Subgroups — #14284151

Order 1920.

432 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#5479	C₂	1	1	yes	maximal, characteristic
2	#11751027	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₄×#681	C₂	1	1	yes	maximal, characteristic, complemented
4	#11745850	C₂	1	2	yes	maximal, complemented
5	#11745847	C₂	1	2	yes	maximal, complemented
6	#10602540	—	3	1	no	maximal, Hall
7	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
8	C₅×#930	C₄	1	1	yes	characteristic
9	C₅×#929	—	2	1	no	—
10	C₄×C₅×#67	—	2	1	no	—
11	C₅×#924	C₄	1	1	yes	characteristic
12	C₅×#876	—	2	1	no	—
13	C₅×#875	C₂²	1	1	yes	characteristic
14	C₄×C₅×D₁₂	C₄	1	1	yes	characteristic
15	C₅×#872	—	2	1	no	—
16	C₅×#870	C₂²	1	1	yes	characteristic
17	C₄×C₅×Dic₆	C₄	1	1	yes	characteristic
18	C₂×C₃×#681	—	2	1	no	—
19	#91114	—	2	1	no	—
20	C₄⋊(C₃⋊#168)	—	2	1	no	—
21	#91087	C₂²	1	2	yes	—
22	C₃×#3935	C₂²	1	2	yes	—
23	C₃×C₄×#168	—	2	1	no	—
24	#69861	—	5	1	no	maximal, Hall
25	C₅×#529	—	3	1	no	—
26	#65215	—	3	1	no	—
27	C₄×#681	D₃	1	1	yes	characteristic, complemented
28	#64763	—	3	2	no	—
29	#64760	—	3	2	no	—
30	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
32	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
33	C₂×C₅×#67	—	2	1	no	—
34	C₂×C₅×D₁₂	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×C₅×D₃	—	2	1	no	—
36	C₂×C₅×Dic₆	C₂×C₄	1	1	yes	characteristic
37	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
38	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
39	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
40	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
41	C₂×C₃⋊#168	—	2	2	no	—
42	C₅×#217	—	2	1	no	—
43	C₅×(C₂×D₃)⋊C₄	—	2	1	no	—
44	C₅×(C₂×D₃)⋊C₄	—	2	1	no	—
45	C₅×(C₂×D₃)⋊C₄	—	2	1	no	—
46	C₅×(C₂×D₃)⋊C₄	—	2	1	no	—
47	C₅×C₄⋊Dic₃	C₂×C₄	1	1	yes	characteristic
48	C₅×C₄⋊Dic₃	C₂×C₄	1	1	yes	characteristic
49	C₅×Dic₃⋊C₄	—	2	1	no	—
50	C₅×Dic₃⋊C₄	—	2	1	no	—
51	C₅×Dic₃⋊C₄	—	2	1	no	—
52	C₅×Dic₃⋊C₄	—	2	1	no	—
53	C₄×C₅×Dic₃	—	2	1	no	—
54	C₃×#681	—	2	4	no	—
55	C₂×C₃×#168	—	2	2	no	—
56	#5479	D₅	1	1	yes	characteristic
57	C₃×C₄×M₁₆	—	5	1	no	—
58	#4926	—	5	1	no	—
59	#4465	—	5	2	no	—
60	#4464	—	5	2	no	—
61	C₅×#126	—	6	1	no	—
62	C₅×#125	—	3	1	no	—
63	C₅×#124	—	3	1	no	—
64	C₅×#123	—	6	1	no	—
65	C₅×C₂²⋊Q₈	—	3	1	no	—
66	C₅×#121	—	3	1	no	—
67	C₄×C₅×Q₈	—	3	1	no	—
68	C₄×C₅×D₄	—	3	1	no	—
