Subgroups — C₃×#10614904 #14329458

Order 1920.

1664 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	4	yes	maximal, complemented
3	C₃×#65510	C₂	1	4	yes	maximal, complemented
4	C₃×#65509	C₂	1	4	yes	maximal, complemented
5	C₃×C₄×C₄⋊Dic₅	C₂	1	2	yes	maximal, complemented
6	#10614904	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₃×C₅×D₄	C₂²	1	4	yes	—
8	C₂²×C₃×C₅×D₄	C₂²	1	2	yes	—
9	C₂³×C₃×Dic₅	C₂²	1	4	yes	—
10	C₃×C₅×#127	C₄	1	8	yes	complemented
11	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₂²⋊Dic₅	C₂²	1	8	yes	—
13	C₂×C₃×C₂²⋊Dic₅	C₂²	1	4	yes	—
14	C₃×D₄×Dic₅	—	2	16	no	—
15	C₂×C₃×C₄⋊Dic₅	C₂²	1	2	yes	—
16	C₂×C₃×C₄⋊Dic₅	C₂²	1	2	yes	—
17	C₂×C₃×C₄×Dic₅	C₂²	1	4	yes	—
18	C₃×#3963	C₂²	1	4	yes	—
19	C₃×#2366	—	5	1	no	maximal, Hall
20	C₂×C₅×#127	C₂×C₃	1	1	yes	characteristic
21	C₂×D₄×Dic₅	C₂×C₃	1	4	yes	complemented
22	#65510	C₂×C₃	1	4	yes	complemented
23	#65509	C₂×C₃	1	4	yes	complemented
24	C₄×C₄⋊Dic₅	C₂×C₃	1	2	yes	complemented
25	C₂⁴×C₃×C₅	C₂³	1	4	yes	—
26	C₂×C₃×C₅×D₄	—	2	8	no	—
27	C₂×C₃×C₅×D₄	—	2	8	no	—
28	C₂×C₃×C₅×D₄	—	4	2	no	—
29	C₂×C₃×C₅×D₄	—	4	2	no	—
30	C₂×C₃×C₅×D₄	C₂×C₄	1	16	yes	complemented
31	C₂×C₃×C₅×D₄	C₂×C₄	1	8	yes	complemented
32	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
33	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
34	C₂²×C₃×Dic₅	—	2	16	no	—
35	C₂²×C₃×Dic₅	—	4	4	no	—
36	C₂²×C₃×Dic₅	C₂³	1	4	yes	—
37	C₂²×C₃×Dic₅	C₂³	1	4	yes	—
38	C₃×C₄²×C₅	C₂×C₄	1	4	yes	complemented
39	C₃×C₂²⋊Dic₅	—	2	16	no	—
40	C₃×C₂²⋊Dic₅	—	4	4	no	—
41	C₃×C₄⋊Dic₅	—	2	4	no	—
42	C₃×C₄⋊Dic₅	D₄	1	8	yes	complemented
43	C₃×C₄×Dic₅	—	2	8	no	—
44	C₂×C₃×#127	D₅	1	1	yes	characteristic
45	C₂×C₃×C₄×D₄	—	5	4	no	—
46	C₃×#390	—	5	4	no	—
47	C₃×#386	—	5	4	no	—
48	C₃×C₄×C₄⋊C₄	—	5	2	no	—
49	C₂²×C₅×D₄	C₂²×C₃	1	4	yes	—
50	C₂²×C₅×D₄	C₂²×C₃	1	2	yes	—
51	C₂³×Dic₅	C₂²×C₃	1	4	yes	—
52	C₅×#127	C₃×C₄	1	8	yes	complemented
53	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
54	C₂×C₂²⋊Dic₅	C₂²×C₃	1	8	yes	—
55	C₂×C₂²⋊Dic₅	C₂²×C₃	1	4	yes	—
56	D₄×Dic₅	—	2	16	no	—
57	C₂×C₄⋊Dic₅	C₂²×C₃	1	2	yes	—
58	C₂×C₄⋊Dic₅	C₂²×C₃	1	2	yes	—
59	C₂×C₄×Dic₅	C₂²×C₃	1	4	yes	—
60	#3963	C₂²×C₃	1	4	yes	—
61	#2366	—	5	1	no	Sylow
62	C₂³×C₃×C₅	—	2	8	no	—
