Subgroups — C₃×#10601601 #14284333

Order 1920.

496 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#4021	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#432	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#65215	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#64684	C₂	1	4	yes	maximal, complemented
5	#10601601	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂²×C₃×D₂₀	C₄	1	1	yes	characteristic
7	C₂²×C₃×Dic₁₀	C₄	1	1	yes	characteristic
8	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
9	C₂×C₃×#685	—	2	1	no	—
10	C₂×C₃×#685	—	2	1	no	—
11	C₃×#4021	—	2	1	no	—
12	C₃×#4021	—	2	1	no	—
13	C₃×#4021	C₄	1	2	yes	—
14	C₃×#4021	C₂²	1	2	yes	—
15	C₃×C₅×C₄⋊C₈	—	2	1	no	—
16	C₃×C₅×C₂²⋊C₈	—	2	1	no	—
17	C₃×C₅×#97	C₂²	1	2	yes	complemented
18	C₃×C₂²⋊#168	—	2	1	no	—
19	C₃×C₄⋊#168	—	2	1	no	—
20	C₃×#3935	C₂²	1	2	yes	complemented
21	C₃×#1533	—	5	1	no	maximal, Hall
22	C₂×#4021	C₂×C₃	1	1	yes	characteristic
23	C₅×#432	C₂×C₃	1	1	yes	characteristic, complemented
24	#65215	C₂×C₃	1	1	yes	characteristic, complemented
25	#64684	C₂×C₃	1	4	yes	complemented
26	C₂³×C₃×D₅	—	2	1	no	—
27	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
29	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
30	C₂²×C₃×Dic₅	—	2	1	no	—
31	C₂×C₃×D₂₀	—	2	1	no	—
32	C₂×C₃×D₂₀	—	4	2	no	—
33	C₂×C₃×D₂₀	C₂×C₄	1	2	yes	—
34	C₂×C₃×Dic₁₀	—	2	1	no	—
35	C₂×C₃×Dic₁₀	—	4	2	no	—
36	C₂×C₃×Dic₁₀	C₂×C₄	1	2	yes	—
37	C₂×C₃×C₅×C₈	—	2	2	no	—
38	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
39	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
40	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
41	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
42	C₃×#685	—	2	2	no	—
43	C₃×#685	—	2	2	no	—
44	C₃×#685	—	2	2	no	—
45	C₃×#685	—	2	2	no	—
46	C₂×C₃×#168	—	2	2	no	—
47	C₂×C₃×#124	—	5	1	no	—
48	C₃×#432	—	5	1	no	—
49	C₃×#432	D₅	1	1	yes	characteristic, complemented
50	C₃×#329	—	5	4	no	—
51	C₂²×D₂₀	C₃×C₄	1	1	yes	characteristic
52	C₂²×Dic₁₀	C₃×C₄	1	1	yes	characteristic
53	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
54	C₂×#685	—	2	1	no	—
55	C₂×#685	—	2	1	no	—
56	#4021	—	2	1	no	—
57	#4021	—	2	1	no	—
58	#4021	C₃×C₄	1	2	yes	—
59	#4021	C₂²×C₃	1	2	yes	—
60	C₅×C₄⋊C₈	—	2	1	no	—
61	C₅×C₂²⋊C₈	—	2	1	no	—
62	C₅×#97	C₂²×C₃	1	2	yes	complemented
63	C₂²⋊#168	—	2	1	no	—
64	C₄⋊#168	—	2	1	no	—
65	#3935	C₂²×C₃	1	2	yes	complemented
66	#1533	—	5	1	no	Sylow
67	C₂³×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₃×D₅	—	2	2	no	—
69	C₂²×C₃×D₅	—	4	2	no	—
70	C₂²×C₃×D₅	—	4	2	no	—
71	C₂²×C₃×D₅	—	4	2	no	—
72	C₂×C₃×C₄×C₅	—	2	2	no	—
73	C₂×C₃×C₄×C₅	—	4	1	no	—
74	C₂×C₃×C₄×C₅	—	4	1	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	C₂×D₄	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	—	4	2	no	—
83	C₃×D₂₀	—	4	2	no	—
84	C₃×D₂₀	—	4	2	no	—
