Subgroups — #14285052

Order 1920.

504 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	#11752550	C₂	1	1	yes	maximal, characteristic
2	C₅×#4835	C₂	1	1	yes	maximal, characteristic, complemented
3	D₃×C₄⋊#168	C₂	1	1	yes	maximal, characteristic, complemented
4	#11746306	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#64701	C₂	1	1	yes	maximal, characteristic, complemented
6	#11745861	C₂	1	1	yes	maximal, characteristic, complemented
7	#11745854	C₂	1	1	yes	maximal, characteristic, complemented
8	#10601609	—	3	1	no	maximal, Hall
9	C₂×#6769	C₄	1	2	yes	—
10	#91763	C₂²	1	1	yes	characteristic
11	C₄²×C₅×D₃	C₂²	1	1	yes	characteristic
12	C₃×#4015	C₂²	1	1	yes	characteristic
13	#91460	—	2	1	no	—
14	#91460	—	2	1	no	—
15	#91428	—	2	1	no	—
16	#91428	—	2	1	no	—
17	#91374	C₄	1	2	yes	—
18	C₂×D₃×#168	—	2	1	no	—
19	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
20	C₄⋊(C₃⋊#168)	C₂²	1	1	yes	characteristic
21	C₅×(C₂×D₃)⋊C₈	—	2	1	no	—
22	C₅×#815	—	2	1	no	—
23	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
24	C₃×C₄⋊#168	C₂²	1	1	yes	characteristic
25	Dic₃⋊#168	—	2	1	no	—
26	#68485	—	5	1	no	maximal, Hall
27	#65811	—	3	1	no	—
28	C₅×#432	—	3	1	no	—
29	C₂×C₄⋊#168	—	3	1	no	—
30	#64705	—	3	1	no	—
31	#64705	—	3	1	no	—
32	#64701	—	3	1	no	—
33	#64701	D₃	1	1	yes	characteristic, complemented
34	C₂×D₆₀	—	2	1	no	—
35	C₂×D₆₀	—	2	1	no	—
36	C₂×D₆₀	C₂×C₄	1	2	yes	—
37	C₂×C₄×C₅×D₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×C₅×D₃	D₄	1	1	yes	characteristic
39	C₂×C₄×C₅×D₃	D₄	1	1	yes	characteristic
40	C₂×C₃×Dic₁₀	C₂×C₄	1	2	yes	—
41	C₂×#1203	—	2	2	no	—
42	C₂×#1200	—	2	2	no	—
43	#6769	—	4	4	no	—
44	C₂×C₃×C₅×C₈	—	2	1	no	—
45	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
46	C₂×C₃⋊#168	—	2	1	no	—
47	C₄×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
48	C₄×C₅×Dic₃	D₄	1	1	yes	characteristic
49	C₄×C₅×Dic₃	D₄	1	1	yes	characteristic
50	C₂×C₅×C₃⋊C₈	—	2	1	no	—
51	C₃×C₄⋊Dic₅	—	2	1	no	—
52	C₃×C₄⋊Dic₅	—	2	1	no	—
53	C₂×C₃×#168	—	2	1	no	—
54	#6667	—	2	2	no	—
55	#6667	—	2	2	no	—
56	D₃×#168	—	4	2	no	—
57	#5725	—	5	1	no	—
58	#4835	D₅	1	1	yes	characteristic, complemented
59	D₃×C₄⋊C₈	—	5	1	no	—
60	C₃×#333	—	5	1	no	—
61	#4546	—	5	1	no	—
62	#4485	—	5	1	no	—
63	#4485	—	5	1	no	—
64	C₂×#709	—	3	2	no	—
65	C₂×C₄²×C₅	—	3	1	no	—
66	#4076	—	6	1	no	—
67	#4076	—	6	1	no	—
68	C₂²×#168	—	6	1	no	—
