Subgroups — C₃×#10608508 #14331620

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×#4027	C₂	1	3	yes	maximal, complemented
3	C₃×C₅×#393	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#65518	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#65249	C₂	1	3	yes	maximal, complemented
6	C₃×#64961	C₂	1	3	yes	maximal, complemented
7	C₃×#64955	C₂	1	3	yes	maximal, complemented
8	#10608508	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂²×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
10	C₂²×C₃×C₂²⋊D₅	C₂²	1	3	yes	—
11	C₂³×C₃×Dic₅	C₂²	1	1	yes	characteristic
12	C₂×C₃×#713	—	2	12	no	—
13	C₂²×C₃×D₂₀	C₂²	1	3	yes	—
14	C₂²×C₃×C₄×D₅	C₂²	1	3	yes	—
15	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	3	yes	—
16	C₂×C₃×#685	C₂²	1	6	yes	—
17	C₂×C₃×#685	C₂²	1	3	yes	—
18	C₂×C₃×#683	C₂²	1	3	yes	—
19	C₂×C₃×C₄×Dic₅	C₂²	1	3	yes	—
20	C₃×#4027	—	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×#4027	C₂×C₃	1	3	yes	complemented
26	C₅×#393	C₂×C₃	1	1	yes	characteristic, complemented
27	#65518	C₂×C₃	1	1	yes	characteristic, complemented
28	#65249	C₂×C₃	1	3	yes	complemented
29	#64961	C₂×C₃	1	3	yes	complemented
30	#64955	C₂×C₃	1	3	yes	complemented
31	C₂⁴×C₃×C₅	C₂³	1	1	yes	characteristic
32	C₂³×C₃×D₅	C₂³	1	3	yes	—
33	C₂×C₃×C₅×Q₈	—	2	6	no	—
34	C₂²×C₃×C₄×C₅	C₂³	1	3	yes	—
35	C₂²×C₃×C₄×C₅	C₂³	1	3	yes	—
36	C₂×C₃×C₂²⋊D₅	—	2	6	no	—
37	C₂×C₃×C₂²⋊D₅	—	2	6	no	—
38	C₂×C₃×C₂²⋊D₅	—	2	6	no	—
39	C₂²×C₃×Dic₅	—	2	6	no	—
40	C₂²×C₃×Dic₅	C₂³	1	1	yes	characteristic
41	C₂²×C₃×Dic₅	C₂³	1	3	yes	—
42	C₂²×C₃×Dic₅	C₂³	1	1	yes	characteristic
43	C₃×#713	—	2	32	no	—
44	C₂×C₃×D₂₀	—	2	12	no	—
45	C₂×C₃×D₂₀	—	2	6	no	—
46	C₂×C₃×C₄×D₅	—	2	12	no	—
47	C₂×C₃×C₄×D₅	D₄	1	12	yes	complemented
48	C₃×C₅×C₂²⋊C₄	—	2	6	no	—
49	C₃×#685	—	2	6	no	—
50	C₃×#685	—	2	12	no	—
51	C₃×#683	—	2	6	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	—	5	1	no	—
56	C₃×#394	—	5	3	no	—
57	C₃×#393	D₅	1	1	yes	characteristic, complemented
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	C₂²×C₂²⋊D₅	C₂²×C₃	1	3	yes	—
62	C₂³×Dic₅	C₂²×C₃	1	1	yes	characteristic
63	C₂×#713	—	2	12	no	—