85	C₃×Dic₁₀	—	4	2	no	—
86	C₃×Dic₁₀	—	4	2	no	—
87	C₃×C₅×C₈	—	4	2	no	—
88	C₃×#168	—	4	2	no	—
89	C₂²×C₃×Q₈	—	5	1	no	—
90	C₂²×C₃×D₄	—	5	1	no	—
91	C₃×#124	—	5	2	no	—
92	C₃×#124	—	5	2	no	—
93	C₃×#124	—	10	1	no	—
94	C₃×#124	—	10	1	no	—
95	C₂×C₃×C₂²⋊C₄	—	10	1	no	—
96	C₂×C₃×C₂²⋊C₄	—	10	1	no	—
97	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
98	C₃×C₄⋊C₈	—	2	1	no	—
99	C₃×C₄⋊C₈	—	10	1	no	—
100	C₃×C₂²⋊C₈	—	2	1	no	—
101	C₃×C₂²⋊C₈	—	10	1	no	—
102	C₃×#97	—	5	2	no	—
103	C₃×#97	C₂×D₅	1	2	yes	complemented
104	C₂³×D₅	—	2	1	no	—
105	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
106	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
107	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
108	C₂²×Dic₅	—	2	1	no	—
109	C₂×D₂₀	—	2	1	no	—
110	C₂×D₂₀	—	4	2	no	—
111	C₂×D₂₀	C₂×C₃×C₄	1	2	yes	—
112	C₂×Dic₁₀	—	2	1	no	—
113	C₂×Dic₁₀	—	4	2	no	—
114	C₂×Dic₁₀	C₂×C₃×C₄	1	2	yes	—
115	C₂×C₅×C₈	—	2	2	no	—
116	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
117	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
118	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
119	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
120	#685	—	2	2	no	—
121	#685	—	2	2	no	—
122	#685	—	2	2	no	—
123	#685	—	2	2	no	—
124	C₂×#168	—	2	2	no	—
125	C₂×#124	—	5	1	no	—
126	#432	—	5	1	no	—
127	#432	C₃×D₅	1	1	yes	characteristic, complemented
128	#329	—	5	4	no	—
129	C₂²×C₃×C₅	—	2	1	no	—
130	C₂²×C₃×C₅	—	2	1	no	—
131	C₂²×C₃×C₅	#100	1	2	yes	—
132	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
133	C₂×C₃×D₅	—	4	2	no	—
134	C₂×C₃×D₅	—	4	2	no	—
135	C₂×C₃×D₅	—	4	2	no	—
136	C₂×C₃×D₅	—	8	4	no	—
137	C₃×C₄×C₅	—	2	1	no	—
138	C₃×C₄×C₅	—	2	1	no	—
139	C₃×C₄×C₅	—	2	1	no	—
140	C₃×C₄×C₅	—	2	1	no	—
141	C₃×C₄×C₅	—	2	1	no	—
142	C₃×C₄×C₅	—	2	1	no	—
143	C₃×Dic₅	—	4	2	no	—
144	C₂⁴×C₃	—	10	1	no	—
145	C₂×C₃×Q₈	—	5	2	no	—
146	C₂×C₃×Q₈	—	10	1	no	—
147	C₂×C₃×Q₈	—	20	2	no	—
148	C₂×C₃×D₄	—	5	2	no	—
149	C₂×C₃×D₄	—	10	1	no	—
150	C₂×C₃×D₄	—	20	2	no	—
151	C₂²×C₃×C₄	—	10	1	no	—
152	C₂²×C₃×C₄	C₄×D₅	1	1	yes	characteristic
153	C₂²×C₃×C₄	D₂₀	1	1	yes	characteristic
154	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
155	C₂×C₃×C₈	—	2	2	no	—
156	C₂×C₃×C₈	—	10	2	no	—
157	C₃×C₂²⋊C₄	—	10	2	no	—
158	C₃×C₂²⋊C₄	—	10	2	no	—
159	C₃×C₂²⋊C₄	—	10	2	no	—
160	C₃×C₂²⋊C₄	—	10	2	no	—
161	C₃×C₄²	C₄×D₅	1	1	yes	characteristic
162	C₃×C₄²	D₂₀	1	1	yes	characteristic
163	C₃×C₄²	C₂²⋊D₅	1	1	yes	characteristic
164	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
165	C₂³×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
166	C₂²×D₅	—	2	2	no	—
167	C₂²×D₅	—	4	2	no	—
168	C₂²×D₅	—	4	2	no	—
169	C₂²×D₅	—	4	2	no	—
170	C₂×C₄×C₅	—	2	2	no	—
171	C₂×C₄×C₅	—	4	1	no	—
172	C₂×C₄×C₅	—	4	1	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	C₂²×C₃×C₄	1	1	yes	characteristic
