Subgroups — C₃×#10614311 #14330638

Order 1920.

1376 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×#120	C₂	1	1	yes	maximal, characteristic
2	C₃×#65536	C₂	1	2	yes	maximal, complemented
3	C₃×#65510	C₂	1	3	yes	maximal, complemented
4	C₃×#65509	C₂	1	3	yes	maximal, complemented
5	C₃×#65240	C₂	1	6	yes	maximal, complemented
6	#10614311	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂⁵×C₃×C₅	C₂²	1	1	yes	characteristic
8	C₂²×C₃×C₅×D₄	C₂²	1	3	yes	—
9	C₂³×C₃×Dic₅	C₂²	1	1	yes	characteristic, complemented
10	C₃×C₅×#120	C₄	1	8	yes	complemented
11	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	3	yes	—
12	C₂×C₃×C₂²⋊Dic₅	C₂²	1	6	yes	—
13	C₂×C₃×C₂²⋊Dic₅	C₂²	1	6	yes	—
14	C₂×C₃×C₂²⋊Dic₅	C₂²	1	3	yes	complemented
15	C₂×C₃×C₄⋊Dic₅	C₂²	1	3	yes	complemented
16	C₂×C₃×C₄×Dic₅	C₂²	1	3	yes	complemented
17	C₃×#3963	C₂²	1	6	yes	complemented
18	C₃×#2375	—	5	1	no	maximal, Hall
19	C₂×C₅×#120	C₂×C₃	1	1	yes	characteristic
20	#65536	C₂×C₃	1	2	yes	complemented
21	#65510	C₂×C₃	1	3	yes	complemented
22	#65509	C₂×C₃	1	3	yes	complemented
23	#65240	C₂×C₃	1	6	yes	complemented
24	C₂⁴×C₃×C₅	—	4	3	no	—
25	C₂⁴×C₃×C₅	—	4	3	no	—
26	C₂⁴×C₃×C₅	C₂×C₄	1	4	yes	complemented
27	C₂⁴×C₃×C₅	C₂³	1	3	yes	—
28	C₂⁴×C₃×C₅	C₂³	1	1	yes	characteristic
29	C₂×C₃×C₅×D₄	—	4	3	no	—
30	C₂×C₃×C₅×D₄	—	4	3	no	—
31	C₂×C₃×C₅×D₄	C₂×C₄	1	12	yes	—
32	C₂²×C₃×C₄×C₅	C₂³	1	3	yes	—
33	C₂²×C₃×Dic₅	—	4	3	no	—
34	C₂²×C₃×Dic₅	C₂³	1	6	yes	—
35	C₂²×C₃×Dic₅	C₂³	1	2	yes	—
36	C₃×C₅×C₂²⋊C₄	C₂×C₄	1	12	yes	complemented
37	C₃×C₂²⋊Dic₅	—	4	6	no	—
38	C₃×C₂²⋊Dic₅	—	4	6	no	—
39	C₃×C₂²⋊Dic₅	—	4	3	no	—
40	C₃×C₄⋊Dic₅	—	4	3	no	—
41	C₃×C₄×Dic₅	—	4	3	no	—
42	C₂×C₃×#120	D₅	1	1	yes	characteristic
43	C₃×#390	—	5	3	no	—
44	C₃×#388	—	5	6	no	—
45	C₃×#386	—	5	3	no	—
46	C₃×#379	—	5	2	no	—
47	C₂⁵×C₅	C₂²×C₃	1	1	yes	characteristic
48	C₂²×C₅×D₄	C₂²×C₃	1	3	yes	—
49	C₂³×Dic₅	C₂²×C₃	1	1	yes	characteristic, complemented
50	C₅×#120	C₃×C₄	1	8	yes	complemented
51	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	3	yes	—
52	C₂×C₂²⋊Dic₅	C₂²×C₃	1	6	yes	—
53	C₂×C₂²⋊Dic₅	C₂²×C₃	1	6	yes	—
54	C₂×C₂²⋊Dic₅	C₂²×C₃	1	3	yes	complemented
