Subgroups — #14413466

Order 1920.

952 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#4047	C₂	1	1	yes	maximal, characteristic
2	#11754499	C₂	1	2	yes	maximal
3	#11751207	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅×#817	C₂	1	1	yes	maximal, characteristic, complemented
5	#11748459	C₂	1	2	yes	maximal, complemented
6	#11747671	C₂	1	2	yes	maximal, complemented
7	#11747618	C₂	1	4	yes	maximal, complemented
8	D₅×#817	C₂	1	2	yes	maximal, complemented
9	#10610847	—	3	1	no	maximal, Hall
10	C₂×C₃×#709	C₂²	1	1	yes	characteristic
11	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
12	#91815	C₂²	1	2	yes	—
13	C₂×C₅×C₄⋊Dic₃	C₂²	1	2	yes	—
14	C₂×C₃×#681	C₂²	1	1	yes	characteristic
15	C₃×#4047	C₄	1	8	yes	complemented
16	C₂×C₃×#676	C₂²	1	2	yes	—
17	C₂×C₃×C₈×D₅	C₂²	1	2	yes	—
18	Dic₃×C₂²⋊D₅	—	2	4	no	—
19	#91434	—	2	2	no	—
20	D₅×C₄⋊Dic₃	C₂²	1	4	yes	—
21	#91358	C₂²	1	4	yes	—
22	#91331	—	2	2	no	—
23	#91102	C₂²	1	2	yes	—
24	#91101	C₂²	1	1	yes	characteristic
25	#91101	C₂²	1	1	yes	characteristic
26	C₅×#817	C₂²	1	2	yes	—
27	C₅×#817	C₂²	1	2	yes	—
28	#90988	C₂²	1	4	yes	—
29	#90985	C₂²	1	4	yes	—
30	#83784	—	5	1	no	maximal, Hall
31	C₂×#4047	D₃	1	1	yes	characteristic
32	#65838	—	3	2	no	—
33	C₂×C₅×#106	—	3	1	no	—
34	#65395	—	3	1	no	—
35	#65253	—	3	2	no	—
36	#65150	—	3	4	no	—
37	#65149	—	3	2	no	—
38	D₅×#106	—	3	2	no	—
39	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
40	C₂²×C₅×Dic₃	—	2	2	no	—
41	C₂×C₃×C₂²⋊D₅	D₄	1	2	yes	—
42	C₃×#709	C₂×C₄	1	4	yes	—
43	C₃×#709	Q₈	1	4	yes	—
44	C₂×C₃×D₂₀	D₄	1	1	yes	characteristic
45	C₂×C₃×C₄×D₅	C₂³	1	2	yes	—
46	C₂×C₃×Dic₁₀	D₄	1	1	yes	characteristic
47	C₂×D₅×Dic₃	—	2	4	no	—
48	C₂×C₃×C₅×C₈	C₂×C₄	1	4	yes	complemented
49	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
50	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
51	#6719	—	2	2	no	—
52	C₄⋊Dic₁₅	C₂³	1	2	yes	—
53	C₄⋊Dic₁₅	C₂³	1	2	yes	—
54	C₄×Dic₁₅	—	2	2	no	—
55	C₅×C₄⋊Dic₃	—	2	2	no	—
56	C₅×C₄⋊Dic₃	C₂³	1	4	yes	—
57	C₃×#681	C₂×C₄	1	4	yes	complemented
58	C₂×C₃×#168	C₂³	1	2	yes	—
59	C₃×#676	C₂×C₄	1	8	yes	complemented
60	C₃×C₈×D₅	C₂×C₄	1	8	yes	complemented
61	#6668	—	2	4	no	—
62	#6665	—	2	4	no	—
63	Dic₃×Dic₅	—	2	4	no	—
64	C₂×C₃×#131	—	5	1	no	—
65	#5827	—	5	2	no	—
66	#5121	—	5	2	no	—
67	#5121	—	5	4	no	—
68	#5106	—	5	1	no	—
69	#5106	—	5	2	no	—
70	C₂×#817	—	5	2	no	—
71	C₂×#817	D₅	1	1	yes	characteristic, complemented
72	C₂×#709	C₂×D₃	1	1	yes	characteristic
