Subgroups — #14280911

Order 1920.

624 conjugacy classes of subgroups.

The subgroup lattice has 360 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	#11751026	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#4925	C₂	1	1	yes	maximal, characteristic, complemented
4	#11745948	C₂	1	4	yes	maximal, complemented
5	#10601576	—	3	1	no	maximal, Hall
6	C₂²×C₃×D₂₀	C₄	1	1	yes	characteristic
7	C₂²×C₃×C₄×D₅	C₄	1	2	yes	—
8	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
9	C₂×C₃×#685	C₂²	1	2	yes	—
10	C₂×C₃×C₄⋊Dic₅	C₄	1	1	yes	characteristic
11	C₃×C₄×D₂₀	—	2	1	no	—
12	C₃×C₄×D₂₀	—	2	1	no	—
13	C₃×C₄×D₂₀	—	2	1	no	—
14	C₃×C₄×D₂₀	—	2	1	no	—
15	#91114	C₂²	1	2	yes	—
16	C₄⋊(C₃⋊#168)	—	2	1	no	—
17	C₄×C₃⋊#168	—	2	1	no	—
18	C₅×#830	C₂²	1	2	yes	—
19	C₅×C₄⋊(C₃⋊C₈)	—	2	1	no	—
20	C₄×C₅×C₃⋊C₈	—	2	1	no	—
21	#71601	—	5	1	no	maximal, Hall
22	C₂×C₄×D₂₀	D₃	1	1	yes	characteristic
23	C₅×#431	—	3	1	no	—
24	#65214	—	3	1	no	—
25	#64688	—	3	4	no	—
26	C₂³×C₃×D₅	C₂×C₄	1	2	yes	—
27	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
30	C₂²×C₃×Dic₅	C₂×C₄	1	2	yes	—
31	C₂×C₃×D₂₀	—	2	1	no	—
32	C₂×C₃×D₂₀	—	2	1	no	—
33	C₂×C₃×D₂₀	—	2	2	no	—
34	C₂×C₃×D₂₀	C₈	1	4	yes	complemented
35	C₂×C₃×C₄×D₅	—	2	2	no	—
36	C₂×C₃×C₄×D₅	—	2	2	no	—
37	C₂×C₃×C₄×D₅	—	4	4	no	—
38	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
39	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
40	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
41	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
42	C₂×C₃⋊#168	—	2	2	no	—
43	C₂×C₅×C₃⋊C₈	—	2	2	no	—
44	C₃×#685	—	2	2	no	—
45	C₃×#685	—	2	2	no	—
46	C₃×C₄⋊Dic₅	—	2	1	no	—
47	C₃×C₄⋊Dic₅	—	2	1	no	—
48	C₂×C₃×C₄×D₄	—	5	1	no	—
49	#4925	—	5	1	no	—
50	#4925	D₅	1	1	yes	characteristic, complemented
51	#4525	—	5	4	no	—
52	C₂²×D₂₀	Dic₃	1	1	yes	characteristic
53	C₂²×C₄×D₅	Dic₃	1	2	yes	—
54	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
55	C₂×#685	C₂×D₃	1	2	yes	—
56	C₂×C₄⋊Dic₅	Dic₃	1	1	yes	characteristic
57	C₄×D₂₀	—	2	1	no	—
58	C₄×D₂₀	—	2	1	no	—
59	C₄×D₂₀	—	2	1	no	—
60	C₄×D₂₀	—	2	1	no	—
61	C₅×C₄⋊C₈	—	6	1	no	—
62	C₅×C₂²⋊C₈	—	3	2	no	—
63	C₄×C₅×C₈	—	6	1	no	—
64	C₂²⋊#168	—	3	2	no	—
65	C₄⋊#168	—	6	1	no	—
66	C₄×#168	—	6	1	no	—
67	#1460	—	15	1	no	Sylow
68	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₃×D₅	—	2	2	no	—
