Subgroups — #14285144

Order 1920.

484 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#5716	C₂	1	1	yes	maximal, characteristic
2	#11746875	C₂	1	1	yes	maximal, characteristic, complemented
3	#11746853	C₂	1	1	yes	maximal, characteristic, complemented
4	#11746365	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#64764	C₂	1	1	yes	maximal, characteristic, complemented
6	#11745858	C₂	1	1	yes	maximal, characteristic, complemented
7	#11745857	C₂	1	1	yes	maximal, characteristic, complemented
8	#10602601	—	3	1	no	maximal, Hall
9	C₂×C₅×#237	—	2	1	no	—
10	C₃×C₅×#127	C₂²	1	1	yes	characteristic
11	C₅×#938	—	2	1	no	—
12	C₅×#938	C₄	1	2	yes	—
13	C₅×#936	—	2	1	no	—
14	C₅×D₄×Dic₃	—	2	1	no	—
15	C₅×D₄×Dic₃	C₄	1	2	yes	—
16	C₄²×C₅×D₃	C₂²	1	1	yes	characteristic
17	C₅×#869	C₂²	1	1	yes	characteristic
18	C₄⋊(C₃⋊#168)	C₂²	1	1	yes	characteristic
19	C₄⋊(C₃⋊#168)	C₂²	1	1	yes	characteristic
20	C₃×C₄⋊#168	C₂²	1	1	yes	characteristic
21	C₃×C₄⋊#168	C₂²	1	1	yes	characteristic
22	(C₂×D₃)⋊#168	—	2	1	no	—
23	(C₂×D₃)⋊#168	—	2	1	no	—
24	#90953	—	2	1	no	—
25	Dic₃×#168	—	2	1	no	—
26	#68386	—	5	1	no	maximal, Hall
27	C₅×#532	—	3	1	no	—
28	#65215	—	3	1	no	—
29	#65214	—	3	1	no	—
30	#64767	—	3	1	no	—
31	#64767	—	3	1	no	—
32	#64764	—	3	1	no	—
33	#64764	D₃	1	1	yes	characteristic, complemented
34	C₂×C₃×C₅×D₄	—	2	1	no	—
35	C₂×C₃×C₅×D₄	—	2	1	no	—
36	C₂×C₃×C₅×D₄	C₂×C₄	1	2	yes	—
37	C₂×C₅×#67	—	2	2	no	—
38	C₂²×C₅×Dic₃	—	2	2	no	—
39	C₅×#237	—	4	2	no	—
40	C₅×#237	—	4	2	no	—
41	C₂×C₄×C₅×D₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄×C₅×D₃	D₄	1	1	yes	characteristic
43	C₂×C₄×C₅×D₃	D₄	1	1	yes	characteristic
44	C₂×C₅×Dic₆	—	2	1	no	—
45	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
46	C₂×C₃⋊#168	—	2	1	no	—
47	C₂×C₃⋊#168	—	2	1	no	—
48	C₅×#217	—	2	2	no	—
49	C₅×#217	—	2	2	no	—
50	C₅×C₄⋊Dic₃	—	2	1	no	—
51	C₅×C₄⋊Dic₃	C₂×C₄	1	2	yes	—
52	C₄×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
53	C₄×C₅×Dic₃	D₄	1	1	yes	characteristic
54	C₄×C₅×Dic₃	D₄	1	1	yes	characteristic
55	C₂×C₃×#168	—	2	1	no	—
56	C₂×C₃×#168	—	2	1	no	—
57	#5716	D₅	1	1	yes	characteristic
58	#4835	—	5	1	no	—
59	#4833	—	5	1	no	—
60	C₃×#331	—	5	1	no	—
61	#4544	—	5	1	no	—
62	#4486	—	5	1	no	—
63	#4482	—	5	1	no	—
64	C₂×C₅×#41	—	6	1	no	—
65	C₅×C₄⋊Q₈	—	3	1	no	—
66	C₅×#127	C₂×D₃	1	1	yes	characteristic
67	C₅×#124	—	6	1	no	—
68	C₅×#121	—	3	2	no	—
69	C₅×#121	—	6	1	no	—
70	C₄×C₅×D₄	—	3	2	no	—
