Subgroups — C₃×#10600264 #14282919

Order 1920.

480 conjugacy classes of subgroups.

The subgroup lattice has 320 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×#65214	C₂	1	1	yes	maximal, characteristic
4	C₃×#64672	C₂	1	4	yes	maximal
5	#10600264	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂×C₃×C₅×C₄⋊C₄	—	2	1	no	—
7	C₂×C₃×C₄²×C₅	—	2	1	no	—
8	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
9	C₃×#4072	—	2	1	no	—
10	C₃×#4070	—	2	1	no	—
11	C₃×#4070	C₂²	1	2	yes	—
12	C₃×#4016	—	2	1	no	—
13	C₃×#4015	C₄	1	1	yes	characteristic, complemented
14	C₃×#4015	C₄	1	1	yes	characteristic, complemented
15	C₃×C₄×Dic₁₀	—	2	1	no	—
16	C₃×C₄×Dic₁₀	C₄	1	2	yes	complemented
17	C₃×C₅×#95	C₂²	1	2	yes	—
18	C₃×C₂²⋊#168	C₂²	1	2	yes	—
19	C₃×C₄⋊#168	—	2	1	no	—
20	C₃×C₄×#168	—	2	1	no	—
21	C₃×#1503	—	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	#65214	C₂×C₃	1	1	yes	characteristic
25	#64672	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	C₂×C₄	1	2	yes	—
32	C₃×C₅×C₄⋊C₄	—	2	4	no	—
33	C₃×C₄²×C₅	—	4	2	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	—	2	1	no	—
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	C₂×C₃×#168	—	2	2	no	—
47	C₃×#533	—	5	1	no	—
48	C₃×#431	—	5	1	no	—
49	C₃×C₄×C₄⋊C₄	D₅	1	1	yes	characteristic
50	C₃×#328	—	5	4	no	—
51	C₂×C₅×C₄⋊C₄	—	2	1	no	—
52	C₂×C₄²×C₅	—	2	1	no	—
53	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
54	#4072	—	2	1	no	—
55	#4070	—	2	1	no	—
56	#4070	C₂²×C₃	1	2	yes	—
57	#4016	—	2	1	no	—
58	#4015	C₃×C₄	1	1	yes	characteristic, complemented
59	#4015	C₃×C₄	1	1	yes	characteristic, complemented
60	C₄×Dic₁₀	—	2	1	no	—
61	C₄×Dic₁₀	C₃×C₄	1	2	yes	complemented
62	C₅×#95	C₂²×C₃	1	2	yes	—
63	C₂²⋊#168	C₂²×C₃	1	2	yes	—
64	C₄⋊#168	—	2	1	no	—
65	C₄×#168	—	2	1	no	—
66	#1503	—	5	1	no	Sylow
67	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
68	C₂×C₃×C₄×C₅	—	2	2	no	—
69	C₂×C₃×C₄×C₅	—	2	4	no	—
70	C₂×C₃×C₄×C₅	—	4	1	no	—
71	C₂×C₃×C₄×C₅	—	4	2	no	—
72	C₂×C₃×C₄×C₅	—	4	2	no	—
73	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
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	SD₁₆	1	2	yes	—
78	C₂×C₃×C₄×C₅	Q₁₆	1	2	yes	—
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×Dic₅	—	2	2	no	—
82	C₂×C₃×Dic₅	—	2	2	no	—
83	C₃×Dic₁₀	—	4	2	no	—
84	C₃×Dic₁₀	—	4	2	no	—
85	C₃×#168	—	4	2	no	—
86	C₃×C₄⋊Q₈	—	5	1	no	—
87	C₃×C₄⋊Q₈	—	5	1	no	—
88	C₃×#125	—	10	1	no	—
89	C₃×C₂²⋊Q₈	—	5	2	no	—
90	C₃×C₂²⋊Q₈	—	10	1	no	—
91	C₃×C₄×Q₈	—	5	2	no	—
92	C₃×C₄×Q₈	—	10	1	no	—
93	C₃×#117	—	10	1	no	—
94	C₂×C₃×C₄⋊C₄	—	2	1	no	—
95	C₂×C₃×C₄²	—	2	1	no	—
96	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
