Subgroups — #14301784

Order 1920.

760 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×C₅×#924	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₃×#168	C₂	1	1	yes	maximal, characteristic
3	#11751313	C₂	1	1	yes	maximal, characteristic
4	C₂×#90973	C₂	1	2	yes	maximal, complemented
5	#11745981	C₂	1	2	yes	maximal, complemented
6	#10602489	—	3	1	no	maximal, Hall
7	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
8	C₂²×C₅×Dic₆	C₂²	1	1	yes	characteristic
9	C₂×#6719	—	2	1	no	—
10	C₂×C₄⋊Dic₁₅	C₂²	1	2	yes	—
11	C₂×C₅×#217	—	2	1	no	—
12	C₂×C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
13	C₅×#924	—	2	2	no	—
14	C₅×#924	C₄	1	4	yes	complemented
15	C₂×C₅×Dic₃⋊C₄	—	2	1	no	—
16	C₂²×C₃×#168	—	2	2	no	—
17	C₂²×C₃×#168	—	2	4	no	—
18	C₂²×C₃×#168	C₂²	1	2	yes	—
19	#91115	—	2	1	no	—
20	#90973	—	2	4	no	—
21	#71972	—	5	1	no	maximal, Hall
22	C₂×C₅×C₂²⋊Q₈	—	3	1	no	—
23	C₂³×#168	D₃	1	1	yes	characteristic
24	C₂×#3967	—	3	2	no	—
25	#65501	—	3	1	no	—
26	#64749	—	3	2	no	—
27	C₂⁴×C₃×C₅	D₄	1	1	yes	characteristic
28	C₂²×C₃×C₄×C₅	—	2	2	no	—
29	C₂²×C₃×C₄×C₅	—	2	2	no	—
30	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
31	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
32	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
33	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
34	C₂²×Dic₁₅	—	2	2	no	—
35	C₂²×C₅×Dic₃	—	2	1	no	—
36	C₂²×C₅×Dic₃	—	2	1	no	—
37	C₂×C₅×Dic₆	—	2	2	no	—
38	C₂×C₅×Dic₆	—	2	2	no	—
39	C₂×C₅×Dic₆	—	2	1	no	—
40	C₂×C₅×Dic₆	C₂×C₄	1	2	yes	—
41	#6719	—	4	2	no	—
42	C₄⋊Dic₁₅	—	2	2	no	—
43	C₄⋊Dic₁₅	—	2	2	no	—
44	C₅×#217	—	2	4	no	—
45	C₅×C₄⋊Dic₃	—	2	1	no	—
46	C₅×C₄⋊Dic₃	C₂×C₄	1	2	yes	—
47	C₅×Dic₃⋊C₄	—	2	4	no	—
48	C₂×C₃×#168	—	2	8	no	—
49	C₂×C₃×#168	—	2	2	no	—
50	C₂×C₃×#168	—	2	2	no	—
51	C₂×C₃×#168	D₄	1	4	yes	—
52	C₂³×C₃×C₈	—	5	1	no	—
53	C₂×#924	D₅	1	1	yes	characteristic
54	#5212	—	5	1	no	—
55	C₂×#816	—	5	2	no	—
56	#4551	—	5	2	no	—
57	C₂²×C₅×Q₈	—	3	1	no	—
58	C₂³×C₄×C₅	C₂×D₃	1	1	yes	characteristic
59	C₅×C₂²⋊Q₈	—	3	4	no	—
60	C₅×C₂²⋊Q₈	—	6	2	no	—
61	C₂×C₅×C₄⋊C₄	—	3	1	no	—
62	C₂×C₅×C₄⋊C₄	—	6	1	no	—
63	C₂×C₅×C₂²⋊C₄	—	6	1	no	—
64	C₂×C₂²⋊Dic₅	—	6	1	no	—
65	C₂×C₄⋊Dic₅	—	3	2	no	—
66	C₂²×#168	—	2	2	no	—
67	C₂²×#168	—	2	4	no	—
68	C₂²×#168	C₂×D₃	1	2	yes	—
69	#3967	—	6	4	no	—
70	#3963	—	6	1	no	—
71	#1892	—	15	1	no	Sylow
72	C₂³×C₃×C₅	—	2	2	no	—
73	C₂³×C₃×C₅	—	2	2	no	—
74	C₂³×C₃×C₅	C₂²⋊C₄	1	2	yes	—
75	C₂³×C₃×C₅	SD₁₆	1	2	yes	—