63	C₂³×C₃×C₅	—	2	8	no	—
64	C₂³×C₃×C₅	—	2	4	no	—
65	C₂³×C₃×C₅	—	2	4	no	—
66	C₂³×C₃×C₅	C₂²×C₄	1	8	yes	—
67	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
68	C₃×C₅×D₄	—	2	32	no	—
69	C₃×C₅×D₄	—	4	8	no	—
70	C₂×C₃×C₄×C₅	—	2	1	no	—
71	C₂×C₃×C₄×C₅	—	2	1	no	—
72	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
73	C₂×C₃×C₄×C₅	C₂²×C₄	1	4	yes	—
74	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
75	C₂×C₃×C₄×C₅	#41	1	4	yes	—
76	C₂×C₃×Dic₅	—	2	4	no	—
77	C₂×C₃×Dic₅	—	2	4	no	—
78	C₂×C₃×Dic₅	—	2	4	no	—
79	C₂×C₃×Dic₅	—	2	4	no	—
80	C₂×C₃×Dic₅	—	2	4	no	—
81	C₂×C₃×Dic₅	—	4	16	no	—
82	C₂×C₃×Dic₅	C₂×D₄	1	8	yes	—
83	C₂²×C₃×D₄	C₂×D₅	1	4	yes	—
84	C₂²×C₃×D₄	C₂×D₅	1	2	yes	—
85	C₂³×C₃×C₄	—	5	4	no	—
86	C₃×#127	Dic₅	1	8	yes	complemented
87	C₃×C₄×D₄	—	10	16	no	—
88	C₂×C₃×C₄⋊C₄	—	5	2	no	—
89	C₂×C₃×C₄⋊C₄	—	5	2	no	—
90	C₂×C₃×C₂²⋊C₄	—	5	8	no	—
91	C₂×C₃×C₂²⋊C₄	—	5	4	no	—
92	C₂×C₃×C₄²	—	5	4	no	—
93	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
94	C₃×#95	—	5	4	no	—
95	C₂⁴×C₅	C₂³×C₃	1	4	yes	—
96	C₂×C₅×D₄	—	2	8	no	—
97	C₂×C₅×D₄	—	2	8	no	—
98	C₂×C₅×D₄	—	4	2	no	—
99	C₂×C₅×D₄	—	4	2	no	—
100	C₂×C₅×D₄	C₂×C₃×C₄	1	16	yes	complemented
101	C₂×C₅×D₄	C₂×C₃×C₄	1	8	yes	complemented
102	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
103	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
104	C₂²×Dic₅	—	2	16	no	—
105	C₂²×Dic₅	—	4	4	no	—
106	C₂²×Dic₅	C₂³×C₃	1	4	yes	—
107	C₂²×Dic₅	C₂³×C₃	1	4	yes	—
108	C₄²×C₅	C₂×C₃×C₄	1	4	yes	complemented
109	C₂²⋊Dic₅	—	2	16	no	—
110	C₂²⋊Dic₅	—	4	4	no	—
111	C₄⋊Dic₅	—	2	4	no	—
112	C₄⋊Dic₅	C₃×D₄	1	8	yes	complemented
113	C₄×Dic₅	—	2	8	no	—
114	C₂×#127	C₃×D₅	1	1	yes	characteristic
115	C₂×C₄×D₄	—	5	4	no	—
116	#390	—	5	4	no	—
117	#386	—	5	4	no	—
118	C₄×C₄⋊C₄	—	5	2	no	—
119	C₂²×C₃×C₅	—	2	16	no	—
120	C₂²×C₃×C₅	—	2	8	no	—
121	C₂²×C₃×C₅	—	4	4	no	—
122	C₂²×C₃×C₅	—	4	8	no	—
123	C₂²×C₃×C₅	—	4	4	no	—
124	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
125	C₂²×C₃×C₅	C₂²×D₄	1	2	yes	—
126	C₂²×C₃×C₅	C₂×#41	1	2	yes	—
127	C₂²×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
128	C₂²×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
129	C₃×C₄×C₅	—	2	2	no	—
130	C₃×C₄×C₅	C₄×D₄	1	8	yes	complemented
131	C₃×Dic₅	—	2	8	no	—
