Subgroups — C₃×#10600286 #14307124

Order 1920.

520 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#65804	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×C₅×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₃×C₄×#681	C₂	1	1	yes	maximal, characteristic
4	C₃×#64743	C₂	1	4	yes	maximal
5	#10600286	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂×C₃×C₅×C₄⋊C₄	—	2	1	no	—
7	C₂×C₃×C₄²×C₅	C₂²	1	2	yes	—
8	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
9	C₃×#4072	C₂²	1	2	yes	—
10	C₃×#4070	—	2	1	no	—
11	C₃×#4070	—	2	1	no	—
12	C₂×C₃×#681	C₂²	1	2	yes	—
13	C₃×#4016	C₄	1	2	yes	complemented
14	C₃×#4015	C₄	1	1	yes	characteristic, complemented
15	C₃×#4015	C₄	1	1	yes	characteristic, complemented
16	C₃×C₄×Dic₁₀	—	2	1	no	—
17	C₃×C₄×Dic₁₀	—	2	1	no	—
18	C₃×C₅×#95	—	2	1	no	—
19	C₃×#3935	—	2	1	no	—
20	C₃×C₄×#168	—	2	1	no	—
21	C₃×#1986	—	5	1	no	maximal, Hall
22	#65804	C₂×C₃	1	1	yes	characteristic
23	C₄×C₅×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
24	C₄×#681	C₂×C₃	1	1	yes	characteristic
25	#64743	C₂×C₃	1	4	yes	—
26	C₂²×C₃×C₄×C₅	—	2	2	no	—
27	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
28	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
29	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
30	C₂×C₃×Dic₁₀	—	2	1	no	—
31	C₂×C₃×Dic₁₀	—	2	1	no	—
32	C₃×C₅×C₄⋊C₄	—	2	4	no	—
33	C₃×C₄²×C₅	—	2	4	no	—
34	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
35	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
36	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
37	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
38	C₃×C₂²⋊Dic₅	—	2	2	no	—
39	C₃×C₄⋊Dic₅	—	2	1	no	—
40	C₃×C₄⋊Dic₅	—	2	1	no	—
41	C₃×C₄⋊Dic₅	C₂×C₄	1	2	yes	—
42	C₃×C₄⋊Dic₅	C₂×C₄	1	2	yes	—
43	C₃×#683	—	2	2	no	—
44	C₃×#683	—	2	2	no	—
45	C₃×C₄×Dic₅	—	2	2	no	—
46	C₃×#681	—	2	2	no	—
47	C₃×#681	Q₈	1	4	yes	complemented
48	C₂×C₃×#168	—	2	2	no	—
49	C₃×#533	—	5	1	no	—
50	C₃×C₄×M₁₆	—	5	1	no	—
51	C₃×C₄×C₄⋊C₄	D₅	1	1	yes	characteristic
52	C₃×#339	—	5	4	no	—
53	C₂×C₅×C₄⋊C₄	—	2	1	no	—
54	C₂×C₄²×C₅	C₂²×C₃	1	2	yes	—
55	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
56	#4072	C₂²×C₃	1	2	yes	—
57	#4070	—	2	1	no	—
58	#4070	—	2	1	no	—
59	C₂×#681	C₂²×C₃	1	2	yes	—
60	#4016	C₃×C₄	1	2	yes	complemented
61	#4015	C₃×C₄	1	1	yes	characteristic, complemented
62	#4015	C₃×C₄	1	1	yes	characteristic, complemented
63	C₄×Dic₁₀	—	2	1	no	—
64	C₄×Dic₁₀	—	2	1	no	—
65	C₅×#95	—	2	1	no	—
66	#3935	—	2	1	no	—
67	C₄×#168	—	2	1	no	—
68	#1986	—	5	1	no	Sylow
69	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
70	C₂×C₃×C₄×C₅	—	2	2	no	—
71	C₂×C₃×C₄×C₅	—	2	4	no	—
72	C₂×C₃×C₄×C₅	—	4	1	no	—
73	C₂×C₃×C₄×C₅	—	4	2	no	—
74	C₂×C₃×C₄×C₅	—	4	2	no	—
75	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
81	C₂×C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
82	C₂×C₃×C₄×C₅	#41	1	2	yes	—
83	C₂×C₃×Dic₅	—	2	2	no	—
