Subgroups — C₃×#10610269 #14439214

Order 1920.

1104 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#4055	C₂	1	4	yes	maximal, complemented
2	C₃×#65811	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#4015	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₄×C₅×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#65004	C₂	1	4	yes	maximal, complemented
6	C₃×#64988	C₂	1	2	yes	maximal, complemented
7	C₃×#64982	C₂	1	2	yes	maximal, complemented
8	#10610269	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂×C₃×#709	C₂²	1	2	yes	—
10	C₂²×C₃×Dic₁₀	C₂²	1	2	yes	—
11	C₂×C₃×C₅×M₁₆	C₂²	1	2	yes	—
12	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
13	C₃×#4076	—	2	2	no	—
14	C₂×C₃×C₄⋊Dic₅	—	2	2	no	—
15	C₃×#4055	—	2	16	no	—
16	C₂×C₃×Dic₂₀	C₂²	1	8	yes	—
17	C₂×C₃×#677	C₂²	1	8	yes	—
18	C₃×#4021	C₂²	1	2	yes	—
19	C₃×#4020	C₂²	1	1	yes	characteristic
20	C₃×C₄×D₂₀	—	2	2	no	—
21	C₃×#4015	—	2	2	no	—
22	C₃×#4015	C₂²	1	2	yes	—
23	C₃×#4015	C₂²	1	2	yes	—
24	C₃×#4015	C₂²	1	1	yes	characteristic
25	C₃×C₅×#97	C₂²	1	2	yes	—
26	C₃×C₄×C₅×C₈	C₂²	1	2	yes	—
27	C₃×#3152	—	5	1	no	maximal, Hall
28	C₂×#4055	C₂×C₃	1	4	yes	complemented
29	#65811	C₂×C₃	1	1	yes	characteristic
30	C₂×#4015	C₂×C₃	1	1	yes	characteristic, complemented
31	C₄×C₅×M₁₆	C₂×C₃	1	1	yes	characteristic, complemented
32	#65004	C₂×C₃	1	4	yes	complemented
33	#64988	C₂×C₃	1	2	yes	complemented
34	#64982	C₂×C₃	1	2	yes	complemented
35	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
36	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
37	C₂×C₃×C₂²⋊D₅	—	2	2	no	—
38	C₂²×C₃×Dic₅	—	2	2	no	—
39	C₃×#709	—	2	8	no	—
40	C₂×C₃×D₂₀	—	2	2	no	—
41	C₂×C₃×D₂₀	C₂³	1	2	yes	—
42	C₂×C₃×C₄×D₅	—	2	2	no	—
43	C₂×C₃×Dic₁₀	—	2	8	no	—
44	C₂×C₃×Dic₁₀	—	2	2	no	—
45	C₂×C₃×Dic₁₀	C₂³	1	4	yes	—
46	C₂×C₃×Dic₁₀	C₂³	1	2	yes	—
47	C₃×C₅×M₁₆	D₄	1	8	yes	complemented
48	C₂×C₃×C₅×C₈	C₂³	1	4	yes	—
49	C₃×C₄²×C₅	D₄	1	2	yes	—
50	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
51	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
52	C₃×#685	—	2	4	no	—
53	C₃×C₄⋊Dic₅	—	2	4	no	—
54	C₃×C₄⋊Dic₅	—	2	4	no	—
55	C₃×C₄⋊Dic₅	—	2	2	no	—
56	C₃×Dic₂₀	—	2	16	no	—
57	C₃×#677	—	2	16	no	—
58	C₂×C₃×#137	—	5	4	no	—
59	C₃×#532	—	5	1	no	—
60	C₂×C₃×C₄⋊Q₈	—	5	1	no	—
61	C₃×#497	—	5	4	no	—
62	C₃×#494	—	5	2	no	—
63	C₃×#492	—	5	2	no	—
