Subgroups — C₃×#10604052 #14305187

Order 1920.

480 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×#3994	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#3992	C₂	1	1	yes	maximal, characteristic
3	C₃×#65712	C₂	1	2	yes	maximal
4	C₃×D₅×#106	C₂	1	1	yes	maximal, characteristic
5	C₃×#64874	C₂	1	2	yes	maximal
6	#10604052	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₄⋊AGL(1,𝔽₅)	—	2	2	no	—
8	C₂×C₃×C₄⋊AGL(1,𝔽₅)	C₂²	1	2	yes	—
9	C₂×C₃×C₄×AGL(1,𝔽₅)	C₂²	1	1	yes	characteristic
10	C₂×C₃×D₅⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₈×D₅	C₂²	1	1	yes	characteristic
12	C₃×D₅×C₄⋊C₄	C₂²	1	2	yes	—
13	C₃×(C₂×AGL(1,𝔽₅))⋊C₄	—	2	2	no	—
14	C₃×#3994	—	2	2	no	—
15	C₃×#3992	—	2	2	no	—
16	C₃×C₅×#106	C₄	1	1	yes	characteristic, complemented
17	C₃×#3949	C₄	1	1	yes	characteristic, complemented
18	C₃×#3940	C₄	1	2	yes	complemented
19	C₃×#1944	—	5	1	no	maximal, Hall
20	C₂×#3994	C₂×C₃	1	1	yes	characteristic
21	C₂×#3992	C₂×C₃	1	1	yes	characteristic
22	#65712	C₂×C₃	1	2	yes	—
23	D₅×#106	C₂×C₃	1	1	yes	characteristic
24	#64874	C₂×C₃	1	2	yes	—
25	C₂²×C₃×AGL(1,𝔽₅)	—	2	2	no	—
26	C₂²×C₃×AGL(1,𝔽₅)	D₄	1	1	yes	characteristic
27	C₂²×C₃×AGL(1,𝔽₅)	D₄	1	1	yes	characteristic
28	C₂×C₃×C₄×D₅	—	2	2	no	—
29	C₂×C₃×C₄×D₅	C₂³	1	1	yes	characteristic
30	C₂×C₃×#170	—	2	1	no	—
31	C₃×C₄⋊AGL(1,𝔽₅)	—	2	2	no	—
32	C₃×C₄⋊AGL(1,𝔽₅)	—	2	2	no	—
33	C₃×C₄⋊AGL(1,𝔽₅)	—	4	4	no	—
34	C₃×C₄×AGL(1,𝔽₅)	—	2	1	no	—
35	C₃×C₄×AGL(1,𝔽₅)	—	2	1	no	—
36	C₃×D₅⋊C₈	—	2	1	no	—
37	C₃×D₅⋊C₈	Q₈	1	2	yes	—
38	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
39	C₃×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
40	C₃×C₄⋊Dic₅	C₂×C₄	1	2	yes	—
41	C₃×#683	—	2	2	no	—
42	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
43	C₃×C₈×D₅	D₄	1	2	yes	—
44	C₃×C₈×D₅	Q₈	1	2	yes	—
45	C₂×C₃×#107	—	5	1	no	—
46	C₂×C₃×#106	—	5	1	no	—
47	C₂×C₃×#97	—	5	1	no	—
48	C₃×#389	—	5	2	no	—
49	C₃×#340	—	5	2	no	—
50	C₂×C₄⋊AGL(1,𝔽₅)	—	2	2	no	—
51	C₂×C₄⋊AGL(1,𝔽₅)	C₂²×C₃	1	2	yes	—
52	C₂×C₄×AGL(1,𝔽₅)	C₂²×C₃	1	1	yes	characteristic
53	C₂×D₅⋊C₈	C₂²×C₃	1	1	yes	characteristic
54	C₂×C₈×D₅	C₂²×C₃	1	1	yes	characteristic
55	D₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
56	(C₂×AGL(1,𝔽₅))⋊C₄	—	2	2	no	—
57	#3994	—	2	2	no	—
58	#3992	—	2	2	no	—