73	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
74	C₂×C₅×C₄⋊C₄	—	3	2	no	—
75	C₄×C₂²⋊D₅	—	6	4	no	—
76	#4072	—	3	2	no	—
77	C₂×#681	C₂×D₃	1	1	yes	characteristic
78	#4047	Dic₃	1	8	yes	complemented
79	C₂×#676	C₂×D₃	1	2	yes	—
80	C₂×C₈×D₅	C₂×D₃	1	2	yes	—
81	#4039	—	6	2	no	—
82	#4038	—	3	4	no	—
83	D₅×C₄⋊C₄	—	3	4	no	—
84	#4033	—	6	2	no	—
85	C₅×#106	—	3	2	no	—
86	C₅×#106	—	3	2	no	—
87	#3950	—	3	2	no	—
88	#3949	—	3	1	no	—
89	#3949	—	3	1	no	—
90	#3940	—	3	4	no	—
91	#3939	—	3	4	no	—
92	#2912	—	15	1	no	Sylow
93	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₃×D₅	C₂×D₄	1	2	yes	—
95	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
96	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
97	C₂×C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
98	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
99	C₂×Dic₁₅	—	2	2	no	—
100	C₂×Dic₁₅	—	2	2	no	—
101	C₂×C₅×Dic₃	—	2	4	no	—
102	C₂×C₅×Dic₃	—	8	2	no	—
103	C₃×C₂²⋊D₅	C₄⋊C₄	1	8	yes	—
104	C₂×C₃×Dic₅	C₂×D₄	1	2	yes	—
105	C₃×D₂₀	C₄⋊C₄	1	4	yes	—
106	C₃×C₄×D₅	C₂²×C₄	1	4	yes	—
107	C₃×C₄×D₅	C₂×Q₈	1	4	yes	—
108	C₃×Dic₁₀	C₄⋊C₄	1	4	yes	—
109	D₅×Dic₃	—	8	4	no	—
110	C₃×C₅×C₈	C₂²×C₄	1	2	yes	—
111	C₃×C₅×C₈	C₂²×C₄	1	2	yes	—
112	C₃×#168	C₂²×C₄	1	4	yes	—
113	C₂×C₃×#41	—	5	1	no	—
114	C₃×#131	—	5	8	no	—
115	C₂×C₃×M₁₆	—	5	1	no	—
116	C₂×C₃×M₁₆	—	5	2	no	—
117	C₂²×C₃×C₈	—	5	2	no	—
118	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
119	Q₈×Dic₃	—	10	2	no	—
120	D₄×Dic₃	—	10	2	no	—
121	D₄×Dic₃	—	10	4	no	—
122	#926	—	5	2	no	—
123	#926	—	5	4	no	—
124	C₂×C₄⋊Dic₃	—	5	4	no	—
125	C₂×C₄⋊Dic₃	C₂×D₅	1	2	yes	—
126	#818	—	5	2	no	—
127	#818	—	5	4	no	—
128	#817	—	5	1	no	—
129	#817	—	5	1	no	—
130	#817	—	5	4	no	—
131	#817	C₂×D₅	1	2	yes	—
132	#817	C₂×D₅	1	2	yes	—
133	C₂²×C₄×C₅	—	6	2	no	—
134	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
135	C₂×C₂²⋊D₅	D₁₂	1	2	yes	—
136	#709	Dic₆	1	4	yes	—
137	#709	C₂×Dic₃	1	4	yes	—
138	C₂×D₂₀	D₁₂	1	1	yes	characteristic
139	C₂×C₄×D₅	—	6	4	no	—
140	C₂×C₄×D₅	C₂²×D₃	1	2	yes	—
141	C₂×Dic₁₀	D₁₂	1	1	yes	characteristic
142	C₂×C₅×C₈	C₂×Dic₃	1	4	yes	complemented
143	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
144	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
145	C₅×C₄⋊C₄	—	3	4	no	—
146	C₅×C₄⋊C₄	—	6	2	no	—
147	C₂²⋊Dic₅	—	6	2	no	—
148	#685	—	6	4	no	—
149	C₄⋊Dic₅	—	3	2	no	—
150	C₄⋊Dic₅	—	3	2	no	—
151	#683	—	6	4	no	—
152	C₄×Dic₅	—	6	2	no	—