69	#4021	—	3	2	no	—
70	#4020	—	3	1	no	—
71	C₄×D₂₀	—	6	1	no	—
72	C₄×D₂₀	—	6	1	no	—
73	#4015	C₂×D₃	1	1	yes	characteristic
74	C₅×C₄⋊C₈	—	3	1	no	—
75	C₅×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
76	C₅×C₂²⋊C₈	—	6	1	no	—
77	C₅×#97	—	6	1	no	—
78	C₄⋊#168	—	3	1	no	—
79	C₄⋊#168	—	6	1	no	—
80	C₄⋊#168	C₂×D₃	1	1	yes	characteristic
81	#1551	—	15	1	no	Sylow
82	C₂²×D₁₅	—	2	2	no	—
83	C₂²×C₅×D₃	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
86	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
87	D₆₀	—	4	2	no	—
88	D₆₀	—	4	1	no	—
89	D₆₀	—	4	1	no	—
90	D₆₀	—	4	1	no	—
91	D₆₀	—	4	1	no	—
92	C₂×C₅×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₅×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₅×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
95	C₄×C₅×D₃	—	2	2	no	—
96	C₄×C₅×D₃	—	2	1	no	—
97	C₄×C₅×D₃	—	2	1	no	—
98	C₄×C₅×D₃	—	2	1	no	—
99	C₄×C₅×D₃	—	2	1	no	—
100	C₂×C₃×Dic₅	—	2	2	no	—
101	C₃×Dic₁₀	—	4	2	no	—
102	#1203	—	4	4	no	—
103	#1200	—	4	4	no	—
104	C₃×C₅×C₈	—	4	1	no	—
105	C₃⋊#168	—	4	1	no	—
106	C₃×#168	—	4	1	no	—
107	C₅×C₃⋊C₈	—	4	1	no	—
108	C₂×#239	—	5	2	no	—
109	C₃×C₄⋊Q₈	—	5	1	no	—
110	#947	—	5	2	no	—
111	C₂×D₃×C₈	—	10	1	no	—
112	#895	—	10	1	no	—
113	#895	—	10	1	no	—
114	#893	—	10	1	no	—
115	#893	—	10	1	no	—
116	#874	—	5	1	no	—
117	C₄²×D₃	C₂×D₅	1	1	yes	characteristic
118	C₃×C₄⋊C₈	—	5	1	no	—
119	C₃×C₄⋊C₈	C₂×D₅	1	1	yes	characteristic
120	(C₂×D₃)⋊C₈	—	2	1	no	—
121	#815	—	2	1	no	—
122	Dic₃⋊C₈	—	10	1	no	—
123	C₄⋊(C₃⋊C₈)	—	5	1	no	—
124	C₄⋊(C₃⋊C₈)	C₂×D₅	1	1	yes	characteristic
125	C₂²×C₄×C₅	—	3	1	no	—
126	C₂²×C₄×C₅	—	3	1	no	—
127	C₂²×C₄×C₅	—	3	1	no	—
128	C₂×C₂²⋊D₅	—	6	2	no	—
129	#709	—	12	4	no	—
130	C₂×D₂₀	—	3	2	no	—
131	C₂×D₂₀	—	6	1	no	—
132	C₂×D₂₀	—	6	1	no	—
133	C₂×C₄×D₅	—	6	2	no	—
134	C₂×Dic₁₀	C₄×D₃	1	2	yes	—
135	C₂×C₅×C₈	—	2	1	no	—
136	C₂×C₅×C₈	—	6	1	no	—
137	C₄²×C₅	—	3	1	no	—
138	C₄²×C₅	—	3	1	no	—
139	C₄²×C₅	—	3	1	no	—
140	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
141	#685	—	6	2	no	—
142	#685	—	6	2	no	—
143	C₄⋊Dic₅	—	2	1	no	—
144	C₄⋊Dic₅	—	2	1	no	—
145	C₂×#168	—	2	1	no	—
146	C₂×#168	—	6	1	no	—
147	C₂×#168	—	12	2	no	—
148	#532	—	15	1	no	—
149	#432	—	3	1	no	—
150	C₂×C₄⋊C₈	—	15	1	no	—
151	#333	—	5	1	no	—
152	#333	—	15	1	no	—
153	#331	—	15	1	no	—
154	#331	—	15	1	no	—
155	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
