Subgroups — C₃×#10608900 #14331623

Order 1920.

1888 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×#713	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×#4039	C₂	1	3	yes	maximal, complemented
3	C₃×C₅×#395	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#65520	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#65280	C₂	1	3	yes	maximal, complemented
6	C₃×#64960	C₂	1	3	yes	maximal, complemented
7	C₃×#64955	C₂	1	3	yes	maximal, complemented
8	#10608900	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂²×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
10	C₂×C₃×#713	—	2	12	no	—
11	C₂²×C₃×D₂₀	C₂²	1	3	yes	—
12	C₂²×C₃×D₂₀	C₂²	1	3	yes	—
13	C₂²×C₃×C₄×D₅	C₂²	1	1	yes	characteristic
14	C₂²×C₃×C₄×D₅	C₂²	1	3	yes	—
15	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	3	yes	—
16	C₂×C₃×#685	C₂²	1	3	yes	—
17	C₂×C₃×#685	C₂²	1	6	yes	—
18	C₂×C₃×#685	C₂²	1	3	yes	—
19	C₂×C₃×C₄×Dic₅	C₂²	1	3	yes	—
20	C₃×#4039	—	2	12	no	—
21	C₃×C₅×#95	C₂²	1	3	yes	—
22	C₃×#3963	C₂²	1	3	yes	—
23	C₃×#2388	—	5	1	no	maximal, Hall
24	C₂²×#713	C₂×C₃	1	1	yes	characteristic, complemented
25	C₂×#4039	C₂×C₃	1	3	yes	complemented
26	C₅×#395	C₂×C₃	1	1	yes	characteristic, complemented
27	#65520	C₂×C₃	1	1	yes	characteristic, complemented
28	#65280	C₂×C₃	1	3	yes	complemented
29	#64960	C₂×C₃	1	3	yes	complemented
30	#64955	C₂×C₃	1	3	yes	complemented
31	C₂³×C₃×D₅	C₂³	1	3	yes	—
32	C₂³×C₃×D₅	C₂³	1	1	yes	characteristic
33	C₂×C₃×C₅×Q₈	—	2	6	no	—
34	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
35	C₂²×C₃×C₄×C₅	C₂³	1	3	yes	—
36	C₂²×C₃×C₄×C₅	C₂³	1	3	yes	—
37	C₂²×C₃×Dic₅	C₂³	1	3	yes	—
38	C₂²×C₃×Dic₅	C₂³	1	1	yes	characteristic
39	C₃×#713	—	2	32	no	—
40	C₂×C₃×D₂₀	—	2	12	no	—
41	C₂×C₃×D₂₀	—	2	6	no	—
42	C₂×C₃×D₂₀	—	2	6	no	—
43	C₂×C₃×D₂₀	—	2	6	no	—
44	C₂×C₃×D₂₀	—	2	6	no	—
45	C₂×C₃×C₄×D₅	—	2	12	no	—
46	C₂×C₃×C₄×D₅	—	2	6	no	—
47	C₂×C₃×C₄×D₅	D₄	1	12	yes	complemented
48	C₃×C₅×C₄⋊C₄	—	2	6	no	—
