Subgroups — C₃×#10617348 #14331291

Order 1920.

816 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×D₅×#125	C₂	1	1	yes	maximal, characteristic
2	C₃×#65712	C₂	1	4	yes	maximal
3	C₃×#65711	C₂	1	4	yes	maximal
4	C₃×#65711	C₂	1	4	yes	maximal
5	C₃×#65689	C₂	1	2	yes	maximal
6	#10617348	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₄⋊AGL(1,𝔽₅)	C₂²	1	2	yes	—
8	C₂×C₃×C₄⋊AGL(1,𝔽₅)	C₂²	1	2	yes	—
9	C₂×C₃×C₄⋊AGL(1,𝔽₅)	C₂²	1	4	yes	—
10	C₂×C₃×C₄×AGL(1,𝔽₅)	C₂²	1	4	yes	—
11	C₃×C₅×#125	C₄	1	1	yes	characteristic, complemented
12	C₃×D₅×C₄⋊C₄	C₂²	1	2	yes	—
13	C₃×D₅×C₄⋊C₄	C₂²	1	4	yes	—
14	C₃×#4036	C₄	1	2	yes	complemented
15	C₃×#4035	C₄	1	4	yes	complemented
16	C₃×C₄²×D₅	C₂²	1	1	yes	characteristic
17	C₃×#4016	C₄	1	1	yes	characteristic, complemented
18	C₃×(C₂×AGL(1,𝔽₅))⋊C₄	C₂²	1	4	yes	—
19	C₃×(C₂×AGL(1,𝔽₅))⋊C₄	C₂²	1	4	yes	—
20	C₃×(C₂×AGL(1,𝔽₅))⋊C₄	C₂²	1	4	yes	—
21	C₃×(C₂×AGL(1,𝔽₅))⋊C₄	C₂²	1	4	yes	—
22	C₃×#2382	—	5	1	no	maximal, Hall
23	D₅×#125	C₂×C₃	1	1	yes	characteristic
24	#65712	C₂×C₃	1	4	yes	—
25	#65711	C₂×C₃	1	4	yes	—
26	#65711	C₂×C₃	1	4	yes	—
27	#65689	C₂×C₃	1	2	yes	—
28	C₂²×C₃×AGL(1,𝔽₅)	C₂³	1	4	yes	—
29	C₂²×C₃×AGL(1,𝔽₅)	C₂³	1	4	yes	—
30	C₂×C₃×C₄×D₅	C₂³	1	4	yes	—
31	C₂×C₃×C₄×D₅	C₂³	1	2	yes	—
32	C₂×C₃×C₄×D₅	C₂³	1	1	yes	characteristic
33	C₃×C₄⋊AGL(1,𝔽₅)	—	4	4	no	—
34	C₃×C₄⋊AGL(1,𝔽₅)	—	4	2	no	—
35	C₃×C₄⋊AGL(1,𝔽₅)	—	4	2	no	—
36	C₃×C₄×AGL(1,𝔽₅)	—	4	4	no	—
37	C₃×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
38	C₃×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
39	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
40	C₃×C₄⋊Dic₅	C₂×C₄	1	2	yes	—
41	C₃×C₄⋊Dic₅	C₂×C₄	1	4	yes	—
42	C₃×#683	C₂×C₄	1	4	yes	—
43	C₃×#683	C₂×C₄	1	4	yes	—
44	C₃×#683	C₂×C₄	1	4	yes	—
45	C₃×C₄×Dic₅	C₂×C₄	1	2	yes	—
46	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
47	C₂×C₃×#125	—	5	1	no	—
48	C₃×#389	—	5	4	no	—
49	C₃×#387	—	5	4	no	—
50	C₃×#387	—	5	4	no	—
51	C₃×#382	—	5	2	no	—
52	C₂×C₄⋊AGL(1,𝔽₅)	C₂²×C₃	1	2	yes	—
53	C₂×C₄⋊AGL(1,𝔽₅)	C₂²×C₃	1	2	yes	—
54	C₂×C₄⋊AGL(1,𝔽₅)	C₂²×C₃	1	4	yes	—
55	C₂×C₄×AGL(1,𝔽₅)	C₂²×C₃	1	4	yes	—
56	C₅×#125	C₃×C₄	1	1	yes	characteristic, complemented
