Subgroups — C₃×#10601568 #14285121

Order 1920.

484 conjugacy classes of subgroups.

The subgroup lattice has 324 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	C₃×C₅×#431	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#65215	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#64705	C₂	1	2	yes	maximal, complemented
5	C₃×#64702	C₂	1	2	yes	maximal, complemented
6	#10601568	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×#709	—	2	1	no	—
8	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
9	C₃×#4076	—	2	1	no	—
10	C₃×#4076	C₄	1	2	yes	—
11	C₃×#4021	—	2	1	no	—
12	C₃×#4020	C₂²	1	1	yes	characteristic
13	C₃×C₄×D₂₀	—	2	1	no	—
14	C₃×C₄×D₂₀	C₄	1	2	yes	—
15	C₃×#4015	C₂²	1	1	yes	characteristic
16	C₃×C₅×C₄⋊C₈	C₂²	1	2	yes	—
17	C₃×C₅×C₂²⋊C₈	—	2	1	no	—
18	C₃×C₄×C₅×C₈	—	2	1	no	—
19	C₃×C₂²⋊#168	—	2	1	no	—
20	C₃×C₄⋊#168	C₂²	1	2	yes	—
21	C₃×#3935	—	2	1	no	—
22	C₃×#1553	—	5	1	no	maximal, Hall
23	#65811	C₂×C₃	1	1	yes	characteristic
24	C₅×#431	C₂×C₃	1	1	yes	characteristic, complemented
25	#65215	C₂×C₃	1	1	yes	characteristic, complemented
26	#64705	C₂×C₃	1	2	yes	complemented
27	#64702	C₂×C₃	1	2	yes	complemented
28	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
30	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
31	C₂×C₃×C₂²⋊D₅	—	2	2	no	—
32	C₃×#709	—	4	4	no	—
33	C₂×C₃×D₂₀	—	2	1	no	—
34	C₂×C₃×D₂₀	—	2	1	no	—
35	C₂×C₃×D₂₀	C₂×C₄	1	2	yes	—
36	C₂×C₃×C₄×D₅	—	2	2	no	—
37	C₂×C₃×Dic₁₀	—	2	1	no	—
38	C₂×C₃×C₅×C₈	—	2	2	no	—
39	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
40	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
41	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
42	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
43	C₃×#685	—	2	2	no	—
44	C₃×#685	—	2	2	no	—
45	C₃×C₄⋊Dic₅	—	2	1	no	—
46	C₃×C₄⋊Dic₅	C₂×C₄	1	2	yes	—
47	C₂×C₃×#168	—	2	2	no	—
48	C₃×#532	—	5	1	no	—
49	C₃×#432	—	5	1	no	—
50	C₃×#431	D₅	1	1	yes	characteristic, complemented
51	C₃×#332	—	5	2	no	—
52	C₃×#331	—	5	2	no	—
53	C₂×#709	—	2	1	no	—
54	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
55	#4076	—	2	1	no	—
56	#4076	C₃×C₄	1	2	yes	—
57	#4021	—	2	1	no	—
58	#4020	C₂²×C₃	1	1	yes	characteristic
59	C₄×D₂₀	—	2	1	no	—
60	C₄×D₂₀	C₃×C₄	1	2	yes	—
61	#4015	C₂²×C₃	1	1	yes	characteristic
62	C₅×C₄⋊C₈	C₂²×C₃	1	2	yes	—
63	C₅×C₂²⋊C₈	—	2	1	no	—
64	C₄×C₅×C₈	—	2	1	no	—
65	C₂²⋊#168	—	2	1	no	—
66	C₄⋊#168	C₂²×C₃	1	2	yes	—
67	#3935	—	2	1	no	—
68	#1553	—	5	1	no	Sylow
69	C₂³×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₃×D₅	—	2	2	no	—
71	C₂×C₃×C₄×C₅	—	2	2	no	—
72	C₂×C₃×C₄×C₅	—	2	2	no	—
73	C₂×C₃×C₄×C₅	—	4	1	no	—
74	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
79	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
80	C₃×C₂²⋊D₅	—	4	4	no	—
81	C₂×C₃×Dic₅	—	2	2	no	—