59	C₅×#106	C₃×C₄	1	1	yes	characteristic, complemented
60	#3949	C₃×C₄	1	1	yes	characteristic, complemented
61	#3940	C₃×C₄	1	2	yes	complemented
62	#1944	—	5	1	no	Sylow
63	C₂²×C₃×D₅	C₂×D₄	1	1	yes	characteristic
64	C₂×C₃×AGL(1,𝔽₅)	—	2	1	no	—
65	C₂×C₃×AGL(1,𝔽₅)	—	2	1	no	—
66	C₂×C₃×AGL(1,𝔽₅)	—	2	2	no	—
67	C₂×C₃×AGL(1,𝔽₅)	—	2	2	no	—
68	C₂×C₃×AGL(1,𝔽₅)	—	4	2	no	—
69	C₂×C₃×AGL(1,𝔽₅)	—	4	2	no	—
70	C₂×C₃×AGL(1,𝔽₅)	—	4	2	no	—
71	C₂×C₃×C₄×C₅	—	2	2	no	—
72	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₃×Dic₅	—	2	2	no	—
74	C₂×C₃×Dic₅	#41	1	1	yes	characteristic
75	C₃×C₄×D₅	—	4	2	no	—
76	C₃×C₄×D₅	—	4	2	no	—
77	C₃×C₄×D₅	C₂×D₄	1	1	yes	characteristic
78	C₃×C₄×D₅	C₂×Q₈	1	1	yes	characteristic
79	C₃×C₄×D₅	C₂×Q₈	1	1	yes	characteristic
80	C₃×C₄×D₅	#41	1	1	yes	characteristic
81	C₃×#170	—	2	2	no	—
82	C₃×C₅×C₈	C₄⋊C₄	1	2	yes	complemented
83	C₃×#168	C₄⋊C₄	1	2	yes	complemented
84	C₂²×C₃×C₈	—	5	1	no	—
85	C₂²×C₃×C₈	—	5	1	no	—
86	C₂×C₃×C₄⋊C₄	—	5	2	no	—
87	C₂×C₃×C₄⋊C₄	—	5	2	no	—
88	C₂×C₃×C₄⋊C₄	—	10	2	no	—
89	C₂×C₃×C₄²	—	5	1	no	—
90	C₃×#107	—	10	2	no	—
91	C₃×#106	—	5	1	no	—
92	C₃×#106	—	5	2	no	—
93	C₃×#106	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
94	C₃×#97	—	10	2	no	—
95	C₃×#95	—	10	2	no	—
96	C₂²×AGL(1,𝔽₅)	—	2	2	no	—
97	C₂²×AGL(1,𝔽₅)	C₃×D₄	1	1	yes	characteristic
98	C₂²×AGL(1,𝔽₅)	C₃×D₄	1	1	yes	characteristic
99	C₂×C₄×D₅	—	2	2	no	—
100	C₂×C₄×D₅	C₂³×C₃	1	1	yes	characteristic
101	C₂×#170	—	2	1	no	—
102	C₄⋊AGL(1,𝔽₅)	—	2	2	no	—
103	C₄⋊AGL(1,𝔽₅)	—	2	2	no	—
104	C₄⋊AGL(1,𝔽₅)	—	4	4	no	—
105	C₄×AGL(1,𝔽₅)	—	2	1	no	—
106	C₄×AGL(1,𝔽₅)	—	2	1	no	—
107	D₅⋊C₈	—	2	1	no	—
108	D₅⋊C₈	C₃×Q₈	1	2	yes	—
109	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
110	C₅×C₄⋊C₄	C₂×C₃×C₄	1	2	yes	—
111	C₄⋊Dic₅	C₂×C₃×C₄	1	2	yes	—
112	#683	—	2	2	no	—
113	C₂×#168	C₂×C₃×C₄	1	1	yes	characteristic
114	C₈×D₅	C₃×D₄	1	2	yes	—
115	C₈×D₅	C₃×Q₈	1	2	yes	—
116	C₂×#107	—	5	1	no	—
117	C₂×#106	—	5	1	no	—
118	C₂×#97	—	5	1	no	—
119	#389	—	5	2	no	—
120	#340	—	5	2	no	—
121	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
122	C₂×C₃×D₅	#121	1	1	yes	characteristic
123	C₂×C₃×D₅	C₄⋊Q₈	1	1	yes	characteristic
124	C₂×C₃×D₅	AGL(1,ℤ/8)	1	2	yes	—
125	C₂×C₃×D₅	#137	1	2	yes	—
126	C₃×AGL(1,𝔽₅)	—	4	1	no	—
127	C₃×AGL(1,𝔽₅)	—	4	1	no	—
128	C₃×AGL(1,𝔽₅)	—	8	2	no	—
