Subgroups — #14282281

Order 1920.

632 conjugacy classes of subgroups.

The subgroup lattice has 344 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×#432	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#6710	C₂	1	1	yes	maximal, characteristic, complemented
4	#11746304	C₂	1	4	yes	maximal, complemented
5	#10601592	—	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	C₂×#6710	—	2	1	no	—
12	C₄×D₆₀	—	2	1	no	—
13	C₄×D₆₀	—	2	1	no	—
14	C₄×D₆₀	C₄	1	2	yes	—
15	C₄×D₆₀	C₂²	1	2	yes	—
16	C₃×C₅×C₄⋊C₈	C₂²	1	2	yes	complemented
17	C₃×C₅×C₂²⋊C₈	—	2	1	no	—
18	C₃×C₅×#97	—	2	1	no	—
19	#91087	—	2	1	no	—
20	C₄×C₃⋊#168	C₂²	1	2	yes	complemented
21	#70799	—	5	1	no	maximal, Hall
22	C₂×C₄×D₂₀	—	3	1	no	—
23	C₅×#432	D₃	1	1	yes	characteristic, complemented
24	C₄×#681	—	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	4	no	—
33	C₂×D₆₀	C₂×C₄	1	2	yes	—
34	C₂×C₄×D₁₅	—	2	2	no	—
35	C₂×C₄×D₁₅	—	2	2	no	—
36	C₂×C₄×D₁₅	—	4	4	no	—
37	C₂×C₃×C₅×C₈	—	2	2	no	—
38	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
39	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
40	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
41	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
42	#6714	—	2	2	no	—
43	#6714	—	2	2	no	—
44	C₄⋊Dic₁₅	—	2	1	no	—
45	C₄⋊Dic₁₅	C₂×C₄	1	2	yes	—
46	#6710	—	2	4	no	—
47	C₂×C₃⋊#168	—	2	2	no	—
48	C₂×C₄×D₁₂	—	5	1	no	—
49	C₃×#432	D₅	1	1	yes	characteristic, complemented
50	C₄×#208	—	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	C₂×#681	—	6	1	no	—
58	C₄×D₂₀	—	3	2	no	—
59	C₄×D₂₀	—	3	2	no	—
60	C₄×D₂₀	—	6	1	no	—
61	C₄×D₂₀	—	6	1	no	—
62	C₅×C₄⋊C₈	C₂×D₃	1	2	yes	complemented
63	C₅×C₂²⋊C₈	—	2	1	no	—
64	C₅×#97	—	2	1	no	—
65	#3935	—	6	1	no	—
66	C₄×#168	—	3	2	no	—
67	#1490	—	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	—	2	2	no	—
75	C₂×C₃×C₄×C₅	—	2	2	no	—
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	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
81	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
82	C₂×Dic₁₅	—	2	2	no	—
83	C₂×Dic₁₅	—	4	2	no	—
84	D₆₀	—	2	4	no	—
85	D₆₀	—	2	2	no	—
86	D₆₀	M₁₆	1	4	yes	complemented
87	C₄×D₁₅	—	4	4	no	—
88	C₄×D₁₅	—	4	4	no	—
89	C₃×C₅×C₈	—	4	2	no	—
90	C₃⋊#168	—	2	4	no	—
91	C₂²×D₁₂	—	5	1	no	—
92	C₂²×C₄×D₃	—	10	1	no	—
93	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
94	C₂×(C₂×D₃)⋊C₄	—	10	1	no	—
95	C₂×C₄⋊Dic₃	—	5	1	no	—
96	C₂×#208	—	10	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×#97	—	2	1	no	—
104	#803	—	10	1	no	—
105	C₄×C₃⋊C₈	—	5	2	no	—
106	C₂³×D₅	—	6	1	no	—
107	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