84	C₂×C₃×Dic₅	—	2	2	no	—
85	C₃×Dic₁₀	—	4	2	no	—
86	C₃×Dic₁₀	—	4	2	no	—
87	C₃×#168	—	2	4	no	—
88	C₂×C₃×M₁₆	—	5	2	no	—
89	C₃×C₄⋊Q₈	—	5	1	no	—
90	C₃×C₄⋊Q₈	—	5	1	no	—
91	C₃×#125	—	5	2	no	—
92	C₃×C₂²⋊Q₈	—	10	1	no	—
93	C₃×C₂²⋊Q₈	—	10	1	no	—
94	C₃×C₄×Q₈	—	10	1	no	—
95	C₃×C₄×Q₈	—	10	1	no	—
96	C₃×#117	—	5	2	no	—
97	C₂×C₃×C₄⋊C₄	—	2	1	no	—
98	C₂×C₃×C₄²	C₂×D₅	1	2	yes	—
99	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
100	C₃×#97	—	10	1	no	—
101	C₃×C₄×C₈	—	10	1	no	—
102	C₃×#95	—	2	1	no	—
103	C₂²×C₄×C₅	—	2	2	no	—
104	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
105	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
106	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
107	C₂×Dic₁₀	—	2	1	no	—
108	C₂×Dic₁₀	—	2	1	no	—
109	C₅×C₄⋊C₄	—	2	4	no	—
110	C₄²×C₅	—	2	4	no	—
111	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
112	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
113	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
114	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
115	C₂²⋊Dic₅	—	2	2	no	—
116	C₄⋊Dic₅	—	2	1	no	—
117	C₄⋊Dic₅	—	2	1	no	—
118	C₄⋊Dic₅	C₂×C₃×C₄	1	2	yes	—
119	C₄⋊Dic₅	C₂×C₃×C₄	1	2	yes	—
120	#683	—	2	2	no	—
121	#683	—	2	2	no	—
122	C₄×Dic₅	—	2	2	no	—
123	#681	—	2	2	no	—
124	#681	C₃×Q₈	1	4	yes	complemented
125	C₂×#168	—	2	2	no	—
126	#533	—	5	1	no	—
127	C₄×M₁₆	—	5	1	no	—
128	C₄×C₄⋊C₄	C₃×D₅	1	1	yes	characteristic
129	#339	—	5	4	no	—
130	C₂²×C₃×C₅	—	2	1	no	—
131	C₂²×C₃×C₅	—	2	1	no	—
132	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
133	C₂²×C₃×C₅	#124	1	1	yes	characteristic
134	C₂²×C₃×C₅	C₄⋊Q₈	1	1	yes	characteristic
135	C₃×C₄×C₅	—	2	1	no	—
136	C₃×C₄×C₅	—	2	1	no	—
137	C₃×C₄×C₅	—	2	1	no	—
138	C₃×C₄×C₅	—	4	4	no	—
139	C₃×C₄×C₅	#104	1	2	yes	—
140	C₃×C₄×C₅	C₄×Q₈	1	2	yes	—
141	C₃×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
142	C₃×Dic₅	—	4	2	no	—
143	C₃×Dic₅	—	4	2	no	—
144	C₂×C₃×Q₈	—	10	1	no	—
145	C₂×C₃×Q₈	—	10	1	no	—
146	C₂²×C₃×C₄	—	2	2	no	—
147	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
148	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
149	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
150	C₃×M₁₆	—	5	4	no	—
151	C₃×M₁₆	—	10	2	no	—
152	C₂×C₃×C₈	—	10	2	no	—
153	C₃×C₄⋊C₄	—	2	4	no	—
154	C₃×C₄⋊C₄	—	5	2	no	—
155	C₃×C₄⋊C₄	—	5	2	no	—
156	C₃×C₄⋊C₄	—	10	2	no	—
157	C₃×C₄⋊C₄	—	10	2	no	—
158	C₃×C₄⋊C₄	—	10	1	no	—
159	C₃×C₄⋊C₄	—	10	1	no	—
160	C₃×C₂²⋊C₄	—	10	2	no	—
161	C₃×C₄²	—	2	4	no	—
162	C₃×C₄²	—	10	2	no	—
163	C₃×C₄²	C₄×D₅	1	1	yes	characteristic
164	C₃×C₄²	C₄×D₅	1	1	yes	characteristic
165	C₃×C₄²	D₂₀	1	1	yes	characteristic
166	C₃×C₄²	D₂₀	1	1	yes	characteristic
167	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
168	C₂×C₄×C₅	—	2	2	no	—
169	C₂×C₄×C₅	—	2	4	no	—
170	C₂×C₄×C₅	—	4	1	no	—
171	C₂×C₄×C₅	—	4	2	no	—
172	C₂×C₄×C₅	—	4	2	no	—
173	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