82	C₃×D₂₀	—	4	2	no	—
83	C₃×D₂₀	—	4	2	no	—
84	C₃×D₂₀	—	4	2	no	—
85	C₃×C₄×D₅	—	4	4	no	—
86	C₃×Dic₁₀	—	4	2	no	—
87	C₃×C₅×C₈	—	4	2	no	—
88	C₃×#168	—	4	2	no	—
89	C₂×C₃×#41	—	10	1	no	—
90	C₃×C₄⋊Q₈	—	5	1	no	—
91	C₃×#127	—	5	1	no	—
92	C₃×#124	—	10	1	no	—
93	C₃×#121	—	5	2	no	—
94	C₃×#121	—	10	1	no	—
95	C₃×C₄×D₄	—	5	2	no	—
96	C₃×C₄×D₄	—	10	1	no	—
97	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
98	C₃×C₄⋊C₈	—	5	2	no	—
99	C₃×C₄⋊C₈	C₂×D₅	1	2	yes	—
100	C₃×C₂²⋊C₈	—	2	1	no	—
101	C₃×C₂²⋊C₈	—	10	1	no	—
102	C₃×#97	—	10	1	no	—
103	C₃×C₄×C₈	—	2	1	no	—
104	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
105	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
106	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
107	C₂×C₂²⋊D₅	—	2	2	no	—
108	#709	—	4	4	no	—
109	C₂×D₂₀	—	2	1	no	—
110	C₂×D₂₀	—	2	1	no	—
111	C₂×D₂₀	C₂×C₃×C₄	1	2	yes	—
112	C₂×C₄×D₅	—	2	2	no	—
113	C₂×Dic₁₀	—	2	1	no	—
114	C₂×C₅×C₈	—	2	2	no	—
115	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
116	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
117	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
118	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
119	#685	—	2	2	no	—
120	#685	—	2	2	no	—
121	C₄⋊Dic₅	—	2	1	no	—
122	C₄⋊Dic₅	C₂×C₃×C₄	1	2	yes	—
123	C₂×#168	—	2	2	no	—
124	#532	—	5	1	no	—
125	#432	—	5	1	no	—
126	#431	C₃×D₅	1	1	yes	characteristic, complemented
127	#332	—	5	2	no	—
128	#331	—	5	2	no	—
129	C₂²×C₃×C₅	—	2	1	no	—
130	C₂²×C₃×C₅	—	2	1	no	—
131	C₂²×C₃×C₅	—	2	1	no	—
132	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
133	C₂×C₃×D₅	—	4	2	no	—
134	C₂×C₃×D₅	—	4	2	no	—
135	C₂×C₃×D₅	—	4	2	no	—
136	C₃×C₄×C₅	—	2	1	no	—
137	C₃×C₄×C₅	—	2	1	no	—
138	C₃×C₄×C₅	—	2	1	no	—
139	C₃×C₄×C₅	—	2	1	no	—
140	C₃×C₄×C₅	#135	1	2	yes	—
141	C₃×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
142	C₃×Dic₅	—	4	2	no	—
143	C₃×#41	—	20	4	no	—
144	C₂×C₃×Q₈	—	10	1	no	—
145	C₂×C₃×D₄	—	5	2	no	—
146	C₂×C₃×D₄	—	10	2	no	—
147	C₂×C₃×D₄	—	10	1	no	—
148	C₂×C₃×D₄	—	10	1	no	—
149	C₂²×C₃×C₄	—	10	2	no	—
150	C₂²×C₃×C₄	C₄×D₅	1	1	yes	characteristic
151	C₂²×C₃×C₄	D₂₀	1	1	yes	characteristic
152	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
153	C₂×C₃×C₈	—	2	2	no	—
154	C₂×C₃×C₈	—	10	2	no	—
155	C₃×C₄⋊C₄	—	5	2	no	—
156	C₃×C₄⋊C₄	—	10	1	no	—
157	C₃×C₂²⋊C₄	—	10	2	no	—
158	C₃×C₂²⋊C₄	—	10	2	no	—
159	C₃×C₄²	C₄×D₅	1	1	yes	characteristic
160	C₃×C₄²	D₂₀	1	1	yes	characteristic
161	C₃×C₄²	C₂²⋊D₅	1	1	yes	characteristic
162	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
163	C₂³×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
164	C₂²×D₅	—	2	2	no	—
165	C₂×C₄×C₅	—	2	2	no	—
166	C₂×C₄×C₅	—	2	2	no	—
167	C₂×C₄×C₅	—	4	1	no	—
168	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
