Subgroups — C₃×#10604604 #14281780

Order 1920.

520 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#66050	C₂	1	1	yes	maximal, characteristic
2	C₃×#65680	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×D₅⋊(C₄⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×D₄⋊#170	C₂	1	4	yes	maximal, complemented
5	#10604604	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂×C₃×#711	—	2	1	no	—
7	C₂×C₃×D₅⋊C₈	—	2	1	no	—
8	C₃×C₅×#127	C₄	1	1	yes	characteristic
9	C₃×#4084	—	2	1	no	—
10	C₃×#4084	C₄	1	2	yes	—
11	C₃×#4082	—	2	1	no	—
12	C₃×D₄×Dic₅	—	2	1	no	—
13	C₃×D₄×Dic₅	C₂²	1	2	yes	—
14	C₃×C₄²×D₅	C₂²	1	1	yes	characteristic
15	C₃×#4015	C₄	1	1	yes	characteristic
16	C₃×#4004	—	2	1	no	—
17	C₃×#4003	—	2	1	no	—
18	C₃×C₄⋊#170	—	2	1	no	—
19	C₃×C₄⋊#170	C₂²	1	2	yes	—
20	C₃×C₄×#170	C₂²	1	2	yes	—
21	C₃×#1480	—	5	1	no	maximal, Hall
22	#66050	C₂×C₃	1	1	yes	characteristic
23	#65680	C₂×C₃	1	1	yes	characteristic, complemented
24	D₅⋊(C₄⋊C₈)	C₂×C₃	1	1	yes	characteristic, complemented
25	D₄⋊#170	C₂×C₃	1	4	yes	complemented
26	C₂×C₃×C₅×D₄	—	2	1	no	—
27	C₂×C₃×C₅×D₄	—	2	1	no	—
28	C₂×C₃×C₅×D₄	C₂×C₄	1	2	yes	—
29	C₂×C₃×C₂²⋊D₅	—	2	2	no	—
30	C₂²×C₃×Dic₅	—	2	2	no	—
31	C₃×#711	—	4	2	no	—
32	C₃×#711	—	4	2	no	—
33	C₂×C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
34	C₂×C₃×C₄×D₅	D₄	1	1	yes	characteristic
35	C₂×C₃×C₄×D₅	D₄	1	1	yes	characteristic
36	C₂×C₃×Dic₁₀	—	2	1	no	—
37	C₂×C₃×#170	—	2	2	no	—
38	C₂×C₃×#170	—	2	2	no	—
39	C₃×D₅⋊C₈	—	4	2	no	—
40	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
41	C₃×C₂²⋊Dic₅	—	2	2	no	—
42	C₃×C₂²⋊Dic₅	—	2	2	no	—
43	C₃×C₄⋊Dic₅	—	2	1	no	—
44	C₃×C₄⋊Dic₅	C₂×C₄	1	2	yes	—
45	C₃×C₄×Dic₅	D₄	1	1	yes	characteristic
46	C₃×C₄×Dic₅	D₄	1	1	yes	characteristic
47	C₃×C₄×Dic₅	C₂³	1	1	yes	characteristic
48	C₃×#532	—	5	1	no	—
49	C₃×#431	—	5	1	no	—
50	C₂×C₃×C₄⋊C₈	—	5	1	no	—
51	C₃×D₄⋊C₈	—	5	4	no	—
52	C₂×#711	—	2	1	no	—
53	C₂×D₅⋊C₈	—	2	1	no	—
54	C₅×#127	C₃×C₄	1	1	yes	characteristic
55	#4084	—	2	1	no	—
56	#4084	C₃×C₄	1	2	yes	—
57	#4082	—	2	1	no	—
58	D₄×Dic₅	—	2	1	no	—
59	D₄×Dic₅	C₂²×C₃	1	2	yes	—
60	C₄²×D₅	C₂²×C₃	1	1	yes	characteristic
61	#4015	C₃×C₄	1	1	yes	characteristic
62	#4004	—	2	1	no	—
63	#4003	—	2	1	no	—
64	C₄⋊#170	—	2	1	no	—
