Subgroups — #14301405

Order 1920.

664 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11756578	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#5832	C₂	1	1	yes	maximal, characteristic
3	C₃×#65529	C₂	1	1	yes	maximal, characteristic, complemented
4	#11748425	C₂	1	2	yes	maximal, complemented
5	#11745974	C₂	1	2	yes	maximal, complemented
6	#10602610	—	3	1	no	maximal, Hall
7	C₂×C₃×C₅×#41	C₂²	1	1	yes	characteristic
8	C₂×C₅×#235	C₂²	1	1	yes	characteristic
9	#91837	—	2	4	no	—
10	C₂×#6710	—	2	2	no	—
11	C₂×#6710	C₂²	1	2	yes	—
12	C₅×#947	C₄	1	1	yes	characteristic, complemented
13	C₅×#944	C₄	1	1	yes	characteristic, complemented
14	C₅×#940	C₄	1	1	yes	characteristic, complemented
15	C₅×#939	—	2	2	no	—
16	C₅×#936	C₄	1	1	yes	characteristic, complemented
17	C₄×C₅×#67	—	2	2	no	—
18	C₂×C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
19	C₃×#4072	C₂²	1	2	yes	—
20	C₃×#3966	—	2	2	no	—
21	#90979	—	2	4	no	—
22	#70045	—	5	1	no	maximal, Hall
23	C₅×#532	—	3	1	no	—
24	#66153	—	3	1	no	—
25	#65529	D₃	1	1	yes	characteristic, complemented
26	#65525	—	3	2	no	—
27	#64743	—	3	2	no	—
28	C₃×C₅×#41	—	2	2	no	—
29	C₃×C₅×#41	—	2	2	no	—
30	C₂×C₃×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
31	C₂×C₃×C₅×D₄	C₂×C₄	1	1	yes	characteristic
32	C₂×C₃×C₅×D₄	D₄	1	2	yes	—
33	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
34	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
35	C₂×C₅×#67	—	2	2	no	—
36	C₂×C₅×#67	—	2	1	no	—
37	C₂²×C₅×Dic₃	—	2	1	no	—
38	C₅×#235	—	2	2	no	—
39	C₅×#235	—	2	2	no	—
40	C₂×C₅×D₁₂	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄×C₅×D₃	—	2	1	no	—
42	C₂×C₅×Dic₆	C₂×C₄	1	1	yes	characteristic
43	#6710	—	2	4	no	—
44	#6710	—	2	2	no	—
45	#6710	D₄	1	4	yes	complemented
46	C₂×C₃⋊#168	—	2	4	no	—
47	C₂×C₃⋊#168	—	2	2	no	—
48	C₅×#217	—	2	2	no	—
49	C₅×(C₂×D₃)⋊C₄	—	2	2	no	—
50	C₅×Dic₃⋊C₄	—	2	2	no	—
51	C₄×C₅×Dic₃	—	2	1	no	—
52	C₄×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
53	C₄×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
54	C₃×C₂²⋊Dic₅	—	2	2	no	—
55	C₃×C₄⋊Dic₅	—	2	2	no	—
56	C₃×C₄×Dic₅	—	2	2	no	—
57	#5832	D₅	1	1	yes	characteristic
58	#5819	—	5	1	no	—
59	C₃×#410	—	5	1	no	—
60	#5005	—	5	2	no	—
61	#4532	—	5	2	no	—
62	C₂×C₅×#41	—	3	1	no	—
63	C₂×C₅×#41	C₂×D₃	1	1	yes	characteristic
64	C₅×C₄⋊Q₈	—	3	1	no	—
65	C₅×#127	—	3	1	no	—
66	C₅×#124	—	3	1	no	—
67	C₅×#124	—	3	1	no	—
68	C₅×#121	—	6	2	no	—
69	C₄×C₅×D₄	—	6	2	no	—
