Subgroups — C₃×#10601527 #14299657

Order 1920.

504 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#66079	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#430	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#65421	C₂	1	1	yes	maximal, characteristic
4	C₃×#65418	C₂	1	1	yes	maximal, characteristic
5	C₃×#65413	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#64778	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#64778	C₂	1	1	yes	maximal, characteristic, complemented
8	#10601527	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂×C₃×#709	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₅×M₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₅×M₁₆	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₅×M₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₃×#681	C₂²	1	1	yes	characteristic
14	C₂×C₃×#681	C₂²	1	1	yes	characteristic
15	C₂²×C₃×#168	C₂²	1	1	yes	characteristic
16	C₃×#4053	—	2	2	no	—
17	C₃×D₅×M₁₆	—	2	2	no	—
18	C₃×#4047	—	2	2	no	—
19	C₃×C₅×#108	—	2	2	no	—
20	C₃×#3956	C₄	1	2	yes	—
21	C₃×#3955	C₄	1	2	yes	—
22	C₃×#3954	—	2	2	no	—
23	C₃×#3952	—	2	2	no	—
24	C₃×#1858	—	5	1	no	maximal, Hall
25	#66079	C₂×C₃	1	1	yes	characteristic
26	C₅×#430	C₂×C₃	1	1	yes	characteristic, complemented
27	#65421	C₂×C₃	1	1	yes	characteristic
28	#65418	C₂×C₃	1	1	yes	characteristic
29	#65413	C₂×C₃	1	1	yes	characteristic, complemented
30	#64778	C₂×C₃	1	1	yes	characteristic, complemented
31	#64778	C₂×C₃	1	1	yes	characteristic, complemented
32	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
33	C₂×C₃×C₂²⋊D₅	Q₈	1	2	yes	—
34	C₃×#709	—	2	1	no	—
35	C₃×#709	—	2	1	no	—
36	C₃×#709	—	2	1	no	—
37	C₃×#709	—	2	1	no	—
38	C₂×C₃×D₂₀	C₂×C₄	1	1	yes	characteristic
39	C₂×C₃×C₄×D₅	D₄	1	2	yes	—
40	C₂×C₃×Dic₁₀	C₂×C₄	1	1	yes	characteristic
41	C₃×C₅×M₁₆	—	2	1	no	—
42	C₃×C₅×M₁₆	—	2	1	no	—
43	C₃×C₅×M₁₆	—	2	1	no	—
44	C₃×C₅×M₁₆	—	2	1	no	—
45	C₃×C₅×M₁₆	C₂×C₄	1	2	yes	—
46	C₃×C₅×M₁₆	D₄	1	2	yes	—
47	C₂×C₃×C₅×C₈	—	2	1	no	—
48	C₂×C₃×C₅×C₈	—	2	1	no	—
49	C₂×C₃×C₅×C₈	—	2	1	no	—
50	C₃×#681	—	2	1	no	—
51	C₃×#681	—	2	1	no	—
52	C₃×#681	C₂×C₄	1	2	yes	—
53	C₃×#681	D₄	1	2	yes	—
54	C₂×C₃×#168	—	2	1	no	—
55	C₂×C₃×#168	—	2	1	no	—
56	C₂×C₃×#168	—	2	1	no	—
57	C₂×C₃×#168	—	2	1	no	—
58	C₂×C₃×#168	—	2	1	no	—
59	C₃×#676	—	2	4	no	—
60	C₃×C₈×D₅	—	2	4	no	—
61	C₃×#568	—	5	1	no	—
62	C₃×#430	—	5	1	no	—
63	C₃×#430	D₅	1	1	yes	characteristic, complemented
64	C₃×#413	—	5	1	no	—
65	C₃×#408	—	5	1	no	—
66	C₃×#341	—	5	1	no	—