57	D₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
58	D₅×C₄⋊C₄	C₂²×C₃	1	4	yes	—
59	#4036	C₃×C₄	1	2	yes	complemented
60	#4035	C₃×C₄	1	4	yes	complemented
61	C₄²×D₅	C₂²×C₃	1	1	yes	characteristic
62	#4016	C₃×C₄	1	1	yes	characteristic, complemented
63	(C₂×AGL(1,𝔽₅))⋊C₄	C₂²×C₃	1	4	yes	—
64	(C₂×AGL(1,𝔽₅))⋊C₄	C₂²×C₃	1	4	yes	—
65	(C₂×AGL(1,𝔽₅))⋊C₄	C₂²×C₃	1	4	yes	—
66	(C₂×AGL(1,𝔽₅))⋊C₄	C₂²×C₃	1	4	yes	—
67	#2382	—	5	1	no	Sylow
68	C₂²×C₃×D₅	C₂⁴	1	1	yes	characteristic
69	C₂×C₃×AGL(1,𝔽₅)	—	2	4	no	—
70	C₂×C₃×AGL(1,𝔽₅)	—	2	4	no	—
71	C₂×C₃×AGL(1,𝔽₅)	—	2	4	no	—
72	C₂×C₃×AGL(1,𝔽₅)	—	2	4	no	—
73	C₂×C₃×AGL(1,𝔽₅)	—	2	4	no	—
74	C₂×C₃×AGL(1,𝔽₅)	—	2	4	no	—
75	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
76	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₃×C₄×C₅	C₂²×C₄	1	4	yes	—
78	C₂×C₃×Dic₅	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₃×Dic₅	C₂²×C₄	1	2	yes	—
80	C₂×C₃×Dic₅	C₂²×C₄	1	4	yes	—
81	C₃×C₄×D₅	—	2	4	no	—
82	C₃×C₄×D₅	—	2	2	no	—
83	C₃×C₄×D₅	—	2	1	no	—
84	C₃×C₄×D₅	—	2	1	no	—
85	C₃×C₄×D₅	—	2	4	no	—
86	C₃×C₄×D₅	—	2	2	no	—
87	C₃×#125	—	5	4	no	—
88	C₃×#125	—	5	2	no	—
89	C₃×#125	—	5	1	no	—
90	C₃×#125	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
91	C₂×C₃×C₄⋊C₄	—	5	2	no	—
92	C₂×C₃×C₄⋊C₄	—	5	4	no	—
93	C₂×C₃×C₄⋊C₄	—	5	2	no	—
94	C₂×C₃×C₄⋊C₄	—	5	2	no	—
95	C₂×C₃×C₄⋊C₄	—	5	4	no	—
96	C₂×C₃×C₄²	—	5	1	no	—
97	C₂×C₃×C₄²	—	5	4	no	—
98	C₃×#95	—	5	4	no	—
99	C₃×#95	—	5	4	no	—
100	C₃×#95	—	5	4	no	—
101	C₃×#95	—	5	4	no	—
102	C₂²×AGL(1,𝔽₅)	C₂³×C₃	1	4	yes	—
103	C₂²×AGL(1,𝔽₅)	C₂³×C₃	1	4	yes	—
104	C₂×C₄×D₅	C₂³×C₃	1	4	yes	—
105	C₂×C₄×D₅	C₂³×C₃	1	2	yes	—
106	C₂×C₄×D₅	C₂³×C₃	1	1	yes	characteristic
107	C₄⋊AGL(1,𝔽₅)	—	4	4	no	—
108	C₄⋊AGL(1,𝔽₅)	—	4	2	no	—
109	C₄⋊AGL(1,𝔽₅)	—	4	2	no	—
110	C₄×AGL(1,𝔽₅)	—	4	4	no	—
111	C₅×C₄⋊C₄	C₂×C₃×C₄	1	4	yes	—
112	C₅×C₄⋊C₄	C₂×C₃×C₄	1	2	yes	—
113	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
114	C₄⋊Dic₅	C₂×C₃×C₄	1	2	yes	—
115	C₄⋊Dic₅	C₂×C₃×C₄	1	4	yes	—
116	#683	C₂×C₃×C₄	1	4	yes	—
117	#683	C₂×C₃×C₄	1	4	yes	—
118	#683	C₂×C₃×C₄	1	4	yes	—
119	C₄×Dic₅	C₂×C₃×C₄	1	2	yes	—
120	C₄×Dic₅	C₂×C₃×C₄	1	1	yes	characteristic
