Subgroups — C₃×#10600651 #14283923

Order 1920.

468 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#65813	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×C₅×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×#3935	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#64685	C₂	1	2	yes	maximal, complemented
5	C₃×#64684	C₂	1	2	yes	maximal, complemented
6	#10600651	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₅×M₁₆	—	2	1	no	—
8	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
9	C₃×#4076	C₄	1	1	yes	characteristic
10	C₃×#4075	—	2	1	no	—
11	C₃×C₄×C₂²⋊D₅	—	2	1	no	—
12	C₃×#4070	C₄	1	1	yes	characteristic
13	C₂²×C₃×#168	—	2	1	no	—
14	C₃×#4022	—	2	1	no	—
15	C₃×#4021	C₂²	1	1	yes	characteristic
16	C₃×C₄×D₂₀	C₄	1	1	yes	characteristic
17	C₃×#4018	—	2	1	no	—
18	C₃×#4016	C₂²	1	1	yes	characteristic
19	C₃×C₄×Dic₁₀	C₄	1	1	yes	characteristic
20	C₃×C₅×#97	C₂²	1	2	yes	—
21	C₃×C₄×C₅×C₈	—	2	1	no	—
22	C₃×#3935	—	2	1	no	—
23	C₃×#3935	C₂²	1	2	yes	—
24	C₃×#1524	—	5	1	no	maximal, Hall
25	#65813	C₂×C₃	1	1	yes	characteristic
26	C₄×C₅×M₁₆	C₂×C₃	1	1	yes	characteristic, complemented
27	C₂×#3935	C₂×C₃	1	1	yes	characteristic, complemented
28	#64685	C₂×C₃	1	2	yes	complemented
29	#64684	C₂×C₃	1	2	yes	complemented
30	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
32	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
33	C₂×C₃×C₂²⋊D₅	—	2	1	no	—
34	C₂×C₃×D₂₀	C₂×C₄	1	1	yes	characteristic
35	C₂×C₃×C₄×D₅	—	2	1	no	—
36	C₂×C₃×Dic₁₀	C₂×C₄	1	1	yes	characteristic
37	C₃×C₅×M₁₆	—	2	4	no	—
38	C₂×C₃×C₅×C₈	—	2	2	no	—
39	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
40	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
41	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
42	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
43	C₃×C₂²⋊Dic₅	—	2	1	no	—
44	C₃×#685	—	2	1	no	—
45	C₃×#685	—	2	1	no	—
46	C₃×#685	—	2	1	no	—
47	C₃×#685	—	2	1	no	—
48	C₃×C₄⋊Dic₅	C₂×C₄	1	1	yes	characteristic
49	C₃×C₄⋊Dic₅	C₂×C₄	1	1	yes	characteristic
50	C₃×#683	—	2	1	no	—
51	C₃×#683	—	2	1	no	—
52	C₃×#683	—	2	1	no	—
53	C₃×#683	—	2	1	no	—
54	C₃×C₄×Dic₅	—	2	1	no	—
55	C₂×C₃×#168	—	2	4	no	—
56	C₂×C₃×#168	—	2	2	no	—
57	C₃×#529	—	5	1	no	—
58	C₃×C₄×M₁₆	D₅	1	1	yes	characteristic, complemented
59	C₂×C₃×#97	—	5	1	no	—
60	C₃×#330	—	5	2	no	—
61	C₃×#329	—	5	2	no	—
62	C₂×C₅×M₁₆	—	2	1	no	—
63	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
64	#4076	C₃×C₄	1	1	yes	characteristic
65	#4075	—	2	1	no	—
66	C₄×C₂²⋊D₅	—	2	1	no	—