70	C₂²×C₃×D₅	—	2	2	no	—
71	C₂²×C₃×D₅	—	4	2	no	—
72	C₂²×C₃×D₅	—	4	2	no	—
73	C₂²×C₃×D₅	C₂×C₈	1	4	yes	—
74	C₂×C₃×C₄×C₅	—	2	2	no	—
75	C₂×C₃×C₄×C₅	—	4	1	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×C₈	1	2	yes	—
81	C₂×C₃×C₄×C₅	M₁₆	1	2	yes	—
82	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
83	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
84	C₂×C₃×Dic₅	—	2	2	no	—
85	C₂×C₃×Dic₅	—	4	2	no	—
86	C₃×D₂₀	—	2	4	no	—
87	C₃×D₂₀	—	2	4	no	—
88	C₃×C₄×D₅	—	4	4	no	—
89	C₃×C₄×D₅	—	4	4	no	—
90	C₃⋊#168	—	4	2	no	—
91	C₅×C₃⋊C₈	—	4	2	no	—
92	C₂²×C₃×D₄	—	5	1	no	—
93	C₂³×C₃×C₄	—	5	2	no	—
94	C₃×C₄×D₄	—	10	1	no	—
95	C₃×C₄×D₄	—	10	1	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	—	5	2	no	—
100	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
101	#830	—	5	2	no	—
102	#830	C₂×D₅	1	2	yes	—
103	C₄⋊(C₃⋊C₈)	—	2	1	no	—
104	C₄⋊(C₃⋊C₈)	—	10	1	no	—
105	C₄×C₃⋊C₈	—	2	1	no	—
106	C₄×C₃⋊C₈	—	10	1	no	—
107	C₂³×D₅	C₂×Dic₃	1	2	yes	—
108	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
109	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
110	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
111	C₂²×Dic₅	C₂×Dic₃	1	2	yes	—
112	C₂×D₂₀	—	2	2	no	—
113	C₂×D₂₀	—	2	1	no	—
114	C₂×D₂₀	—	2	1	no	—
115	C₂×D₂₀	C₃⋊C₈	1	4	yes	complemented
116	C₂×C₄×D₅	—	2	2	no	—
117	C₂×C₄×D₅	—	2	2	no	—
118	C₂×C₄×D₅	—	4	4	no	—
119	C₂×C₅×C₈	—	6	2	no	—
120	C₄²×C₅	#67	1	1	yes	characteristic
121	C₄²×C₅	#67	1	1	yes	characteristic
122	C₄²×C₅	#67	1	1	yes	characteristic
123	C₄²×C₅	#67	1	1	yes	characteristic
124	#685	—	2	2	no	—
125	#685	—	2	2	no	—
126	C₄⋊Dic₅	—	2	1	no	—
127	C₄⋊Dic₅	—	2	1	no	—
128	C₂×#168	—	6	2	no	—
129	C₂×C₄×D₄	—	5	1	no	—
130	#431	—	3	1	no	—
131	#431	—	15	1	no	—
132	#323	—	15	4	no	—
133	C₂²×C₃×C₅	—	2	1	no	—
134	C₂²×C₃×C₅	—	2	1	no	—
135	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
136	C₂²×C₃×C₅	C₂²×C₈	1	1	yes	characteristic
137	C₂²×C₃×C₅	C₂×M₁₆	1	1	yes	characteristic
138	C₂×C₃×D₅	—	2	4	no	—
139	C₂×C₃×D₅	—	4	4	no	—
140	C₂×C₃×D₅	—	4	2	no	—
141	C₂×C₃×D₅	—	4	2	no	—
142	C₂×C₃×D₅	—	4	4	no	—
143	C₃×C₄×C₅	—	2	1	no	—
144	C₃×C₄×C₅	—	2	1	no	—
145	C₃×C₄×C₅	—	2	1	no	—
146	C₃×C₄×C₅	C₂²⋊C₈	1	2	yes	—
147	C₃×C₄×C₅	C₂²⋊C₈	1	2	yes	—
148	C₃×C₄×C₅	#100	1	2	yes	—
149	C₃×Dic₅	—	4	2	no	—
150	C₂⁴×C₃	—	5	2	no	—
