Subgroups — #14281685

Order 1920.

624 conjugacy classes of subgroups.

The subgroup lattice has 352 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×#431	C₂	1	1	yes	maximal, characteristic, complemented
3	#11751026	C₂	1	1	yes	maximal, characteristic, complemented
4	#11746304	C₂	1	4	yes	maximal, complemented
5	#10601549	—	3	1	no	maximal, Hall
6	C₂²×D₆₀	C₄	1	1	yes	characteristic
7	C₂²×C₄×D₁₅	—	2	1	no	—
8	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
9	C₂×#6714	—	2	1	no	—
10	C₂×C₄⋊Dic₁₅	C₄	1	1	yes	characteristic
11	C₄×D₆₀	—	2	1	no	—
12	C₄×D₆₀	—	2	1	no	—
13	C₄×D₆₀	C₄	1	2	yes	—
14	C₄×D₆₀	C₂²	1	2	yes	—
15	C₃×C₅×C₄⋊C₈	C₂²	1	2	yes	complemented
16	C₃×C₅×C₂²⋊C₈	—	2	1	no	—
17	C₃×C₄×C₅×C₈	—	2	1	no	—
18	#91114	—	2	1	no	—
19	C₄⋊(C₃⋊#168)	—	2	1	no	—
20	C₄×C₃⋊#168	C₂²	1	2	yes	complemented
21	#70756	—	5	1	no	maximal, Hall
22	C₂×C₄×D₂₀	—	3	1	no	—
23	C₅×#431	D₃	1	1	yes	characteristic, complemented
24	#65214	—	3	1	no	—
25	#64703	—	3	4	no	—
26	C₂³×D₁₅	—	2	1	no	—
27	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
29	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
30	C₂²×Dic₁₅	—	2	1	no	—
31	C₂×D₆₀	—	2	1	no	—
32	C₂×D₆₀	—	2	2	no	—
33	C₂×D₆₀	C₈	1	4	yes	complemented
34	C₂×D₆₀	C₂×C₄	1	2	yes	—
35	C₂×C₄×D₁₅	—	2	2	no	—
36	C₂×C₄×D₁₅	—	2	2	no	—
37	C₂×C₄×D₁₅	—	4	4	no	—
38	C₂×C₃×C₅×C₈	—	2	2	no	—
39	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
40	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
41	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
42	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
43	#6714	—	2	2	no	—
44	#6714	—	2	2	no	—
45	C₄⋊Dic₁₅	—	2	1	no	—
46	C₄⋊Dic₁₅	C₂×C₄	1	2	yes	—
47	C₂×C₃⋊#168	—	2	2	no	—
48	C₂×C₄×D₁₂	—	5	1	no	—
49	C₃×#431	D₅	1	1	yes	characteristic, complemented
50	#4925	—	5	1	no	—
51	#4483	—	5	4	no	—
52	C₂²×D₂₀	—	3	1	no	—
53	C₂²×C₄×D₅	—	6	1	no	—
54	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
55	C₂×#685	—	6	1	no	—
56	C₂×C₄⋊Dic₅	—	3	1	no	—
57	C₄×D₂₀	—	3	2	no	—
58	C₄×D₂₀	—	3	2	no	—
59	C₄×D₂₀	—	6	1	no	—
60	C₄×D₂₀	—	6	1	no	—
61	C₅×C₄⋊C₈	C₂×D₃	1	2	yes	complemented
62	C₅×C₂²⋊C₈	—	2	1	no	—
63	C₄×C₅×C₈	—	2	1	no	—
64	C₂²⋊#168	—	6	1	no	—
65	C₄⋊#168	—	6	1	no	—
66	C₄×#168	—	3	2	no	—
67	#1478	—	15	1	no	Sylow
68	C₂³×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×D₁₅	—	2	4	no	—