67	C₃×#341	—	5	1	no	—
68	C₂×#709	C₂²×C₃	1	1	yes	characteristic
69	C₂×C₅×M₁₆	C₂²×C₃	1	1	yes	characteristic
70	C₂×C₅×M₁₆	C₂²×C₃	1	1	yes	characteristic
71	C₂×C₅×M₁₆	C₂²×C₃	1	1	yes	characteristic
72	C₂×#681	C₂²×C₃	1	1	yes	characteristic
73	C₂×#681	C₂²×C₃	1	1	yes	characteristic
74	C₂²×#168	C₂²×C₃	1	1	yes	characteristic
75	#4053	—	2	2	no	—
76	D₅×M₁₆	—	2	2	no	—
77	#4047	—	2	2	no	—
78	C₅×#108	—	2	2	no	—
79	#3956	C₃×C₄	1	2	yes	—
80	#3955	C₃×C₄	1	2	yes	—
81	#3954	—	2	2	no	—
82	#3952	—	2	2	no	—
83	#1858	—	5	1	no	Sylow
84	C₂³×C₃×C₅	C₂×Q₈	1	1	yes	characteristic
85	C₂²×C₃×D₅	C₄⋊C₄	1	2	yes	—
86	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
88	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
89	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
90	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
91	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
92	C₃×C₂²⋊D₅	—	2	2	no	—
93	C₃×C₂²⋊D₅	—	2	2	no	—
94	C₂×C₃×Dic₅	C₄⋊C₄	1	2	yes	—
95	C₃×D₂₀	—	2	1	no	—
96	C₃×D₂₀	—	2	1	no	—
97	C₃×C₄×D₅	—	2	2	no	—
98	C₃×C₄×D₅	—	2	2	no	—
99	C₃×Dic₁₀	—	2	1	no	—
100	C₃×Dic₁₀	—	2	1	no	—
101	C₃×C₅×C₈	—	2	2	no	—
102	C₃×C₅×C₈	—	2	2	no	—
103	C₃×C₅×C₈	—	4	1	no	—
104	C₃×#168	—	2	2	no	—
105	C₃×#168	—	2	2	no	—
106	C₃×#168	—	4	1	no	—
107	C₂×C₃×#41	—	5	1	no	—
108	C₃×#131	—	10	2	no	—
109	C₃×#131	—	10	2	no	—
110	C₂×C₃×M₁₆	—	5	1	no	—
111	C₂×C₃×M₁₆	—	5	1	no	—
112	C₂×C₃×M₁₆	—	10	2	no	—
113	C₂×C₃×M₁₆	C₂×D₅	1	1	yes	characteristic
114	C₂×C₃×M₁₆	C₂×D₅	1	1	yes	characteristic
115	C₂×C₃×M₁₆	C₂×D₅	1	1	yes	characteristic
116	C₂²×C₃×C₈	—	5	1	no	—
117	C₃×#108	—	2	2	no	—
118	C₃×#108	—	10	2	no	—
119	C₃×#101	—	5	2	no	—
120	C₃×#100	—	5	2	no	—
121	C₃×C₂²⋊C₈	—	10	2	no	—
122	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
123	C₂×C₂²⋊D₅	C₃×Q₈	1	2	yes	—
124	#709	—	2	1	no	—
125	#709	—	2	1	no	—
126	#709	—	2	1	no	—
127	#709	—	2	1	no	—
128	C₂×D₂₀	C₂×C₃×C₄	1	1	yes	characteristic
129	C₂×C₄×D₅	C₃×D₄	1	2	yes	—
130	C₂×Dic₁₀	C₂×C₃×C₄	1	1	yes	characteristic
131	C₅×M₁₆	—	2	1	no	—
132	C₅×M₁₆	—	2	1	no	—
133	C₅×M₁₆	—	2	1	no	—
134	C₅×M₁₆	—	2	1	no	—
135	C₅×M₁₆	C₂×C₃×C₄	1	2	yes	—
136	C₅×M₁₆	C₃×D₄	1	2	yes	—
137	C₂×C₅×C₈	—	2	1	no	—
138	C₂×C₅×C₈	—	2	1	no	—
139	C₂×C₅×C₈	—	2	1	no	—
140	#681	—	2	1	no	—
141	#681	—	2	1	no	—
142	#681	C₂×C₃×C₄	1	2	yes	—
143	#681	C₃×D₄	1	2	yes	—
144	C₂×#168	—	2	1	no	—
145	C₂×#168	—	2	1	no	—
146	C₂×#168	—	2	1	no	—
147	C₂×#168	—	2	1	no	—
148	C₂×#168	—	2	1	no	—
149	#676	—	2	4	no	—
150	C₈×D₅	—	2	4	no	—
