Subgroups — #14297737

Order 1920.

712 conjugacy classes of subgroups.

The subgroup lattice has 380 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	C₂×C₄⋊(C₃⋊#168)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#4936	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#90964	C₂	1	2	yes	maximal
5	#11745979	C₂	1	2	yes	maximal, complemented
6	#10600349	—	3	1	no	maximal, Hall
7	C₂²×C₃×D₂₀	C₂²	1	1	yes	characteristic
8	C₂²×C₃×C₄×D₅	—	2	1	no	—
9	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
10	C₂²×C₃⋊#168	C₂²	1	2	yes	—
11	C₂×C₅×C₄⋊Dic₃	C₂²	1	2	yes	—
12	C₂×C₃×#685	—	2	1	no	—
13	C₂×C₃×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
14	C₃×C₄×D₂₀	—	2	2	no	—
15	C₃×C₄×D₂₀	C₄	1	4	yes	complemented
16	C₄⋊(C₃⋊#168)	—	2	2	no	—
17	C₅×C₃⋊#95	—	2	1	no	—
18	C₅×C₃⋊#95	—	2	1	no	—
19	#90964	—	2	4	no	—
20	#71605	—	5	1	no	maximal, Hall
21	C₂×C₄×D₂₀	D₃	1	1	yes	characteristic
22	C₅×#382	—	3	1	no	—
23	C₂×#3941	—	3	2	no	—
24	C₂×C₄⋊#168	—	3	1	no	—
25	#64749	—	3	2	no	—
26	C₂³×C₃×D₅	—	2	1	no	—
27	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
28	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
29	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
30	C₂²×C₅×Dic₃	—	2	2	no	—
31	C₂²×C₃×Dic₅	—	2	1	no	—
32	C₂×C₃×D₂₀	—	2	1	no	—
33	C₂×C₃×D₂₀	—	2	1	no	—
34	C₂×C₃×D₂₀	—	2	1	no	—
35	C₂×C₃×D₂₀	C₂×C₄	1	2	yes	—
36	C₂×C₃×D₂₀	D₄	1	2	yes	—
37	C₂×C₃×D₂₀	Q₈	1	2	yes	—
38	C₂×C₃×C₄×D₅	—	2	4	no	—
39	C₂×C₃×C₄×D₅	—	4	4	no	—
40	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
41	C₃×C₄²×C₅	D₄	1	2	yes	—
42	C₂×C₃⋊#168	—	2	2	no	—
43	C₂×C₃⋊#168	—	2	2	no	—
44	C₂×C₃⋊#168	—	2	2	no	—
45	C₅×C₄⋊Dic₃	—	2	2	no	—
46	C₅×C₄⋊Dic₃	—	2	2	no	—
47	C₃×#685	—	2	4	no	—
48	C₃×C₄⋊Dic₅	—	2	1	no	—
49	C₃×C₄⋊Dic₅	C₂×C₄	1	2	yes	—
50	C₂×C₃×C₄×D₄	—	5	1	no	—
51	C₂×#832	—	5	2	no	—
52	#4936	D₅	1	1	yes	characteristic, complemented
53	C₂×C₄⋊(C₃⋊C₈)	—	5	1	no	—
54	#4529	—	5	2	no	—
55	C₂²×D₂₀	C₂×D₃	1	1	yes	characteristic
56	C₂²×C₄×D₅	—	2	1	no	—
57	C₂×C₅×C₄⋊C₄	—	3	2	no	—
58	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
59	C₂×#685	—	2	1	no	—
60	C₂×C₄⋊Dic₅	C₂×D₃	1	1	yes	characteristic
61	C₂²×#168	—	3	2	no	—
62	C₄×D₂₀	—	2	2	no	—
63	C₄×D₂₀	Dic₃	1	4	yes	complemented
64	C₅×#95	—	6	1	no	—
65	C₅×#95	—	6	1	no	—
66	#3941	—	6	4	no	—
67	C₄⋊#168	—	6	2	no	—
68	#1804	—	15	1	no	Sylow
69	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₃×D₅	—	2	2	no	—
71	C₂²×C₃×D₅	—	2	2	no	—
72	C₂²×C₃×D₅	—	2	2	no	—