70	C₂²×D₁₅	—	2	2	no	—
71	C₂²×D₁₅	—	4	2	no	—
72	C₂²×D₁₅	—	4	2	no	—
73	C₂×C₃×C₄×C₅	—	2	2	no	—
74	C₂×C₃×C₄×C₅	—	4	1	no	—
75	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₃×C₄×C₅	C₂×C₈	1	2	yes	—
79	C₂×C₃×C₄×C₅	M₁₆	1	2	yes	—
80	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
82	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
83	C₂×Dic₁₅	—	2	2	no	—
84	C₂×Dic₁₅	—	4	2	no	—
85	D₆₀	—	2	4	no	—
86	D₆₀	—	2	2	no	—
87	D₆₀	C₂×C₈	1	4	yes	—
88	C₄×D₁₅	—	4	4	no	—
89	C₄×D₁₅	—	4	4	no	—
90	C₃×C₅×C₈	—	4	2	no	—
91	C₃⋊#168	—	4	2	no	—
92	C₂²×D₁₂	—	5	1	no	—
93	C₂²×C₄×D₃	—	10	1	no	—
94	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
95	C₂×(C₂×D₃)⋊C₄	—	10	1	no	—
96	C₂×C₄⋊Dic₃	—	5	1	no	—
97	C₄×D₁₂	—	5	2	no	—
98	C₄×D₁₂	—	5	2	no	—
99	C₄×D₁₂	—	10	1	no	—
100	C₄×D₁₂	—	10	1	no	—
101	C₃×C₄⋊C₈	C₂×D₅	1	2	yes	complemented
102	C₃×C₂²⋊C₈	—	2	1	no	—
103	C₃×C₄×C₈	—	2	1	no	—
104	#830	—	10	1	no	—
105	C₄⋊(C₃⋊C₈)	—	10	1	no	—
106	C₄×C₃⋊C₈	—	5	2	no	—
107	C₂³×D₅	—	6	1	no	—
108	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
109	C₂²×C₄×C₅	D₁₂	1	1	yes	characteristic
110	C₂²×C₄×C₅	#67	1	1	yes	characteristic
111	C₂²×Dic₅	—	6	1	no	—
112	C₂×D₂₀	—	3	4	no	—
113	C₂×D₂₀	—	3	2	no	—
114	C₂×D₂₀	—	6	2	no	—
115	C₂×D₂₀	—	6	1	no	—
116	C₂×C₄×D₅	—	6	2	no	—
117	C₂×C₄×D₅	—	6	2	no	—
118	C₂×C₄×D₅	—	12	4	no	—
119	C₂×C₅×C₈	—	2	2	no	—
120	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
121	C₄²×C₅	D₁₂	1	1	yes	characteristic
122	C₄²×C₅	#67	1	1	yes	characteristic
123	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
124	#685	—	6	2	no	—
125	#685	—	6	2	no	—
126	C₄⋊Dic₅	—	3	2	no	—
127	C₄⋊Dic₅	—	6	1	no	—
128	C₂×#168	—	6	2	no	—
129	C₂×C₄×D₄	—	15	1	no	—
130	#431	—	15	1	no	—
131	#431	D₁₅	1	1	yes	characteristic, complemented
132	D₄⋊C₈	—	15	4	no	—
133	C₂²×C₃×C₅	—	2	1	no	—
134	C₂²×C₃×C₅	—	2	1	no	—
135	C₂²×C₃×C₅	C₂²⋊C₈	1	2	yes	—
136	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
137	C₂×D₁₅	—	2	4	no	—
138	C₂×D₁₅	—	4	8	no	—
139	C₂×D₁₅	—	4	2	no	—
140	C₂×D₁₅	—	4	2	no	—
141	C₃×C₄×C₅	—	2	1	no	—
142	C₃×C₄×C₅	—	2	1	no	—
143	C₃×C₄×C₅	—	2	1	no	—
144	C₃×C₄×C₅	C₂²⋊C₈	1	2	yes	—
145	C₃×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
146	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
147	C₃×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
148	Dic₁₅	—	4	2	no	—