69	C₄×C₅×D₄	—	6	1	no	—
70	C₅×#117	—	6	1	no	—
71	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
72	C₂×#681	—	2	1	no	—
73	C₂²⋊#168	—	6	1	no	—
74	C₄⋊#168	—	6	1	no	—
75	#3935	—	3	2	no	—
76	#3935	C₂×D₃	1	2	yes	—
77	C₄×#168	—	2	1	no	—
78	#1529	—	15	1	no	Sylow
79	C₂³×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
80	C₂²×C₅×D₃	—	2	1	no	—
81	C₂×C₃×C₄×C₅	—	2	2	no	—
82	C₂×C₃×C₄×C₅	—	4	1	no	—
83	C₂×C₃×C₄×C₅	—	4	1	no	—
84	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
89	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
90	C₅×#67	—	4	2	no	—
91	C₂×C₅×Dic₃	—	2	1	no	—
92	C₂×C₅×Dic₃	—	2	1	no	—
93	C₂×C₅×Dic₃	—	2	1	no	—
94	C₅×D₁₂	—	4	1	no	—
95	C₄×C₅×D₃	—	4	2	no	—
96	C₅×Dic₆	—	4	1	no	—
97	C₃⋊#168	—	4	2	no	—
98	C₃×#168	—	2	4	no	—
99	C₂×C₃×M₁₆	—	10	1	no	—
100	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
101	#930	Dic₅	1	1	yes	characteristic
102	#929	—	2	1	no	—
103	C₄×#67	—	2	1	no	—
104	#924	Dic₅	1	1	yes	characteristic
105	#876	—	2	1	no	—
106	#875	C₂×D₅	1	1	yes	characteristic
107	C₄×D₁₂	Dic₅	1	1	yes	characteristic
108	#872	—	2	1	no	—
109	#870	C₂×D₅	1	1	yes	characteristic
110	C₄×Dic₆	Dic₅	1	1	yes	characteristic
111	C₃×#97	—	5	2	no	—
112	C₃×C₄×C₈	—	10	1	no	—
113	#830	—	10	1	no	—
114	C₄⋊(C₃⋊C₈)	—	10	1	no	—
115	#803	—	5	2	no	—
116	C₂×C₅×Q₈	—	3	1	no	—
117	C₂×C₅×D₄	—	3	1	no	—
118	C₂×C₅×D₄	—	6	1	no	—
119	C₂²×C₄×C₅	—	6	1	no	—
120	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
121	C₂²×C₄×C₅	D₁₂	1	1	yes	characteristic
122	C₂²×C₄×C₅	#67	1	1	yes	characteristic
123	C₅×C₄⋊C₄	—	3	1	no	—
124	C₅×C₄⋊C₄	—	3	1	no	—
125	C₅×C₄⋊C₄	—	6	1	no	—
126	C₅×C₄⋊C₄	—	6	1	no	—
127	C₅×C₄⋊C₄	—	6	1	no	—
128	C₅×C₄⋊C₄	—	6	1	no	—
129	C₅×C₂²⋊C₄	—	6	1	no	—
130	C₅×C₂²⋊C₄	—	6	1	no	—
131	C₅×C₂²⋊C₄	—	6	1	no	—
132	C₅×C₂²⋊C₄	—	6	1	no	—
133	C₅×C₂²⋊C₄	—	6	1	no	—
134	C₄²×C₅	—	6	1	no	—
135	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
136	C₄²×C₅	D₁₂	1	1	yes	characteristic
137	C₄²×C₅	#67	1	1	yes	characteristic
138	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
139	#681	—	2	4	no	—
140	C₂×#168	—	2	2	no	—
141	C₂×#168	—	6	2	no	—
142	#529	—	3	1	no	—
143	#432	—	15	1	no	—
144	C₄×M₁₆	—	5	1	no	—
145	#330	—	15	2	no	—
146	#329	—	15	2	no	—
147	C₂²×C₃×C₅	—	2	1	no	—
148	C₂²×C₃×C₅	—	2	1	no	—
149	C₂²×C₃×C₅	—	2	1	no	—
150	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