108	C₂²×C₄×C₅	D₁₂	1	1	yes	characteristic
109	C₂²×C₄×C₅	#67	1	1	yes	characteristic
110	C₂²×Dic₅	—	6	1	no	—
111	C₂×D₂₀	—	3	2	no	—
112	C₂×D₂₀	—	6	4	no	—
113	C₂×D₂₀	—	6	1	no	—
114	C₂×C₄×D₅	—	6	2	no	—
115	C₂×C₄×D₅	—	6	2	no	—
116	C₂×C₄×D₅	—	12	4	no	—
117	C₂×C₅×C₈	—	2	2	no	—
118	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
119	C₄²×C₅	D₁₂	1	1	yes	characteristic
120	C₄²×C₅	#67	1	1	yes	characteristic
121	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
122	#685	—	6	2	no	—
123	#685	—	6	2	no	—
124	C₄⋊Dic₅	—	3	2	no	—
125	C₄⋊Dic₅	—	6	1	no	—
126	#681	—	6	4	no	—
127	C₂×#168	—	6	2	no	—
128	C₂×C₄×D₄	—	15	1	no	—
129	#432	D₁₅	1	1	yes	characteristic, complemented
130	C₄×M₁₆	—	15	1	no	—
131	D₄⋊C₈	—	15	4	no	—
132	C₂²×C₃×C₅	—	2	1	no	—
133	C₂²×C₃×C₅	—	2	1	no	—
134	C₂²×C₃×C₅	#104	1	2	yes	—
135	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
136	C₂×D₁₅	—	2	4	no	—
137	C₂×D₁₅	—	4	8	no	—
138	C₂×D₁₅	—	4	2	no	—
139	C₂×D₁₅	—	4	2	no	—
140	C₃×C₄×C₅	—	2	1	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	C₂×M₁₆	1	2	yes	—
145	C₃×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
146	Dic₁₅	—	4	2	no	—
147	C₂³×D₃	—	10	1	no	—
148	C₂²×C₃×C₄	C₄×D₅	1	1	yes	characteristic
149	C₂²×C₃×C₄	D₂₀	1	1	yes	characteristic
150	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
151	C₂²×Dic₃	—	10	1	no	—
152	C₂×D₁₂	—	5	2	no	—
153	C₂×D₁₂	—	10	4	no	—
154	C₂×D₁₂	—	10	1	no	—
155	C₂×C₄×D₃	—	10	2	no	—
156	C₂×C₄×D₃	—	10	2	no	—
157	C₂×C₄×D₃	—	20	4	no	—
158	C₂×C₃×C₈	—	2	2	no	—
159	C₃×C₄²	C₄×D₅	1	1	yes	characteristic
160	C₃×C₄²	D₂₀	1	1	yes	characteristic
161	C₃×C₄²	C₂²⋊D₅	1	1	yes	characteristic
162	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
163	(C₂×D₃)⋊C₄	—	10	2	no	—
164	(C₂×D₃)⋊C₄	—	10	2	no	—
165	C₄⋊Dic₃	—	5	2	no	—
166	C₄⋊Dic₃	—	10	1	no	—
167	#208	—	10	4	no	—
168	C₂×C₃⋊C₈	—	10	2	no	—
169	C₂³×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
170	C₂²×D₅	—	6	4	no	—
171	C₂²×D₅	—	6	2	no	—
172	C₂²×D₅	—	12	2	no	—
173	C₂²×D₅	—	12	2	no	—
174	C₂×C₄×C₅	—	2	2	no	—
175	C₂×C₄×C₅	—	2	2	no	—
176	C₂×C₄×C₅	—	2	2	no	—
177	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
178	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
179	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
180	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
181	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
182	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
183	C₂×Dic₅	—	6	2	no	—
184	C₂×Dic₅	—	12	2	no	—
185	D₂₀	—	3	4	no	—
186	D₂₀	—	6	4	no	—
187	D₂₀	—	6	2	no	—
188	C₄×D₅	—	12	4	no	—
189	C₄×D₅	—	12	4	no	—
190	C₅×C₈	—	4	2	no	—
