Subgroups — C₃×#10604721 #14294563

Order 1920.

848 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#4130	C₂	1	2	yes	maximal
2	C₂²×C₃×C₈×D₅	C₂	1	1	yes	maximal, characteristic
3	C₃×#64893	C₂	1	4	yes	maximal, complemented
4	#10604721	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂²×C₃×C₄×D₅	C₂²	1	1	yes	characteristic
6	C₂×C₃×C₂²⋊AGL(1,𝔽₅)	—	2	2	no	—
7	C₃×#4130	—	2	4	no	—
8	C₃×#4130	C₄	1	8	yes	complemented
9	C₂×C₃×C₄⋊AGL(1,𝔽₅)	C₂²	1	4	yes	—
10	C₂×C₃×C₄×AGL(1,𝔽₅)	—	2	2	no	—
11	C₂²×C₃×C₅×C₈	C₄	1	1	yes	characteristic, complemented
12	C₂²×C₃×#168	C₄	1	1	yes	characteristic, complemented
13	C₂×C₃×C₈×D₅	—	2	4	no	—
14	C₂×C₃×C₈×D₅	C₄	1	2	yes	complemented
15	C₂×C₃×C₈×D₅	C₂²	1	2	yes	—
16	C₃×#1735	—	5	1	no	maximal, Hall
17	C₂×#4130	C₂×C₃	1	2	yes	—
18	C₂²×C₈×D₅	C₂×C₃	1	1	yes	characteristic
19	#64893	C₂×C₃	1	4	yes	complemented
20	C₂³×C₃×D₅	D₄	1	1	yes	characteristic
21	C₂²×C₃×AGL(1,𝔽₅)	—	2	4	no	—
22	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₃×Dic₅	Q₈	1	1	yes	characteristic
24	C₂×C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
25	C₂×C₃×C₄×D₅	C₂×C₄	1	4	yes	—
26	C₂×C₃×C₄×D₅	D₄	1	1	yes	characteristic
27	C₂×C₃×C₄×D₅	D₄	1	4	yes	—
28	C₂×C₃×C₄×D₅	Q₈	1	1	yes	characteristic
29	C₂×C₃×C₄×D₅	C₂³	1	1	yes	characteristic
30	C₃×C₂²⋊AGL(1,𝔽₅)	—	2	8	no	—
31	C₃×C₄⋊AGL(1,𝔽₅)	—	2	4	no	—
32	C₃×C₄⋊AGL(1,𝔽₅)	C₂×C₄	1	8	yes	—
33	C₃×C₄×AGL(1,𝔽₅)	—	2	8	no	—
34	C₂×C₃×C₅×C₈	—	2	2	no	—
35	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
36	C₂×C₃×#168	—	2	2	no	—
37	C₂×C₃×#168	C₂×C₄	1	2	yes	—
38	C₃×C₈×D₅	—	2	4	no	—
39	C₃×C₈×D₅	—	2	4	no	—
40	C₂³×C₃×C₈	—	5	1	no	—
41	C₂×C₃×#117	—	5	2	no	—
42	C₃×#340	—	5	4	no	—
43	C₂²×C₄×D₅	C₂²×C₃	1	1	yes	characteristic
44	C₂×C₂²⋊AGL(1,𝔽₅)	—	2	2	no	—
45	#4130	—	2	4	no	—
46	#4130	C₃×C₄	1	8	yes	complemented
47	C₂×C₄⋊AGL(1,𝔽₅)	C₂²×C₃	1	4	yes	—
48	C₂×C₄×AGL(1,𝔽₅)	—	2	2	no	—
49	C₂²×C₅×C₈	C₃×C₄	1	1	yes	characteristic, complemented
50	C₂²×#168	C₃×C₄	1	1	yes	characteristic, complemented
51	C₂×C₈×D₅	—	2	4	no	—
52	C₂×C₈×D₅	C₃×C₄	1	2	yes	complemented
53	C₂×C₈×D₅	C₂²×C₃	1	2	yes	—
54	#1735	—	5	1	no	Sylow
55	C₂³×C₃×C₅	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₃×D₅	—	2	4	no	—
57	C₂²×C₃×D₅	C₂²⋊C₄	1	4	yes	—