132	C₃×Dic₅	—	4	4	no	—
133	C₂⁴×C₃	C₂²×D₅	1	4	yes	—
134	C₂×C₃×D₄	—	2	8	no	—
135	C₂×C₃×D₄	—	2	8	no	—
136	C₂×C₃×D₄	—	4	2	no	—
137	C₂×C₃×D₄	—	4	2	no	—
138	C₂×C₃×D₄	C₂×Dic₅	1	16	yes	complemented
139	C₂×C₃×D₄	C₂×Dic₅	1	8	yes	complemented
140	C₂²×C₃×C₄	—	5	4	no	—
141	C₂²×C₃×C₄	—	5	4	no	—
142	C₂²×C₃×C₄	—	10	16	no	—
143	C₂²×C₃×C₄	—	20	4	no	—
144	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
145	C₂²×C₃×C₄	C₂²×D₅	1	2	yes	—
146	C₃×C₄⋊C₄	—	5	8	no	—
147	C₃×C₄⋊C₄	—	10	4	no	—
148	C₃×C₂²⋊C₄	—	10	16	no	—
149	C₃×C₂²⋊C₄	—	20	4	no	—
150	C₃×C₄²	—	10	8	no	—
151	C₃×C₄²	C₂×Dic₅	1	4	yes	complemented
152	C₂³×C₅	—	2	8	no	—
153	C₂³×C₅	—	2	8	no	—
154	C₂³×C₅	—	2	4	no	—
155	C₂³×C₅	—	2	4	no	—
156	C₂³×C₅	C₂²×C₃×C₄	1	8	yes	—
157	C₂³×C₅	C₂⁴×C₃	1	1	yes	characteristic
158	C₅×D₄	—	2	32	no	—
159	C₅×D₄	—	4	8	no	—
160	C₂×C₄×C₅	—	2	1	no	—
161	C₂×C₄×C₅	—	2	1	no	—
162	C₂×C₄×C₅	C₂²×C₃×C₄	1	2	yes	—
163	C₂×C₄×C₅	C₂²×C₃×C₄	1	4	yes	—
164	C₂×C₄×C₅	C₂×C₃×D₄	1	4	yes	—
165	C₂×C₄×C₅	C₃×#41	1	4	yes	—
166	C₂×Dic₅	—	2	4	no	—
167	C₂×Dic₅	—	2	4	no	—
168	C₂×Dic₅	—	2	4	no	—
169	C₂×Dic₅	—	2	4	no	—
170	C₂×Dic₅	—	2	4	no	—
171	C₂×Dic₅	—	4	16	no	—
172	C₂×Dic₅	C₂×C₃×D₄	1	8	yes	—
173	C₂²×D₄	C₂×C₃×D₅	1	4	yes	—
174	C₂²×D₄	C₂×C₃×D₅	1	2	yes	—
175	C₂³×C₄	—	5	4	no	—
176	#127	C₃×Dic₅	1	8	yes	complemented
177	C₄×D₄	—	10	16	no	—
178	C₂×C₄⋊C₄	—	5	2	no	—
179	C₂×C₄⋊C₄	—	5	2	no	—
180	C₂×C₂²⋊C₄	—	5	8	no	—
181	C₂×C₂²⋊C₄	—	5	4	no	—
182	C₂×C₄²	—	5	4	no	—
183	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
184	#95	—	5	4	no	—
185	C₂×C₃×C₅	—	4	8	no	—
186	C₂×C₃×C₅	C₂×C₄×D₄	1	2	yes	—
187	C₂×C₃×C₅	#518	1	1	yes	characteristic
188	C₂×C₃×C₅	D₄²	1	1	yes	characteristic
189	C₂×C₃×C₅	#550	1	2	yes	—
190	C₂×C₃×C₅	#559	1	1	yes	characteristic
191	C₂³×C₃	—	2	8	no	—
192	C₂³×C₃	—	2	8	no	—
193	C₂³×C₃	—	2	4	no	—
194	C₂³×C₃	—	2	4	no	—
195	C₂³×C₃	C₂²×Dic₅	1	8	yes	—
196	C₂³×C₃	C₂³×D₅	1	1	yes	characteristic
197	C₃×D₄	—	2	32	no	—
198	C₃×D₄	—	4	8	no	—
199	C₂×C₃×C₄	—	2	1	no	—
200	C₂×C₃×C₄	—	2	1	no	—
201	C₂×C₃×C₄	—	5	8	no	—
202	C₂×C₃×C₄	—	10	4	no	—
203	C₂×C₃×C₄	—	10	4	no	—
204	C₂×C₃×C₄	—	10	4	no	—
205	C₂×C₃×C₄	—	10	4	no	—
206	C₂×C₃×C₄	—	10	4	no	—
207	C₂×C₃×C₄	—	20	16	no	—
208	C₂×C₃×C₄	D₄×D₅	1	4	yes	—
209	C₂×C₃×C₄	#711	1	4	yes	—