49	C₃×#685	—	2	6	no	—
50	C₃×#685	—	2	6	no	—
51	C₃×#685	—	2	12	no	—
52	C₃×C₄×Dic₅	—	2	6	no	—
53	C₂²×C₃×#41	—	5	1	no	—
54	C₂×C₃×C₄×D₄	—	5	3	no	—
55	C₃×#395	D₅	1	1	yes	characteristic, complemented
56	C₃×#394	—	5	3	no	—
57	C₃×#393	—	5	1	no	—
58	C₃×#392	—	5	3	no	—
59	C₃×C₄×C₂²⋊C₄	—	5	3	no	—
60	C₂²×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic
61	C₂×#713	—	2	12	no	—
62	C₂²×D₂₀	C₂²×C₃	1	3	yes	—
63	C₂²×D₂₀	C₂²×C₃	1	3	yes	—
64	C₂²×C₄×D₅	C₂²×C₃	1	1	yes	characteristic
65	C₂²×C₄×D₅	C₂²×C₃	1	3	yes	—
66	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	3	yes	—
67	C₂×#685	C₂²×C₃	1	3	yes	—
68	C₂×#685	C₂²×C₃	1	6	yes	—
69	C₂×#685	C₂²×C₃	1	3	yes	—
70	C₂×C₄×Dic₅	C₂²×C₃	1	3	yes	—
71	#4039	—	2	12	no	—
72	C₅×#95	C₂²×C₃	1	3	yes	—
73	#3963	C₂²×C₃	1	3	yes	—
74	#2388	—	5	1	no	Sylow
75	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
76	C₂²×C₃×D₅	—	2	3	no	—
77	C₂²×C₃×D₅	—	2	6	no	—
78	C₂²×C₃×D₅	—	2	6	no	—
79	C₂²×C₃×D₅	—	2	3	no	—
80	C₂²×C₃×D₅	—	2	3	no	—
81	C₂²×C₃×D₅	—	2	3	no	—
82	C₂²×C₃×D₅	C₂×D₄	1	6	yes	—
83	C₂²×C₃×D₅	#41	1	2	yes	complemented
84	C₃×C₅×Q₈	—	2	8	no	—
85	C₂×C₃×C₄×C₅	—	2	3	no	—
86	C₂×C₃×C₄×C₅	—	2	6	no	—
87	C₂×C₃×C₄×C₅	—	2	3	no	—
88	C₂×C₃×C₄×C₅	—	2	3	no	—
89	C₂×C₃×C₄×C₅	C₂×D₄	1	6	yes	—
90	C₂×C₃×C₄×C₅	#41	1	6	yes	complemented
91	C₂×C₃×Dic₅	—	2	3	no	—
92	C₂×C₃×Dic₅	—	2	6	no	—
93	C₂×C₃×Dic₅	C₂×D₄	1	6	yes	—
94	C₃×D₂₀	—	2	24	no	—
95	C₃×D₂₀	—	2	24	no	—
96	C₃×C₄×D₅	—	2	24	no	—
97	C₃×C₄×D₅	—	4	4	no	—
98	C₂×C₃×#41	—	10	12	no	—
99	C₂²×C₃×Q₈	C₂×D₅	1	1	yes	characteristic
100	C₂²×C₃×D₄	—	5	3	no	—
101	C₂²×C₃×D₄	—	5	3	no	—
102	C₂³×C₃×C₄	—	5	3	no	—
103	C₂³×C₃×C₄	—	5	1	no	—
104	C₃×C₄×D₄	—	10	12	no	—
105	C₂×C₃×C₄⋊C₄	C₂×D₅	1	3	yes	—
106	C₂×C₃×C₂²⋊C₄	—	5	6	no	—
107	C₂×C₃×C₂²⋊C₄	—	5	3	no	—
108	C₂×C₃×C₂²⋊C₄	—	5	3	no	—
109	C₂×C₃×C₄²	—	5	3	no	—
110	C₃×#95	—	5	3	no	—
111	C₃×#95	C₂×D₅	1	3	yes	—
112	C₂³×D₅	C₂³×C₃	1	3	yes	—
113	C₂³×D₅	C₂³×C₃	1	1	yes	characteristic
