Subgroups — C₃×#10614903 #14324381

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×C₅×#127	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×D₄×Dic₅	C₂	1	2	yes	maximal, complemented
3	C₃×#65510	C₂	1	2	yes	maximal, complemented
4	C₃×#65509	C₂	1	8	yes	maximal, complemented
5	C₃×#65225	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#65221	C₂	1	1	yes	maximal, characteristic, complemented
7	#10614903	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₂²×C₃×C₅×D₄	C₂²	1	2	yes	—
9	C₂²×C₃×C₅×D₄	C₂²	1	4	yes	—
10	C₂³×C₃×Dic₅	C₂²	1	4	yes	—
11	C₃×C₅×#127	C₄	1	8	yes	complemented
12	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₂²⋊Dic₅	C₂²	1	4	yes	—
14	C₂×C₃×C₂²⋊Dic₅	C₂²	1	8	yes	—
15	C₃×D₄×Dic₅	—	2	8	no	—
16	C₂×C₃×C₄⋊Dic₅	C₂²	1	2	yes	—
17	C₂×C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
18	C₂×C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
19	C₃×#3963	C₂²	1	4	yes	—
20	C₃×#3963	C₂²	1	4	yes	—
21	C₃×#2328	—	5	1	no	maximal, Hall
22	C₂×C₅×#127	C₂×C₃	1	1	yes	characteristic
23	C₂×D₄×Dic₅	C₂×C₃	1	2	yes	complemented
24	#65510	C₂×C₃	1	2	yes	complemented
25	#65509	C₂×C₃	1	8	yes	complemented
26	#65225	C₂×C₃	1	1	yes	characteristic, complemented
27	#65221	C₂×C₃	1	1	yes	characteristic, complemented
28	C₂⁴×C₃×C₅	C₂³	1	4	yes	—
29	C₂×C₃×C₅×D₄	—	2	4	no	—
30	C₂×C₃×C₅×D₄	—	2	4	no	—
31	C₂×C₃×C₅×D₄	—	4	8	no	—
32	C₂×C₃×C₅×D₄	C₂×C₄	1	8	yes	complemented
33	C₂×C₃×C₅×D₄	C₂×C₄	1	16	yes	complemented
34	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
35	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
36	C₂²×C₃×Dic₅	—	2	8	no	—
37	C₂²×C₃×Dic₅	—	4	8	no	—
38	C₂²×C₃×Dic₅	C₂³	1	2	yes	—
39	C₂²×C₃×Dic₅	C₂³	1	4	yes	—
40	C₂²×C₃×Dic₅	C₂³	1	2	yes	—
41	C₃×C₄²×C₅	C₂×C₄	1	4	yes	complemented
42	C₃×C₂²⋊Dic₅	—	2	8	no	—
43	C₃×C₂²⋊Dic₅	—	4	8	no	—
44	C₃×C₄⋊Dic₅	—	2	4	no	—
45	C₃×C₄×Dic₅	—	2	2	no	—
46	C₃×C₄×Dic₅	D₄	1	4	yes	complemented
47	C₂×C₃×#127	D₅	1	1	yes	characteristic
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	—	5	1	no	—
53	C₂²×C₅×D₄	C₂²×C₃	1	2	yes	—
54	C₂²×C₅×D₄	C₂²×C₃	1	4	yes	—
55	C₂³×Dic₅	C₂²×C₃	1	4	yes	—
56	C₅×#127	C₃×C₄	1	8	yes	complemented
57	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