64	C₃×C₄×M₁₆	D₅	1	1	yes	characteristic, complemented
65	C₂×#709	C₂²×C₃	1	2	yes	—
66	C₂²×Dic₁₀	C₂²×C₃	1	2	yes	—
67	C₂×C₅×M₁₆	C₂²×C₃	1	2	yes	—
68	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
69	#4076	—	2	2	no	—
70	C₂×C₄⋊Dic₅	—	2	2	no	—
71	#4055	—	2	16	no	—
72	C₂×Dic₂₀	C₂²×C₃	1	8	yes	—
73	C₂×#677	C₂²×C₃	1	8	yes	—
74	#4021	C₂²×C₃	1	2	yes	—
75	#4020	C₂²×C₃	1	1	yes	characteristic
76	C₄×D₂₀	—	2	2	no	—
77	#4015	—	2	2	no	—
78	#4015	C₂²×C₃	1	2	yes	—
79	#4015	C₂²×C₃	1	2	yes	—
80	#4015	C₂²×C₃	1	1	yes	characteristic
81	C₅×#97	C₂²×C₃	1	2	yes	—
82	C₄×C₅×C₈	C₂²×C₃	1	2	yes	—
83	#3152	—	5	1	no	Sylow
84	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₃×D₅	—	2	2	no	—
86	C₂×C₃×C₄×C₅	—	2	4	no	—
87	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
88	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
89	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
90	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
91	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
92	C₃×C₂²⋊D₅	—	4	4	no	—
93	C₂×C₃×Dic₅	—	2	4	no	—
94	C₂×C₃×Dic₅	—	2	2	no	—
95	C₂×C₃×Dic₅	—	4	4	no	—
96	C₃×D₂₀	—	2	4	no	—
97	C₃×D₂₀	—	4	4	no	—
98	C₃×C₄×D₅	—	4	4	no	—
99	C₃×Dic₁₀	—	2	8	no	—
100	C₃×Dic₁₀	—	2	4	no	—
101	C₃×Dic₁₀	—	4	4	no	—
102	C₃×C₅×C₈	C₂×D₄	1	8	yes	—
103	C₂×C₃×#41	—	5	2	no	—
104	C₂²×C₃×Q₈	—	5	2	no	—
105	C₃×#137	—	10	16	no	—
106	C₂×C₃×Q₁₆	—	5	8	no	—
107	C₂×C₃×SD₁₆	—	5	8	no	—
108	C₂×C₃×M₁₆	C₂×D₅	1	2	yes	—
109	C₃×C₄⋊Q₈	—	5	2	no	—
110	C₃×C₄⋊Q₈	—	5	2	no	—
111	C₃×C₄⋊Q₈	—	5	1	no	—
112	C₃×C₄⋊Q₈	—	10	2	no	—
113	C₃×#127	—	5	1	no	—
114	C₃×#124	—	5	2	no	—
115	C₃×#121	—	10	2	no	—
116	C₃×C₄×D₄	—	10	2	no	—
117	C₂×C₃×C₄⋊C₄	—	10	2	no	—
118	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
119	C₃×#97	C₂×D₅	1	2	yes	—
120	C₃×C₄×C₈	C₂×D₅	1	2	yes	—
121	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
122	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
123	C₂×C₂²⋊D₅	—	2	2	no	—
124	C₂²×Dic₅	—	2	2	no	—
125	#709	—	2	8	no	—
126	C₂×D₂₀	—	2	2	no	—
127	C₂×D₂₀	C₂³×C₃	1	2	yes	—
128	C₂×C₄×D₅	—	2	2	no	—
129	C₂×Dic₁₀	—	2	8	no	—
130	C₂×Dic₁₀	—	2	2	no	—
131	C₂×Dic₁₀	C₂³×C₃	1	4	yes	—
132	C₂×Dic₁₀	C₂³×C₃	1	2	yes	—
133	C₅×M₁₆	C₃×D₄	1	8	yes	complemented
134	C₂×C₅×C₈	C₂³×C₃	1	4	yes	—
135	C₄²×C₅	C₃×D₄	1	2	yes	—
136	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
137	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