210	C₂×C₃×C₄	C₂²×Dic₅	1	2	yes	—
211	C₂×C₃×C₄	C₂²×Dic₅	1	4	yes	—
212	C₂²×C₅	—	2	16	no	—
213	C₂²×C₅	—	2	8	no	—
214	C₂²×C₅	—	4	4	no	—
215	C₂²×C₅	—	4	4	no	—
216	C₂²×C₅	—	4	8	no	—
217	C₂²×C₅	C₂³×C₃×C₄	1	1	yes	characteristic
218	C₂²×C₅	C₂²×C₃×D₄	1	2	yes	—
219	C₂²×C₅	C₂×C₃×#41	1	2	yes	—
220	C₂²×C₅	C₃×H(5,𝔽₂)	1	1	yes	characteristic
221	C₂²×C₅	C₃×H(5,𝔽₂)	1	1	yes	characteristic
222	C₄×C₅	—	2	2	no	—
223	C₄×C₅	C₃×C₄×D₄	1	8	yes	complemented
224	Dic₅	—	2	8	no	—
225	Dic₅	—	4	4	no	—
226	C₂⁴	C₂²×C₃×D₅	1	4	yes	—
227	C₂×D₄	—	2	8	no	—
228	C₂×D₄	—	2	8	no	—
229	C₂×D₄	—	4	2	no	—
230	C₂×D₄	—	4	2	no	—
231	C₂×D₄	C₂×C₃×Dic₅	1	16	yes	complemented
232	C₂×D₄	C₂×C₃×Dic₅	1	8	yes	complemented
233	C₂²×C₄	—	5	4	no	—
234	C₂²×C₄	—	5	4	no	—
235	C₂²×C₄	—	10	16	no	—
236	C₂²×C₄	—	20	4	no	—
237	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
238	C₂²×C₄	C₂²×C₃×D₅	1	2	yes	—
239	C₄⋊C₄	—	5	8	no	—
240	C₄⋊C₄	—	10	4	no	—
241	C₂²⋊C₄	—	10	16	no	—
242	C₂²⋊C₄	—	20	4	no	—
243	C₄²	—	10	8	no	—
244	C₄²	C₂×C₃×Dic₅	1	4	yes	complemented
245	C₃×C₅	#2366	1	1	yes	characteristic, complemented, Hall
246	C₂²×C₃	—	2	16	no	—
247	C₂²×C₃	—	2	8	no	—
248	C₂²×C₃	—	4	4	no	—
249	C₂²×C₃	—	4	8	no	—
250	C₂²×C₃	—	4	4	no	—
251	C₂²×C₃	C₂×D₄×D₅	1	2	yes	—
252	C₂²×C₃	C₂×#711	1	2	yes	—
253	C₂²×C₃	#4144	1	1	yes	characteristic
254	C₂²×C₃	#4144	1	1	yes	characteristic
255	C₂²×C₃	C₂³×Dic₅	1	1	yes	characteristic
256	C₃×C₄	—	2	2	no	—
257	C₃×C₄	—	10	8	no	—
258	C₃×C₄	—	20	4	no	—
259	C₃×C₄	D₄×Dic₅	1	8	yes	complemented
260	C₂×C₅	—	4	8	no	—
261	C₂×C₅	C₂×C₃×C₄×D₄	1	2	yes	—
262	C₂×C₅	C₃×#518	1	1	yes	characteristic
263	C₂×C₅	C₃×D₄²	1	1	yes	characteristic
264	C₂×C₅	C₃×#550	1	2	yes	—
265	C₂×C₅	C₃×#559	1	1	yes	characteristic
266	C₂³	—	2	8	no	—
267	C₂³	—	2	8	no	—
268	C₂³	—	2	4	no	—
269	C₂³	—	2	4	no	—
270	C₂³	C₂²×C₃×Dic₅	1	8	yes	—
271	C₂³	C₂³×C₃×D₅	1	1	yes	characteristic
272	D₄	—	2	32	no	—
273	D₄	—	4	8	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	5	8	no	—
277	C₂×C₄	—	10	4	no	—
278	C₂×C₄	—	10	4	no	—
279	C₂×C₄	—	10	4	no	—
280	C₂×C₄	—	10	4	no	—
281	C₂×C₄	—	10	4	no	—
282	C₂×C₄	—	20	16	no	—
283	C₂×C₄	C₃×D₄×D₅	1	4	yes	—
284	C₂×C₄	C₃×#711	1	4	yes	—
285	C₂×C₄	C₂²×C₃×Dic₅	1	2	yes	—
286	C₂×C₄	C₂²×C₃×Dic₅	1	4	yes	—
