Subgroups — #14283945

Order 1920.

468 conjugacy classes of subgroups.

The subgroup lattice has 368 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	C₂×#91087	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	#10602539	—	3	1	no	maximal, Hall
7	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
8	C₂²×C₃⋊#168	—	2	1	no	—
9	C₅×#930	C₄	1	1	yes	characteristic
10	C₅×#929	—	2	1	no	—
11	C₄×C₅×#67	—	2	1	no	—
12	C₅×#924	C₄	1	1	yes	characteristic
13	C₅×#876	—	2	1	no	—
14	C₅×#875	C₂²	1	1	yes	characteristic
15	C₄×C₅×D₁₂	C₄	1	1	yes	characteristic
16	C₅×#872	—	2	1	no	—
17	C₅×#870	C₂²	1	1	yes	characteristic
18	C₄×C₅×Dic₆	C₄	1	1	yes	characteristic
19	C₂×C₃×#681	—	2	1	no	—
20	#91087	—	2	1	no	—
21	#91087	C₂²	1	2	yes	—
22	C₃×#3935	C₂²	1	2	yes	—
23	C₃×C₄×#168	—	2	1	no	—
24	#69858	—	5	1	no	maximal, Hall
25	C₅×#529	—	3	1	no	—
26	C₄×#681	D₃	1	1	yes	characteristic, complemented
27	C₂×#3935	—	3	1	no	—
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	4	no	—
42	C₂×C₃⋊#168	—	2	2	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	C₃×#681	—	2	4	no	—
56	C₂×C₃×#168	—	2	2	no	—
57	#5479	D₅	1	1	yes	characteristic
58	C₃×C₄×M₁₆	—	5	1	no	—
59	C₂×#803	—	5	1	no	—
60	#4465	—	5	2	no	—
61	#4464	—	5	2	no	—
62	C₅×#126	—	6	1	no	—
63	C₅×#125	—	3	1	no	—
64	C₅×#124	—	3	1	no	—
65	C₅×#123	—	6	1	no	—
66	C₅×C₂²⋊Q₈	—	3	1	no	—
67	C₅×#121	—	3	1	no	—
68	C₄×C₅×Q₈	—	3	1	no	—
69	C₄×C₅×D₄	—	3	1	no	—
70	C₄×C₅×D₄	—	6	1	no	—
71	C₅×#117	—	6	1	no	—
72	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
73	C₂×#681	—	2	1	no	—
74	C₂²×#168	—	6	1	no	—
75	#3935	—	3	2	no	—
76	#3935	—	6	1	no	—
77	#3935	C₂×D₃	1	2	yes	—
78	C₄×#168	—	2	1	no	—
79	#1524	—	15	1	no	Sylow
80	C₂³×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
81	C₂²×C₅×D₃	—	2	1	no	—
82	C₂×C₃×C₄×C₅	—	2	2	no	—
83	C₂×C₃×C₄×C₅	—	2	2	no	—
84	C₂×C₃×C₄×C₅	—	4	1	no	—
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×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
91	C₅×#67	—	4	2	no	—
92	C₂×C₅×Dic₃	—	2	1	no	—
93	C₂×C₅×Dic₃	—	2	1	no	—
94	C₂×C₅×Dic₃	—	2	1	no	—
95	C₅×D₁₂	—	4	1	no	—
96	C₄×C₅×D₃	—	4	2	no	—
97	C₅×Dic₆	—	4	1	no	—
98	C₃⋊#168	—	2	4	no	—
99	C₃×#168	—	2	4	no	—
100	C₂×C₃×M₁₆	—	10	1	no	—
101	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
102	#930	Dic₅	1	1	yes	characteristic
103	#929	—	2	1	no	—
104	C₄×#67	—	2	1	no	—
105	#924	Dic₅	1	1	yes	characteristic
106	C₂²×C₃⋊C₈	—	10	1	no	—
107	#876	—	2	1	no	—