121	C₂×#125	—	5	1	no	—
122	#389	—	5	4	no	—
123	#387	—	5	4	no	—
124	#387	—	5	4	no	—
125	#382	—	5	2	no	—
126	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
127	C₂×C₃×D₅	H(5,𝔽₂)	1	2	yes	—
128	C₂×C₃×D₅	2₋¹⁺⁴	1	1	yes	characteristic
129	C₂×C₃×D₅	2₋¹⁺⁴	1	1	yes	characteristic
130	C₂×C₃×D₅	2₋¹⁺⁴	1	2	yes	—
131	C₃×AGL(1,𝔽₅)	—	4	4	no	—
132	C₃×AGL(1,𝔽₅)	—	4	4	no	—
133	C₃×C₄×C₅	—	2	1	no	—
134	C₃×C₄×C₅	—	2	4	no	—
135	C₃×C₄×C₅	—	2	2	no	—
136	C₃×Dic₅	—	2	1	no	—
137	C₃×Dic₅	—	2	4	no	—
138	C₃×Dic₅	—	2	2	no	—
139	C₂²×C₃×C₄	—	5	1	no	—
140	C₂²×C₃×C₄	—	5	2	no	—
141	C₂²×C₃×C₄	—	5	4	no	—
142	C₂²×C₃×C₄	—	5	4	no	—
143	C₂²×C₃×C₄	—	5	4	no	—
144	C₃×C₄⋊C₄	—	5	4	no	—
145	C₃×C₄⋊C₄	—	5	4	no	—
146	C₃×C₄⋊C₄	—	5	4	no	—
147	C₃×C₄⋊C₄	—	5	4	no	—
148	C₃×C₄⋊C₄	—	5	2	no	—
149	C₃×C₄⋊C₄	—	20	4	no	—
150	C₃×C₄⋊C₄	—	20	2	no	—
151	C₃×C₄⋊C₄	—	20	2	no	—
152	C₃×C₄⋊C₄	C₂×AGL(1,𝔽₅)	1	4	yes	—
153	C₃×C₄⋊C₄	C₂×AGL(1,𝔽₅)	1	2	yes	—
154	C₃×C₄²	—	5	2	no	—
155	C₃×C₄²	—	5	1	no	—
156	C₃×C₄²	—	20	4	no	—
157	C₃×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
158	C₂²×D₅	C₂⁴×C₃	1	1	yes	characteristic
159	C₂×AGL(1,𝔽₅)	—	2	4	no	—
160	C₂×AGL(1,𝔽₅)	—	2	4	no	—
161	C₂×AGL(1,𝔽₅)	—	2	4	no	—
162	C₂×AGL(1,𝔽₅)	—	2	4	no	—
163	C₂×AGL(1,𝔽₅)	—	2	4	no	—
164	C₂×AGL(1,𝔽₅)	—	2	4	no	—
165	C₂×C₄×C₅	C₂²×C₃×C₄	1	2	yes	—
166	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
167	C₂×C₄×C₅	C₂²×C₃×C₄	1	4	yes	—
168	C₂×Dic₅	C₂²×C₃×C₄	1	1	yes	characteristic
169	C₂×Dic₅	C₂²×C₃×C₄	1	2	yes	—
170	C₂×Dic₅	C₂²×C₃×C₄	1	4	yes	—
171	C₄×D₅	—	2	4	no	—
172	C₄×D₅	—	2	2	no	—
173	C₄×D₅	—	2	1	no	—
174	C₄×D₅	—	2	1	no	—
175	C₄×D₅	—	2	4	no	—
176	C₄×D₅	—	2	2	no	—
177	#125	—	5	4	no	—
178	#125	—	5	2	no	—
179	#125	—	5	1	no	—
180	#125	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
181	C₂×C₄⋊C₄	—	5	2	no	—
182	C₂×C₄⋊C₄	—	5	4	no	—
183	C₂×C₄⋊C₄	—	5	2	no	—
184	C₂×C₄⋊C₄	—	5	2	no	—
185	C₂×C₄⋊C₄	—	5	4	no	—
186	C₂×C₄²	—	5	1	no	—
187	C₂×C₄²	—	5	4	no	—
188	#95	—	5	4	no	—
189	#95	—	5	4	no	—
190	#95	—	5	4	no	—
191	#95	—	5	4	no	—
192	C₂×C₃×C₅	#519	1	1	yes	characteristic
193	C₂×C₃×C₅	#520	1	2	yes	—
194	C₃×D₅	#562	1	1	yes	characteristic
195	C₃×D₅	#563	1	2	yes	—