129	C₃×C₄×C₅	—	4	2	no	—
130	C₃×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
131	C₃×C₄×C₅	C₄×Q₈	1	1	yes	characteristic
132	C₃×Dic₅	—	4	2	no	—
133	C₃×Dic₅	C₂²⋊Q₈	1	1	yes	characteristic
134	C₃×Dic₅	#125	1	1	yes	characteristic
135	C₂²×C₃×C₄	—	5	1	no	—
136	C₂²×C₃×C₄	—	5	1	no	—
137	C₂²×C₃×C₄	—	5	1	no	—
138	C₂²×C₃×C₄	—	10	2	no	—
139	C₂²×C₃×C₄	—	10	2	no	—
140	C₂×C₃×C₈	—	5	1	no	—
141	C₂×C₃×C₈	—	5	2	no	—
142	C₂×C₃×C₈	—	5	2	no	—
143	C₂×C₃×C₈	—	5	2	no	—
144	C₂×C₃×C₈	—	10	1	no	—
145	C₂×C₃×C₈	—	10	1	no	—
146	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
147	C₃×C₄⋊C₄	—	5	2	no	—
148	C₃×C₄⋊C₄	—	10	2	no	—
149	C₃×C₄⋊C₄	—	10	2	no	—
150	C₃×C₄⋊C₄	—	10	2	no	—
151	C₃×C₄⋊C₄	—	20	4	no	—
152	C₃×C₄⋊C₄	C₂×AGL(1,𝔽₅)	1	2	yes	—
153	C₃×C₄²	—	10	1	no	—
154	C₃×C₄²	—	10	1	no	—
155	C₂²×D₅	C₂×C₃×D₄	1	1	yes	characteristic
156	C₂×AGL(1,𝔽₅)	—	2	1	no	—
157	C₂×AGL(1,𝔽₅)	—	2	1	no	—
158	C₂×AGL(1,𝔽₅)	—	2	2	no	—
159	C₂×AGL(1,𝔽₅)	—	2	2	no	—
160	C₂×AGL(1,𝔽₅)	—	4	2	no	—
161	C₂×AGL(1,𝔽₅)	—	4	2	no	—
162	C₂×AGL(1,𝔽₅)	—	4	2	no	—
163	C₂×C₄×C₅	—	2	2	no	—
164	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
165	C₂×Dic₅	—	2	2	no	—
166	C₂×Dic₅	C₃×#41	1	1	yes	characteristic
167	C₄×D₅	—	4	2	no	—
168	C₄×D₅	—	4	2	no	—
169	C₄×D₅	C₂×C₃×D₄	1	1	yes	characteristic
170	C₄×D₅	C₂×C₃×Q₈	1	1	yes	characteristic
171	C₄×D₅	C₂×C₃×Q₈	1	1	yes	characteristic
172	C₄×D₅	C₃×#41	1	1	yes	characteristic
173	#170	—	2	2	no	—
174	C₅×C₈	C₃×C₄⋊C₄	1	2	yes	complemented
175	#168	C₃×C₄⋊C₄	1	2	yes	complemented
176	C₂²×C₈	—	5	1	no	—
177	C₂²×C₈	—	5	1	no	—
178	C₂×C₄⋊C₄	—	5	2	no	—
179	C₂×C₄⋊C₄	—	5	2	no	—
180	C₂×C₄⋊C₄	—	10	2	no	—
181	C₂×C₄²	—	5	1	no	—
182	#107	—	10	2	no	—
183	#106	—	5	1	no	—
184	#106	—	5	2	no	—
185	#106	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
186	#97	—	10	2	no	—
187	#95	—	10	2	no	—
188	C₂×C₃×C₅	#389	1	1	yes	characteristic
189	C₂×C₃×C₅	#440	1	2	yes	—
190	C₃×D₅	#469	1	1	yes	characteristic
191	C₃×D₅	#470	1	1	yes	characteristic
192	C₃×D₅	#501	1	2	yes	—
193	C₂³×C₃	—	5	1	no	—
194	C₂×C₃×C₄	—	2	2	no	—
195	C₂×C₃×C₄	—	5	1	no	—
196	C₂×C₃×C₄	—	5	1	no	—
197	C₂×C₃×C₄	—	5	1	no	—
198	C₂×C₃×C₄	—	5	1	no	—
199	C₂×C₃×C₄	—	5	1	no	—
200	C₂×C₃×C₄	—	10	1	no	—
201	C₂×C₃×C₄	—	10	1	no	—
202	C₂×C₃×C₄	—	10	2	no	—
203	C₂×C₃×C₄	—	10	2	no	—