151	C₂×C₅×D₃	—	4	1	no	—
152	C₂×C₅×D₃	—	4	1	no	—
153	C₂×C₅×D₃	—	4	1	no	—
154	C₃×C₄×C₅	—	2	1	no	—
155	C₃×C₄×C₅	—	2	1	no	—
156	C₃×C₄×C₅	—	2	1	no	—
157	C₃×C₄×C₅	—	2	1	no	—
158	C₃×C₄×C₅	—	2	1	no	—
159	C₃×C₄×C₅	#104	1	2	yes	—
160	C₅×Dic₃	—	4	1	no	—
161	C₅×Dic₃	—	4	1	no	—
162	C₅×Dic₃	—	4	1	no	—
163	C₂²×C₃×C₄	C₂×Dic₅	1	1	yes	characteristic
164	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
165	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
166	C₂×#67	—	2	1	no	—
167	C₂×D₁₂	C₂×Dic₅	1	1	yes	characteristic
168	C₂×C₄×D₃	—	2	1	no	—
169	C₂×Dic₆	C₂×Dic₅	1	1	yes	characteristic
170	C₃×M₁₆	—	10	4	no	—
171	C₂×C₃×C₈	—	10	2	no	—
172	C₃×C₄²	C₂×Dic₅	1	1	yes	characteristic
173	C₃×C₄²	C₂²⋊D₅	1	1	yes	characteristic
174	C₃×C₄²	C₂²⋊D₅	1	1	yes	characteristic
175	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
176	#217	—	2	1	no	—
177	(C₂×D₃)⋊C₄	—	2	1	no	—
178	(C₂×D₃)⋊C₄	—	2	1	no	—
179	(C₂×D₃)⋊C₄	—	2	1	no	—
180	(C₂×D₃)⋊C₄	—	2	1	no	—
181	C₄⋊Dic₃	C₂×Dic₅	1	1	yes	characteristic
182	C₄⋊Dic₃	C₂×Dic₅	1	1	yes	characteristic
183	Dic₃⋊C₄	—	2	1	no	—
184	Dic₃⋊C₄	—	2	1	no	—
185	Dic₃⋊C₄	—	2	1	no	—
186	Dic₃⋊C₄	—	2	1	no	—
187	C₄×Dic₃	—	2	1	no	—
188	C₂×C₃⋊C₈	—	10	2	no	—
189	C₂³×C₅	—	6	1	no	—
190	C₂³×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
191	C₅×Q₈	—	12	1	no	—
192	C₅×D₄	—	12	1	no	—
193	C₅×D₄	—	12	2	no	—
194	C₂×C₄×C₅	—	2	2	no	—
195	C₂×C₄×C₅	—	4	1	no	—
196	C₂×C₄×C₅	—	4	1	no	—
197	C₂×C₄×C₅	—	6	1	no	—
198	C₂×C₄×C₅	—	6	1	no	—
199	C₂×C₄×C₅	—	6	1	no	—
200	C₂×C₄×C₅	—	12	2	no	—
201	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
202	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
203	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
204	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
205	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
206	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
207	#168	—	2	4	no	—
208	#168	—	12	2	no	—
209	C₂×M₁₆	—	10	1	no	—
210	#126	—	6	1	no	—
211	#125	—	3	1	no	—
212	#124	—	3	1	no	—
213	#123	—	6	1	no	—
214	C₂²⋊Q₈	—	3	1	no	—
215	#121	—	3	1	no	—
216	C₄×Q₈	—	3	1	no	—
217	C₄×D₄	—	3	1	no	—
218	C₄×D₄	—	6	1	no	—
219	#117	—	6	1	no	—
220	C₂×C₄²	D₃×D₅	1	1	yes	characteristic
221	C₄⋊C₈	—	30	1	no	—
222	C₂²⋊C₈	—	30	1	no	—
223	#97	—	5	2	no	—
224	#97	—	15	2	no	—
225	C₄×C₈	—	10	1	no	—
226	C₂×C₃×C₅	—	4	1	no	—