55	C₂×C₄⋊Dic₅	C₂²×C₃	1	3	yes	complemented
56	C₂×C₄×Dic₅	C₂²×C₃	1	3	yes	complemented
57	#3963	C₂²×C₃	1	6	yes	complemented
58	#2375	—	5	1	no	Sylow
59	C₂³×C₃×C₅	—	2	3	no	—
60	C₂³×C₃×C₅	—	2	6	no	—
61	C₂³×C₃×C₅	—	2	6	no	—
62	C₂³×C₃×C₅	—	2	3	no	—
63	C₂³×C₃×C₅	—	2	3	no	—
64	C₂³×C₃×C₅	—	2	3	no	—
65	C₂³×C₃×C₅	—	4	12	no	—
66	C₂³×C₃×C₅	—	4	12	no	—
67	C₂³×C₃×C₅	—	4	4	no	—
68	C₂³×C₃×C₅	C₂²×C₄	1	6	yes	—
69	C₂³×C₃×C₅	C₂²×C₄	1	2	yes	—
70	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
71	C₃×C₅×D₄	—	4	12	no	—
72	C₂×C₃×C₄×C₅	—	2	3	no	—
73	C₂×C₃×C₄×C₅	—	2	3	no	—
74	C₂×C₃×C₄×C₅	C₂²×C₄	1	6	yes	—
75	C₂×C₃×Dic₅	—	2	6	no	—
76	C₂×C₃×Dic₅	—	2	6	no	—
77	C₂×C₃×Dic₅	—	2	6	no	—
78	C₂×C₃×Dic₅	—	2	6	no	—
79	C₂×C₃×Dic₅	—	4	4	no	—
80	C₂⁵×C₃	C₂×D₅	1	1	yes	characteristic
81	C₂²×C₃×D₄	C₂×D₅	1	3	yes	—
82	C₂³×C₃×C₄	—	5	1	no	—
83	C₃×#120	Dic₅	1	8	yes	complemented
84	C₂×C₃×C₄⋊C₄	—	5	3	no	—
85	C₂×C₃×C₂²⋊C₄	—	5	6	no	—
86	C₂×C₃×C₂²⋊C₄	—	5	6	no	—
87	C₂×C₃×C₂²⋊C₄	—	5	3	no	—
88	C₂×C₃×C₂²⋊C₄	C₂×D₅	1	3	yes	—
89	C₂×C₃×C₄²	—	5	3	no	—
90	C₃×#95	—	5	6	no	—
91	C₂⁴×C₅	—	4	3	no	—
92	C₂⁴×C₅	—	4	3	no	—
93	C₂⁴×C₅	C₂×C₃×C₄	1	4	yes	complemented
94	C₂⁴×C₅	C₂³×C₃	1	3	yes	—
95	C₂⁴×C₅	C₂³×C₃	1	1	yes	characteristic
96	C₂×C₅×D₄	—	4	3	no	—
97	C₂×C₅×D₄	—	4	3	no	—
98	C₂×C₅×D₄	C₂×C₃×C₄	1	12	yes	—
99	C₂²×C₄×C₅	C₂³×C₃	1	3	yes	—
100	C₂²×Dic₅	—	4	3	no	—
101	C₂²×Dic₅	C₂³×C₃	1	6	yes	—
102	C₂²×Dic₅	C₂³×C₃	1	2	yes	—
103	C₅×C₂²⋊C₄	C₂×C₃×C₄	1	12	yes	complemented
104	C₂²⋊Dic₅	—	4	6	no	—
105	C₂²⋊Dic₅	—	4	6	no	—
106	C₂²⋊Dic₅	—	4	3	no	—
107	C₄⋊Dic₅	—	4	3	no	—
108	C₄×Dic₅	—	4	3	no	—
109	C₂×#120	C₃×D₅	1	1	yes	characteristic
110	#390	—	5	3	no	—
111	#388	—	5	6	no	—
112	#386	—	5	3	no	—
113	#379	—	5	2	no	—
114	C₂²×C₃×C₅	—	2	6	no	—
115	C₂²×C₃×C₅	—	2	6	no	—
116	C₂²×C₃×C₅	—	2	12	no	—
117	C₂²×C₃×C₅	—	4	16	no	—
118	C₂²×C₃×C₅	—	4	3	no	—
119	C₂²×C₃×C₅	—	4	3	no	—
120	C₂²×C₃×C₅	—	4	6	no	—
121	C₂²×C₃×C₅	—	4	1	no	—
122	C₂²×C₃×C₅	—	4	3	no	—
123	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
124	C₂²×C₃×C₅	H(5,𝔽₂)	1	3	yes	—
125	C₂²×C₃×C₅	H(5,𝔽₂)	1	3	yes	—
126	C₃×C₄×C₅	—	2	6	no	—
127	C₃×Dic₅	—	4	6	no	—