174	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
175	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
176	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
177	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
178	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
179	C₂×C₄×C₅	C₂×C₃×Q₈	1	2	yes	—
180	C₂×C₄×C₅	C₃×#41	1	2	yes	—
181	C₂×Dic₅	—	2	2	no	—
182	C₂×Dic₅	—	2	2	no	—
183	Dic₁₀	—	4	2	no	—
184	Dic₁₀	—	4	2	no	—
185	#168	—	2	4	no	—
186	C₂×M₁₆	—	5	2	no	—
187	C₄⋊Q₈	—	5	1	no	—
188	C₄⋊Q₈	—	5	1	no	—
189	#125	—	5	2	no	—
190	C₂²⋊Q₈	—	10	1	no	—
191	C₂²⋊Q₈	—	10	1	no	—
192	C₄×Q₈	—	10	1	no	—
193	C₄×Q₈	—	10	1	no	—
194	#117	—	5	2	no	—
195	C₂×C₄⋊C₄	—	2	1	no	—
196	C₂×C₄²	C₂×C₃×D₅	1	2	yes	—
197	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
198	#97	—	10	1	no	—
199	C₄×C₈	—	10	1	no	—
200	#95	—	2	1	no	—
201	C₂×C₃×C₅	—	2	1	no	—
202	C₂×C₃×C₅	—	2	1	no	—
203	C₂×C₃×C₅	#391	1	1	yes	characteristic
204	C₂×C₃×C₅	C₂×#104	1	1	yes	characteristic
205	C₂×C₃×C₅	#421	1	1	yes	characteristic
206	C₂³×C₃	C₂×C₂²⋊D₅	1	1	yes	characteristic
207	C₃×Q₈	—	20	2	no	—
208	C₃×Q₈	—	20	2	no	—
209	C₂×C₃×C₄	—	2	2	no	—
210	C₂×C₃×C₄	—	2	4	no	—
211	C₂×C₃×C₄	—	4	1	no	—
212	C₂×C₃×C₄	—	4	2	no	—
213	C₂×C₃×C₄	—	4	2	no	—
214	C₂×C₃×C₄	—	10	2	no	—
215	C₂×C₃×C₄	—	10	2	no	—
216	C₂×C₃×C₄	#685	1	1	yes	characteristic
217	C₂×C₃×C₄	#685	1	1	yes	characteristic
218	C₂×C₃×C₄	#685	1	1	yes	characteristic
219	C₂×C₃×C₄	#685	1	1	yes	characteristic
220	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
221	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
222	C₂×C₃×C₄	#711	1	2	yes	—
223	C₂×C₃×C₄	Q₈×D₅	1	2	yes	—
224	C₃×C₈	—	10	4	no	—
225	C₂²×C₅	—	2	1	no	—
226	C₂²×C₅	—	2	1	no	—
227	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
228	C₂²×C₅	C₃×#124	1	1	yes	characteristic
229	C₂²×C₅	C₃×C₄⋊Q₈	1	1	yes	characteristic
230	C₄×C₅	—	2	1	no	—
231	C₄×C₅	—	2	1	no	—
232	C₄×C₅	—	2	1	no	—
233	C₄×C₅	—	4	4	no	—
234	C₄×C₅	C₃×#104	1	2	yes	—
235	C₄×C₅	C₃×C₄×Q₈	1	2	yes	—
236	C₄×C₅	C₃×C₂²⋊Q₈	1	2	yes	—
237	Dic₅	—	4	2	no	—
238	Dic₅	—	4	2	no	—
239	C₂×Q₈	—	10	1	no	—
240	C₂×Q₈	—	10	1	no	—
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	C₃×C₂²⋊D₅	1	1	yes	characteristic
243	C₂²×C₄	C₃×C₂²⋊D₅	1	1	yes	characteristic
244	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
245	M₁₆	—	5	4	no	—
246	M₁₆	—	10	2	no	—
247	C₂×C₈	—	10	2	no	—
248	C₄⋊C₄	—	2	4	no	—
249	C₄⋊C₄	—	5	2	no	—
250	C₄⋊C₄	—	5	2	no	—
251	C₄⋊C₄	—	10	2	no	—
252	C₄⋊C₄	—	10	2	no	—
253	C₄⋊C₄	—	10	1	no	—
254	C₄⋊C₄	—	10	1	no	—
255	C₂²⋊C₄	—	10	2	no	—
256	C₄²	—	2	4	no	—
257	C₄²	—	10	2	no	—
258	C₄²	C₃×C₄×D₅	1	1	yes	characteristic
259	C₄²	C₃×C₄×D₅	1	1	yes	characteristic
260	C₄²	C₃×D₂₀	1	1	yes	characteristic
261	C₄²	C₃×D₂₀	1	1	yes	characteristic
262	C₃×C₅	#1986	1	1	yes	characteristic, complemented, Hall
263	C₂²×C₃	—	2	1	no	—
264	C₂²×C₃	—	2	1	no	—