156	C₂×D₁₅	—	4	2	no	—
157	C₂×D₁₅	—	4	2	no	—
158	C₂×D₁₅	—	4	2	no	—
159	C₂×C₅×D₃	—	2	1	no	—
160	C₂×C₅×D₃	—	2	1	no	—
161	C₂×C₅×D₃	#100	1	2	yes	—
162	C₃×C₄×C₅	—	2	1	no	—
163	C₃×C₄×C₅	#135	1	1	yes	characteristic
164	C₃×C₄×C₅	#135	1	1	yes	characteristic
165	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
166	C₃×C₄×C₅	#137	1	1	yes	characteristic
167	C₃×Dic₅	—	4	2	no	—
168	C₅×Dic₃	—	2	1	no	—
169	C₅×Dic₃	—	2	1	no	—
170	C₅×Dic₃	—	2	1	no	—
171	C₂×C₃×Q₈	—	5	2	no	—
172	#239	—	20	4	no	—
173	C₂×D₁₂	—	5	2	no	—
174	C₂×D₁₂	—	10	2	no	—
175	C₂×D₁₂	—	10	1	no	—
176	C₂×D₁₂	—	10	1	no	—
177	C₂×C₄×D₃	—	10	2	no	—
178	C₂×C₄×D₃	C₄×D₅	1	1	yes	characteristic
179	C₂×C₄×D₃	D₂₀	1	1	yes	characteristic
180	C₂×C₄×D₃	C₂²⋊D₅	1	1	yes	characteristic
181	C₂×C₃×C₈	—	2	1	no	—
182	C₂×C₃×C₈	—	10	1	no	—
183	C₃×C₄⋊C₄	—	10	1	no	—
184	C₃×C₄⋊C₄	—	10	1	no	—
185	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
186	(C₂×D₃)⋊C₄	—	10	2	no	—
187	(C₂×D₃)⋊C₄	—	10	2	no	—
188	C₄×Dic₃	C₄×D₅	1	1	yes	characteristic
189	C₄×Dic₃	D₂₀	1	1	yes	characteristic
190	C₄×Dic₃	C₂²⋊D₅	1	1	yes	characteristic
191	C₂×C₃⋊C₈	—	2	1	no	—
192	C₂×C₃⋊C₈	—	10	1	no	—
193	D₃×C₈	—	20	2	no	—
194	C₂³×C₅	—	3	1	no	—
195	C₂²×D₅	—	6	2	no	—
196	C₂×C₄×C₅	—	3	1	no	—
197	C₂×C₄×C₅	—	3	1	no	—
198	C₂×C₄×C₅	—	3	1	no	—
199	C₂×C₄×C₅	—	6	2	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	—	6	1	no	—
204	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
205	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
206	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
207	C₂²⋊D₅	—	12	4	no	—
208	C₂×Dic₅	—	2	2	no	—
209	D₂₀	—	12	2	no	—
210	D₂₀	—	12	1	no	—
211	D₂₀	—	12	1	no	—
212	D₂₀	—	12	1	no	—
213	D₂₀	—	12	1	no	—
214	C₄×D₅	—	12	4	no	—
215	Dic₁₀	—	4	2	no	—
216	C₅×C₈	—	4	1	no	—
217	C₅×C₈	—	12	1	no	—
218	#168	—	4	1	no	—
219	#168	—	12	1	no	—
220	C₂×#41	—	15	2	no	—
221	C₂²×C₈	—	30	1	no	—
222	C₄⋊Q₈	—	5	1	no	—
223	#127	—	15	1	no	—
224	#124	—	15	2	no	—
225	#121	—	30	1	no	—
226	#121	—	30	1	no	—
227	C₄×D₄	—	30	1	no	—
228	C₄×D₄	—	30	1	no	—
229	C₂×C₄²	—	3	1	no	—
230	C₄⋊C₈	—	3	1	no	—
231	C₄⋊C₈	—	5	1	no	—
232	C₄⋊C₈	—	15	1	no	—
233	C₄⋊C₈	—	30	1	no	—
234	C₄⋊C₈	D₃×D₅	1	1	yes	characteristic
235	C₂²⋊C₈	—	6	1	no	—
236	#97	—	6	1	no	—
237	C₂×C₃×C₅	C₂×#100	1	1	yes	characteristic
238	C₂×C₃×C₅	#416	1	1	yes	characteristic
239	C₂×C₃×C₅	#417	1	1	yes	characteristic
240	D₁₅	—	8	2	no	—
