Subgroups — #14284590

Order 1920.

512 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11752672	C₂	1	1	yes	maximal, characteristic
2	C₃×#65122	C₂	1	1	yes	maximal, characteristic, complemented
3	D₅×C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
4	#11746307	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#4485	C₂	1	1	yes	maximal, characteristic, complemented
6	#11745859	C₂	1	1	yes	maximal, characteristic, complemented
7	#11745853	C₂	1	1	yes	maximal, characteristic, complemented
8	#10599174	—	3	1	no	maximal, Hall
9	C₂×#6772	—	2	1	no	—
10	#91758	C₂²	1	1	yes	characteristic
11	C₅×#874	C₂²	1	1	yes	characteristic
12	C₃×C₄²×D₅	C₂²	1	1	yes	characteristic
13	#91449	—	2	1	no	—
14	#91449	C₄	1	2	yes	—
15	Dic₅×D₁₂	—	2	1	no	—
16	Dic₅×D₁₂	C₄	1	2	yes	—
17	#91373	—	2	1	no	—
18	C₂×D₅×C₃⋊C₈	—	2	1	no	—
19	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
20	C₄⋊(C₃⋊#168)	C₂²	1	1	yes	characteristic
21	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
22	C₃×#3952	—	2	1	no	—
23	C₃×C₈×Dic₅	—	2	1	no	—
24	C₃×C₄⋊#168	C₂²	1	1	yes	characteristic
25	#90982	—	2	1	no	—
26	#70772	—	5	1	no	maximal, Hall
27	#66050	—	3	1	no	—
28	#65122	D₃	1	1	yes	characteristic, complemented
29	D₅×C₄⋊C₈	—	3	1	no	—
30	C₅×#331	—	3	1	no	—
31	#64764	—	3	1	no	—
32	#64706	—	3	1	no	—
33	#64702	—	3	1	no	—
34	C₂×Dic₃₀	—	2	1	no	—
35	C₂×C₅×D₁₂	—	2	1	no	—
36	C₂×C₅×D₁₂	—	2	1	no	—
37	C₂×C₅×D₁₂	C₂×C₄	1	2	yes	—
38	C₂×C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
39	C₂×C₃×C₄×D₅	D₄	1	1	yes	characteristic
40	C₂×C₃×C₄×D₅	D₄	1	1	yes	characteristic
41	C₂×#1201	—	2	2	no	—
42	C₂×D₃×Dic₅	—	2	2	no	—
43	#6772	—	4	2	no	—
44	#6772	—	4	2	no	—
45	C₂×C₃×C₅×C₈	—	2	1	no	—
46	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
47	C₄⋊Dic₁₅	—	2	1	no	—
48	C₄⋊Dic₁₅	C₂×C₄	1	2	yes	—
49	C₂×C₃⋊#168	—	2	1	no	—
50	C₂×C₅×C₃⋊C₈	—	2	1	no	—
51	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
52	C₃×C₄×Dic₅	D₄	1	1	yes	characteristic
53	C₃×C₄×Dic₅	D₄	1	1	yes	characteristic
54	C₂×C₃×#168	—	2	1	no	—
55	(C₂×D₃)⋊Dic₅	—	2	2	no	—
56	(C₂×D₃)⋊Dic₅	—	2	2	no	—
57	D₅×C₃⋊C₈	—	4	2	no	—
58	#5477	—	5	1	no	—
59	C₃×#431	—	5	1	no	—
60	C₂×C₄⋊(C₃⋊C₈)	—	5	1	no	—
61	#4486	—	5	1	no	—
62	#4486	—	5	1	no	—
63	#4485	—	5	1	no	—
64	#4485	D₅	1	1	yes	characteristic, complemented
65	C₂×#711	—	6	1	no	—
66	C₅×#127	—	3	1	no	—
67	#4084	—	3	2	no	—
68	#4084	—	6	1	no	—
69	#4082	—	6	1	no	—
70	D₄×Dic₅	—	3	2	no	—
71	D₄×Dic₅	—	6	1	no	—
72	C₂×C₈×D₅	—	6	1	no	—
