Subgroups — #14282900

Order 1920.

568 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	#11756236	C₂	1	1	yes	maximal, characteristic
2	C₅×#4925	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₄×C₄⋊Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
4	#11745958	C₂	1	4	yes	maximal, complemented
5	#10601582	—	3	1	no	maximal, Hall
6	C₂×#6817	—	2	1	no	—
7	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
8	#91819	—	2	1	no	—
9	#91819	C₂²	1	2	yes	—
10	#91764	—	2	1	no	—
11	#91763	C₄	1	1	yes	characteristic, complemented
12	C₄×D₆₀	—	2	1	no	—
13	C₄×D₆₀	C₄	1	2	yes	complemented
14	#91758	C₄	1	1	yes	characteristic, complemented
15	C₂×C₃×C₄⋊Dic₅	—	2	1	no	—
16	C₂×C₃×C₄×Dic₅	—	2	1	no	—
17	C₅×#830	C₂²	1	2	yes	—
18	C₅×C₄⋊(C₃⋊C₈)	—	2	1	no	—
19	C₄×C₅×C₃⋊C₈	—	2	1	no	—
20	C₃×#3963	C₂²	1	2	yes	—
21	#69469	—	5	1	no	maximal, Hall
22	#65811	—	3	1	no	—
23	C₅×#431	—	3	1	no	—
24	C₄×C₄⋊Dic₅	D₃	1	1	yes	characteristic, complemented
25	#64690	—	3	4	no	—
26	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
27	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
28	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
29	C₂×#1220	—	2	2	no	—
30	#6817	—	4	2	no	—
31	#6817	—	4	2	no	—
32	C₂×D₆₀	—	2	1	no	—
33	C₂×D₆₀	—	2	1	no	—
34	C₂×D₆₀	C₂×C₄	1	2	yes	—
35	C₂×C₄×D₁₅	—	2	2	no	—
36	C₂×Dic₃₀	—	2	1	no	—
37	C₂²×C₃×Dic₅	—	2	2	no	—
38	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
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	#6714	—	2	2	no	—
43	#6714	—	2	2	no	—
44	C₄⋊Dic₁₅	—	2	1	no	—
45	C₄⋊Dic₁₅	C₂×C₄	1	2	yes	—
46	C₂×C₅×C₃⋊C₈	—	2	2	no	—
47	C₃×C₄⋊Dic₅	—	2	4	no	—
48	C₃×C₄×Dic₅	—	4	2	no	—
49	#5477	—	5	1	no	—
50	C₃×C₄×C₄⋊C₄	—	5	1	no	—
51	#4925	D₅	1	1	yes	characteristic, complemented
52	#4451	—	5	4	no	—
53	C₂×#709	—	6	1	no	—
54	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
55	#4076	—	3	2	no	—
56	#4076	—	6	1	no	—
57	C₂×C₄⋊Dic₅	—	2	1	no	—
58	C₂×C₄×Dic₅	—	2	1	no	—
59	#4021	—	6	1	no	—
60	#4020	—	3	1	no	—
61	C₄×D₂₀	—	3	2	no	—
62	C₄×D₂₀	—	6	1	no	—
63	#4015	—	3	1	no	—
64	C₅×C₄⋊C₈	—	6	1	no	—
65	C₅×C₂²⋊C₈	—	3	2	no	—
66	C₄×C₅×C₈	—	6	1	no	—
67	#3963	C₂×D₃	1	2	yes	—
68	#1502	—	15	1	no	Sylow
69	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
70	C₂²×D₁₅	—	2	2	no	—
71	C₂×C₃×C₄×C₅	—	2	2	no	—
72	C₂×C₃×C₄×C₅	—	4	1	no	—
73	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₃×C₄×C₅	D₈	1	2	yes	—
78	C₂×C₃×C₄×C₅	SD₁₆	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	#1220	—	4	2	no	—
82	#1220	—	4	2	no	—
83	C₂×Dic₁₅	—	2	2	no	—
84	D₆₀	—	4	2	no	—
85	D₆₀	—	4	2	no	—
86	D₆₀	—	4	2	no	—
87	C₄×D₁₅	—	4	4	no	—
88	Dic₃₀	—	4	2	no	—
89	C₂×C₃×Dic₅	—	2	4	no	—
90	C₂×C₃×Dic₅	—	4	2	no	—
91	C₂×C₃×Dic₅	—	4	2	no	—
92	C₅×C₃⋊C₈	—	4	2	no	—
93	C₂×#235	—	10	1	no	—
94	C₂×C₃×C₄⋊C₄	—	10	1	no	—
95	C₂×C₃×C₄²	—	10	1	no	—
96	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