76	C₂³×C₃×C₅	Q₁₆	1	2	yes	—
77	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
78	C₂×C₃×C₄×C₅	—	2	2	no	—
79	C₂×C₃×C₄×C₅	—	2	4	no	—
80	C₂×C₃×C₄×C₅	—	2	2	no	—
81	C₂×C₃×C₄×C₅	—	2	2	no	—
82	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
83	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
85	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
86	C₂×C₃×C₄×C₅	#41	1	2	yes	—
87	C₂×Dic₁₅	—	4	2	no	—
88	C₂×Dic₁₅	—	4	2	no	—
89	C₂×Dic₁₅	—	4	2	no	—
90	C₂×C₅×Dic₃	—	2	2	no	—
91	C₂×C₅×Dic₃	—	2	2	no	—
92	C₂×C₅×Dic₃	—	4	2	no	—
93	C₂×C₅×Dic₃	—	4	2	no	—
94	C₅×Dic₆	—	2	4	no	—
95	C₅×Dic₆	—	4	1	no	—
96	C₅×Dic₆	—	4	1	no	—
97	C₃×#168	—	2	4	no	—
98	C₂³×C₃×C₄	C₂×D₅	1	1	yes	characteristic
99	C₂²×Dic₆	C₂×D₅	1	1	yes	characteristic
100	C₂²×C₃×C₈	—	5	2	no	—
101	C₂²×C₃×C₈	—	10	4	no	—
102	C₂²×C₃×C₈	—	10	2	no	—
103	C₂×#217	—	2	1	no	—
104	C₂×#217	—	10	1	no	—
105	C₂×C₄⋊Dic₃	—	5	2	no	—
106	C₂×C₄⋊Dic₃	C₂×D₅	1	1	yes	characteristic
107	#924	—	2	2	no	—
108	#924	Dic₅	1	4	yes	complemented
109	C₂×Dic₃⋊C₄	—	2	1	no	—
110	C₃⋊#95	—	10	1	no	—
111	#816	—	10	4	no	—
112	C₂⁴×C₅	D₁₂	1	1	yes	characteristic
113	C₂×C₅×Q₈	—	3	2	no	—
114	C₂×C₅×Q₈	—	6	1	no	—
115	C₂×C₅×Q₈	—	6	2	no	—
116	C₂×C₅×Q₈	—	6	2	no	—
117	C₂²×C₄×C₅	—	2	2	no	—
118	C₂²×C₄×C₅	—	2	2	no	—
119	C₂²×C₄×C₅	—	6	1	no	—
120	C₂²×C₄×C₅	—	6	1	no	—
121	C₂²×C₄×C₅	C₄×D₃	1	2	yes	—
122	C₂²×C₄×C₅	D₁₂	1	1	yes	characteristic
123	C₂²×C₄×C₅	#67	1	2	yes	—
124	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
125	C₂²×Dic₅	—	6	2	no	—
126	C₅×C₄⋊C₄	—	3	2	no	—
127	C₅×C₄⋊C₄	—	6	4	no	—
128	C₅×C₄⋊C₄	—	6	1	no	—
129	C₅×C₂²⋊C₄	—	6	4	no	—
130	C₂²⋊Dic₅	—	12	2	no	—
131	C₄⋊Dic₅	—	6	2	no	—
132	C₄⋊Dic₅	—	6	2	no	—
133	C₂×#168	—	2	8	no	—
134	C₂×#168	—	2	2	no	—
135	C₂×#168	—	2	2	no	—
136	C₂×#168	#67	1	4	yes	—
137	C₂³×C₈	—	5	1	no	—
138	C₂×C₂²⋊Q₈	—	3	1	no	—
139	C₂×#103	—	15	2	no	—
140	#380	—	15	1	no	—
141	#340	—	15	2	no	—
142	C₂²×C₃×C₅	—	2	2	no	—
143	C₂²×C₃×C₅	—	2	2	no	—
144	C₂²×C₃×C₅	—	2	4	no	—
145	C₂²×C₃×C₅	—	2	4	no	—
146	C₂²×C₃×C₅	#103	1	4	yes	complemented
147	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
148	C₂²×C₃×C₅	#121	1	2	yes	—
149	C₂²×C₃×C₅	C₂×SD₁₆	1	1	yes	characteristic
150	C₂²×C₃×C₅	C₂×Q₁₆	1	1	yes	characteristic
151	C₂²×C₃×C₅	#135	1	2	yes	—
152	C₃×C₄×C₅	—	2	2	no	—
153	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
154	C₃×C₄×C₅	#120	1	1	yes	characteristic
155	C₃×C₄×C₅	#123	1	1	yes	characteristic
156	Dic₁₅	—	8	2	no	—
157	C₅×Dic₃	—	4	2	no	—
