Subgroups — #14282781

Order 1920.

476 conjugacy classes of subgroups.

The subgroup lattice has 376 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	#11751033	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#65214	C₂	1	1	yes	maximal, characteristic, complemented
4	#11745963	C₂	1	2	yes	maximal, complemented
5	#11745957	C₂	1	2	yes	maximal
6	#10602547	—	3	1	no	maximal, Hall
7	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₁₅	—	2	1	no	—
9	C₅×#930	C₂²	1	1	yes	characteristic
10	C₅×#929	—	2	1	no	—
11	C₄×C₅×#67	—	2	1	no	—
12	C₅×#924	C₂²	1	1	yes	characteristic
13	C₅×#876	—	2	1	no	—
14	C₅×#875	C₄	1	1	yes	characteristic, complemented
15	C₄×C₅×D₁₂	C₄	1	1	yes	characteristic, complemented
16	C₅×#872	—	2	1	no	—
17	C₅×#870	C₄	1	1	yes	characteristic, complemented
18	C₄×C₅×Dic₆	C₄	1	1	yes	characteristic, complemented
19	#91115	—	2	1	no	—
20	#91115	C₂²	1	2	yes	—
21	C₃×C₂²⋊#168	C₂²	1	2	yes	—
22	C₃×C₄⋊#168	—	2	1	no	—
23	C₃×C₄×#168	—	2	1	no	—
24	#70788	—	5	1	no	maximal, Hall
25	C₅×#529	—	3	1	no	—
26	#65221	—	3	1	no	—
27	#65214	D₃	1	1	yes	characteristic, complemented
28	#64758	—	3	2	no	—
29	#64757	—	3	2	no	—
30	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
31	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
32	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
33	C₂²×Dic₁₅	—	2	2	no	—
34	C₂×C₅×#67	—	2	1	no	—
35	C₂×C₅×D₁₂	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×C₅×D₃	—	2	1	no	—
37	C₂×C₅×Dic₆	C₂×C₄	1	1	yes	characteristic
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	C₄×Dic₁₅	—	2	4	no	—
43	C₅×#217	—	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×(C₂×D₃)⋊C₄	—	2	1	no	—
48	C₅×C₄⋊Dic₃	C₂×C₄	1	1	yes	characteristic
49	C₅×C₄⋊Dic₃	C₂×C₄	1	1	yes	characteristic
50	C₅×Dic₃⋊C₄	—	2	1	no	—
51	C₅×Dic₃⋊C₄	—	2	1	no	—
52	C₅×Dic₃⋊C₄	—	2	1	no	—
53	C₅×Dic₃⋊C₄	—	2	1	no	—
54	C₄×C₅×Dic₃	—	2	1	no	—
55	C₂×C₃×#168	—	2	2	no	—
56	#5479	D₅	1	1	yes	characteristic
57	C₃×#431	—	5	1	no	—
58	#4932	—	5	1	no	—
59	#4470	—	5	2	no	—
60	#4467	—	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×C₄×Dic₅	—	6	1	no	—
73	#3963	—	3	2	no	—
74	#3963	—	6	1	no	—
75	C₂²⋊#168	C₂×D₃	1	2	yes	—
76	C₄⋊#168	—	2	1	no	—
77	C₄×#168	—	2	1	no	—
78	#1500	—	15	1	no	Sylow
79	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₅×D₃	—	2	1	no	—