73	C₄²×D₅	C₂×D₃	1	1	yes	characteristic
74	#4015	—	3	1	no	—
75	C₅×C₄⋊C₈	—	3	1	no	—
76	C₅×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
77	#3952	—	2	1	no	—
78	#3946	—	6	1	no	—
79	C₈×Dic₅	—	2	1	no	—
80	C₄⋊#168	—	3	1	no	—
81	C₄⋊#168	C₂×D₃	1	1	yes	characteristic
82	#1540	—	15	1	no	Sylow
83	C₂²×C₅×D₃	—	2	2	no	—
84	C₂²×C₃×D₅	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
87	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
88	C₂×Dic₁₅	—	2	2	no	—
89	Dic₃₀	—	4	1	no	—
90	Dic₃₀	—	4	1	no	—
91	C₅×D₁₂	—	4	2	no	—
92	C₅×D₁₂	—	4	1	no	—
93	C₅×D₁₂	—	4	1	no	—
94	C₅×D₁₂	—	4	1	no	—
95	C₅×D₁₂	—	4	1	no	—
96	C₂×C₃×Dic₅	C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₃×Dic₅	C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₃×Dic₅	C₂²⋊C₄	1	1	yes	characteristic
99	C₃×C₄×D₅	—	2	1	no	—
100	C₃×C₄×D₅	—	2	1	no	—
101	C₃×C₄×D₅	—	4	1	no	—
102	C₃×C₄×D₅	D₈	1	2	yes	—
103	C₃×C₄×D₅	SD₁₆	1	2	yes	—
104	#1201	—	4	4	no	—
105	D₃×Dic₅	—	4	4	no	—
106	C₃×C₅×C₈	—	4	1	no	—
107	C₃⋊#168	—	4	1	no	—
108	C₃×#168	—	4	1	no	—
109	C₅×C₃⋊C₈	—	4	1	no	—
110	C₂×#235	—	10	1	no	—
111	C₂×C₃×C₄²	—	5	1	no	—
112	#930	—	5	2	no	—
113	#930	—	10	1	no	—
114	C₂²×C₃⋊C₈	—	10	1	no	—
115	#875	—	10	1	no	—
116	#874	C₂×D₅	1	1	yes	characteristic
117	C₄×D₁₂	—	5	2	no	—
118	C₄×D₁₂	—	10	1	no	—
119	#869	—	5	1	no	—
120	C₃×C₄⋊C₈	—	5	1	no	—
121	C₃×C₄⋊C₈	C₂×D₅	1	1	yes	characteristic
122	C₃×C₂²⋊C₈	—	10	1	no	—
123	C₃×C₄×C₈	—	10	1	no	—
124	C₄⋊(C₃⋊C₈)	—	5	1	no	—
125	C₄⋊(C₃⋊C₈)	—	10	1	no	—
126	C₄⋊(C₃⋊C₈)	C₂×D₅	1	1	yes	characteristic
127	C₂×C₅×D₄	—	3	2	no	—
128	C₂×C₅×D₄	—	6	1	no	—
129	C₂×C₅×D₄	—	6	1	no	—
130	C₂×C₂²⋊D₅	—	6	2	no	—
131	C₂²×Dic₅	—	6	2	no	—
132	#711	—	12	2	no	—
133	#711	—	12	2	no	—
134	C₂×C₄×D₅	C₄×D₃	1	1	yes	characteristic
135	C₂×C₄×D₅	D₁₂	1	1	yes	characteristic
136	C₂×C₄×D₅	#67	1	1	yes	characteristic
137	C₂×Dic₁₀	—	6	1	no	—
138	C₂×C₅×C₈	—	2	1	no	—
139	C₂×C₅×C₈	—	6	1	no	—
140	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
141	C₂²⋊Dic₅	—	6	2	no	—
142	C₂²⋊Dic₅	—	6	2	no	—
143	C₄⋊Dic₅	—	3	2	no	—
144	C₄⋊Dic₅	—	6	1	no	—
145	C₄×Dic₅	C₄×D₃	1	1	yes	characteristic
146	C₄×Dic₅	D₁₂	1	1	yes	characteristic
147	C₄×Dic₅	#67	1	1	yes	characteristic
148	C₂×#168	—	2	1	no	—
149	C₂×#168	—	6	1	no	—
150	C₈×D₅	—	12	2	no	—
151	#532	—	15	1	no	—
152	#431	—	5	1	no	—
153	C₂×C₄⋊C₈	—	15	1	no	—
154	#332	—	15	1	no	—
155	#332	—	15	1	no	—
156	#331	—	3	1	no	—
157	#331	—	15	1	no	—
158	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
