Subgroups — #14297716

Order 1920.

712 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₆₀	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#382	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#91093	C₂	1	2	yes	maximal
4	C₂×C₄⋊(C₃⋊#168)	C₂	1	1	yes	maximal, characteristic, complemented
5	#11746350	C₂	1	2	yes	maximal, complemented
6	#10600349	—	3	1	no	maximal, Hall
7	C₂²×D₆₀	C₂²	1	1	yes	characteristic
8	C₂²×C₄×D₁₅	—	2	1	no	—
9	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
10	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
11	C₂×#6714	—	2	1	no	—
12	C₂×C₄⋊Dic₁₅	C₂²	1	1	yes	characteristic
13	C₂²×C₃⋊#168	C₂²	1	2	yes	—
14	C₄×D₆₀	—	2	2	no	—
15	C₄×D₆₀	C₄	1	4	yes	complemented
16	C₃×C₅×#95	—	2	1	no	—
17	C₃×C₅×#95	—	2	1	no	—
18	#91093	—	2	4	no	—
19	C₄⋊(C₃⋊#168)	—	2	2	no	—
20	#69556	—	5	1	no	maximal, Hall
21	C₂×C₄×D₂₀	—	3	1	no	—
22	C₅×#382	D₃	1	1	yes	characteristic, complemented
23	C₂×#3941	—	3	2	no	—
24	C₂×C₄⋊#168	—	3	1	no	—
25	#64749	—	3	2	no	—
26	C₂³×D₁₅	—	2	1	no	—
27	C₂²×C₃×C₄×C₅	—	2	2	no	—
28	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
29	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
30	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
31	C₂²×Dic₁₅	—	2	1	no	—
32	C₂×D₆₀	—	2	1	no	—
33	C₂×D₆₀	—	2	1	no	—
34	C₂×D₆₀	—	2	1	no	—
35	C₂×D₆₀	C₂×C₄	1	2	yes	—
36	C₂×D₆₀	D₄	1	2	yes	—
37	C₂×D₆₀	Q₈	1	2	yes	—
38	C₂×C₄×D₁₅	—	2	4	no	—
39	C₂×C₄×D₁₅	—	4	4	no	—
40	C₃×C₅×C₄⋊C₄	—	2	2	no	—
41	C₃×C₅×C₄⋊C₄	—	2	2	no	—
42	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
43	C₃×C₄²×C₅	D₄	1	2	yes	—
44	#6714	—	2	4	no	—
45	C₄⋊Dic₁₅	—	2	1	no	—
46	C₄⋊Dic₁₅	C₂×C₄	1	2	yes	—
47	C₂×C₃⋊#168	—	2	2	no	—
48	C₂×C₃⋊#168	—	2	2	no	—
49	C₂×C₃⋊#168	—	2	2	no	—
50	C₂×C₄×D₁₂	—	5	1	no	—
51	C₃×#382	D₅	1	1	yes	characteristic, complemented
52	C₂×#809	—	5	2	no	—
53	C₂×C₄⋊(C₃⋊C₈)	—	5	1	no	—
54	#4529	—	5	2	no	—
55	C₂²×D₂₀	—	3	1	no	—
56	C₂²×C₄×D₅	—	6	1	no	—
57	C₂×C₅×C₄⋊C₄	C₂×D₃	1	2	yes	—
58	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
59	C₂×#685	—	6	1	no	—
60	C₂×C₄⋊Dic₅	—	3	1	no	—
61	C₂²×#168	—	3	2	no	—
62	C₄×D₂₀	—	3	4	no	—
63	C₄×D₂₀	—	6	2	no	—
64	C₅×#95	—	2	1	no	—
65	C₅×#95	—	2	1	no	—
66	#3941	—	6	4	no	—
67	C₄⋊#168	—	6	2	no	—
68	#1804	—	15	1	no	Sylow
69	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
70	C₂²×D₁₅	—	2	2	no	—
71	C₂²×D₁₅	—	2	2	no	—
72	C₂²×D₁₅	—	2	2	no	—
73	C₂²×D₁₅	—	4	2	no	—
74	C₂²×D₁₅	—	4	2	no	—
75	C₂×C₃×C₄×C₅	—	2	2	no	—
76	C₂×C₃×C₄×C₅	—	2	2	no	—
