Subgroups — C₃×#10599659 #14294553

Order 1920.

848 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	C₂²×C₃×C₈×D₅	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#4038	C₂	1	2	yes	maximal
3	C₃×C₅×#340	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#64771	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#64749	C₂	1	2	yes	maximal, complemented
6	#10599659	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₃×C₄×D₅	C₂²	1	1	yes	characteristic
8	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
10	C₂×C₃×#685	—	2	2	no	—
11	C₂×C₃×C₄⋊Dic₅	C₂²	1	2	yes	—
12	C₂×C₃×C₄×Dic₅	—	2	2	no	—
13	C₂²×C₃×#168	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₈×D₅	—	2	4	no	—
15	C₂×C₃×C₈×D₅	C₄	1	4	yes	complemented
16	C₃×#4038	—	2	4	no	—
17	C₃×#4038	C₄	1	8	yes	complemented
18	C₃×#1735	—	5	1	no	maximal, Hall
19	C₂²×C₈×D₅	C₂×C₃	1	1	yes	characteristic
20	C₂×#4038	C₂×C₃	1	2	yes	—
21	C₅×#340	C₂×C₃	1	1	yes	characteristic, complemented
22	#64771	C₂×C₃	1	1	yes	characteristic, complemented
23	#64749	C₂×C₃	1	2	yes	complemented
24	C₂³×C₃×D₅	D₄	1	1	yes	characteristic
25	C₂²×C₃×C₄×C₅	—	2	2	no	—
26	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
27	C₂²×C₃×Dic₅	—	2	2	no	—
28	C₂²×C₃×Dic₅	D₄	1	1	yes	characteristic
29	C₂×C₃×C₄×D₅	C₂×C₄	1	2	yes	—
30	C₂×C₃×C₄×D₅	C₂×C₄	1	4	yes	—
31	C₂×C₃×C₄×D₅	D₄	1	4	yes	—
32	C₂×C₃×C₄×D₅	Q₈	1	2	yes	—
33	C₂×C₃×C₅×C₈	—	2	2	no	—
34	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
35	C₃×C₅×C₄⋊C₄	—	2	2	no	—
36	C₃×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
37	C₃×#685	—	2	8	no	—
38	C₃×C₄⋊Dic₅	—	2	2	no	—
39	C₃×C₄⋊Dic₅	C₂×C₄	1	4	yes	—
40	C₃×C₄×Dic₅	—	2	8	no	—
41	C₂×C₃×#168	—	2	2	no	—
42	C₂×C₃×#168	C₂×C₄	1	2	yes	—
43	C₃×C₈×D₅	—	2	8	no	—
44	C₂³×C₃×C₈	—	5	1	no	—
45	C₂×C₃×#117	—	5	2	no	—
46	C₃×#340	—	5	1	no	—
47	C₃×#340	—	5	2	no	—
48	C₃×#340	D₅	1	1	yes	characteristic, complemented
49	C₂²×C₄×D₅	C₂²×C₃	1	1	yes	characteristic
50	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
51	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
52	C₂×#685	—	2	2	no	—
53	C₂×C₄⋊Dic₅	C₂²×C₃	1	2	yes	—
54	C₂×C₄×Dic₅	—	2	2	no	—
55	C₂²×#168	C₂²×C₃	1	1	yes	characteristic
56	C₂×C₈×D₅	—	2	4	no	—
57	C₂×C₈×D₅	C₃×C₄	1	4	yes	complemented
58	#4038	—	2	4	no	—
59	#4038	C₃×C₄	1	8	yes	complemented
60	#1735	—	5	1	no	Sylow