159	C₂×C₅×D₃	—	4	2	no	—
160	C₂×C₅×D₃	—	4	2	no	—
161	C₂×C₅×D₃	—	4	2	no	—
162	C₂×C₃×D₅	—	2	1	no	—
163	C₂×C₃×D₅	—	2	1	no	—
164	C₂×C₃×D₅	#102	1	2	yes	—
165	C₃×C₄×C₅	—	2	1	no	—
166	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
167	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
168	C₃×C₄×C₅	#135	1	1	yes	characteristic
169	C₃×C₄×C₅	#135	1	1	yes	characteristic
170	Dic₁₅	—	4	2	no	—
171	C₃×Dic₅	—	2	1	no	—
172	C₃×Dic₅	—	2	1	no	—
173	C₃×Dic₅	#102	1	2	yes	—
174	C₂²×C₃×C₄	—	5	1	no	—
175	C₂²×C₃×C₄	—	5	1	no	—
176	C₂²×C₃×C₄	—	5	1	no	—
177	C₂×#67	—	10	2	no	—
178	#235	—	20	2	no	—
179	#235	—	20	2	no	—
180	C₂×D₁₂	—	2	1	no	—
181	C₂×D₁₂	—	2	1	no	—
182	C₂×D₁₂	C₄×D₅	1	2	yes	—
183	C₂×C₄×D₃	—	10	2	no	—
184	C₂×Dic₆	—	10	1	no	—
185	C₂×C₃×C₈	—	2	1	no	—
186	C₂×C₃×C₈	—	10	1	no	—
187	C₃×C₄²	—	5	1	no	—
188	C₃×C₄²	—	5	1	no	—
189	C₃×C₄²	—	5	1	no	—
190	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
191	(C₂×D₃)⋊C₄	—	10	2	no	—
192	(C₂×D₃)⋊C₄	—	10	2	no	—
193	C₄⋊Dic₃	—	5	2	no	—
194	C₄⋊Dic₃	—	10	1	no	—
195	C₂×C₃⋊C₈	—	2	1	no	—
196	C₂×C₃⋊C₈	—	10	1	no	—
197	C₂×C₃⋊C₈	—	20	2	no	—
198	C₂³×C₅	—	6	2	no	—
199	C₂²×D₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
200	C₅×D₄	—	12	2	no	—
201	C₅×D₄	—	12	1	no	—
202	C₅×D₄	—	12	1	no	—
203	C₅×D₄	—	12	1	no	—
204	C₅×D₄	—	12	1	no	—
205	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
206	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
207	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
208	C₂²⋊D₅	—	12	4	no	—
209	C₂×Dic₅	—	6	2	no	—
210	C₂×Dic₅	—	12	4	no	—
211	C₂×Dic₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
212	C₂×Dic₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
213	C₂×Dic₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
214	C₄×D₅	—	2	1	no	—
215	C₄×D₅	—	2	1	no	—
216	C₄×D₅	—	4	1	no	—
217	C₄×D₅	#204	1	2	yes	—
218	C₄×D₅	D₂₄	1	2	yes	—
219	Dic₁₀	—	12	1	no	—
220	Dic₁₀	—	12	1	no	—
221	C₅×C₈	—	4	1	no	—
222	C₅×C₈	—	12	1	no	—
223	#168	—	4	1	no	—
224	#168	—	12	1	no	—
225	C₂×#41	—	30	1	no	—
226	C₂²×C₈	—	30	1	no	—
227	C₄⋊Q₈	—	15	1	no	—
228	#127	—	3	1	no	—
229	#124	—	30	1	no	—
230	#121	—	15	2	no	—
231	#121	—	30	1	no	—
232	C₄×D₄	—	15	2	no	—
233	C₄×D₄	—	30	1	no	—
234	C₂×C₄²	—	5	1	no	—
235	C₄⋊C₈	—	3	1	no	—
236	C₄⋊C₈	—	5	1	no	—
237	C₄⋊C₈	—	15	1	no	—
238	C₄⋊C₈	—	30	1	no	—
239	C₄⋊C₈	D₃×D₅	1	1	yes	characteristic
240	C₂²⋊C₈	—	10	1	no	—
241	C₄×C₈	—	10	1	no	—
242	C₂×C₃×C₅	#413	1	1	yes	characteristic