73	C₂²×C₃×D₅	—	4	2	no	—
74	C₂²×C₃×D₅	—	4	2	no	—
75	C₂×C₃×C₄×C₅	—	2	2	no	—
76	C₂×C₃×C₄×C₅	—	2	2	no	—
77	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
78	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
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×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
83	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
84	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
85	C₂×C₅×Dic₃	—	4	2	no	—
86	C₂×C₅×Dic₃	—	4	2	no	—
87	C₂×C₅×Dic₃	—	4	2	no	—
88	C₂×C₃×Dic₅	—	2	2	no	—
89	C₂×C₃×Dic₅	—	4	2	no	—
90	C₃×D₂₀	—	2	2	no	—
91	C₃×D₂₀	—	2	2	no	—
92	C₃×D₂₀	—	2	2	no	—
93	C₃×D₂₀	C₄⋊C₄	1	4	yes	—
94	C₃×C₄×D₅	—	4	8	no	—
95	C₃⋊#168	—	4	2	no	—
96	C₂²×C₃×D₄	—	5	1	no	—
97	C₂³×C₃×C₄	—	10	1	no	—
98	C₃×C₄×D₄	—	5	4	no	—
99	C₃×C₄×D₄	—	10	2	no	—
100	C₂×C₃×C₄⋊C₄	—	5	1	no	—
101	C₂×C₃×C₂²⋊C₄	—	10	1	no	—
102	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
103	C₂×C₄⋊Dic₃	C₂×D₅	1	2	yes	—
104	C₂²×C₃⋊C₈	—	5	2	no	—
105	#832	—	10	4	no	—
106	C₃⋊#95	—	2	1	no	—
107	C₃⋊#95	—	2	1	no	—
108	C₄⋊(C₃⋊C₈)	—	10	2	no	—
109	C₂³×D₅	—	2	1	no	—
110	C₂²×C₄×C₅	—	6	2	no	—
111	C₂²×C₄×C₅	#67	1	1	yes	characteristic
112	C₂²×C₄×C₅	#67	1	1	yes	characteristic
113	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
114	C₂²×Dic₅	—	2	1	no	—
115	C₂×D₂₀	—	2	1	no	—
116	C₂×D₂₀	—	2	1	no	—
117	C₂×D₂₀	—	2	1	no	—
118	C₂×D₂₀	Dic₆	1	2	yes	—
119	C₂×D₂₀	D₁₂	1	2	yes	—
120	C₂×D₂₀	C₂×Dic₃	1	2	yes	—
121	C₂×C₄×D₅	—	2	4	no	—
122	C₂×C₄×D₅	—	4	4	no	—
123	C₅×C₄⋊C₄	—	6	2	no	—
124	C₅×C₄⋊C₄	—	6	2	no	—
125	C₄²×C₅	C₂×Dic₃	1	2	yes	—
126	C₄²×C₅	#67	1	2	yes	—
127	#685	—	2	4	no	—
128	C₄⋊Dic₅	—	2	1	no	—
129	C₄⋊Dic₅	C₂×Dic₃	1	2	yes	—
130	C₂×#168	—	6	2	no	—
131	C₂×#168	—	6	2	no	—
132	C₂×#168	—	6	2	no	—
133	C₂×C₄×D₄	—	5	1	no	—
134	C₂×C₄⋊C₈	—	15	1	no	—
135	C₂×#102	—	15	2	no	—
136	#382	—	3	1	no	—
137	#340	—	15	2	no	—
138	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
139	C₂²×C₃×C₅	#121	1	1	yes	characteristic
140	C₂²×C₃×C₅	C₂²⋊Q₈	1	1	yes	characteristic
141	C₂²×C₃×C₅	#135	1	2	yes	—
142	C₂²×C₃×C₅	AGL(1,ℤ/8)	1	2	yes	—
143	C₂×C₃×D₅	—	2	4	no	—
144	C₂×C₃×D₅	—	4	2	no	—
145	C₂×C₃×D₅	—	4	2	no	—
146	C₂×C₃×D₅	—	4	4	no	—
147	C₂×C₃×D₅	—	4	4	no	—
148	C₃×C₄×C₅	—	2	2	no	—
149	C₃×C₄×C₅	—	2	2	no	—
150	C₃×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
151	C₃×C₄×C₅	#117	1	1	yes	characteristic
152	C₃×C₄×C₅	#121	1	1	yes	characteristic
153	C₃×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