64	C₂²×D₂₀	C₂²×C₃	1	3	yes	—
65	C₂²×C₄×D₅	C₂²×C₃	1	3	yes	—
66	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	3	yes	—
67	C₂×#685	C₂²×C₃	1	3	yes	—
68	C₂×#685	C₂²×C₃	1	6	yes	—
69	C₂×#683	C₂²×C₃	1	3	yes	—
70	C₂×C₄×Dic₅	C₂²×C₃	1	3	yes	—
71	#4027	—	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	—	2	3	no	—
76	C₂³×C₃×C₅	—	2	3	no	—
77	C₂³×C₃×C₅	#41	1	2	yes	complemented
78	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
79	C₂²×C₃×D₅	—	2	6	no	—
80	C₂²×C₃×D₅	—	2	6	no	—
81	C₂²×C₃×D₅	—	2	3	no	—
82	C₂²×C₃×D₅	—	2	3	no	—
83	C₂²×C₃×D₅	C₂×D₄	1	6	yes	—
84	C₃×C₅×Q₈	—	2	8	no	—
85	C₂×C₃×C₄×C₅	—	2	6	no	—
86	C₂×C₃×C₄×C₅	—	2	3	no	—
87	C₂×C₃×C₄×C₅	—	2	6	no	—
88	C₂×C₃×C₄×C₅	C₂×D₄	1	6	yes	—
89	C₃×C₂²⋊D₅	—	2	24	no	—
90	C₂×C₃×Dic₅	—	2	3	no	—
91	C₂×C₃×Dic₅	—	2	3	no	—
92	C₂×C₃×Dic₅	—	2	3	no	—
93	C₂×C₃×Dic₅	—	4	4	no	—
94	C₂×C₃×Dic₅	C₂×D₄	1	6	yes	—
95	C₂×C₃×Dic₅	#41	1	6	yes	complemented
96	C₃×D₂₀	—	2	24	no	—
97	C₃×C₄×D₅	—	2	24	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	—	5	3	no	—
106	C₂×C₃×C₂²⋊C₄	—	5	6	no	—
107	C₂×C₃×C₂²⋊C₄	—	5	3	no	—
108	C₂×C₃×C₂²⋊C₄	C₂×D₅	1	3	yes	—
109	C₂×C₃×C₄²	—	5	3	no	—
110	C₃×#95	—	5	3	no	—
111	C₃×#95	C₂×D₅	1	3	yes	—
112	C₂⁴×C₅	C₂³×C₃	1	1	yes	characteristic
113	C₂³×D₅	C₂³×C₃	1	3	yes	—
114	C₂×C₅×Q₈	—	2	6	no	—
115	C₂²×C₄×C₅	C₂³×C₃	1	3	yes	—
116	C₂²×C₄×C₅	C₂³×C₃	1	3	yes	—
117	C₂×C₂²⋊D₅	—	2	6	no	—
118	C₂×C₂²⋊D₅	—	2	6	no	—
119	C₂×C₂²⋊D₅	—	2	6	no	—
120	C₂²×Dic₅	—	2	6	no	—
121	C₂²×Dic₅	C₂³×C₃	1	1	yes	characteristic
122	C₂²×Dic₅	C₂³×C₃	1	3	yes	—
123	C₂²×Dic₅	C₂³×C₃	1	1	yes	characteristic
124	#713	—	2	32	no	—
125	C₂×D₂₀	—	2	12	no	—
126	C₂×D₂₀	—	2	6	no	—
127	C₂×C₄×D₅	—	2	12	no	—
128	C₂×C₄×D₅	C₃×D₄	1	12	yes	complemented
129	C₅×C₂²⋊C₄	—	2	6	no	—
130	#685	—	2	6	no	—
131	#685	—	2	12	no	—
132	#683	—	2	6	no	—
133	C₄×Dic₅	—	2	6	no	—
134	C₂²×#41	—	5	1	no	—
135	C₂×C₄×D₄	—	5	3	no	—
136	#395	—	5	1	no	—
137	#394	—	5	3	no	—
138	#393	C₃×D₅	1	1	yes	characteristic, complemented
139	#392	—	5	3	no	—
140	C₄×C₂²⋊C₄	—	5	3	no	—
141	C₂²×C₃×C₅	—	2	6	no	—
142	C₂²×C₃×C₅	—	4	1	no	—
143	C₂²×C₃×C₅	—	4	3	no	—
144	C₂²×C₃×C₅	C₂²×D₄	1	3	yes	—