108	#875	C₂×D₅	1	1	yes	characteristic
109	C₄×D₁₂	Dic₅	1	1	yes	characteristic
110	#872	—	2	1	no	—
111	#870	C₂×D₅	1	1	yes	characteristic
112	C₄×Dic₆	Dic₅	1	1	yes	characteristic
113	C₃×#97	—	5	2	no	—
114	C₃×C₄×C₈	—	10	1	no	—
115	#803	—	5	2	no	—
116	#803	—	10	1	no	—
117	C₂×C₅×Q₈	—	3	1	no	—
118	C₂×C₅×D₄	—	3	1	no	—
119	C₂×C₅×D₄	—	6	1	no	—
120	C₂²×C₄×C₅	—	6	1	no	—
121	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
122	C₂²×C₄×C₅	D₁₂	1	1	yes	characteristic
123	C₂²×C₄×C₅	#67	1	1	yes	characteristic
124	C₅×C₄⋊C₄	—	3	1	no	—
125	C₅×C₄⋊C₄	—	3	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×C₄⋊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	C₅×C₂²⋊C₄	—	6	1	no	—
135	C₄²×C₅	—	6	1	no	—
136	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
137	C₄²×C₅	D₁₂	1	1	yes	characteristic
138	C₄²×C₅	#67	1	1	yes	characteristic
139	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
140	#681	—	2	4	no	—
141	C₂×#168	—	2	2	no	—
142	C₂×#168	—	6	4	no	—
143	C₂×#168	—	6	2	no	—
144	#529	—	3	1	no	—
145	C₄×M₁₆	—	5	1	no	—
146	C₂×#97	—	15	1	no	—
147	#330	—	15	2	no	—
148	#329	—	15	2	no	—
149	C₂²×C₃×C₅	—	2	1	no	—
150	C₂²×C₃×C₅	—	2	1	no	—
151	C₂²×C₃×C₅	#104	1	2	yes	—
152	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
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	C₅×Dic₃	—	4	1	no	—
163	C₅×Dic₃	—	4	1	no	—
164	C₅×Dic₃	—	4	1	no	—
165	C₂²×C₃×C₄	C₂×Dic₅	1	1	yes	characteristic
166	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
167	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
168	C₂×#67	—	2	1	no	—
169	C₂×D₁₂	C₂×Dic₅	1	1	yes	characteristic
170	C₂×C₄×D₃	—	2	1	no	—
171	C₂×Dic₆	C₂×Dic₅	1	1	yes	characteristic
172	C₃×M₁₆	—	10	4	no	—
173	C₂×C₃×C₈	—	10	2	no	—
174	C₃×C₄²	C₂×Dic₅	1	1	yes	characteristic
175	C₃×C₄²	C₂²⋊D₅	1	1	yes	characteristic
176	C₃×C₄²	C₂²⋊D₅	1	1	yes	characteristic
177	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
178	#217	—	2	1	no	—
179	(C₂×D₃)⋊C₄	—	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	C₄⋊Dic₃	C₂×Dic₅	1	1	yes	characteristic
184	C₄⋊Dic₃	C₂×Dic₅	1	1	yes	characteristic
185	Dic₃⋊C₄	—	2	1	no	—
186	Dic₃⋊C₄	—	2	1	no	—
187	Dic₃⋊C₄	—	2	1	no	—
188	Dic₃⋊C₄	—	2	1	no	—
189	C₄×Dic₃	—	2	1	no	—
190	C₂×C₃⋊C₈	—	10	4	no	—
191	C₂×C₃⋊C₈	—	10	2	no	—
192	C₂³×C₅	—	6	1	no	—
193	C₂³×C₅	(C₂×D₃)⋊C₄	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×C₄×D₃	1	1	yes	characteristic
208	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