158	C₅×Dic₃	—	4	2	no	—
159	C₂⁴×C₃	C₂²⋊D₅	1	1	yes	characteristic
160	C₂²×C₃×C₄	—	2	2	no	—
161	C₂²×C₃×C₄	—	2	2	no	—
162	C₂²×C₃×C₄	C₂×Dic₅	1	2	yes	—
163	C₂²×C₃×C₄	C₂²⋊D₅	1	2	yes	—
164	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
165	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
166	C₂²×Dic₃	—	2	1	no	—
167	C₂²×Dic₃	—	2	1	no	—
168	C₂²×Dic₃	—	10	2	no	—
169	C₂×Dic₆	—	2	2	no	—
170	C₂×Dic₆	—	2	2	no	—
171	C₂×Dic₆	—	2	1	no	—
172	C₂×Dic₆	C₂×Dic₅	1	2	yes	—
173	C₂×C₃×C₈	—	5	4	no	—
174	C₂×C₃×C₈	—	10	8	no	—
175	C₂×C₃×C₈	—	10	2	no	—
176	C₂×C₃×C₈	—	10	2	no	—
177	#217	—	2	4	no	—
178	#217	—	20	2	no	—
179	C₄⋊Dic₃	—	2	1	no	—
180	C₄⋊Dic₃	—	10	2	no	—
181	C₄⋊Dic₃	—	10	2	no	—
182	C₄⋊Dic₃	C₂×Dic₅	1	2	yes	—
183	Dic₃⋊C₄	—	2	4	no	—
184	C₂³×C₅	—	2	2	no	—
185	C₂³×C₅	—	2	2	no	—
186	C₂³×C₅	#204	1	2	yes	—
187	C₂³×C₅	Dic₁₂	1	2	yes	—
188	C₂³×C₅	(C₂×D₃)⋊C₄	1	2	yes	—
189	C₂³×C₅	C₂×D₁₂	1	1	yes	characteristic
190	C₅×Q₈	—	6	4	no	—
191	C₅×Q₈	—	12	1	no	—
192	C₅×Q₈	—	12	1	no	—
193	C₂×C₄×C₅	—	2	2	no	—
194	C₂×C₄×C₅	—	2	4	no	—
195	C₂×C₄×C₅	—	2	2	no	—
196	C₂×C₄×C₅	—	2	2	no	—
197	C₂×C₄×C₅	—	6	2	no	—
198	C₂×C₄×C₅	—	6	2	no	—
199	C₂×C₄×C₅	—	12	2	no	—
200	C₂×C₄×C₅	—	12	2	no	—
201	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	2	yes	—
202	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
203	C₂×C₄×C₅	#235	1	2	yes	—
204	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
205	C₂×C₄×C₅	C₂×#67	1	2	yes	—
206	C₂×Dic₅	—	12	2	no	—
207	C₂×Dic₅	—	12	2	no	—
208	C₂×Dic₅	—	12	2	no	—
209	#168	—	2	4	no	—
210	C₂²×Q₈	—	3	1	no	—
211	C₂³×C₄	D₃×D₅	1	1	yes	characteristic
212	C₂²×C₈	—	5	2	no	—
213	C₂²×C₈	—	10	4	no	—
214	C₂²×C₈	—	10	2	no	—
215	C₂²⋊Q₈	—	3	4	no	—
216	C₂²⋊Q₈	—	6	2	no	—
217	C₂×C₄⋊C₄	—	3	1	no	—
218	C₂×C₄⋊C₄	—	6	1	no	—
219	C₂×C₄⋊C₄	—	15	2	no	—
220	C₂×C₂²⋊C₄	—	6	1	no	—
221	C₂×C₂²⋊C₄	—	30	1	no	—
222	#103	—	30	4	no	—
223	#95	—	30	1	no	—
224	C₂×C₃×C₅	—	2	4	no	—
225	C₂×C₃×C₅	#386	1	1	yes	characteristic
226	C₂×C₃×C₅	C₂×#103	1	1	yes	characteristic
227	C₂×C₃×C₅	#416	1	1	yes	characteristic
228	C₂×C₃×C₅	#465	1	2	yes	—
229	C₂×C₃×C₅	#467	1	2	yes	—
230	C₂³×C₃	—	2	2	no	—
231	C₂³×C₃	—	2	2	no	—
232	C₂³×C₃	#688	1	2	yes	—
233	C₂³×C₃	#689	1	2	yes	—
234	C₂³×C₃	C₂²⋊Dic₅	1	2	yes	—
235	C₂³×C₃	C₂×C₂²⋊D₅	1	1	yes	characteristic
236	C₂×C₃×C₄	—	2	2	no	—
237	C₂×C₃×C₄	—	2	4	no	—
238	C₂×C₃×C₄	—	2	2	no	—
239	C₂×C₃×C₄	—	2	2	no	—
240	C₂×C₃×C₄	C₂²⋊Dic₅	1	2	yes	—
241	C₂×C₃×C₄	D₄×D₅	1	2	yes	—