138	#685	—	2	4	no	—
139	C₄⋊Dic₅	—	2	4	no	—
140	C₄⋊Dic₅	—	2	4	no	—
141	C₄⋊Dic₅	—	2	2	no	—
142	Dic₂₀	—	2	16	no	—
143	#677	—	2	16	no	—
144	C₂×#137	—	5	4	no	—
145	#532	—	5	1	no	—
146	C₂×C₄⋊Q₈	—	5	1	no	—
147	#497	—	5	4	no	—
148	#494	—	5	2	no	—
149	#492	—	5	2	no	—
150	C₄×M₁₆	C₃×D₅	1	1	yes	characteristic, complemented
151	C₂²×C₃×C₅	—	2	2	no	—
152	C₂²×C₃×C₅	#127	1	2	yes	—
153	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
154	C₂×C₃×D₅	—	4	2	no	—
155	C₂×C₃×D₅	—	4	4	no	—
156	C₃×C₄×C₅	—	2	2	no	—
157	C₃×C₄×C₅	#127	1	2	yes	—
158	C₃×C₄×C₅	#137	1	4	yes	complemented
159	C₃×C₄×C₅	C₂²×D₄	1	2	yes	—
160	C₃×Dic₅	—	4	4	no	—
161	C₃×Dic₅	—	4	2	no	—
162	C₃×#41	—	10	8	no	—
163	C₂×C₃×Q₈	—	5	2	no	—
164	C₂×C₃×Q₈	—	5	4	no	—
165	C₂×C₃×Q₈	—	10	2	no	—
166	C₂×C₃×Q₈	—	10	8	no	—
167	C₂×C₃×D₄	—	5	2	no	—
168	C₂×C₃×D₄	—	10	2	no	—
169	C₂×C₃×D₄	—	10	2	no	—
170	C₂²×C₃×C₄	—	10	2	no	—
171	C₂²×C₃×C₄	—	10	2	no	—
172	C₂²×C₃×C₄	D₂₀	1	2	yes	—
173	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
174	C₃×Q₁₆	—	10	16	no	—
175	C₃×SD₁₆	—	10	16	no	—
176	C₃×M₁₆	D₂₀	1	8	yes	complemented
177	C₂×C₃×C₈	C₂²×D₅	1	4	yes	—
178	C₃×C₄⋊C₄	—	10	4	no	—
179	C₃×C₄⋊C₄	—	10	4	no	—
180	C₃×C₄⋊C₄	—	10	2	no	—
181	C₃×C₂²⋊C₄	—	10	4	no	—
182	C₃×C₄²	D₂₀	1	2	yes	—
183	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
184	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
185	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
186	C₂²×D₅	—	2	2	no	—
187	C₂×C₄×C₅	—	2	4	no	—
188	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
189	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
190	C₂×C₄×C₅	C₂×C₃×D₄	1	4	yes	—
191	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
192	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
193	C₂²⋊D₅	—	4	4	no	—
194	C₂×Dic₅	—	2	4	no	—
195	C₂×Dic₅	—	2	2	no	—
196	C₂×Dic₅	—	4	4	no	—
197	D₂₀	—	2	4	no	—
198	D₂₀	—	4	4	no	—
199	C₄×D₅	—	4	4	no	—
200	Dic₁₀	—	2	8	no	—
201	Dic₁₀	—	2	4	no	—
202	Dic₁₀	—	4	4	no	—
203	C₅×C₈	C₂×C₃×D₄	1	8	yes	—
204	C₂×#41	—	5	2	no	—
205	C₂²×Q₈	—	5	2	no	—
206	#137	—	10	16	no	—
207	C₂×Q₁₆	—	5	8	no	—
208	C₂×SD₁₆	—	5	8	no	—
209	C₂×M₁₆	C₂×C₃×D₅	1	2	yes	—
210	C₄⋊Q₈	—	5	2	no	—
211	C₄⋊Q₈	—	5	2	no	—
212	C₄⋊Q₈	—	5	1	no	—
213	C₄⋊Q₈	—	10	2	no	—
214	#127	—	5	1	no	—
215	#124	—	5	2	no	—
216	#121	—	10	2	no	—
217	C₄×D₄	—	10	2	no	—
218	C₂×C₄⋊C₄	—	10	2	no	—