145	C₂²×C₃×C₅	C₂×#41	1	1	yes	characteristic
146	C₂²×C₃×C₅	C₂×#41	1	3	yes	—
147	C₂×C₃×D₅	—	2	6	no	—
148	C₂×C₃×D₅	—	2	6	no	—
149	C₂×C₃×D₅	—	2	6	no	—
150	C₂×C₃×D₅	—	4	6	no	—
151	C₂×C₃×D₅	—	4	6	no	—
152	C₃×C₄×C₅	—	2	6	no	—
153	C₃×C₄×C₅	—	4	3	no	—
154	C₃×Dic₅	—	2	6	no	—
155	C₃×Dic₅	—	4	1	no	—
156	C₃×Dic₅	#120	1	4	yes	—
157	C₂⁴×C₃	—	5	3	no	—
158	C₂⁴×C₃	C₂²×D₅	1	1	yes	characteristic
159	C₃×#41	—	10	32	no	—
160	C₂×C₃×Q₈	—	2	6	no	—
161	C₂×C₃×D₄	—	10	12	no	—
162	C₂×C₃×D₄	—	10	6	no	—
163	C₂×C₃×D₄	—	10	6	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	3	no	—
168	C₂²×C₃×C₄	—	5	12	no	—
169	C₂²×C₃×C₄	—	5	1	no	—
170	C₂²×C₃×C₄	—	10	12	no	—
171	C₂²×C₃×C₄	—	10	6	no	—
172	C₂²×C₃×C₄	C₂²×D₅	1	3	yes	—
173	C₂²×C₃×C₄	C₂²×D₅	1	3	yes	—
174	C₃×C₄⋊C₄	—	10	6	no	—
175	C₃×C₂²⋊C₄	—	2	6	no	—
176	C₃×C₂²⋊C₄	—	10	12	no	—
177	C₃×C₂²⋊C₄	—	10	6	no	—
178	C₃×C₄²	—	10	6	no	—
179	C₂³×C₅	—	2	3	no	—
180	C₂³×C₅	—	2	3	no	—
181	C₂³×C₅	C₃×#41	1	2	yes	complemented
182	C₂³×C₅	C₂⁴×C₃	1	1	yes	characteristic
183	C₂²×D₅	—	2	6	no	—
184	C₂²×D₅	—	2	6	no	—
185	C₂²×D₅	—	2	3	no	—
186	C₂²×D₅	—	2	3	no	—
187	C₂²×D₅	C₂×C₃×D₄	1	6	yes	—
188	C₅×Q₈	—	2	8	no	—
189	C₂×C₄×C₅	—	2	6	no	—
190	C₂×C₄×C₅	—	2	3	no	—
191	C₂×C₄×C₅	—	2	6	no	—
192	C₂×C₄×C₅	C₂×C₃×D₄	1	6	yes	—
193	C₂²⋊D₅	—	2	24	no	—
194	C₂×Dic₅	—	2	3	no	—
195	C₂×Dic₅	—	2	3	no	—
196	C₂×Dic₅	—	2	3	no	—
197	C₂×Dic₅	—	4	4	no	—
198	C₂×Dic₅	C₂×C₃×D₄	1	6	yes	—
199	C₂×Dic₅	C₃×#41	1	6	yes	complemented
200	D₂₀	—	2	24	no	—
201	C₄×D₅	—	2	24	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	—	5	3	no	—
210	C₂×C₂²⋊C₄	—	5	3	no	—
211	C₂×C₂²⋊C₄	—	5	6	no	—
212	C₂×C₂²⋊C₄	C₂×C₃×D₅	1	3	yes	—
213	C₂×C₄²	—	5	3	no	—
214	#95	—	5	3	no	—
215	#95	C₂×C₃×D₅	1	3	yes	—
216	C₂×C₃×C₅	—	4	2	no	—
217	C₂×C₃×C₅	C₂×#120	1	1	yes	characteristic
218	C₂×C₃×C₅	#525	1	3	yes	—
219	C₂×C₃×C₅	#532	1	3	yes	—
220	C₃×D₅	—	4	6	no	—
221	C₂³×C₃	—	2	3	no	—
222	C₂³×C₃	—	2	3	no	—
223	C₂³×C₃	—	5	6	no	—
224	C₂³×C₃	—	10	6	no	—
225	C₂³×C₃	—	10	3	no	—
226	C₂³×C₃	—	10	6	no	—
227	C₂³×C₃	—	10	3	no	—
228	C₂³×C₃	#713	1	2	yes	complemented
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	6	no	—