242	C₂×C₃×C₄	#711	1	2	yes	—
243	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
244	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
245	C₂×Dic₃	—	2	2	no	—
246	C₂×Dic₃	—	2	2	no	—
247	C₂×Dic₃	—	4	2	no	—
248	C₂×Dic₃	—	4	2	no	—
249	C₂×Dic₃	—	20	2	no	—
250	C₂×Dic₃	—	20	2	no	—
251	C₂×Dic₃	—	20	2	no	—
252	Dic₆	—	2	4	no	—
253	Dic₆	—	4	1	no	—
254	Dic₆	—	4	1	no	—
255	C₃×C₈	—	10	4	no	—
256	C₂²×C₅	—	2	2	no	—
257	C₂²×C₅	—	2	2	no	—
258	C₂²×C₅	—	2	4	no	—
259	C₂²×C₅	—	2	4	no	—
260	C₂²×C₅	#816	1	4	yes	complemented
261	C₂²×C₅	C₂×#204	1	1	yes	characteristic
262	C₂²×C₅	#904	1	2	yes	—
263	C₂²×C₅	C₂×Dic₁₂	1	1	yes	characteristic
264	C₂²×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
265	C₂²×C₅	#930	1	2	yes	—
266	C₄×C₅	—	2	2	no	—
267	C₄×C₅	—	12	2	no	—
268	C₄×C₅	—	12	2	no	—
269	C₄×C₅	C₄×#67	1	2	yes	—
270	C₄×C₅	#929	1	1	yes	characteristic
271	C₄×C₅	#953	1	1	yes	characteristic
272	Dic₅	—	24	2	no	—
273	C₂⁴	#1203	1	1	yes	characteristic
274	C₂×Q₈	—	3	2	no	—
275	C₂×Q₈	—	6	1	no	—
276	C₂×Q₈	—	6	2	no	—
277	C₂×Q₈	—	6	2	no	—
278	C₂²×C₄	—	2	2	no	—
279	C₂²×C₄	—	2	2	no	—
280	C₂²×C₄	—	6	1	no	—
281	C₂²×C₄	—	6	1	no	—
282	C₂²×C₄	—	30	2	no	—
283	C₂²×C₄	D₃×Dic₅	1	2	yes	—
284	C₂²×C₄	#1201	1	2	yes	—
285	C₂²×C₄	#1203	1	1	yes	characteristic
286	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
287	C₂×C₈	—	5	4	no	—
288	C₂×C₈	—	10	8	no	—
289	C₂×C₈	—	10	2	no	—
290	C₂×C₈	—	10	2	no	—
291	C₄⋊C₄	—	3	2	no	—
292	C₄⋊C₄	—	6	4	no	—
293	C₄⋊C₄	—	6	1	no	—
294	C₄⋊C₄	—	30	2	no	—
295	C₄⋊C₄	—	30	2	no	—
296	C₂²⋊C₄	—	6	4	no	—
297	C₂²⋊C₄	—	60	2	no	—
298	C₃×C₅	#1892	1	1	yes	characteristic, complemented, Hall
299	C₂²×C₃	—	2	4	no	—
300	C₂²×C₃	—	2	4	no	—
301	C₂²×C₃	—	2	2	no	—
302	C₂²×C₃	—	2	2	no	—
303	C₂²×C₃	#3967	1	4	yes	complemented
304	C₂²×C₃	#4085	1	2	yes	—
305	C₂²×C₃	C₂×#688	1	1	yes	characteristic
306	C₂²×C₃	C₂×#689	1	1	yes	characteristic
307	C₂²×C₃	#4096	1	2	yes	—
308	C₂²×C₃	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
309	C₃×C₄	—	2	2	no	—
310	C₃×C₄	D₄×Dic₅	1	2	yes	—
311	C₃×C₄	#4081	1	1	yes	characteristic
312	C₃×C₄	#4083	1	1	yes	characteristic
313	Dic₃	—	4	2	no	—
314	Dic₃	—	4	2	no	—
315	Dic₃	—	40	2	no	—
316	C₂×C₅	—	2	4	no	—
317	C₂×C₅	C₂×#816	1	1	yes	characteristic
318	C₂×C₅	#5113	1	1	yes	characteristic
319	C₂×C₅	#5114	1	2	yes	—
320	C₂×C₅	#5116	1	2	yes	—
321	C₂×C₅	#5213	1	1	yes	characteristic
322	C₂³	—	2	2	no	—
323	C₂³	—	2	2	no	—
324	C₂³	#6660	1	2	yes	—