71	C₄×C₅×D₄	—	6	1	no	—
72	C₂×C₄²×C₅	—	3	1	no	—
73	C₂²⋊#168	—	6	1	no	—
74	C₂²⋊#168	—	6	1	no	—
75	C₄⋊#168	—	3	1	no	—
76	C₄⋊#168	—	3	1	no	—
77	C₄⋊#168	C₂×D₃	1	1	yes	characteristic
78	C₄⋊#168	C₂×D₃	1	1	yes	characteristic
79	#3935	—	6	1	no	—
80	C₄×#168	—	6	1	no	—
81	#1553	—	15	1	no	Sylow
82	C₂³×C₃×C₅	—	2	2	no	—
83	C₂²×C₅×D₃	C₂²⋊C₄	1	1	yes	characteristic
84	C₃×C₅×D₄	—	4	2	no	—
85	C₃×C₅×D₄	—	4	1	no	—
86	C₃×C₅×D₄	—	4	1	no	—
87	C₃×C₅×D₄	—	4	1	no	—
88	C₃×C₅×D₄	—	4	1	no	—
89	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
91	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
92	C₅×#67	—	4	4	no	—
93	C₂×C₅×Dic₃	—	2	2	no	—
94	C₂×C₅×Dic₃	—	4	4	no	—
95	C₂×C₅×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₅×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₅×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
98	C₄×C₅×D₃	—	2	1	no	—
99	C₄×C₅×D₃	—	2	1	no	—
100	C₄×C₅×D₃	—	2	1	no	—
101	C₄×C₅×D₃	—	2	1	no	—
102	C₄×C₅×D₃	—	4	1	no	—
103	C₅×Dic₆	—	4	1	no	—
104	C₅×Dic₆	—	4	1	no	—
105	C₃⋊#168	—	4	1	no	—
106	C₃⋊#168	—	4	1	no	—
107	C₃×#168	—	4	1	no	—
108	C₃×#168	—	4	1	no	—
109	C₂×#237	—	2	1	no	—
110	C₃×#127	C₂×D₅	1	1	yes	characteristic
111	#938	—	2	1	no	—
112	#938	Dic₅	1	2	yes	—
113	#936	—	2	1	no	—
114	D₄×Dic₃	—	2	1	no	—
115	D₄×Dic₃	Dic₅	1	2	yes	—
116	C₄²×D₃	C₂×D₅	1	1	yes	characteristic
117	#869	C₂×D₅	1	1	yes	characteristic
118	C₃×C₄⋊C₈	—	5	1	no	—
119	C₃×C₄⋊C₈	—	5	1	no	—
120	(C₂×D₃)⋊C₈	—	10	1	no	—
121	(C₂×D₃)⋊C₈	—	10	1	no	—
122	#815	—	10	1	no	—
123	C₈×Dic₃	—	10	1	no	—
124	C₄⋊(C₃⋊C₈)	—	5	1	no	—
125	C₄⋊(C₃⋊C₈)	—	5	1	no	—
126	C₅×#41	—	12	2	no	—
127	C₅×#41	—	12	2	no	—
128	C₂×C₅×Q₈	—	6	1	no	—
129	C₂×C₅×D₄	—	2	1	no	—
130	C₂×C₅×D₄	—	2	1	no	—
131	C₂×C₅×D₄	—	6	2	no	—
132	C₂×C₅×D₄	C₄×D₃	1	2	yes	—
133	C₂²×C₄×C₅	—	3	1	no	—
134	C₂²×C₄×C₅	—	3	1	no	—
135	C₂²×C₄×C₅	—	3	1	no	—
136	C₂²×C₄×C₅	—	6	2	no	—
137	C₅×C₄⋊C₄	—	3	2	no	—
138	C₅×C₄⋊C₄	—	6	1	no	—
139	C₅×C₂²⋊C₄	—	6	2	no	—
140	C₅×C₂²⋊C₄	—	6	2	no	—
141	C₄²×C₅	—	3	1	no	—
142	C₄²×C₅	—	3	1	no	—
143	C₄²×C₅	—	3	1	no	—
144	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
145	C₂×#168	—	2	1	no	—
146	C₂×#168	—	2	1	no	—
147	C₂×#168	—	6	1	no	—
148	C₂×#168	—	6	1	no	—
149	#532	—	3	1	no	—
150	#432	—	15	1	no	—
151	#431	—	15	1	no	—
152	#332	—	15	1	no	—
153	#332	—	15	1	no	—
154	#331	—	5	1	no	—
155	#331	—	15	1	no	—
156	C₂²×C₃×C₅	—	4	2	no	—
157	C₂²×C₃×C₅	—	4	2	no	—
158	C₂²×C₃×C₅	—	4	2	no	—