114	C₂×C₅×Q₈	—	2	6	no	—
115	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
116	C₂²×C₄×C₅	C₂³×C₃	1	3	yes	—
117	C₂²×C₄×C₅	C₂³×C₃	1	3	yes	—
118	C₂²×Dic₅	C₂³×C₃	1	3	yes	—
119	C₂²×Dic₅	C₂³×C₃	1	1	yes	characteristic
120	#713	—	2	32	no	—
121	C₂×D₂₀	—	2	12	no	—
122	C₂×D₂₀	—	2	6	no	—
123	C₂×D₂₀	—	2	6	no	—
124	C₂×D₂₀	—	2	6	no	—
125	C₂×D₂₀	—	2	6	no	—
126	C₂×C₄×D₅	—	2	12	no	—
127	C₂×C₄×D₅	—	2	6	no	—
128	C₂×C₄×D₅	C₃×D₄	1	12	yes	complemented
129	C₅×C₄⋊C₄	—	2	6	no	—
130	#685	—	2	6	no	—
131	#685	—	2	6	no	—
132	#685	—	2	12	no	—
133	C₄×Dic₅	—	2	6	no	—
134	C₂²×#41	—	5	1	no	—
135	C₂×C₄×D₄	—	5	3	no	—
136	#395	C₃×D₅	1	1	yes	characteristic, complemented
137	#394	—	5	3	no	—
138	#393	—	5	1	no	—
139	#392	—	5	3	no	—
140	C₄×C₂²⋊C₄	—	5	3	no	—
141	C₂²×C₃×C₅	C₂²×D₄	1	3	yes	—
142	C₂²×C₃×C₅	C₂×#41	1	3	yes	—
143	C₂²×C₃×C₅	C₂×#41	1	1	yes	characteristic
144	C₂×C₃×D₅	—	2	6	no	—
145	C₂×C₃×D₅	—	2	6	no	—
146	C₂×C₃×D₅	—	2	6	no	—
147	C₂×C₃×D₅	—	2	6	no	—
148	C₂×C₃×D₅	—	4	1	no	—
149	C₂×C₃×D₅	—	4	6	no	—
150	C₂×C₃×D₅	—	4	3	no	—
151	C₂×C₃×D₅	—	4	6	no	—
152	C₃×C₄×C₅	—	2	6	no	—
153	C₃×C₄×C₅	—	2	6	no	—
154	C₃×C₄×C₅	—	4	1	no	—
155	C₃×Dic₅	—	4	3	no	—
156	C₃×Dic₅	#120	1	4	yes	—
157	C₂⁴×C₃	—	5	3	no	—
158	C₂⁴×C₃	—	5	1	no	—
159	C₃×#41	—	10	32	no	—
160	C₂×C₃×Q₈	—	2	6	no	—
161	C₂×C₃×D₄	—	10	6	no	—
162	C₂×C₃×D₄	—	10	6	no	—
163	C₂×C₃×D₄	—	10	12	no	—
164	C₂×C₃×D₄	—	10	6	no	—
165	C₂×C₃×D₄	—	10	6	no	—
166	C₂²×C₃×C₄	—	5	1	no	—
167	C₂²×C₃×C₄	—	5	12	no	—
168	C₂²×C₃×C₄	—	5	3	no	—
169	C₂²×C₃×C₄	—	10	12	no	—
170	C₂²×C₃×C₄	—	10	6	no	—
171	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
172	C₂²×C₃×C₄	C₂²×D₅	1	3	yes	—
173	C₂²×C₃×C₄	C₂²×D₅	1	3	yes	—
174	C₃×C₄⋊C₄	—	2	6	no	—
175	C₃×C₂²⋊C₄	—	10	6	no	—
176	C₃×C₂²⋊C₄	—	10	6	no	—
177	C₃×C₂²⋊C₄	—	10	12	no	—
178	C₃×C₄²	—	10	6	no	—
179	C₂³×C₅	C₂⁴×C₃	1	1	yes	characteristic
180	C₂²×D₅	—	2	3	no	—
181	C₂²×D₅	—	2	6	no	—
182	C₂²×D₅	—	2	6	no	—
183	C₂²×D₅	—	2	3	no	—
