Subgroups — C₃×#10610260 #14438659

Order 1920.

784 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#65811	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#4016	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₄×C₅×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#65422	C₂	1	4	yes	maximal, complemented
5	C₃×#65002	C₂	1	2	yes	maximal, complemented
6	C₃×#64992	C₂	1	2	yes	maximal, complemented
7	C₃×#64986	C₂	1	4	yes	maximal, complemented
8	#10610260	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂×C₃×#709	C₂²	1	2	yes	—
10	C₂×C₃×C₅×M₁₆	C₂²	1	2	yes	—
11	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
12	C₃×#4076	—	2	2	no	—
13	C₂×C₃×C₄⋊Dic₅	C₂²	1	2	yes	—
14	C₂×C₃×#683	—	2	2	no	—
15	C₃×#4021	C₂²	1	2	yes	—
16	C₃×#4020	C₂²	1	1	yes	characteristic
17	C₃×C₄×D₂₀	—	2	2	no	—
18	C₃×#4016	—	2	2	no	—
19	C₃×#4016	C₂²	1	2	yes	—
20	C₃×#4016	C₂²	1	2	yes	—
21	C₃×#4015	C₂²	1	1	yes	characteristic
22	C₃×C₅×#97	C₂²	1	2	yes	—
23	C₃×C₄×C₅×C₈	C₂²	1	2	yes	—
24	C₃×#3953	C₂²	1	8	yes	—
25	C₃×#3948	C₂²	1	8	yes	—
26	C₃×#3142	—	5	1	no	maximal, Hall
27	#65811	C₂×C₃	1	1	yes	characteristic
28	C₂×#4016	C₂×C₃	1	1	yes	characteristic, complemented
29	C₄×C₅×M₁₆	C₂×C₃	1	1	yes	characteristic, complemented
30	#65422	C₂×C₃	1	4	yes	complemented
31	#65002	C₂×C₃	1	2	yes	complemented
32	#64992	C₂×C₃	1	2	yes	complemented
33	#64986	C₂×C₃	1	4	yes	complemented
34	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
35	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
36	C₂×C₃×C₂²⋊D₅	—	2	2	no	—
37	C₂²×C₃×Dic₅	—	2	2	no	—
38	C₃×#709	—	2	8	no	—
39	C₂×C₃×D₂₀	—	2	2	no	—
40	C₂×C₃×D₂₀	C₂³	1	2	yes	—
41	C₂×C₃×C₄×D₅	—	2	2	no	—
42	C₂×C₃×Dic₁₀	C₂³	1	2	yes	—
43	C₃×C₅×M₁₆	—	2	4	no	—
44	C₂×C₃×C₅×C₈	C₂³	1	4	yes	—
45	C₃×C₄²×C₅	D₄	1	2	yes	—
46	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
47	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
48	C₃×#685	—	2	4	no	—
49	C₃×C₄⋊Dic₅	—	2	2	no	—
50	C₃×C₄⋊Dic₅	—	2	2	no	—
51	C₃×C₄⋊Dic₅	C₂³	1	4	yes	—
52	C₃×#683	—	2	4	no	—
53	C₃×#683	—	2	4	no	—
54	C₃×#532	—	5	1	no	—
55	C₂×C₃×#125	—	5	1	no	—
56	C₃×#491	—	5	2	no	—
57	C₃×#490	—	5	2	no	—
58	C₃×#488	—	5	4	no	—
59	C₃×#417	—	5	4	no	—
60	C₃×C₄×M₁₆	D₅	1	1	yes	characteristic, complemented
61	C₂×#709	C₂²×C₃	1	2	yes	—
62	C₂×C₅×M₁₆	C₂²×C₃	1	2	yes	—
63	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
64	#4076	—	2	2	no	—
65	C₂×C₄⋊Dic₅	C₂²×C₃	1	2	yes	—
66	C₂×#683	—	2	2	no	—
67	#4021	C₂²×C₃	1	2	yes	—
68	#4020	C₂²×C₃	1	1	yes	characteristic
69	C₄×D₂₀	—	2	2	no	—
70	#4016	—	2	2	no	—
71	#4016	C₂²×C₃	1	2	yes	—
72	#4016	C₂²×C₃	1	2	yes	—
73	#4015	C₂²×C₃	1	1	yes	characteristic
74	C₅×#97	C₂²×C₃	1	2	yes	—
75	C₄×C₅×C₈	C₂²×C₃	1	2	yes	—