153	C₄×Dic₅	—	6	4	no	—
154	#681	C₂×Dic₃	1	4	yes	complemented
155	C₂×#168	C₂²×D₃	1	2	yes	—
156	#676	C₂×Dic₃	1	8	yes	complemented
157	C₈×D₅	C₂×Dic₃	1	8	yes	complemented
158	C₂×#131	—	5	1	no	—
159	#520	—	15	2	no	—
160	#428	—	15	4	no	—
161	#428	—	15	2	no	—
162	#427	—	15	2	no	—
163	#427	—	15	1	no	—
164	C₂×#106	—	3	1	no	—
165	C₂×#106	—	15	2	no	—
166	C₂²×C₃×C₅	—	2	2	no	—
167	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	2	yes	—
168	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
169	C₂×C₃×D₅	—	2	4	no	—
170	C₂×C₃×D₅	C₂×C₄⋊C₄	1	4	yes	—
171	C₃×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
172	C₃×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
173	C₃×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
174	Dic₁₅	—	4	2	no	—
175	Dic₁₅	—	4	2	no	—
176	C₃×Dic₅	C₂×C₄⋊C₄	1	4	yes	—
177	C₅×Dic₃	—	4	4	no	—
178	C₃×#41	—	5	4	no	—
179	C₃×#41	—	5	4	no	—
180	C₂×C₃×Q₈	—	5	1	no	—
181	C₂×C₃×D₄	—	5	2	no	—
182	C₂×C₃×D₄	—	5	1	no	—
183	C₂²×C₃×C₄	—	5	2	no	—
184	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
185	C₂²×Dic₃	—	2	2	no	—
186	C₂²×Dic₃	—	10	4	no	—
187	C₃×M₁₆	—	5	4	no	—
188	C₃×M₁₆	—	5	8	no	—
189	C₂×C₃×C₈	—	5	2	no	—
190	C₂×C₃×C₈	—	5	8	no	—
191	C₂×C₃×C₈	C₄×D₅	1	4	yes	complemented
192	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
193	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
194	#217	—	10	4	no	—
195	#217	—	10	2	no	—
196	C₄⋊Dic₃	—	2	2	no	—
197	C₄⋊Dic₃	—	5	2	no	—
198	C₄⋊Dic₃	—	5	2	no	—
199	C₄⋊Dic₃	—	10	4	no	—
200	C₄⋊Dic₃	C₂²×D₅	1	4	yes	—
201	C₄×Dic₃	—	10	4	no	—
202	C₄×Dic₃	—	10	2	no	—
203	C₂³×C₅	C₂×D₁₂	1	1	yes	characteristic
204	C₂²×D₅	C₂×D₁₂	1	2	yes	—
205	C₂×C₄×C₅	—	6	4	no	—
206	C₂×C₄×C₅	—	24	2	no	—
207	C₂×C₄×C₅	C₂×Dic₆	1	2	yes	—
208	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
209	C₂×C₄×C₅	C₂²×Dic₃	1	2	yes	—
210	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
211	C₂²⋊D₅	C₄⋊Dic₃	1	8	yes	—
212	C₂×Dic₅	—	6	2	no	—
213	C₂×Dic₅	—	6	2	no	—
214	C₂×Dic₅	C₂×D₁₂	1	2	yes	—
215	D₂₀	C₄⋊Dic₃	1	4	yes	—
216	C₄×D₅	—	24	4	no	—
217	C₄×D₅	C₂×Dic₆	1	4	yes	—
218	C₄×D₅	C₂²×Dic₃	1	4	yes	—
219	Dic₁₀	C₄⋊Dic₃	1	4	yes	—
220	C₅×C₈	C₂²×Dic₃	1	2	yes	—
221	C₅×C₈	C₂²×Dic₃	1	2	yes	—
222	#168	C₂²×Dic₃	1	4	yes	—
223	C₂×#41	—	5	1	no	—
224	#131	—	5	8	no	—
225	C₂×M₁₆	—	5	1	no	—
226	C₂×M₁₆	—	5	2	no	—
227	C₂²×C₈	—	5	2	no	—
228	C₂²×C₈	D₃×D₅	1	1	yes	characteristic
229	C₄×Q₈	—	30	2	no	—
230	C₄×D₄	—	30	2	no	—