287	C₂×C₃	—	4	8	no	—
288	C₂×C₃	#66047	1	1	yes	characteristic
289	C₂×C₃	#66051	1	1	yes	characteristic
290	C₂×C₃	#66052	1	2	yes	—
291	C₂×C₃	C₂×D₄×Dic₅	1	2	yes	—
292	C₂×C₃	#66132	1	1	yes	characteristic
293	C₅	C₃×#2366	1	1	yes	characteristic, complemented, Sylow
294	C₂²	—	2	16	no	—
295	C₂²	—	2	8	no	—
296	C₂²	—	4	4	no	—
297	C₂²	—	4	4	no	—
298	C₂²	—	4	8	no	—
299	C₂²	C₂×C₃×D₄×D₅	1	2	yes	—
300	C₂²	C₂×C₃×#711	1	2	yes	—
301	C₂²	C₃×#4144	1	1	yes	characteristic
302	C₂²	C₃×#4144	1	1	yes	characteristic
303	C₂²	C₂³×C₃×Dic₅	1	1	yes	characteristic
304	C₄	—	2	2	no	—
305	C₄	—	10	8	no	—
306	C₄	—	20	4	no	—
307	C₄	C₃×D₄×Dic₅	1	8	yes	complemented
308	C₃	#10614904	1	1	yes	characteristic, direct product factor, Sylow
309	C₂	—	4	8	no	—
310	C₂	C₃×#66047	1	1	yes	characteristic
311	C₂	C₃×#66051	1	1	yes	characteristic
312	C₂	C₃×#66052	1	2	yes	—
313	C₂	C₂×C₃×D₄×Dic₅	1	2	yes	—
314	C₂	C₃×#66132	1	1	yes	characteristic

	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×#2366
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×C₄×C₄⋊Dic₅	C₂	semidirect
6. #10614904	C₃	direct (N × H)
10. C₃×C₅×#127	C₄	semidirect
21. C₂×D₄×Dic₅	C₂×C₃	semidirect
22. #65510	C₂×C₃	semidirect
23. #65509	C₂×C₃	semidirect
24. C₄×C₄⋊Dic₅	C₂×C₃	semidirect
30. C₂×C₃×C₅×D₄	C₂×C₄	semidirect
31. C₂×C₃×C₅×D₄	C₂×C₄	semidirect
38. C₃×C₄²×C₅	C₂×C₄	semidirect
42. C₃×C₄⋊Dic₅	D₄	semidirect
52. C₅×#127	C₃×C₄	semidirect
86. C₃×#127	Dic₅	semidirect
100. C₂×C₅×D₄	C₂×C₃×C₄	semidirect
101. C₂×C₅×D₄	C₂×C₃×C₄	semidirect
108. C₄²×C₅	C₂×C₃×C₄	semidirect
112. C₄⋊Dic₅	C₃×D₄	semidirect
130. C₃×C₄×C₅	C₄×D₄	semidirect
138. C₂×C₃×D₄	C₂×Dic₅	semidirect
139. C₂×C₃×D₄	C₂×Dic₅	semidirect
151. C₃×C₄²	C₂×Dic₅	semidirect
176. #127	C₃×Dic₅	semidirect
223. C₄×C₅	C₃×C₄×D₄	semidirect
231. C₂×D₄	C₂×C₃×Dic₅	semidirect
232. C₂×D₄	C₂×C₃×Dic₅	semidirect
244. C₄²	C₂×C₃×Dic₅	semidirect
245. C₃×C₅	#2366	semidirect
259. C₃×C₄	D₄×Dic₅	semidirect
293. C₅	C₃×#2366	semidirect
307. C₄	C₃×D₄×Dic₅	semidirect
308. C₃	#10614904	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×C₄×C₄⋊Dic₅	2
#10614904	3
C₃×#2366	5

Subgroups — C3×#10614904 #14329458

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10614904 #14329458