151	#568	—	5	1	no	—
152	#430	—	5	1	no	—
153	#430	C₃×D₅	1	1	yes	characteristic, complemented
154	#413	—	5	1	no	—
155	#408	—	5	1	no	—
156	#341	—	5	1	no	—
157	#341	—	5	1	no	—
158	C₂²×C₃×C₅	—	4	1	no	—
159	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
160	C₂²×C₃×C₅	C₂²⋊Q₈	1	1	yes	characteristic
161	C₂²×C₃×C₅	C₂²⋊Q₈	1	1	yes	characteristic
162	C₂×C₃×D₅	—	2	2	no	—
163	C₂×C₃×D₅	—	4	2	no	—
164	C₃×C₄×C₅	C₄×D₄	1	1	yes	characteristic
165	C₃×C₄×C₅	C₄×D₄	1	1	yes	characteristic
166	C₃×C₄×C₅	#120	1	1	yes	characteristic
167	C₃×C₄×C₅	#123	1	1	yes	characteristic
168	C₃×Dic₅	—	2	2	no	—
169	C₃×#41	—	10	1	no	—
170	C₃×#41	—	10	1	no	—
171	C₃×#41	—	10	1	no	—
172	C₃×#41	—	10	1	no	—
173	C₂×C₃×Q₈	—	5	1	no	—
174	C₂×C₃×D₄	—	5	2	no	—
175	C₂×C₃×D₄	—	5	1	no	—
176	C₂²×C₃×C₄	—	5	2	no	—
177	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
178	C₃×M₁₆	—	2	1	no	—
179	C₃×M₁₆	—	2	1	no	—
180	C₃×M₁₆	—	2	1	no	—
181	C₃×M₁₆	—	2	1	no	—
182	C₃×M₁₆	—	5	2	no	—
183	C₃×M₁₆	—	5	2	no	—
184	C₃×M₁₆	—	10	4	no	—
185	C₃×M₁₆	—	10	1	no	—
186	C₃×M₁₆	—	10	1	no	—
187	C₃×M₁₆	C₄×D₅	1	2	yes	—
188	C₃×M₁₆	C₂²⋊D₅	1	2	yes	—
189	C₂×C₃×C₈	—	2	1	no	—
190	C₂×C₃×C₈	—	2	1	no	—
191	C₂×C₃×C₈	—	2	1	no	—
192	C₂×C₃×C₈	—	10	4	no	—
193	C₂×C₃×C₈	—	10	1	no	—
194	C₂×C₃×C₈	—	10	1	no	—
195	C₂×C₃×C₈	—	10	1	no	—
196	C₂×C₃×C₈	—	10	1	no	—
197	C₂×C₃×C₈	—	10	1	no	—
198	C₂³×C₅	C₂×C₃×Q₈	1	1	yes	characteristic
199	C₂²×D₅	C₃×C₄⋊C₄	1	2	yes	—
200	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
201	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
202	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
203	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
204	C₂×C₄×C₅	C₃×#41	1	1	yes	characteristic
205	C₂×C₄×C₅	C₃×#41	1	1	yes	characteristic
206	C₂²⋊D₅	—	2	2	no	—
207	C₂²⋊D₅	—	2	2	no	—
208	C₂×Dic₅	C₃×C₄⋊C₄	1	2	yes	—
209	D₂₀	—	2	1	no	—
210	D₂₀	—	2	1	no	—
211	C₄×D₅	—	2	2	no	—
212	C₄×D₅	—	2	2	no	—
213	Dic₁₀	—	2	1	no	—
214	Dic₁₀	—	2	1	no	—
215	C₅×C₈	—	2	2	no	—
216	C₅×C₈	—	2	2	no	—
217	C₅×C₈	—	4	1	no	—
218	#168	—	2	2	no	—
219	#168	—	2	2	no	—
220	#168	—	4	1	no	—
221	C₂×#41	—	5	1	no	—
222	#131	—	10	2	no	—
223	#131	—	10	2	no	—
224	C₂×M₁₆	—	5	1	no	—
225	C₂×M₁₆	—	5	1	no	—
226	C₂×M₁₆	—	10	2	no	—
227	C₂×M₁₆	C₂×C₃×D₅	1	1	yes	characteristic
228	C₂×M₁₆	C₂×C₃×D₅	1	1	yes	characteristic
229	C₂×M₁₆	C₂×C₃×D₅	1	1	yes	characteristic
230	C₂²×C₈	—	5	1	no	—
231	#108	—	2	2	no	—
232	#108	—	10	2	no	—
233	#101	—	5	2	no	—
234	#100	—	5	2	no	—
235	C₂²⋊C₈	—	10	2	no	—