97	#930	—	5	2	no	—
98	#930	—	10	1	no	—
99	#875	—	10	1	no	—
100	#874	—	5	1	no	—
101	C₄×D₁₂	—	5	2	no	—
102	C₄×D₁₂	—	10	1	no	—
103	#869	—	5	1	no	—
104	C₃×#95	—	5	2	no	—
105	#830	C₂×D₅	1	2	yes	—
106	C₄⋊(C₃⋊C₈)	—	2	1	no	—
107	C₄×C₃⋊C₈	—	2	1	no	—
108	C₂²×C₄×C₅	#67	1	1	yes	characteristic
109	C₂²×C₄×C₅	#67	1	1	yes	characteristic
110	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
111	C₂×C₂²⋊D₅	—	6	2	no	—
112	C₂²×Dic₅	—	2	2	no	—
113	#709	—	12	2	no	—
114	#709	—	12	2	no	—
115	C₂×D₂₀	—	3	2	no	—
116	C₂×D₂₀	—	6	1	no	—
117	C₂×D₂₀	—	6	1	no	—
118	C₂×C₄×D₅	—	6	2	no	—
119	C₂×Dic₁₀	—	6	1	no	—
120	C₂×C₅×C₈	—	6	2	no	—
121	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
122	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
123	C₄²×C₅	D₁₂	1	1	yes	characteristic
124	C₄²×C₅	D₁₂	1	1	yes	characteristic
125	#685	—	6	2	no	—
126	#685	—	6	2	no	—
127	C₄⋊Dic₅	—	2	4	no	—
128	C₄⋊Dic₅	—	3	2	no	—
129	C₄⋊Dic₅	—	6	1	no	—
130	C₄×Dic₅	—	4	2	no	—
131	#532	—	15	1	no	—
132	#431	—	3	1	no	—
133	C₄×C₄⋊C₄	—	5	1	no	—
134	#327	—	15	4	no	—
135	C₂²×C₃×C₅	—	2	1	no	—
136	C₂²×C₃×C₅	—	2	1	no	—
137	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
138	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
139	C₂²×C₃×C₅	C₂×SD₁₆	1	1	yes	characteristic
140	C₂×D₁₅	—	4	2	no	—
141	C₂×D₁₅	—	4	2	no	—
142	C₂×D₁₅	—	4	2	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	#102	1	2	yes	—
147	C₃×C₄×C₅	#102	1	2	yes	—
148	C₃×C₄×C₅	#104	1	2	yes	—
149	Dic₁₅	—	4	2	no	—
150	C₃×Dic₅	—	4	4	no	—
151	C₂²×C₃×C₄	—	10	2	no	—
152	C₂²×C₃×C₄	D₂₀	1	1	yes	characteristic
153	C₂²×C₃×C₄	D₂₀	1	1	yes	characteristic
154	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
155	C₂×#67	—	10	2	no	—
156	#235	—	20	2	no	—
157	#235	—	20	2	no	—
158	C₂×D₁₂	—	5	2	no	—
159	C₂×D₁₂	—	10	1	no	—
160	C₂×D₁₂	—	10	1	no	—
161	C₂×C₄×D₃	—	10	2	no	—
162	C₂×Dic₆	—	10	1	no	—
163	C₃×C₄⋊C₄	—	10	4	no	—
164	C₃×C₄²	—	20	2	no	—
165	C₃×C₄²	C₄×D₅	1	1	yes	characteristic
166	C₃×C₄²	C₄×D₅	1	1	yes	characteristic
167	C₃×C₄²	C₂²⋊D₅	1	1	yes	characteristic
168	C₃×C₄²	C₂²⋊D₅	1	1	yes	characteristic
169	(C₂×D₃)⋊C₄	—	10	2	no	—
170	(C₂×D₃)⋊C₄	—	10	2	no	—
171	C₄⋊Dic₃	—	5	2	no	—
172	C₄⋊Dic₃	—	10	1	no	—
173	C₂×C₃⋊C₈	—	2	2	no	—
174	C₂³×C₅	C₂×#67	1	1	yes	characteristic
175	C₂²×D₅	—	6	2	no	—
176	C₂×C₄×C₅	—	2	2	no	—
177	C₂×C₄×C₅	—	4	1	no	—
178	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
179	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
180	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
181	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
182	C₂×C₄×C₅	#213	1	2	yes	—
183	C₂×C₄×C₅	#215	1	2	yes	—
184	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
185	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
186	C₂²⋊D₅	—	12	2	no	—
187	C₂²⋊D₅	—	12	2	no	—
188	C₂×Dic₅	—	2	4	no	—
189	C₂×Dic₅	—	4	2	no	—