58	C₂×C₂²⋊Dic₅	C₂²×C₃	1	4	yes	—
59	C₂×C₂²⋊Dic₅	C₂²×C₃	1	8	yes	—
60	D₄×Dic₅	—	2	8	no	—
61	C₂×C₄⋊Dic₅	C₂²×C₃	1	2	yes	—
62	C₂×C₄×Dic₅	C₂²×C₃	1	1	yes	characteristic
63	C₂×C₄×Dic₅	C₂²×C₃	1	1	yes	characteristic
64	#3963	C₂²×C₃	1	4	yes	—
65	#3963	C₂²×C₃	1	4	yes	—
66	#2328	—	5	1	no	Sylow
67	C₂³×C₃×C₅	—	2	4	no	—
68	C₂³×C₃×C₅	—	2	4	no	—
69	C₂³×C₃×C₅	—	2	8	no	—
70	C₂³×C₃×C₅	—	2	8	no	—
71	C₂³×C₃×C₅	C₂²×C₄	1	8	yes	—
72	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
73	C₃×C₅×D₄	—	2	16	no	—
74	C₃×C₅×D₄	—	4	16	no	—
75	C₂×C₃×C₄×C₅	—	2	4	no	—
76	C₂×C₃×C₄×C₅	C₂²×C₄	1	4	yes	—
77	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
78	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
79	C₂×C₃×C₄×C₅	#41	1	2	yes	—
80	C₂×C₃×Dic₅	—	2	4	no	—
81	C₂×C₃×Dic₅	—	2	4	no	—
82	C₂×C₃×Dic₅	—	2	2	no	—
83	C₂×C₃×Dic₅	—	2	4	no	—
84	C₂×C₃×Dic₅	—	2	8	no	—
85	C₂×C₃×Dic₅	—	4	16	no	—
86	C₂×C₃×Dic₅	C₂×D₄	1	4	yes	—
87	C₂²×C₃×D₄	C₂×D₅	1	2	yes	—
88	C₂²×C₃×D₄	C₂×D₅	1	4	yes	—
89	C₂³×C₃×C₄	—	5	4	no	—
90	C₃×#127	Dic₅	1	8	yes	complemented
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	—	5	1	no	—
96	C₂×C₃×C₄²	—	5	1	no	—
97	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
98	C₃×#95	—	5	4	no	—
99	C₃×#95	—	5	4	no	—
100	C₂⁴×C₅	C₂³×C₃	1	4	yes	—
101	C₂×C₅×D₄	—	2	4	no	—
102	C₂×C₅×D₄	—	2	4	no	—
103	C₂×C₅×D₄	—	4	8	no	—
104	C₂×C₅×D₄	C₂×C₃×C₄	1	8	yes	complemented
105	C₂×C₅×D₄	C₂×C₃×C₄	1	16	yes	complemented
106	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
107	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
108	C₂²×Dic₅	—	2	8	no	—
109	C₂²×Dic₅	—	4	8	no	—
110	C₂²×Dic₅	C₂³×C₃	1	2	yes	—
111	C₂²×Dic₅	C₂³×C₃	1	4	yes	—
112	C₂²×Dic₅	C₂³×C₃	1	2	yes	—
113	C₄²×C₅	C₂×C₃×C₄	1	4	yes	complemented
114	C₂²⋊Dic₅	—	2	8	no	—
115	C₂²⋊Dic₅	—	4	8	no	—
116	C₄⋊Dic₅	—	2	4	no	—
117	C₄×Dic₅	—	2	2	no	—
118	C₄×Dic₅	C₃×D₄	1	4	yes	complemented
119	C₂×#127	C₃×D₅	1	1	yes	characteristic
120	C₂×C₄×D₄	—	5	2	no	—
121	#390	—	5	2	no	—
122	#386	—	5	8	no	—
123	#382	—	5	1	no	—
124	#376	—	5	1	no	—
125	C₂²×C₃×C₅	—	2	8	no	—
126	C₂²×C₃×C₅	—	2	16	no	—
127	C₂²×C₃×C₅	—	4	4	no	—
128	C₂²×C₃×C₅	—	4	4	no	—
129	C₂²×C₃×C₅	—	4	8	no	—
130	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