65	C₄⋊#170	C₂²×C₃	1	2	yes	—
66	C₄×#170	C₂²×C₃	1	2	yes	—
67	#1480	—	5	1	no	Sylow
68	C₂³×C₃×C₅	—	2	2	no	—
69	C₂²×C₃×D₅	C₂²⋊C₄	1	1	yes	characteristic
70	C₃×C₅×D₄	—	2	4	no	—
71	C₃×C₅×D₄	—	4	2	no	—
72	C₃×C₅×D₄	—	4	2	no	—
73	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
76	C₃×C₂²⋊D₅	—	4	4	no	—
77	C₂×C₃×Dic₅	—	2	2	no	—
78	C₂×C₃×Dic₅	—	4	2	no	—
79	C₂×C₃×Dic₅	—	4	2	no	—
80	C₂×C₃×Dic₅	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₃×Dic₅	C₂×D₄	1	1	yes	characteristic
82	C₂×C₃×Dic₅	C₂×D₄	1	1	yes	characteristic
83	C₃×C₄×D₅	—	2	2	no	—
84	C₃×C₄×D₅	—	4	1	no	—
85	C₃×C₄×D₅	D₈	1	2	yes	—
86	C₃×C₄×D₅	SD₁₆	1	2	yes	—
87	C₃×Dic₁₀	—	4	2	no	—
88	C₃×#170	—	4	2	no	—
89	C₃×#170	—	4	2	no	—
90	C₂×C₃×#41	—	10	1	no	—
91	C₂²×C₃×C₈	—	10	1	no	—
92	C₃×C₄⋊Q₈	—	5	1	no	—
93	C₃×#127	AGL(1,𝔽₅)	1	1	yes	characteristic
94	C₃×#124	—	10	1	no	—
95	C₃×#121	—	5	2	no	—
96	C₃×#121	—	10	1	no	—
97	C₃×C₄×D₄	—	5	2	no	—
98	C₃×C₄×D₄	—	10	1	no	—
99	C₂×C₃×C₄²	—	5	1	no	—
100	C₃×C₄⋊C₈	—	5	2	no	—
101	C₃×C₄⋊C₈	—	10	1	no	—
102	C₃×C₄⋊C₈	—	10	1	no	—
103	C₃×C₂²⋊C₈	—	10	1	no	—
104	C₃×C₄×C₈	—	5	2	no	—
105	C₂×C₅×D₄	—	2	1	no	—
106	C₂×C₅×D₄	—	2	1	no	—
107	C₂×C₅×D₄	C₂×C₃×C₄	1	2	yes	—
108	C₂×C₂²⋊D₅	—	2	2	no	—
109	C₂²×Dic₅	—	2	2	no	—
110	#711	—	4	2	no	—
111	#711	—	4	2	no	—
112	C₂×C₄×D₅	C₂×C₃×C₄	1	1	yes	characteristic
113	C₂×C₄×D₅	C₃×D₄	1	1	yes	characteristic
114	C₂×C₄×D₅	C₃×D₄	1	1	yes	characteristic
115	C₂×Dic₁₀	—	2	1	no	—
116	C₂×#170	—	2	2	no	—
117	C₂×#170	—	2	2	no	—
118	D₅⋊C₈	—	4	2	no	—
119	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
120	C₂²⋊Dic₅	—	2	2	no	—
121	C₂²⋊Dic₅	—	2	2	no	—
122	C₄⋊Dic₅	—	2	1	no	—
123	C₄⋊Dic₅	C₂×C₃×C₄	1	2	yes	—
124	C₄×Dic₅	C₃×D₄	1	1	yes	characteristic
125	C₄×Dic₅	C₃×D₄	1	1	yes	characteristic
126	C₄×Dic₅	C₂³×C₃	1	1	yes	characteristic
127	#532	—	5	1	no	—
128	#431	—	5	1	no	—
129	C₂×C₄⋊C₈	—	5	1	no	—
130	D₄⋊C₈	—	5	4	no	—
131	C₂²×C₃×C₅	—	4	2	no	—
132	C₂²×C₃×C₅	—	4	2	no	—
133	C₂²×C₃×C₅	—	4	2	no	—
134	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
135	C₂×C₃×D₅	—	2	1	no	—
136	C₂×C₃×D₅	—	2	1	no	—
137	C₂×C₃×D₅	#102	1	2	yes	—
138	C₃×C₄×C₅	—	2	1	no	—
139	C₃×C₄×C₅	#102	1	2	yes	—
140	C₃×C₄×C₅	#131	1	2	yes	—
141	C₃×Dic₅	—	2	1	no	—
142	C₃×Dic₅	—	2	1	no	—