81	C₂×C₃×C₄×C₅	—	2	2	no	—
82	C₂×C₃×C₄×C₅	—	2	2	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	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
90	C₂×Dic₁₅	—	2	4	no	—
91	C₂×Dic₁₅	—	4	2	no	—
92	C₂×Dic₁₅	—	4	2	no	—
93	C₅×#67	—	4	2	no	—
94	C₂×C₅×Dic₃	—	2	1	no	—
95	C₂×C₅×Dic₃	—	2	1	no	—
96	C₂×C₅×Dic₃	—	2	1	no	—
97	C₅×D₁₂	—	4	1	no	—
98	C₄×C₅×D₃	—	4	2	no	—
99	C₅×Dic₆	—	4	1	no	—
100	C₃×#168	—	4	2	no	—
101	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
102	#930	C₂×D₅	1	1	yes	characteristic
103	#929	—	2	1	no	—
104	C₄×#67	—	2	1	no	—
105	#924	C₂×D₅	1	1	yes	characteristic
106	C₂×C₄×Dic₃	—	10	1	no	—
107	#876	—	2	1	no	—
108	#875	Dic₅	1	1	yes	characteristic, complemented
109	C₄×D₁₂	Dic₅	1	1	yes	characteristic, complemented
110	#872	—	2	1	no	—
111	#870	Dic₅	1	1	yes	characteristic, complemented
112	C₄×Dic₆	Dic₅	1	1	yes	characteristic, complemented
113	C₃×C₄⋊C₈	—	10	1	no	—
114	C₃×C₂²⋊C₈	—	5	2	no	—
115	C₃×C₄×C₈	—	10	1	no	—
116	C₃⋊#95	—	5	2	no	—
117	C₃⋊#95	—	10	1	no	—
118	C₂×C₅×Q₈	—	3	1	no	—
119	C₂×C₅×D₄	—	3	1	no	—
120	C₂×C₅×D₄	—	6	1	no	—
121	C₂²×C₄×C₅	—	6	1	no	—
122	C₂²×C₄×C₅	D₁₂	1	1	yes	characteristic
123	C₂²×C₄×C₅	D₁₂	1	1	yes	characteristic
124	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
125	C₂²×Dic₅	—	6	2	no	—
126	C₅×C₄⋊C₄	—	3	1	no	—
127	C₅×C₄⋊C₄	—	3	1	no	—
128	C₅×C₄⋊C₄	—	6	1	no	—
129	C₅×C₄⋊C₄	—	6	1	no	—
130	C₅×C₄⋊C₄	—	6	1	no	—
131	C₅×C₄⋊C₄	—	6	1	no	—
132	C₅×C₂²⋊C₄	—	6	1	no	—
133	C₅×C₂²⋊C₄	—	6	1	no	—
134	C₅×C₂²⋊C₄	—	6	1	no	—
135	C₅×C₂²⋊C₄	—	6	1	no	—
136	C₅×C₂²⋊C₄	—	6	1	no	—
137	C₄²×C₅	—	6	1	no	—
138	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
139	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
140	C₄²×C₅	#67	1	1	yes	characteristic
141	C₄²×C₅	#67	1	1	yes	characteristic
142	C₄×Dic₅	—	6	4	no	—
143	C₂×#168	—	2	2	no	—
144	#529	—	3	1	no	—
145	#431	—	5	1	no	—
146	#376	—	15	1	no	—
147	#328	—	15	2	no	—
148	#327	—	15	2	no	—
149	C₂²×C₃×C₅	—	2	1	no	—
150	C₂²×C₃×C₅	—	2	1	no	—
151	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
152	C₂²×C₃×C₅	#135	1	2	yes	—
153	C₂×C₅×D₃	—	4	1	no	—
154	C₂×C₅×D₃	—	4	1	no	—
155	C₂×C₅×D₃	—	4	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	—	2	1	no	—
160	C₃×C₄×C₅	—	2	1	no	—
161	C₃×C₄×C₅	#104	1	2	yes	—
162	Dic₁₅	—	4	4	no	—
163	C₅×Dic₃	—	4	1	no	—