241	C₅×D₃	—	2	2	no	—
242	C₂²×D₃	—	10	2	no	—
243	C₂²×D₃	#685	1	1	yes	characteristic
244	C₃×Q₈	—	20	2	no	—
245	C₂×C₃×C₄	—	10	2	no	—
246	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
247	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
248	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
249	C₂×Dic₃	#685	1	1	yes	characteristic
250	C₂×Dic₃	#685	1	1	yes	characteristic
251	C₂×Dic₃	#685	1	1	yes	characteristic
252	D₁₂	—	20	2	no	—
253	D₁₂	—	20	4	no	—
254	D₁₂	—	20	1	no	—
255	D₁₂	—	20	1	no	—
256	D₁₂	—	20	1	no	—
257	D₁₂	—	20	1	no	—
258	C₄×D₃	—	2	2	no	—
259	C₄×D₃	—	2	1	no	—
260	C₄×D₃	—	2	1	no	—
261	C₄×D₃	—	2	1	no	—
262	C₄×D₃	—	2	1	no	—
263	C₄×D₃	—	20	4	no	—
264	C₃×C₈	—	4	1	no	—
265	C₃×C₈	—	20	1	no	—
266	C₃⋊C₈	—	4	1	no	—
267	C₃⋊C₈	—	20	1	no	—
268	C₂²×C₅	—	3	2	no	—
269	C₂²×C₅	—	6	1	no	—
270	C₂²×C₅	—	6	1	no	—
271	C₂²×C₅	D₃×C₂²⋊C₄	1	1	yes	characteristic
272	C₂×D₅	—	12	2	no	—
273	C₂×D₅	—	12	2	no	—
274	C₂×D₅	—	12	2	no	—
275	C₄×C₅	—	2	1	no	—
276	C₄×C₅	—	6	1	no	—
277	C₄×C₅	—	6	1	no	—
278	C₄×C₅	—	6	1	no	—
279	C₄×C₅	#914	1	1	yes	characteristic
280	C₄×C₅	#916	1	1	yes	characteristic
281	C₄×C₅	#918	1	1	yes	characteristic
282	C₄×C₅	#919	1	1	yes	characteristic
283	Dic₅	—	4	2	no	—
284	#41	—	60	4	no	—
285	C₂×Q₈	—	5	2	no	—
286	C₂×D₄	—	15	2	no	—
287	C₂×D₄	—	30	2	no	—
288	C₂×D₄	—	30	1	no	—
289	C₂×D₄	—	30	1	no	—
290	C₂²×C₄	—	3	1	no	—
291	C₂²×C₄	—	3	1	no	—
292	C₂²×C₄	—	3	1	no	—
293	C₂²×C₄	—	30	2	no	—
294	C₂×C₈	—	2	1	no	—
295	C₂×C₈	—	6	1	no	—
296	C₂×C₈	—	10	1	no	—
297	C₂×C₈	—	30	1	no	—
298	C₂×C₈	—	60	2	no	—
299	C₄⋊C₄	—	10	1	no	—
300	C₄⋊C₄	—	10	1	no	—
301	C₂²⋊C₄	—	30	2	no	—
302	C₂²⋊C₄	—	30	2	no	—
303	C₄²	—	3	1	no	—
304	C₄²	—	3	1	no	—
305	C₄²	—	3	1	no	—
306	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
307	C₃×C₅	#1551	1	1	yes	characteristic, complemented, Hall
308	C₂²×C₃	C₂×#685	1	1	yes	characteristic
309	C₂×D₃	—	2	1	no	—
310	C₂×D₃	—	2	1	no	—
311	C₂×D₃	—	20	2	no	—
312	C₂×D₃	—	20	2	no	—
313	C₂×D₃	—	20	2	no	—
314	C₂×D₃	#3955	1	2	yes	—
315	C₃×C₄	—	2	1	no	—
316	C₃×C₄	—	20	2	no	—
317	C₃×C₄	#4054	1	1	yes	characteristic
318	C₃×C₄	#4055	1	1	yes	characteristic
319	C₃×C₄	#4096	1	1	yes	characteristic
320	C₃×C₄	#4096	1	1	yes	characteristic
321	Dic₃	—	2	1	no	—
322	Dic₃	—	2	1	no	—
323	Dic₃	—	2	1	no	—
324	C₂×C₅	—	6	2	no	—
325	C₂×C₅	D₃×#100	1	1	yes	characteristic
326	C₂×C₅	#4803	1	1	yes	characteristic