227	C₂×C₃×C₅	#413	1	1	yes	characteristic
228	C₂×C₃×C₅	C₂×#104	1	1	yes	characteristic
229	C₂×C₃×C₅	#421	1	1	yes	characteristic
230	C₅×D₃	—	8	1	no	—
231	C₂³×C₃	C₂²⋊Dic₅	1	1	yes	characteristic
232	C₂²×D₃	—	2	1	no	—
233	C₂×C₃×C₄	—	2	2	no	—
234	C₂×C₃×C₄	—	4	1	no	—
235	C₂×C₃×C₄	—	4	1	no	—
236	C₂×C₃×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
237	C₂×C₃×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
238	C₂×C₃×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
239	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
240	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
241	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
242	#67	—	4	2	no	—
243	C₂×Dic₃	—	2	1	no	—
244	C₂×Dic₃	—	2	1	no	—
245	C₂×Dic₃	—	2	1	no	—
246	D₁₂	—	4	1	no	—
247	C₄×D₃	—	4	2	no	—
248	Dic₆	—	4	1	no	—
249	C₃×C₈	—	10	4	no	—
250	C₃⋊C₈	—	20	2	no	—
251	C₂²×C₅	—	2	1	no	—
252	C₂²×C₅	—	2	1	no	—
253	C₂²×C₅	—	2	1	no	—
254	C₂²×C₅	—	12	1	no	—
255	C₂²×C₅	—	12	1	no	—
256	C₂²×C₅	—	12	1	no	—
257	C₂²×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
258	C₄×C₅	—	2	1	no	—
259	C₄×C₅	—	2	1	no	—
260	C₄×C₅	—	2	1	no	—
261	C₄×C₅	—	2	1	no	—
262	C₄×C₅	—	2	1	no	—
263	C₄×C₅	—	12	1	no	—
264	C₄×C₅	—	12	1	no	—
265	C₄×C₅	—	12	1	no	—
266	C₄×C₅	#805	1	2	yes	—
267	C₂×Q₈	—	3	1	no	—
268	C₂×D₄	—	3	1	no	—
269	C₂×D₄	—	6	1	no	—
270	C₂²×C₄	—	6	1	no	—
271	C₂²×C₄	D₃×Dic₅	1	1	yes	characteristic
272	C₂²×C₄	#1201	1	1	yes	characteristic
273	C₂²×C₄	#1203	1	1	yes	characteristic
274	M₁₆	—	10	4	no	—
275	C₂×C₈	—	10	2	no	—
276	C₂×C₈	—	30	2	no	—
277	C₄⋊C₄	—	3	1	no	—
278	C₄⋊C₄	—	3	1	no	—
279	C₄⋊C₄	—	6	1	no	—
280	C₄⋊C₄	—	6	1	no	—
281	C₄⋊C₄	—	6	1	no	—
282	C₄⋊C₄	—	6	1	no	—
283	C₂²⋊C₄	—	6	1	no	—
284	C₂²⋊C₄	—	6	1	no	—
285	C₂²⋊C₄	—	6	1	no	—
286	C₂²⋊C₄	—	6	1	no	—
287	C₂²⋊C₄	—	6	1	no	—
288	C₄²	—	6	1	no	—
289	C₄²	D₃×Dic₅	1	1	yes	characteristic
290	C₄²	#1201	1	1	yes	characteristic
291	C₄²	#1203	1	1	yes	characteristic
292	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
293	C₃×C₅	#1529	1	1	yes	characteristic, complemented, Hall
294	C₂²×C₃	—	2	1	no	—
295	C₂²×C₃	—	2	1	no	—
296	C₂²×C₃	—	2	1	no	—
297	C₂²×C₃	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
298	C₂×D₃	—	4	1	no	—
299	C₂×D₃	—	4	1	no	—
300	C₂×D₃	—	4	1	no	—
301	C₃×C₄	—	2	1	no	—
302	C₃×C₄	—	2	1	no	—
303	C₃×C₄	—	2	1	no	—
304	C₃×C₄	—	2	1	no	—
305	C₃×C₄	—	2	1	no	—