159	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
160	C₂×C₅×D₃	—	2	1	no	—
161	C₂×C₅×D₃	—	2	1	no	—
162	C₂×C₅×D₃	—	2	1	no	—
163	C₃×C₄×C₅	—	2	1	no	—
164	C₃×C₄×C₅	#135	1	1	yes	characteristic
165	C₃×C₄×C₅	#135	1	1	yes	characteristic
166	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
167	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
168	C₅×Dic₃	—	2	1	no	—
169	C₅×Dic₃	—	2	1	no	—
170	C₅×Dic₃	—	2	1	no	—
171	C₅×Dic₃	—	4	2	no	—
172	C₂×C₃×D₄	—	2	1	no	—
173	C₂×C₃×D₄	—	2	1	no	—
174	C₂×C₃×D₄	C₂×Dic₅	1	2	yes	—
175	C₂×#67	—	2	2	no	—
176	C₂²×Dic₃	—	2	2	no	—
177	#237	—	4	2	no	—
178	#237	—	4	2	no	—
179	C₂×C₄×D₃	C₂×Dic₅	1	1	yes	characteristic
180	C₂×C₄×D₃	C₂²⋊D₅	1	1	yes	characteristic
181	C₂×C₄×D₃	C₂²⋊D₅	1	1	yes	characteristic
182	C₂×Dic₆	—	2	1	no	—
183	C₂×C₃×C₈	—	10	1	no	—
184	C₂×C₃×C₈	—	10	1	no	—
185	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
186	#217	—	2	2	no	—
187	#217	—	2	2	no	—
188	C₄⋊Dic₃	—	2	1	no	—
189	C₄⋊Dic₃	C₂×Dic₅	1	2	yes	—
190	C₄×Dic₃	C₂×Dic₅	1	1	yes	characteristic
191	C₄×Dic₃	C₂²⋊D₅	1	1	yes	characteristic
192	C₄×Dic₃	C₂²⋊D₅	1	1	yes	characteristic
193	C₂×C₃⋊C₈	—	10	1	no	—
194	C₂×C₃⋊C₈	—	10	1	no	—
195	C₂³×C₅	—	2	2	no	—
196	C₂³×C₅	—	3	1	no	—
197	C₅×Q₈	—	12	1	no	—
198	C₅×Q₈	—	12	1	no	—
199	C₅×D₄	—	4	2	no	—
200	C₅×D₄	—	4	1	no	—
201	C₅×D₄	—	4	1	no	—
202	C₅×D₄	—	4	1	no	—
203	C₅×D₄	—	4	1	no	—
204	C₅×D₄	—	12	4	no	—
205	C₂×C₄×C₅	—	3	1	no	—
206	C₂×C₄×C₅	—	3	1	no	—
207	C₂×C₄×C₅	—	3	1	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	—	6	1	no	—
212	C₂×C₄×C₅	—	6	2	no	—
213	C₂×C₄×C₅	—	12	1	no	—
214	C₂×C₄×C₅	—	12	4	no	—
215	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
216	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
217	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
218	#168	—	4	1	no	—
219	#168	—	4	1	no	—
220	#168	—	12	1	no	—
221	#168	—	12	1	no	—
222	C₂×#41	—	6	1	no	—
223	C₄⋊Q₈	—	3	1	no	—
224	#127	D₃×D₅	1	1	yes	characteristic
225	#124	—	6	1	no	—
226	#121	—	3	2	no	—
227	#121	—	6	1	no	—
228	C₄×D₄	—	3	2	no	—
229	C₄×D₄	—	6	1	no	—
230	C₂×C₄²	—	3	1	no	—
231	C₄⋊C₈	—	5	1	no	—
232	C₄⋊C₈	—	5	1	no	—
233	C₄⋊C₈	—	15	1	no	—
234	C₄⋊C₈	—	15	1	no	—
235	C₂²⋊C₈	—	30	1	no	—
236	C₂²⋊C₈	—	30	1	no	—
237	#97	—	30	1	no	—
238	C₄×C₈	—	30	1	no	—
239	C₂×C₃×C₅	—	8	2	no	—
240	C₂×C₃×C₅	#413	1	1	yes	characteristic
241	C₂×C₃×C₅	#416	1	1	yes	characteristic
242	C₂×C₃×C₅	#418	1	1	yes	characteristic
243	C₅×D₃	—	4	1	no	—