143	C₃×Dic₅	—	4	2	no	—
144	C₃×Dic₅	C₂×D₈	1	1	yes	characteristic
145	C₃×Dic₅	C₂×SD₁₆	1	1	yes	characteristic
146	C₃×#41	—	20	2	no	—
147	C₃×#41	—	20	2	no	—
148	C₂×C₃×Q₈	—	10	1	no	—
149	C₂×C₃×D₄	—	2	1	no	—
150	C₂×C₃×D₄	—	2	1	no	—
151	C₂×C₃×D₄	—	10	2	no	—
152	C₂×C₃×D₄	C₂×AGL(1,𝔽₅)	1	2	yes	—
153	C₂²×C₃×C₄	—	5	1	no	—
154	C₂²×C₃×C₄	—	5	1	no	—
155	C₂²×C₃×C₄	—	5	1	no	—
156	C₂²×C₃×C₄	—	10	2	no	—
157	C₂×C₃×C₈	—	10	2	no	—
158	C₂×C₃×C₈	—	10	2	no	—
159	C₂×C₃×C₈	—	20	2	no	—
160	C₃×C₄⋊C₄	—	5	2	no	—
161	C₃×C₄⋊C₄	—	10	1	no	—
162	C₃×C₂²⋊C₄	—	10	2	no	—
163	C₃×C₂²⋊C₄	—	10	2	no	—
164	C₃×C₄²	—	5	1	no	—
165	C₃×C₄²	—	5	1	no	—
166	C₃×C₄²	—	5	1	no	—
167	C₃×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
168	C₂³×C₅	—	2	2	no	—
169	C₂²×D₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
170	C₅×D₄	—	2	4	no	—
171	C₅×D₄	—	4	2	no	—
172	C₅×D₄	—	4	2	no	—
173	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
174	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
175	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
176	C₂²⋊D₅	—	4	4	no	—
177	C₂×Dic₅	—	2	2	no	—
178	C₂×Dic₅	—	4	2	no	—
179	C₂×Dic₅	—	4	2	no	—
180	C₂×Dic₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
181	C₂×Dic₅	C₂×C₃×D₄	1	1	yes	characteristic
182	C₂×Dic₅	C₂×C₃×D₄	1	1	yes	characteristic
183	C₄×D₅	—	2	2	no	—
184	C₄×D₅	—	4	1	no	—
185	C₄×D₅	C₃×D₈	1	2	yes	—
186	C₄×D₅	C₃×SD₁₆	1	2	yes	—
187	Dic₁₀	—	4	2	no	—
188	#170	—	4	2	no	—
189	#170	—	4	2	no	—
190	C₂×#41	—	10	1	no	—
191	C₂²×C₈	—	10	1	no	—
192	C₄⋊Q₈	—	5	1	no	—
193	#127	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
194	#124	—	10	1	no	—
195	#121	—	5	2	no	—
196	#121	—	10	1	no	—
197	C₄×D₄	—	5	2	no	—
198	C₄×D₄	—	10	1	no	—
199	C₂×C₄²	—	5	1	no	—
200	C₄⋊C₈	—	5	2	no	—
201	C₄⋊C₈	—	10	1	no	—
202	C₄⋊C₈	—	10	1	no	—
203	C₂²⋊C₈	—	10	1	no	—
204	C₄×C₈	—	5	2	no	—
205	C₂×C₃×C₅	—	8	2	no	—
206	C₂×C₃×C₅	#408	1	1	yes	characteristic
207	C₂×C₃×C₅	C₂×#102	1	1	yes	characteristic
208	C₂×C₃×C₅	#421	1	1	yes	characteristic
209	C₃×D₅	—	2	2	no	—
210	C₂³×C₃	—	2	2	no	—
211	C₂³×C₃	—	5	1	no	—
212	C₃×Q₈	—	20	2	no	—
213	C₃×D₄	—	2	4	no	—
214	C₃×D₄	—	4	2	no	—
215	C₃×D₄	—	4	2	no	—
216	C₃×D₄	—	20	4	no	—
217	C₂×C₃×C₄	—	5	1	no	—
218	C₂×C₃×C₄	—	5	2	no	—