154	C₃×Dic₅	—	4	2	no	—
155	C₅×Dic₃	—	8	2	no	—
156	C₂⁴×C₃	—	10	1	no	—
157	C₂×C₃×D₄	—	5	2	no	—
158	C₂×C₃×D₄	—	5	2	no	—
159	C₂×C₃×D₄	—	5	2	no	—
160	C₂×C₃×D₄	—	10	1	no	—
161	C₂×C₃×D₄	—	10	1	no	—
162	C₂×C₃×D₄	—	10	1	no	—
163	C₂²×C₃×C₄	—	10	1	no	—
164	C₂²×C₃×C₄	—	10	4	no	—
165	C₂²×C₃×C₄	—	20	4	no	—
166	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
167	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
168	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
169	C₂²×Dic₃	—	2	2	no	—
170	C₃×C₄⋊C₄	—	5	2	no	—
171	C₃×C₄⋊C₄	—	10	1	no	—
172	C₃×C₂²⋊C₄	—	10	4	no	—
173	C₃×C₄²	C₄×D₅	1	2	yes	—
174	C₃×C₄²	D₂₀	1	2	yes	—
175	C₄⋊Dic₃	—	2	2	no	—
176	C₄⋊Dic₃	—	2	2	no	—
177	C₂×C₃⋊C₈	—	10	2	no	—
178	C₂×C₃⋊C₈	—	10	2	no	—
179	C₂×C₃⋊C₈	—	10	2	no	—
180	C₂³×C₅	C₂×#67	1	1	yes	characteristic
181	C₂²×D₅	—	2	2	no	—
182	C₂²×D₅	—	2	2	no	—
183	C₂²×D₅	—	2	2	no	—
184	C₂²×D₅	—	4	2	no	—
185	C₂²×D₅	—	4	2	no	—
186	C₂×C₄×C₅	—	2	2	no	—
187	C₂×C₄×C₅	—	2	2	no	—
188	C₂×C₄×C₅	—	12	2	no	—
189	C₂×C₄×C₅	—	12	2	no	—
190	C₂×C₄×C₅	—	12	2	no	—
191	C₂×C₄×C₅	#217	1	2	yes	—
192	C₂×C₄×C₅	#217	1	2	yes	—
193	C₂×C₄×C₅	C₂×Dic₆	1	1	yes	characteristic
194	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
195	C₂×C₄×C₅	#235	1	1	yes	characteristic
196	C₂×C₄×C₅	#235	1	1	yes	characteristic
197	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
198	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
199	C₂×Dic₅	—	2	2	no	—
200	C₂×Dic₅	—	4	2	no	—
201	D₂₀	—	2	2	no	—
202	D₂₀	—	2	2	no	—
203	D₂₀	—	2	2	no	—
204	D₂₀	C₄⋊Dic₃	1	4	yes	—
205	C₄×D₅	—	4	8	no	—
206	#168	—	12	2	no	—
207	C₂²×D₄	—	5	1	no	—
208	C₂³×C₄	—	10	1	no	—
209	C₂²×C₈	—	15	2	no	—
210	C₄×D₄	—	5	4	no	—
211	C₄×D₄	—	10	2	no	—
212	C₂×C₄⋊C₄	—	3	2	no	—
213	C₂×C₄⋊C₄	—	5	1	no	—
214	C₂×C₂²⋊C₄	—	10	1	no	—
215	C₂×C₄²	D₃×D₅	1	1	yes	characteristic
216	C₄⋊C₈	—	30	2	no	—
217	#102	—	30	4	no	—
218	#95	—	6	1	no	—
219	#95	—	6	1	no	—
220	C₂×C₃×C₅	#380	1	1	yes	characteristic
221	C₂×C₃×C₅	#416	1	1	yes	characteristic
222	C₂×C₃×C₅	#418	1	1	yes	characteristic
223	C₂×C₃×C₅	#463	1	2	yes	—
224	C₂×C₃×C₅	#478	1	2	yes	—
225	C₃×D₅	—	4	4	no	—
226	C₂³×C₃	—	10	2	no	—
227	C₂³×C₃	—	10	2	no	—
228	C₂³×C₃	—	10	2	no	—
229	C₂³×C₃	—	20	2	no	—
230	C₂³×C₃	—	20	2	no	—
231	C₂³×C₃	C₂×C₂²⋊D₅	1	1	yes	characteristic
232	C₃×D₄	—	5	4	no	—
233	C₃×D₄	—	10	2	no	—
234	C₃×D₄	—	10	2	no	—
235	C₃×D₄	—	10	2	no	—
236	C₂×C₃×C₄	—	2	2	no	—
237	C₂×C₃×C₄	—	2	2	no	—
238	C₂×C₃×C₄	—	10	2	no	—
239	C₂×C₃×C₄	—	20	8	no	—