128	C₃×Dic₅	—	4	2	no	—
129	C₂⁴×C₃	—	4	3	no	—
130	C₂⁴×C₃	—	4	3	no	—
131	C₂⁴×C₃	C₂×Dic₅	1	4	yes	complemented
132	C₂⁴×C₃	C₂²×D₅	1	3	yes	—
133	C₂⁴×C₃	C₂²×D₅	1	1	yes	characteristic
134	C₂×C₃×D₄	—	4	3	no	—
135	C₂×C₃×D₄	—	4	3	no	—
136	C₂×C₃×D₄	C₂×Dic₅	1	12	yes	—
137	C₂²×C₃×C₄	—	5	2	no	—
138	C₂²×C₃×C₄	—	5	6	no	—
139	C₂²×C₃×C₄	—	20	3	no	—
140	C₂²×C₃×C₄	C₂²×D₅	1	3	yes	—
141	C₃×C₄⋊C₄	—	20	3	no	—
142	C₃×C₂²⋊C₄	—	20	6	no	—
143	C₃×C₂²⋊C₄	—	20	3	no	—
144	C₃×C₂²⋊C₄	—	20	6	no	—
145	C₃×C₂²⋊C₄	C₂×Dic₅	1	12	yes	complemented
146	C₃×C₄²	—	20	3	no	—
147	C₂³×C₅	—	2	3	no	—
148	C₂³×C₅	—	2	6	no	—
149	C₂³×C₅	—	2	6	no	—
150	C₂³×C₅	—	2	3	no	—
151	C₂³×C₅	—	2	3	no	—
152	C₂³×C₅	—	2	3	no	—
153	C₂³×C₅	—	4	12	no	—
154	C₂³×C₅	—	4	12	no	—
155	C₂³×C₅	—	4	4	no	—
156	C₂³×C₅	C₂²×C₃×C₄	1	6	yes	—
157	C₂³×C₅	C₂²×C₃×C₄	1	2	yes	—
158	C₂³×C₅	C₂⁴×C₃	1	1	yes	characteristic
159	C₅×D₄	—	4	12	no	—
160	C₂×C₄×C₅	—	2	3	no	—
161	C₂×C₄×C₅	—	2	3	no	—
162	C₂×C₄×C₅	C₂²×C₃×C₄	1	6	yes	—
163	C₂×Dic₅	—	2	6	no	—
164	C₂×Dic₅	—	2	6	no	—
165	C₂×Dic₅	—	2	6	no	—
166	C₂×Dic₅	—	2	6	no	—
167	C₂×Dic₅	—	4	4	no	—
168	C₂⁵	C₂×C₃×D₅	1	1	yes	characteristic
169	C₂²×D₄	C₂×C₃×D₅	1	3	yes	—
170	C₂³×C₄	—	5	1	no	—
171	#120	C₃×Dic₅	1	8	yes	complemented
172	C₂×C₄⋊C₄	—	5	3	no	—
173	C₂×C₂²⋊C₄	—	5	6	no	—
174	C₂×C₂²⋊C₄	—	5	6	no	—
175	C₂×C₂²⋊C₄	—	5	3	no	—
176	C₂×C₂²⋊C₄	C₂×C₃×D₅	1	3	yes	—
177	C₂×C₄²	—	5	3	no	—
178	#95	—	5	6	no	—
179	C₂×C₃×C₅	—	4	2	no	—
180	C₂×C₃×C₅	—	4	6	no	—
181	C₂×C₃×C₅	#518	1	3	yes	—
182	C₂×C₃×C₅	#560	1	3	yes	—
183	C₂×C₃×C₅	H(3,𝔽₄)	1	1	yes	characteristic
184	C₂³×C₃	—	2	3	no	—
185	C₂³×C₃	—	2	6	no	—
186	C₂³×C₃	—	2	6	no	—
187	C₂³×C₃	—	2	3	no	—
188	C₂³×C₃	—	2	3	no	—
189	C₂³×C₃	—	2	3	no	—
190	C₂³×C₃	—	4	12	no	—
191	C₂³×C₃	—	4	4	no	—
192	C₂³×C₃	—	4	12	no	—
193	C₂³×C₃	C₂²×Dic₅	1	6	yes	—
194	C₂³×C₃	C₂²×Dic₅	1	2	yes	—
195	C₂³×C₃	C₂³×D₅	1	1	yes	characteristic
196	C₃×D₄	—	4	12	no	—
197	C₂×C₃×C₄	—	2	3	no	—
198	C₂×C₃×C₄	—	2	3	no	—
199	C₂×C₃×C₄	—	10	6	no	—
200	C₂×C₃×C₄	—	10	6	no	—
201	C₂×C₃×C₄	—	10	6	no	—
202	C₂×C₃×C₄	—	10	6	no	—
203	C₂×C₃×C₄	—	20	4	no	—
204	C₂×C₃×C₄	C₂²×Dic₅	1	6	yes	—
205	C₂²×C₅	—	2	6	no	—
206	C₂²×C₅	—	2	6	no	—