219	C₂×C₃×C₄	—	5	2	no	—
220	C₂×C₃×C₄	—	5	1	no	—
221	C₂×C₃×C₄	—	5	1	no	—
222	C₂×C₃×C₄	—	10	2	no	—
223	C₂×C₃×C₄	—	10	2	no	—
224	C₂×C₃×C₄	—	20	1	no	—
225	C₂×C₃×C₄	—	20	2	no	—
226	C₂×C₃×C₄	—	20	2	no	—
227	C₂×C₃×C₄	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
228	C₂×C₃×C₄	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
229	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
230	C₃×C₈	—	20	2	no	—
231	C₃×C₈	—	20	2	no	—
232	C₂²×C₅	—	4	2	no	—
233	C₂²×C₅	—	4	2	no	—
234	C₂²×C₅	—	4	2	no	—
235	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
236	C₂×D₅	—	2	1	no	—
237	C₂×D₅	—	2	1	no	—
238	C₂×D₅	C₃×#102	1	2	yes	—
239	C₄×C₅	—	2	1	no	—
240	C₄×C₅	C₃×#102	1	2	yes	—
241	C₄×C₅	C₃×#131	1	2	yes	—
242	Dic₅	—	2	1	no	—
243	Dic₅	—	2	1	no	—
244	Dic₅	—	4	2	no	—
245	Dic₅	C₂×C₃×D₈	1	1	yes	characteristic
246	Dic₅	C₂×C₃×SD₁₆	1	1	yes	characteristic
247	#41	—	20	2	no	—
248	#41	—	20	2	no	—
249	C₂×Q₈	—	10	1	no	—
250	C₂×D₄	—	2	1	no	—
251	C₂×D₄	—	2	1	no	—
252	C₂×D₄	—	10	2	no	—
253	C₂×D₄	C₂×C₃×AGL(1,𝔽₅)	1	2	yes	—
254	C₂²×C₄	—	5	1	no	—
255	C₂²×C₄	—	5	1	no	—
256	C₂²×C₄	—	5	1	no	—
257	C₂²×C₄	—	10	2	no	—
258	C₂×C₈	—	10	2	no	—
259	C₂×C₈	—	10	2	no	—
260	C₂×C₈	—	20	2	no	—
261	C₄⋊C₄	—	5	2	no	—
262	C₄⋊C₄	—	10	1	no	—
263	C₂²⋊C₄	—	10	2	no	—
264	C₂²⋊C₄	—	10	2	no	—
265	C₄²	—	5	1	no	—
266	C₄²	—	5	1	no	—
267	C₄²	—	5	1	no	—
268	C₄²	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
269	C₃×C₅	#1480	1	1	yes	characteristic, complemented, Hall
270	C₂²×C₃	—	4	2	no	—
271	C₂²×C₃	—	4	2	no	—
272	C₂²×C₃	—	4	2	no	—
273	C₂²×C₃	—	5	2	no	—
274	C₂²×C₃	—	10	1	no	—
275	C₂²×C₃	—	10	1	no	—
276	C₂²×C₃	C₂×C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
277	C₃×C₄	—	2	1	no	—
278	C₃×C₄	—	5	1	no	—
279	C₃×C₄	—	5	1	no	—
280	C₃×C₄	—	10	1	no	—
281	C₃×C₄	—	10	1	no	—
282	C₃×C₄	—	20	2	no	—
283	C₃×C₄	#4007	1	2	yes	—
284	C₃×C₄	#4131	1	2	yes	—
285	C₂×C₅	—	8	2	no	—
286	C₂×C₅	C₃×#408	1	1	yes	characteristic
287	C₂×C₅	C₂×C₃×#102	1	1	yes	characteristic
288	C₂×C₅	C₃×#421	1	1	yes	characteristic
289	D₅	—	2	2	no	—
290	C₂³	—	2	2	no	—
291	C₂³	—	5	1	no	—
292	Q₈	—	20	2	no	—
293	D₄	—	2	4	no	—
294	D₄	—	4	2	no	—
295	D₄	—	4	2	no	—
296	D₄	—	20	4	no	—
297	C₂×C₄	—	5	1	no	—
298	C₂×C₄	—	5	2	no	—
299	C₂×C₄	—	5	2	no	—