151	C₂×C₃×D₄	—	5	4	no	—
152	C₂×C₃×D₄	—	10	2	no	—
153	C₂×C₃×D₄	—	10	1	no	—
154	C₂×C₃×D₄	—	10	1	no	—
155	C₂²×C₃×C₄	—	5	2	no	—
156	C₂²×C₃×C₄	—	10	2	no	—
157	C₂²×C₃×C₄	—	10	2	no	—
158	C₂²×C₃×C₄	—	20	4	no	—
159	C₂²×C₃×C₄	C₄×D₅	1	1	yes	characteristic
160	C₂²×C₃×C₄	C₄×D₅	1	1	yes	characteristic
161	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
162	C₃×C₄⋊C₄	—	10	1	no	—
163	C₃×C₄⋊C₄	—	10	1	no	—
164	C₃×C₂²⋊C₄	—	10	2	no	—
165	C₃×C₂²⋊C₄	—	10	2	no	—
166	C₃×C₄²	D₂₀	1	1	yes	characteristic
167	C₃×C₄²	D₂₀	1	1	yes	characteristic
168	C₃×C₄²	C₂²⋊D₅	1	1	yes	characteristic
169	C₃×C₄²	C₂²⋊D₅	1	1	yes	characteristic
170	C₂×C₃⋊C₈	—	2	2	no	—
171	C₂×C₃⋊C₈	—	10	2	no	—
172	C₂³×C₅	C₂²×Dic₃	1	1	yes	characteristic
173	C₂²×D₅	—	2	2	no	—
174	C₂²×D₅	—	2	2	no	—
175	C₂²×D₅	—	4	2	no	—
176	C₂²×D₅	—	4	2	no	—
177	C₂²×D₅	C₂×C₃⋊C₈	1	4	yes	—
178	C₂×C₄×C₅	—	2	2	no	—
179	C₂×C₄×C₅	—	4	1	no	—
180	C₂×C₄×C₅	C₂×C₃⋊C₈	1	2	yes	—
181	C₂×C₄×C₅	#208	1	2	yes	—
182	C₂×C₄×C₅	#217	1	1	yes	characteristic
183	C₂×C₄×C₅	#217	1	1	yes	characteristic
184	C₂×C₄×C₅	#217	1	1	yes	characteristic
185	C₂×C₄×C₅	#217	1	1	yes	characteristic
186	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
187	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
188	C₂×Dic₅	—	2	2	no	—
189	C₂×Dic₅	—	4	2	no	—
190	D₂₀	—	2	4	no	—
191	D₂₀	—	2	4	no	—
192	C₄×D₅	—	4	4	no	—
193	C₄×D₅	—	4	4	no	—
194	C₅×C₈	—	12	2	no	—
195	#168	—	12	2	no	—
196	C₂²×D₄	—	5	1	no	—
197	C₂³×C₄	—	5	2	no	—
198	C₄×D₄	—	10	1	no	—
199	C₄×D₄	—	10	1	no	—
200	C₄×D₄	—	10	1	no	—
201	C₄×D₄	—	10	1	no	—
202	C₂×C₄⋊C₄	—	5	1	no	—
203	C₂×C₂²⋊C₄	—	5	2	no	—
204	C₂×C₄²	D₃×D₅	1	1	yes	characteristic
205	C₄⋊C₈	—	6	1	no	—
206	C₄⋊C₈	—	30	1	no	—
207	C₂²⋊C₈	—	3	2	no	—
208	C₂²⋊C₈	—	15	2	no	—
209	C₄×C₈	—	6	1	no	—
210	C₄×C₈	—	30	1	no	—
211	C₂×C₃×C₅	—	2	1	no	—
212	C₂×C₃×C₅	—	2	1	no	—
213	C₂×C₃×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
214	C₂×C₃×C₅	#410	1	1	yes	characteristic
215	C₂×C₃×C₅	C₂×#100	1	1	yes	characteristic
216	C₃×D₅	—	4	4	no	—
217	C₂³×C₃	—	5	4	no	—
218	C₂³×C₃	—	10	2	no	—
219	C₂³×C₃	—	10	2	no	—
220	C₂³×C₃	—	20	2	no	—
221	C₂³×C₃	—	20	2	no	—
222	C₂³×C₃	C₂×C₄×D₅	1	1	yes	characteristic
223	C₃×D₄	—	10	4	no	—
224	C₃×D₄	—	10	4	no	—
225	C₂×C₃×C₄	—	2	2	no	—
226	C₂×C₃×C₄	—	4	1	no	—
227	C₂×C₃×C₄	—	10	2	no	—