76	#3953	C₂²×C₃	1	8	yes	—
77	#3948	C₂²×C₃	1	8	yes	—
78	#3142	—	5	1	no	Sylow
79	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₃×D₅	—	2	2	no	—
81	C₂×C₃×C₄×C₅	—	4	2	no	—
82	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
83	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
84	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
85	C₂×C₃×C₄×C₅	#41	1	4	yes	—
86	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
87	C₃×C₂²⋊D₅	—	4	4	no	—
88	C₂×C₃×Dic₅	—	2	4	no	—
89	C₂×C₃×Dic₅	—	2	2	no	—
90	C₂×C₃×Dic₅	—	4	4	no	—
91	C₃×D₂₀	—	2	4	no	—
92	C₃×D₂₀	—	4	4	no	—
93	C₃×C₄×D₅	—	4	4	no	—
94	C₃×Dic₁₀	—	2	4	no	—
95	C₃×C₅×C₈	—	2	4	no	—
96	C₂×C₃×#41	—	5	2	no	—
97	C₂×C₃×M₁₆	C₂×D₅	1	2	yes	—
98	C₃×C₄⋊Q₈	—	5	1	no	—
99	C₃×#127	—	5	1	no	—
100	C₃×#125	—	5	2	no	—
101	C₃×#125	—	5	2	no	—
102	C₃×#125	—	10	2	no	—
103	C₃×#124	—	5	2	no	—
104	C₃×#121	—	10	2	no	—
105	C₃×C₄×D₄	—	10	2	no	—
106	C₂×C₃×C₄⋊C₄	—	5	2	no	—
107	C₂×C₃×C₄⋊C₄	—	10	2	no	—
108	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
109	C₃×#103	—	5	8	no	—
110	C₃×#102	—	5	8	no	—
111	C₃×#97	C₂×D₅	1	2	yes	—
112	C₃×C₄×C₈	C₂×D₅	1	2	yes	—
113	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
114	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
115	C₂×C₂²⋊D₅	—	2	2	no	—
116	C₂²×Dic₅	—	2	2	no	—
117	#709	—	2	8	no	—
118	C₂×D₂₀	—	2	2	no	—
119	C₂×D₂₀	C₂³×C₃	1	2	yes	—
120	C₂×C₄×D₅	—	2	2	no	—
121	C₂×Dic₁₀	C₂³×C₃	1	2	yes	—
122	C₅×M₁₆	—	2	4	no	—
123	C₂×C₅×C₈	C₂³×C₃	1	4	yes	—
124	C₄²×C₅	C₃×D₄	1	2	yes	—
125	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
126	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
127	#685	—	2	4	no	—
128	C₄⋊Dic₅	—	2	2	no	—
129	C₄⋊Dic₅	—	2	2	no	—
130	C₄⋊Dic₅	C₂³×C₃	1	4	yes	—
131	#683	—	2	4	no	—
132	#683	—	2	4	no	—
133	#532	—	5	1	no	—
134	C₂×#125	—	5	1	no	—
135	#491	—	5	2	no	—
136	#490	—	5	2	no	—
137	#488	—	5	4	no	—
138	#417	—	5	4	no	—
139	C₄×M₁₆	C₃×D₅	1	1	yes	characteristic, complemented
140	C₂²×C₃×C₅	—	2	2	no	—
141	C₂²×C₃×C₅	#124	1	2	yes	—
142	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
143	C₂×C₃×D₅	—	4	2	no	—
144	C₂×C₃×D₅	—	4	4	no	—
145	C₃×C₄×C₅	—	2	2	no	—
146	C₃×C₄×C₅	—	2	2	no	—
147	C₃×C₄×C₅	#124	1	2	yes	—
148	C₃×C₄×C₅	C₂×#41	1	2	yes	—
149	C₃×Dic₅	—	4	2	no	—
150	C₃×Dic₅	—	4	4	no	—
151	C₃×#41	—	10	8	no	—
152	C₂×C₃×Q₈	—	5	2	no	—
153	C₂×C₃×D₄	—	5	2	no	—
154	C₂×C₃×D₄	—	10	2	no	—
155	C₂×C₃×D₄	—	10	2	no	—
156	C₂²×C₃×C₄	—	10	2	no	—
157	C₂²×C₃×C₄	—	10	2	no	—
158	C₂²×C₃×C₄	D₂₀	1	2	yes	—
159	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
160	C₃×M₁₆	—	2	4	no	—
161	C₂×C₃×C₈	C₂²×D₅	1	4	yes	—
162	C₃×C₄⋊C₄	—	5	4	no	—
163	C₃×C₄⋊C₄	—	10	4	no	—
164	C₃×C₄⋊C₄	—	10	4	no	—
165	C₃×C₄⋊C₄	—	10	2	no	—