234	C₂×C₃×C₄	—	2	3	no	—
235	C₂×C₃×C₄	—	2	6	no	—
236	C₂×C₃×C₄	—	5	6	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	3	no	—
241	C₂×C₃×C₄	—	10	3	no	—
242	C₂×C₃×C₄	—	20	4	no	—
243	C₂×C₃×C₄	D₄×D₅	1	6	yes	—
244	C₂²×C₅	—	2	6	no	—
245	C₂²×C₅	—	4	1	no	—
246	C₂²×C₅	—	4	3	no	—
247	C₂²×C₅	C₂²×C₃×D₄	1	3	yes	—
248	C₂²×C₅	C₂×C₃×#41	1	1	yes	characteristic
249	C₂²×C₅	C₂×C₃×#41	1	3	yes	—
250	C₂×D₅	—	2	6	no	—
251	C₂×D₅	—	2	6	no	—
252	C₂×D₅	—	2	6	no	—
253	C₂×D₅	—	4	6	no	—
254	C₂×D₅	—	4	6	no	—
255	C₄×C₅	—	2	6	no	—
256	C₄×C₅	—	4	3	no	—
257	Dic₅	—	2	6	no	—
258	Dic₅	—	4	1	no	—
259	Dic₅	C₃×#120	1	4	yes	—
260	C₂⁴	—	5	3	no	—
261	C₂⁴	C₂²×C₃×D₅	1	1	yes	characteristic
262	#41	—	10	32	no	—
263	C₂×Q₈	—	2	6	no	—
264	C₂×D₄	—	10	12	no	—
265	C₂×D₄	—	10	6	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	3	no	—
271	C₂²×C₄	—	5	12	no	—
272	C₂²×C₄	—	5	1	no	—
273	C₂²×C₄	—	10	12	no	—
274	C₂²×C₄	—	10	6	no	—
275	C₂²×C₄	C₂²×C₃×D₅	1	3	yes	—
276	C₂²×C₄	C₂²×C₃×D₅	1	3	yes	—
277	C₄⋊C₄	—	10	6	no	—
278	C₂²⋊C₄	—	2	6	no	—
279	C₂²⋊C₄	—	10	12	no	—
280	C₂²⋊C₄	—	10	6	no	—
281	C₄²	—	10	6	no	—
282	C₃×C₅	#2388	1	1	yes	characteristic, complemented, Hall
283	C₂²×C₃	—	2	6	no	—
284	C₂²×C₃	—	4	1	no	—
285	C₂²×C₃	—	4	3	no	—
286	C₂²×C₃	—	10	6	no	—
287	C₂²×C₃	—	10	6	no	—
288	C₂²×C₃	—	10	6	no	—
289	C₂²×C₃	—	20	6	no	—
290	C₂²×C₃	—	20	6	no	—
291	C₂²×C₃	C₂×D₄×D₅	1	3	yes	—
292	C₂²×C₃	C₂×#713	1	1	yes	characteristic
293	C₂²×C₃	D₅×#41	1	3	yes	—
294	C₃×C₄	—	2	6	no	—
295	C₃×C₄	—	4	3	no	—
296	C₃×C₄	—	5	4	no	—
297	C₃×C₄	—	10	6	no	—
298	C₃×C₄	—	20	1	no	—
299	C₂×C₅	—	4	2	no	—
300	C₂×C₅	C₂×C₃×#120	1	1	yes	characteristic
301	C₂×C₅	C₃×#525	1	3	yes	—
302	C₂×C₅	C₃×#532	1	3	yes	—
303	D₅	—	4	6	no	—
304	C₂³	—	2	3	no	—
305	C₂³	—	2	3	no	—
306	C₂³	—	5	6	no	—
307	C₂³	—	10	6	no	—
308	C₂³	—	10	3	no	—
309	C₂³	—	10	6	no	—
310	C₂³	—	10	3	no	—
311	C₂³	C₃×#713	1	2	yes	complemented
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	6	no	—
317	C₂×C₄	—	2	3	no	—
318	C₂×C₄	—	2	6	no	—
319	C₂×C₄	—	5	6	no	—
320	C₂×C₄	—	5	6	no	—
321	C₂×C₄	—	10	24	no	—
322	C₂×C₄	—	10	3	no	—
323	C₂×C₄	—	10	3	no	—