244	C₂³×C₃	—	2	2	no	—
245	C₂²×D₃	C₂²⋊Dic₅	1	1	yes	characteristic
246	C₃×D₄	—	4	2	no	—
247	C₃×D₄	—	4	1	no	—
248	C₃×D₄	—	4	1	no	—
249	C₃×D₄	—	4	1	no	—
250	C₃×D₄	—	4	1	no	—
251	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
252	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
253	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
254	#67	—	4	4	no	—
255	C₂×Dic₃	—	2	2	no	—
256	C₂×Dic₃	—	4	4	no	—
257	C₂×Dic₃	C₂²⋊Dic₅	1	1	yes	characteristic
258	C₂×Dic₃	C₂²⋊Dic₅	1	1	yes	characteristic
259	C₂×Dic₃	C₂²⋊Dic₅	1	1	yes	characteristic
260	C₄×D₃	—	2	1	no	—
261	C₄×D₃	—	2	1	no	—
262	C₄×D₃	—	2	1	no	—
263	C₄×D₃	—	2	1	no	—
264	C₄×D₃	—	4	1	no	—
265	Dic₆	—	4	1	no	—
266	Dic₆	—	4	1	no	—
267	C₃×C₈	—	20	1	no	—
268	C₃×C₈	—	20	1	no	—
269	C₃⋊C₈	—	20	1	no	—
270	C₃⋊C₈	—	20	1	no	—
271	C₂²×C₅	—	4	2	no	—
272	C₂²×C₅	—	4	2	no	—
273	C₂²×C₅	—	4	2	no	—
274	C₂²×C₅	—	6	1	no	—
275	C₂²×C₅	—	6	1	no	—
276	C₂²×C₅	—	6	1	no	—
277	C₂²×C₅	D₃×C₂²⋊C₄	1	1	yes	characteristic
278	C₄×C₅	—	2	1	no	—
279	C₄×C₅	—	6	1	no	—
280	C₄×C₅	—	6	1	no	—
281	C₄×C₅	—	6	1	no	—
282	C₄×C₅	—	12	2	no	—
283	C₄×C₅	#911	1	1	yes	characteristic
284	C₄×C₅	#912	1	1	yes	characteristic
285	C₄×C₅	#914	1	1	yes	characteristic
286	C₄×C₅	#916	1	1	yes	characteristic
287	#41	—	12	2	no	—
288	#41	—	12	2	no	—
289	C₂×Q₈	—	6	1	no	—
290	C₂×D₄	—	2	1	no	—
291	C₂×D₄	—	2	1	no	—
292	C₂×D₄	—	6	2	no	—
293	C₂×D₄	D₃×Dic₅	1	2	yes	—
294	C₂²×C₄	—	3	1	no	—
295	C₂²×C₄	—	3	1	no	—
296	C₂²×C₄	—	3	1	no	—
297	C₂²×C₄	—	6	2	no	—
298	C₂×C₈	—	10	1	no	—
299	C₂×C₈	—	10	1	no	—
300	C₂×C₈	—	30	1	no	—
301	C₂×C₈	—	30	1	no	—
302	C₄⋊C₄	—	3	2	no	—
303	C₄⋊C₄	—	6	1	no	—
304	C₂²⋊C₄	—	6	2	no	—
305	C₂²⋊C₄	—	6	2	no	—
306	C₄²	—	3	1	no	—
307	C₄²	—	3	1	no	—
308	C₄²	—	3	1	no	—
309	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
310	C₃×C₅	#1553	1	1	yes	characteristic, complemented, Hall
311	C₂²×C₃	—	4	2	no	—
312	C₂²×C₃	—	4	2	no	—
313	C₂²×C₃	—	4	2	no	—
314	C₂²×C₃	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
315	C₂×D₃	—	2	1	no	—
316	C₂×D₃	—	2	1	no	—
317	C₂×D₃	—	2	1	no	—
318	C₃×C₄	—	2	1	no	—
319	C₃×C₄	#4078	1	1	yes	characteristic
320	C₃×C₄	#4078	1	1	yes	characteristic
321	C₃×C₄	#4096	1	1	yes	characteristic
322	C₃×C₄	#4096	1	1	yes	characteristic
323	Dic₃	—	2	1	no	—
324	Dic₃	—	2	1	no	—
325	Dic₃	—	2	1	no	—
326	Dic₃	—	4	2	no	—
327	C₂×C₅	—	8	2	no	—
328	C₂×C₅	—	12	1	no	—
329	C₂×C₅	#4745	1	1	yes	characteristic
330	C₂×C₅	#4770	1	1	yes	characteristic