204	C₂×C₃×C₄	—	10	2	no	—
205	C₂×C₃×C₄	—	20	2	no	—
206	C₂×C₃×C₄	—	20	2	no	—
207	C₂×C₃×C₄	—	20	2	no	—
208	C₂×C₃×C₄	—	20	2	no	—
209	C₂×C₃×C₄	—	20	2	no	—
210	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
211	C₃×C₈	—	5	2	no	—
212	C₃×C₈	—	10	2	no	—
213	C₃×C₈	C₄⋊AGL(1,𝔽₅)	1	2	yes	complemented
214	C₂²×C₅	C₃×C₄×D₄	1	1	yes	characteristic
215	C₂×D₅	C₃×#121	1	1	yes	characteristic
216	C₂×D₅	C₃×C₄⋊Q₈	1	1	yes	characteristic
217	C₂×D₅	C₃×AGL(1,ℤ/8)	1	2	yes	—
218	C₂×D₅	C₃×#137	1	2	yes	—
219	AGL(1,𝔽₅)	—	4	1	no	—
220	AGL(1,𝔽₅)	—	4	1	no	—
221	AGL(1,𝔽₅)	—	8	2	no	—
222	C₄×C₅	—	4	2	no	—
223	C₄×C₅	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
224	C₄×C₅	C₃×C₄×Q₈	1	1	yes	characteristic
225	Dic₅	—	4	2	no	—
226	Dic₅	C₃×C₂²⋊Q₈	1	1	yes	characteristic
227	Dic₅	C₃×#125	1	1	yes	characteristic
228	C₂²×C₄	—	5	1	no	—
229	C₂²×C₄	—	5	1	no	—
230	C₂²×C₄	—	5	1	no	—
231	C₂²×C₄	—	10	2	no	—
232	C₂²×C₄	—	10	2	no	—
233	C₂×C₈	—	5	1	no	—
234	C₂×C₈	—	5	2	no	—
235	C₂×C₈	—	5	2	no	—
236	C₂×C₈	—	5	2	no	—
237	C₂×C₈	—	10	1	no	—
238	C₂×C₈	—	10	1	no	—
239	C₂×C₈	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
240	C₄⋊C₄	—	5	2	no	—
241	C₄⋊C₄	—	10	2	no	—
242	C₄⋊C₄	—	10	2	no	—
243	C₄⋊C₄	—	10	2	no	—
244	C₄⋊C₄	—	20	4	no	—
245	C₄⋊C₄	C₂×C₃×AGL(1,𝔽₅)	1	2	yes	—
246	C₄²	—	10	1	no	—
247	C₄²	—	10	1	no	—
248	C₃×C₅	#1944	1	1	yes	characteristic, complemented, Hall
249	C₂²×C₃	—	5	1	no	—
250	C₂²×C₃	—	5	1	no	—
251	C₂²×C₃	—	5	2	no	—
252	C₂²×C₃	—	5	2	no	—
253	C₂²×C₃	D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
254	C₃×C₄	—	4	2	no	—
255	C₃×C₄	—	5	1	no	—
256	C₃×C₄	—	5	1	no	—
257	C₃×C₄	—	20	1	no	—
258	C₃×C₄	—	20	1	no	—
259	C₃×C₄	—	20	2	no	—
260	C₃×C₄	—	40	2	no	—
261	C₃×C₄	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
262	C₃×C₄	Q₈×AGL(1,𝔽₅)	1	1	yes	characteristic
263	C₂×C₅	C₃×#389	1	1	yes	characteristic
264	C₂×C₅	C₃×#440	1	2	yes	—
265	D₅	C₃×#469	1	1	yes	characteristic
266	D₅	C₃×#470	1	1	yes	characteristic
267	D₅	C₃×#501	1	2	yes	—
268	C₂³	—	5	1	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	5	1	no	—
271	C₂×C₄	—	5	1	no	—
272	C₂×C₄	—	5	1	no	—
273	C₂×C₄	—	5	1	no	—
274	C₂×C₄	—	5	1	no	—
275	C₂×C₄	—	10	1	no	—
276	C₂×C₄	—	10	1	no	—
277	C₂×C₄	—	10	2	no	—
278	C₂×C₄	—	10	2	no	—
279	C₂×C₄	—	10	2	no	—
280	C₂×C₄	—	20	2	no	—
