₅	1	1	yes	characteristic
324	C₄	C₃×#4072	1	1	yes	characteristic
325	C₄	C₃×C₄×C₂²⋊D₅	1	2	yes	—
326	C₃	#10602436	1	1	yes	characteristic, direct product factor, Sylow
327	C₂	—	2	2	no	—
328	C₂	—	4	2	no	—
329	C₂	C₃×C₄×C₂²⋊Dic₅	1	1	yes	characteristic
330	C₂	C₃×#65525	1	2	yes	—

	subgroup	G/H
center	C₂×C₃×C₄	C₂×C₂²⋊D₅
commutator	C₄×C₅	C₂×C₃×C₄²
Frattini	C₂²×C₄	C₂²×C₃×D₅
Fitting	C₃×C₄×C₅×#41	C₂
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₂×C₃×C₅×#41	C₂²
cosocle	C₂²×C₄×C₅	C₂³×C₃
hypercenter	C₃×C₄×#41	D₅
hypocenter	C₅	C₃×#1759
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₄² ◁ C₃×C₄×#41

Complements

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

N	H ≅ G/N	type
3. C₃×#65221	C₂	semidirect
4. C₃×C₄×#681	C₂	semidirect
5. C₃×#64752	C₂	semidirect
6. #10602436	C₃	direct (N × H)
8. C₃×C₄×C₅×Q₈	C₄	semidirect
10. C₃×C₄×C₅×D₄	C₄	semidirect
17. C₃×#3969	C₄	semidirect
24. #65221	C₂×C₃	semidirect
25. C₄×#681	C₂×C₃	semidirect
26. #64752	C₂×C₃	semidirect
47. C₃×C₄×Dic₅	C₂×C₄	semidirect
49. C₃×#681	C₂×C₄	semidirect
57. C₄×C₅×Q₈	C₃×C₄	semidirect
59. C₄×C₅×D₄	C₃×C₄	semidirect
66. #3969	C₃×C₄	semidirect
94. C₃×C₄×Q₈	Dic₅	semidirect
96. C₃×C₄×D₄	Dic₅	semidirect
126. C₄×Dic₅	C₂×C₃×C₄	semidirect
128. #681	C₂×C₃×C₄	semidirect
195. C₄×Q₈	C₃×Dic₅	semidirect
197. C₄×D₄	C₃×Dic₅	semidirect
269. C₃×C₅	#1759	semidirect
312. C₅	C₃×#1759	semidirect
326. C₃	#10602436	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×C₅×#41	2
C₂×C₃×#3969	2
C₃×#65221	2
C₃×C₄×#681	2
C₃×#64752	2
#10602436	3
C₃×#1759	5

Subgroups — C3×#10602436 #14295631

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10602436 #14295631