149	C₂³×D₃	—	10	1	no	—
150	C₂²×C₃×C₄	C₄×D₅	1	1	yes	characteristic
151	C₂²×C₃×C₄	D₂₀	1	1	yes	characteristic
152	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
153	C₂²×Dic₃	—	10	1	no	—
154	C₂×D₁₂	—	5	4	no	—
155	C₂×D₁₂	—	5	2	no	—
156	C₂×D₁₂	—	10	2	no	—
157	C₂×D₁₂	—	10	1	no	—
158	C₂×C₄×D₃	—	10	2	no	—
159	C₂×C₄×D₃	—	10	2	no	—
160	C₂×C₄×D₃	—	20	4	no	—
161	C₂×C₃×C₈	—	2	2	no	—
162	C₃×C₄²	C₄×D₅	1	1	yes	characteristic
163	C₃×C₄²	D₂₀	1	1	yes	characteristic
164	C₃×C₄²	C₂²⋊D₅	1	1	yes	characteristic
165	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
166	(C₂×D₃)⋊C₄	—	10	2	no	—
167	(C₂×D₃)⋊C₄	—	10	2	no	—
168	C₄⋊Dic₃	—	5	2	no	—
169	C₄⋊Dic₃	—	10	1	no	—
170	C₂×C₃⋊C₈	—	10	2	no	—
171	C₂³×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
172	C₂²×D₅	—	6	4	no	—
173	C₂²×D₅	—	6	2	no	—
174	C₂²×D₅	—	12	2	no	—
175	C₂²×D₅	—	12	2	no	—
176	C₂×C₄×C₅	—	2	2	no	—
177	C₂×C₄×C₅	—	4	1	no	—
178	C₂×C₄×C₅	D₃×C₈	1	2	yes	—
179	C₂×C₄×C₅	ΓL(1,𝔽₂₅)	1	2	yes	—
180	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
181	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
182	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
183	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
184	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
185	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
186	C₂×Dic₅	—	6	2	no	—
187	C₂×Dic₅	—	12	2	no	—
188	D₂₀	—	3	4	no	—
189	D₂₀	—	6	4	no	—
190	D₂₀	—	6	2	no	—
191	C₄×D₅	—	12	4	no	—
192	C₄×D₅	—	12	4	no	—
193	C₅×C₈	—	4	2	no	—
194	#168	—	12	2	no	—
195	C₂²×D₄	—	15	1	no	—
196	C₂³×C₄	—	30	1	no	—
197	C₄×D₄	—	15	2	no	—
198	C₄×D₄	—	15	2	no	—
199	C₄×D₄	—	30	1	no	—
200	C₄×D₄	—	30	1	no	—
201	C₂×C₄⋊C₄	—	15	1	no	—
202	C₂×C₂²⋊C₄	—	30	1	no	—
203	C₂×C₄²	C₂×D₁₅	1	1	yes	characteristic
204	C₄⋊C₈	—	30	1	no	—
205	C₄⋊C₈	C₂×D₁₅	1	2	yes	complemented
206	C₂²⋊C₈	—	2	1	no	—
207	C₂²⋊C₈	—	30	1	no	—
208	C₄×C₈	—	2	1	no	—
209	C₄×C₈	—	15	2	no	—
210	C₂×C₃×C₅	—	2	2	no	—
211	C₂×C₃×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
212	C₂×C₃×C₅	#418	1	1	yes	characteristic
213	C₂×C₃×C₅	#421	1	1	yes	characteristic
214	D₁₅	—	4	4	no	—
215	C₂³×C₃	#685	1	1	yes	characteristic
216	C₂²×D₃	—	10	4	no	—
217	C₂²×D₃	—	10	2	no	—
218	C₂²×D₃	—	20	2	no	—
219	C₂²×D₃	—	20	2	no	—
220	C₂×C₃×C₄	—	2	2	no	—
221	C₂×C₃×C₄	—	4	1	no	—
222	C₂×C₃×C₄	C₈×D₅	1	2	yes	—