228	C₂×C₃×C₄	—	20	4	no	—
229	C₂×C₃×C₄	—	20	4	no	—
230	C₂×C₃×C₄	—	20	2	no	—
231	C₂×C₃×C₄	C₈×D₅	1	2	yes	—
232	C₂×C₃×C₄	#676	1	2	yes	—
233	C₂×C₃×C₄	#685	1	1	yes	characteristic
234	C₂×C₃×C₄	#685	1	1	yes	characteristic
235	C₂×C₃×C₄	#685	1	1	yes	characteristic
236	C₂×C₃×C₄	#685	1	1	yes	characteristic
237	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
238	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
239	C₃⋊C₈	—	4	2	no	—
240	C₃⋊C₈	—	20	2	no	—
241	C₂²×C₅	—	2	1	no	—
242	C₂²×C₅	—	2	1	no	—
243	C₂²×C₅	C₂²×C₃⋊C₈	1	1	yes	characteristic
244	C₂²×C₅	C₂×#208	1	1	yes	characteristic
245	C₂²×C₅	C₂×#217	1	1	yes	characteristic
246	C₂×D₅	—	2	4	no	—
247	C₂×D₅	—	4	4	no	—
248	C₂×D₅	—	4	2	no	—
249	C₂×D₅	—	4	2	no	—
250	C₂×D₅	—	4	4	no	—
251	C₄×C₅	—	2	1	no	—
252	C₄×C₅	—	2	1	no	—
253	C₄×C₅	—	2	1	no	—
254	C₄×C₅	#830	1	2	yes	—
255	C₄×C₅	#830	1	2	yes	—
256	C₄×C₅	#833	1	2	yes	—
257	Dic₅	—	4	2	no	—
258	C₂⁴	—	5	2	no	—
259	C₂×D₄	—	5	4	no	—
260	C₂×D₄	—	10	1	no	—
261	C₂×D₄	—	10	1	no	—
262	C₂×D₄	—	10	2	no	—
263	C₂²×C₄	—	5	2	no	—
264	C₂²×C₄	—	10	2	no	—
265	C₂²×C₄	—	10	2	no	—
266	C₂²×C₄	—	20	4	no	—
267	C₂²×C₄	D₅×Dic₃	1	1	yes	characteristic
268	C₂²×C₄	D₅×Dic₃	1	1	yes	characteristic
269	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
270	C₂×C₈	—	6	2	no	—
271	C₂×C₈	—	30	2	no	—
272	C₄⋊C₄	—	10	1	no	—
273	C₄⋊C₄	—	10	1	no	—
274	C₂²⋊C₄	—	10	2	no	—
275	C₂²⋊C₄	—	10	2	no	—
276	C₄²	#1201	1	1	yes	characteristic
277	C₄²	#1201	1	1	yes	characteristic
278	C₄²	#1202	1	1	yes	characteristic
279	C₄²	#1202	1	1	yes	characteristic
280	C₃×C₅	#1460	1	1	yes	characteristic, complemented, Hall
281	C₂²×C₃	—	2	1	no	—
282	C₂²×C₃	—	2	1	no	—
283	C₂²×C₃	—	10	4	no	—
284	C₂²×C₃	—	20	4	no	—
285	C₂²×C₃	—	20	4	no	—
286	C₂²×C₃	—	20	2	no	—
287	C₂²×C₃	—	20	2	no	—
288	C₂²×C₃	C₂×C₈×D₅	1	1	yes	characteristic
289	C₂²×C₃	C₂×#676	1	1	yes	characteristic
290	C₂²×C₃	C₂×#685	1	1	yes	characteristic
291	C₃×C₄	—	2	1	no	—
292	C₃×C₄	—	2	1	no	—
293	C₃×C₄	—	2	1	no	—
294	C₃×C₄	—	20	2	no	—
295	C₃×C₄	#3952	1	2	yes	—
296	C₃×C₄	#3952	1	2	yes	—
297	C₃×C₄	#3955	1	2	yes	—
298	C₂×C₅	—	2	1	no	—
299	C₂×C₅	—	2	1	no	—
300	C₂×C₅	C₂×#830	1	1	yes	characteristic
301	C₂×C₅	C₂×#833	1	1	yes	characteristic
302	C₂×C₅	#5240	1	1	yes	characteristic
303	D₅	—	4	4	no	—
304	C₂³	—	5	4	no	—