70	C₂×C₄²×C₅	—	3	1	no	—
71	#4094	—	6	4	no	—
72	#4072	C₂×D₃	1	2	yes	—
73	C₂×#681	—	3	2	no	—
74	C₂×#681	—	6	2	no	—
75	#3969	—	6	4	no	—
76	#3966	—	2	2	no	—
77	#1884	—	15	1	no	Sylow
78	C₂³×C₃×C₅	C₂²⋊C₄	1	2	yes	—
79	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₅×D₃	—	2	1	no	—
81	C₃×C₅×Q₈	—	2	2	no	—
82	C₃×C₅×D₄	—	2	4	no	—
83	C₃×C₅×D₄	—	2	2	no	—
84	C₂×C₃×C₄×C₅	—	2	2	no	—
85	C₂×C₃×C₄×C₅	—	2	2	no	—
86	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
87	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
89	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
90	C₂×C₃×C₄×C₅	#41	1	2	yes	—
91	C₅×#67	—	4	4	no	—
92	C₅×#67	—	4	2	no	—
93	C₂×C₅×Dic₃	—	2	1	no	—
94	C₂×C₅×Dic₃	—	2	1	no	—
95	C₂×C₅×Dic₃	—	2	1	no	—
96	C₂×C₅×Dic₃	—	4	2	no	—
97	C₅×D₁₂	—	2	2	no	—
98	C₄×C₅×D₃	—	4	1	no	—
99	C₄×C₅×D₃	—	4	1	no	—
100	C₅×Dic₆	—	2	2	no	—
101	C₂×C₃×Dic₅	—	4	2	no	—
102	C₃⋊#168	—	2	4	no	—
103	C₂×C₃×#41	C₂×D₅	1	1	yes	characteristic
104	C₂×#235	C₂×D₅	1	1	yes	characteristic
105	C₃×#117	—	5	2	no	—
106	#948	—	10	4	no	—
107	#947	Dic₅	1	1	yes	characteristic, complemented
108	#944	Dic₅	1	1	yes	characteristic, complemented
109	#940	Dic₅	1	1	yes	characteristic, complemented
110	#939	—	2	2	no	—
111	#936	Dic₅	1	1	yes	characteristic, complemented
112	C₄×#67	—	2	2	no	—
113	C₂×C₄×Dic₃	C₂×D₅	1	1	yes	characteristic
114	C₂×#208	—	5	2	no	—
115	C₂×#208	—	10	2	no	—
116	C₃×#99	—	10	2	no	—
117	#805	—	10	4	no	—
118	C₅×#41	—	2	2	no	—
119	C₅×#41	—	2	2	no	—
120	C₅×#41	—	6	2	no	—
121	C₅×#41	—	6	2	no	—
122	C₂×C₅×Q₈	—	3	1	no	—
123	C₂×C₅×Q₈	C₄×D₃	1	1	yes	characteristic
124	C₂×C₅×D₄	—	3	1	no	—
125	C₂×C₅×D₄	—	6	2	no	—
126	C₂×C₅×D₄	—	6	1	no	—
127	C₂×C₅×D₄	C₄×D₃	1	1	yes	characteristic
128	C₂×C₅×D₄	#67	1	2	yes	—
129	C₂²×C₄×C₅	—	6	1	no	—
130	C₂²×C₄×C₅	—	6	1	no	—
131	C₂²×C₄×C₅	D₁₂	1	2	yes	—
132	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
133	C₅×C₄⋊C₄	—	6	2	no	—
134	C₅×C₂²⋊C₄	—	6	2	no	—
135	C₅×C₂²⋊C₄	—	6	2	no	—
136	C₄²×C₅	—	3	1	no	—
137	C₄²×C₅	—	3	1	no	—
138	C₄²×C₅	—	6	1	no	—
139	C₂²⋊Dic₅	—	2	2	no	—
140	C₄⋊Dic₅	—	2	2	no	—
141	C₄×Dic₅	—	2	2	no	—
142	#681	—	3	4	no	—
143	#681	—	6	4	no	—
144	#681	—	6	2	no	—
145	C₂×#168	—	6	4	no	—
146	C₂×#168	—	6	2	no	—
147	#568	—	15	1	no	—
148	#532	—	3	1	no	—
149	#421	—	15	2	no	—
150	#410	—	5	1	no	—
151	#339	—	15	2	no	—
152	C₂²×C₃×C₅	—	2	2	no	—
153	C₂²×C₃×C₅	—	4	1	no	—
154	C₂²×C₃×C₅	—	4	2	no	—
155	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