97	C₃×C₄⋊C₈	—	10	1	no	—
98	C₃×C₂²⋊C₈	—	5	2	no	—
99	C₃×C₄×C₈	—	10	1	no	—
100	C₃×#95	C₂×D₅	1	2	yes	—
101	C₂²×C₄×C₅	—	2	2	no	—
102	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
103	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
104	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
105	C₂×Dic₁₀	—	2	1	no	—
106	C₂×Dic₁₀	C₂×C₃×C₄	1	2	yes	—
107	C₅×C₄⋊C₄	—	2	4	no	—
108	C₄²×C₅	—	4	2	no	—
109	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
110	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
111	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
112	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
113	C₂²⋊Dic₅	—	2	2	no	—
114	C₄⋊Dic₅	—	2	1	no	—
115	C₄⋊Dic₅	—	2	1	no	—
116	C₄⋊Dic₅	—	2	1	no	—
117	C₄⋊Dic₅	C₂×C₃×C₄	1	2	yes	—
118	#683	—	2	2	no	—
119	#683	—	2	2	no	—
120	C₄×Dic₅	—	2	2	no	—
121	C₂×#168	—	2	2	no	—
122	#533	—	5	1	no	—
123	#431	—	5	1	no	—
124	C₄×C₄⋊C₄	C₃×D₅	1	1	yes	characteristic
125	#328	—	5	4	no	—
126	C₂²×C₃×C₅	—	2	1	no	—
127	C₂²×C₃×C₅	—	2	1	no	—
128	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
129	C₂²×C₃×C₅	C₂×SD₁₆	1	1	yes	characteristic
130	C₂²×C₃×C₅	C₂×Q₁₆	1	1	yes	characteristic
131	C₃×C₄×C₅	—	2	1	no	—
132	C₃×C₄×C₅	—	2	1	no	—
133	C₃×C₄×C₅	—	2	1	no	—
134	C₃×C₄×C₅	—	4	4	no	—
135	C₃×C₄×C₅	#103	1	2	yes	—
136	C₃×C₄×C₅	#103	1	2	yes	—
137	C₃×C₄×C₅	#104	1	2	yes	—
138	C₃×Dic₅	—	4	2	no	—
139	C₃×Dic₅	—	4	2	no	—
140	C₂×C₃×Q₈	—	5	2	no	—
141	C₂×C₃×Q₈	—	10	1	no	—
142	C₂²×C₃×C₄	—	2	2	no	—
143	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
144	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
145	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
146	C₂×C₃×C₈	—	10	2	no	—
147	C₃×C₄⋊C₄	—	2	4	no	—
148	C₃×C₄⋊C₄	—	5	2	no	—
149	C₃×C₄⋊C₄	—	10	2	no	—
150	C₃×C₄⋊C₄	—	10	1	no	—
151	C₃×C₄⋊C₄	—	10	1	no	—
152	C₃×C₄⋊C₄	—	10	1	no	—
153	C₃×C₄⋊C₄	—	10	2	no	—
154	C₃×C₂²⋊C₄	—	10	2	no	—
155	C₃×C₄²	—	4	2	no	—
156	C₃×C₄²	—	10	2	no	—
157	C₃×C₄²	C₄×D₅	1	1	yes	characteristic
158	C₃×C₄²	C₄×D₅	1	1	yes	characteristic
159	C₃×C₄²	D₂₀	1	1	yes	characteristic
160	C₃×C₄²	D₂₀	1	1	yes	characteristic
161	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
162	C₂×C₄×C₅	—	2	2	no	—
163	C₂×C₄×C₅	—	2	4	no	—
164	C₂×C₄×C₅	—	4	1	no	—
165	C₂×C₄×C₅	—	4	2	no	—
166	C₂×C₄×C₅	—	4	2	no	—
167	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
168	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
169	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
170	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
171	C₂×C₄×C₅	C₃×SD₁₆	1	2	yes	—
172	C₂×C₄×C₅	C₃×Q₁₆	1	2	yes	—