196	C₃×D₅	#564	1	1	yes	characteristic
197	C₂³×C₃	—	5	1	no	—
198	C₂×C₃×C₄	—	5	1	no	—
199	C₂×C₃×C₄	—	5	2	no	—
200	C₂×C₃×C₄	—	5	4	no	—
201	C₂×C₃×C₄	—	10	4	no	—
202	C₂×C₃×C₄	—	10	2	no	—
203	C₂×C₃×C₄	—	10	1	no	—
204	C₂×C₃×C₄	—	10	1	no	—
205	C₂×C₃×C₄	—	10	4	no	—
206	C₂×C₃×C₄	—	10	2	no	—
207	C₂×C₃×C₄	—	10	4	no	—
208	C₂×C₃×C₄	—	10	4	no	—
209	C₂×C₃×C₄	—	10	4	no	—
210	C₂×C₃×C₄	—	10	4	no	—
211	C₂×C₃×C₄	—	10	4	no	—
212	C₂×C₃×C₄	—	10	4	no	—
213	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
214	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	2	yes	—
215	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	4	yes	—
216	C₂²×C₅	C₂³×C₃×C₄	1	1	yes	characteristic
217	C₂×D₅	C₃×H(5,𝔽₂)	1	2	yes	—
218	C₂×D₅	C₃×2₋¹⁺⁴	1	1	yes	characteristic
219	C₂×D₅	C₃×2₋¹⁺⁴	1	1	yes	characteristic
220	C₂×D₅	C₃×2₋¹⁺⁴	1	2	yes	—
221	AGL(1,𝔽₅)	—	4	4	no	—
222	AGL(1,𝔽₅)	—	4	4	no	—
223	C₄×C₅	—	2	1	no	—
224	C₄×C₅	—	2	4	no	—
225	C₄×C₅	—	2	2	no	—
226	Dic₅	—	2	1	no	—
227	Dic₅	—	2	4	no	—
228	Dic₅	—	2	2	no	—
229	C₂²×C₄	—	5	1	no	—
230	C₂²×C₄	—	5	2	no	—
231	C₂²×C₄	—	5	4	no	—
232	C₂²×C₄	—	5	4	no	—
233	C₂²×C₄	—	5	4	no	—
234	C₄⋊C₄	—	5	4	no	—
235	C₄⋊C₄	—	5	4	no	—
236	C₄⋊C₄	—	5	4	no	—
237	C₄⋊C₄	—	5	4	no	—
238	C₄⋊C₄	—	5	2	no	—
239	C₄⋊C₄	—	20	4	no	—
240	C₄⋊C₄	—	20	2	no	—
241	C₄⋊C₄	—	20	2	no	—
242	C₄⋊C₄	C₂×C₃×AGL(1,𝔽₅)	1	2	yes	—
243	C₄⋊C₄	C₂×C₃×AGL(1,𝔽₅)	1	4	yes	—
244	C₄²	—	5	2	no	—
245	C₄²	—	5	1	no	—
246	C₄²	—	20	4	no	—
247	C₄²	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
248	C₃×C₅	#2382	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	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic
254	C₃×C₄	—	2	1	no	—
255	C₃×C₄	—	2	4	no	—
256	C₃×C₄	—	2	2	no	—
257	C₃×C₄	—	10	1	no	—
258	C₃×C₄	—	10	2	no	—
259	C₃×C₄	—	10	4	no	—
260	C₃×C₄	—	20	4	no	—
261	C₃×C₄	—	20	4	no	—
262	C₂×C₅	C₃×#519	1	1	yes	characteristic
263	C₂×C₅	C₃×#520	1	2	yes	—
264	D₅	C₃×#562	1	1	yes	characteristic
265	D₅	C₃×#563	1	2	yes	—
266	D₅	C₃×#564	1	1	yes	characteristic
267	C₂³	—	5	1	no	—
268	C₂×C₄	—	5	1	no	—
269	C₂×C₄	—	5	2	no	—
270	C₂×C₄	—	5	4	no	—
271	C₂×C₄	—	10	4	no	—
272	C₂×C₄	—	10	2	no	—
273	C₂×C₄	—	10	1	no	—