156	C₂²×C₃×C₅	#121	1	2	yes	—
157	C₂×C₅×D₃	—	4	2	no	—
158	C₂×C₅×D₃	—	4	1	no	—
159	C₃×C₄×C₅	—	2	2	no	—
160	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
161	C₃×C₄×C₅	#120	1	1	yes	characteristic
162	C₃×C₄×C₅	#123	1	1	yes	characteristic
163	C₃×Dic₅	—	8	2	no	—
164	C₅×Dic₃	—	2	2	no	—
165	C₅×Dic₃	—	4	1	no	—
166	C₅×Dic₃	—	4	1	no	—
167	C₃×#41	—	2	2	no	—
168	C₃×#41	—	2	2	no	—
169	C₂×C₃×Q₈	C₂×Dic₅	1	1	yes	characteristic
170	C₂×C₃×D₄	C₂×Dic₅	1	1	yes	characteristic
171	C₂×C₃×D₄	C₂²⋊D₅	1	2	yes	—
172	C₂²×C₃×C₄	C₂²⋊D₅	1	2	yes	—
173	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
174	C₂×#67	—	2	2	no	—
175	C₂×#67	—	2	1	no	—
176	C₂²×Dic₃	—	2	1	no	—
177	#235	—	2	2	no	—
178	#235	—	2	2	no	—
179	C₂×D₁₂	C₂×Dic₅	1	1	yes	characteristic
180	C₂×C₄×D₃	—	2	1	no	—
181	C₂×Dic₆	C₂×Dic₅	1	1	yes	characteristic
182	C₃×C₄⋊C₄	—	10	2	no	—
183	C₃×C₂²⋊C₄	—	10	2	no	—
184	C₃×C₄²	—	10	2	no	—
185	#217	—	2	2	no	—
186	(C₂×D₃)⋊C₄	—	2	2	no	—
187	Dic₃⋊C₄	—	2	2	no	—
188	C₄×Dic₃	—	2	1	no	—
189	C₄×Dic₃	C₂×Dic₅	1	1	yes	characteristic
190	C₄×Dic₃	C₂×Dic₅	1	1	yes	characteristic
191	#208	—	5	4	no	—
192	#208	—	10	4	no	—
193	#208	—	10	2	no	—
194	C₂×C₃⋊C₈	—	10	4	no	—
195	C₂×C₃⋊C₈	—	10	2	no	—
196	C₂³×C₅	—	6	1	no	—
197	C₂³×C₅	(C₂×D₃)⋊C₄	1	2	yes	—
198	C₂³×C₅	C₂×#67	1	1	yes	characteristic
199	C₅×Q₈	—	2	2	no	—
200	C₅×Q₈	—	6	2	no	—
201	C₅×D₄	—	2	4	no	—
202	C₅×D₄	—	2	2	no	—
203	C₅×D₄	—	6	2	no	—
204	C₅×D₄	—	12	4	no	—
205	C₅×D₄	—	12	2	no	—
206	C₂×C₄×C₅	—	2	2	no	—
207	C₂×C₄×C₅	—	2	2	no	—
208	C₂×C₄×C₅	—	6	1	no	—
209	C₂×C₄×C₅	—	6	1	no	—
210	C₂×C₄×C₅	—	6	1	no	—
211	C₂×C₄×C₅	—	12	2	no	—
212	C₂×C₄×C₅	—	12	1	no	—
213	C₂×C₄×C₅	—	12	1	no	—
214	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	2	yes	—
215	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
216	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
217	C₂×C₄×C₅	D₃×D₄	1	2	yes	—
218	C₂×C₄×C₅	#237	1	2	yes	—
219	C₂×Dic₅	—	4	2	no	—
220	#168	—	6	4	no	—
221	C₂×#41	—	3	1	no	—
222	C₂×#41	D₃×D₅	1	1	yes	characteristic
223	#131	—	30	4	no	—
224	C₂×M₁₆	—	15	2	no	—
225	C₂×M₁₆	—	30	2	no	—
226	C₄⋊Q₈	—	3	1	no	—
227	#127	—	3	1	no	—
228	#124	—	3	1	no	—
229	#124	—	3	1	no	—
230	#121	—	6	2	no	—
231	C₄×D₄	—	6	2	no	—
232	#117	—	5	2	no	—
233	C₂×C₄²	—	3	1	no	—
234	#104	—	30	4	no	—
235	#99	—	10	2	no	—
236	C₂×C₃×C₅	—	2	1	no	—
237	C₂×C₃×C₅	—	2	2	no	—
238	C₂×C₃×C₅	—	4	2	no	—
239	C₂×C₃×C₅	#386	1	1	yes	characteristic
