Subgroups — C₃×#10600614 #14305885

Order 1920.

688 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#4021	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₅×#97	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×#3953	C₂	1	2	yes	maximal
4	C₂×C₃×#3948	C₂	1	2	yes	maximal, complemented
5	C₃×#65224	C₂	1	1	yes	maximal, characteristic, complemented
6	#10600614	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₃×D₂₀	C₂²	1	1	yes	characteristic
8	C₂²×C₃×Dic₁₀	C₂²	1	1	yes	characteristic
9	C₂²×C₃×C₅×C₈	C₂²	1	2	yes	—
10	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
11	C₂×C₃×#685	—	2	2	no	—
12	C₂×C₃×C₄⋊Dic₅	—	2	2	no	—
13	C₂×C₃×C₄⋊Dic₅	C₂²	1	2	yes	—
14	C₃×#4021	—	2	2	no	—
15	C₃×#4021	C₄	1	4	yes	complemented
16	C₃×C₅×#97	—	2	2	no	—
17	C₃×#3953	—	2	4	no	—
18	C₃×#3948	—	2	4	no	—
19	C₃×#1959	—	5	1	no	maximal, Hall
20	C₂×#4021	C₂×C₃	1	1	yes	characteristic
21	C₂×C₅×#97	C₂×C₃	1	1	yes	characteristic, complemented
22	C₂×#3953	C₂×C₃	1	2	yes	—
23	C₂×#3948	C₂×C₃	1	2	yes	complemented
24	#65224	C₂×C₃	1	1	yes	characteristic, complemented
25	C₂³×C₃×D₅	—	2	1	no	—
26	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
27	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
28	C₂²×C₃×Dic₅	—	2	1	no	—
29	C₂²×C₃×Dic₅	—	2	2	no	—
30	C₂×C₃×D₂₀	—	2	2	no	—
31	C₂×C₃×D₂₀	—	2	2	no	—
32	C₂×C₃×D₂₀	—	2	1	no	—
33	C₂×C₃×D₂₀	C₂×C₄	1	2	yes	—
34	C₂×C₃×Dic₁₀	—	2	2	no	—
35	C₂×C₃×Dic₁₀	—	2	2	no	—
36	C₂×C₃×Dic₁₀	—	2	1	no	—
37	C₂×C₃×Dic₁₀	C₂×C₄	1	2	yes	—
38	C₂×C₃×C₅×C₈	—	2	2	no	—
39	C₂×C₃×C₅×C₈	—	2	2	no	—
40	C₂×C₃×C₅×C₈	D₄	1	4	yes	—
41	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
42	C₃×C₄²×C₅	D₄	1	2	yes	—
43	C₃×#685	—	2	8	no	—
44	C₃×C₄⋊Dic₅	—	2	2	no	—
45	C₃×C₄⋊Dic₅	—	2	2	no	—
46	C₃×C₄⋊Dic₅	—	4	4	no	—
47	C₂×C₃×#124	—	5	1	no	—
48	C₂×C₃×#103	—	5	2	no	—
49	C₂×C₃×#102	—	5	2	no	—
50	C₂×C₃×#97	D₅	1	1	yes	characteristic, complemented
51	C₃×#384	—	5	1	no	—
52	C₂²×D₂₀	C₂²×C₃	1	1	yes	characteristic
53	C₂²×Dic₁₀	C₂²×C₃	1	1	yes	characteristic
54	C₂²×C₅×C₈	C₂²×C₃	1	2	yes	—
55	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
56	C₂×#685	—	2	2	no	—
57	C₂×C₄⋊Dic₅	—	2	2	no	—
58	C₂×C₄⋊Dic₅	C₂²×C₃	1	2	yes	—
59	#4021	—	2	2	no	—
60	#4021	C₃×C₄	1	4	yes	complemented
61	C₅×#97	—	2	2	no	—