173	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
174	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
175	C₂×Dic₅	—	2	2	no	—
176	C₂×Dic₅	—	2	2	no	—
177	Dic₁₀	—	4	2	no	—
178	Dic₁₀	—	4	2	no	—
179	#168	—	4	2	no	—
180	C₄⋊Q₈	—	5	1	no	—
181	C₄⋊Q₈	—	5	1	no	—
182	#125	—	10	1	no	—
183	C₂²⋊Q₈	—	5	2	no	—
184	C₂²⋊Q₈	—	10	1	no	—
185	C₄×Q₈	—	5	2	no	—
186	C₄×Q₈	—	10	1	no	—
187	#117	—	10	1	no	—
188	C₂×C₄⋊C₄	—	2	1	no	—
189	C₂×C₄²	—	2	1	no	—
190	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
191	C₄⋊C₈	—	10	1	no	—
192	C₂²⋊C₈	—	5	2	no	—
193	C₄×C₈	—	10	1	no	—
194	#95	C₂×C₃×D₅	1	2	yes	—
195	C₂×C₃×C₅	—	2	1	no	—
196	C₂×C₃×C₅	—	2	1	no	—
197	C₂×C₃×C₅	#410	1	1	yes	characteristic
198	C₂×C₃×C₅	C₂×#103	1	1	yes	characteristic
199	C₂×C₃×C₅	C₂×#104	1	1	yes	characteristic
200	C₂³×C₃	C₂×C₂²⋊D₅	1	1	yes	characteristic
201	C₃×Q₈	—	20	2	no	—
202	C₃×Q₈	—	20	2	no	—
203	C₂×C₃×C₄	—	2	2	no	—
204	C₂×C₃×C₄	—	2	4	no	—
205	C₂×C₃×C₄	—	4	1	no	—
206	C₂×C₃×C₄	—	4	2	no	—
207	C₂×C₃×C₄	—	4	2	no	—
208	C₂×C₃×C₄	—	10	2	no	—
209	C₂×C₃×C₄	—	10	2	no	—
210	C₂×C₃×C₄	#685	1	1	yes	characteristic
211	C₂×C₃×C₄	#685	1	1	yes	characteristic
212	C₂×C₃×C₄	#685	1	1	yes	characteristic
213	C₂×C₃×C₄	#685	1	1	yes	characteristic
214	C₂×C₃×C₄	#687	1	2	yes	—
215	C₂×C₃×C₄	#689	1	2	yes	—
216	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
217	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
218	C₃×C₈	—	20	2	no	—
219	C₂²×C₅	—	2	1	no	—
220	C₂²×C₅	—	2	1	no	—
221	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
222	C₂²×C₅	C₂×C₃×SD₁₆	1	1	yes	characteristic
223	C₂²×C₅	C₂×C₃×Q₁₆	1	1	yes	characteristic
224	C₄×C₅	—	2	1	no	—
225	C₄×C₅	—	2	1	no	—
226	C₄×C₅	—	2	1	no	—
227	C₄×C₅	—	4	4	no	—
228	C₄×C₅	C₃×#103	1	2	yes	—
229	C₄×C₅	C₃×#103	1	2	yes	—
230	C₄×C₅	C₃×#104	1	2	yes	—
231	Dic₅	—	4	2	no	—
232	Dic₅	—	4	2	no	—
233	C₂×Q₈	—	5	2	no	—
234	C₂×Q₈	—	10	1	no	—
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	C₃×C₂²⋊D₅	1	1	yes	characteristic
237	C₂²×C₄	C₃×C₂²⋊D₅	1	1	yes	characteristic
238	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
239	C₂×C₈	—	10	2	no	—
240	C₄⋊C₄	—	2	4	no	—
241	C₄⋊C₄	—	5	2	no	—
242	C₄⋊C₄	—	10	1	no	—
243	C₄⋊C₄	—	10	1	no	—
244	C₄⋊C₄	—	10	2	no	—
245	C₄⋊C₄	—	10	1	no	—
246	C₄⋊C₄	—	10	2	no	—
247	C₂²⋊C₄	—	10	2	no	—
248	C₄²	—	4	2	no	—
249	C₄²	—	10	2	no	—
250	C₄²	C₃×C₄×D₅	1	1	yes	characteristic
251	C₄²	C₃×C₄×D₅	1	1	yes	characteristic
252	C₄²	C₃×D₂₀	1	1	yes	characteristic
253	C₄²	C₃×D₂₀	1	1	yes	characteristic
254	C₃×C₅	#1503	1	1	yes	characteristic, complemented, Hall
255	C₂²×C₃	—	2	1	no	—
256	C₂²×C₃	—	2	1	no	—