231	C₄×D₄	—	30	4	no	—
232	#117	—	15	4	no	—
233	#117	—	15	2	no	—
234	C₂×C₄⋊C₄	—	3	2	no	—
235	C₂×C₄⋊C₄	—	15	4	no	—
236	#107	—	15	4	no	—
237	#107	—	15	2	no	—
238	#106	—	3	2	no	—
239	#106	—	3	2	no	—
240	#106	—	15	4	no	—
241	#106	—	15	1	no	—
242	#106	—	15	1	no	—
243	C₂×C₃×C₅	—	2	2	no	—
244	C₂×C₃×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
245	C₂×C₃×C₅	#577	1	2	yes	—
246	C₃×D₅	—	2	4	no	—
247	C₂³×C₃	—	5	2	no	—
248	C₂³×C₃	D₄×D₅	1	1	yes	characteristic
249	C₃×Q₈	—	5	4	no	—
250	C₃×D₄	—	5	8	no	—
251	C₃×D₄	—	5	4	no	—
252	C₂×C₃×C₄	—	5	2	no	—
253	C₂×C₃×C₄	—	5	4	no	—
254	C₂×C₃×C₄	—	5	4	no	—
255	C₂×C₃×C₄	C₂×C₄×D₅	1	2	yes	—
256	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
257	C₂×C₃×C₄	Q₈×D₅	1	2	yes	—
258	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
259	C₂×Dic₃	—	2	4	no	—
260	C₂×Dic₃	—	8	2	no	—
261	C₂×Dic₃	—	10	2	no	—
262	C₂×Dic₃	—	10	2	no	—
263	C₂×Dic₃	—	40	4	no	—
264	C₃×C₈	—	5	4	no	—
265	C₃×C₈	C₂×C₄×D₅	1	2	yes	—
266	C₃×C₈	C₂×C₄×D₅	1	2	yes	—
267	C₂²×C₅	—	2	2	no	—
268	C₂²×C₅	C₂×C₄⋊Dic₃	1	2	yes	—
269	C₂²×C₅	C₂²×D₁₂	1	1	yes	characteristic
270	C₂×D₅	—	2	4	no	—
271	C₂×D₅	C₂×C₄⋊Dic₃	1	4	yes	—
272	C₄×C₅	—	12	4	no	—
273	C₄×C₅	C₂×C₄⋊Dic₃	1	2	yes	—
274	C₄×C₅	C₂²×Dic₆	1	1	yes	characteristic
275	C₄×C₅	C₂³×Dic₃	1	1	yes	characteristic
276	Dic₅	—	12	2	no	—
277	Dic₅	—	12	2	no	—
278	Dic₅	C₂×C₄⋊Dic₃	1	4	yes	—
279	#41	—	5	4	no	—
280	#41	—	5	4	no	—
281	C₂×Q₈	—	5	1	no	—
282	C₂×D₄	—	5	2	no	—
283	C₂×D₄	—	5	1	no	—
284	C₂²×C₄	—	5	2	no	—
285	C₂²×C₄	—	6	2	no	—
286	C₂²×C₄	—	30	4	no	—
287	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
288	M₁₆	—	5	4	no	—
289	M₁₆	—	5	8	no	—
290	C₂×C₈	—	5	2	no	—
291	C₂×C₈	—	5	8	no	—
292	C₂×C₈	D₅×Dic₃	1	4	yes	complemented
293	C₂×C₈	C₂×D₃×D₅	1	1	yes	characteristic
294	C₂×C₈	C₂×D₃×D₅	1	1	yes	characteristic
295	C₄⋊C₄	—	3	4	no	—
296	C₄⋊C₄	—	6	2	no	—
297	C₄⋊C₄	—	15	2	no	—
298	C₄⋊C₄	—	15	2	no	—
299	C₄⋊C₄	—	30	4	no	—
300	C₂²⋊C₄	—	30	4	no	—
301	C₂²⋊C₄	—	30	2	no	—
302	C₄²	—	30	4	no	—
303	C₄²	—	30	2	no	—
304	C₃×C₅	#2912	1	1	yes	characteristic, complemented, Hall
305	C₂²×C₃	—	2	2	no	—
306	C₂²×C₃	—	5	4	no	—
307	C₂²×C₃	—	10	4	no	—
308	C₂²×C₃	D₅×C₄⋊C₄	1	2	yes	—
309	C₂²×C₃	C₂×D₄×D₅	1	1	yes	characteristic
310	C₃×C₄	—	5	4	no	—
311	C₃×C₄	D₅×C₄⋊C₄	1	2	yes	—
312	C₃×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
313	C₃×C₄	C₂×Q₈×D₅	1	1	yes	characteristic
314	Dic₃	—	4	4	no	—
315	Dic₃	—	20	2	no	—
316	Dic₃	—	20	2	no	—