58	C₂²×C₃×D₅	C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₃×D₅	C₂×D₄	1	1	yes	characteristic
60	C₂×C₃×AGL(1,𝔽₅)	—	2	8	no	—
61	C₂×C₃×AGL(1,𝔽₅)	—	4	8	no	—
62	C₂×C₃×C₄×C₅	C₄²	1	2	yes	—
63	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
64	C₂×C₃×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₃×Dic₅	C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₃×Dic₅	C₄⋊C₄	1	4	yes	—
68	C₂×C₃×Dic₅	C₂×Q₈	1	1	yes	characteristic
69	C₃×C₄×D₅	C₄²	1	2	yes	—
70	C₃×C₄×D₅	C₂²⋊C₄	1	4	yes	—
71	C₃×C₄×D₅	C₂²⋊C₄	1	2	yes	—
72	C₃×C₄×D₅	C₄⋊C₄	1	4	yes	—
73	C₃×C₄×D₅	C₂²×C₄	1	2	yes	—
74	C₃×C₄×D₅	C₂×D₄	1	2	yes	—
75	C₃×C₅×C₈	—	2	2	no	—
76	C₃×#168	—	2	2	no	—
77	C₂³×C₃×C₄	—	5	1	no	—
78	C₂²×C₃×C₈	—	5	1	no	—
79	C₂²×C₃×C₈	—	5	2	no	—
80	C₂²×C₃×C₈	—	5	2	no	—
81	C₂²×C₃×C₈	—	10	4	no	—
82	C₂²×C₃×C₈	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
83	C₃×#117	—	5	8	no	—
84	C₃×#117	—	10	4	no	—
85	C₂×C₃×C₄⋊C₄	—	5	4	no	—
86	C₂×C₃×C₂²⋊C₄	—	10	2	no	—
87	C₂×C₃×C₄²	—	10	2	no	—
88	C₂³×D₅	C₃×D₄	1	1	yes	characteristic
89	C₂²×AGL(1,𝔽₅)	—	2	4	no	—
90	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
91	C₂²×Dic₅	C₃×Q₈	1	1	yes	characteristic
92	C₂×C₄×D₅	C₂×C₃×C₄	1	1	yes	characteristic
93	C₂×C₄×D₅	C₂×C₃×C₄	1	4	yes	—
94	C₂×C₄×D₅	C₃×D₄	1	1	yes	characteristic
95	C₂×C₄×D₅	C₃×D₄	1	4	yes	—
96	C₂×C₄×D₅	C₃×Q₈	1	1	yes	characteristic
97	C₂×C₄×D₅	C₂³×C₃	1	1	yes	characteristic
98	C₂²⋊AGL(1,𝔽₅)	—	2	8	no	—
99	C₄⋊AGL(1,𝔽₅)	—	2	4	no	—
100	C₄⋊AGL(1,𝔽₅)	C₂×C₃×C₄	1	8	yes	—
101	C₄×AGL(1,𝔽₅)	—	2	8	no	—
102	C₂×C₅×C₈	—	2	2	no	—
103	C₂×C₅×C₈	C₂×C₃×C₄	1	2	yes	—
104	C₂×#168	—	2	2	no	—
105	C₂×#168	C₂×C₃×C₄	1	2	yes	—
106	C₈×D₅	—	2	4	no	—
107	C₈×D₅	—	2	4	no	—
108	C₂³×C₈	—	5	1	no	—
109	C₂×#117	—	5	2	no	—
110	#340	—	5	4	no	—
111	C₂²×C₃×C₅	—	2	2	no	—
112	C₂²×C₃×C₅	#95	1	2	yes	—
113	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₃×D₅	—	2	2	no	—
115	C₂×C₃×D₅	—	2	8	no	—
116	C₂×C₃×D₅	#95	1	2	yes	—
117	C₂×C₃×D₅	C₂×C₂²⋊C₄	1	2	yes	—
118	C₂×C₃×D₅	#135	1	4	yes	—
119	C₃×AGL(1,𝔽₅)	—	4	8	no	—
120	C₃×C₄×C₅	#95	1	2	yes	—
121	C₃×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
122	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₃×Dic₅	#95	1	2	yes	—
124	C₃×Dic₅	C₂×C₄⋊C₄	1	2	yes	—
125	C₂⁴×C₃	—	5	1	no	—
126	C₂²×C₃×C₄	—	5	1	no	—
127	C₂²×C₃×C₄	—	5	4	no	—