236	C₂×C₃×C₅	—	2	1	no	—
237	C₂×C₃×C₅	—	2	1	no	—
238	C₂×C₃×C₅	—	2	1	no	—
239	C₂×C₃×C₅	#385	1	1	yes	characteristic
240	C₃×D₅	—	4	2	no	—
241	C₂³×C₃	—	5	2	no	—
242	C₂³×C₃	Q₈×D₅	1	1	yes	characteristic
243	C₃×Q₈	—	10	1	no	—
244	C₃×Q₈	—	10	1	no	—
245	C₃×D₄	—	10	2	no	—
246	C₃×D₄	—	10	2	no	—
247	C₃×D₄	—	10	1	no	—
248	C₃×D₄	—	10	1	no	—
249	C₂×C₃×C₄	—	5	2	no	—
250	C₂×C₃×C₄	—	10	2	no	—
251	C₂×C₃×C₄	—	10	2	no	—
252	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
253	C₂×C₃×C₄	#709	1	1	yes	characteristic
254	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
255	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
256	C₂×C₃×C₄	#713	1	1	yes	characteristic
257	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
258	C₃×C₈	—	2	2	no	—
259	C₃×C₈	—	2	2	no	—
260	C₃×C₈	—	4	1	no	—
261	C₃×C₈	—	10	2	no	—
262	C₃×C₈	—	10	2	no	—
263	C₃×C₈	—	20	1	no	—
264	C₂²×C₅	—	4	1	no	—
265	C₂²×C₅	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
266	C₂²×C₅	C₃×C₂²⋊Q₈	1	1	yes	characteristic
267	C₂²×C₅	C₃×C₂²⋊Q₈	1	1	yes	characteristic
268	C₂×D₅	—	2	2	no	—
269	C₂×D₅	—	4	2	no	—
270	C₄×C₅	C₃×C₄×D₄	1	1	yes	characteristic
271	C₄×C₅	C₃×C₄×D₄	1	1	yes	characteristic
272	C₄×C₅	C₃×#120	1	1	yes	characteristic
273	C₄×C₅	C₃×#123	1	1	yes	characteristic
274	Dic₅	—	2	2	no	—
275	#41	—	10	1	no	—
276	#41	—	10	1	no	—
277	#41	—	10	1	no	—
278	#41	—	10	1	no	—
279	C₂×Q₈	—	5	1	no	—
280	C₂×D₄	—	5	2	no	—
281	C₂×D₄	—	5	1	no	—
282	C₂²×C₄	—	5	2	no	—
283	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
284	M₁₆	—	2	1	no	—
285	M₁₆	—	2	1	no	—
286	M₁₆	—	2	1	no	—
287	M₁₆	—	2	1	no	—
288	M₁₆	—	5	2	no	—
289	M₁₆	—	5	2	no	—
290	M₁₆	—	10	4	no	—
291	M₁₆	—	10	1	no	—
292	M₁₆	—	10	1	no	—
293	M₁₆	C₃×C₄×D₅	1	2	yes	—
294	M₁₆	C₃×C₂²⋊D₅	1	2	yes	—
295	C₂×C₈	—	2	1	no	—
296	C₂×C₈	—	2	1	no	—
297	C₂×C₈	—	2	1	no	—
298	C₂×C₈	—	10	4	no	—
299	C₂×C₈	—	10	1	no	—
300	C₂×C₈	—	10	1	no	—
301	C₂×C₈	—	10	1	no	—
302	C₂×C₈	—	10	1	no	—
303	C₂×C₈	—	10	1	no	—
304	C₃×C₅	#1858	1	1	yes	characteristic, complemented, Hall
305	C₂²×C₃	—	4	1	no	—
306	C₂²×C₃	—	10	2	no	—
307	C₂²×C₃	—	20	2	no	—
308	C₂²×C₃	D₅×C₄⋊C₄	1	1	yes	characteristic
309	C₂²×C₃	#4042	1	1	yes	characteristic
310	C₂²×C₃	#4092	1	1	yes	characteristic
311	C₃×C₄	—	10	2	no	—
312	C₃×C₄	#4039	1	1	yes	characteristic
313	C₃×C₄	#4040	1	1	yes	characteristic
314	C₃×C₄	C₄×C₂²⋊D₅	1	1	yes	characteristic
315	C₃×C₄	#4083	1	1	yes	characteristic
316	C₂×C₅	—	2	1	no	—
317	C₂×C₅	—	2	1	no	—
318	C₂×C₅	—	2	1	no	—