77	C₂×C₃×C₄×C₅	—	4	2	no	—
78	C₂×C₃×C₄×C₅	—	4	2	no	—
79	C₂×C₃×C₄×C₅	—	4	2	no	—
80	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
81	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
82	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
84	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
85	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
86	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
87	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
88	C₂×Dic₁₅	—	2	2	no	—
89	C₂×Dic₁₅	—	4	2	no	—
90	D₆₀	—	2	2	no	—
91	D₆₀	—	2	2	no	—
92	D₆₀	—	2	2	no	—
93	D₆₀	C₄⋊C₄	1	4	yes	—
94	C₄×D₁₅	—	4	8	no	—
95	C₃⋊#168	—	4	2	no	—
96	C₂²×D₁₂	—	5	1	no	—
97	C₂²×C₄×D₃	—	10	1	no	—
98	C₂×C₃×C₄⋊C₄	C₂×D₅	1	2	yes	—
99	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
100	C₂×(C₂×D₃)⋊C₄	—	10	1	no	—
101	C₂×C₄⋊Dic₃	—	5	1	no	—
102	C₂²×C₃⋊C₈	—	5	2	no	—
103	C₄×D₁₂	—	5	4	no	—
104	C₄×D₁₂	—	10	2	no	—
105	C₃×#95	—	2	1	no	—
106	C₃×#95	—	2	1	no	—
107	#809	—	10	4	no	—
108	C₄⋊(C₃⋊C₈)	—	10	2	no	—
109	C₂³×D₅	—	6	1	no	—
110	C₂²×C₄×C₅	—	2	2	no	—
111	C₂²×C₄×C₅	#67	1	1	yes	characteristic
112	C₂²×C₄×C₅	#67	1	1	yes	characteristic
113	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
114	C₂²×Dic₅	—	6	1	no	—
115	C₂×D₂₀	—	3	2	no	—
116	C₂×D₂₀	—	3	2	no	—
117	C₂×D₂₀	—	3	2	no	—
118	C₂×D₂₀	—	6	1	no	—
119	C₂×D₂₀	—	6	1	no	—
120	C₂×D₂₀	—	6	1	no	—
121	C₂×C₄×D₅	—	6	4	no	—
122	C₂×C₄×D₅	—	12	4	no	—
123	C₅×C₄⋊C₄	—	2	2	no	—
124	C₅×C₄⋊C₄	—	2	2	no	—
125	C₄²×C₅	C₄×D₃	1	2	yes	—
126	C₄²×C₅	D₁₂	1	2	yes	—
127	#685	—	6	4	no	—
128	C₄⋊Dic₅	—	3	2	no	—
129	C₄⋊Dic₅	—	6	1	no	—
130	C₂×#168	—	6	2	no	—
131	C₂×#168	—	6	2	no	—
132	C₂×#168	—	6	2	no	—
133	C₂×C₄×D₄	—	15	1	no	—
134	C₂×C₄⋊C₈	—	15	1	no	—
135	C₂×#102	—	15	2	no	—
136	#382	D₁₅	1	1	yes	characteristic, complemented
137	#340	—	15	2	no	—
138	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
139	C₂²×C₃×C₅	#121	1	1	yes	characteristic
140	C₂²×C₃×C₅	C₂²⋊Q₈	1	1	yes	characteristic
141	C₂²×C₃×C₅	#135	1	2	yes	—
142	C₂²×C₃×C₅	AGL(1,ℤ/8)	1	2	yes	—
143	C₂×D₁₅	—	2	4	no	—
144	C₂×D₁₅	—	4	2	no	—
145	C₂×D₁₅	—	4	2	no	—
146	C₂×D₁₅	—	4	4	no	—
147	C₂×D₁₅	—	4	4	no	—
148	C₃×C₄×C₅	—	2	2	no	—
149	C₃×C₄×C₅	—	2	2	no	—
150	C₃×C₄×C₅	—	8	2	no	—
151	C₃×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
152	C₃×C₄×C₅	#117	1	1	yes	characteristic
153	C₃×C₄×C₅	#121	1	1	yes	characteristic
154	C₃×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
155	Dic₁₅	—	4	2	no	—