67	#4070	C₃×C₄	1	1	yes	characteristic
68	C₂²×#168	—	2	1	no	—
69	#4022	—	2	1	no	—
70	#4021	C₂²×C₃	1	1	yes	characteristic
71	C₄×D₂₀	C₃×C₄	1	1	yes	characteristic
72	#4018	—	2	1	no	—
73	#4016	C₂²×C₃	1	1	yes	characteristic
74	C₄×Dic₁₀	C₃×C₄	1	1	yes	characteristic
75	C₅×#97	C₂²×C₃	1	2	yes	—
76	C₄×C₅×C₈	—	2	1	no	—
77	#3935	—	2	1	no	—
78	#3935	C₂²×C₃	1	2	yes	—
79	#1524	—	5	1	no	Sylow
80	C₂³×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
81	C₂²×C₃×D₅	—	2	1	no	—
82	C₂×C₃×C₄×C₅	—	2	2	no	—
83	C₂×C₃×C₄×C₅	—	2	2	no	—
84	C₂×C₃×C₄×C₅	—	4	1	no	—
85	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
90	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
91	C₃×C₂²⋊D₅	—	4	2	no	—
92	C₂×C₃×Dic₅	—	2	1	no	—
93	C₂×C₃×Dic₅	—	2	1	no	—
94	C₂×C₃×Dic₅	—	2	1	no	—
95	C₃×D₂₀	—	4	1	no	—
96	C₃×C₄×D₅	—	4	2	no	—
97	C₃×Dic₁₀	—	4	1	no	—
98	C₃×C₅×C₈	—	2	4	no	—
99	C₃×#168	—	2	4	no	—
100	C₂×C₃×M₁₆	—	2	1	no	—
101	C₂²×C₃×C₈	—	10	1	no	—
102	C₃×#126	—	10	1	no	—
103	C₃×#125	—	5	1	no	—
104	C₃×#124	—	5	1	no	—
105	C₃×#123	—	10	1	no	—
106	C₃×C₂²⋊Q₈	—	5	1	no	—
107	C₃×#121	—	5	1	no	—
108	C₃×C₄×Q₈	—	5	1	no	—
109	C₃×C₄×D₄	—	5	1	no	—
110	C₃×C₄×D₄	—	10	1	no	—
111	C₃×#117	—	10	1	no	—
112	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
113	C₃×#97	—	5	2	no	—
114	C₃×#97	—	10	1	no	—
115	C₃×#97	C₂×D₅	1	2	yes	—
116	C₃×C₄×C₈	—	2	1	no	—
117	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
118	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
119	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
120	C₂×C₂²⋊D₅	—	2	1	no	—
121	C₂×D₂₀	C₂×C₃×C₄	1	1	yes	characteristic
122	C₂×C₄×D₅	—	2	1	no	—
123	C₂×Dic₁₀	C₂×C₃×C₄	1	1	yes	characteristic
124	C₅×M₁₆	—	2	4	no	—
125	C₂×C₅×C₈	—	2	2	no	—
126	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
127	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
128	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
129	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
130	C₂²⋊Dic₅	—	2	1	no	—
131	#685	—	2	1	no	—
132	#685	—	2	1	no	—
133	#685	—	2	1	no	—
134	#685	—	2	1	no	—
135	C₄⋊Dic₅	C₂×C₃×C₄	1	1	yes	characteristic
136	C₄⋊Dic₅	C₂×C₃×C₄	1	1	yes	characteristic
137	#683	—	2	1	no	—
138	#683	—	2	1	no	—
139	#683	—	2	1	no	—
140	#683	—	2	1	no	—
141	C₄×Dic₅	—	2	1	no	—
142	C₂×#168	—	2	4	no	—
143	C₂×#168	—	2	2	no	—
144	#529	—	5	1	no	—
145	C₄×M₁₆	C₃×D₅	1	1	yes	characteristic, complemented
146	C₂×#97	—	5	1	no	—
147	#330	—	5	2	no	—
148	#329	—	5	2	no	—
149	C₂²×C₃×C₅	—	2	1	no	—