257	C₂²×C₃	C₂×#685	1	1	yes	characteristic
258	C₂²×C₃	C₂×#687	1	1	yes	characteristic
259	C₂²×C₃	C₂×#689	1	1	yes	characteristic
260	C₃×C₄	—	2	1	no	—
261	C₃×C₄	—	2	1	no	—
262	C₃×C₄	—	2	1	no	—
263	C₃×C₄	—	4	4	no	—
264	C₃×C₄	—	20	2	no	—
265	C₃×C₄	—	20	2	no	—
266	C₃×C₄	#3937	1	2	yes	—
267	C₃×C₄	#3942	1	2	yes	—
268	C₃×C₄	#3942	1	2	yes	—
269	C₂×C₅	—	2	1	no	—
270	C₂×C₅	—	2	1	no	—
271	C₂×C₅	C₃×#410	1	1	yes	characteristic
272	C₂×C₅	C₂×C₃×#103	1	1	yes	characteristic
273	C₂×C₅	C₂×C₃×#104	1	1	yes	characteristic
274	C₂³	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
275	Q₈	—	20	2	no	—
276	Q₈	—	20	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	4	1	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	10	2	no	—
283	C₂×C₄	—	10	2	no	—
284	C₂×C₄	C₃×#685	1	1	yes	characteristic
285	C₂×C₄	C₃×#685	1	1	yes	characteristic
286	C₂×C₄	C₃×#685	1	1	yes	characteristic
287	C₂×C₄	C₃×#685	1	1	yes	characteristic
288	C₂×C₄	C₃×#687	1	2	yes	—
289	C₂×C₄	C₃×#689	1	2	yes	—
290	C₂×C₄	C₂×C₃×C₄×D₅	1	1	yes	characteristic
291	C₂×C₄	C₂×C₃×D₂₀	1	1	yes	characteristic
292	C₈	—	20	2	no	—
293	C₂×C₃	—	2	1	no	—
294	C₂×C₃	—	2	1	no	—
295	C₂×C₃	C₂×#3937	1	1	yes	characteristic
296	C₂×C₃	C₂×#3942	1	1	yes	characteristic
297	C₂×C₃	#65295	1	1	yes	characteristic
298	C₅	C₃×#1503	1	1	yes	characteristic, complemented, Sylow
299	C₂²	—	2	1	no	—
300	C₂²	—	2	1	no	—
301	C₂²	C₂×C₃×#685	1	1	yes	characteristic
302	C₂²	C₂×C₃×#687	1	1	yes	characteristic
303	C₂²	C₂×C₃×#689	1	1	yes	characteristic
304	C₄	—	2	1	no	—
305	C₄	—	2	1	no	—
306	C₄	—	2	1	no	—
307	C₄	—	4	4	no	—
308	C₄	—	20	2	no	—
309	C₄	—	20	2	no	—
310	C₄	C₃×#3937	1	2	yes	—
311	C₄	C₃×#3942	1	2	yes	—
312	C₄	C₃×#3942	1	2	yes	—
313	C₃	#10600264	1	1	yes	characteristic, direct product factor, Sylow
314	C₂	—	2	1	no	—
315	C₂	—	2	1	no	—
316	C₂	C₂×C₃×#3937	1	1	yes	characteristic
317	C₂	C₂×C₃×#3942	1	1	yes	characteristic
318	C₂	C₃×#65295	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	#685
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×#1503
LCS	G ▷ C₂×C₄×C₅ ▷ C₂²×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. #10600264	C₃	direct (N × H)
13. C₃×#4015	C₄	semidirect
14. C₃×#4015	C₄	semidirect
16. C₃×C₄×Dic₁₀	C₄	semidirect
58. #4015	C₃×C₄	semidirect
59. #4015	C₃×C₄	semidirect
61. C₄×Dic₁₀	C₃×C₄	semidirect
254. C₃×C₅	#1503	semidirect
298. C₅	C₃×#1503	semidirect
313. C₃	#10600264	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×#65214	2
C₃×#64672	2
#10600264	3
C₃×#1503	5

Subgroups — C3×#10600264 #14282919

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10600264 #14282919