184	C₂²×D₅	—	2	3	no	—
185	C₂²×D₅	—	2	3	no	—
186	C₂²×D₅	C₂×C₃×D₄	1	6	yes	—
187	C₂²×D₅	C₃×#41	1	2	yes	complemented
188	C₅×Q₈	—	2	8	no	—
189	C₂×C₄×C₅	—	2	3	no	—
190	C₂×C₄×C₅	—	2	6	no	—
191	C₂×C₄×C₅	—	2	3	no	—
192	C₂×C₄×C₅	—	2	3	no	—
193	C₂×C₄×C₅	C₂×C₃×D₄	1	6	yes	—
194	C₂×C₄×C₅	C₃×#41	1	6	yes	complemented
195	C₂×Dic₅	—	2	3	no	—
196	C₂×Dic₅	—	2	6	no	—
197	C₂×Dic₅	C₂×C₃×D₄	1	6	yes	—
198	D₂₀	—	2	24	no	—
199	D₂₀	—	2	24	no	—
200	C₄×D₅	—	2	24	no	—
201	C₄×D₅	—	4	4	no	—
202	C₂×#41	—	10	12	no	—
203	C₂²×Q₈	C₂×C₃×D₅	1	1	yes	characteristic
204	C₂²×D₄	—	5	3	no	—
205	C₂²×D₄	—	5	3	no	—
206	C₂³×C₄	—	5	3	no	—
207	C₂³×C₄	—	5	1	no	—
208	C₄×D₄	—	10	12	no	—
209	C₂×C₄⋊C₄	C₂×C₃×D₅	1	3	yes	—
210	C₂×C₂²⋊C₄	—	5	3	no	—
211	C₂×C₂²⋊C₄	—	5	3	no	—
212	C₂×C₂²⋊C₄	—	5	6	no	—
213	C₂×C₄²	—	5	3	no	—
214	#95	—	5	3	no	—
215	#95	C₂×C₃×D₅	1	3	yes	—
216	C₂×C₃×C₅	C₂×#120	1	1	yes	characteristic
217	C₂×C₃×C₅	#525	1	3	yes	—
218	C₂×C₃×C₅	#532	1	3	yes	—
219	C₃×D₅	—	4	6	no	—
220	C₃×D₅	—	4	2	no	—
221	C₂³×C₃	—	5	6	no	—
222	C₂³×C₃	—	5	2	no	—
223	C₂³×C₃	—	10	3	no	—
224	C₂³×C₃	—	10	6	no	—
225	C₂³×C₃	—	10	6	no	—
226	C₂³×C₃	—	10	3	no	—
227	C₂³×C₃	—	10	3	no	—
228	C₂³×C₃	—	10	3	no	—
229	C₂³×C₃	C₂³×D₅	1	1	yes	characteristic
230	C₃×Q₈	—	2	8	no	—
231	C₃×D₄	—	10	24	no	—
232	C₃×D₄	—	10	24	no	—
233	C₂×C₃×C₄	—	2	3	no	—
234	C₂×C₃×C₄	—	2	6	no	—
235	C₂×C₃×C₄	—	2	3	no	—
236	C₂×C₃×C₄	—	2	3	no	—
237	C₂×C₃×C₄	—	5	6	no	—
238	C₂×C₃×C₄	—	10	24	no	—
239	C₂×C₃×C₄	—	10	3	no	—
240	C₂×C₃×C₄	—	10	6	no	—
241	C₂×C₃×C₄	—	20	4	no	—
242	C₂×C₃×C₄	D₄×D₅	1	6	yes	—
243	C₂×C₃×C₄	#713	1	6	yes	complemented
244	C₂²×C₅	C₂²×C₃×D₄	1	3	yes	—
245	C₂²×C₅	C₂×C₃×#41	1	3	yes	—
246	C₂²×C₅	C₂×C₃×#41	1	1	yes	characteristic
247	C₂×D₅	—	2	6	no	—
248	C₂×D₅	—	2	6	no	—
249	C₂×D₅	—	2	6	no	—
250	C₂×D₅	—	2	6	no	—
251	C₂×D₅	—	4	1	no	—
252	C₂×D₅	—	4	6	no	—
253	C₂×D₅	—	4	3	no	—
254	C₂×D₅	—	4	6	no	—
255	C₄×C₅	—	2	6	no	—
256	C₄×C₅	—	2	6	no	—
257	C₄×C₅	—	4	1	no	—
258	Dic₅	—	4	3	no	—