62	#3953	—	2	4	no	—
63	#3948	—	2	4	no	—
64	#1959	—	5	1	no	Sylow
65	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₃×D₅	—	2	2	no	—
67	C₂²×C₃×D₅	—	4	1	no	—
68	C₂²×C₃×D₅	—	4	1	no	—
69	C₂²×C₃×D₅	—	4	2	no	—
70	C₂²×C₃×D₅	—	4	2	no	—
71	C₂×C₃×C₄×C₅	—	2	2	no	—
72	C₂×C₃×C₄×C₅	—	2	2	no	—
73	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	4	yes	—
74	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
76	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
77	C₂×C₃×C₄×C₅	#41	1	2	yes	—
78	C₂×C₃×Dic₅	—	2	2	no	—
79	C₂×C₃×Dic₅	—	4	1	no	—
80	C₂×C₃×Dic₅	—	4	1	no	—
81	C₂×C₃×Dic₅	—	4	2	no	—
82	C₂×C₃×Dic₅	—	4	4	no	—
83	C₃×D₂₀	—	2	4	no	—
84	C₃×D₂₀	—	4	2	no	—
85	C₃×Dic₁₀	—	2	4	no	—
86	C₃×Dic₁₀	—	4	2	no	—
87	C₃×C₅×C₈	—	2	4	no	—
88	C₂²×C₃×Q₈	—	5	1	no	—
89	C₂²×C₃×D₄	—	5	1	no	—
90	C₂²×C₃×C₈	C₂×D₅	1	2	yes	—
91	C₃×#124	—	5	4	no	—
92	C₃×#124	—	10	2	no	—
93	C₂×C₃×C₄⋊C₄	—	5	2	no	—
94	C₂×C₃×C₄⋊C₄	—	10	2	no	—
95	C₂×C₃×C₂²⋊C₄	—	10	2	no	—
96	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
97	C₃×#103	—	10	4	no	—
98	C₃×#102	—	10	4	no	—
99	C₃×#97	—	2	2	no	—
100	C₂³×D₅	—	2	1	no	—
101	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
102	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
103	C₂²×Dic₅	—	2	1	no	—
104	C₂²×Dic₅	—	2	2	no	—
105	C₂×D₂₀	—	2	2	no	—
106	C₂×D₂₀	—	2	2	no	—
107	C₂×D₂₀	—	2	1	no	—
108	C₂×D₂₀	C₂×C₃×C₄	1	2	yes	—
109	C₂×Dic₁₀	—	2	2	no	—
110	C₂×Dic₁₀	—	2	2	no	—
111	C₂×Dic₁₀	—	2	1	no	—
112	C₂×Dic₁₀	C₂×C₃×C₄	1	2	yes	—
113	C₂×C₅×C₈	—	2	2	no	—
114	C₂×C₅×C₈	—	2	2	no	—
115	C₂×C₅×C₈	C₃×D₄	1	4	yes	—
116	C₄²×C₅	C₂×C₃×C₄	1	2	yes	—
117	C₄²×C₅	C₃×D₄	1	2	yes	—
118	#685	—	2	8	no	—
119	C₄⋊Dic₅	—	2	2	no	—
120	C₄⋊Dic₅	—	2	2	no	—
121	C₄⋊Dic₅	—	4	4	no	—
122	C₂×#124	—	5	1	no	—
123	C₂×#103	—	5	2	no	—
124	C₂×#102	—	5	2	no	—
125	C₂×#97	C₃×D₅	1	1	yes	characteristic, complemented
126	#384	—	5	1	no	—
127	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
128	C₂²×C₃×C₅	#124	1	1	yes	characteristic
129	C₂²×C₃×C₅	#127	1	1	yes	characteristic
130	C₂²×C₃×C₅	AGL(1,ℤ/8)	1	2	yes	—
131	C₂²×C₃×C₅	#137	1	2	yes	—
132	C₂×C₃×D₅	—	4	2	no	—
133	C₂×C₃×D₅	—	4	4	no	—
134	C₂×C₃×D₅	—	8	2	no	—
135	C₂×C₃×D₅	—	8	1	no	—
136	C₂×C₃×D₅	—	8	1	no	—
137	C₃×C₄×C₅	—	2	4	no	—
