Subgroups — C₃×#10600648 #14282286

Order 1920.

632 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×D₂₀	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×C₅×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#65215	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#64679	C₂	1	4	yes	maximal, complemented
5	#10600648	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂²×C₃×D₂₀	C₄	1	1	yes	characteristic
7	C₂²×C₃×C₄×D₅	—	2	1	no	—
8	C₂×C₃×C₅×M₁₆	—	2	1	no	—
9	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
10	C₂×C₃×#685	—	2	1	no	—
11	C₂×C₃×C₄⋊Dic₅	C₄	1	1	yes	characteristic
12	C₃×C₄×D₂₀	—	2	1	no	—
13	C₃×C₄×D₂₀	—	2	1	no	—
14	C₃×C₄×D₂₀	C₄	1	2	yes	—
15	C₃×C₄×D₂₀	C₂²	1	2	yes	—
16	C₃×C₅×#97	—	2	1	no	—
17	C₃×C₄×C₅×C₈	C₂²	1	2	yes	complemented
18	C₃×C₂²⋊#168	—	2	1	no	—
19	C₃×C₄⋊#168	C₂²	1	2	yes	complemented
20	C₃×#3935	—	2	1	no	—
21	C₃×#1490	—	5	1	no	maximal, Hall
22	C₂×C₄×D₂₀	C₂×C₃	1	1	yes	characteristic
23	C₄×C₅×M₁₆	C₂×C₃	1	1	yes	characteristic, complemented
24	#65215	C₂×C₃	1	1	yes	characteristic, complemented
25	#64679	C₂×C₃	1	4	yes	complemented
26	C₂³×C₃×D₅	—	2	1	no	—
27	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
29	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
30	C₂²×C₃×Dic₅	—	2	1	no	—
31	C₂×C₃×D₂₀	—	2	1	no	—
32	C₂×C₃×D₂₀	—	2	4	no	—
33	C₂×C₃×D₂₀	C₂×C₄	1	2	yes	—
34	C₂×C₃×C₄×D₅	—	2	2	no	—
35	C₂×C₃×C₄×D₅	—	2	2	no	—
36	C₂×C₃×C₄×D₅	—	4	4	no	—
37	C₃×C₅×M₁₆	—	2	4	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	C₂×C₃×C₄×D₄	—	5	1	no	—
49	C₃×#432	—	5	1	no	—
50	C₃×C₄×M₁₆	D₅	1	1	yes	characteristic, complemented
51	C₃×D₄⋊C₈	—	5	4	no	—
52	C₂²×D₂₀	C₃×C₄	1	1	yes	characteristic
53	C₂²×C₄×D₅	—	2	1	no	—
54	C₂×C₅×M₁₆	—	2	1	no	—
55	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
56	C₂×#685	—	2	1	no	—
57	C₂×C₄⋊Dic₅	C₃×C₄	1	1	yes	characteristic
58	C₄×D₂₀	—	2	1	no	—
59	C₄×D₂₀	—	2	1	no	—
60	C₄×D₂₀	C₃×C₄	1	2	yes	—
61	C₄×D₂₀	C₂²×C₃	1	2	yes	—
62	C₅×#97	—	2	1	no	—
63	C₄×C₅×C₈	C₂²×C₃	1	2	yes	complemented
64	C₂²⋊#168	—	2	1	no	—
65	C₄⋊#168	C₂²×C₃	1	2	yes	complemented
66	#3935	—	2	1	no	—
67	#1490	—	5	1	no	Sylow
68	C₂³×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₃×D₅	—	2	4	no	—
70	C₂²×C₃×D₅	—	2	2	no	—
71	C₂²×C₃×D₅	—	4	2	no	—
72	C₂²×C₃×D₅	—	4	2	no	—
73	C₂×C₃×C₄×C₅	—	2	2	no	—
74	C₂×C₃×C₄×C₅	—	2	2	no	—