128	C₂²×C₃×C₄	—	5	1	no	—
129	C₂²×C₃×C₄	—	5	1	no	—
130	C₂²×C₃×C₄	—	5	4	no	—
131	C₂²×C₃×C₄	—	5	1	no	—
132	C₂²×C₃×C₄	—	5	1	no	—
133	C₂²×C₃×C₄	—	10	4	no	—
134	C₂²×C₃×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
135	C₂×C₃×C₈	—	2	2	no	—
136	C₂×C₃×C₈	—	5	2	no	—
137	C₂×C₃×C₈	—	10	4	no	—
138	C₂×C₃×C₈	—	10	4	no	—
139	C₂×C₃×C₈	—	10	2	no	—
140	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
141	C₃×C₄⋊C₄	—	5	8	no	—
142	C₃×C₄⋊C₄	—	10	4	no	—
143	C₃×C₂²⋊C₄	—	10	8	no	—
144	C₃×C₄²	—	10	8	no	—
145	C₂³×C₅	C₃×C₄⋊C₄	1	1	yes	characteristic
146	C₂²×D₅	—	2	4	no	—
147	C₂²×D₅	C₃×C₂²⋊C₄	1	4	yes	—
148	C₂²×D₅	C₃×C₄⋊C₄	1	1	yes	characteristic
149	C₂²×D₅	C₂×C₃×D₄	1	1	yes	characteristic
150	C₂×AGL(1,𝔽₅)	—	2	8	no	—
151	C₂×AGL(1,𝔽₅)	—	4	8	no	—
152	C₂×C₄×C₅	C₃×C₄²	1	2	yes	—
153	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	2	yes	—
154	C₂×C₄×C₅	C₃×C₄⋊C₄	1	1	yes	characteristic
155	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
156	C₂×Dic₅	C₃×C₄⋊C₄	1	1	yes	characteristic
157	C₂×Dic₅	C₃×C₄⋊C₄	1	4	yes	—
158	C₂×Dic₅	C₂×C₃×Q₈	1	1	yes	characteristic
159	C₄×D₅	C₃×C₄²	1	2	yes	—
160	C₄×D₅	C₃×C₂²⋊C₄	1	4	yes	—
161	C₄×D₅	C₃×C₂²⋊C₄	1	2	yes	—
162	C₄×D₅	C₃×C₄⋊C₄	1	4	yes	—
163	C₄×D₅	C₂²×C₃×C₄	1	2	yes	—
164	C₄×D₅	C₂×C₃×D₄	1	2	yes	—
165	C₅×C₈	—	2	2	no	—
166	#168	—	2	2	no	—
167	C₂³×C₄	—	5	1	no	—
168	C₂²×C₈	—	5	1	no	—
169	C₂²×C₈	—	5	2	no	—
170	C₂²×C₈	—	5	2	no	—
171	C₂²×C₈	—	10	4	no	—
172	C₂²×C₈	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
173	#117	—	5	8	no	—
174	#117	—	10	4	no	—
175	C₂×C₄⋊C₄	—	5	4	no	—
176	C₂×C₂²⋊C₄	—	10	2	no	—
177	C₂×C₄²	—	10	2	no	—
178	C₂×C₃×C₅	—	2	2	no	—
179	C₂×C₃×C₅	C₂×#95	1	1	yes	characteristic
180	C₂×C₃×C₅	#427	1	2	yes	—
181	C₃×D₅	—	2	2	no	—
182	C₃×D₅	#416	1	4	yes	—
183	C₂³×C₃	—	5	1	no	—
184	C₂³×C₃	—	5	1	no	—
185	C₂³×C₃	—	5	4	no	—
186	C₂³×C₃	—	10	4	no	—
187	C₂³×C₃	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
188	C₂×C₃×C₄	—	5	1	no	—
189	C₂×C₃×C₄	—	5	1	no	—
190	C₂×C₃×C₄	—	5	4	no	—
191	C₂×C₃×C₄	—	5	2	no	—
192	C₂×C₃×C₄	—	5	2	no	—
193	C₂×C₃×C₄	—	5	2	no	—
194	C₂×C₃×C₄	—	5	2	no	—
195	C₂×C₃×C₄	—	5	4	no	—
196	C₂×C₃×C₄	—	5	4	no	—
197	C₂×C₃×C₄	—	10	8	no	—
198	C₂×C₃×C₄	—	20	8	no	—
199	C₂×C₃×C₄	C₄×AGL(1,𝔽₅)	1	2	yes	—