207	C₂²×C₅	—	2	12	no	—
208	C₂²×C₅	—	4	16	no	—
209	C₂²×C₅	—	4	3	no	—
210	C₂²×C₅	—	4	3	no	—
211	C₂²×C₅	—	4	6	no	—
212	C₂²×C₅	—	4	1	no	—
213	C₂²×C₅	—	4	3	no	—
214	C₂²×C₅	C₂³×C₃×C₄	1	1	yes	characteristic
215	C₂²×C₅	C₃×H(5,𝔽₂)	1	3	yes	—
216	C₂²×C₅	C₃×H(5,𝔽₂)	1	3	yes	—
217	C₄×C₅	—	2	6	no	—
218	Dic₅	—	4	6	no	—
219	Dic₅	—	4	2	no	—
220	C₂⁴	—	4	3	no	—
221	C₂⁴	—	4	3	no	—
222	C₂⁴	C₂×C₃×Dic₅	1	4	yes	complemented
223	C₂⁴	C₂²×C₃×D₅	1	3	yes	—
224	C₂⁴	C₂²×C₃×D₅	1	1	yes	characteristic
225	C₂×D₄	—	4	3	no	—
226	C₂×D₄	—	4	3	no	—
227	C₂×D₄	C₂×C₃×Dic₅	1	12	yes	—
228	C₂²×C₄	—	5	2	no	—
229	C₂²×C₄	—	5	6	no	—
230	C₂²×C₄	—	20	3	no	—
231	C₂²×C₄	C₂²×C₃×D₅	1	3	yes	—
232	C₄⋊C₄	—	20	3	no	—
233	C₂²⋊C₄	—	20	6	no	—
234	C₂²⋊C₄	—	20	3	no	—
235	C₂²⋊C₄	—	20	6	no	—
236	C₂²⋊C₄	C₂×C₃×Dic₅	1	12	yes	complemented
237	C₄²	—	20	3	no	—
238	C₃×C₅	#2375	1	1	yes	characteristic, complemented, Hall
239	C₂²×C₃	—	2	6	no	—
240	C₂²×C₃	—	2	6	no	—
241	C₂²×C₃	—	2	12	no	—
242	C₂²×C₃	—	4	16	no	—
243	C₂²×C₃	—	4	1	no	—
244	C₂²×C₃	—	4	3	no	—
245	C₂²×C₃	—	4	3	no	—
246	C₂²×C₃	—	4	6	no	—
247	C₂²×C₃	—	4	3	no	—
248	C₂²×C₃	#4144	1	3	yes	—
249	C₂²×C₃	#4144	1	3	yes	—
250	C₂²×C₃	C₂³×Dic₅	1	1	yes	characteristic
251	C₃×C₄	—	2	6	no	—
252	C₃×C₄	—	20	2	no	—
253	C₃×C₄	—	20	6	no	—
254	C₂×C₅	—	4	6	no	—
255	C₂×C₅	—	4	2	no	—
256	C₂×C₅	C₃×#518	1	3	yes	—
257	C₂×C₅	C₃×#560	1	3	yes	—
258	C₂×C₅	C₃×H(3,𝔽₄)	1	1	yes	characteristic
259	C₂³	—	2	3	no	—
260	C₂³	—	2	6	no	—
261	C₂³	—	2	6	no	—
262	C₂³	—	2	3	no	—
263	C₂³	—	2	3	no	—
264	C₂³	—	2	3	no	—
265	C₂³	—	4	12	no	—
266	C₂³	—	4	4	no	—
267	C₂³	—	4	12	no	—
268	C₂³	C₂²×C₃×Dic₅	1	6	yes	—
269	C₂³	C₂²×C₃×Dic₅	1	2	yes	—
270	C₂³	C₂³×C₃×D₅	1	1	yes	characteristic
271	D₄	—	4	12	no	—
272	C₂×C₄	—	2	3	no	—
273	C₂×C₄	—	2	3	no	—
274	C₂×C₄	—	10	6	no	—
275	C₂×C₄	—	10	6	no	—
276	C₂×C₄	—	10	6	no	—
277	C₂×C₄	—	10	6	no	—
278	C₂×C₄	—	20	4	no	—
279	C₂×C₄	C₂²×C₃×Dic₅	1	6	yes	—
280	C₂×C₃	—	4	6	no	—
281	C₂×C₃	—	4	2	no	—
282	C₂×C₃	#65928	1	1	yes	characteristic
283	C₂×C₃	#65929	1	3	yes	—
284	C₂×C₃	#66132	1	3	yes	—
285	C₅	C₃×#2375	1	1	yes	characteristic, complemented, Sylow
286	C₂²	—	2	6	no	—