191	#168	—	6	4	no	—
192	C₂²×D₄	—	15	1	no	—
193	C₂³×C₄	—	30	1	no	—
194	C₂×M₁₆	—	30	1	no	—
195	C₄×D₄	—	15	2	no	—
196	C₄×D₄	—	15	2	no	—
197	C₄×D₄	—	30	1	no	—
198	C₄×D₄	—	30	1	no	—
199	C₂×C₄⋊C₄	—	15	1	no	—
200	C₂×C₂²⋊C₄	—	30	1	no	—
201	C₂×C₄²	C₂×D₁₅	1	1	yes	characteristic
202	C₄⋊C₈	C₂×D₁₅	1	2	yes	complemented
203	C₂²⋊C₈	—	2	1	no	—
204	#97	—	2	1	no	—
205	#97	—	30	1	no	—
206	C₄×C₈	—	15	2	no	—
207	C₂×C₃×C₅	—	2	2	no	—
208	C₂×C₃×C₅	#407	1	1	yes	characteristic
209	C₂×C₃×C₅	#418	1	1	yes	characteristic
210	C₂×C₃×C₅	C₂×#104	1	1	yes	characteristic
211	D₁₅	—	4	4	no	—
212	C₂³×C₃	#685	1	1	yes	characteristic
213	C₂²×D₃	—	10	4	no	—
214	C₂²×D₃	—	10	2	no	—
215	C₂²×D₃	—	20	2	no	—
216	C₂²×D₃	—	20	2	no	—
217	C₂×C₃×C₄	—	2	2	no	—
218	C₂×C₃×C₄	—	2	2	no	—
219	C₂×C₃×C₄	—	2	2	no	—
220	C₂×C₃×C₄	#685	1	1	yes	characteristic
221	C₂×C₃×C₄	#685	1	1	yes	characteristic
222	C₂×C₃×C₄	#685	1	1	yes	characteristic
223	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
224	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
225	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
226	C₂×Dic₃	—	10	2	no	—
227	C₂×Dic₃	—	20	2	no	—
228	D₁₂	—	5	4	no	—
229	D₁₂	—	10	4	no	—
230	D₁₂	—	10	2	no	—
231	C₄×D₃	—	20	4	no	—
232	C₄×D₃	—	20	4	no	—
233	C₃×C₈	—	4	2	no	—
234	C₃⋊C₈	—	10	4	no	—
235	C₂²×C₅	—	2	1	no	—
236	C₂²×C₅	—	2	1	no	—
237	C₂²×C₅	#825	1	2	yes	—
238	C₂²×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
239	C₂×D₅	—	6	4	no	—
240	C₂×D₅	—	12	8	no	—
241	C₂×D₅	—	12	2	no	—
242	C₂×D₅	—	12	2	no	—
243	C₄×C₅	—	2	1	no	—
244	C₄×C₅	—	2	1	no	—
245	C₄×C₅	—	2	1	no	—
246	C₄×C₅	—	2	1	no	—
247	C₄×C₅	D₃×M₁₆	1	2	yes	—
248	C₄×C₅	#949	1	2	yes	—
249	Dic₅	—	12	2	no	—
250	C₂⁴	—	30	1	no	—
251	C₂×D₄	—	15	2	no	—
252	C₂×D₄	—	30	4	no	—
253	C₂×D₄	—	30	1	no	—
254	C₂²×C₄	—	30	1	no	—
255	C₂²×C₄	—	30	2	no	—
256	C₂²×C₄	—	30	2	no	—
257	C₂²×C₄	—	60	4	no	—
258	C₂²×C₄	C₄×D₁₅	1	1	yes	characteristic
259	C₂²×C₄	D₆₀	1	1	yes	characteristic
260	C₂²×C₄	#1220	1	1	yes	characteristic
261	M₁₆	—	30	4	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	30	2	no	—
264	C₄⋊C₄	—	15	2	no	—
265	C₄⋊C₄	—	30	1	no	—
266	C₂²⋊C₄	—	30	2	no	—
267	C₂²⋊C₄	—	30	2	no	—
268	C₄²	C₄×D₁₅	1	1	yes	characteristic
269	C₄²	D₆₀	1	1	yes	characteristic
270	C₄²	#1220	1	1	yes	characteristic
271	C₄²	C₂²×D₁₅	1	1	yes	characteristic
272	C₃×C₅	#1490	1	1	yes	characteristic, complemented, Hall
273	C₂²×C₃	—	2	1	no	—
274	C₂²×C₃	—	2	1	no	—
275	C₂²×C₃	#3957	1	2	yes	—
276	C₂²×C₃	C₂×#685	1	1	yes	characteristic
277	C₂×D₃	—	10	4	no	—
278	C₂×D₃	—	20	8	no	—
279	C₂×D₃	—	20	2	no	—
280	C₂×D₃	—	20	2	no	—
281	C₃×C₄	—	2	1	no	—