200	C₂×C₃×C₄	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
201	C₂×C₃×C₄	C₂²⋊AGL(1,𝔽₅)	1	2	yes	—
202	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
203	C₃×C₈	—	2	2	no	—
204	C₃×C₈	—	10	2	no	—
205	C₂²×C₅	—	2	2	no	—
206	C₂²×C₅	C₃×#95	1	2	yes	—
207	C₂²×C₅	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
208	C₂×D₅	—	2	2	no	—
209	C₂×D₅	—	2	8	no	—
210	C₂×D₅	C₃×#95	1	2	yes	—
211	C₂×D₅	C₂×C₃×C₂²⋊C₄	1	2	yes	—
212	C₂×D₅	C₃×#135	1	4	yes	—
213	AGL(1,𝔽₅)	—	4	8	no	—
214	C₄×C₅	C₃×#95	1	2	yes	—
215	C₄×C₅	C₂×C₃×C₄²	1	1	yes	characteristic
216	C₄×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
217	Dic₅	C₃×#95	1	2	yes	—
218	Dic₅	C₂×C₃×C₄⋊C₄	1	2	yes	—
219	C₂⁴	—	5	1	no	—
220	C₂²×C₄	—	5	1	no	—
221	C₂²×C₄	—	5	4	no	—
222	C₂²×C₄	—	5	1	no	—
223	C₂²×C₄	—	5	1	no	—
224	C₂²×C₄	—	5	4	no	—
225	C₂²×C₄	—	5	1	no	—
226	C₂²×C₄	—	5	1	no	—
227	C₂²×C₄	—	10	4	no	—
228	C₂²×C₄	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	5	2	no	—
231	C₂×C₈	—	10	4	no	—
232	C₂×C₈	—	10	4	no	—
233	C₂×C₈	—	10	2	no	—
234	C₂×C₈	C₂×C₃×AGL(1,𝔽₅)	1	2	yes	—
235	C₄⋊C₄	—	5	8	no	—
236	C₄⋊C₄	—	10	4	no	—
237	C₂²⋊C₄	—	10	8	no	—
238	C₄²	—	10	8	no	—
239	C₃×C₅	#1735	1	1	yes	characteristic, complemented, Hall
240	C₂²×C₃	—	2	2	no	—
241	C₂²×C₃	—	5	4	no	—
242	C₂²×C₃	—	5	2	no	—
243	C₂²×C₃	—	5	2	no	—
244	C₂²×C₃	—	10	8	no	—
245	C₂²×C₃	—	10	2	no	—
246	C₂²×C₃	(C₂×AGL(1,𝔽₅))⋊C₄	1	2	yes	—
247	C₂²×C₃	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
248	C₃×C₄	—	5	2	no	—
249	C₃×C₄	—	5	2	no	—
250	C₃×C₄	—	20	8	no	—
251	C₃×C₄	(C₂×AGL(1,𝔽₅))⋊C₄	1	2	yes	—
252	C₃×C₄	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
253	C₃×C₄	C₂×C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
254	C₂×C₅	—	2	2	no	—
255	C₂×C₅	C₂×C₃×#95	1	1	yes	characteristic
256	C₂×C₅	C₃×#427	1	2	yes	—
257	D₅	—	2	2	no	—
258	D₅	C₃×#416	1	4	yes	—
259	C₂³	—	5	1	no	—
260	C₂³	—	5	1	no	—
261	C₂³	—	5	4	no	—
262	C₂³	—	10	4	no	—
263	C₂³	C₃×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
264	C₂×C₄	—	5	1	no	—
265	C₂×C₄	—	5	1	no	—
266	C₂×C₄	—	5	4	no	—
267	C₂×C₄	—	5	2	no	—
268	C₂×C₄	—	5	2	no	—
269	C₂×C₄	—	5	4	no	—
270	C₂×C₄	—	5	4	no	—
271	C₂×C₄	—	5	2	no	—
272	C₂×C₄	—	5	2	no	—
273	C₂×C₄	—	10	8	no	—
274	C₂×C₄	—	20	8	no	—
275	C₂×C₄	C₃×C₄×AGL(1,𝔽₅)	1	2	yes	—