164	C₅×Dic₃	—	4	1	no	—
165	C₅×Dic₃	—	4	1	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	C₂×#67	—	2	1	no	—
170	C₂²×Dic₃	—	10	2	no	—
171	C₂×D₁₂	C₂×Dic₅	1	1	yes	characteristic
172	C₂×C₄×D₃	—	2	1	no	—
173	C₂×Dic₆	C₂×Dic₅	1	1	yes	characteristic
174	C₂×C₃×C₈	—	10	2	no	—
175	C₃×C₄²	C₂×Dic₅	1	1	yes	characteristic
176	C₃×C₄²	C₂×Dic₅	1	1	yes	characteristic
177	C₃×C₄²	C₂²⋊D₅	1	1	yes	characteristic
178	C₃×C₄²	C₂²⋊D₅	1	1	yes	characteristic
179	#217	—	2	1	no	—
180	(C₂×D₃)⋊C₄	—	2	1	no	—
181	(C₂×D₃)⋊C₄	—	2	1	no	—
182	(C₂×D₃)⋊C₄	—	2	1	no	—
183	(C₂×D₃)⋊C₄	—	2	1	no	—
184	C₄⋊Dic₃	C₂×Dic₅	1	1	yes	characteristic
185	C₄⋊Dic₃	C₂×Dic₅	1	1	yes	characteristic
186	Dic₃⋊C₄	—	2	1	no	—
187	Dic₃⋊C₄	—	2	1	no	—
188	Dic₃⋊C₄	—	2	1	no	—
189	Dic₃⋊C₄	—	2	1	no	—
190	C₄×Dic₃	—	2	1	no	—
191	C₄×Dic₃	—	10	4	no	—
192	C₂³×C₅	—	6	1	no	—
193	C₂³×C₅	C₂×D₁₂	1	1	yes	characteristic
194	C₅×Q₈	—	12	1	no	—
195	C₅×D₄	—	12	1	no	—
196	C₅×D₄	—	12	2	no	—
197	C₂×C₄×C₅	—	2	2	no	—
198	C₂×C₄×C₅	—	2	2	no	—
199	C₂×C₄×C₅	—	4	1	no	—
200	C₂×C₄×C₅	—	6	1	no	—
201	C₂×C₄×C₅	—	6	1	no	—
202	C₂×C₄×C₅	—	6	1	no	—
203	C₂×C₄×C₅	—	12	2	no	—
204	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
205	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
206	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
207	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
208	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
209	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
210	C₂×Dic₅	—	6	4	no	—
211	C₂×Dic₅	—	12	2	no	—
212	C₂×Dic₅	—	12	2	no	—
213	#168	—	4	2	no	—
214	#126	—	6	1	no	—
215	#125	—	3	1	no	—
216	#124	—	3	1	no	—
217	#123	—	6	1	no	—
218	C₂²⋊Q₈	—	3	1	no	—
219	#121	—	3	1	no	—
220	C₄×Q₈	—	3	1	no	—
221	C₄×D₄	—	3	1	no	—
222	C₄×D₄	—	6	1	no	—
223	#117	—	6	1	no	—
224	C₂×C₄²	—	30	1	no	—
225	C₂×C₄²	D₃×D₅	1	1	yes	characteristic
226	C₄⋊C₈	—	10	1	no	—
227	C₂²⋊C₈	—	5	2	no	—
228	C₄×C₈	—	10	1	no	—
229	#95	—	15	2	no	—
230	#95	—	30	1	no	—
231	C₂×C₃×C₅	—	2	2	no	—
232	C₂×C₃×C₅	#410	1	1	yes	characteristic
233	C₂×C₃×C₅	#416	1	1	yes	characteristic
234	C₂×C₃×C₅	C₂×#104	1	1	yes	characteristic
235	C₅×D₃	—	8	1	no	—
236	C₂³×C₃	C₂×C₂²⋊D₅	1	1	yes	characteristic
237	C₂²×D₃	—	2	1	no	—
238	C₂×C₃×C₄	—	2	2	no	—
239	C₂×C₃×C₄	—	2	2	no	—
240	C₂×C₃×C₄	—	4	1	no	—
241	C₂×C₃×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