240	C₂×C₃×C₄	—	20	2	no	—
241	C₂×C₃×C₄	#685	1	2	yes	—
242	C₂×C₃×C₄	#685	1	2	yes	—
243	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
244	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
245	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
246	C₂×C₃×C₄	#711	1	1	yes	characteristic
247	C₂×C₃×C₄	Q₈×D₅	1	1	yes	characteristic
248	C₂×C₃×C₄	#713	1	1	yes	characteristic
249	C₂×Dic₃	—	4	2	no	—
250	C₂×Dic₃	—	4	2	no	—
251	C₂×Dic₃	—	4	2	no	—
252	C₃⋊C₈	—	20	2	no	—
253	C₂²×C₅	#924	1	1	yes	characteristic
254	C₂²×C₅	#930	1	1	yes	characteristic
255	C₂²×C₅	#932	1	2	yes	—
256	C₂²×C₅	#950	1	2	yes	—
257	C₂²×C₅	C₂×#217	1	1	yes	characteristic
258	C₂×D₅	—	2	4	no	—
259	C₂×D₅	—	4	2	no	—
260	C₂×D₅	—	4	2	no	—
261	C₂×D₅	—	4	4	no	—
262	C₂×D₅	—	4	4	no	—
263	C₄×C₅	—	2	2	no	—
264	C₄×C₅	—	2	2	no	—
265	C₄×C₅	—	24	2	no	—
266	C₄×C₅	#924	1	1	yes	characteristic
267	C₄×C₅	C₂×C₄⋊Dic₃	1	1	yes	characteristic
268	C₄×C₅	#926	1	1	yes	characteristic
269	C₄×C₅	#930	1	1	yes	characteristic
270	Dic₅	—	4	2	no	—
271	C₂⁴	—	10	1	no	—
272	C₂×D₄	—	5	2	no	—
273	C₂×D₄	—	5	2	no	—
274	C₂×D₄	—	5	2	no	—
275	C₂×D₄	—	10	1	no	—
276	C₂×D₄	—	10	1	no	—
277	C₂×D₄	—	10	1	no	—
278	C₂²×C₄	—	6	2	no	—
279	C₂²×C₄	—	10	1	no	—
280	C₂²×C₄	—	10	4	no	—
281	C₂²×C₄	—	20	4	no	—
282	C₂²×C₄	#1201	1	1	yes	characteristic
283	C₂²×C₄	#1201	1	1	yes	characteristic
284	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
285	C₂×C₈	—	30	2	no	—
286	C₂×C₈	—	30	2	no	—
287	C₂×C₈	—	30	2	no	—
288	C₄⋊C₄	—	5	2	no	—
289	C₄⋊C₄	—	6	2	no	—
290	C₄⋊C₄	—	6	2	no	—
291	C₄⋊C₄	—	10	1	no	—
292	C₂²⋊C₄	—	10	4	no	—
293	C₄²	D₅×Dic₃	1	2	yes	—
294	C₄²	#1202	1	2	yes	—
295	C₃×C₅	#1804	1	1	yes	characteristic, complemented, Hall
296	C₂²×C₃	—	10	4	no	—
297	C₂²×C₃	—	20	2	no	—
298	C₂²×C₃	—	20	2	no	—
299	C₂²×C₃	—	20	4	no	—
300	C₂²×C₃	—	20	4	no	—
301	C₂²×C₃	C₂×#685	1	1	yes	characteristic
302	C₂²×C₃	#4084	1	1	yes	characteristic
303	C₂²×C₃	#4092	1	1	yes	characteristic
304	C₂²×C₃	#4095	1	2	yes	—
305	C₂²×C₃	#4096	1	2	yes	—
306	C₃×C₄	—	2	2	no	—
307	C₃×C₄	—	2	2	no	—
308	C₃×C₄	—	20	2	no	—
309	C₃×C₄	D₅×C₄⋊C₄	1	1	yes	characteristic
310	C₃×C₄	#4038	1	1	yes	characteristic
311	C₃×C₄	#4041	1	1	yes	characteristic
312	C₃×C₄	#4043	1	1	yes	characteristic
313	Dic₃	—	8	2	no	—
314	C₂×C₅	#5009	1	2	yes	—
315	C₂×C₅	#5016	1	2	yes	—
316	C₂×C₅	#5212	1	1	yes	characteristic
317	C₂×C₅	#5215	1	1	yes	characteristic
318	C₂×C₅	#5233	1	1	yes	characteristic
319	D₅	—	4	4	no	—
320	C₂³	—	10	2	no	—