156	C₂³×D₃	—	10	1	no	—
157	C₂²×C₃×C₄	—	2	2	no	—
158	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
159	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
160	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
161	C₂²×Dic₃	—	10	1	no	—
162	C₂×D₁₂	—	5	2	no	—
163	C₂×D₁₂	—	5	2	no	—
164	C₂×D₁₂	—	5	2	no	—
165	C₂×D₁₂	—	10	1	no	—
166	C₂×D₁₂	—	10	1	no	—
167	C₂×D₁₂	—	10	1	no	—
168	C₂×C₄×D₃	—	10	4	no	—
169	C₂×C₄×D₃	—	20	4	no	—
170	C₃×C₄⋊C₄	—	2	2	no	—
171	C₃×C₄⋊C₄	—	2	2	no	—
172	C₃×C₄²	C₄×D₅	1	2	yes	—
173	C₃×C₄²	D₂₀	1	2	yes	—
174	(C₂×D₃)⋊C₄	—	10	4	no	—
175	C₄⋊Dic₃	—	5	2	no	—
176	C₄⋊Dic₃	—	10	1	no	—
177	C₂×C₃⋊C₈	—	10	2	no	—
178	C₂×C₃⋊C₈	—	10	2	no	—
179	C₂×C₃⋊C₈	—	10	2	no	—
180	C₂³×C₅	C₂×#67	1	1	yes	characteristic
181	C₂²×D₅	—	6	2	no	—
182	C₂²×D₅	—	6	2	no	—
183	C₂²×D₅	—	6	2	no	—
184	C₂²×D₅	—	12	2	no	—
185	C₂²×D₅	—	12	2	no	—
186	C₂×C₄×C₅	—	2	2	no	—
187	C₂×C₄×C₅	—	2	2	no	—
188	C₂×C₄×C₅	—	4	2	no	—
189	C₂×C₄×C₅	—	4	2	no	—
190	C₂×C₄×C₅	—	4	2	no	—
191	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	2	yes	—
192	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	2	yes	—
193	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
194	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
195	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
196	C₂×C₄×C₅	#237	1	1	yes	characteristic
197	C₂×C₄×C₅	D₃×Q₈	1	1	yes	characteristic
198	C₂×C₄×C₅	#239	1	1	yes	characteristic
199	C₂×Dic₅	—	6	2	no	—
200	C₂×Dic₅	—	12	2	no	—
201	D₂₀	—	3	4	no	—
202	D₂₀	—	6	2	no	—
203	D₂₀	—	6	2	no	—
204	D₂₀	—	6	2	no	—
205	C₄×D₅	—	12	8	no	—
206	#168	—	12	2	no	—
207	C₂²×D₄	—	15	1	no	—
208	C₂³×C₄	—	30	1	no	—
209	C₂²×C₈	—	15	2	no	—
210	C₄×D₄	—	15	4	no	—
211	C₄×D₄	—	30	2	no	—
212	C₂×C₄⋊C₄	—	15	1	no	—
213	C₂×C₄⋊C₄	C₂×D₁₅	1	2	yes	—
214	C₂×C₂²⋊C₄	—	30	1	no	—
215	C₂×C₄²	C₂×D₁₅	1	1	yes	characteristic
216	C₄⋊C₈	—	30	2	no	—
217	#102	—	30	4	no	—
218	#95	—	2	1	no	—
219	#95	—	2	1	no	—
220	C₂×C₃×C₅	#380	1	1	yes	characteristic
221	C₂×C₃×C₅	#416	1	1	yes	characteristic
222	C₂×C₃×C₅	#418	1	1	yes	characteristic
223	C₂×C₃×C₅	#463	1	2	yes	—
224	C₂×C₃×C₅	#478	1	2	yes	—
225	D₁₅	—	4	4	no	—
226	C₂³×C₃	C₂×C₂²⋊D₅	1	1	yes	characteristic
227	C₂²×D₃	—	10	2	no	—
228	C₂²×D₃	—	10	2	no	—
229	C₂²×D₃	—	10	2	no	—
230	C₂²×D₃	—	20	2	no	—
231	C₂²×D₃	—	20	2	no	—
232	C₂×C₃×C₄	—	2	2	no	—
233	C₂×C₃×C₄	—	2	2	no	—
234	C₂×C₃×C₄	—	4	2	no	—
235	C₂×C₃×C₄	—	4	2	no	—
236	C₂×C₃×C₄	—	4	2	no	—
237	C₂×C₃×C₄	#685	1	2	yes	—
238	C₂×C₃×C₄	#685	1	2	yes	—