325	C₂³	#6663	1	2	yes	—
326	C₂³	(C₂×D₃)⋊Dic₅	1	2	yes	—
327	C₂³	C₂×#1203	1	1	yes	characteristic
328	Q₈	—	6	4	no	—
329	Q₈	—	12	1	no	—
330	Q₈	—	12	1	no	—
331	C₂×C₄	—	2	2	no	—
332	C₂×C₄	—	2	2	no	—
333	C₂×C₄	—	2	4	no	—
334	C₂×C₄	—	2	2	no	—
335	C₂×C₄	—	6	2	no	—
336	C₂×C₄	—	6	2	no	—
337	C₂×C₄	—	12	2	no	—
338	C₂×C₄	—	12	2	no	—
339	C₂×C₄	—	60	2	no	—
340	C₂×C₄	—	60	2	no	—
341	C₂×C₄	—	60	2	no	—
342	C₂×C₄	(C₂×D₃)⋊Dic₅	1	2	yes	—
343	C₂×C₄	C₂×D₃×Dic₅	1	1	yes	characteristic
344	C₂×C₄	#6782	1	2	yes	—
345	C₂×C₄	C₂×#1201	1	1	yes	characteristic
346	C₂×C₄	D₅×#67	1	2	yes	—
347	C₈	—	10	4	no	—
348	C₂×C₃	—	2	4	no	—
349	C₂×C₃	#65337	1	2	yes	—
350	C₂×C₃	#65340	1	2	yes	—
351	C₂×C₃	#65509	1	1	yes	characteristic
352	C₂×C₃	C₂×#3967	1	1	yes	characteristic
353	C₂×C₃	#65522	1	1	yes	characteristic
354	C₅	#71972	1	1	yes	characteristic, complemented, Sylow
355	C₂²	—	2	2	no	—
356	C₂²	—	2	2	no	—
357	C₂²	—	2	4	no	—
358	C₂²	—	2	4	no	—
359	C₂²	#90973	1	4	yes	complemented
360	C₂²	#91303	1	2	yes	—
361	C₂²	C₂×#6660	1	1	yes	characteristic
362	C₂²	C₂×#6663	1	1	yes	characteristic
363	C₂²	C₂×(C₂×D₃)⋊Dic₅	1	1	yes	characteristic
364	C₂²	#91566	1	2	yes	—
365	C₄	—	2	2	no	—
366	C₄	—	12	2	no	—
367	C₄	—	12	2	no	—
368	C₄	—	120	2	no	—
369	C₄	#91539	1	1	yes	characteristic
370	C₄	Dic₅×#67	1	2	yes	—
371	C₄	#91563	1	1	yes	characteristic
372	C₃	#10602489	1	1	yes	characteristic, complemented, Sylow
373	C₂	—	2	4	no	—
374	C₂	#11748388	1	1	yes	characteristic
375	C₂	C₂×#90973	1	1	yes	characteristic
376	C₂	#11748535	1	2	yes	—
377	C₂	#11748557	1	2	yes	—
378	C₂	#11749492	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#1203
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₂²×C₄	C₂×D₃×D₅
Fitting	C₂³×C₃×C₄×C₅	C₂²
socle	C₂³×C₃×C₅	C₂×D₄
prime radical	C₂⁴×C₃×C₅	D₄
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	C₂³×C₄	D₃×D₅
hypocenter	C₃×C₅	#1892
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄

Complements

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

N	H ≅ G/N	type
4. C₂×#90973	C₂	semidirect
5. #11745981	C₂	semidirect
14. C₅×#924	C₄	semidirect
108. #924	Dic₅	semidirect
146. C₂²×C₃×C₅	#103	semidirect
260. C₂²×C₅	#816	semidirect
298. C₃×C₅	#1892	semidirect
303. C₂²×C₃	#3967	semidirect
354. C₅	#71972	semidirect
359. C₂²	#90973	semidirect
372. C₃	#10602489	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#924	2
C₂³×C₃×#168	2
#11751313	2
C₂×#90973	2
#11745981	2
#10602489	3
#71972	5