331	C₂×C₅	#4772	1	1	yes	characteristic
332	C₂³	—	2	2	no	—
333	C₂³	—	3	1	no	—
334	Q₈	—	12	1	no	—
335	Q₈	—	12	1	no	—
336	D₄	—	4	2	no	—
337	D₄	—	4	1	no	—
338	D₄	—	4	1	no	—
339	D₄	—	4	1	no	—
340	D₄	—	4	1	no	—
341	D₄	—	12	4	no	—
342	C₂×C₄	—	3	1	no	—
343	C₂×C₄	—	3	1	no	—
344	C₂×C₄	—	3	1	no	—
345	C₂×C₄	—	6	1	no	—
346	C₂×C₄	—	6	1	no	—
347	C₂×C₄	—	6	1	no	—
348	C₂×C₄	—	6	1	no	—
349	C₂×C₄	—	6	2	no	—
350	C₂×C₄	—	12	1	no	—
351	C₂×C₄	—	12	4	no	—
352	C₂×C₄	C₂×D₃×Dic₅	1	1	yes	characteristic
353	C₂×C₄	D₃×C₂²⋊D₅	1	1	yes	characteristic
354	C₂×C₄	D₃×C₂²⋊D₅	1	1	yes	characteristic
355	C₈	—	20	1	no	—
356	C₈	—	20	1	no	—
357	C₈	—	60	1	no	—
358	C₈	—	60	1	no	—
359	C₂×C₃	—	8	2	no	—
360	C₂×C₃	#65504	1	1	yes	characteristic
361	C₂×C₃	#65522	1	1	yes	characteristic
362	C₂×C₃	#65526	1	1	yes	characteristic
363	D₃	—	4	1	no	—
364	C₅	#68386	1	1	yes	characteristic, complemented, Sylow
365	C₂²	—	4	2	no	—
366	C₂²	—	4	2	no	—
367	C₂²	—	4	2	no	—
368	C₂²	—	6	1	no	—
369	C₂²	—	6	1	no	—
370	C₂²	—	6	1	no	—
371	C₂²	D₃×C₂²⋊Dic₅	1	1	yes	characteristic
372	C₄	—	2	1	no	—
373	C₄	—	6	1	no	—
374	C₄	—	6	1	no	—
375	C₄	—	6	1	no	—
376	C₄	—	12	2	no	—
377	C₄	#91493	1	1	yes	characteristic
378	C₄	#91494	1	1	yes	characteristic
379	C₄	#91495	1	1	yes	characteristic
380	C₄	#91496	1	1	yes	characteristic
381	C₃	#10602601	1	1	yes	characteristic, complemented, Sylow
382	C₂	—	8	2	no	—
383	C₂	—	12	1	no	—
384	C₂	#11748965	1	1	yes	characteristic
385	C₂	#11748967	1	1	yes	characteristic
386	C₂	#11749126	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₃×C₂²⋊Dic₅
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₄²	C₂×D₃×D₅
Fitting	C₃×C₅×#127	C₂²
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₅×#5716	C₂
cosocle	C₃×C₄²×C₅	C₂³
hypercenter	#127	D₃×D₅
hypocenter	C₃×C₅	#1553
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #127

Complements

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

N	H ≅ G/N	type
2. #11746875	C₂	semidirect
3. #11746853	C₂	semidirect
4. #11746365	C₂	semidirect
5. C₃×#64764	C₂	semidirect
6. #11745858	C₂	semidirect
7. #11745857	C₂	semidirect
33. #64764	D₃	semidirect
310. C₃×C₅	#1553	semidirect
364. C₅	#68386	semidirect
381. C₃	#10602601	semidirect

Maximal subgroups

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

subgroup	index
C₅×#5716	2
#11746875	2
#11746853	2
#11746365	2
C₃×#64764	2
#11745858	2
#11745857	2
#10602601	3
#68386	5