242	C₂×C₃×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
243	C₂×C₃×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
244	C₂×C₃×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
245	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
246	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
247	#67	—	4	2	no	—
248	C₂×Dic₃	—	2	1	no	—
249	C₂×Dic₃	—	2	1	no	—
250	C₂×Dic₃	—	2	1	no	—
251	C₂×Dic₃	—	10	4	no	—
252	C₂×Dic₃	—	20	2	no	—
253	C₂×Dic₃	—	20	2	no	—
254	D₁₂	—	4	1	no	—
255	C₄×D₃	—	4	2	no	—
256	Dic₆	—	4	1	no	—
257	C₃×C₈	—	20	2	no	—
258	C₂²×C₅	—	2	1	no	—
259	C₂²×C₅	—	2	1	no	—
260	C₂²×C₅	—	12	1	no	—
261	C₂²×C₅	—	12	1	no	—
262	C₂²×C₅	—	12	1	no	—
263	C₂²×C₅	#904	1	2	yes	—
264	C₂²×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
265	C₄×C₅	—	2	1	no	—
266	C₄×C₅	—	2	1	no	—
267	C₄×C₅	—	2	1	no	—
268	C₄×C₅	—	2	1	no	—
269	C₄×C₅	—	2	1	no	—
270	C₄×C₅	—	12	1	no	—
271	C₄×C₅	—	12	1	no	—
272	C₄×C₅	—	12	1	no	—
273	C₄×C₅	#825	1	2	yes	—
274	Dic₅	—	12	4	no	—
275	C₂×Q₈	—	3	1	no	—
276	C₂×D₄	—	3	1	no	—
277	C₂×D₄	—	6	1	no	—
278	C₂²×C₄	—	6	1	no	—
279	C₂²×C₄	—	30	2	no	—
280	C₂²×C₄	#1203	1	1	yes	characteristic
281	C₂²×C₄	#1203	1	1	yes	characteristic
282	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
283	C₂×C₈	—	10	2	no	—
284	C₄⋊C₄	—	3	1	no	—
285	C₄⋊C₄	—	3	1	no	—
286	C₄⋊C₄	—	6	1	no	—
287	C₄⋊C₄	—	6	1	no	—
288	C₄⋊C₄	—	6	1	no	—
289	C₄⋊C₄	—	6	1	no	—
290	C₂²⋊C₄	—	6	1	no	—
291	C₂²⋊C₄	—	6	1	no	—
292	C₂²⋊C₄	—	6	1	no	—
293	C₂²⋊C₄	—	6	1	no	—
294	C₂²⋊C₄	—	6	1	no	—
295	C₄²	—	6	1	no	—
296	C₄²	—	30	4	no	—
297	C₄²	D₃×Dic₅	1	1	yes	characteristic
298	C₄²	D₃×Dic₅	1	1	yes	characteristic
299	C₄²	#1201	1	1	yes	characteristic
300	C₄²	#1201	1	1	yes	characteristic
301	C₃×C₅	#1500	1	1	yes	characteristic, complemented, Hall
302	C₂²×C₃	—	2	1	no	—
303	C₂²×C₃	—	2	1	no	—
304	C₂²×C₃	#4096	1	2	yes	—
305	C₂²×C₃	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
306	C₂×D₃	—	4	1	no	—
307	C₂×D₃	—	4	1	no	—
308	C₂×D₃	—	4	1	no	—
309	C₃×C₄	—	2	1	no	—
310	C₃×C₄	—	2	1	no	—
311	C₃×C₄	—	2	1	no	—
312	C₃×C₄	—	2	1	no	—
313	C₃×C₄	—	2	1	no	—
314	C₃×C₄	#3969	1	2	yes	—
315	Dic₃	—	4	1	no	—
316	Dic₃	—	4	1	no	—
317	Dic₃	—	4	1	no	—
318	Dic₃	—	20	4	no	—
319	C₂×C₅	—	2	2	no	—
320	C₂×C₅	—	24	1	no	—
321	C₂×C₅	#5006	1	1	yes	characteristic
322	C₂×C₅	#5113	1	1	yes	characteristic