259	Dic₅	C₃×#120	1	4	yes	—
260	C₂⁴	—	5	3	no	—
261	C₂⁴	—	5	1	no	—
262	#41	—	10	32	no	—
263	C₂×Q₈	—	2	6	no	—
264	C₂×D₄	—	10	6	no	—
265	C₂×D₄	—	10	12	no	—
266	C₂×D₄	—	10	6	no	—
267	C₂×D₄	—	10	6	no	—
268	C₂×D₄	—	10	6	no	—
269	C₂²×C₄	—	5	1	no	—
270	C₂²×C₄	—	5	12	no	—
271	C₂²×C₄	—	5	3	no	—
272	C₂²×C₄	—	10	12	no	—
273	C₂²×C₄	—	10	6	no	—
274	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
275	C₂²×C₄	C₂²×C₃×D₅	1	3	yes	—
276	C₂²×C₄	C₂²×C₃×D₅	1	3	yes	—
277	C₄⋊C₄	—	2	6	no	—
278	C₂²⋊C₄	—	10	6	no	—
279	C₂²⋊C₄	—	10	6	no	—
280	C₂²⋊C₄	—	10	12	no	—
281	C₄²	—	10	6	no	—
282	C₃×C₅	#2388	1	1	yes	characteristic, complemented, Hall
283	C₂²×C₃	—	10	6	no	—
284	C₂²×C₃	—	10	6	no	—
285	C₂²×C₃	—	10	6	no	—
286	C₂²×C₃	—	10	6	no	—
287	C₂²×C₃	—	20	3	no	—
288	C₂²×C₃	—	20	6	no	—
289	C₂²×C₃	—	20	1	no	—
290	C₂²×C₃	—	20	6	no	—
291	C₂²×C₃	C₂×D₄×D₅	1	3	yes	—
292	C₂²×C₃	C₂×#713	1	3	yes	—
293	C₂²×C₃	D₅×#41	1	1	yes	characteristic
294	C₃×C₄	—	2	6	no	—
295	C₃×C₄	—	2	6	no	—
296	C₃×C₄	—	4	1	no	—
297	C₃×C₄	—	5	4	no	—
298	C₃×C₄	—	20	3	no	—
299	C₂×C₅	C₂×C₃×#120	1	1	yes	characteristic
300	C₂×C₅	C₃×#525	1	3	yes	—
301	C₂×C₅	C₃×#532	1	3	yes	—
302	D₅	—	4	6	no	—
303	D₅	—	4	2	no	—
304	C₂³	—	5	6	no	—
305	C₂³	—	5	2	no	—
306	C₂³	—	10	3	no	—
307	C₂³	—	10	6	no	—
308	C₂³	—	10	6	no	—
309	C₂³	—	10	3	no	—
310	C₂³	—	10	3	no	—
311	C₂³	—	10	3	no	—
312	C₂³	C₂³×C₃×D₅	1	1	yes	characteristic
313	Q₈	—	2	8	no	—
314	D₄	—	10	24	no	—
315	D₄	—	10	24	no	—
316	C₂×C₄	—	2	3	no	—
317	C₂×C₄	—	2	6	no	—
318	C₂×C₄	—	2	3	no	—
319	C₂×C₄	—	2	3	no	—
320	C₂×C₄	—	5	6	no	—
321	C₂×C₄	—	10	24	no	—
322	C₂×C₄	—	10	3	no	—
323	C₂×C₄	—	10	6	no	—
324	C₂×C₄	—	20	4	no	—
325	C₂×C₄	C₃×D₄×D₅	1	6	yes	—
326	C₂×C₄	C₃×#713	1	6	yes	complemented
327	C₂×C₃	—	20	6	no	—
328	C₂×C₃	—	20	2	no	—
329	C₂×C₃	D₅×#120	1	1	yes	characteristic
330	C₂×C₃	#65961	1	3	yes	—
331	C₂×C₃	#66059	1	3	yes	—
332	C₅	C₃×#2388	1	1	yes	characteristic, complemented, Sylow