150	C₂²×C₃×C₅	—	2	1	no	—
151	C₂²×C₃×C₅	#104	1	2	yes	—
152	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
153	C₂×C₃×D₅	—	4	1	no	—
154	C₂×C₃×D₅	—	4	1	no	—
155	C₂×C₃×D₅	—	4	1	no	—
156	C₃×C₄×C₅	—	2	1	no	—
157	C₃×C₄×C₅	—	2	1	no	—
158	C₃×C₄×C₅	—	2	1	no	—
159	C₃×C₄×C₅	—	2	1	no	—
160	C₃×C₄×C₅	—	2	1	no	—
161	C₃×C₄×C₅	#104	1	2	yes	—
162	C₃×Dic₅	—	4	1	no	—
163	C₃×Dic₅	—	4	1	no	—
164	C₃×Dic₅	—	4	1	no	—
165	C₂×C₃×Q₈	—	5	1	no	—
166	C₂×C₃×D₄	—	5	1	no	—
167	C₂×C₃×D₄	—	10	1	no	—
168	C₂²×C₃×C₄	—	10	1	no	—
169	C₂²×C₃×C₄	C₄×D₅	1	1	yes	characteristic
170	C₂²×C₃×C₄	D₂₀	1	1	yes	characteristic
171	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
172	C₃×M₁₆	—	2	4	no	—
173	C₂×C₃×C₈	—	2	2	no	—
174	C₂×C₃×C₈	—	10	4	no	—
175	C₂×C₃×C₈	—	10	2	no	—
176	C₃×C₄⋊C₄	—	5	1	no	—
177	C₃×C₄⋊C₄	—	5	1	no	—
178	C₃×C₄⋊C₄	—	10	1	no	—
179	C₃×C₄⋊C₄	—	10	1	no	—
180	C₃×C₄⋊C₄	—	10	1	no	—
181	C₃×C₄⋊C₄	—	10	1	no	—
182	C₃×C₂²⋊C₄	—	10	1	no	—
183	C₃×C₂²⋊C₄	—	10	1	no	—
184	C₃×C₂²⋊C₄	—	10	1	no	—
185	C₃×C₂²⋊C₄	—	10	1	no	—
186	C₃×C₂²⋊C₄	—	10	1	no	—
187	C₃×C₄²	—	10	1	no	—
188	C₃×C₄²	C₄×D₅	1	1	yes	characteristic
189	C₃×C₄²	D₂₀	1	1	yes	characteristic
190	C₃×C₄²	C₂²⋊D₅	1	1	yes	characteristic
191	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
192	C₂³×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
193	C₂²×D₅	—	2	1	no	—
194	C₂×C₄×C₅	—	2	2	no	—
195	C₂×C₄×C₅	—	2	2	no	—
196	C₂×C₄×C₅	—	4	1	no	—
197	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
198	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
199	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
200	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
201	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
202	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
203	C₂²⋊D₅	—	4	2	no	—
204	C₂×Dic₅	—	2	1	no	—
205	C₂×Dic₅	—	2	1	no	—
206	C₂×Dic₅	—	2	1	no	—
207	D₂₀	—	4	1	no	—
208	C₄×D₅	—	4	2	no	—
209	Dic₁₀	—	4	1	no	—
210	C₅×C₈	—	2	4	no	—
211	#168	—	2	4	no	—
212	C₂×M₁₆	—	2	1	no	—
213	C₂²×C₈	—	10	1	no	—
214	#126	—	10	1	no	—
215	#125	—	5	1	no	—
216	#124	—	5	1	no	—
217	#123	—	10	1	no	—
218	C₂²⋊Q₈	—	5	1	no	—
219	#121	—	5	1	no	—
220	C₄×Q₈	—	5	1	no	—
221	C₄×D₄	—	5	1	no	—
222	C₄×D₄	—	10	1	no	—
223	#117	—	10	1	no	—
224	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
225	#97	—	5	2	no	—
226	#97	—	10	1	no	—
227	#97	C₂×C₃×D₅	1	2	yes	—
228	C₄×C₈	—	2	1	no	—
229	C₂×C₃×C₅	—	2	2	no	—
230	C₂×C₃×C₅	#413	1	1	yes	characteristic
231	C₂×C₃×C₅	C₂×#104	1	1	yes	characteristic