219	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
220	#97	C₂×C₃×D₅	1	2	yes	—
221	C₄×C₈	C₂×C₃×D₅	1	2	yes	—
222	C₂×C₃×C₅	—	2	2	no	—
223	C₂×C₃×C₅	C₂×#127	1	1	yes	characteristic
224	C₂×C₃×C₅	C₂×#137	1	2	yes	—
225	C₃×D₅	—	8	2	no	—
226	C₂³×C₃	—	10	2	no	—
227	C₂³×C₃	C₂×D₂₀	1	1	yes	characteristic
228	C₃×Q₈	—	10	8	no	—
229	C₃×Q₈	—	10	4	no	—
230	C₃×Q₈	—	20	4	no	—
231	C₃×D₄	—	10	4	no	—
232	C₃×D₄	—	20	4	no	—
233	C₃×D₄	—	20	4	no	—
234	C₂×C₃×C₄	—	2	4	no	—
235	C₂×C₃×C₄	—	10	2	no	—
236	C₂×C₃×C₄	—	10	4	no	—
237	C₂×C₃×C₄	—	20	4	no	—
238	C₂×C₃×C₄	—	20	4	no	—
239	C₂×C₃×C₄	C₂×D₂₀	1	2	yes	—
240	C₂×C₃×C₄	C₂×D₂₀	1	4	yes	—
241	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
242	C₂×C₃×C₄	C₂×D₂₀	1	2	yes	—
243	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
244	C₃×C₈	C₂×D₂₀	1	8	yes	—
245	C₂²×C₅	—	2	2	no	—
246	C₂²×C₅	C₃×#127	1	2	yes	—
247	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
248	C₂×D₅	—	4	4	no	—
249	C₂×D₅	—	4	2	no	—
250	C₄×C₅	—	2	2	no	—
251	C₄×C₅	C₃×#127	1	2	yes	—
252	C₄×C₅	C₃×#137	1	4	yes	complemented
253	C₄×C₅	C₂²×C₃×D₄	1	2	yes	—
254	Dic₅	—	4	4	no	—
255	Dic₅	—	4	2	no	—
256	#41	—	10	8	no	—
257	C₂×Q₈	—	5	2	no	—
258	C₂×Q₈	—	5	4	no	—
259	C₂×Q₈	—	10	2	no	—
260	C₂×Q₈	—	10	8	no	—
261	C₂×D₄	—	5	2	no	—
262	C₂×D₄	—	10	2	no	—
263	C₂×D₄	—	10	2	no	—
264	C₂²×C₄	—	10	2	no	—
265	C₂²×C₄	—	10	2	no	—
266	C₂²×C₄	C₃×D₂₀	1	2	yes	—
267	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
268	Q₁₆	—	10	16	no	—
269	SD₁₆	—	10	16	no	—
270	M₁₆	C₃×D₂₀	1	8	yes	complemented
271	C₂×C₈	C₂²×C₃×D₅	1	4	yes	—
272	C₄⋊C₄	—	10	4	no	—
273	C₄⋊C₄	—	10	4	no	—
274	C₄⋊C₄	—	10	2	no	—
275	C₂²⋊C₄	—	10	4	no	—
276	C₄²	C₃×D₂₀	1	2	yes	—
277	C₄²	C₂²×C₃×D₅	1	1	yes	characteristic
278	C₄²	C₂²×C₃×D₅	1	1	yes	characteristic
279	C₃×C₅	#3152	1	1	yes	characteristic, complemented, Hall
280	C₂²×C₃	—	2	2	no	—
281	C₂²×C₃	—	20	2	no	—
282	C₂²×C₃	—	20	4	no	—
283	C₂²×C₃	#4020	1	2	yes	—
284	C₂²×C₃	C₂²×D₂₀	1	1	yes	characteristic
285	C₃×C₄	—	2	2	no	—
286	C₃×C₄	—	20	2	no	—
287	C₃×C₄	—	20	4	no	—
288	C₃×C₄	#4020	1	2	yes	—
289	C₃×C₄	#4055	1	4	yes	complemented
290	C₃×C₄	C₂²×D₂₀	1	2	yes	—
291	C₂×C₅	—	2	2	no	—
292	C₂×C₅	C₂×C₃×#127	1	1	yes	characteristic
293	C₂×C₅	C₂×C₃×#137	1	2	yes	—
294	D₅	—	8	2	no	—
295	C₂³	—	10	2	no	—
296	C₂³	C₂×C₃×D₂₀	1	1	yes	characteristic
297	Q₈	—	10	8	no	—
298	Q₈	—	10	4	no	—