281	C₂×C₄	—	20	2	no	—
282	C₂×C₄	—	20	2	no	—
283	C₂×C₄	—	20	2	no	—
284	C₂×C₄	—	20	2	no	—
285	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
286	C₈	—	5	2	no	—
287	C₈	—	10	2	no	—
288	C₈	C₃×C₄⋊AGL(1,𝔽₅)	1	2	yes	complemented
289	C₂×C₃	—	5	1	no	—
290	C₂×C₃	—	5	1	no	—
291	C₂×C₃	—	5	2	no	—
292	C₂×C₃	#65712	1	1	yes	characteristic
293	C₂×C₃	#65735	1	2	yes	—
294	C₅	C₃×#1944	1	1	yes	characteristic, complemented, Sylow
295	C₂²	—	5	1	no	—
296	C₂²	—	5	1	no	—
297	C₂²	—	5	2	no	—
298	C₂²	—	5	2	no	—
299	C₂²	C₃×D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
300	C₄	—	4	2	no	—
301	C₄	—	5	1	no	—
302	C₄	—	5	1	no	—
303	C₄	—	20	1	no	—
304	C₄	—	20	1	no	—
305	C₄	—	20	2	no	—
306	C₄	—	40	2	no	—
307	C₄	C₂×C₃×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
308	C₄	C₃×Q₈×AGL(1,𝔽₅)	1	1	yes	characteristic
309	C₃	#10604052	1	1	yes	characteristic, direct product factor, Sylow
310	C₂	—	5	1	no	—
311	C₂	—	5	1	no	—
312	C₂	—	5	2	no	—
313	C₂	C₃×#65712	1	1	yes	characteristic
314	C₂	C₃×#65735	1	2	yes	—

	subgroup	G/H
center	C₂²×C₃	D₄×AGL(1,𝔽₅)
commutator	C₂×C₄×C₅	C₂²×C₃×C₄
Frattini	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)
Fitting	C₃×C₅×#106	C₄
socle	C₂²×C₃×C₅	C₄×D₄
prime radical	C₂²×C₃×D₅	C₂×D₄
cosocle	C₂×C₄×D₅	C₂³×C₃
hypercenter	C₃×#106	AGL(1,𝔽₅)
hypocenter	C₅	C₃×#1944
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄ ◁ C₃×#106

Complements

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

N	H ≅ G/N	type
6. #10604052	C₃	direct (N × H)
16. C₃×C₅×#106	C₄	semidirect
17. C₃×#3949	C₄	semidirect
18. C₃×#3940	C₄	semidirect
59. C₅×#106	C₃×C₄	semidirect
60. #3949	C₃×C₄	semidirect
61. #3940	C₃×C₄	semidirect
82. C₃×C₅×C₈	C₄⋊C₄	semidirect
83. C₃×#168	C₄⋊C₄	semidirect
93. C₃×#106	AGL(1,𝔽₅)	semidirect
174. C₅×C₈	C₃×C₄⋊C₄	semidirect
175. #168	C₃×C₄⋊C₄	semidirect
185. #106	C₃×AGL(1,𝔽₅)	semidirect
213. C₃×C₈	C₄⋊AGL(1,𝔽₅)	semidirect
248. C₃×C₅	#1944	semidirect
288. C₈	C₃×C₄⋊AGL(1,𝔽₅)	semidirect
294. C₅	C₃×#1944	semidirect
309. C₃	#10604052	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×#3994	2
C₂×C₃×#3992	2
C₃×#65712	2
C₃×D₅×#106	2
C₃×#64874	2
#10604052	3
C₃×#1944	5

Subgroups — C3×#10604052 #14305187

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10604052 #14305187