327	C₂×C₅	#4804	1	1	yes	characteristic
328	D₅	—	24	2	no	—
329	C₂³	—	3	1	no	—
330	C₂³	—	30	2	no	—
331	Q₈	—	20	2	no	—
332	D₄	—	60	2	no	—
333	D₄	—	60	4	no	—
334	D₄	—	60	1	no	—
335	D₄	—	60	1	no	—
336	D₄	—	60	1	no	—
337	D₄	—	60	1	no	—
338	C₂×C₄	—	3	1	no	—
339	C₂×C₄	—	3	1	no	—
340	C₂×C₄	—	3	1	no	—
341	C₂×C₄	—	6	2	no	—
342	C₂×C₄	—	6	1	no	—
343	C₂×C₄	—	6	1	no	—
344	C₂×C₄	—	6	1	no	—
345	C₂×C₄	—	6	1	no	—
346	C₂×C₄	—	10	2	no	—
347	C₂×C₄	—	60	4	no	—
348	C₂×C₄	C₄×D₃×D₅	1	1	yes	characteristic
349	C₂×C₄	D₃×D₂₀	1	1	yes	characteristic
350	C₂×C₄	D₃×C₂²⋊D₅	1	1	yes	characteristic
351	C₈	—	4	1	no	—
352	C₈	—	12	1	no	—
353	C₈	—	20	1	no	—
354	C₈	—	60	1	no	—
355	C₂×C₃	#65291	1	1	yes	characteristic
356	C₂×C₃	C₂×#3955	1	1	yes	characteristic
357	C₂×C₃	#65422	1	1	yes	characteristic
358	D₃	—	2	2	no	—
359	D₃	—	40	2	no	—
360	C₅	#68485	1	1	yes	characteristic, complemented, Sylow
361	C₂²	—	3	2	no	—
362	C₂²	—	6	1	no	—
363	C₂²	—	6	1	no	—
364	C₂²	—	60	2	no	—
365	C₂²	—	60	2	no	—
366	C₂²	—	60	2	no	—
367	C₂²	D₃×#685	1	1	yes	characteristic
368	C₄	—	2	1	no	—
369	C₄	—	6	1	no	—
370	C₄	—	6	1	no	—
371	C₄	—	6	1	no	—
372	C₄	—	20	2	no	—
373	C₄	#91253	1	1	yes	characteristic
374	C₄	#91263	1	1	yes	characteristic
375	C₄	#91495	1	1	yes	characteristic
376	C₄	#91519	1	1	yes	characteristic
377	C₃	#10601609	1	1	yes	characteristic, complemented, Sylow
378	C₂	—	6	2	no	—
379	C₂	—	120	2	no	—
380	C₂	#11747504	1	1	yes	characteristic
381	C₂	#11747608	1	1	yes	characteristic
382	C₂	D₃×#3955	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₃×#685
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₄²	C₂×D₃×D₅
Fitting	C₃×C₅×C₄⋊C₈	C₂²
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	#11752550	C₂
cosocle	C₃×C₄²×C₅	C₂³
hypercenter	C₄⋊C₈	D₃×D₅
hypocenter	C₃×C₅	#1551
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
2. C₅×#4835	C₂	semidirect
3. D₃×C₄⋊#168	C₂	semidirect
4. #11746306	C₂	semidirect
5. C₃×#64701	C₂	semidirect
6. #11745861	C₂	semidirect
7. #11745854	C₂	semidirect
33. #64701	D₃	semidirect
58. #4835	D₅	semidirect
307. C₃×C₅	#1551	semidirect
360. C₅	#68485	semidirect
377. C₃	#10601609	semidirect

Maximal subgroups

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

subgroup	index
#11752550	2
C₅×#4835	2
D₃×C₄⋊#168	2
#11746306	2
C₃×#64701	2
#11745861	2
#11745854	2
#10601609	3
#68485	5