276	C₂×C₄	C₃×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
277	C₂×C₄	C₃×C₂²⋊AGL(1,𝔽₅)	1	2	yes	—
278	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
279	C₈	—	2	2	no	—
280	C₈	—	10	2	no	—
281	C₂×C₃	—	2	2	no	—
282	C₂×C₃	—	5	4	no	—
283	C₂×C₃	—	10	2	no	—
284	C₂×C₃	#65721	1	2	yes	—
285	C₂×C₃	C₂×(C₂×AGL(1,𝔽₅))⋊C₄	1	1	yes	characteristic
286	C₅	C₃×#1735	1	1	yes	characteristic, complemented, Sylow
287	C₂²	—	2	2	no	—
288	C₂²	—	5	4	no	—
289	C₂²	—	5	2	no	—
290	C₂²	—	5	2	no	—
291	C₂²	—	10	8	no	—
292	C₂²	—	10	2	no	—
293	C₂²	C₃×(C₂×AGL(1,𝔽₅))⋊C₄	1	2	yes	—
294	C₂²	C₂×C₃×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
295	C₄	—	5	2	no	—
296	C₄	—	5	2	no	—
297	C₄	—	20	8	no	—
298	C₄	C₃×(C₂×AGL(1,𝔽₅))⋊C₄	1	2	yes	—
299	C₄	C₂×C₃×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
300	C₄	C₂×C₃×C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
301	C₃	#10604721	1	1	yes	characteristic, direct product factor, Sylow
302	C₂	—	2	2	no	—
303	C₂	—	5	4	no	—
304	C₂	—	10	2	no	—
305	C₂	C₃×#65721	1	2	yes	—
306	C₂	C₂×C₃×(C₂×AGL(1,𝔽₅))⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₄⋊AGL(1,𝔽₅)
commutator	C₄×C₅	C₂×C₃×C₄²
Frattini	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)
Fitting	C₂²×C₃×C₅×C₈	C₄
socle	C₂²×C₃×C₅	C₂×C₄⋊C₄
prime radical	C₂³×C₃×D₅	D₄
cosocle	C₂×C₄×D₅	C₂³×C₃
hypercenter	C₂²×C₃×C₈	AGL(1,𝔽₅)
hypocenter	C₅	C₃×#1735
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂²×C₃×C₄ ◁ C₂²×C₃×C₈

Complements

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

N	H ≅ G/N	type
3. C₃×#64893	C₂	semidirect
4. #10604721	C₃	direct (N × H)
8. C₃×#4130	C₄	semidirect
11. C₂²×C₃×C₅×C₈	C₄	semidirect
12. C₂²×C₃×#168	C₄	semidirect
14. C₂×C₃×C₈×D₅	C₄	semidirect
19. #64893	C₂×C₃	semidirect
46. #4130	C₃×C₄	semidirect
49. C₂²×C₅×C₈	C₃×C₄	semidirect
50. C₂²×#168	C₃×C₄	semidirect
52. C₂×C₈×D₅	C₃×C₄	semidirect
82. C₂²×C₃×C₈	AGL(1,𝔽₅)	semidirect
172. C₂²×C₈	C₃×AGL(1,𝔽₅)	semidirect
239. C₃×C₅	#1735	semidirect
286. C₅	C₃×#1735	semidirect
301. C₃	#10604721	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×#4130	2
C₂²×C₃×C₈×D₅	2
C₃×#64893	2
#10604721	3
C₃×#1735	5

Subgroups — C3×#10604721 #14294563

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10604721 #14294563