177	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
178	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
179	C₂×Dic₅	—	2	2	no	—
180	C₂×Dic₅	—	4	2	no	—
181	D₂₀	—	4	2	no	—
182	D₂₀	—	4	2	no	—
183	Dic₁₀	—	4	2	no	—
184	Dic₁₀	—	4	2	no	—
185	C₅×C₈	—	4	2	no	—
186	#168	—	4	2	no	—
187	C₂²×Q₈	—	5	1	no	—
188	C₂²×D₄	—	5	1	no	—
189	#124	—	5	2	no	—
190	#124	—	5	2	no	—
191	#124	—	10	1	no	—
192	#124	—	10	1	no	—
193	C₂×C₂²⋊C₄	—	10	1	no	—
194	C₂×C₂²⋊C₄	—	10	1	no	—
195	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
196	C₄⋊C₈	—	2	1	no	—
197	C₄⋊C₈	—	10	1	no	—
198	C₂²⋊C₈	—	2	1	no	—
199	C₂²⋊C₈	—	10	1	no	—
200	#97	—	5	2	no	—
201	#97	C₂×C₃×D₅	1	2	yes	complemented
202	C₂×C₃×C₅	—	2	2	no	—
203	C₂×C₃×C₅	C₂×#100	1	1	yes	characteristic
204	C₂×C₃×C₅	#421	1	1	yes	characteristic
205	C₂×C₃×C₅	#421	1	1	yes	characteristic
206	C₃×D₅	—	8	2	no	—
207	C₂³×C₃	—	10	2	no	—
208	C₂³×C₃	—	20	2	no	—
209	C₂³×C₃	—	20	2	no	—
210	C₂³×C₃	—	20	2	no	—
211	C₂³×C₃	#685	1	1	yes	characteristic
212	C₃×Q₈	—	20	2	no	—
213	C₃×Q₈	—	20	2	no	—
214	C₃×D₄	—	20	2	no	—
215	C₃×D₄	—	20	2	no	—
216	C₂×C₃×C₄	—	2	2	no	—
217	C₂×C₃×C₄	—	4	1	no	—
218	C₂×C₃×C₄	—	4	1	no	—
219	C₂×C₃×C₄	—	10	2	no	—
220	C₂×C₃×C₄	—	20	2	no	—
221	C₂×C₃×C₄	#685	1	1	yes	characteristic
222	C₂×C₃×C₄	#685	1	1	yes	characteristic
223	C₂×C₃×C₄	#685	1	1	yes	characteristic
224	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
225	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
226	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
227	C₃×C₈	—	4	2	no	—
228	C₃×C₈	—	20	2	no	—
229	C₂²×C₅	—	2	1	no	—
230	C₂²×C₅	—	2	1	no	—
231	C₂²×C₅	C₃×#100	1	2	yes	—
232	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
233	C₂×D₅	—	4	2	no	—
234	C₂×D₅	—	4	2	no	—
235	C₂×D₅	—	4	2	no	—
236	C₂×D₅	—	8	4	no	—
237	C₄×C₅	—	2	1	no	—
238	C₄×C₅	—	2	1	no	—
239	C₄×C₅	—	2	1	no	—
240	C₄×C₅	—	2	1	no	—
241	C₄×C₅	—	2	1	no	—
242	C₄×C₅	—	2	1	no	—
243	Dic₅	—	4	2	no	—
244	C₂⁴	—	10	1	no	—
245	C₂×Q₈	—	5	2	no	—
246	C₂×Q₈	—	10	1	no	—
247	C₂×Q₈	—	20	2	no	—
248	C₂×D₄	—	5	2	no	—
249	C₂×D₄	—	10	1	no	—
250	C₂×D₄	—	20	2	no	—
251	C₂²×C₄	—	10	1	no	—
252	C₂²×C₄	C₃×C₄×D₅	1	1	yes	characteristic
253	C₂²×C₄	C₃×D₂₀	1	1	yes	characteristic
254	C₂²×C₄	C₃×C₂²⋊D₅	1	1	yes	characteristic
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	10	2	no	—
257	C₂²⋊C₄	—	10	2	no	—
258	C₂²⋊C₄	—	10	2	no	—
259	C₂²⋊C₄	—	10	2	no	—
260	C₂²⋊C₄	—	10	2	no	—
261	C₄²	C₃×C₄×D₅	1	1	yes	characteristic
262	C₄²	C₃×D₂₀	1	1	yes	characteristic
263	C₄²	C₃×C₂²⋊D₅	1	1	yes	characteristic
264	C₄²	C₂²×C₃×D₅	1	1	yes	characteristic
265	C₃×C₅	#1533	1	1	yes	characteristic, complemented, Hall
266	C₂²×C₃	—	2	1	no	—
267	C₂²×C₃	—	2	1	no	—
268	C₂²×C₃	—	20	2	no	—
269	C₂²×C₃	—	20	2	no	—
270	C₂²×C₃	—	20	2	no	—
271	C₂²×C₃	—	40	4	no	—
272	C₂²×C₃	#3955	1	2	yes	—