232	C₂×C₃×C₅	C₂×#104	1	1	yes	characteristic
233	C₃×D₅	—	8	1	no	—
234	C₂³×C₃	—	10	1	no	—
235	C₂³×C₃	#685	1	1	yes	characteristic
236	C₃×Q₈	—	20	1	no	—
237	C₃×D₄	—	20	1	no	—
238	C₃×D₄	—	20	2	no	—
239	C₂×C₃×C₄	—	2	2	no	—
240	C₂×C₃×C₄	—	2	2	no	—
241	C₂×C₃×C₄	—	4	1	no	—
242	C₂×C₃×C₄	—	10	1	no	—
243	C₂×C₃×C₄	—	10	1	no	—
244	C₂×C₃×C₄	—	10	1	no	—
245	C₂×C₃×C₄	—	20	2	no	—
246	C₂×C₃×C₄	#685	1	1	yes	characteristic
247	C₂×C₃×C₄	#685	1	1	yes	characteristic
248	C₂×C₃×C₄	#685	1	1	yes	characteristic
249	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
250	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
251	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
252	C₃×C₈	—	2	4	no	—
253	C₃×C₈	—	10	4	no	—
254	C₂²×C₅	—	2	1	no	—
255	C₂²×C₅	—	2	1	no	—
256	C₂²×C₅	C₃×#104	1	2	yes	—
257	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
258	C₂×D₅	—	4	1	no	—
259	C₂×D₅	—	4	1	no	—
260	C₂×D₅	—	4	1	no	—
261	C₄×C₅	—	2	1	no	—
262	C₄×C₅	—	2	1	no	—
263	C₄×C₅	—	2	1	no	—
264	C₄×C₅	—	2	1	no	—
265	C₄×C₅	—	2	1	no	—
266	C₄×C₅	C₃×#104	1	2	yes	—
267	Dic₅	—	4	1	no	—
268	Dic₅	—	4	1	no	—
269	Dic₅	—	4	1	no	—
270	C₂×Q₈	—	5	1	no	—
271	C₂×D₄	—	5	1	no	—
272	C₂×D₄	—	10	1	no	—
273	C₂²×C₄	—	10	1	no	—
274	C₂²×C₄	C₃×C₄×D₅	1	1	yes	characteristic
275	C₂²×C₄	C₃×D₂₀	1	1	yes	characteristic
276	C₂²×C₄	C₃×C₂²⋊D₅	1	1	yes	characteristic
277	M₁₆	—	2	4	no	—
278	C₂×C₈	—	2	2	no	—
279	C₂×C₈	—	10	4	no	—
280	C₂×C₈	—	10	2	no	—
281	C₄⋊C₄	—	5	1	no	—
282	C₄⋊C₄	—	5	1	no	—
283	C₄⋊C₄	—	10	1	no	—
284	C₄⋊C₄	—	10	1	no	—
285	C₄⋊C₄	—	10	1	no	—
286	C₄⋊C₄	—	10	1	no	—
287	C₂²⋊C₄	—	10	1	no	—
288	C₂²⋊C₄	—	10	1	no	—
289	C₂²⋊C₄	—	10	1	no	—
290	C₂²⋊C₄	—	10	1	no	—
291	C₂²⋊C₄	—	10	1	no	—
292	C₄²	—	10	1	no	—
293	C₄²	C₃×C₄×D₅	1	1	yes	characteristic
294	C₄²	C₃×D₂₀	1	1	yes	characteristic
295	C₄²	C₃×C₂²⋊D₅	1	1	yes	characteristic
296	C₄²	C₂²×C₃×D₅	1	1	yes	characteristic
297	C₃×C₅	#1524	1	1	yes	characteristic, complemented, Hall
298	C₂²×C₃	—	2	1	no	—
299	C₂²×C₃	—	2	1	no	—
300	C₂²×C₃	—	20	1	no	—
301	C₂²×C₃	—	20	1	no	—
302	C₂²×C₃	—	20	1	no	—
303	C₂²×C₃	#3937	1	2	yes	—
304	C₂²×C₃	C₂×#685	1	1	yes	characteristic
305	C₃×C₄	—	2	1	no	—
306	C₃×C₄	—	2	1	no	—
307	C₃×C₄	—	2	1	no	—
308	C₃×C₄	—	2	1	no	—
309	C₃×C₄	—	2	1	no	—
310	C₃×C₄	—	20	1	no	—
311	C₃×C₄	—	20	1	no	—
312	C₃×C₄	—	20	1	no	—
313	C₃×C₄	#3957	1	2	yes	—
314	C₂×C₅	—	2	2	no	—
315	C₂×C₅	C₃×#413	1	1	yes	characteristic
316	C₂×C₅	C₂×C₃×#104	1	1	yes	characteristic
317	C₂×C₅	C₂×C₃×#104	1	1	yes	characteristic