223	C₂×C₃×C₄	#676	1	2	yes	—
224	C₂×C₃×C₄	#685	1	1	yes	characteristic
225	C₂×C₃×C₄	#685	1	1	yes	characteristic
226	C₂×C₃×C₄	#685	1	1	yes	characteristic
227	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
228	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
229	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
230	C₂×Dic₃	—	10	2	no	—
231	C₂×Dic₃	—	20	2	no	—
232	D₁₂	—	5	4	no	—
233	D₁₂	—	10	4	no	—
234	D₁₂	—	10	2	no	—
235	C₄×D₃	—	20	4	no	—
236	C₄×D₃	—	20	4	no	—
237	C₃×C₈	—	4	2	no	—
238	C₃⋊C₈	—	20	2	no	—
239	C₂²×C₅	—	2	1	no	—
240	C₂²×C₅	—	2	1	no	—
241	C₂²×C₅	(C₂×D₃)⋊C₈	1	2	yes	—
242	C₂²×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
243	C₂×D₅	—	6	4	no	—
244	C₂×D₅	—	12	8	no	—
245	C₂×D₅	—	12	2	no	—
246	C₂×D₅	—	12	2	no	—
247	C₄×C₅	—	2	1	no	—
248	C₄×C₅	—	2	1	no	—
249	C₄×C₅	—	2	1	no	—
250	C₄×C₅	(C₂×D₃)⋊C₈	1	2	yes	—
251	C₄×C₅	C₂×D₃×C₈	1	1	yes	characteristic
252	C₄×C₅	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
253	C₄×C₅	#949	1	2	yes	—
254	Dic₅	—	12	2	no	—
255	C₂⁴	—	30	1	no	—
256	C₂×D₄	—	15	4	no	—
257	C₂×D₄	—	15	2	no	—
258	C₂×D₄	—	30	2	no	—
259	C₂×D₄	—	30	1	no	—
260	C₂²×C₄	—	30	1	no	—
261	C₂²×C₄	—	30	2	no	—
262	C₂²×C₄	—	30	2	no	—
263	C₂²×C₄	—	60	4	no	—
264	C₂²×C₄	C₄×D₁₅	1	1	yes	characteristic
265	C₂²×C₄	D₆₀	1	1	yes	characteristic
266	C₂²×C₄	#1220	1	1	yes	characteristic
267	C₂×C₈	—	2	2	no	—
268	C₂×C₈	—	30	2	no	—
269	C₄⋊C₄	—	15	2	no	—
270	C₄⋊C₄	—	30	1	no	—
271	C₂²⋊C₄	—	30	2	no	—
272	C₂²⋊C₄	—	30	2	no	—
273	C₄²	C₄×D₁₅	1	1	yes	characteristic
274	C₄²	D₆₀	1	1	yes	characteristic
275	C₄²	#1220	1	1	yes	characteristic
276	C₄²	C₂²×D₁₅	1	1	yes	characteristic
277	C₃×C₅	#1478	1	1	yes	characteristic, complemented, Hall
278	C₂²×C₃	—	2	1	no	—
279	C₂²×C₃	—	2	1	no	—
280	C₂²×C₃	#3952	1	2	yes	—
281	C₂²×C₃	C₂×#685	1	1	yes	characteristic
282	C₂×D₃	—	10	4	no	—
283	C₂×D₃	—	20	8	no	—
284	C₂×D₃	—	20	2	no	—
285	C₂×D₃	—	20	2	no	—
286	C₃×C₄	—	2	1	no	—
287	C₃×C₄	—	2	1	no	—
288	C₃×C₄	—	2	1	no	—
289	C₃×C₄	#3952	1	2	yes	—
290	C₃×C₄	C₂×C₈×D₅	1	1	yes	characteristic
291	C₃×C₄	C₂×#676	1	1	yes	characteristic
292	C₃×C₄	#4095	1	2	yes	—
293	Dic₃	—	20	2	no	—
294	C₂×C₅	—	2	2	no	—
295	C₂×C₅	#5001	1	1	yes	characteristic
296	C₂×C₅	C₂×(C₂×D₃)⋊C₈	1	1	yes	characteristic
297	C₂×C₅	#5139	1	1	yes	characteristic
298	D₅	—	12	4	no	—
299	C₂³	—	30	2	no	—
300	C₂³	—	30	4	no	—