299	Q₈	—	20	4	no	—
300	D₄	—	10	4	no	—
301	D₄	—	20	4	no	—
302	D₄	—	20	4	no	—
303	C₂×C₄	—	2	4	no	—
304	C₂×C₄	—	10	2	no	—
305	C₂×C₄	—	10	4	no	—
306	C₂×C₄	—	20	4	no	—
307	C₂×C₄	—	20	4	no	—
308	C₂×C₄	C₂×C₃×D₂₀	1	2	yes	—
309	C₂×C₄	C₂×C₃×D₂₀	1	4	yes	—
310	C₂×C₄	C₂×C₃×D₂₀	1	1	yes	characteristic
311	C₂×C₄	C₂×C₃×D₂₀	1	2	yes	—
312	C₂×C₄	C₂³×C₃×D₅	1	1	yes	characteristic
313	C₈	C₂×C₃×D₂₀	1	8	yes	—
314	C₂×C₃	—	2	2	no	—
315	C₂×C₃	—	40	2	no	—
316	C₂×C₃	C₂×#4020	1	1	yes	characteristic
317	C₂×C₃	C₂×#4055	1	2	yes	—
318	C₅	C₃×#3152	1	1	yes	characteristic, complemented, Sylow
319	C₂²	—	2	2	no	—
320	C₂²	—	20	2	no	—
321	C₂²	—	20	4	no	—
322	C₂²	C₃×#4020	1	2	yes	—
323	C₂²	C₂²×C₃×D₂₀	1	1	yes	characteristic
324	C₄	—	2	2	no	—
325	C₄	—	20	2	no	—
326	C₄	—	20	4	no	—
327	C₄	C₃×#4020	1	2	yes	—
328	C₄	C₃×#4055	1	4	yes	complemented
329	C₄	C₂²×C₃×D₂₀	1	2	yes	—
330	C₃	#10610269	1	1	yes	characteristic, direct product factor, Sylow
331	C₂	—	2	2	no	—
332	C₂	—	40	2	no	—
333	C₂	C₂×C₃×#4020	1	1	yes	characteristic
334	C₂	C₂×C₃×#4055	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×C₄×C₅×M₁₆	C₂
socle	C₂²×C₃×C₅	C₂²×D₄
prime radical	C₃×#65811	C₂
hypercenter	C₃×C₄×M₁₆	D₅
hypocenter	C₅	C₃×#3152
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄² ◁ C₃×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×#4055	C₂	semidirect
3. C₂×C₃×#4015	C₂	semidirect
4. C₃×C₄×C₅×M₁₆	C₂	semidirect
5. C₃×#65004	C₂	semidirect
6. C₃×#64988	C₂	semidirect
7. C₃×#64982	C₂	semidirect
8. #10610269	C₃	direct (N × H)
28. C₂×#4055	C₂×C₃	semidirect
30. C₂×#4015	C₂×C₃	semidirect
31. C₄×C₅×M₁₆	C₂×C₃	semidirect
32. #65004	C₂×C₃	semidirect
33. #64988	C₂×C₃	semidirect
34. #64982	C₂×C₃	semidirect
47. C₃×C₅×M₁₆	D₄	semidirect
64. C₃×C₄×M₁₆	D₅	semidirect
133. C₅×M₁₆	C₃×D₄	semidirect
150. C₄×M₁₆	C₃×D₅	semidirect
158. C₃×C₄×C₅	#137	semidirect
176. C₃×M₁₆	D₂₀	semidirect
252. C₄×C₅	C₃×#137	semidirect
270. M₁₆	C₃×D₂₀	semidirect
279. C₃×C₅	#3152	semidirect
289. C₃×C₄	#4055	semidirect
318. C₅	C₃×#3152	semidirect
328. C₄	C₃×#4055	semidirect
330. C₃	#10610269	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×#4055	2
C₃×#65811	2
C₂×C₃×#4015	2
C₃×C₄×C₅×M₁₆	2
C₃×#65004	2
C₃×#64988	2
C₃×#64982	2
#10610269	3
C₃×#3152	5

Subgroups — C3×#10610269 #14439214

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10610269 #14439214