317	C₂×C₅	—	2	2	no	—
318	C₂×C₅	#5751	1	2	yes	—
319	C₂×C₅	C₂²×C₄⋊Dic₃	1	1	yes	characteristic
320	D₅	—	2	4	no	—
321	C₂³	—	5	2	no	—
322	C₂³	D₅×D₁₂	1	1	yes	characteristic
323	Q₈	—	5	4	no	—
324	D₄	—	5	8	no	—
325	D₄	—	5	4	no	—
326	C₂×C₄	—	5	2	no	—
327	C₂×C₄	—	5	4	no	—
328	C₂×C₄	—	5	4	no	—
329	C₂×C₄	—	6	4	no	—
330	C₂×C₄	—	24	2	no	—
331	C₂×C₄	—	30	2	no	—
332	C₂×C₄	—	30	2	no	—
333	C₂×C₄	—	120	4	no	—
334	C₂×C₄	D₅×Dic₆	1	2	yes	—
335	C₂×C₄	D₅×D₁₂	1	1	yes	characteristic
336	C₂×C₄	C₂×D₅×Dic₃	1	2	yes	—
337	C₂×C₄	C₂²×D₃×D₅	1	1	yes	characteristic
338	C₈	—	5	4	no	—
339	C₈	C₂×D₅×Dic₃	1	2	yes	—
340	C₈	C₂×D₅×Dic₃	1	2	yes	—
341	C₂×C₃	—	2	2	no	—
342	C₂×C₃	—	10	4	no	—
343	C₂×C₃	C₂×D₅×C₄⋊C₄	1	1	yes	characteristic
344	C₂×C₃	#66096	1	2	yes	—
345	C₅	#83784	1	1	yes	characteristic, complemented, Sylow
346	C₂²	—	2	2	no	—
347	C₂²	—	5	4	no	—
348	C₂²	—	10	4	no	—
349	C₂²	D₅×C₄⋊Dic₃	1	2	yes	—
350	C₂²	C₂×D₅×D₁₂	1	1	yes	characteristic
351	C₄	—	5	4	no	—
352	C₄	—	12	4	no	—
353	C₄	—	60	2	no	—
354	C₄	—	60	2	no	—
355	C₄	D₅×C₄⋊Dic₃	1	2	yes	—
356	C₄	C₂×D₅×Dic₆	1	1	yes	characteristic
357	C₄	C₂²×D₅×Dic₃	1	1	yes	characteristic
358	C₃	#10610847	1	1	yes	characteristic, complemented, Sylow
359	C₂	—	2	2	no	—
360	C₂	—	10	4	no	—
361	C₂	#11753960	1	2	yes	—
362	C₂	C₂×D₅×C₄⋊Dic₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₅×D₁₂
commutator	C₃×C₄×C₅	C₂³×C₄
Frattini	C₂×C₄	C₂²×D₃×D₅
Fitting	C₂²×C₃×C₅×C₈	C₂²
socle	C₂²×C₃×C₅	C₂²×D₄
prime radical	C₂×C₃×#709	C₂²
cosocle	C₂×C₃×C₄×C₅	C₂⁴
hypercenter	C₂²×C₈	D₃×D₅
hypocenter	C₃×C₅	#2912
LCS	G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
3. #11751207	C₂	semidirect
4. C₂×C₅×#817	C₂	semidirect
5. #11748459	C₂	semidirect
6. #11747671	C₂	semidirect
7. #11747618	C₂	semidirect
8. D₅×#817	C₂	semidirect
15. C₃×#4047	C₄	semidirect
48. C₂×C₃×C₅×C₈	C₂×C₄	semidirect
57. C₃×#681	C₂×C₄	semidirect
59. C₃×#676	C₂×C₄	semidirect
60. C₃×C₈×D₅	C₂×C₄	semidirect
71. C₂×#817	D₅	semidirect
78. #4047	Dic₃	semidirect
142. C₂×C₅×C₈	C₂×Dic₃	semidirect
154. #681	C₂×Dic₃	semidirect
156. #676	C₂×Dic₃	semidirect
157. C₈×D₅	C₂×Dic₃	semidirect
191. C₂×C₃×C₈	C₄×D₅	semidirect
292. C₂×C₈	D₅×Dic₃	semidirect
304. C₃×C₅	#2912	semidirect
345. C₅	#83784	semidirect
358. C₃	#10610847	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×#4047	2
#11754499	2
#11751207	2
C₂×C₅×#817	2
#11748459	2
#11747671	2
#11747618	2
D₅×#817	2
#10610847	3
#83784	5