138	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
139	C₃×C₄×C₅	#121	1	2	yes	—
140	C₃×Dic₅	—	4	2	no	—
141	C₃×Dic₅	—	8	2	no	—
142	C₂⁴×C₃	—	10	1	no	—
143	C₂×C₃×Q₈	—	5	2	no	—
144	C₂×C₃×Q₈	—	10	1	no	—
145	C₂×C₃×Q₈	—	10	2	no	—
146	C₂×C₃×Q₈	—	10	2	no	—
147	C₂×C₃×D₄	—	5	2	no	—
148	C₂×C₃×D₄	—	10	2	no	—
149	C₂×C₃×D₄	—	10	2	no	—
150	C₂×C₃×D₄	—	10	1	no	—
151	C₂²×C₃×C₄	—	10	1	no	—
152	C₂²×C₃×C₄	—	10	2	no	—
153	C₂²×C₃×C₄	D₂₀	1	2	yes	—
154	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
155	C₂×C₃×C₈	—	2	2	no	—
156	C₂×C₃×C₈	—	2	2	no	—
157	C₂×C₃×C₈	D₂₀	1	4	yes	—
158	C₃×C₄⋊C₄	—	10	2	no	—
159	C₃×C₄⋊C₄	—	10	2	no	—
160	C₃×C₄⋊C₄	—	20	4	no	—
161	C₃×C₂²⋊C₄	—	10	8	no	—
162	C₃×C₄²	C₄×D₅	1	2	yes	—
163	C₃×C₄²	C₂²⋊D₅	1	2	yes	—
164	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
165	C₂²×D₅	—	2	2	no	—
166	C₂²×D₅	—	4	1	no	—
167	C₂²×D₅	—	4	1	no	—
168	C₂²×D₅	—	4	2	no	—
169	C₂²×D₅	—	4	2	no	—
170	C₂×C₄×C₅	—	2	2	no	—
171	C₂×C₄×C₅	—	2	2	no	—
172	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	4	yes	—
173	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
174	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
175	C₂×C₄×C₅	C₂×C₃×D₄	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	—	4	1	no	—
180	C₂×Dic₅	—	4	2	no	—
181	C₂×Dic₅	—	4	4	no	—
182	D₂₀	—	2	4	no	—
183	D₂₀	—	4	2	no	—
184	Dic₁₀	—	2	4	no	—
185	Dic₁₀	—	4	2	no	—
186	C₅×C₈	—	2	4	no	—
187	C₂²×Q₈	—	5	1	no	—
188	C₂²×D₄	—	5	1	no	—
189	C₂²×C₈	C₂×C₃×D₅	1	2	yes	—
190	#124	—	5	4	no	—
191	#124	—	10	2	no	—
192	C₂×C₄⋊C₄	—	5	2	no	—
193	C₂×C₄⋊C₄	—	10	2	no	—
194	C₂×C₂²⋊C₄	—	10	2	no	—
195	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
196	#103	—	10	4	no	—
197	#102	—	10	4	no	—
198	#97	—	2	2	no	—
199	C₂×C₃×C₅	#390	1	1	yes	characteristic
200	C₂×C₃×C₅	#417	1	2	yes	—
201	C₂×C₃×C₅	#489	1	2	yes	—
202	C₂×C₃×C₅	#496	1	1	yes	characteristic
203	C₂×C₃×C₅	#497	1	1	yes	characteristic
204	C₃×D₅	—	8	2	no	—
205	C₂³×C₃	—	10	2	no	—
206	C₂³×C₃	—	20	1	no	—
207	C₂³×C₃	—	20	1	no	—
208	C₂³×C₃	—	20	2	no	—
209	C₂³×C₃	—	20	2	no	—
210	C₂³×C₃	C₂×D₂₀	1	1	yes	characteristic
211	C₃×Q₈	—	10	4	no	—
212	C₃×Q₈	—	20	2	no	—
213	C₃×D₄	—	10	4	no	—
214	C₃×D₄	—	20	2	no	—
215	C₂×C₃×C₄	—	2	2	no	—
216	C₂×C₃×C₄	—	2	2	no	—
217	C₂×C₃×C₄	—	10	2	no	—