324	C₂×C₄	—	10	3	no	—
325	C₂×C₄	—	20	4	no	—
326	C₂×C₄	C₃×D₄×D₅	1	6	yes	—
327	C₂×C₃	—	4	2	no	—
328	C₂×C₃	—	20	6	no	—
329	C₂×C₃	D₅×#120	1	1	yes	characteristic
330	C₂×C₃	#65961	1	3	yes	—
331	C₂×C₃	#66002	1	3	yes	—
332	C₅	C₃×#2388	1	1	yes	characteristic, complemented, Sylow
333	C₂²	—	2	6	no	—
334	C₂²	—	4	1	no	—
335	C₂²	—	4	3	no	—
336	C₂²	—	10	6	no	—
337	C₂²	—	10	6	no	—
338	C₂²	—	10	6	no	—
339	C₂²	—	20	6	no	—
340	C₂²	—	20	6	no	—
341	C₂²	C₂×C₃×D₄×D₅	1	3	yes	—
342	C₂²	C₂×C₃×#713	1	1	yes	characteristic
343	C₂²	C₃×D₅×#41	1	3	yes	—
344	C₄	—	2	6	no	—
345	C₄	—	4	3	no	—
346	C₄	—	5	4	no	—
347	C₄	—	10	6	no	—
348	C₄	—	20	1	no	—
349	C₃	#10608508	1	1	yes	characteristic, direct product factor, Sylow
350	C₂	—	4	2	no	—
351	C₂	—	20	6	no	—
352	C₂	C₃×D₅×#120	1	1	yes	characteristic
353	C₂	C₃×#65961	1	3	yes	—
354	C₂	C₃×#66002	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×#393	C₂
socle	C₂³×C₃×C₅	C₂⁴
hypercenter	C₃×#393	D₅
hypocenter	C₅	C₃×#2388
LCS	G ▷ C₂³×C₅ ▷ C₅
UCS	C₁ ◁ C₂³×C₃ ◁ C₃×#393

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×#4027	C₂	semidirect
3. C₃×C₅×#393	C₂	semidirect
4. C₃×#65518	C₂	semidirect
5. C₃×#65249	C₂	semidirect
6. C₃×#64961	C₂	semidirect
7. C₃×#64955	C₂	semidirect
8. #10608508	C₃	direct (N × H)
24. C₂²×#713	C₂×C₃	semidirect
25. C₂×#4027	C₂×C₃	semidirect
26. C₅×#393	C₂×C₃	semidirect
27. #65518	C₂×C₃	semidirect
28. #65249	C₂×C₃	semidirect
29. #64961	C₂×C₃	semidirect
30. #64955	C₂×C₃	semidirect
47. C₂×C₃×C₄×D₅	D₄	semidirect
57. C₃×#393	D₅	semidirect
77. C₂³×C₃×C₅	#41	semidirect
95. C₂×C₃×Dic₅	#41	semidirect
128. C₂×C₄×D₅	C₃×D₄	semidirect
138. #393	C₃×D₅	semidirect
181. C₂³×C₅	C₃×#41	semidirect
199. C₂×Dic₅	C₃×#41	semidirect
228. C₂³×C₃	#713	semidirect
282. C₃×C₅	#2388	semidirect
311. C₂³	C₃×#713	semidirect
332. C₅	C₃×#2388	semidirect
349. C₃	#10608508	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×#4027	2
C₃×C₅×#393	2
C₃×#65518	2
C₃×#65249	2
C₃×#64961	2
C₃×#64955	2
#10608508	3
C₃×#2388	5

Subgroups — C3×#10608508 #14331620

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10608508 #14331620