75	C₂×C₃×C₄×C₅	—	2	2	no	—
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	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
81	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
82	C₂×C₃×Dic₅	—	2	2	no	—
83	C₂×C₃×Dic₅	—	4	2	no	—
84	C₃×D₂₀	—	2	4	no	—
85	C₃×D₂₀	—	2	2	no	—
86	C₃×D₂₀	M₁₆	1	4	yes	complemented
87	C₃×C₄×D₅	—	4	4	no	—
88	C₃×C₄×D₅	—	4	4	no	—
89	C₃×C₅×C₈	—	2	4	no	—
90	C₃×#168	—	4	2	no	—
91	C₂²×C₃×D₄	—	5	1	no	—
92	C₂³×C₃×C₄	—	10	1	no	—
93	C₂×C₃×M₁₆	—	2	1	no	—
94	C₃×C₄×D₄	—	5	2	no	—
95	C₃×C₄×D₄	—	5	2	no	—
96	C₃×C₄×D₄	—	10	1	no	—
97	C₃×C₄×D₄	—	10	1	no	—
98	C₂×C₃×C₄⋊C₄	—	5	1	no	—
99	C₂×C₃×C₂²⋊C₄	—	10	1	no	—
100	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
101	C₃×C₄⋊C₈	—	5	2	no	—
102	C₃×C₂²⋊C₈	—	10	1	no	—
103	C₃×#97	—	2	1	no	—
104	C₃×#97	—	10	1	no	—
105	C₃×C₄×C₈	C₂×D₅	1	2	yes	complemented
106	C₂³×D₅	—	2	1	no	—
107	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
108	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
109	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
110	C₂²×Dic₅	—	2	1	no	—
111	C₂×D₂₀	—	2	1	no	—
112	C₂×D₂₀	—	2	4	no	—
113	C₂×D₂₀	C₂×C₃×C₄	1	2	yes	—
114	C₂×C₄×D₅	—	2	2	no	—
115	C₂×C₄×D₅	—	2	2	no	—
116	C₂×C₄×D₅	—	4	4	no	—
117	C₅×M₁₆	—	2	4	no	—
118	C₂×C₅×C₈	—	2	2	no	—
119	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
120	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
121	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
122	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
123	#685	—	2	2	no	—
124	#685	—	2	2	no	—
125	C₄⋊Dic₅	—	2	1	no	—
126	C₄⋊Dic₅	C₂×C₃×C₄	1	2	yes	—
127	C₂×#168	—	2	2	no	—
128	C₂×C₄×D₄	—	5	1	no	—
129	#432	—	5	1	no	—
130	C₄×M₁₆	C₃×D₅	1	1	yes	characteristic, complemented
131	D₄⋊C₈	—	5	4	no	—
132	C₂²×C₃×C₅	—	2	1	no	—
133	C₂²×C₃×C₅	—	2	1	no	—
134	C₂²×C₃×C₅	#104	1	2	yes	—
135	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
136	C₂×C₃×D₅	—	2	4	no	—
137	C₂×C₃×D₅	—	4	8	no	—
138	C₂×C₃×D₅	—	4	2	no	—
139	C₂×C₃×D₅	—	4	2	no	—
140	C₃×C₄×C₅	—	2	1	no	—
141	C₃×C₄×C₅	—	2	1	no	—
142	C₃×C₄×C₅	—	2	1	no	—
143	C₃×C₄×C₅	—	2	1	no	—
144	C₃×C₄×C₅	C₂×M₁₆	1	2	yes	—
145	C₃×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
146	C₃×Dic₅	—	4	2	no	—
147	C₂⁴×C₃	—	10	1	no	—
148	C₂×C₃×D₄	—	5	2	no	—
149	C₂×C₃×D₄	—	10	4	no	—
150	C₂×C₃×D₄	—	10	1	no	—
151	C₂²×C₃×C₄	—	10	1	no	—