166	C₃×C₄⋊C₄	—	10	2	no	—
167	C₃×C₂²⋊C₄	—	10	4	no	—
168	C₃×C₄²	D₂₀	1	2	yes	—
169	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
170	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
171	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
172	C₂²×D₅	—	2	2	no	—
173	C₂×C₄×C₅	—	4	2	no	—
174	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
175	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
176	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
177	C₂×C₄×C₅	C₃×#41	1	4	yes	—
178	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
179	C₂²⋊D₅	—	4	4	no	—
180	C₂×Dic₅	—	2	4	no	—
181	C₂×Dic₅	—	2	2	no	—
182	C₂×Dic₅	—	4	4	no	—
183	D₂₀	—	2	4	no	—
184	D₂₀	—	4	4	no	—
185	C₄×D₅	—	4	4	no	—
186	Dic₁₀	—	2	4	no	—
187	C₅×C₈	—	2	4	no	—
188	C₂×#41	—	5	2	no	—
189	C₂×M₁₆	C₂×C₃×D₅	1	2	yes	—
190	C₄⋊Q₈	—	5	1	no	—
191	#127	—	5	1	no	—
192	#125	—	5	2	no	—
193	#125	—	5	2	no	—
194	#125	—	10	2	no	—
195	#124	—	5	2	no	—
196	#121	—	10	2	no	—
197	C₄×D₄	—	10	2	no	—
198	C₂×C₄⋊C₄	—	5	2	no	—
199	C₂×C₄⋊C₄	—	10	2	no	—
200	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
201	#103	—	5	8	no	—
202	#102	—	5	8	no	—
203	#97	C₂×C₃×D₅	1	2	yes	—
204	C₄×C₈	C₂×C₃×D₅	1	2	yes	—
205	C₂×C₃×C₅	—	2	2	no	—
206	C₂×C₃×C₅	C₂×#124	1	1	yes	characteristic
207	C₂×C₃×C₅	#575	1	2	yes	—
208	C₃×D₅	—	8	2	no	—
209	C₂³×C₃	—	10	2	no	—
210	C₂³×C₃	C₂×D₂₀	1	1	yes	characteristic
211	C₃×Q₈	—	10	4	no	—
212	C₃×D₄	—	10	4	no	—
213	C₃×D₄	—	20	4	no	—
214	C₃×D₄	—	20	4	no	—
215	C₂×C₃×C₄	—	4	2	no	—
216	C₂×C₃×C₄	—	10	2	no	—
217	C₂×C₃×C₄	—	10	4	no	—
218	C₂×C₃×C₄	—	20	4	no	—
219	C₂×C₃×C₄	—	20	4	no	—
220	C₂×C₃×C₄	C₂×D₂₀	1	2	yes	—
221	C₂×C₃×C₄	C₂×D₂₀	1	2	yes	—
222	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
223	C₂×C₃×C₄	#709	1	4	yes	—
224	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
225	C₃×C₈	—	2	4	no	—
226	C₂²×C₅	—	2	2	no	—
227	C₂²×C₅	C₃×#124	1	2	yes	—
228	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
229	C₂×D₅	—	4	4	no	—
230	C₂×D₅	—	4	2	no	—
231	C₄×C₅	—	2	2	no	—
232	C₄×C₅	—	2	2	no	—
233	C₄×C₅	C₃×#124	1	2	yes	—
234	C₄×C₅	C₂×C₃×#41	1	2	yes	—
235	Dic₅	—	4	4	no	—
236	Dic₅	—	4	2	no	—
237	#41	—	10	8	no	—
238	C₂×Q₈	—	5	2	no	—
239	C₂×D₄	—	5	2	no	—
240	C₂×D₄	—	10	2	no	—
241	C₂×D₄	—	10	2	no	—
242	C₂²×C₄	—	10	2	no	—
243	C₂²×C₄	—	10	2	no	—
244	C₂²×C₄	C₃×D₂₀	1	2	yes	—
245	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
246	M₁₆	—	2	4	no	—
247	C₂×C₈	C₂²×C₃×D₅	1	4	yes	—
248	C₄⋊C₄	—	5	4	no	—
249	C₄⋊C₄	—	10	2	no	—
250	C₄⋊C₄	—	10	4	no	—
251	C₄⋊C₄	—	10	4	no	—
252	C₄⋊C₄	—	10	2	no	—
253	C₂²⋊C₄	—	10	4	no	—
254	C₄²	C₃×D₂₀	1	2	yes	—
255	C₄²	C₂²×C₃×D₅	1	1	yes	characteristic
256	C₄²	C₂²×C₃×D₅	1	1	yes	characteristic