265	C₂²×C₃	C₂×#685	1	1	yes	characteristic
266	C₂²×C₃	#4082	1	1	yes	characteristic
267	C₂²×C₃	#4090	1	1	yes	characteristic
268	C₃×C₄	—	2	1	no	—
269	C₃×C₄	—	2	1	no	—
270	C₃×C₄	—	2	1	no	—
271	C₃×C₄	—	4	4	no	—
272	C₃×C₄	—	20	2	no	—
273	C₃×C₄	—	20	2	no	—
274	C₃×C₄	#3937	1	2	yes	—
275	C₃×C₄	#4033	1	2	yes	—
276	C₃×C₄	#4043	1	2	yes	—
277	C₂×C₅	—	2	1	no	—
278	C₂×C₅	—	2	1	no	—
279	C₂×C₅	C₃×#391	1	1	yes	characteristic
280	C₂×C₅	C₂×C₃×#104	1	1	yes	characteristic
281	C₂×C₅	C₃×#421	1	1	yes	characteristic
282	C₂³	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
283	Q₈	—	20	2	no	—
284	Q₈	—	20	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	4	no	—
287	C₂×C₄	—	4	1	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	10	2	no	—
291	C₂×C₄	—	10	2	no	—
292	C₂×C₄	C₃×#685	1	1	yes	characteristic
293	C₂×C₄	C₃×#685	1	1	yes	characteristic
294	C₂×C₄	C₃×#685	1	1	yes	characteristic
295	C₂×C₄	C₃×#685	1	1	yes	characteristic
296	C₂×C₄	C₂×C₃×C₄×D₅	1	1	yes	characteristic
297	C₂×C₄	C₂×C₃×D₂₀	1	1	yes	characteristic
298	C₂×C₄	C₃×#711	1	2	yes	—
299	C₂×C₄	C₃×Q₈×D₅	1	2	yes	—
300	C₈	—	10	4	no	—
301	C₂×C₃	—	2	1	no	—
302	C₂×C₃	—	2	1	no	—
303	C₂×C₃	C₂×#3937	1	1	yes	characteristic
304	C₂×C₃	#65263	1	1	yes	characteristic
305	C₂×C₃	#65294	1	1	yes	characteristic
306	C₅	C₃×#1986	1	1	yes	characteristic, complemented, Sylow
307	C₂²	—	2	1	no	—
308	C₂²	—	2	1	no	—
309	C₂²	C₂×C₃×#685	1	1	yes	characteristic
310	C₂²	C₃×#4082	1	1	yes	characteristic
311	C₂²	C₃×#4090	1	1	yes	characteristic
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	1	no	—
315	C₄	—	4	4	no	—
316	C₄	—	20	2	no	—
317	C₄	—	20	2	no	—
318	C₄	C₃×#3937	1	2	yes	—
319	C₄	C₃×#4033	1	2	yes	—
320	C₄	C₃×#4043	1	2	yes	—
321	C₃	#10600286	1	1	yes	characteristic, direct product factor, Sylow
322	C₂	—	2	1	no	—
323	C₂	—	2	1	no	—
324	C₂	C₂×C₃×#3937	1	1	yes	characteristic
325	C₂	C₃×#65263	1	1	yes	characteristic
326	C₂	C₃×#65294	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₂×D₂₀
commutator	C₂×C₄×C₅	C₂²×C₃×C₄
Frattini	C₂²×C₄	C₂²×C₃×D₅
Fitting	C₃×C₄×C₅×C₄⋊C₄	C₂
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₂³×C₃×C₅	C₂×D₄
cosocle	C₂²×C₄×C₅	C₂³×C₃
hypercenter	C₃×C₄×C₄⋊C₄	D₅
hypocenter	C₅	C₃×#1986
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₄² ◁ C₃×C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
5. #10600286	C₃	direct (N × H)
13. C₃×#4016	C₄	semidirect
14. C₃×#4015	C₄	semidirect
15. C₃×#4015	C₄	semidirect
47. C₃×#681	Q₈	semidirect
60. #4016	C₃×C₄	semidirect
61. #4015	C₃×C₄	semidirect
62. #4015	C₃×C₄	semidirect
124. #681	C₃×Q₈	semidirect
262. C₃×C₅	#1986	semidirect
306. C₅	C₃×#1986	semidirect
321. C₃	#10600286	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×#65804	2
C₃×C₄×C₅×C₄⋊C₄	2
C₃×C₄×#681	2
C₃×#64743	2
#10600286	3
C₃×#1986	5

Subgroups — C3×#10600286 #14307124

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10600286 #14307124