Subgroups — C₃×#10605120 #14324378

Order 1920.

1480 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#4020	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₄×D₂₀	C₂	1	2	yes	maximal, complemented
3	C₃×C₅×#376	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#65228	C₂	1	2	yes	maximal, complemented
5	C₃×#65225	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#64954	C₂	1	8	yes	maximal, complemented
7	#10605120	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₂²×C₃×D₂₀	C₂²	1	2	yes	—
9	C₂²×C₃×D₂₀	C₂²	1	4	yes	—
10	C₂²×C₃×C₄×D₅	C₂²	1	4	yes	—
11	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
14	C₂×C₃×#685	C₂²	1	4	yes	—
15	C₂×C₃×#685	C₂²	1	8	yes	—
16	C₂×C₃×C₄⋊Dic₅	C₂²	1	2	yes	—
17	C₃×#4020	C₄	1	8	yes	complemented
18	C₃×C₄×D₂₀	—	2	8	no	—
19	C₃×C₅×#95	C₂²	1	4	yes	—
20	C₃×#3963	C₂²	1	4	yes	—
21	C₃×#2328	—	5	1	no	maximal, Hall
22	C₂×#4020	C₂×C₃	1	1	yes	characteristic
23	C₂×C₄×D₂₀	C₂×C₃	1	2	yes	complemented
24	C₅×#376	C₂×C₃	1	1	yes	characteristic, complemented
25	#65228	C₂×C₃	1	2	yes	complemented
26	#65225	C₂×C₃	1	1	yes	characteristic, complemented
27	#64954	C₂×C₃	1	8	yes	complemented
28	C₂³×C₃×D₅	C₂³	1	4	yes	—
29	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
30	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
31	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
32	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
33	C₂²×C₃×Dic₅	C₂³	1	4	yes	—
34	C₂×C₃×D₂₀	—	2	4	no	—
35	C₂×C₃×D₂₀	—	2	4	no	—
36	C₂×C₃×D₂₀	—	4	8	no	—
37	C₂×C₃×D₂₀	C₂×C₄	1	8	yes	complemented
38	C₂×C₃×D₂₀	C₂×C₄	1	16	yes	complemented
39	C₂×C₃×C₄×D₅	—	2	8	no	—
40	C₂×C₃×C₄×D₅	—	4	8	no	—
41	C₃×C₄²×C₅	—	2	2	no	—
42	C₃×C₄²×C₅	C₂×C₄	1	4	yes	complemented
43	C₃×C₄²×C₅	D₄	1	4	yes	complemented
44	C₃×#685	—	2	8	no	—
45	C₃×#685	—	4	8	no	—
46	C₃×C₄⋊Dic₅	—	2	4	no	—
47	C₂×C₃×#127	—	5	1	no	—
48	C₂×C₃×C₄×D₄	—	5	2	no	—
49	C₃×#390	—	5	2	no	—
50	C₃×#386	—	5	8	no	—
51	C₃×#382	—	5	1	no	—
52	C₃×#376	D₅	1	1	yes	characteristic, complemented
53	C₂²×D₂₀	C₂²×C₃	1	2	yes	—
54	C₂²×D₂₀	C₂²×C₃	1	4	yes	—
55	C₂²×C₄×D₅	C₂²×C₃	1	4	yes	—
56	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
57	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
58	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