209	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
210	#168	—	2	4	no	—
211	#168	—	6	4	no	—
212	C₂×M₁₆	—	10	1	no	—
213	C₂²×C₈	—	30	1	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	D₃×D₅	1	1	yes	characteristic
225	#97	—	5	2	no	—
226	#97	—	15	2	no	—
227	#97	—	30	1	no	—
228	C₄×C₈	—	10	1	no	—
229	C₂×C₃×C₅	—	2	2	no	—
230	C₂×C₃×C₅	#413	1	1	yes	characteristic
231	C₂×C₃×C₅	C₂×#104	1	1	yes	characteristic
232	C₂×C₃×C₅	C₂×#104	1	1	yes	characteristic
233	C₅×D₃	—	8	1	no	—
234	C₂³×C₃	C₂²⋊Dic₅	1	1	yes	characteristic
235	C₂²×D₃	—	2	1	no	—
236	C₂×C₃×C₄	—	2	2	no	—
237	C₂×C₃×C₄	—	2	2	no	—
238	C₂×C₃×C₄	—	4	1	no	—
239	C₂×C₃×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
240	C₂×C₃×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
241	C₂×C₃×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
242	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
243	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
244	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
245	#67	—	4	2	no	—
246	C₂×Dic₃	—	2	1	no	—
247	C₂×Dic₃	—	2	1	no	—
248	C₂×Dic₃	—	2	1	no	—
249	D₁₂	—	4	1	no	—
250	C₄×D₃	—	4	2	no	—
251	Dic₆	—	4	1	no	—
252	C₃×C₈	—	10	4	no	—
253	C₃⋊C₈	—	10	4	no	—
254	C₂²×C₅	—	2	1	no	—
255	C₂²×C₅	—	2	1	no	—
256	C₂²×C₅	—	12	1	no	—
257	C₂²×C₅	—	12	1	no	—
258	C₂²×C₅	—	12	1	no	—
259	C₂²×C₅	#805	1	2	yes	—
260	C₂²×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
261	C₄×C₅	—	2	1	no	—
262	C₄×C₅	—	2	1	no	—
263	C₄×C₅	—	2	1	no	—
264	C₄×C₅	—	2	1	no	—
265	C₄×C₅	—	2	1	no	—
266	C₄×C₅	—	12	1	no	—
267	C₄×C₅	—	12	1	no	—
268	C₄×C₅	—	12	1	no	—
269	C₄×C₅	#825	1	2	yes	—
270	C₂×Q₈	—	3	1	no	—
271	C₂×D₄	—	3	1	no	—
272	C₂×D₄	—	6	1	no	—
273	C₂²×C₄	—	6	1	no	—
274	C₂²×C₄	D₃×Dic₅	1	1	yes	characteristic
275	C₂²×C₄	#1201	1	1	yes	characteristic
276	C₂²×C₄	#1203	1	1	yes	characteristic
277	M₁₆	—	10	4	no	—
278	C₂×C₈	—	10	2	no	—
279	C₂×C₈	—	30	4	no	—
280	C₂×C₈	—	30	2	no	—
281	C₄⋊C₄	—	3	1	no	—
282	C₄⋊C₄	—	3	1	no	—
283	C₄⋊C₄	—	6	1	no	—
284	C₄⋊C₄	—	6	1	no	—
285	C₄⋊C₄	—	6	1	no	—
286	C₄⋊C₄	—	6	1	no	—
287	C₂²⋊C₄	—	6	1	no	—
288	C₂²⋊C₄	—	6	1	no	—
289	C₂²⋊C₄	—	6	1	no	—
290	C₂²⋊C₄	—	6	1	no	—
291	C₂²⋊C₄	—	6	1	no	—
292	C₄²	—	6	1	no	—
293	C₄²	D₃×Dic₅	1	1	yes	characteristic
294	C₄²	#1201	1	1	yes	characteristic
295	C₄²	#1203	1	1	yes	characteristic
296	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
297	C₃×C₅	#1524	1	1	yes	characteristic, complemented, Hall
298	C₂²×C₃	—	2	1	no	—