240	C₅×D₃	—	8	1	no	—
241	C₂³×C₃	C₂²⋊Dic₅	1	2	yes	—
242	C₂³×C₃	C₂×C₂²⋊D₅	1	1	yes	characteristic
243	C₂²×D₃	—	2	1	no	—
244	C₃×Q₈	—	2	2	no	—
245	C₃×D₄	—	2	4	no	—
246	C₃×D₄	—	2	2	no	—
247	C₂×C₃×C₄	—	2	2	no	—
248	C₂×C₃×C₄	—	2	2	no	—
249	C₂×C₃×C₄	—	20	2	no	—
250	C₂×C₃×C₄	C₂²⋊Dic₅	1	2	yes	—
251	C₂×C₃×C₄	D₄×D₅	1	2	yes	—
252	C₂×C₃×C₄	#711	1	2	yes	—
253	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
254	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
255	#67	—	4	4	no	—
256	#67	—	4	2	no	—
257	C₂×Dic₃	—	2	1	no	—
258	C₂×Dic₃	—	2	1	no	—
259	C₂×Dic₃	—	2	1	no	—
260	C₂×Dic₃	—	4	2	no	—
261	D₁₂	—	2	2	no	—
262	C₄×D₃	—	4	1	no	—
263	C₄×D₃	—	4	1	no	—
264	Dic₆	—	2	2	no	—
265	C₃⋊C₈	—	10	4	no	—
266	C₂²×C₅	—	2	2	no	—
267	C₂²×C₅	—	4	1	no	—
268	C₂²×C₅	—	4	2	no	—
269	C₂²×C₅	—	12	2	no	—
270	C₂²×C₅	—	12	1	no	—
271	C₂²×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
272	C₂²×C₅	#939	1	2	yes	—
273	C₄×C₅	—	2	2	no	—
274	C₄×C₅	—	6	2	no	—
275	C₄×C₅	—	12	1	no	—
276	C₄×C₅	—	12	1	no	—
277	C₄×C₅	#67⋊C₄	1	2	yes	—
278	C₄×C₅	#882	1	1	yes	characteristic
279	C₄×C₅	#886	1	1	yes	characteristic
280	Dic₅	—	8	2	no	—
281	#41	—	2	2	no	—
282	#41	—	2	2	no	—
283	#41	—	6	2	no	—
284	#41	—	6	2	no	—
285	C₂×Q₈	—	3	1	no	—
286	C₂×Q₈	D₃×Dic₅	1	1	yes	characteristic
287	C₂×D₄	—	3	1	no	—
288	C₂×D₄	—	6	2	no	—
289	C₂×D₄	—	6	1	no	—
290	C₂×D₄	D₃×Dic₅	1	1	yes	characteristic
291	C₂×D₄	#1201	1	2	yes	—
292	C₂²×C₄	—	6	1	no	—
293	C₂²×C₄	—	6	1	no	—
294	C₂²×C₄	#1203	1	2	yes	—
295	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
296	M₁₆	—	15	4	no	—
297	M₁₆	—	30	4	no	—
298	M₁₆	—	30	2	no	—
299	C₂×C₈	—	30	4	no	—
300	C₂×C₈	—	30	2	no	—
301	C₄⋊C₄	—	6	2	no	—
302	C₄⋊C₄	—	10	2	no	—
303	C₂²⋊C₄	—	6	2	no	—
304	C₂²⋊C₄	—	6	2	no	—
305	C₂²⋊C₄	—	10	2	no	—
306	C₄²	—	3	1	no	—
307	C₄²	—	3	1	no	—
308	C₄²	—	6	1	no	—
309	C₄²	—	10	2	no	—
310	C₃×C₅	#1884	1	1	yes	characteristic, complemented, Hall
311	C₂²×C₃	—	2	2	no	—
312	C₂²×C₃	—	4	1	no	—
313	C₂²×C₃	—	4	2	no	—
314	C₂²×C₃	#4085	1	2	yes	—
315	C₂²×C₃	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
316	C₂×D₃	—	4	2	no	—
317	C₂×D₃	—	4	1	no	—
318	C₃×C₄	—	2	2	no	—
319	C₃×C₄	—	40	2	no	—
320	C₃×C₄	D₄×Dic₅	1	2	yes	—
321	C₃×C₄	#4081	1	1	yes	characteristic
322	C₃×C₄	#4083	1	1	yes	characteristic
323	Dic₃	—	2	2	no	—
324	Dic₃	—	4	1	no	—
325	Dic₃	—	4	1	no	—
326	C₂×C₅	—	2	2	no	—
327	C₂×C₅	—	2	1	no	—
328	C₂×C₅	—	4	2	no	—
329	C₂×C₅	—	24	1	no	—
330	C₂×C₅	#4958	1	1	yes	characteristic