152	C₂²×C₃×C₄	—	10	2	no	—
153	C₂²×C₃×C₄	—	10	2	no	—
154	C₂²×C₃×C₄	—	20	4	no	—
155	C₂²×C₃×C₄	C₄×D₅	1	1	yes	characteristic
156	C₂²×C₃×C₄	D₂₀	1	1	yes	characteristic
157	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
158	C₃×M₁₆	—	2	4	no	—
159	C₂×C₃×C₈	—	2	2	no	—
160	C₂×C₃×C₈	—	10	2	no	—
161	C₃×C₄⋊C₄	—	5	2	no	—
162	C₃×C₄⋊C₄	—	10	1	no	—
163	C₃×C₂²⋊C₄	—	10	2	no	—
164	C₃×C₂²⋊C₄	—	10	2	no	—
165	C₃×C₄²	C₄×D₅	1	1	yes	characteristic
166	C₃×C₄²	D₂₀	1	1	yes	characteristic
167	C₃×C₄²	C₂²⋊D₅	1	1	yes	characteristic
168	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
169	C₂³×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
170	C₂²×D₅	—	2	4	no	—
171	C₂²×D₅	—	2	2	no	—
172	C₂²×D₅	—	4	2	no	—
173	C₂²×D₅	—	4	2	no	—
174	C₂×C₄×C₅	—	2	2	no	—
175	C₂×C₄×C₅	—	2	2	no	—
176	C₂×C₄×C₅	—	2	2	no	—
177	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
178	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
179	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
180	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
181	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
182	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
183	C₂×Dic₅	—	2	2	no	—
184	C₂×Dic₅	—	4	2	no	—
185	D₂₀	—	2	4	no	—
186	D₂₀	—	2	2	no	—
187	D₂₀	C₃×M₁₆	1	4	yes	complemented
188	C₄×D₅	—	4	4	no	—
189	C₄×D₅	—	4	4	no	—
190	C₅×C₈	—	2	4	no	—
191	#168	—	4	2	no	—
192	C₂²×D₄	—	5	1	no	—
193	C₂³×C₄	—	10	1	no	—
194	C₂×M₁₆	—	2	1	no	—
195	C₄×D₄	—	5	2	no	—
196	C₄×D₄	—	5	2	no	—
197	C₄×D₄	—	10	1	no	—
198	C₄×D₄	—	10	1	no	—
199	C₂×C₄⋊C₄	—	5	1	no	—
200	C₂×C₂²⋊C₄	—	10	1	no	—
201	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
202	C₄⋊C₈	—	5	2	no	—
203	C₂²⋊C₈	—	10	1	no	—
204	#97	—	2	1	no	—
205	#97	—	10	1	no	—
206	C₄×C₈	C₂×C₃×D₅	1	2	yes	complemented
207	C₂×C₃×C₅	—	2	2	no	—
208	C₂×C₃×C₅	#407	1	1	yes	characteristic
209	C₂×C₃×C₅	#418	1	1	yes	characteristic
210	C₂×C₃×C₅	C₂×#104	1	1	yes	characteristic
211	C₃×D₅	—	4	4	no	—
212	C₂³×C₃	—	10	4	no	—
213	C₂³×C₃	—	10	2	no	—
214	C₂³×C₃	—	20	2	no	—
215	C₂³×C₃	—	20	2	no	—
216	C₂³×C₃	#685	1	1	yes	characteristic
217	C₃×D₄	—	5	4	no	—
218	C₃×D₄	—	10	4	no	—
219	C₃×D₄	—	10	2	no	—
220	C₂×C₃×C₄	—	2	2	no	—
221	C₂×C₃×C₄	—	2	2	no	—
222	C₂×C₃×C₄	—	2	2	no	—
223	C₂×C₃×C₄	—	10	2	no	—
224	C₂×C₃×C₄	—	20	4	no	—
225	C₂×C₃×C₄	—	20	2	no	—
226	C₂×C₃×C₄	—	20	4	no	—