287	C₂²	—	2	6	no	—
288	C₂²	—	2	12	no	—
289	C₂²	—	4	16	no	—
290	C₂²	—	4	1	no	—
291	C₂²	—	4	3	no	—
292	C₂²	—	4	6	no	—
293	C₂²	—	4	3	no	—
294	C₂²	—	4	3	no	—
295	C₂²	C₃×#4144	1	3	yes	—
296	C₂²	C₃×#4144	1	3	yes	—
297	C₂²	C₂³×C₃×Dic₅	1	1	yes	characteristic
298	C₄	—	2	6	no	—
299	C₄	—	20	2	no	—
300	C₄	—	20	6	no	—
301	C₃	#10614311	1	1	yes	characteristic, direct product factor, Sylow
302	C₂	—	4	6	no	—
303	C₂	—	4	2	no	—
304	C₂	C₃×#65928	1	1	yes	characteristic
305	C₂	C₃×#65929	1	3	yes	—
306	C₂	C₃×#66132	1	3	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×#120	C₂
socle	C₂³×C₃×C₅	C₂⁴
cosocle	C₂³×C₅	C₂⁴×C₃
hypercenter	C₂×C₃×#120	D₅
hypocenter	C₅	C₃×#2375
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×#120

Complements

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

N	H ≅ G/N	type
2. C₃×#65536	C₂	semidirect
3. C₃×#65510	C₂	semidirect
4. C₃×#65509	C₂	semidirect
5. C₃×#65240	C₂	semidirect
6. #10614311	C₃	direct (N × H)
9. C₂³×C₃×Dic₅	C₂²	semidirect
10. C₃×C₅×#120	C₄	semidirect
14. C₂×C₃×C₂²⋊Dic₅	C₂²	semidirect
15. C₂×C₃×C₄⋊Dic₅	C₂²	semidirect
16. C₂×C₃×C₄×Dic₅	C₂²	semidirect
17. C₃×#3963	C₂²	semidirect
20. #65536	C₂×C₃	semidirect
21. #65510	C₂×C₃	semidirect
22. #65509	C₂×C₃	semidirect
23. #65240	C₂×C₃	semidirect
26. C₂⁴×C₃×C₅	C₂×C₄	semidirect
36. C₃×C₅×C₂²⋊C₄	C₂×C₄	semidirect
49. C₂³×Dic₅	C₂²×C₃	semidirect
50. C₅×#120	C₃×C₄	semidirect
54. C₂×C₂²⋊Dic₅	C₂²×C₃	semidirect
55. C₂×C₄⋊Dic₅	C₂²×C₃	semidirect
56. C₂×C₄×Dic₅	C₂²×C₃	semidirect
57. #3963	C₂²×C₃	semidirect
83. C₃×#120	Dic₅	semidirect
93. C₂⁴×C₅	C₂×C₃×C₄	semidirect
103. C₅×C₂²⋊C₄	C₂×C₃×C₄	semidirect
131. C₂⁴×C₃	C₂×Dic₅	semidirect
145. C₃×C₂²⋊C₄	C₂×Dic₅	semidirect
171. #120	C₃×Dic₅	semidirect
222. C₂⁴	C₂×C₃×Dic₅	semidirect
236. C₂²⋊C₄	C₂×C₃×Dic₅	semidirect
238. C₃×C₅	#2375	semidirect
285. C₅	C₃×#2375	semidirect
301. C₃	#10614311	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×#120	2
C₃×#65536	2
C₃×#65510	2
C₃×#65509	2
C₃×#65240	2
#10614311	3
C₃×#2375	5

Subgroups — C3×#10614311 #14330638

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10614311 #14330638