333	C₂²	—	10	6	no	—
334	C₂²	—	10	6	no	—
335	C₂²	—	10	6	no	—
336	C₂²	—	10	6	no	—
337	C₂²	—	20	1	no	—
338	C₂²	—	20	6	no	—
339	C₂²	—	20	3	no	—
340	C₂²	—	20	6	no	—
341	C₂²	C₂×C₃×D₄×D₅	1	3	yes	—
342	C₂²	C₂×C₃×#713	1	3	yes	—
343	C₂²	C₃×D₅×#41	1	1	yes	characteristic
344	C₄	—	2	6	no	—
345	C₄	—	2	6	no	—
346	C₄	—	4	1	no	—
347	C₄	—	5	4	no	—
348	C₄	—	20	3	no	—
349	C₃	#10608900	1	1	yes	characteristic, direct product factor, Sylow
350	C₂	—	20	6	no	—
351	C₂	—	20	2	no	—
352	C₂	C₃×D₅×#120	1	1	yes	characteristic
353	C₂	C₃×#65961	1	3	yes	—
354	C₂	C₃×#66059	1	3	yes	—

	subgroup	G/H
center	C₂³×C₃	C₂³×D₅
commutator, cosocle	C₂³×C₅	C₂⁴×C₃
Frattini	C₂³	C₂³×C₃×D₅
Fitting	C₃×C₅×#395	C₂
socle	C₂³×C₃×C₅	C₂⁴
hypercenter	C₃×#395	D₅
hypocenter	C₅	C₃×#2388
LCS	G ▷ C₂³×C₅ ▷ C₅
UCS	C₁ ◁ C₂³×C₃ ◁ C₃×#395

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×#713	C₂	semidirect
2. C₂×C₃×#4039	C₂	semidirect
3. C₃×C₅×#395	C₂	semidirect
4. C₃×#65520	C₂	semidirect
5. C₃×#65280	C₂	semidirect
6. C₃×#64960	C₂	semidirect
7. C₃×#64955	C₂	semidirect
8. #10608900	C₃	direct (N × H)
24. C₂²×#713	C₂×C₃	semidirect
25. C₂×#4039	C₂×C₃	semidirect
26. C₅×#395	C₂×C₃	semidirect
27. #65520	C₂×C₃	semidirect
28. #65280	C₂×C₃	semidirect
29. #64960	C₂×C₃	semidirect
30. #64955	C₂×C₃	semidirect
47. C₂×C₃×C₄×D₅	D₄	semidirect
55. C₃×#395	D₅	semidirect
83. C₂²×C₃×D₅	#41	semidirect
90. C₂×C₃×C₄×C₅	#41	semidirect
128. C₂×C₄×D₅	C₃×D₄	semidirect
136. #395	C₃×D₅	semidirect
187. C₂²×D₅	C₃×#41	semidirect
194. C₂×C₄×C₅	C₃×#41	semidirect
243. C₂×C₃×C₄	#713	semidirect
282. C₃×C₅	#2388	semidirect
326. C₂×C₄	C₃×#713	semidirect
332. C₅	C₃×#2388	semidirect
349. C₃	#10608900	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₃×#713	2
C₂×C₃×#4039	2
C₃×C₅×#395	2
C₃×#65520	2
C₃×#65280	2
C₃×#64960	2
C₃×#64955	2
#10608900	3
C₃×#2388	5

Subgroups — C3×#10608900 #14331623

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10608900 #14331623