61	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₃×D₅	—	2	1	no	—
63	C₂²×C₃×D₅	—	2	2	no	—
64	C₂²×C₃×D₅	—	2	1	no	—
65	C₂²×C₃×D₅	C₂²⋊C₄	1	4	yes	—
66	C₂²×C₃×D₅	C₄⋊C₄	1	2	yes	—
67	C₂×C₃×C₄×C₅	—	2	4	no	—
68	C₂×C₃×C₄×C₅	—	4	2	no	—
69	C₂×C₃×C₄×C₅	—	4	2	no	—
70	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
72	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
73	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
74	C₂×C₃×Dic₅	—	2	4	no	—
75	C₂×C₃×Dic₅	—	4	2	no	—
76	C₂×C₃×Dic₅	—	4	2	no	—
77	C₂×C₃×Dic₅	C₂²⋊C₄	1	4	yes	—
78	C₂×C₃×Dic₅	C₄⋊C₄	1	2	yes	—
79	C₃×C₄×D₅	C₄²	1	4	yes	—
80	C₃×C₄×D₅	C₂²⋊C₄	1	4	yes	—
81	C₃×C₄×D₅	C₄⋊C₄	1	8	yes	—
82	C₃×C₅×C₈	—	2	2	no	—
83	C₃×#168	—	2	2	no	—
84	C₂³×C₃×C₄	—	5	1	no	—
85	C₂²×C₃×C₈	—	5	1	no	—
86	C₂²×C₃×C₈	—	5	4	no	—
87	C₂²×C₃×C₈	—	10	4	no	—
88	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
89	C₃×#117	—	5	8	no	—
90	C₃×#117	—	10	4	no	—
91	C₂×C₃×C₄⋊C₄	—	5	2	no	—
92	C₂×C₃×C₄⋊C₄	C₂×D₅	1	2	yes	—
93	C₂×C₃×C₂²⋊C₄	—	10	2	no	—
94	C₂×C₃×C₄²	—	10	2	no	—
95	C₂³×D₅	C₃×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₅	—	2	2	no	—
97	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
98	C₂²×Dic₅	—	2	2	no	—
99	C₂²×Dic₅	C₃×D₄	1	1	yes	characteristic
100	C₂×C₄×D₅	C₂×C₃×C₄	1	2	yes	—
101	C₂×C₄×D₅	C₂×C₃×C₄	1	4	yes	—
102	C₂×C₄×D₅	C₃×D₄	1	4	yes	—
103	C₂×C₄×D₅	C₃×Q₈	1	2	yes	—
104	C₂×C₅×C₈	—	2	2	no	—
105	C₂×C₅×C₈	C₂×C₃×C₄	1	2	yes	—
106	C₅×C₄⋊C₄	—	2	2	no	—
107	C₅×C₄⋊C₄	C₂×C₃×C₄	1	4	yes	—
108	#685	—	2	8	no	—
109	C₄⋊Dic₅	—	2	2	no	—
110	C₄⋊Dic₅	C₂×C₃×C₄	1	4	yes	—
111	C₄×Dic₅	—	2	8	no	—
112	C₂×#168	—	2	2	no	—
113	C₂×#168	C₂×C₃×C₄	1	2	yes	—
114	C₈×D₅	—	2	8	no	—
115	C₂³×C₈	—	5	1	no	—
116	C₂×#117	—	5	2	no	—
117	#340	—	5	1	no	—
118	#340	—	5	2	no	—
119	#340	C₃×D₅	1	1	yes	characteristic, complemented
120	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	2	yes	—
121	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂²×C₃×C₅	#135	1	1	yes	characteristic
123	C₂²×C₃×C₅	#135	1	2	yes	—
124	C₂²×C₃×C₅	#135	1	1	yes	characteristic
125	C₂×C₃×D₅	—	2	4	no	—
126	C₂×C₃×D₅	—	2	4	no	—
127	C₂×C₃×D₅	—	2	2	no	—
128	C₂×C₃×D₅	—	2	2	no	—
129	C₂×C₃×D₅	#95	1	4	yes	—
130	C₃×C₄×C₅	—	4	4	no	—
131	C₃×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
132	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