243	C₂×C₃×C₅	C₂×#102	1	1	yes	characteristic
244	C₂×C₃×C₅	#416	1	1	yes	characteristic
245	C₃×D₅	—	2	2	no	—
246	C₅×D₃	—	8	2	no	—
247	C₂³×C₃	—	5	1	no	—
248	C₂²×D₃	—	2	2	no	—
249	C₂×C₃×C₄	—	5	1	no	—
250	C₂×C₃×C₄	—	5	2	no	—
251	C₂×C₃×C₄	—	5	2	no	—
252	C₂×C₃×C₄	—	5	1	no	—
253	C₂×C₃×C₄	—	5	1	no	—
254	C₂×C₃×C₄	—	10	1	no	—
255	C₂×C₃×C₄	—	10	1	no	—
256	C₂×C₃×C₄	—	20	1	no	—
257	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
258	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
259	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
260	#67	—	20	4	no	—
261	C₂×Dic₃	—	10	2	no	—
262	D₁₂	—	4	2	no	—
263	D₁₂	—	4	1	no	—
264	D₁₂	—	4	1	no	—
265	D₁₂	—	4	1	no	—
266	D₁₂	—	4	1	no	—
267	C₄×D₃	—	20	4	no	—
268	Dic₆	—	20	1	no	—
269	Dic₆	—	20	1	no	—
270	C₃×C₈	—	4	1	no	—
271	C₃×C₈	—	20	1	no	—
272	C₃⋊C₈	—	4	1	no	—
273	C₃⋊C₈	—	20	1	no	—
274	C₂²×C₅	—	12	2	no	—
275	C₂²×C₅	—	12	2	no	—
276	C₂²×C₅	—	12	2	no	—
277	C₂²×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
278	C₂×D₅	—	2	1	no	—
279	C₂×D₅	—	2	1	no	—
280	C₂×D₅	#821	1	2	yes	—
281	C₄×C₅	—	2	1	no	—
282	C₄×C₅	C₂×#204	1	1	yes	characteristic
283	C₄×C₅	C₂×D₂₄	1	1	yes	characteristic
284	C₄×C₅	#950	1	1	yes	characteristic
285	C₄×C₅	#950	1	1	yes	characteristic
286	Dic₅	—	2	1	no	—
287	Dic₅	—	2	1	no	—
288	Dic₅	—	12	2	no	—
289	Dic₅	#821	1	2	yes	—
290	#41	—	60	2	no	—
291	#41	—	60	2	no	—
292	C₂×Q₈	—	30	1	no	—
293	C₂×D₄	—	3	2	no	—
294	C₂×D₄	—	6	1	no	—
295	C₂×D₄	—	6	1	no	—
296	C₂×D₄	—	30	2	no	—
297	C₂²×C₄	—	5	1	no	—
298	C₂²×C₄	—	5	1	no	—
299	C₂²×C₄	—	5	1	no	—
300	C₂²×C₄	—	30	2	no	—
301	C₂×C₈	—	2	1	no	—
302	C₂×C₈	—	6	1	no	—
303	C₂×C₈	—	10	1	no	—
304	C₂×C₈	—	30	1	no	—
305	C₂×C₈	—	60	2	no	—
306	C₄⋊C₄	—	15	2	no	—
307	C₄⋊C₄	—	30	1	no	—
308	C₂²⋊C₄	—	30	2	no	—
309	C₂²⋊C₄	—	30	2	no	—
310	C₄²	—	5	1	no	—
311	C₄²	—	5	1	no	—
312	C₄²	—	5	1	no	—
313	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
314	C₃×C₅	#1540	1	1	yes	characteristic, complemented, Hall
315	C₂²×C₃	—	5	2	no	—
316	C₂²×C₃	—	10	1	no	—
317	C₂²×C₃	—	10	1	no	—
318	C₂²×C₃	D₅×C₂²⋊C₄	1	1	yes	characteristic
319	C₂×D₃	—	4	2	no	—
320	C₂×D₃	—	4	2	no	—
321	C₂×D₃	—	4	2	no	—
322	C₃×C₄	—	2	1	no	—
323	C₃×C₄	—	5	2	no	—
324	C₃×C₄	—	10	1	no	—
325	C₃×C₄	—	10	1	no	—
326	C₃×C₄	D₅×D₈	1	1	yes	characteristic
327	C₃×C₄	#4058	1	1	yes	characteristic
328	C₃×C₄	D₅×SD₁₆	1	1	yes	characteristic
329	C₃×C₄	#4062	1	1	yes	characteristic
330	Dic₃	—	20	2	no	—
331	C₂×C₅	—	24	2	no	—
332	C₂×C₅	#5002	1	1	yes	characteristic