190	C₂×Dic₅	—	4	2	no	—
191	C₂×Dic₅	—	6	2	no	—
192	D₂₀	—	12	2	no	—
193	D₂₀	—	12	2	no	—
194	D₂₀	—	12	2	no	—
195	C₄×D₅	—	12	4	no	—
196	Dic₁₀	—	12	2	no	—
197	C₅×C₈	—	12	2	no	—
198	C₂×#41	—	30	1	no	—
199	C₄⋊Q₈	—	15	1	no	—
200	#127	—	15	1	no	—
201	#124	—	30	1	no	—
202	#121	—	15	2	no	—
203	#121	—	30	1	no	—
204	C₄×D₄	—	15	2	no	—
205	C₄×D₄	—	30	1	no	—
206	C₂×C₄⋊C₄	—	10	1	no	—
207	C₂×C₄²	—	10	1	no	—
208	C₂×C₄²	D₃×D₅	1	1	yes	characteristic
209	C₄⋊C₈	—	6	1	no	—
210	C₂²⋊C₈	—	3	2	no	—
211	C₄×C₈	—	6	1	no	—
212	#95	—	5	2	no	—
213	C₂×C₃×C₅	—	2	1	no	—
214	C₂×C₃×C₅	—	2	1	no	—
215	C₂×C₃×C₅	#410	1	1	yes	characteristic
216	C₂×C₃×C₅	C₂×#102	1	1	yes	characteristic
217	C₂×C₃×C₅	C₂×#104	1	1	yes	characteristic
218	D₁₅	—	8	2	no	—
219	C₂³×C₃	C₂×D₂₀	1	1	yes	characteristic
220	C₂²×D₃	—	10	2	no	—
221	C₂×C₃×C₄	—	2	2	no	—
222	C₂×C₃×C₄	—	4	1	no	—
223	C₂×C₃×C₄	—	10	4	no	—
224	C₂×C₃×C₄	—	20	2	no	—
225	C₂×C₃×C₄	—	20	2	no	—
226	C₂×C₃×C₄	#677	1	2	yes	—
227	C₂×C₃×C₄	D₄₀	1	2	yes	—
228	C₂×C₃×C₄	#685	1	1	yes	characteristic
229	C₂×C₃×C₄	#685	1	1	yes	characteristic
230	C₂×C₃×C₄	#685	1	1	yes	characteristic
231	C₂×C₃×C₄	#685	1	1	yes	characteristic
232	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
233	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
234	#67	—	20	2	no	—
235	#67	—	20	2	no	—
236	C₂×Dic₃	—	10	2	no	—
237	D₁₂	—	20	2	no	—
238	D₁₂	—	20	2	no	—
239	D₁₂	—	20	2	no	—
240	C₄×D₃	—	20	4	no	—
241	Dic₆	—	20	2	no	—
242	C₃⋊C₈	—	4	2	no	—
243	C₂²×C₅	—	2	1	no	—
244	C₂²×C₅	—	2	1	no	—
245	C₂²×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
246	C₂²×C₅	C₂×#213	1	1	yes	characteristic
247	C₂²×C₅	C₂×#215	1	1	yes	characteristic
248	C₂×D₅	—	12	2	no	—
249	C₂×D₅	—	12	2	no	—
250	C₂×D₅	—	12	2	no	—
251	C₄×C₅	—	2	1	no	—
252	C₄×C₅	—	2	1	no	—
253	C₄×C₅	—	2	1	no	—
254	C₄×C₅	#805	1	2	yes	—
255	C₄×C₅	#809	1	2	yes	—
256	C₄×C₅	#809	1	2	yes	—
257	Dic₅	—	4	4	no	—
258	Dic₅	—	12	2	no	—
259	#41	—	60	2	no	—
260	#41	—	60	2	no	—
261	C₂×Q₈	—	30	1	no	—
262	C₂×D₄	—	15	2	no	—
263	C₂×D₄	—	30	2	no	—
264	C₂×D₄	—	30	1	no	—
265	C₂×D₄	—	30	1	no	—
266	C₂²×C₄	—	10	2	no	—
267	C₂²×C₄	—	30	2	no	—
268	C₂²×C₄	#1202	1	1	yes	characteristic
269	C₂²×C₄	#1202	1	1	yes	characteristic
270	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
271	C₂×C₈	—	6	2	no	—
272	C₄⋊C₄	—	10	4	no	—
273	C₄⋊C₄	—	15	2	no	—
274	C₄⋊C₄	—	30	1	no	—
275	C₂²⋊C₄	—	30	2	no	—
276	C₂²⋊C₄	—	30	2	no	—
277	C₄²	—	20	2	no	—
278	C₄²	#1200	1	1	yes	characteristic
279	C₄²	#1200	1	1	yes	characteristic
280	C₄²	#1203	1	1	yes	characteristic
281	C₄²	#1203	1	1	yes	characteristic
282	C₃×C₅	#1502	1	1	yes	characteristic, complemented, Hall
283	C₂²×C₃	—	2	1	no	—
284	C₂²×C₃	—	2	1	no	—
285	C₂²×C₃	C₂×#677	1	1	yes	characteristic
286	C₂²×C₃	C₂×D₄₀	1	1	yes	characteristic
287	C₂²×C₃	C₂×#685	1	1	yes	characteristic
288	C₂×D₃	—	20	2	no	—
289	C₂×D₃	—	20	2	no	—