133	C₃×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
134	C₃×Dic₅	—	4	4	no	—
135	C₃×Dic₅	#95	1	4	yes	—
136	C₂⁴×C₃	—	5	1	no	—
137	C₂²×C₃×C₄	—	2	2	no	—
138	C₂²×C₃×C₄	—	5	1	no	—
139	C₂²×C₃×C₄	—	5	4	no	—
140	C₂²×C₃×C₄	—	5	2	no	—
141	C₂²×C₃×C₄	—	5	4	no	—
142	C₂²×C₃×C₄	—	5	2	no	—
143	C₂²×C₃×C₄	—	10	2	no	—
144	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
145	C₂×C₃×C₈	—	2	2	no	—
146	C₂×C₃×C₈	—	5	2	no	—
147	C₂×C₃×C₈	—	10	8	no	—
148	C₂×C₃×C₈	—	10	2	no	—
149	C₂×C₃×C₈	C₄×D₅	1	2	yes	—
150	C₃×C₄⋊C₄	—	2	2	no	—
151	C₃×C₄⋊C₄	—	5	4	no	—
152	C₃×C₄⋊C₄	—	10	2	no	—
153	C₃×C₄⋊C₄	C₄×D₅	1	4	yes	—
154	C₃×C₂²⋊C₄	—	10	8	no	—
155	C₃×C₄²	—	10	8	no	—
156	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
157	C₂²×D₅	—	2	1	no	—
158	C₂²×D₅	—	2	2	no	—
159	C₂²×D₅	—	2	1	no	—
160	C₂²×D₅	C₃×C₂²⋊C₄	1	4	yes	—
161	C₂²×D₅	C₃×C₄⋊C₄	1	2	yes	—
162	C₂×C₄×C₅	—	2	4	no	—
163	C₂×C₄×C₅	—	4	2	no	—
164	C₂×C₄×C₅	—	4	2	no	—
165	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
166	C₂×C₄×C₅	C₂²×C₃×C₄	1	2	yes	—
167	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
168	C₂×C₄×C₅	C₂×C₃×Q₈	1	1	yes	characteristic
169	C₂×Dic₅	—	2	4	no	—
170	C₂×Dic₅	—	4	2	no	—
171	C₂×Dic₅	—	4	2	no	—
172	C₂×Dic₅	C₃×C₂²⋊C₄	1	4	yes	—
173	C₂×Dic₅	C₃×C₄⋊C₄	1	2	yes	—
174	C₄×D₅	C₃×C₄²	1	4	yes	—
175	C₄×D₅	C₃×C₂²⋊C₄	1	4	yes	—
176	C₄×D₅	C₃×C₄⋊C₄	1	8	yes	—
177	C₅×C₈	—	2	2	no	—
178	#168	—	2	2	no	—
179	C₂³×C₄	—	5	1	no	—
180	C₂²×C₈	—	5	1	no	—
181	C₂²×C₈	—	5	4	no	—
182	C₂²×C₈	—	10	4	no	—
183	C₂²×C₈	C₂×C₃×D₅	1	1	yes	characteristic
184	#117	—	5	8	no	—
185	#117	—	10	4	no	—
186	C₂×C₄⋊C₄	—	5	2	no	—
187	C₂×C₄⋊C₄	C₂×C₃×D₅	1	2	yes	—
188	C₂×C₂²⋊C₄	—	10	2	no	—
189	C₂×C₄²	—	10	2	no	—
190	C₂×C₃×C₅	C₂×#95	1	1	yes	characteristic
191	C₂×C₃×C₅	#416	1	2	yes	—
192	C₂×C₃×C₅	#416	1	2	yes	—
193	C₂×C₃×C₅	#427	1	1	yes	characteristic
194	C₂×C₃×C₅	#427	1	1	yes	characteristic
195	C₃×D₅	—	2	4	no	—
196	C₂³×C₃	—	5	4	no	—
197	C₂³×C₃	—	5	2	no	—
198	C₂³×C₃	—	10	1	no	—
199	C₂³×C₃	—	10	2	no	—
200	C₂³×C₃	—	10	1	no	—
201	C₂³×C₃	D₄×D₅	1	1	yes	characteristic
202	C₂×C₃×C₄	—	2	4	no	—
203	C₂×C₃×C₄	—	4	2	no	—
204	C₂×C₃×C₄	—	4	2	no	—
205	C₂×C₃×C₄	—	5	4	no	—
206	C₂×C₃×C₄	—	5	2	no	—
207	C₂×C₃×C₄	—	5	4	no	—
208	C₂×C₃×C₄	—	5	4	no	—
209	C₂×C₃×C₄	—	5	8	no	—
210	C₂×C₃×C₄	—	10	4	no	—
211	C₂×C₃×C₄	—	20	2	no	—
212	C₂×C₃×C₄	—	20	2	no	—
213	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