274	C₂×C₄	—	10	1	no	—
275	C₂×C₄	—	10	4	no	—
276	C₂×C₄	—	10	2	no	—
277	C₂×C₄	—	10	4	no	—
278	C₂×C₄	—	10	4	no	—
279	C₂×C₄	—	10	4	no	—
280	C₂×C₄	—	10	4	no	—
281	C₂×C₄	—	10	4	no	—
282	C₂×C₄	—	10	4	no	—
283	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
284	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)	1	2	yes	—
285	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)	1	4	yes	—
286	C₂×C₃	—	5	1	no	—
287	C₂×C₃	—	5	2	no	—
288	C₂×C₃	—	5	1	no	—
289	C₂×C₃	#66262	1	1	yes	characteristic
290	C₂×C₃	#66266	1	2	yes	—
291	C₅	C₃×#2382	1	1	yes	characteristic, complemented, Sylow
292	C₂²	—	5	1	no	—
293	C₂²	—	5	1	no	—
294	C₂²	—	5	2	no	—
295	C₂²	—	5	2	no	—
296	C₂²	C₂³×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
297	C₄	—	2	1	no	—
298	C₄	—	2	4	no	—
299	C₄	—	2	2	no	—
300	C₄	—	10	1	no	—
301	C₄	—	10	2	no	—
302	C₄	—	10	4	no	—
303	C₄	—	20	4	no	—
304	C₄	—	20	4	no	—
305	C₃	#10617348	1	1	yes	characteristic, direct product factor, Sylow
306	C₂	—	5	1	no	—
307	C₂	—	5	2	no	—
308	C₂	—	5	1	no	—
309	C₂	C₃×#66262	1	1	yes	characteristic
310	C₂	C₃×#66266	1	2	yes	—

	subgroup	G/H
center	C₂²×C₃	C₂³×AGL(1,𝔽₅)
commutator	C₂²×C₅	C₂³×C₃×C₄
Frattini	C₂²	C₂³×C₃×AGL(1,𝔽₅)
Fitting	C₃×C₅×#125	C₄
socle	C₂²×C₃×C₅	C₂³×C₄
prime radical	C₂²×C₃×D₅	C₂⁴
cosocle	C₂²×D₅	C₂⁴×C₃
hypercenter	C₃×#125	AGL(1,𝔽₅)
hypocenter	C₅	C₃×#2382
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×#125

Complements

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

N	H ≅ G/N	type
6. #10617348	C₃	direct (N × H)
11. C₃×C₅×#125	C₄	semidirect
14. C₃×#4036	C₄	semidirect
15. C₃×#4035	C₄	semidirect
17. C₃×#4016	C₄	semidirect
56. C₅×#125	C₃×C₄	semidirect
59. #4036	C₃×C₄	semidirect
60. #4035	C₃×C₄	semidirect
62. #4016	C₃×C₄	semidirect
90. C₃×#125	AGL(1,𝔽₅)	semidirect
180. #125	C₃×AGL(1,𝔽₅)	semidirect
248. C₃×C₅	#2382	semidirect
291. C₅	C₃×#2382	semidirect
305. C₃	#10617348	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×D₅×#125	2
C₃×#65712	2
C₃×#65711	2
C₃×#65689	2
#10617348	3
C₃×#2382	5

Subgroups — C3×#10617348 #14331291

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10617348 #14331291