59	C₂×#685	C₂²×C₃	1	4	yes	—
60	C₂×#685	C₂²×C₃	1	8	yes	—
61	C₂×C₄⋊Dic₅	C₂²×C₃	1	2	yes	—
62	#4020	C₃×C₄	1	8	yes	complemented
63	C₄×D₂₀	—	2	8	no	—
64	C₅×#95	C₂²×C₃	1	4	yes	—
65	#3963	C₂²×C₃	1	4	yes	—
66	#2328	—	5	1	no	Sylow
67	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
68	C₂²×C₃×D₅	—	2	4	no	—
69	C₂²×C₃×D₅	—	2	4	no	—
70	C₂²×C₃×D₅	—	2	8	no	—
71	C₂²×C₃×D₅	—	2	8	no	—
72	C₂²×C₃×D₅	C₂²×C₄	1	8	yes	—
73	C₂×C₃×C₄×C₅	—	2	4	no	—
74	C₂×C₃×C₄×C₅	—	2	4	no	—
75	C₂×C₃×C₄×C₅	—	2	2	no	—
76	C₂×C₃×C₄×C₅	—	2	4	no	—
77	C₂×C₃×C₄×C₅	C₂²×C₄	1	4	yes	—
78	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
79	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
80	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
81	C₂×C₃×C₄×C₅	#41	1	2	yes	—
82	C₂×C₃×Dic₅	—	2	4	no	—
83	C₂×C₃×Dic₅	—	2	8	no	—
84	C₃×D₂₀	—	2	16	no	—
85	C₃×D₂₀	—	4	16	no	—
86	C₃×C₄×D₅	—	4	16	no	—
87	C₂²×C₃×D₄	—	5	2	no	—
88	C₂²×C₃×D₄	—	5	4	no	—
89	C₂³×C₃×C₄	—	5	4	no	—
90	C₃×#127	—	5	8	no	—
91	C₃×C₄×D₄	—	10	8	no	—
92	C₂×C₃×C₄⋊C₄	—	5	2	no	—
93	C₂×C₃×C₂²⋊C₄	—	5	4	no	—
94	C₂×C₃×C₂²⋊C₄	—	5	8	no	—
95	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
96	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
97	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
98	C₃×#95	—	5	4	no	—
99	C₃×#95	C₂×D₅	1	4	yes	—
100	C₂³×D₅	C₂³×C₃	1	4	yes	—
101	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
102	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
103	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
104	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
105	C₂²×Dic₅	C₂³×C₃	1	4	yes	—
106	C₂×D₂₀	—	2	4	no	—
107	C₂×D₂₀	—	2	4	no	—
108	C₂×D₂₀	—	4	8	no	—
109	C₂×D₂₀	C₂×C₃×C₄	1	8	yes	complemented
110	C₂×D₂₀	C₂×C₃×C₄	1	16	yes	complemented
111	C₂×C₄×D₅	—	2	8	no	—
112	C₂×C₄×D₅	—	4	8	no	—
113	C₄²×C₅	—	2	2	no	—
114	C₄²×C₅	C₂×C₃×C₄	1	4	yes	complemented
115	C₄²×C₅	C₃×D₄	1	4	yes	complemented
116	#685	—	2	8	no	—
117	#685	—	4	8	no	—
118	C₄⋊Dic₅	—	2	4	no	—
119	C₂×#127	—	5	1	no	—
120	C₂×C₄×D₄	—	5	2	no	—
121	#390	—	5	2	no	—
122	#386	—	5	8	no	—
123	#382	—	5	1	no	—
124	#376	C₃×D₅	1	1	yes	characteristic, complemented
125	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
126	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