323	C₂×C₅	C₂×#825	1	1	yes	characteristic
324	C₂³	—	6	1	no	—
325	C₂³	C₂×#1203	1	1	yes	characteristic
326	Q₈	—	12	1	no	—
327	D₄	—	12	1	no	—
328	D₄	—	12	2	no	—
329	C₂×C₄	—	2	2	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	4	1	no	—
332	C₂×C₄	—	6	1	no	—
333	C₂×C₄	—	6	1	no	—
334	C₂×C₄	—	6	1	no	—
335	C₂×C₄	—	12	2	no	—
336	C₂×C₄	—	30	4	no	—
337	C₂×C₄	—	60	2	no	—
338	C₂×C₄	—	60	2	no	—
339	C₂×C₄	(C₂×D₃)⋊Dic₅	1	1	yes	characteristic
340	C₂×C₄	(C₂×D₃)⋊Dic₅	1	1	yes	characteristic
341	C₂×C₄	(C₂×D₃)⋊Dic₅	1	1	yes	characteristic
342	C₂×C₄	(C₂×D₃)⋊Dic₅	1	1	yes	characteristic
343	C₂×C₄	C₂×D₃×Dic₅	1	1	yes	characteristic
344	C₂×C₄	C₂×#1201	1	1	yes	characteristic
345	C₈	—	20	2	no	—
346	C₂×C₃	—	2	2	no	—
347	C₂×C₃	#65522	1	1	yes	characteristic
348	C₂×C₃	C₂×#3969	1	1	yes	characteristic
349	C₂×C₃	#65529	1	1	yes	characteristic
350	D₃	—	8	1	no	—
351	C₅	#70788	1	1	yes	characteristic, complemented, Sylow
352	C₂²	—	2	1	no	—
353	C₂²	—	2	1	no	—
354	C₂²	—	12	1	no	—
355	C₂²	—	12	1	no	—
356	C₂²	—	12	1	no	—
357	C₂²	#91303	1	2	yes	—
358	C₂²	C₂×(C₂×D₃)⋊Dic₅	1	1	yes	characteristic
359	C₄	—	2	1	no	—
360	C₄	—	2	1	no	—
361	C₄	—	2	1	no	—
362	C₄	—	2	1	no	—
363	C₄	—	2	1	no	—
364	C₄	—	12	1	no	—
365	C₄	—	12	1	no	—
366	C₄	—	12	1	no	—
367	C₄	—	60	4	no	—
368	C₄	#90978	1	2	yes	—
369	C₃	#10602547	1	1	yes	characteristic, complemented, Sylow
370	C₂	—	2	2	no	—
371	C₂	—	24	1	no	—
372	C₂	#11748388	1	1	yes	characteristic
373	C₂	C₂×#90978	1	1	yes	characteristic
374	C₂	#11748625	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	(C₂×D₃)⋊Dic₅
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	C₅×#930	C₂²
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	C₂×C₄²	D₃×D₅
hypocenter	C₃×C₅	#1500
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. #11751033	C₂	semidirect
3. C₃×#65214	C₂	semidirect
4. #11745963	C₂	semidirect
14. C₅×#875	C₄	semidirect
15. C₄×C₅×D₁₂	C₄	semidirect
17. C₅×#870	C₄	semidirect
18. C₄×C₅×Dic₆	C₄	semidirect
27. #65214	D₃	semidirect
108. #875	Dic₅	semidirect
109. C₄×D₁₂	Dic₅	semidirect
111. #870	Dic₅	semidirect
112. C₄×Dic₆	Dic₅	semidirect
301. C₃×C₅	#1500	semidirect
351. C₅	#70788	semidirect
369. C₃	#10602547	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
#11751033	2
C₃×#65214	2
#11745963	2
#11745957	2
#10602547	3
#70788	5