218	C₂×C₃×C₄	—	20	1	no	—
219	C₂×C₃×C₄	—	20	1	no	—
220	C₂×C₃×C₄	—	20	2	no	—
221	C₂×C₃×C₄	—	20	4	no	—
222	C₂×C₃×C₄	#685	1	4	yes	—
223	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
224	C₂×C₃×C₄	C₂×D₂₀	1	2	yes	—
225	C₂×C₃×C₄	#709	1	2	yes	—
226	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
227	C₃×C₈	—	2	4	no	—
228	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
229	C₂²×C₅	C₃×#124	1	1	yes	characteristic
230	C₂²×C₅	C₃×#127	1	1	yes	characteristic
231	C₂²×C₅	C₃×AGL(1,ℤ/8)	1	2	yes	—
232	C₂²×C₅	C₃×#137	1	2	yes	—
233	C₂×D₅	—	4	2	no	—
234	C₂×D₅	—	4	4	no	—
235	C₂×D₅	—	8	2	no	—
236	C₂×D₅	—	8	1	no	—
237	C₂×D₅	—	8	1	no	—
238	C₄×C₅	—	2	4	no	—
239	C₄×C₅	C₃×C₄×D₄	1	2	yes	—
240	C₄×C₅	C₃×#121	1	2	yes	—
241	Dic₅	—	4	2	no	—
242	Dic₅	—	8	2	no	—
243	C₂⁴	—	10	1	no	—
244	C₂×Q₈	—	5	2	no	—
245	C₂×Q₈	—	10	1	no	—
246	C₂×Q₈	—	10	2	no	—
247	C₂×Q₈	—	10	2	no	—
248	C₂×D₄	—	5	2	no	—
249	C₂×D₄	—	10	2	no	—
250	C₂×D₄	—	10	2	no	—
251	C₂×D₄	—	10	1	no	—
252	C₂²×C₄	—	10	1	no	—
253	C₂²×C₄	—	10	2	no	—
254	C₂²×C₄	C₃×D₂₀	1	2	yes	—
255	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	C₃×D₂₀	1	4	yes	—
259	C₄⋊C₄	—	10	2	no	—
260	C₄⋊C₄	—	10	2	no	—
261	C₄⋊C₄	—	20	4	no	—
262	C₂²⋊C₄	—	10	8	no	—
263	C₄²	C₃×C₄×D₅	1	2	yes	—
264	C₄²	C₃×C₂²⋊D₅	1	2	yes	—
265	C₃×C₅	#1959	1	1	yes	characteristic, complemented, Hall
266	C₂²×C₃	—	20	4	no	—
267	C₂²×C₃	—	20	2	no	—
268	C₂²×C₃	—	40	1	no	—
269	C₂²×C₃	—	40	2	no	—
270	C₂²×C₃	—	40	1	no	—
271	C₂²×C₃	#4020	1	1	yes	characteristic
272	C₂²×C₃	#4021	1	1	yes	characteristic
273	C₂²×C₃	#4054	1	2	yes	—
274	C₂²×C₃	#4055	1	2	yes	—
275	C₂²×C₃	C₂×#685	1	1	yes	characteristic
276	C₃×C₄	—	2	4	no	—
277	C₃×C₄	—	20	2	no	—
278	C₃×C₄	—	40	2	no	—
279	C₃×C₄	C₄×D₂₀	1	2	yes	—
280	C₃×C₄	#4076	1	2	yes	—
281	C₂×C₅	C₃×#390	1	1	yes	characteristic
282	C₂×C₅	C₃×#417	1	2	yes	—
283	C₂×C₅	C₃×#489	1	2	yes	—
284	C₂×C₅	C₃×#496	1	1	yes	characteristic
285	C₂×C₅	C₃×#497	1	1	yes	characteristic
286	D₅	—	8	2	no	—
287	C₂³	—	10	2	no	—
288	C₂³	—	20	1	no	—
289	C₂³	—	20	1	no	—
290	C₂³	—	20	2	no	—
291	C₂³	—	20	2	no	—
292	C₂³	C₂×C₃×D₂₀	1	1	yes	characteristic
293	Q₈	—	10	4	no	—
294	Q₈	—	20	2	no	—
295	D₄	—	10	4	no	—
296	D₄	—	20	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	10	2	no	—
300	C₂×C₄	—	20	1	no	—