319	C₂×C₅	C₃×#385	1	1	yes	characteristic
320	D₅	—	4	2	no	—
321	C₂³	—	5	2	no	—
322	C₂³	C₃×Q₈×D₅	1	1	yes	characteristic
323	Q₈	—	10	1	no	—
324	Q₈	—	10	1	no	—
325	D₄	—	10	2	no	—
326	D₄	—	10	2	no	—
327	D₄	—	10	1	no	—
328	D₄	—	10	1	no	—
329	C₂×C₄	—	5	2	no	—
330	C₂×C₄	—	10	2	no	—
331	C₂×C₄	—	10	2	no	—
332	C₂×C₄	C₂×C₃×C₄×D₅	1	1	yes	characteristic
333	C₂×C₄	C₃×#709	1	1	yes	characteristic
334	C₂×C₄	C₃×D₄×D₅	1	1	yes	characteristic
335	C₂×C₄	C₃×D₄×D₅	1	1	yes	characteristic
336	C₂×C₄	C₃×#713	1	1	yes	characteristic
337	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
338	C₈	—	2	2	no	—
339	C₈	—	2	2	no	—
340	C₈	—	4	1	no	—
341	C₈	—	10	2	no	—
342	C₈	—	10	2	no	—
343	C₈	—	20	1	no	—
344	C₂×C₃	—	2	1	no	—
345	C₂×C₃	—	2	1	no	—
346	C₂×C₃	—	2	1	no	—
347	C₂×C₃	—	20	2	no	—
348	C₂×C₃	#65278	1	1	yes	characteristic
349	C₅	C₃×#1858	1	1	yes	characteristic, complemented, Sylow
350	C₂²	—	4	1	no	—
351	C₂²	—	10	2	no	—
352	C₂²	—	20	2	no	—
353	C₂²	C₃×D₅×C₄⋊C₄	1	1	yes	characteristic
354	C₂²	C₃×#4042	1	1	yes	characteristic
355	C₂²	C₃×#4092	1	1	yes	characteristic
356	C₄	—	10	2	no	—
357	C₄	C₃×#4039	1	1	yes	characteristic
358	C₄	C₃×#4040	1	1	yes	characteristic
359	C₄	C₃×C₄×C₂²⋊D₅	1	1	yes	characteristic
360	C₄	C₃×#4083	1	1	yes	characteristic
361	C₃	#10601527	1	1	yes	characteristic, direct product factor, Sylow
362	C₂	—	2	1	no	—
363	C₂	—	2	1	no	—
364	C₂	—	2	1	no	—
365	C₂	—	20	2	no	—
366	C₂	C₃×#65278	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₄	#4083
commutator	C₂×C₄×C₅	C₂²×C₃×C₄
Frattini	C₂²×C₄	C₂²×C₃×D₅
Fitting	C₃×C₅×#430	C₂
socle	C₂×C₃×C₅	#385
prime radical	C₂×C₃×#709	C₂²
cosocle	C₂²×C₄×C₅	C₂³×C₃
hypercenter	C₃×#430	D₅
hypocenter	C₅	C₃×#1858
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₃×C₄ ◁ C₂²×C₃×C₄ ◁ C₃×#430

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×#430	C₂	semidirect
5. C₃×#65413	C₂	semidirect
6. C₃×#64778	C₂	semidirect
7. C₃×#64778	C₂	semidirect
8. #10601527	C₃	direct (N × H)
26. C₅×#430	C₂×C₃	semidirect
29. #65413	C₂×C₃	semidirect
30. #64778	C₂×C₃	semidirect
31. #64778	C₂×C₃	semidirect
63. C₃×#430	D₅	semidirect
153. #430	C₃×D₅	semidirect
304. C₃×C₅	#1858	semidirect
349. C₅	C₃×#1858	semidirect
361. C₃	#10601527	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×#66079	2
C₃×C₅×#430	2
C₃×#65421	2
C₃×#65418	2
C₃×#65413	2
C₃×#64778	2
#10601527	3
C₃×#1858	5

Subgroups — C3×#10601527 #14299657

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10601527 #14299657