331	C₂³	—	6	1	no	—
332	C₂³	(C₂×D₃)⋊Dic₅	1	2	yes	—
333	C₂³	C₂×#1201	1	1	yes	characteristic
334	Q₈	—	2	2	no	—
335	Q₈	—	6	2	no	—
336	D₄	—	2	4	no	—
337	D₄	—	2	2	no	—
338	D₄	—	6	2	no	—
339	D₄	—	12	4	no	—
340	D₄	—	12	2	no	—
341	C₂×C₄	—	2	2	no	—
342	C₂×C₄	—	2	2	no	—
343	C₂×C₄	—	6	1	no	—
344	C₂×C₄	—	6	1	no	—
345	C₂×C₄	—	6	1	no	—
346	C₂×C₄	—	12	2	no	—
347	C₂×C₄	—	12	1	no	—
348	C₂×C₄	—	12	1	no	—
349	C₂×C₄	—	20	2	no	—
350	C₂×C₄	(C₂×D₃)⋊Dic₅	1	2	yes	—
351	C₂×C₄	#6780	1	2	yes	—
352	C₂×C₄	C₂×D₃×Dic₅	1	1	yes	characteristic
353	C₂×C₄	C₂×#1203	1	1	yes	characteristic
354	C₂×C₄	#6790	1	2	yes	—
355	C₈	—	30	4	no	—
356	C₂×C₃	—	2	1	no	—
357	C₂×C₃	—	2	2	no	—
358	C₂×C₃	—	4	2	no	—
359	C₂×C₃	#65509	1	1	yes	characteristic
360	D₃	—	8	1	no	—
361	C₅	#70045	1	1	yes	characteristic, complemented, Sylow
362	C₂²	—	2	2	no	—
363	C₂²	—	4	1	no	—
364	C₂²	—	4	2	no	—
365	C₂²	—	12	2	no	—
366	C₂²	—	12	1	no	—
367	C₂²	C₂×(C₂×D₃)⋊Dic₅	1	1	yes	characteristic
368	C₂²	#91571	1	2	yes	—
369	C₄	—	2	2	no	—
370	C₄	—	6	2	no	—
371	C₄	—	12	1	no	—
372	C₄	—	12	1	no	—
373	C₄	—	40	2	no	—
374	C₄	#91543	1	1	yes	characteristic
375	C₄	#67⋊Dic₅	1	2	yes	—
376	C₄	#91573	1	1	yes	characteristic
377	C₃	#10602610	1	1	yes	characteristic, complemented, Sylow
378	C₂	—	2	1	no	—
379	C₂	—	2	2	no	—
380	C₂	—	4	2	no	—
381	C₂	—	24	1	no	—
382	C₂	#11749501	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#91573
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₂²×C₄	C₂×D₃×D₅
Fitting	C₂×C₃×C₅×#41	C₂²
socle	C₂×C₃×C₅	#386
prime radical	C₅×#5832	C₂
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	C₂×#41	D₃×D₅
hypocenter	C₃×C₅	#1884
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#41

Complements

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

N	H ≅ G/N	type
1. #11756578	C₂	semidirect
3. C₃×#65529	C₂	semidirect
4. #11748425	C₂	semidirect
5. #11745974	C₂	semidirect
12. C₅×#947	C₄	semidirect
13. C₅×#944	C₄	semidirect
14. C₅×#940	C₄	semidirect
16. C₅×#936	C₄	semidirect
25. #65529	D₃	semidirect
45. #6710	D₄	semidirect
107. #947	Dic₅	semidirect
108. #944	Dic₅	semidirect
109. #940	Dic₅	semidirect
111. #936	Dic₅	semidirect
310. C₃×C₅	#1884	semidirect
361. C₅	#70045	semidirect
377. C₃	#10602610	semidirect

Maximal subgroups

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

subgroup	index
#11756578	2
C₅×#5832	2
C₃×#65529	2
#11748425	2
#11745974	2
#10602610	3
#70045	5