131	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
132	C₂²×C₃×C₅	C₂×#41	1	1	yes	characteristic
133	C₂²×C₃×C₅	H(5,𝔽₂)	1	4	yes	—
134	C₃×C₄×C₅	—	2	4	no	—
135	C₃×C₄×C₅	C₄×D₄	1	4	yes	—
136	C₃×Dic₅	—	2	4	no	—
137	C₃×Dic₅	—	4	4	no	—
138	C₃×Dic₅	—	4	2	no	—
139	C₂⁴×C₃	C₂²×D₅	1	4	yes	—
140	C₂×C₃×D₄	—	2	4	no	—
141	C₂×C₃×D₄	—	2	4	no	—
142	C₂×C₃×D₄	—	4	8	no	—
143	C₂×C₃×D₄	C₂×Dic₅	1	8	yes	complemented
144	C₂×C₃×D₄	C₂×Dic₅	1	16	yes	complemented
145	C₂²×C₃×C₄	—	5	2	no	—
146	C₂²×C₃×C₄	—	5	2	no	—
147	C₂²×C₃×C₄	—	5	4	no	—
148	C₂²×C₃×C₄	—	10	8	no	—
149	C₂²×C₃×C₄	—	20	8	no	—
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	—	5	4	no	—
156	C₃×C₄²	—	10	2	no	—
157	C₃×C₄²	C₂×Dic₅	1	4	yes	complemented
158	C₂³×C₅	—	2	4	no	—
159	C₂³×C₅	—	2	4	no	—
160	C₂³×C₅	—	2	8	no	—
161	C₂³×C₅	—	2	8	no	—
162	C₂³×C₅	C₂²×C₃×C₄	1	8	yes	—
163	C₂³×C₅	C₂⁴×C₃	1	1	yes	characteristic
164	C₅×D₄	—	2	16	no	—
165	C₅×D₄	—	4	16	no	—
166	C₂×C₄×C₅	—	2	4	no	—
167	C₂×C₄×C₅	C₂²×C₃×C₄	1	4	yes	—
168	C₂×C₄×C₅	C₂²×C₃×C₄	1	2	yes	—
169	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
170	C₂×C₄×C₅	C₃×#41	1	2	yes	—
171	C₂×Dic₅	—	2	4	no	—
172	C₂×Dic₅	—	2	4	no	—
173	C₂×Dic₅	—	2	2	no	—
174	C₂×Dic₅	—	2	4	no	—
175	C₂×Dic₅	—	2	8	no	—
176	C₂×Dic₅	—	4	16	no	—
177	C₂×Dic₅	C₂×C₃×D₄	1	4	yes	—
178	C₂²×D₄	C₂×C₃×D₅	1	2	yes	—
179	C₂²×D₄	C₂×C₃×D₅	1	4	yes	—
180	C₂³×C₄	—	5	4	no	—
181	#127	C₃×Dic₅	1	8	yes	complemented
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	—	5	1	no	—
187	C₂×C₄²	—	5	1	no	—
188	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
189	#95	—	5	4	no	—
190	#95	—	5	4	no	—
191	C₂×C₃×C₅	—	4	8	no	—
192	C₂×C₃×C₅	C₂×C₄×D₄	1	1	yes	characteristic
193	C₂×C₃×C₅	#518	1	2	yes	—
194	C₂×C₃×C₅	#535	1	2	yes	—
195	C₂×C₃×C₅	#540	1	2	yes	—
196	C₂³×C₃	—	2	4	no	—
197	C₂³×C₃	—	2	4	no	—
198	C₂³×C₃	—	2	8	no	—
199	C₂³×C₃	—	2	8	no	—
200	C₂³×C₃	C₂²×Dic₅	1	8	yes	—
201	C₂³×C₃	C₂³×D₅	1	1	yes	characteristic
202	C₃×D₄	—	2	16	no	—
203	C₃×D₄	—	4	16	no	—
204	C₂×C₃×C₄	—	2	4	no	—
205	C₂×C₃×C₄	—	5	4	no	—
206	C₂×C₃×C₄	—	10	4	no	—
207	C₂×C₃×C₄	—	10	4	no	—
208	C₂×C₃×C₄	—	10	2	no	—
209	C₂×C₃×C₄	—	10	4	no	—
210	C₂×C₃×C₄	—	10	8	no	—
211	C₂×C₃×C₄	—	20	16	no	—
212	C₂×C₃×C₄	D₄×D₅	1	2	yes	—