239	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
240	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
241	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
242	C₂×C₃×C₄	#711	1	1	yes	characteristic
243	C₂×C₃×C₄	Q₈×D₅	1	1	yes	characteristic
244	C₂×C₃×C₄	#713	1	1	yes	characteristic
245	C₂×Dic₃	—	10	2	no	—
246	C₂×Dic₃	—	20	2	no	—
247	D₁₂	—	5	4	no	—
248	D₁₂	—	10	2	no	—
249	D₁₂	—	10	2	no	—
250	D₁₂	—	10	2	no	—
251	C₄×D₃	—	20	8	no	—
252	C₃⋊C₈	—	20	2	no	—
253	C₂²×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
254	C₂²×C₅	#938	1	1	yes	characteristic
255	C₂²×C₅	#946	1	1	yes	characteristic
256	C₂²×C₅	#949	1	2	yes	—
257	C₂²×C₅	#950	1	2	yes	—
258	C₂×D₅	—	6	4	no	—
259	C₂×D₅	—	12	2	no	—
260	C₂×D₅	—	12	2	no	—
261	C₂×D₅	—	12	4	no	—
262	C₂×D₅	—	12	4	no	—
263	C₄×C₅	—	2	2	no	—
264	C₄×C₅	—	2	2	no	—
265	C₄×C₅	—	8	2	no	—
266	C₄×C₅	D₃×C₄⋊C₄	1	1	yes	characteristic
267	C₄×C₅	#892	1	1	yes	characteristic
268	C₄×C₅	#895	1	1	yes	characteristic
269	C₄×C₅	#897	1	1	yes	characteristic
270	Dic₅	—	12	2	no	—
271	C₂⁴	—	30	1	no	—
272	C₂×D₄	—	15	2	no	—
273	C₂×D₄	—	15	2	no	—
274	C₂×D₄	—	15	2	no	—
275	C₂×D₄	—	30	1	no	—
276	C₂×D₄	—	30	1	no	—
277	C₂×D₄	—	30	1	no	—
278	C₂²×C₄	—	2	2	no	—
279	C₂²×C₄	—	30	1	no	—
280	C₂²×C₄	—	30	4	no	—
281	C₂²×C₄	—	60	4	no	—
282	C₂²×C₄	#1220	1	1	yes	characteristic
283	C₂²×C₄	#1220	1	1	yes	characteristic
284	C₂²×C₄	C₂²×D₁₅	1	1	yes	characteristic
285	C₂×C₈	—	30	2	no	—
286	C₂×C₈	—	30	2	no	—
287	C₂×C₈	—	30	2	no	—
288	C₄⋊C₄	—	2	2	no	—
289	C₄⋊C₄	—	2	2	no	—
290	C₄⋊C₄	—	15	2	no	—
291	C₄⋊C₄	—	30	1	no	—
292	C₂²⋊C₄	—	30	4	no	—
293	C₄²	C₄×D₁₅	1	2	yes	—
294	C₄²	D₆₀	1	2	yes	—
295	C₃×C₅	#1804	1	1	yes	characteristic, complemented, Hall
296	C₂²×C₃	C₂×#685	1	1	yes	characteristic
297	C₂²×C₃	#4084	1	1	yes	characteristic
298	C₂²×C₃	#4092	1	1	yes	characteristic
299	C₂²×C₃	#4095	1	2	yes	—
300	C₂²×C₃	#4096	1	2	yes	—
301	C₂×D₃	—	10	4	no	—
302	C₂×D₃	—	20	2	no	—
303	C₂×D₃	—	20	2	no	—
304	C₂×D₃	—	20	4	no	—
305	C₂×D₃	—	20	4	no	—
306	C₃×C₄	—	2	2	no	—
307	C₃×C₄	—	2	2	no	—
308	C₃×C₄	—	8	2	no	—
309	C₃×C₄	D₅×C₄⋊C₄	1	1	yes	characteristic
310	C₃×C₄	#4038	1	1	yes	characteristic
311	C₃×C₄	#4041	1	1	yes	characteristic
312	C₃×C₄	#4043	1	1	yes	characteristic
313	Dic₃	—	20	2	no	—
314	C₂×C₅	#4987	1	1	yes	characteristic
315	C₂×C₅	#5001	1	1	yes	characteristic
316	C₂×C₅	#5002	1	1	yes	characteristic
317	C₂×C₅	#5057	1	2	yes	—
318	C₂×C₅	#5066	1	2	yes	—
319	D₅	—	12	4	no	—
320	C₂³	—	30	2	no	—