290	C₂×D₃	—	20	2	no	—
291	C₃×C₄	—	2	1	no	—
292	C₃×C₄	—	2	1	no	—
293	C₃×C₄	—	2	1	no	—
294	C₃×C₄	—	20	4	no	—
295	C₃×C₄	#3953	1	2	yes	—
296	C₃×C₄	#3953	1	2	yes	—
297	C₃×C₄	#3957	1	2	yes	—
298	Dic₃	—	20	2	no	—
299	C₂×C₅	—	2	1	no	—
300	C₂×C₅	—	2	1	no	—
301	C₂×C₅	C₂×#805	1	1	yes	characteristic
302	C₂×C₅	C₂×#809	1	1	yes	characteristic
303	C₂×C₅	#5006	1	1	yes	characteristic
304	D₅	—	24	2	no	—
305	C₂³	—	30	2	no	—
306	C₂³	C₂×#1202	1	1	yes	characteristic
307	Q₈	—	60	2	no	—
308	D₄	—	60	2	no	—
309	D₄	—	60	2	no	—
310	D₄	—	60	2	no	—
311	D₄	—	60	2	no	—
312	D₄	—	60	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	4	1	no	—
315	C₂×C₄	—	10	4	no	—
316	C₂×C₄	—	20	2	no	—
317	C₂×C₄	—	20	2	no	—
318	C₂×C₄	—	30	2	no	—
319	C₂×C₄	—	60	4	no	—
320	C₂×C₄	#6653	1	2	yes	—
321	C₂×C₄	#6658	1	2	yes	—
322	C₂×C₄	#6667	1	1	yes	characteristic
323	C₂×C₄	#6667	1	1	yes	characteristic
324	C₂×C₄	#6667	1	1	yes	characteristic
325	C₂×C₄	#6667	1	1	yes	characteristic
326	C₂×C₄	C₂×#1200	1	1	yes	characteristic
327	C₂×C₄	C₂×#1203	1	1	yes	characteristic
328	C₈	—	12	2	no	—
329	C₂×C₃	—	2	1	no	—
330	C₂×C₃	—	2	1	no	—
331	C₂×C₃	#65295	1	1	yes	characteristic
332	C₂×C₃	C₂×#3953	1	1	yes	characteristic
333	C₂×C₃	C₂×#3957	1	1	yes	characteristic
334	D₃	—	40	2	no	—
335	C₅	#69469	1	1	yes	characteristic, complemented, Sylow
336	C₂²	—	2	1	no	—
337	C₂²	—	2	1	no	—
338	C₂²	—	60	2	no	—
339	C₂²	—	60	2	no	—
340	C₂²	—	60	2	no	—
341	C₂²	C₂×#6653	1	1	yes	characteristic
342	C₂²	C₂×#6658	1	1	yes	characteristic
343	C₂²	C₂×#6667	1	1	yes	characteristic
344	C₄	—	2	1	no	—
345	C₄	—	2	1	no	—
346	C₄	—	2	1	no	—
347	C₄	—	20	4	no	—
348	C₄	—	60	2	no	—
349	C₄	#90968	1	2	yes	—
350	C₄	#90968	1	2	yes	—
351	C₄	#90981	1	2	yes	—
352	C₃	#10601582	1	1	yes	characteristic, complemented, Sylow
353	C₂	—	2	1	no	—
354	C₂	—	2	1	no	—
355	C₂	—	120	2	no	—
356	C₂	C₂×#90968	1	1	yes	characteristic
357	C₂	C₂×#90981	1	1	yes	characteristic
358	C₂	#11748626	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#6667
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₂²×C₄	C₂×D₃×D₅
Fitting	C₂×C₃×C₄²×C₅	C₂²
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	#11756236	C₂
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	C₂×C₄²	D₃×D₅
hypocenter	C₃×C₅	#1502
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂²×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. C₅×#4925	C₂	semidirect
3. C₃×C₄×C₄⋊Dic₅	C₂	semidirect
4. #11745958	C₂	semidirect
11. #91763	C₄	semidirect
13. C₄×D₆₀	C₄	semidirect
14. #91758	C₄	semidirect
24. C₄×C₄⋊Dic₅	D₃	semidirect
51. #4925	D₅	semidirect
282. C₃×C₅	#1502	semidirect
335. C₅	#69469	semidirect
352. C₃	#10601582	semidirect

Maximal subgroups

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

subgroup	index
#11756236	2
C₅×#4925	2
C₃×C₄×C₄⋊Dic₅	2
#11745958	2
#10601582	3
#69469	5