301	C₂³	—	60	2	no	—
302	C₂³	—	60	2	no	—
303	C₂³	#6714	1	1	yes	characteristic
304	D₄	—	15	4	no	—
305	D₄	—	30	2	no	—
306	D₄	—	30	4	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	4	1	no	—
309	C₂×C₄	—	30	2	no	—
310	C₂×C₄	—	60	4	no	—
311	C₂×C₄	—	60	2	no	—
312	C₂×C₄	—	60	4	no	—
313	C₂×C₄	C₈×D₁₅	1	2	yes	—
314	C₂×C₄	#6705	1	2	yes	—
315	C₂×C₄	#6714	1	1	yes	characteristic
316	C₂×C₄	#6714	1	1	yes	characteristic
317	C₂×C₄	#6714	1	1	yes	characteristic
318	C₂×C₄	C₂×C₄×D₁₅	1	1	yes	characteristic
319	C₂×C₄	C₂×D₆₀	1	1	yes	characteristic
320	C₂×C₄	C₂×#1220	1	1	yes	characteristic
321	C₈	—	4	2	no	—
322	C₈	—	60	2	no	—
323	C₂×C₃	—	2	2	no	—
324	C₂×C₃	#65290	1	1	yes	characteristic
325	C₂×C₃	C₂×#3952	1	1	yes	characteristic
326	C₂×C₃	#65428	1	1	yes	characteristic
327	D₃	—	20	4	no	—
328	C₅	#70756	1	1	yes	characteristic, complemented, Sylow
329	C₂²	—	2	1	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	30	4	no	—
332	C₂²	—	60	8	no	—
333	C₂²	—	60	2	no	—
334	C₂²	—	60	2	no	—
335	C₂²	#91104	1	2	yes	—
336	C₂²	C₂×#6714	1	1	yes	characteristic
337	C₄	—	2	1	no	—
338	C₄	—	2	1	no	—
339	C₄	—	2	1	no	—
340	C₄	—	60	2	no	—
341	C₄	#91104	1	2	yes	—
342	C₄	C₂×C₈×D₁₅	1	1	yes	characteristic
343	C₄	C₂×#6705	1	1	yes	characteristic
344	C₄	#91838	1	2	yes	—
345	C₃	#10601549	1	1	yes	characteristic, complemented, Sylow
346	C₂	—	2	2	no	—
347	C₂	—	60	4	no	—
348	C₂	#11751102	1	1	yes	characteristic
349	C₂	C₂×#91104	1	1	yes	characteristic
350	C₂	#11751240	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#1220
commutator	C₃×C₄×C₅	C₂²×C₈
Frattini	C₄²	C₂²×D₁₅
Fitting	C₃×C₅×#431	C₂
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₂²×D₆₀	C₄
cosocle	C₃×C₄²×C₅	C₂³
hypercenter	#431	D₁₅
hypocenter	C₃×C₅	#1478
LCS	G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×#431	C₂	semidirect
3. #11751026	C₂	semidirect
4. #11746304	C₂	semidirect
15. C₃×C₅×C₄⋊C₈	C₂²	semidirect
20. C₄×C₃⋊#168	C₂²	semidirect
23. C₅×#431	D₃	semidirect
33. C₂×D₆₀	C₈	semidirect
49. C₃×#431	D₅	semidirect
61. C₅×C₄⋊C₈	C₂×D₃	semidirect
101. C₃×C₄⋊C₈	C₂×D₅	semidirect
131. #431	D₁₅	semidirect
205. C₄⋊C₈	C₂×D₁₅	semidirect
277. C₃×C₅	#1478	semidirect
328. C₅	#70756	semidirect
345. C₃	#10601549	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×#431	2
#11751026	2
#11746304	2
#10601549	3
#70756	5