300	C₂×C₄	—	5	1	no	—
301	C₂×C₄	—	5	1	no	—
302	C₂×C₄	—	10	2	no	—
303	C₂×C₄	—	10	2	no	—
304	C₂×C₄	—	20	1	no	—
305	C₂×C₄	—	20	2	no	—
306	C₂×C₄	—	20	2	no	—
307	C₂×C₄	C₃×C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
308	C₂×C₄	C₃×C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
309	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
310	C₈	—	20	2	no	—
311	C₈	—	20	2	no	—
312	C₂×C₃	—	8	2	no	—
313	C₂×C₃	—	10	2	no	—
314	C₂×C₃	C₂×#4007	1	1	yes	characteristic
315	C₂×C₃	#65769	1	1	yes	characteristic
316	C₂×C₃	#65776	1	1	yes	characteristic
317	C₅	C₃×#1480	1	1	yes	characteristic, complemented, Sylow
318	C₂²	—	4	2	no	—
319	C₂²	—	4	2	no	—
320	C₂²	—	4	2	no	—
321	C₂²	—	5	2	no	—
322	C₂²	—	10	1	no	—
323	C₂²	—	10	1	no	—
324	C₂²	C₂×C₃×C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
325	C₄	—	2	1	no	—
326	C₄	—	5	1	no	—
327	C₄	—	5	1	no	—
328	C₄	—	10	1	no	—
329	C₄	—	10	1	no	—
330	C₄	—	20	2	no	—
331	C₄	C₃×#4007	1	2	yes	—
332	C₄	C₃×#4131	1	2	yes	—
333	C₃	#10604604	1	1	yes	characteristic, direct product factor, Sylow
334	C₂	—	8	2	no	—
335	C₂	—	10	2	no	—
336	C₂	C₂×C₃×#4007	1	1	yes	characteristic
337	C₂	C₃×#65769	1	1	yes	characteristic
338	C₂	C₃×#65776	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂×C₂²⋊AGL(1,𝔽₅)
commutator	C₂×C₄×C₅	C₂²×C₃×C₄
Frattini	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)
Fitting	C₃×C₅×#127	C₄
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₃×#66050	C₂
cosocle	C₄×Dic₅	C₂³×C₃
hypercenter	C₃×#127	AGL(1,𝔽₅)
hypocenter	C₅	C₃×#1480
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×#127

Complements

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

N	H ≅ G/N	type
2. C₃×#65680	C₂	semidirect
3. C₃×D₅⋊(C₄⋊C₈)	C₂	semidirect
4. C₃×D₄⋊#170	C₂	semidirect
5. #10604604	C₃	direct (N × H)
23. #65680	C₂×C₃	semidirect
24. D₅⋊(C₄⋊C₈)	C₂×C₃	semidirect
25. D₄⋊#170	C₂×C₃	semidirect
269. C₃×C₅	#1480	semidirect
317. C₅	C₃×#1480	semidirect
333. C₃	#10604604	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×#66050	2
C₃×#65680	2
C₃×D₅⋊(C₄⋊C₈)	2
C₃×D₄⋊#170	2
#10604604	3
C₃×#1480	5

Subgroups — C3×#10604604 #14281780

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10604604 #14281780