213	C₂×C₃×C₄	#711	1	2	yes	—
214	C₂×C₃×C₄	C₂²×Dic₅	1	4	yes	—
215	C₂×C₃×C₄	C₂²×Dic₅	1	2	yes	—
216	C₂²×C₅	—	2	8	no	—
217	C₂²×C₅	—	2	16	no	—
218	C₂²×C₅	—	4	4	no	—
219	C₂²×C₅	—	4	4	no	—
220	C₂²×C₅	—	4	8	no	—
221	C₂²×C₅	C₂³×C₃×C₄	1	1	yes	characteristic
222	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
223	C₂²×C₅	C₂×C₃×#41	1	1	yes	characteristic
224	C₂²×C₅	C₃×H(5,𝔽₂)	1	4	yes	—
225	C₄×C₅	—	2	4	no	—
226	C₄×C₅	C₃×C₄×D₄	1	4	yes	—
227	Dic₅	—	2	4	no	—
228	Dic₅	—	4	4	no	—
229	Dic₅	—	4	2	no	—
230	C₂⁴	C₂²×C₃×D₅	1	4	yes	—
231	C₂×D₄	—	2	4	no	—
232	C₂×D₄	—	2	4	no	—
233	C₂×D₄	—	4	8	no	—
234	C₂×D₄	C₂×C₃×Dic₅	1	8	yes	complemented
235	C₂×D₄	C₂×C₃×Dic₅	1	16	yes	complemented
236	C₂²×C₄	—	5	2	no	—
237	C₂²×C₄	—	5	2	no	—
238	C₂²×C₄	—	5	4	no	—
239	C₂²×C₄	—	10	8	no	—
240	C₂²×C₄	—	20	8	no	—
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	—	5	4	no	—
247	C₄²	—	10	2	no	—
248	C₄²	C₂×C₃×Dic₅	1	4	yes	complemented
249	C₃×C₅	#2328	1	1	yes	characteristic, complemented, Hall
250	C₂²×C₃	—	2	8	no	—
251	C₂²×C₃	—	2	16	no	—
252	C₂²×C₃	—	4	4	no	—
253	C₂²×C₃	—	4	4	no	—
254	C₂²×C₃	—	4	8	no	—
255	C₂²×C₃	C₂×D₄×D₅	1	1	yes	characteristic
256	C₂²×C₃	C₂×#711	1	1	yes	characteristic
257	C₂²×C₃	#4144	1	4	yes	—
258	C₂²×C₃	C₂³×Dic₅	1	1	yes	characteristic
259	C₃×C₄	—	2	4	no	—
260	C₃×C₄	—	10	4	no	—
261	C₃×C₄	—	20	4	no	—
262	C₃×C₄	—	20	2	no	—
263	C₃×C₄	D₄×Dic₅	1	4	yes	—
264	C₂×C₅	—	4	8	no	—
265	C₂×C₅	C₂×C₃×C₄×D₄	1	1	yes	characteristic
266	C₂×C₅	C₃×#518	1	2	yes	—
267	C₂×C₅	C₃×#535	1	2	yes	—
268	C₂×C₅	C₃×#540	1	2	yes	—
269	C₂³	—	2	4	no	—
270	C₂³	—	2	4	no	—
271	C₂³	—	2	8	no	—
272	C₂³	—	2	8	no	—
273	C₂³	C₂²×C₃×Dic₅	1	8	yes	—
274	C₂³	C₂³×C₃×D₅	1	1	yes	characteristic
275	D₄	—	2	16	no	—
276	D₄	—	4	16	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	5	4	no	—
279	C₂×C₄	—	10	4	no	—
280	C₂×C₄	—	10	4	no	—
281	C₂×C₄	—	10	2	no	—
282	C₂×C₄	—	10	4	no	—
283	C₂×C₄	—	10	8	no	—
284	C₂×C₄	—	20	16	no	—
285	C₂×C₄	C₃×D₄×D₅	1	2	yes	—
286	C₂×C₄	C₃×#711	1	2	yes	—
287	C₂×C₄	C₂²×C₃×Dic₅	1	4	yes	—
288	C₂×C₄	C₂²×C₃×Dic₅	1	2	yes	—
289	C₂×C₃	—	4	8	no	—
290	C₂×C₃	#66049	1	2	yes	—
291	C₂×C₃	#66053	1	2	yes	—
292	C₂×C₃	C₂×D₄×Dic₅	1	1	yes	characteristic
293	C₂×C₃	#66132	1	2	yes	—