333	C₂×C₅	C₂×#821	1	1	yes	characteristic
334	C₂×C₅	#5132	1	1	yes	characteristic
335	D₅	—	2	2	no	—
336	C₂³	—	5	1	no	—
337	C₂³	—	6	2	no	—
338	Q₈	—	60	1	no	—
339	Q₈	—	60	1	no	—
340	D₄	—	12	2	no	—
341	D₄	—	12	1	no	—
342	D₄	—	12	1	no	—
343	D₄	—	12	1	no	—
344	D₄	—	12	1	no	—
345	D₄	—	60	4	no	—
346	C₂×C₄	—	5	1	no	—
347	C₂×C₄	—	5	2	no	—
348	C₂×C₄	—	5	2	no	—
349	C₂×C₄	—	5	1	no	—
350	C₂×C₄	—	5	1	no	—
351	C₂×C₄	—	10	1	no	—
352	C₂×C₄	—	10	1	no	—
353	C₂×C₄	—	20	1	no	—
354	C₂×C₄	—	30	2	no	—
355	C₂×C₄	—	60	4	no	—
356	C₂×C₄	C₄×D₃×D₅	1	1	yes	characteristic
357	C₂×C₄	D₅×D₁₂	1	1	yes	characteristic
358	C₂×C₄	D₅×#67	1	1	yes	characteristic
359	C₈	—	4	1	no	—
360	C₈	—	12	1	no	—
361	C₈	—	20	1	no	—
362	C₈	—	60	1	no	—
363	C₂×C₃	—	10	2	no	—
364	C₂×C₃	#65034	1	1	yes	characteristic
365	C₂×C₃	D₅×#102	1	1	yes	characteristic
366	C₂×C₃	#65061	1	1	yes	characteristic
367	D₃	—	8	2	no	—
368	C₅	#70772	1	1	yes	characteristic, complemented, Sylow
369	C₂²	—	5	2	no	—
370	C₂²	—	10	1	no	—
371	C₂²	—	10	1	no	—
372	C₂²	—	12	2	no	—
373	C₂²	—	12	2	no	—
374	C₂²	—	12	2	no	—
375	C₂²	D₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
376	C₄	—	2	1	no	—
377	C₄	—	5	2	no	—
378	C₄	—	10	1	no	—
379	C₄	—	10	1	no	—
380	C₄	—	60	2	no	—
381	C₄	D₅×#204	1	1	yes	characteristic
382	C₄	D₅×D₂₄	1	1	yes	characteristic
383	C₄	#91490	1	1	yes	characteristic
384	C₄	#91514	1	1	yes	characteristic
385	C₃	#10599174	1	1	yes	characteristic, complemented, Sylow
386	C₂	—	10	2	no	—
387	C₂	—	24	2	no	—
388	C₂	#11747499	1	1	yes	characteristic
389	C₂	D₅×#821	1	1	yes	characteristic
390	C₂	#11748230	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₅×(C₂×D₃)⋊C₄
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₄²	C₂×D₃×D₅
Fitting	C₃×C₅×C₄⋊C₈	C₂²
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	#11752672	C₂
cosocle	C₃×C₄²×C₅	C₂³
hypercenter	C₄⋊C₈	D₃×D₅
hypocenter	C₃×C₅	#1540
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
2. C₃×#65122	C₂	semidirect
3. D₅×C₄⋊(C₃⋊C₈)	C₂	semidirect
4. #11746307	C₂	semidirect
5. C₅×#4485	C₂	semidirect
6. #11745859	C₂	semidirect
7. #11745853	C₂	semidirect
28. #65122	D₃	semidirect
64. #4485	D₅	semidirect
314. C₃×C₅	#1540	semidirect
368. C₅	#70772	semidirect
385. C₃	#10599174	semidirect

Maximal subgroups

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

subgroup	index
#11752672	2
C₃×#65122	2
D₅×C₄⋊(C₃⋊C₈)	2
#11746307	2
C₅×#4485	2
#11745859	2
#11745853	2
#10599174	3
#70772	5