257	C₃×C₅	#3142	1	1	yes	characteristic, complemented, Hall
258	C₂²×C₃	—	2	2	no	—
259	C₂²×C₃	—	20	2	no	—
260	C₂²×C₃	—	20	4	no	—
261	C₂²×C₃	#4021	1	2	yes	—
262	C₂²×C₃	C₂²×D₂₀	1	1	yes	characteristic
263	C₃×C₄	—	2	2	no	—
264	C₃×C₄	—	2	2	no	—
265	C₃×C₄	—	20	2	no	—
266	C₃×C₄	—	20	4	no	—
267	C₃×C₄	#4021	1	2	yes	—
268	C₃×C₄	C₂×#709	1	2	yes	—
269	C₂×C₅	—	2	2	no	—
270	C₂×C₅	C₂×C₃×#124	1	1	yes	characteristic
271	C₂×C₅	C₃×#575	1	2	yes	—
272	D₅	—	8	2	no	—
273	C₂³	—	10	2	no	—
274	C₂³	C₂×C₃×D₂₀	1	1	yes	characteristic
275	Q₈	—	10	4	no	—
276	D₄	—	10	4	no	—
277	D₄	—	20	4	no	—
278	D₄	—	20	4	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	10	2	no	—
281	C₂×C₄	—	10	4	no	—
282	C₂×C₄	—	20	4	no	—
283	C₂×C₄	—	20	4	no	—
284	C₂×C₄	C₂×C₃×D₂₀	1	2	yes	—
285	C₂×C₄	C₂×C₃×D₂₀	1	2	yes	—
286	C₂×C₄	C₂×C₃×D₂₀	1	1	yes	characteristic
287	C₂×C₄	C₃×#709	1	4	yes	—
288	C₂×C₄	C₂³×C₃×D₅	1	1	yes	characteristic
289	C₈	—	2	4	no	—
290	C₂×C₃	—	2	2	no	—
291	C₂×C₃	—	40	2	no	—
292	C₂×C₃	C₂×#4021	1	1	yes	characteristic
293	C₂×C₃	#66082	1	2	yes	—
294	C₅	C₃×#3142	1	1	yes	characteristic, complemented, Sylow
295	C₂²	—	2	2	no	—
296	C₂²	—	20	2	no	—
297	C₂²	—	20	4	no	—
298	C₂²	C₃×#4021	1	2	yes	—
299	C₂²	C₂²×C₃×D₂₀	1	1	yes	characteristic
300	C₄	—	2	2	no	—
301	C₄	—	2	2	no	—
302	C₄	—	20	2	no	—
303	C₄	—	20	4	no	—
304	C₄	C₃×#4021	1	2	yes	—
305	C₄	C₂×C₃×#709	1	2	yes	—
306	C₃	#10610260	1	1	yes	characteristic, direct product factor, Sylow
307	C₂	—	2	2	no	—
308	C₂	—	40	2	no	—
309	C₂	C₂×C₃×#4021	1	1	yes	characteristic
310	C₂	C₃×#66082	1	2	yes	—

	subgroup	G/H
center	C₂²×C₃	C₂²×D₂₀
commutator, cosocle	C₂×C₄×C₅	C₂⁴×C₃
Frattini	C₂×C₄	C₂³×C₃×D₅
Fitting	C₃×C₄×C₅×M₁₆	C₂
socle	C₂²×C₃×C₅	C₂²×D₄
prime radical	C₃×#65811	C₂
hypercenter	C₃×C₄×M₁₆	D₅
hypocenter	C₅	C₃×#3142
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄² ◁ C₃×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×#4016	C₂	semidirect
3. C₃×C₄×C₅×M₁₆	C₂	semidirect
4. C₃×#65422	C₂	semidirect
5. C₃×#65002	C₂	semidirect
6. C₃×#64992	C₂	semidirect
7. C₃×#64986	C₂	semidirect
8. #10610260	C₃	direct (N × H)
28. C₂×#4016	C₂×C₃	semidirect
29. C₄×C₅×M₁₆	C₂×C₃	semidirect
30. #65422	C₂×C₃	semidirect
31. #65002	C₂×C₃	semidirect
32. #64992	C₂×C₃	semidirect
33. #64986	C₂×C₃	semidirect
60. C₃×C₄×M₁₆	D₅	semidirect
139. C₄×M₁₆	C₃×D₅	semidirect
257. C₃×C₅	#3142	semidirect
294. C₅	C₃×#3142	semidirect
306. C₃	#10610260	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×#65811	2
C₂×C₃×#4016	2
C₃×C₄×C₅×M₁₆	2
C₃×#65422	2
C₃×#65002	2
C₃×#64992	2
C₃×#64986	2
#10610260	3
C₃×#3142	5

Subgroups — C3×#10610260 #14438659

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10610260 #14438659