169	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
170	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
171	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
172	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
173	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
174	C₂²⋊D₅	—	4	4	no	—
175	C₂×Dic₅	—	2	2	no	—
176	D₂₀	—	4	2	no	—
177	D₂₀	—	4	2	no	—
178	D₂₀	—	4	2	no	—
179	C₄×D₅	—	4	4	no	—
180	Dic₁₀	—	4	2	no	—
181	C₅×C₈	—	4	2	no	—
182	#168	—	4	2	no	—
183	C₂×#41	—	10	1	no	—
184	C₄⋊Q₈	—	5	1	no	—
185	#127	—	5	1	no	—
186	#124	—	10	1	no	—
187	#121	—	5	2	no	—
188	#121	—	10	1	no	—
189	C₄×D₄	—	5	2	no	—
190	C₄×D₄	—	10	1	no	—
191	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
192	C₄⋊C₈	—	5	2	no	—
193	C₄⋊C₈	C₂×C₃×D₅	1	2	yes	—
194	C₂²⋊C₈	—	2	1	no	—
195	C₂²⋊C₈	—	10	1	no	—
196	#97	—	10	1	no	—
197	C₄×C₈	—	2	1	no	—
198	C₂×C₃×C₅	—	4	1	no	—
199	C₂×C₃×C₅	#413	1	1	yes	characteristic
200	C₂×C₃×C₅	#416	1	1	yes	characteristic
201	C₂×C₃×C₅	#418	1	1	yes	characteristic
202	C₃×D₅	—	8	2	no	—
203	C₂³×C₃	—	10	2	no	—
204	C₂³×C₃	#685	1	1	yes	characteristic
205	C₃×Q₈	—	20	2	no	—
206	C₃×D₄	—	20	2	no	—
207	C₃×D₄	—	20	4	no	—
208	C₃×D₄	—	20	2	no	—
209	C₃×D₄	—	20	2	no	—
210	C₂×C₃×C₄	—	2	2	no	—
211	C₂×C₃×C₄	—	2	2	no	—
212	C₂×C₃×C₄	—	4	1	no	—
213	C₂×C₃×C₄	—	10	2	no	—
214	C₂×C₃×C₄	—	20	4	no	—
215	C₂×C₃×C₄	#685	1	1	yes	characteristic
216	C₂×C₃×C₄	#685	1	1	yes	characteristic
217	C₂×C₃×C₄	#685	1	1	yes	characteristic
218	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
219	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
220	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
221	C₃×C₈	—	4	2	no	—
222	C₃×C₈	—	20	2	no	—
223	C₂²×C₅	—	2	1	no	—
224	C₂²×C₅	—	2	1	no	—
225	C₂²×C₅	—	2	1	no	—
226	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
227	C₂×D₅	—	4	2	no	—
228	C₂×D₅	—	4	2	no	—
229	C₂×D₅	—	4	2	no	—
230	C₄×C₅	—	2	1	no	—
231	C₄×C₅	—	2	1	no	—
232	C₄×C₅	—	2	1	no	—
233	C₄×C₅	—	2	1	no	—
234	C₄×C₅	C₃×#135	1	2	yes	—
235	C₄×C₅	C₃×AGL(1,ℤ/8)	1	2	yes	—
236	Dic₅	—	4	2	no	—
237	#41	—	20	4	no	—
238	C₂×Q₈	—	10	1	no	—
239	C₂×D₄	—	5	2	no	—
240	C₂×D₄	—	10	2	no	—
241	C₂×D₄	—	10	1	no	—
242	C₂×D₄	—	10	1	no	—
243	C₂²×C₄	—	10	2	no	—
244	C₂²×C₄	C₃×C₄×D₅	1	1	yes	characteristic
245	C₂²×C₄	C₃×D₂₀	1	1	yes	characteristic
246	C₂²×C₄	C₃×C₂²⋊D₅	1	1	yes	characteristic
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	10	2	no	—
249	C₄⋊C₄	—	5	2	no	—
250	C₄⋊C₄	—	10	1	no	—
251	C₂²⋊C₄	—	10	2	no	—
252	C₂²⋊C₄	—	10	2	no	—
253	C₄²	C₃×C₄×D₅	1	1	yes	characteristic
254	C₄²	C₃×D₂₀	1	1	yes	characteristic
255	C₄²	C₃×C₂²⋊D₅	1	1	yes	characteristic
256	C₄²	C₂²×C₃×D₅	1	1	yes	characteristic
257	C₃×C₅	#1553	1	1	yes	characteristic, complemented, Hall
258	C₂²×C₃	—	2	1	no	—
259	C₂²×C₃	—	2	1	no	—
260	C₂²×C₃	—	2	1	no	—