214	C₂×C₃×C₄	C₂×C₄×D₅	1	2	yes	—
215	C₂×C₃×C₄	D₄×D₅	1	2	yes	—
216	C₂×C₃×C₄	Q₈×D₅	1	1	yes	characteristic
217	C₃×C₈	—	2	2	no	—
218	C₃×C₈	—	10	2	no	—
219	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	2	yes	—
220	C₂²×C₅	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
221	C₂²×C₅	C₃×#135	1	1	yes	characteristic
222	C₂²×C₅	C₃×#135	1	2	yes	—
223	C₂²×C₅	C₃×#135	1	1	yes	characteristic
224	C₂×D₅	—	2	4	no	—
225	C₂×D₅	—	2	4	no	—
226	C₂×D₅	—	2	2	no	—
227	C₂×D₅	—	2	2	no	—
228	C₂×D₅	C₃×#95	1	4	yes	—
229	C₄×C₅	—	4	4	no	—
230	C₄×C₅	C₂×C₃×C₄²	1	1	yes	characteristic
231	C₄×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
232	C₄×C₅	C₂×C₃×C₄⋊C₄	1	2	yes	—
233	Dic₅	—	4	4	no	—
234	Dic₅	C₃×#95	1	4	yes	—
235	C₂⁴	—	5	1	no	—
236	C₂²×C₄	—	2	2	no	—
237	C₂²×C₄	—	5	1	no	—
238	C₂²×C₄	—	5	4	no	—
239	C₂²×C₄	—	5	2	no	—
240	C₂²×C₄	—	5	4	no	—
241	C₂²×C₄	—	5	2	no	—
242	C₂²×C₄	—	10	2	no	—
243	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	5	2	no	—
246	C₂×C₈	—	10	8	no	—
247	C₂×C₈	—	10	2	no	—
248	C₂×C₈	C₃×C₄×D₅	1	2	yes	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	5	4	no	—
251	C₄⋊C₄	—	10	2	no	—
252	C₄⋊C₄	C₃×C₄×D₅	1	4	yes	—
253	C₂²⋊C₄	—	10	8	no	—
254	C₄²	—	10	8	no	—
255	C₃×C₅	#1735	1	1	yes	characteristic, complemented, Hall
256	C₂²×C₃	—	5	4	no	—
257	C₂²×C₃	—	10	4	no	—
258	C₂²×C₃	—	10	4	no	—
259	C₂²×C₃	—	10	2	no	—
260	C₂²×C₃	—	10	2	no	—
261	C₂²×C₃	D₅×C₂²⋊C₄	1	2	yes	—
262	C₂²×C₃	D₅×C₄⋊C₄	1	1	yes	characteristic
263	C₂²×C₃	#4058	1	1	yes	characteristic
264	C₂²×C₃	#4062	1	2	yes	—
265	C₂²×C₃	#4065	1	1	yes	characteristic
266	C₃×C₄	—	4	4	no	—
267	C₃×C₄	—	5	4	no	—
268	C₃×C₄	—	20	4	no	—
269	C₃×C₄	C₄²×D₅	1	1	yes	characteristic
270	C₃×C₄	D₅×C₂²⋊C₄	1	1	yes	characteristic
271	C₃×C₄	D₅×C₄⋊C₄	1	2	yes	—
272	C₂×C₅	C₂×C₃×#95	1	1	yes	characteristic
273	C₂×C₅	C₃×#416	1	2	yes	—
274	C₂×C₅	C₃×#416	1	2	yes	—
275	C₂×C₅	C₃×#427	1	1	yes	characteristic
276	C₂×C₅	C₃×#427	1	1	yes	characteristic
277	D₅	—	2	4	no	—
278	C₂³	—	5	4	no	—
279	C₂³	—	5	2	no	—
280	C₂³	—	10	1	no	—
281	C₂³	—	10	2	no	—
282	C₂³	—	10	1	no	—
283	C₂³	C₃×D₄×D₅	1	1	yes	characteristic
284	C₂×C₄	—	2	4	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	5	4	no	—
288	C₂×C₄	—	5	2	no	—
289	C₂×C₄	—	5	4	no	—
290	C₂×C₄	—	5	4	no	—
291	C₂×C₄	—	5	8	no	—
292	C₂×C₄	—	10	4	no	—
293	C₂×C₄	—	20	2	no	—