127	C₂²×C₃×C₅	C₂×#41	1	1	yes	characteristic
128	C₂²×C₃×C₅	H(5,𝔽₂)	1	4	yes	—
129	C₂×C₃×D₅	—	2	8	no	—
130	C₂×C₃×D₅	—	2	16	no	—
131	C₂×C₃×D₅	—	4	4	no	—
132	C₂×C₃×D₅	—	4	4	no	—
133	C₂×C₃×D₅	—	4	8	no	—
134	C₃×C₄×C₅	—	2	4	no	—
135	C₃×C₄×C₅	—	2	4	no	—
136	C₃×C₄×C₅	—	4	2	no	—
137	C₃×C₄×C₅	C₄×D₄	1	4	yes	—
138	C₃×Dic₅	—	4	4	no	—
139	C₂⁴×C₃	—	5	4	no	—
140	C₂×C₃×D₄	—	5	8	no	—
141	C₂×C₃×D₄	—	5	16	no	—
142	C₂×C₃×D₄	—	10	4	no	—
143	C₂×C₃×D₄	—	10	4	no	—
144	C₂×C₃×D₄	—	20	8	no	—
145	C₂²×C₃×C₄	—	5	4	no	—
146	C₂²×C₃×C₄	—	10	8	no	—
147	C₂²×C₃×C₄	—	20	8	no	—
148	C₂²×C₃×C₄	C₂²×D₅	1	2	yes	—
149	C₂²×C₃×C₄	C₂²×D₅	1	2	yes	—
150	C₂²×C₃×C₄	C₂²×D₅	1	2	yes	—
151	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
152	C₃×C₄⋊C₄	—	10	4	no	—
153	C₃×C₂²⋊C₄	—	10	8	no	—
154	C₃×C₂²⋊C₄	—	20	8	no	—
155	C₃×C₄²	—	2	2	no	—
156	C₃×C₄²	C₄×D₅	1	4	yes	complemented
157	C₃×C₄²	D₂₀	1	4	yes	complemented
158	C₂³×C₅	C₂⁴×C₃	1	1	yes	characteristic
159	C₂²×D₅	—	2	4	no	—
160	C₂²×D₅	—	2	4	no	—
161	C₂²×D₅	—	2	8	no	—
162	C₂²×D₅	—	2	8	no	—
163	C₂²×D₅	C₂²×C₃×C₄	1	8	yes	—
164	C₂×C₄×C₅	—	2	4	no	—
165	C₂×C₄×C₅	—	2	4	no	—
166	C₂×C₄×C₅	—	2	2	no	—
167	C₂×C₄×C₅	—	2	4	no	—
168	C₂×C₄×C₅	C₂²×C₃×C₄	1	4	yes	—
169	C₂×C₄×C₅	C₂²×C₃×C₄	1	2	yes	—
170	C₂×C₄×C₅	C₂×C₃×D₄	1	4	yes	—
171	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
172	C₂×C₄×C₅	C₃×#41	1	2	yes	—
173	C₂×Dic₅	—	2	4	no	—
174	C₂×Dic₅	—	2	8	no	—
175	D₂₀	—	2	16	no	—
176	D₂₀	—	4	16	no	—
177	C₄×D₅	—	4	16	no	—
178	C₂²×D₄	—	5	2	no	—
179	C₂²×D₄	—	5	4	no	—
180	C₂³×C₄	—	5	4	no	—
181	#127	—	5	8	no	—
182	C₄×D₄	—	10	8	no	—
183	C₂×C₄⋊C₄	—	5	2	no	—
184	C₂×C₂²⋊C₄	—	5	4	no	—
185	C₂×C₂²⋊C₄	—	5	8	no	—
186	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
187	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
188	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
189	#95	—	5	4	no	—
190	#95	C₂×C₃×D₅	1	4	yes	—
191	C₂×C₃×C₅	C₂×C₄×D₄	1	1	yes	characteristic
192	C₂×C₃×C₅	#518	1	2	yes	—
193	C₂×C₃×C₅	#535	1	2	yes	—
194	C₂×C₃×C₅	#540	1	2	yes	—
195	C₃×D₅	—	4	8	no	—
196	C₂³×C₃	—	5	8	no	—
197	C₂³×C₃	—	10	4	no	—
198	C₂³×C₃	—	10	4	no	—
199	C₂³×C₃	—	10	8	no	—
200	C₂³×C₃	—	10	8	no	—
201	C₂³×C₃	C₂³×D₅	1	1	yes	characteristic
202	C₃×D₄	—	10	16	no	—
203	C₃×D₄	—	20	16	no	—
204	C₂×C₃×C₄	—	2	4	no	—
205	C₂×C₃×C₄	—	2	4	no	—