305	C₂³	—	10	2	no	—
306	C₂³	—	10	2	no	—
307	C₂³	—	20	2	no	—
308	C₂³	—	20	2	no	—
309	C₂³	C₂×D₅×Dic₃	1	1	yes	characteristic
310	D₄	—	10	4	no	—
311	D₄	—	10	4	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	4	1	no	—
314	C₂×C₄	—	10	2	no	—
315	C₂×C₄	—	20	4	no	—
316	C₂×C₄	—	20	4	no	—
317	C₂×C₄	—	20	2	no	—
318	C₂×C₄	D₅×C₃⋊C₈	1	2	yes	—
319	C₂×C₄	#6649	1	2	yes	—
320	C₂×C₄	#6665	1	1	yes	characteristic
321	C₂×C₄	#6665	1	1	yes	characteristic
322	C₂×C₄	#6665	1	1	yes	characteristic
323	C₂×C₄	#6665	1	1	yes	characteristic
324	C₂×C₄	C₂×#1201	1	1	yes	characteristic
325	C₂×C₄	C₂×#1202	1	1	yes	characteristic
326	C₈	—	12	2	no	—
327	C₈	—	60	2	no	—
328	C₂×C₃	—	2	1	no	—
329	C₂×C₃	—	2	1	no	—
330	C₂×C₃	—	20	4	no	—
331	C₂×C₃	#65295	1	1	yes	characteristic
332	C₂×C₃	C₂×#3952	1	1	yes	characteristic
333	C₂×C₃	C₂×#3955	1	1	yes	characteristic
334	C₅	#71601	1	1	yes	characteristic, complemented, Sylow
335	C₂²	—	2	1	no	—
336	C₂²	—	2	1	no	—
337	C₂²	—	10	4	no	—
338	C₂²	—	20	4	no	—
339	C₂²	—	20	4	no	—
340	C₂²	—	20	2	no	—
341	C₂²	—	20	2	no	—
342	C₂²	C₂×D₅×C₃⋊C₈	1	1	yes	characteristic
343	C₂²	C₂×#6649	1	1	yes	characteristic
344	C₂²	C₂×#6665	1	1	yes	characteristic
345	C₄	—	2	1	no	—
346	C₄	—	2	1	no	—
347	C₄	—	2	1	no	—
348	C₄	—	20	2	no	—
349	C₄	#90955	1	2	yes	—
350	C₄	#90955	1	2	yes	—
351	C₄	#90958	1	2	yes	—
352	C₃	#10601576	1	1	yes	characteristic, complemented, Sylow
353	C₂	—	2	1	no	—
354	C₂	—	2	1	no	—
355	C₂	—	20	4	no	—
356	C₂	C₂×#90955	1	1	yes	characteristic
357	C₂	C₂×#90958	1	1	yes	characteristic
358	C₂	#11748624	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#6665
commutator	C₂²×C₃×C₅	C₂²×C₈
Frattini	C₂²×C₄	C₂×D₃×D₅
Fitting	C₂×C₃×C₄²×C₅	C₂²
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₂²×C₃×D₂₀	C₄
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	C₂×C₄²	D₃×D₅
hypocenter	C₃×C₅	#1460
LCS	G ▷ C₂²×C₃×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
2. #11751026	C₂	semidirect
3. C₅×#4925	C₂	semidirect
4. #11745948	C₂	semidirect
34. C₂×C₃×D₂₀	C₈	semidirect
50. #4925	D₅	semidirect
115. C₂×D₂₀	C₃⋊C₈	semidirect
280. C₃×C₅	#1460	semidirect
334. C₅	#71601	semidirect
352. C₃	#10601576	semidirect

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
#11751026	2
C₅×#4925	2
#11745948	2
#10601576	3
#71601	5