227	C₂×C₃×C₄	#685	1	1	yes	characteristic
228	C₂×C₃×C₄	#685	1	1	yes	characteristic
229	C₂×C₃×C₄	#685	1	1	yes	characteristic
230	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
231	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
232	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
233	C₃×C₈	—	2	4	no	—
234	C₃×C₈	—	20	2	no	—
235	C₂²×C₅	—	2	1	no	—
236	C₂²×C₅	—	2	1	no	—
237	C₂²×C₅	C₃×#104	1	2	yes	—
238	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
239	C₂×D₅	—	2	4	no	—
240	C₂×D₅	—	4	8	no	—
241	C₂×D₅	—	4	2	no	—
242	C₂×D₅	—	4	2	no	—
243	C₄×C₅	—	2	1	no	—
244	C₄×C₅	—	2	1	no	—
245	C₄×C₅	—	2	1	no	—
246	C₄×C₅	—	2	1	no	—
247	C₄×C₅	C₂×C₃×M₁₆	1	2	yes	—
248	C₄×C₅	C₃×AGL(1,ℤ/8)	1	2	yes	—
249	Dic₅	—	4	2	no	—
250	C₂⁴	—	10	1	no	—
251	C₂×D₄	—	5	2	no	—
252	C₂×D₄	—	10	4	no	—
253	C₂×D₄	—	10	1	no	—
254	C₂²×C₄	—	10	1	no	—
255	C₂²×C₄	—	10	2	no	—
256	C₂²×C₄	—	10	2	no	—
257	C₂²×C₄	—	20	4	no	—
258	C₂²×C₄	C₃×C₄×D₅	1	1	yes	characteristic
259	C₂²×C₄	C₃×D₂₀	1	1	yes	characteristic
260	C₂²×C₄	C₃×C₂²⋊D₅	1	1	yes	characteristic
261	M₁₆	—	2	4	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	10	2	no	—
264	C₄⋊C₄	—	5	2	no	—
265	C₄⋊C₄	—	10	1	no	—
266	C₂²⋊C₄	—	10	2	no	—
267	C₂²⋊C₄	—	10	2	no	—
268	C₄²	C₃×C₄×D₅	1	1	yes	characteristic
269	C₄²	C₃×D₂₀	1	1	yes	characteristic
270	C₄²	C₃×C₂²⋊D₅	1	1	yes	characteristic
271	C₄²	C₂²×C₃×D₅	1	1	yes	characteristic
272	C₃×C₅	#1490	1	1	yes	characteristic, complemented, Hall
273	C₂²×C₃	—	2	1	no	—
274	C₂²×C₃	—	2	1	no	—
275	C₂²×C₃	—	10	4	no	—
276	C₂²×C₃	—	20	8	no	—
277	C₂²×C₃	—	20	2	no	—
278	C₂²×C₃	—	20	2	no	—
279	C₂²×C₃	#3937	1	2	yes	—
280	C₂²×C₃	C₂×#685	1	1	yes	characteristic
281	C₃×C₄	—	2	1	no	—
282	C₃×C₄	—	2	1	no	—
283	C₃×C₄	—	2	1	no	—
284	C₃×C₄	—	2	1	no	—
285	C₃×C₄	—	20	2	no	—
286	C₃×C₄	D₅×M₁₆	1	2	yes	—
287	C₃×C₄	#4054	1	2	yes	—
288	C₂×C₅	—	2	2	no	—
289	C₂×C₅	C₃×#407	1	1	yes	characteristic
290	C₂×C₅	C₃×#418	1	1	yes	characteristic
291	C₂×C₅	C₂×C₃×#104	1	1	yes	characteristic
292	D₅	—	4	4	no	—
293	C₂³	—	10	4	no	—
294	C₂³	—	10	2	no	—
295	C₂³	—	20	2	no	—
296	C₂³	—	20	2	no	—
297	C₂³	C₃×#685	1	1	yes	characteristic
298	D₄	—	5	4	no	—
299	D₄	—	10	4	no	—
300	D₄	—	10	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	10	2	no	—
305	C₂×C₄	—	20	4	no	—
306	C₂×C₄	—	20	2	no	—
307	C₂×C₄	—	20	4	no	—