299	C₂²×C₃	—	2	1	no	—
300	C₂²×C₃	#3969	1	2	yes	—
301	C₂²×C₃	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
302	C₂×D₃	—	4	1	no	—
303	C₂×D₃	—	4	1	no	—
304	C₂×D₃	—	4	1	no	—
305	C₃×C₄	—	2	1	no	—
306	C₃×C₄	—	2	1	no	—
307	C₃×C₄	—	2	1	no	—
308	C₃×C₄	—	2	1	no	—
309	C₃×C₄	—	2	1	no	—
310	C₃×C₄	#3969	1	2	yes	—
311	Dic₃	—	4	1	no	—
312	Dic₃	—	4	1	no	—
313	Dic₃	—	4	1	no	—
314	C₂×C₅	—	2	2	no	—
315	C₂×C₅	—	24	1	no	—
316	C₂×C₅	C₂×#805	1	1	yes	characteristic
317	C₂×C₅	#5132	1	1	yes	characteristic
318	C₂×C₅	C₂×#825	1	1	yes	characteristic
319	C₂³	—	6	1	no	—
320	C₂³	(C₂×D₃)⋊Dic₅	1	1	yes	characteristic
321	Q₈	—	12	1	no	—
322	D₄	—	12	1	no	—
323	D₄	—	12	2	no	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	2	2	no	—
326	C₂×C₄	—	4	1	no	—
327	C₂×C₄	—	6	1	no	—
328	C₂×C₄	—	6	1	no	—
329	C₂×C₄	—	6	1	no	—
330	C₂×C₄	—	12	2	no	—
331	C₂×C₄	(C₂×D₃)⋊Dic₅	1	1	yes	characteristic
332	C₂×C₄	(C₂×D₃)⋊Dic₅	1	1	yes	characteristic
333	C₂×C₄	(C₂×D₃)⋊Dic₅	1	1	yes	characteristic
334	C₂×C₄	C₂×D₃×Dic₅	1	1	yes	characteristic
335	C₂×C₄	C₂×#1201	1	1	yes	characteristic
336	C₂×C₄	C₂×#1203	1	1	yes	characteristic
337	C₈	—	10	4	no	—
338	C₈	—	30	4	no	—
339	C₂×C₃	—	2	2	no	—
340	C₂×C₃	C₂×#3969	1	1	yes	characteristic
341	C₂×C₃	C₂×#3969	1	1	yes	characteristic
342	C₂×C₃	#65526	1	1	yes	characteristic
343	D₃	—	8	1	no	—
344	C₅	#69858	1	1	yes	characteristic, complemented, Sylow
345	C₂²	—	2	1	no	—
346	C₂²	—	2	1	no	—
347	C₂²	—	12	1	no	—
348	C₂²	—	12	1	no	—
349	C₂²	—	12	1	no	—
350	C₂²	#90979	1	2	yes	—
351	C₂²	C₂×(C₂×D₃)⋊Dic₅	1	1	yes	characteristic
352	C₄	—	2	1	no	—
353	C₄	—	2	1	no	—
354	C₄	—	2	1	no	—
355	C₄	—	2	1	no	—
356	C₄	—	2	1	no	—
357	C₄	—	12	1	no	—
358	C₄	—	12	1	no	—
359	C₄	—	12	1	no	—
360	C₄	#90978	1	2	yes	—
361	C₃	#10602539	1	1	yes	characteristic, complemented, Sylow
362	C₂	—	2	2	no	—
363	C₂	—	24	1	no	—
364	C₂	#11748355	1	1	yes	characteristic
365	C₂	C₂×#90978	1	1	yes	characteristic
366	C₂	C₂×#90979	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	(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	#1524
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. C₂×#91087	C₂	semidirect
3. C₃×C₄×#681	C₂	semidirect
4. #11745850	C₂	semidirect
5. #11745847	C₂	semidirect
26. C₄×#681	D₃	semidirect
297. C₃×C₅	#1524	semidirect
344. C₅	#69858	semidirect
361. C₃	#10602539	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
C₂×#91087	2
C₃×C₄×#681	2
#11745850	2
#11745847	2
#10602539	3
#69858	5