261	C₂²×C₃	—	20	2	no	—
262	C₂²×C₃	—	20	2	no	—
263	C₂²×C₃	—	20	2	no	—
264	C₂²×C₃	C₂×#685	1	1	yes	characteristic
265	C₃×C₄	—	2	1	no	—
266	C₃×C₄	—	2	1	no	—
267	C₃×C₄	—	2	1	no	—
268	C₃×C₄	—	2	1	no	—
269	C₃×C₄	—	20	2	no	—
270	C₃×C₄	#4050	1	2	yes	—
271	C₃×C₄	#4095	1	2	yes	—
272	C₂×C₅	—	4	1	no	—
273	C₂×C₅	C₃×#413	1	1	yes	characteristic
274	C₂×C₅	C₃×#416	1	1	yes	characteristic
275	C₂×C₅	C₃×#418	1	1	yes	characteristic
276	D₅	—	8	2	no	—
277	C₂³	—	10	2	no	—
278	C₂³	C₃×#685	1	1	yes	characteristic
279	Q₈	—	20	2	no	—
280	D₄	—	20	2	no	—
281	D₄	—	20	4	no	—
282	D₄	—	20	2	no	—
283	D₄	—	20	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	4	1	no	—
287	C₂×C₄	—	10	2	no	—
288	C₂×C₄	—	20	4	no	—
289	C₂×C₄	C₃×#685	1	1	yes	characteristic
290	C₂×C₄	C₃×#685	1	1	yes	characteristic
291	C₂×C₄	C₃×#685	1	1	yes	characteristic
292	C₂×C₄	C₂×C₃×C₄×D₅	1	1	yes	characteristic
293	C₂×C₄	C₂×C₃×D₂₀	1	1	yes	characteristic
294	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
295	C₈	—	4	2	no	—
296	C₈	—	20	2	no	—
297	C₂×C₃	—	4	1	no	—
298	C₂×C₃	—	40	2	no	—
299	C₂×C₃	#65290	1	1	yes	characteristic
300	C₂×C₃	#65402	1	1	yes	characteristic
301	C₂×C₃	#65421	1	1	yes	characteristic
302	C₅	C₃×#1553	1	1	yes	characteristic, complemented, Sylow
303	C₂²	—	2	1	no	—
304	C₂²	—	2	1	no	—
305	C₂²	—	2	1	no	—
306	C₂²	—	20	2	no	—
307	C₂²	—	20	2	no	—
308	C₂²	—	20	2	no	—
309	C₂²	C₂×C₃×#685	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	20	2	no	—
315	C₄	C₃×#4050	1	2	yes	—
316	C₄	C₃×#4095	1	2	yes	—
317	C₃	#10601568	1	1	yes	characteristic, direct product factor, Sylow
318	C₂	—	4	1	no	—
319	C₂	—	40	2	no	—
320	C₂	C₃×#65290	1	1	yes	characteristic
321	C₂	C₃×#65402	1	1	yes	characteristic
322	C₂	C₃×#65421	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂×#685
commutator	C₂×C₄×C₅	C₂²×C₃×C₄
Frattini	C₄²	C₂²×C₃×D₅
Fitting	C₃×C₅×#431	C₂
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₃×#65811	C₂
cosocle	C₄²×C₅	C₂³×C₃
hypercenter	C₃×#431	D₅
hypocenter	C₅	C₃×#1553
LCS	G ▷ C₂×C₄×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄² ◁ C₃×#431

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×#431	C₂	semidirect
3. C₃×#65215	C₂	semidirect
4. C₃×#64705	C₂	semidirect
5. C₃×#64702	C₂	semidirect
6. #10601568	C₃	direct (N × H)
24. C₅×#431	C₂×C₃	semidirect
25. #65215	C₂×C₃	semidirect
26. #64705	C₂×C₃	semidirect
27. #64702	C₂×C₃	semidirect
50. C₃×#431	D₅	semidirect
126. #431	C₃×D₅	semidirect
257. C₃×C₅	#1553	semidirect
302. C₅	C₃×#1553	semidirect
317. C₃	#10601568	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
C₃×C₅×#431	2
C₃×#65215	2
C₃×#64705	2
C₃×#64702	2
#10601568	3
C₃×#1553	5

Subgroups — C3×#10601568 #14285121

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10601568 #14285121