318	D₅	—	8	1	no	—
319	C₂³	—	10	1	no	—
320	C₂³	C₃×#685	1	1	yes	characteristic
321	Q₈	—	20	1	no	—
322	D₄	—	20	1	no	—
323	D₄	—	20	2	no	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	2	2	no	—
326	C₂×C₄	—	4	1	no	—
327	C₂×C₄	—	10	1	no	—
328	C₂×C₄	—	10	1	no	—
329	C₂×C₄	—	10	1	no	—
330	C₂×C₄	—	20	2	no	—
331	C₂×C₄	C₃×#685	1	1	yes	characteristic
332	C₂×C₄	C₃×#685	1	1	yes	characteristic
333	C₂×C₄	C₃×#685	1	1	yes	characteristic
334	C₂×C₄	C₂×C₃×C₄×D₅	1	1	yes	characteristic
335	C₂×C₄	C₂×C₃×D₂₀	1	1	yes	characteristic
336	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
337	C₈	—	2	4	no	—
338	C₈	—	10	4	no	—
339	C₂×C₃	—	2	2	no	—
340	C₂×C₃	—	40	1	no	—
341	C₂×C₃	C₂×#3937	1	1	yes	characteristic
342	C₂×C₃	#65421	1	1	yes	characteristic
343	C₂×C₃	C₂×#3957	1	1	yes	characteristic
344	C₅	C₃×#1524	1	1	yes	characteristic, complemented, Sylow
345	C₂²	—	2	1	no	—
346	C₂²	—	2	1	no	—
347	C₂²	—	20	1	no	—
348	C₂²	—	20	1	no	—
349	C₂²	—	20	1	no	—
350	C₂²	C₃×#3937	1	2	yes	—
351	C₂²	C₂×C₃×#685	1	1	yes	characteristic
352	C₄	—	2	1	no	—
353	C₄	—	2	1	no	—
354	C₄	—	2	1	no	—
355	C₄	—	2	1	no	—
356	C₄	—	2	1	no	—
357	C₄	—	20	1	no	—
358	C₄	—	20	1	no	—
359	C₄	—	20	1	no	—
360	C₄	C₃×#3957	1	2	yes	—
361	C₃	#10600651	1	1	yes	characteristic, direct product factor, Sylow
362	C₂	—	2	2	no	—
363	C₂	—	40	1	no	—
364	C₂	C₂×C₃×#3937	1	1	yes	characteristic
365	C₂	C₃×#65421	1	1	yes	characteristic
366	C₂	C₂×C₃×#3957	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	#685
commutator	C₂×C₄×C₅	C₂²×C₃×C₄
Frattini	C₄²	C₂²×C₃×D₅
Fitting	C₃×C₄×C₅×M₁₆	C₂
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₃×#4076	C₄
cosocle	C₄²×C₅	C₂³×C₃
hypercenter	C₃×C₄×M₁₆	D₅
hypocenter	C₅	C₃×#1524
LCS	G ▷ C₂×C₄×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₄² ◁ C₃×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
2. C₃×C₄×C₅×M₁₆	C₂	semidirect
3. C₂×C₃×#3935	C₂	semidirect
4. C₃×#64685	C₂	semidirect
5. C₃×#64684	C₂	semidirect
6. #10600651	C₃	direct (N × H)
26. C₄×C₅×M₁₆	C₂×C₃	semidirect
27. C₂×#3935	C₂×C₃	semidirect
28. #64685	C₂×C₃	semidirect
29. #64684	C₂×C₃	semidirect
58. C₃×C₄×M₁₆	D₅	semidirect
145. C₄×M₁₆	C₃×D₅	semidirect
297. C₃×C₅	#1524	semidirect
344. C₅	C₃×#1524	semidirect
361. C₃	#10600651	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×#65813	2
C₃×C₄×C₅×M₁₆	2
C₂×C₃×#3935	2
C₃×#64685	2
C₃×#64684	2
#10600651	3
C₃×#1524	5

Subgroups — C3×#10600651 #14283923

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10600651 #14283923