308	C₂×C₄	C₃×#685	1	1	yes	characteristic
309	C₂×C₄	C₃×#685	1	1	yes	characteristic
310	C₂×C₄	C₃×#685	1	1	yes	characteristic
311	C₂×C₄	C₂×C₃×C₄×D₅	1	1	yes	characteristic
312	C₂×C₄	C₂×C₃×D₂₀	1	1	yes	characteristic
313	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
314	C₈	—	2	4	no	—
315	C₈	—	20	2	no	—
316	C₂×C₃	—	2	2	no	—
317	C₂×C₃	—	20	4	no	—
318	C₂×C₃	C₂×#3937	1	1	yes	characteristic
319	C₂×C₃	#65415	1	1	yes	characteristic
320	C₂×C₃	#65420	1	1	yes	characteristic
321	C₅	C₃×#1490	1	1	yes	characteristic, complemented, Sylow
322	C₂²	—	2	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	10	4	no	—
325	C₂²	—	20	8	no	—
326	C₂²	—	20	2	no	—
327	C₂²	—	20	2	no	—
328	C₂²	C₃×#3937	1	2	yes	—
329	C₂²	C₂×C₃×#685	1	1	yes	characteristic
330	C₄	—	2	1	no	—
331	C₄	—	2	1	no	—
332	C₄	—	2	1	no	—
333	C₄	—	2	1	no	—
334	C₄	—	20	2	no	—
335	C₄	C₃×D₅×M₁₆	1	2	yes	—
336	C₄	C₃×#4054	1	2	yes	—
337	C₃	#10600648	1	1	yes	characteristic, direct product factor, Sylow
338	C₂	—	2	2	no	—
339	C₂	—	20	4	no	—
340	C₂	C₂×C₃×#3937	1	1	yes	characteristic
341	C₂	C₃×#65415	1	1	yes	characteristic
342	C₂	C₃×#65420	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₂×D₂₀
commutator	C₂×C₄×C₅	C₂²×C₃×C₄
Frattini	C₄²	C₂²×C₃×D₅
Fitting	C₃×C₄×C₅×M₁₆	C₂
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₂²×C₃×D₂₀	C₄
cosocle	C₄²×C₅	C₂³×C₃
hypercenter	C₃×C₄×M₁₆	D₅
hypocenter	C₅	C₃×#1490
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₄² ◁ C₃×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
2. C₃×C₄×C₅×M₁₆	C₂	semidirect
3. C₃×#65215	C₂	semidirect
4. C₃×#64679	C₂	semidirect
5. #10600648	C₃	direct (N × H)
17. C₃×C₄×C₅×C₈	C₂²	semidirect
19. C₃×C₄⋊#168	C₂²	semidirect
23. C₄×C₅×M₁₆	C₂×C₃	semidirect
24. #65215	C₂×C₃	semidirect
25. #64679	C₂×C₃	semidirect
50. C₃×C₄×M₁₆	D₅	semidirect
63. C₄×C₅×C₈	C₂²×C₃	semidirect
65. C₄⋊#168	C₂²×C₃	semidirect
86. C₃×D₂₀	M₁₆	semidirect
105. C₃×C₄×C₈	C₂×D₅	semidirect
130. C₄×M₁₆	C₃×D₅	semidirect
187. D₂₀	C₃×M₁₆	semidirect
206. C₄×C₈	C₂×C₃×D₅	semidirect
272. C₃×C₅	#1490	semidirect
321. C₅	C₃×#1490	semidirect
337. C₃	#10600648	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×C₄×D₂₀	2
C₃×C₄×C₅×M₁₆	2
C₃×#65215	2
C₃×#64679	2
#10600648	3
C₃×#1490	5

Subgroups — C3×#10600648 #14282286

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10600648 #14282286