282	C₃×C₄	—	2	1	no	—
283	C₃×C₄	—	2	1	no	—
284	C₃×C₄	—	2	1	no	—
285	C₃×C₄	D₅×M₁₆	1	2	yes	—
286	C₃×C₄	#4095	1	2	yes	—
287	Dic₃	—	20	2	no	—
288	C₂×C₅	—	2	2	no	—
289	C₂×C₅	#5001	1	1	yes	characteristic
290	C₂×C₅	#5126	1	1	yes	characteristic
291	C₂×C₅	C₂×#825	1	1	yes	characteristic
292	D₅	—	12	4	no	—
293	C₂³	—	30	4	no	—
294	C₂³	—	30	2	no	—
295	C₂³	—	60	2	no	—
296	C₂³	—	60	2	no	—
297	C₂³	#6714	1	1	yes	characteristic
298	D₄	—	15	4	no	—
299	D₄	—	30	4	no	—
300	D₄	—	30	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	30	2	no	—
305	C₂×C₄	—	60	4	no	—
306	C₂×C₄	—	60	2	no	—
307	C₂×C₄	—	60	4	no	—
308	C₂×C₄	#6714	1	1	yes	characteristic
309	C₂×C₄	#6714	1	1	yes	characteristic
310	C₂×C₄	#6714	1	1	yes	characteristic
311	C₂×C₄	C₂×C₄×D₁₅	1	1	yes	characteristic
312	C₂×C₄	C₂×D₆₀	1	1	yes	characteristic
313	C₂×C₄	C₂×#1220	1	1	yes	characteristic
314	C₈	—	4	2	no	—
315	C₈	—	30	4	no	—
316	C₂×C₃	—	2	2	no	—
317	C₂×C₃	#65290	1	1	yes	characteristic
318	C₂×C₃	#65415	1	1	yes	characteristic
319	C₂×C₃	C₂×#3957	1	1	yes	characteristic
320	D₃	—	20	4	no	—
321	C₅	#70799	1	1	yes	characteristic, complemented, Sylow
322	C₂²	—	2	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	30	4	no	—
325	C₂²	—	60	8	no	—
326	C₂²	—	60	2	no	—
327	C₂²	—	60	2	no	—
328	C₂²	#91109	1	2	yes	—
329	C₂²	C₂×#6714	1	1	yes	characteristic
330	C₄	—	2	1	no	—
331	C₄	—	2	1	no	—
332	C₄	—	2	1	no	—
333	C₄	—	2	1	no	—
334	C₄	—	60	2	no	—
335	C₄	M₁₆×D₁₅	1	2	yes	—
336	C₄	#91838	1	2	yes	—
337	C₃	#10601592	1	1	yes	characteristic, complemented, Sylow
338	C₂	—	2	2	no	—
339	C₂	—	60	4	no	—
340	C₂	#11751102	1	1	yes	characteristic
341	C₂	#11751227	1	1	yes	characteristic
342	C₂	C₂×#91109	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#1220
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₄²	C₂²×D₁₅
Fitting	C₃×C₅×#432	C₂
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₂²×D₆₀	C₄
cosocle	C₃×C₄²×C₅	C₂³
hypercenter	#432	D₁₅
hypocenter	C₃×C₅	#1490
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #432

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×#432	C₂	semidirect
3. C₄×#6710	C₂	semidirect
4. #11746304	C₂	semidirect
16. C₃×C₅×C₄⋊C₈	C₂²	semidirect
20. C₄×C₃⋊#168	C₂²	semidirect
23. C₅×#432	D₃	semidirect
49. C₃×#432	D₅	semidirect
62. C₅×C₄⋊C₈	C₂×D₃	semidirect
86. D₆₀	M₁₆	semidirect
101. C₃×C₄⋊C₈	C₂×D₅	semidirect
129. #432	D₁₅	semidirect
202. C₄⋊C₈	C₂×D₁₅	semidirect
272. C₃×C₅	#1490	semidirect
321. C₅	#70799	semidirect
337. C₃	#10601592	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×#432	2
C₄×#6710	2
#11746304	2
#10601592	3
#70799	5