306	C₃×C₄	#3969	1	2	yes	—
307	Dic₃	—	4	1	no	—
308	Dic₃	—	4	1	no	—
309	Dic₃	—	4	1	no	—
310	C₂×C₅	—	4	1	no	—
311	C₂×C₅	—	24	1	no	—
312	C₂×C₅	C₂×#805	1	1	yes	characteristic
313	C₂×C₅	#5132	1	1	yes	characteristic
314	C₂×C₅	#5139	1	1	yes	characteristic
315	C₂³	—	6	1	no	—
316	C₂³	(C₂×D₃)⋊Dic₅	1	1	yes	characteristic
317	Q₈	—	12	1	no	—
318	D₄	—	12	1	no	—
319	D₄	—	12	2	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	4	1	no	—
322	C₂×C₄	—	4	1	no	—
323	C₂×C₄	—	6	1	no	—
324	C₂×C₄	—	6	1	no	—
325	C₂×C₄	—	6	1	no	—
326	C₂×C₄	—	12	2	no	—
327	C₂×C₄	(C₂×D₃)⋊Dic₅	1	1	yes	characteristic
328	C₂×C₄	(C₂×D₃)⋊Dic₅	1	1	yes	characteristic
329	C₂×C₄	(C₂×D₃)⋊Dic₅	1	1	yes	characteristic
330	C₂×C₄	C₂×D₃×Dic₅	1	1	yes	characteristic
331	C₂×C₄	C₂×#1201	1	1	yes	characteristic
332	C₂×C₄	C₂×#1203	1	1	yes	characteristic
333	C₈	—	10	4	no	—
334	C₈	—	60	2	no	—
335	C₂×C₃	—	4	1	no	—
336	C₂×C₃	C₂×#3969	1	1	yes	characteristic
337	C₂×C₃	#65525	1	1	yes	characteristic
338	C₂×C₃	#65526	1	1	yes	characteristic
339	D₃	—	8	1	no	—
340	C₅	#69861	1	1	yes	characteristic, complemented, Sylow
341	C₂²	—	2	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	2	1	no	—
344	C₂²	—	12	1	no	—
345	C₂²	—	12	1	no	—
346	C₂²	—	12	1	no	—
347	C₂²	C₂×(C₂×D₃)⋊Dic₅	1	1	yes	characteristic
348	C₄	—	2	1	no	—
349	C₄	—	2	1	no	—
350	C₄	—	2	1	no	—
351	C₄	—	2	1	no	—
352	C₄	—	2	1	no	—
353	C₄	—	12	1	no	—
354	C₄	—	12	1	no	—
355	C₄	—	12	1	no	—
356	C₄	#90979	1	2	yes	—
357	C₃	#10602540	1	1	yes	characteristic, complemented, Sylow
358	C₂	—	4	1	no	—
359	C₂	—	24	1	no	—
360	C₂	#11748355	1	1	yes	characteristic
361	C₂	C₂×#90979	1	1	yes	characteristic
362	C₂	#11748424	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×(C₂×D₃)⋊Dic₅
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₄²	C₂×D₃×D₅
Fitting	C₂×C₃×C₄²×C₅	C₂²
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₅×#930	C₄
cosocle	C₃×C₄²×C₅	C₂³
hypercenter	C₂×C₄²	D₃×D₅
hypocenter	C₃×C₅	#1529
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
2. #11751027	C₂	semidirect
3. C₃×C₄×#681	C₂	semidirect
4. #11745850	C₂	semidirect
5. #11745847	C₂	semidirect
27. C₄×#681	D₃	semidirect
293. C₃×C₅	#1529	semidirect
340. C₅	#69861	semidirect
357. C₃	#10602540	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₅×#5479	2
#11751027	2
C₃×C₄×#681	2
#11745850	2
#11745847	2
#10602540	3
#69861	5