206	C₂×C₃×C₄	—	2	2	no	—
207	C₂×C₃×C₄	—	2	4	no	—
208	C₂×C₃×C₄	—	10	4	no	—
209	C₂×C₃×C₄	—	10	8	no	—
210	C₂×C₃×C₄	—	20	16	no	—
211	C₂×C₃×C₄	C₂×C₄×D₅	1	4	yes	—
212	C₂×C₃×C₄	C₂×C₄×D₅	1	2	yes	—
213	C₂×C₃×C₄	C₂×D₂₀	1	4	yes	—
214	C₂×C₃×C₄	C₂×D₂₀	1	2	yes	—
215	C₂×C₃×C₄	#709	1	2	yes	—
216	C₂²×C₅	C₂³×C₃×C₄	1	1	yes	characteristic
217	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
218	C₂²×C₅	C₂×C₃×#41	1	1	yes	characteristic
219	C₂²×C₅	C₃×H(5,𝔽₂)	1	4	yes	—
220	C₂×D₅	—	2	8	no	—
221	C₂×D₅	—	2	16	no	—
222	C₂×D₅	—	4	4	no	—
223	C₂×D₅	—	4	4	no	—
224	C₂×D₅	—	4	8	no	—
225	C₄×C₅	—	2	4	no	—
226	C₄×C₅	—	2	4	no	—
227	C₄×C₅	—	4	2	no	—
228	C₄×C₅	C₃×C₄×D₄	1	4	yes	—
229	Dic₅	—	4	4	no	—
230	C₂⁴	—	5	4	no	—
231	C₂×D₄	—	5	8	no	—
232	C₂×D₄	—	5	16	no	—
233	C₂×D₄	—	10	4	no	—
234	C₂×D₄	—	10	4	no	—
235	C₂×D₄	—	20	8	no	—
236	C₂²×C₄	—	5	4	no	—
237	C₂²×C₄	—	10	8	no	—
238	C₂²×C₄	—	20	8	no	—
239	C₂²×C₄	C₂²×C₃×D₅	1	2	yes	—
240	C₂²×C₄	C₂²×C₃×D₅	1	2	yes	—
241	C₂²×C₄	C₂²×C₃×D₅	1	2	yes	—
242	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
243	C₄⋊C₄	—	10	4	no	—
244	C₂²⋊C₄	—	10	8	no	—
245	C₂²⋊C₄	—	20	8	no	—
246	C₄²	—	2	2	no	—
247	C₄²	C₃×C₄×D₅	1	4	yes	complemented
248	C₄²	C₃×D₂₀	1	4	yes	complemented
249	C₃×C₅	#2328	1	1	yes	characteristic, complemented, Hall
250	C₂²×C₃	—	10	8	no	—
251	C₂²×C₃	—	10	16	no	—
252	C₂²×C₃	—	20	4	no	—
253	C₂²×C₃	—	20	4	no	—
254	C₂²×C₃	—	20	8	no	—
255	C₂²×C₃	C₂²×C₄×D₅	1	1	yes	characteristic
256	C₂²×C₃	C₂²×D₂₀	1	1	yes	characteristic
257	C₂²×C₃	C₂×#709	1	1	yes	characteristic
258	C₂²×C₃	#4149	1	4	yes	—
259	C₃×C₄	—	2	4	no	—
260	C₃×C₄	—	2	4	no	—
261	C₃×C₄	—	4	2	no	—
262	C₃×C₄	—	20	4	no	—
263	C₃×C₄	C₄×D₂₀	1	4	yes	—
264	C₂×C₅	C₂×C₃×C₄×D₄	1	1	yes	characteristic
265	C₂×C₅	C₃×#518	1	2	yes	—
266	C₂×C₅	C₃×#535	1	2	yes	—
267	C₂×C₅	C₃×#540	1	2	yes	—
268	D₅	—	4	8	no	—
269	C₂³	—	5	8	no	—
270	C₂³	—	10	4	no	—
271	C₂³	—	10	4	no	—
272	C₂³	—	10	8	no	—
273	C₂³	—	10	8	no	—
274	C₂³	C₂³×C₃×D₅	1	1	yes	characteristic
275	D₄	—	10	16	no	—
276	D₄	—	20	16	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	10	4	no	—
282	C₂×C₄	—	10	8	no	—
283	C₂×C₄	—	20	16	no	—
284	C₂×C₄	C₂×C₃×C₄×D₅	1	4	yes	—
285	C₂×C₄	C₂×C₃×C₄×D₅	1	2	yes	—
286	C₂×C₄	C₂×C₃×D₂₀	1	4	yes	—
287	C₂×C₄	C₂×C₃×D₂₀	1	2	yes	—
288	C₂×C₄	C₃×#709	1	2	yes	—
289	C₂×C₃	—	20	8	no	—