321	C₂³	—	30	2	no	—
322	C₂³	—	30	2	no	—
323	C₂³	—	60	2	no	—
324	C₂³	—	60	2	no	—
325	C₂³	C₂×#1220	1	1	yes	characteristic
326	D₄	—	15	4	no	—
327	D₄	—	30	2	no	—
328	D₄	—	30	2	no	—
329	D₄	—	30	2	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	2	2	no	—
332	C₂×C₄	—	4	2	no	—
333	C₂×C₄	—	4	2	no	—
334	C₂×C₄	—	4	2	no	—
335	C₂×C₄	—	30	2	no	—
336	C₂×C₄	—	60	8	no	—
337	C₂×C₄	—	60	2	no	—
338	C₂×C₄	#6714	1	2	yes	—
339	C₂×C₄	#6714	1	2	yes	—
340	C₂×C₄	C₂×C₄×D₁₅	1	1	yes	characteristic
341	C₂×C₄	C₂×D₆₀	1	1	yes	characteristic
342	C₂×C₄	D₄×D₁₅	1	1	yes	characteristic
343	C₂×C₄	#6819	1	1	yes	characteristic
344	C₂×C₄	Q₈×D₁₅	1	1	yes	characteristic
345	C₂×C₄	#6821	1	1	yes	characteristic
346	C₈	—	60	2	no	—
347	C₂×C₃	#65276	1	1	yes	characteristic
348	C₂×C₃	#65290	1	1	yes	characteristic
349	C₂×C₃	#65291	1	1	yes	characteristic
350	C₂×C₃	#65346	1	2	yes	—
351	C₂×C₃	#65355	1	2	yes	—
352	D₃	—	20	4	no	—
353	C₅	#69556	1	1	yes	characteristic, complemented, Sylow
354	C₂²	—	30	4	no	—
355	C₂²	—	60	2	no	—
356	C₂²	—	60	2	no	—
357	C₂²	—	60	4	no	—
358	C₂²	—	60	4	no	—
359	C₂²	C₂×#6714	1	1	yes	characteristic
360	C₂²	#91827	1	1	yes	characteristic
361	C₂²	#91835	1	1	yes	characteristic
362	C₂²	#91838	1	2	yes	—
363	C₂²	#91839	1	2	yes	—
364	C₄	—	2	2	no	—
365	C₄	—	2	2	no	—
366	C₄	—	8	2	no	—
367	C₄	—	60	2	no	—
368	C₄	C₄⋊C₄×D₁₅	1	1	yes	characteristic
369	C₄	#91781	1	1	yes	characteristic
370	C₄	#91784	1	1	yes	characteristic
371	C₄	#91786	1	1	yes	characteristic
372	C₃	#10600349	1	1	yes	characteristic, complemented, Sylow
373	C₂	—	60	4	no	—
374	C₂	#11751088	1	1	yes	characteristic
375	C₂	#11751102	1	1	yes	characteristic
376	C₂	#11751103	1	1	yes	characteristic
377	C₂	#11751158	1	2	yes	—
378	C₂	#11751167	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#1220
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₅
Fitting	C₃×C₅×#382	C₂
socle	C₂³×C₃×C₅	C₂×D₄
prime radical	C₂²×D₆₀	C₂²
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	#382	D₁₅
hypocenter	C₃×C₅	#1804
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×#382	C₂	semidirect
4. C₂×C₄⋊(C₃⋊#168)	C₂	semidirect
5. #11746350	C₂	semidirect
15. C₄×D₆₀	C₄	semidirect
22. C₅×#382	D₃	semidirect
51. C₃×#382	D₅	semidirect
136. #382	D₁₅	semidirect
295. C₃×C₅	#1804	semidirect
353. C₅	#69556	semidirect
372. C₃	#10600349	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₆₀	2
C₃×C₅×#382	2
C₂×#91093	2
C₂×C₄⋊(C₃⋊#168)	2
#11746350	2
#10600349	3
#69556	5