301	C₂×C₄	—	20	1	no	—
302	C₂×C₄	—	20	2	no	—
303	C₂×C₄	—	20	4	no	—
304	C₂×C₄	C₃×#685	1	4	yes	—
305	C₂×C₄	C₂×C₃×C₄×D₅	1	1	yes	characteristic
306	C₂×C₄	C₂×C₃×D₂₀	1	2	yes	—
307	C₂×C₄	C₃×#709	1	2	yes	—
308	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
309	C₈	—	2	4	no	—
310	C₂×C₃	—	40	2	no	—
311	C₂×C₃	#65001	1	1	yes	characteristic
312	C₂×C₃	#65003	1	2	yes	—
313	C₂×C₃	#65004	1	1	yes	characteristic
314	C₂×C₃	#65228	1	1	yes	characteristic
315	C₂×C₃	#65422	1	2	yes	—
316	C₅	C₃×#1959	1	1	yes	characteristic, complemented, Sylow
317	C₂²	—	20	4	no	—
318	C₂²	—	20	2	no	—
319	C₂²	—	40	1	no	—
320	C₂²	—	40	2	no	—
321	C₂²	—	40	1	no	—
322	C₂²	C₃×#4020	1	1	yes	characteristic
323	C₂²	C₃×#4021	1	1	yes	characteristic
324	C₂²	C₃×#4054	1	2	yes	—
325	C₂²	C₃×#4055	1	2	yes	—
326	C₂²	C₂×C₃×#685	1	1	yes	characteristic
327	C₄	—	2	4	no	—
328	C₄	—	20	2	no	—
329	C₄	—	40	2	no	—
330	C₄	C₃×C₄×D₂₀	1	2	yes	—
331	C₄	C₃×#4076	1	2	yes	—
332	C₃	#10600614	1	1	yes	characteristic, direct product factor, Sylow
333	C₂	—	40	2	no	—
334	C₂	C₃×#65001	1	1	yes	characteristic
335	C₂	C₃×#65003	1	2	yes	—
336	C₂	C₃×#65004	1	1	yes	characteristic
337	C₂	C₃×#65228	1	1	yes	characteristic
338	C₂	C₃×#65422	1	2	yes	—

	subgroup	G/H
center	C₂³×C₃	C₂×D₂₀
commutator	C₂×C₄×C₅	C₂²×C₃×C₄
Frattini	C₂²×C₄	C₂²×C₃×D₅
Fitting	C₂×C₃×C₅×#97	C₂
socle	C₂³×C₃×C₅	C₂×D₄
prime radical	C₂²×C₃×D₂₀	C₂²
cosocle	C₂²×C₄×C₅	C₂³×C₃
hypercenter	C₂×C₃×#97	D₅
hypocenter	C₅	C₃×#1959
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄² ◁ C₂×C₃×#97

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₅×#97	C₂	semidirect
4. C₂×C₃×#3948	C₂	semidirect
5. C₃×#65224	C₂	semidirect
6. #10600614	C₃	direct (N × H)
15. C₃×#4021	C₄	semidirect
21. C₂×C₅×#97	C₂×C₃	semidirect
23. C₂×#3948	C₂×C₃	semidirect
24. #65224	C₂×C₃	semidirect
50. C₂×C₃×#97	D₅	semidirect
60. #4021	C₃×C₄	semidirect
125. C₂×#97	C₃×D₅	semidirect
265. C₃×C₅	#1959	semidirect
316. C₅	C₃×#1959	semidirect
332. C₃	#10600614	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×#4021	2
C₂×C₃×C₅×#97	2
C₂×C₃×#3953	2
C₂×C₃×#3948	2
C₃×#65224	2
#10600614	3
C₃×#1959	5

Subgroups — C3×#10600614 #14305885

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10600614 #14305885