290	C₂×C₃	C₂×C₄×D₂₀	1	1	yes	characteristic
291	C₂×C₃	#65855	1	2	yes	—
292	C₂×C₃	#65859	1	2	yes	—
293	C₂×C₃	#65869	1	2	yes	—
294	C₅	C₃×#2328	1	1	yes	characteristic, complemented, Sylow
295	C₂²	—	10	16	no	—
296	C₂²	—	10	8	no	—
297	C₂²	—	20	4	no	—
298	C₂²	—	20	4	no	—
299	C₂²	—	20	8	no	—
300	C₂²	C₂²×C₃×C₄×D₅	1	1	yes	characteristic
301	C₂²	C₂²×C₃×D₂₀	1	1	yes	characteristic
302	C₂²	C₂×C₃×#709	1	1	yes	characteristic
303	C₂²	C₃×#4149	1	4	yes	—
304	C₄	—	2	4	no	—
305	C₄	—	2	4	no	—
306	C₄	—	4	2	no	—
307	C₄	—	20	4	no	—
308	C₄	C₃×C₄×D₂₀	1	4	yes	—
309	C₃	#10605120	1	1	yes	characteristic, direct product factor, Sylow
310	C₂	—	20	8	no	—
311	C₂	C₂×C₃×C₄×D₂₀	1	1	yes	characteristic
312	C₂	C₃×#65855	1	2	yes	—
313	C₂	C₃×#65859	1	2	yes	—
314	C₂	C₃×#65869	1	2	yes	—

	subgroup	G/H
center	C₂³×C₃	C₂³×D₅
commutator	C₂²×C₅	C₂³×C₃×C₄
Frattini	C₂³	C₂³×C₃×D₅
Fitting	C₃×C₅×#376	C₂
socle	C₂³×C₃×C₅	C₂⁴
prime radical	C₂×C₃×#4020	C₂
cosocle	C₂³×C₅	C₂⁴×C₃
hypercenter	C₃×#376	D₅
hypocenter	C₅	C₃×#2328
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³×C₃ ◁ C₃×#376

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₄×D₂₀	C₂	semidirect
3. C₃×C₅×#376	C₂	semidirect
4. C₃×#65228	C₂	semidirect
5. C₃×#65225	C₂	semidirect
6. C₃×#64954	C₂	semidirect
7. #10605120	C₃	direct (N × H)
17. C₃×#4020	C₄	semidirect
23. C₂×C₄×D₂₀	C₂×C₃	semidirect
24. C₅×#376	C₂×C₃	semidirect
25. #65228	C₂×C₃	semidirect
26. #65225	C₂×C₃	semidirect
27. #64954	C₂×C₃	semidirect
37. C₂×C₃×D₂₀	C₂×C₄	semidirect
38. C₂×C₃×D₂₀	C₂×C₄	semidirect
42. C₃×C₄²×C₅	C₂×C₄	semidirect
43. C₃×C₄²×C₅	D₄	semidirect
52. C₃×#376	D₅	semidirect
62. #4020	C₃×C₄	semidirect
109. C₂×D₂₀	C₂×C₃×C₄	semidirect
110. C₂×D₂₀	C₂×C₃×C₄	semidirect
114. C₄²×C₅	C₂×C₃×C₄	semidirect
115. C₄²×C₅	C₃×D₄	semidirect
124. #376	C₃×D₅	semidirect
156. C₃×C₄²	C₄×D₅	semidirect
157. C₃×C₄²	D₂₀	semidirect
247. C₄²	C₃×C₄×D₅	semidirect
248. C₄²	C₃×D₂₀	semidirect
249. C₃×C₅	#2328	semidirect
294. C₅	C₃×#2328	semidirect
309. C₃	#10605120	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×#4020	2
C₂×C₃×C₄×D₂₀	2
C₃×C₅×#376	2
C₃×#65228	2
C₃×#65225	2
C₃×#64954	2
#10605120	3
C₃×#2328	5

Subgroups — C3×#10605120 #14324378

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10605120 #14324378