273	C₂²×C₃	C₂×#685	1	1	yes	characteristic
274	C₃×C₄	—	2	1	no	—
275	C₃×C₄	—	2	1	no	—
276	C₃×C₄	—	2	1	no	—
277	C₃×C₄	—	2	1	no	—
278	C₃×C₄	—	2	1	no	—
279	C₃×C₄	—	2	1	no	—
280	C₃×C₄	—	20	2	no	—
281	C₂×C₅	—	2	2	no	—
282	C₂×C₅	C₂×C₃×#100	1	1	yes	characteristic
283	C₂×C₅	C₃×#421	1	1	yes	characteristic
284	C₂×C₅	C₃×#421	1	1	yes	characteristic
285	D₅	—	8	2	no	—
286	C₂³	—	10	2	no	—
287	C₂³	—	20	2	no	—
288	C₂³	—	20	2	no	—
289	C₂³	—	20	2	no	—
290	C₂³	C₃×#685	1	1	yes	characteristic
291	Q₈	—	20	2	no	—
292	Q₈	—	20	2	no	—
293	D₄	—	20	2	no	—
294	D₄	—	20	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	10	2	no	—
299	C₂×C₄	—	20	2	no	—
300	C₂×C₄	C₃×#685	1	1	yes	characteristic
301	C₂×C₄	C₃×#685	1	1	yes	characteristic
302	C₂×C₄	C₃×#685	1	1	yes	characteristic
303	C₂×C₄	C₂×C₃×C₄×D₅	1	1	yes	characteristic
304	C₂×C₄	C₂×C₃×D₂₀	1	1	yes	characteristic
305	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
306	C₈	—	4	2	no	—
307	C₈	—	20	2	no	—
308	C₂×C₃	—	2	2	no	—
309	C₂×C₃	—	40	2	no	—
310	C₂×C₃	#65294	1	1	yes	characteristic
311	C₂×C₃	C₂×#3955	1	1	yes	characteristic
312	C₂×C₃	#65428	1	1	yes	characteristic
313	C₅	C₃×#1533	1	1	yes	characteristic, complemented, Sylow
314	C₂²	—	2	1	no	—
315	C₂²	—	2	1	no	—
316	C₂²	—	20	2	no	—
317	C₂²	—	20	2	no	—
318	C₂²	—	20	2	no	—
319	C₂²	—	40	4	no	—
320	C₂²	C₃×#3955	1	2	yes	—
321	C₂²	C₂×C₃×#685	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	20	2	no	—
329	C₃	#10601601	1	1	yes	characteristic, direct product factor, Sylow
330	C₂	—	2	2	no	—
331	C₂	—	40	2	no	—
332	C₂	C₃×#65294	1	1	yes	characteristic
333	C₂	C₂×C₃×#3955	1	1	yes	characteristic
334	C₂	C₃×#65428	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂×#685
commutator	C₂×C₄×C₅	C₂²×C₃×C₄
Frattini	C₄²	C₂²×C₃×D₅
Fitting	C₃×C₅×#432	C₂
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₂²×C₃×D₂₀	C₄
cosocle	C₄²×C₅	C₂³×C₃
hypercenter	C₃×#432	D₅
hypocenter	C₅	C₃×#1533
LCS	G ▷ C₂×C₄×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄² ◁ C₃×#432

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×#432	C₂	semidirect
3. C₃×#65215	C₂	semidirect
4. C₃×#64684	C₂	semidirect
5. #10601601	C₃	direct (N × H)
17. C₃×C₅×#97	C₂²	semidirect
20. C₃×#3935	C₂²	semidirect
23. C₅×#432	C₂×C₃	semidirect
24. #65215	C₂×C₃	semidirect
25. #64684	C₂×C₃	semidirect
49. C₃×#432	D₅	semidirect
62. C₅×#97	C₂²×C₃	semidirect
65. #3935	C₂²×C₃	semidirect
103. C₃×#97	C₂×D₅	semidirect
127. #432	C₃×D₅	semidirect
201. #97	C₂×C₃×D₅	semidirect
265. C₃×C₅	#1533	semidirect
313. C₅	C₃×#1533	semidirect
329. C₃	#10601601	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×#4021	2
C₃×C₅×#432	2
C₃×#65215	2
C₃×#64684	2
#10601601	3
C₃×#1533	5

Subgroups — C3×#10601601 #14284333

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10601601 #14284333