294	C₅	C₃×#2328	1	1	yes	characteristic, complemented, Sylow
295	C₂²	—	2	8	no	—
296	C₂²	—	2	16	no	—
297	C₂²	—	4	4	no	—
298	C₂²	—	4	4	no	—
299	C₂²	—	4	8	no	—
300	C₂²	C₂×C₃×D₄×D₅	1	1	yes	characteristic
301	C₂²	C₂×C₃×#711	1	1	yes	characteristic
302	C₂²	C₃×#4144	1	4	yes	—
303	C₂²	C₂³×C₃×Dic₅	1	1	yes	characteristic
304	C₄	—	2	4	no	—
305	C₄	—	10	4	no	—
306	C₄	—	20	4	no	—
307	C₄	—	20	2	no	—
308	C₄	C₃×D₄×Dic₅	1	4	yes	—
309	C₃	#10614903	1	1	yes	characteristic, direct product factor, Sylow
310	C₂	—	4	8	no	—
311	C₂	C₃×#66049	1	2	yes	—
312	C₂	C₃×#66053	1	2	yes	—
313	C₂	C₂×C₃×D₄×Dic₅	1	1	yes	characteristic
314	C₂	C₃×#66132	1	2	yes	—

	subgroup	G/H
center	C₂³×C₃	C₂³×D₅
commutator	C₂²×C₅	C₂³×C₃×C₄
Frattini	C₂³	C₂³×C₃×D₅
Fitting, prime radical	C₂×C₃×C₅×#127	C₂
socle	C₂³×C₃×C₅	C₂⁴
cosocle	C₂³×C₅	C₂⁴×C₃
hypercenter	C₂×C₃×#127	D₅
hypocenter	C₅	C₃×#2328
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×#127

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×D₄×Dic₅	C₂	semidirect
3. C₃×#65510	C₂	semidirect
4. C₃×#65509	C₂	semidirect
5. C₃×#65225	C₂	semidirect
6. C₃×#65221	C₂	semidirect
7. #10614903	C₃	direct (N × H)
11. C₃×C₅×#127	C₄	semidirect
23. C₂×D₄×Dic₅	C₂×C₃	semidirect
24. #65510	C₂×C₃	semidirect
25. #65509	C₂×C₃	semidirect
26. #65225	C₂×C₃	semidirect
27. #65221	C₂×C₃	semidirect
32. C₂×C₃×C₅×D₄	C₂×C₄	semidirect
33. C₂×C₃×C₅×D₄	C₂×C₄	semidirect
41. C₃×C₄²×C₅	C₂×C₄	semidirect
46. C₃×C₄×Dic₅	D₄	semidirect
56. C₅×#127	C₃×C₄	semidirect
90. C₃×#127	Dic₅	semidirect
104. C₂×C₅×D₄	C₂×C₃×C₄	semidirect
105. C₂×C₅×D₄	C₂×C₃×C₄	semidirect
113. C₄²×C₅	C₂×C₃×C₄	semidirect
118. C₄×Dic₅	C₃×D₄	semidirect
143. C₂×C₃×D₄	C₂×Dic₅	semidirect
144. C₂×C₃×D₄	C₂×Dic₅	semidirect
157. C₃×C₄²	C₂×Dic₅	semidirect
181. #127	C₃×Dic₅	semidirect
234. C₂×D₄	C₂×C₃×Dic₅	semidirect
235. C₂×D₄	C₂×C₃×Dic₅	semidirect
248. C₄²	C₂×C₃×Dic₅	semidirect
249. C₃×C₅	#2328	semidirect
294. C₅	C₃×#2328	semidirect
309. C₃	#10614903	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×C₅×#127	2
C₂×C₃×D₄×Dic₅	2
C₃×#65510	2
C₃×#65509	2
C₃×#65225	2
C₃×#65221	2
#10614903	3
C₃×#2328	5

Subgroups — C3×#10614903 #14324381

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10614903 #14324381