321	C₂³	—	10	2	no	—
322	C₂³	—	10	2	no	—
323	C₂³	—	20	2	no	—
324	C₂³	—	20	2	no	—
325	C₂³	C₂×#1201	1	1	yes	characteristic
326	D₄	—	5	4	no	—
327	D₄	—	10	2	no	—
328	D₄	—	10	2	no	—
329	D₄	—	10	2	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	2	2	no	—
332	C₂×C₄	—	10	2	no	—
333	C₂×C₄	—	12	2	no	—
334	C₂×C₄	—	12	2	no	—
335	C₂×C₄	—	12	2	no	—
336	C₂×C₄	—	20	8	no	—
337	C₂×C₄	—	20	2	no	—
338	C₂×C₄	#6665	1	2	yes	—
339	C₂×C₄	#6665	1	2	yes	—
340	C₂×C₄	D₅×Dic₆	1	1	yes	characteristic
341	C₂×C₄	#6772	1	1	yes	characteristic
342	C₂×C₄	#6773	1	1	yes	characteristic
343	C₂×C₄	D₅×D₁₂	1	1	yes	characteristic
344	C₂×C₄	C₂×D₅×Dic₃	1	1	yes	characteristic
345	C₂×C₄	C₂×#1202	1	1	yes	characteristic
346	C₈	—	60	2	no	—
347	C₂×C₃	—	20	4	no	—
348	C₂×C₃	#65276	1	1	yes	characteristic
349	C₂×C₃	#65290	1	1	yes	characteristic
350	C₂×C₃	#65291	1	1	yes	characteristic
351	C₂×C₃	#65346	1	2	yes	—
352	C₂×C₃	#65355	1	2	yes	—
353	C₅	#71605	1	1	yes	characteristic, complemented, Sylow
354	C₂²	—	10	4	no	—
355	C₂²	—	20	2	no	—
356	C₂²	—	20	2	no	—
357	C₂²	—	20	4	no	—
358	C₂²	—	20	4	no	—
359	C₂²	#91297	1	2	yes	—
360	C₂²	#91298	1	2	yes	—
361	C₂²	#91359	1	1	yes	characteristic
362	C₂²	#91449	1	1	yes	characteristic
363	C₂²	C₂×#6665	1	1	yes	characteristic
364	C₄	—	2	2	no	—
365	C₄	—	2	2	no	—
366	C₄	—	20	2	no	—
367	C₄	—	24	2	no	—
368	C₄	#91358	1	1	yes	characteristic
369	C₄	#91360	1	1	yes	characteristic
370	C₄	D₅×C₄⋊Dic₃	1	1	yes	characteristic
371	C₄	#91450	1	1	yes	characteristic
372	C₃	#10600349	1	1	yes	characteristic, complemented, Sylow
373	C₂	—	20	4	no	—
374	C₂	#11746887	1	2	yes	—
375	C₂	#11747027	1	2	yes	—
376	C₂	#11748364	1	1	yes	characteristic
377	C₂	#11748366	1	1	yes	characteristic
378	C₂	#11748759	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#1201
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₂²×C₄	C₂×D₃×D₅
Fitting	C₂×C₃×C₄²×C₅	C₂²
socle	C₂³×C₃×C₅	C₂×D₄
prime radical	C₂²×C₃×D₂₀	C₂²
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	C₂×C₄²	D₃×D₅
hypocenter	C₃×C₅	#1804
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
2. C₂×C₄⋊(C₃⋊#168)	C₂	semidirect
3. C₅×#4936	C₂	semidirect
5. #11745979	C₂	semidirect
15. C₃×C₄×D₂₀	C₄	semidirect
52. #4936	D₅	semidirect
63. C₄×D₂₀	Dic₃	semidirect
295. C₃×C₅	#1804	semidirect
353. C₅	#71605	semidirect
372. C₃	#10600349	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
C₂×C₄⋊(C₃⋊#168)	2
C₅×#4936	2
C₂×#90964	2
#11745979	2
#10600349	3
#71605	5