294	C₂×C₄	—	20	2	no	—
295	C₂×C₄	C₂×C₃×C₄×D₅	1	1	yes	characteristic
296	C₂×C₄	C₂×C₃×C₄×D₅	1	2	yes	—
297	C₂×C₄	C₃×D₄×D₅	1	2	yes	—
298	C₂×C₄	C₃×Q₈×D₅	1	1	yes	characteristic
299	C₈	—	2	2	no	—
300	C₈	—	10	2	no	—
301	C₂×C₃	—	10	4	no	—
302	C₂×C₃	D₅×#95	1	1	yes	characteristic
303	C₂×C₃	#65061	1	2	yes	—
304	C₂×C₃	#65093	1	2	yes	—
305	C₂×C₃	#65149	1	1	yes	characteristic
306	C₂×C₃	#65169	1	1	yes	characteristic
307	C₅	C₃×#1735	1	1	yes	characteristic, complemented, Sylow
308	C₂²	—	5	4	no	—
309	C₂²	—	10	4	no	—
310	C₂²	—	10	4	no	—
311	C₂²	—	10	2	no	—
312	C₂²	—	10	2	no	—
313	C₂²	C₃×D₅×C₂²⋊C₄	1	2	yes	—
314	C₂²	C₃×D₅×C₄⋊C₄	1	1	yes	characteristic
315	C₂²	C₃×#4058	1	1	yes	characteristic
316	C₂²	C₃×#4062	1	2	yes	—
317	C₂²	C₃×#4065	1	1	yes	characteristic
318	C₄	—	4	4	no	—
319	C₄	—	5	4	no	—
320	C₄	—	20	4	no	—
321	C₄	C₃×C₄²×D₅	1	1	yes	characteristic
322	C₄	C₃×D₅×C₂²⋊C₄	1	1	yes	characteristic
323	C₄	C₃×D₅×C₄⋊C₄	1	2	yes	—
324	C₃	#10599659	1	1	yes	characteristic, direct product factor, Sylow
325	C₂	—	10	4	no	—
326	C₂	C₃×D₅×#95	1	1	yes	characteristic
327	C₂	C₃×#65061	1	2	yes	—
328	C₂	C₃×#65093	1	2	yes	—
329	C₂	C₃×#65149	1	1	yes	characteristic
330	C₂	C₃×#65169	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₃	D₄×D₅
commutator	C₄×C₅	C₂×C₃×C₄²
Frattini	C₂²×C₄	C₂²×C₃×D₅
Fitting	C₃×C₅×#340	C₂
socle	C₂³×C₃×C₅	C₂×D₄
prime radical	C₂³×C₃×D₅	D₄
cosocle	C₂²×C₄×C₅	C₂³×C₃
hypercenter	C₃×#340	D₅
hypocenter	C₅	C₃×#1735
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³×C₃ ◁ C₂²×C₃×C₄ ◁ C₃×#340

Complements

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

N	H ≅ G/N	type
3. C₃×C₅×#340	C₂	semidirect
4. C₃×#64771	C₂	semidirect
5. C₃×#64749	C₂	semidirect
6. #10599659	C₃	direct (N × H)
15. C₂×C₃×C₈×D₅	C₄	semidirect
17. C₃×#4038	C₄	semidirect
21. C₅×#340	C₂×C₃	semidirect
22. #64771	C₂×C₃	semidirect
23. #64749	C₂×C₃	semidirect
48. C₃×#340	D₅	semidirect
57. C₂×C₈×D₅	C₃×C₄	semidirect
59. #4038	C₃×C₄	semidirect
119. #340	C₃×D₅	semidirect
255. C₃×C₅	#1735	semidirect
307. C₅	C₃×#1735	semidirect
324. C₃	#10599659	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₈×D₅	2
C₂×C₃×#4038	2
C₃×C₅×#340	2
C₃×#64771	2
C₃×#64749	2
#10599659	3
C₃×#1735	5

Subgroups — C3×#10599659 #14294553

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10599659 #14294553