Subgroups — #14286494

Order 1920.

364 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#436	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#6715	C₂	1	1	yes	maximal, characteristic
3	C₈×D₆₀	C₂	1	1	yes	maximal, characteristic, complemented
4	D₄⋊(C₃⋊#168)	C₂	1	1	yes	maximal, characteristic, complemented
5	#11746304	C₂	1	1	yes	maximal, characteristic, complemented
6	#11746300	C₂	1	1	yes	maximal, characteristic, complemented
7	#11746276	C₂	1	1	yes	maximal, characteristic, complemented
8	#10601655	—	3	1	no	maximal, Hall
9	C₂×C₃×C₅×M₁₆	—	2	1	no	—
10	C₃×C₄×C₅×D₄	C₂²	1	1	yes	characteristic
11	C₂×#6715	C₄	1	1	yes	characteristic
12	C₂×C₈×D₁₅	—	2	1	no	—
13	C₄×D₆₀	C₂²	1	1	yes	characteristic
14	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₃×C₅×C₂²⋊C₈	—	2	1	no	—
16	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
17	#91116	C₄	1	1	yes	characteristic
18	#91104	—	2	1	no	—
19	#91093	C₄	1	1	yes	characteristic
20	#91091	C₄	1	1	yes	characteristic
21	C₄⋊(C₃⋊#168)	C₂²	1	1	yes	characteristic
22	C₄×C₃⋊#168	C₂²	1	1	yes	characteristic
23	C₄×C₃⋊#168	C₂²	1	1	yes	characteristic
24	#70862	—	5	1	no	maximal, Hall
25	C₅×#436	D₃	1	1	yes	characteristic, complemented
26	C₄×#686	—	3	1	no	—
27	C₈×D₂₀	—	3	1	no	—
28	D₄⋊#168	—	3	1	no	—
29	#64703	—	3	1	no	—
30	#64699	—	3	1	no	—
31	#64675	—	3	1	no	—
32	C₂×C₃×C₅×D₄	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₃×C₄×C₅	—	2	1	no	—
34	C₂×D₆₀	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×D₁₅	—	2	1	no	—
36	C₃×C₅×M₁₆	—	4	2	no	—
37	C₂×C₃×C₅×C₈	—	2	1	no	—
38	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
39	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
40	C₃×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
41	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
42	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
43	#6715	—	2	2	no	—
44	#6714	—	2	1	no	—
45	C₄⋊Dic₁₅	C₂×C₄	1	1	yes	characteristic
46	C₂×C₃⋊#168	—	2	1	no	—
47	C₂×C₃⋊#168	—	2	1	no	—
48	C₂×C₃⋊#168	C₂×C₄	1	1	yes	characteristic
49	C₂×C₃⋊#168	C₂×C₄	1	1	yes	characteristic
50	C₈×D₁₅	—	4	2	no	—
51	C₃×#436	D₅	1	1	yes	characteristic, complemented
52	C₄×#213	—	5	1	no	—
53	C₈×D₁₂	—	5	1	no	—
54	D₄⋊(C₃⋊C₈)	—	5	1	no	—
55	#4483	—	5	1	no	—
56	#4479	—	5	1	no	—
57	#4455	—	5	1	no	—
58	C₂×C₅×M₁₆	—	2	1	no	—
59	C₄×C₅×D₄	C₂×D₃	1	1	yes	characteristic
60	C₂×#686	—	3	1	no	—
61	C₂×C₈×D₅	—	6	1	no	—
62	C₄×D₂₀	—	3	1	no	—
63	C₅×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
64	C₅×C₂²⋊C₈	—	2	1	no	—
65	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
66	#3964	—	3	1	no	—
67	#3952	—	6	1	no	—
68	#3941	—	3	1	no	—
69	#3939	—	3	1	no	—
70	C₄⋊#168	—	3	1	no	—
71	C₄×#168	—	3	1	no	—
72	C₄×#168	—	3	1	no	—
73	#1581	—	15	1	no	Sylow
74	C₂³×C₃×C₅	—	2	1	no	—
75	C₂²×D₁₅	—	2	1	no	—
76	C₃×C₅×D₄	—	2	1	no	—
77	C₃×C₅×D₄	—	2	1	no	—
78	C₂×C₃×C₄×C₅	—	2	1	no	—
79	C₂×C₃×C₄×C₅	—	4	1	no	—
80	C₂×C₃×C₄×C₅	—	4	1	no	—
81	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
83	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
84	C₂×Dic₁₅	—	2	1	no	—
85	D₆₀	—	2	1	no	—
86	D₆₀	M₁₆	1	2	yes	complemented
87	C₄×D₁₅	—	4	1	no	—
88	C₄×D₁₅	—	4	1	no	—
89	C₃×C₅×C₈	—	2	1	no	—
90	C₃×C₅×C₈	—	4	1	no	—
91	C₃×C₅×C₈	D₈	1	2	yes	complemented
92	C₃⋊#168	—	2	1	no	—
93	C₃⋊#168	—	2	1	no	—
94	C₃⋊#168	—	4	1	no	—
95	C₃⋊#168	—	4	1	no	—
96	C₂×C₃×M₁₆	—	2	1	no	—
97	C₃×C₄×D₄	C₂×D₅	1	1	yes	characteristic
98	C₂×#213	—	5	1	no	—
99	C₂×D₃×C₈	—	10	1	no	—
100	C₄×D₁₂	—	5	1	no	—
101	C₃×C₄⋊C₈	C₂×D₅	1	1	yes	characteristic
102	C₃×C₂²⋊C₈	—	2	1	no	—
103	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
104	#832	—	5	1	no	—
105	(C₂×D₃)⋊C₈	—	10	1	no	—
106	#809	—	5	1	no	—
107	#807	—	5	1	no	—
108	C₄⋊(C₃⋊C₈)	—	5	1	no	—
109	C₄×C₃⋊C₈	—	5	1	no	—
110	C₄×C₃⋊C₈	—	5	1	no	—
111	C₂×C₅×D₄	C₄×D₃	1	1	yes	characteristic
112	C₂²×C₄×C₅	—	2	1	no	—
113	C₂×D₂₀	—	3	1	no	—
114	C₂×C₄×D₅	—	6	1	no	—
115	C₅×M₁₆	—	4	2	no	—
116	C₂×C₅×C₈	—	2	1	no	—
117	C₂×C₅×C₈	#67	1	1	yes	characteristic
118	C₂×C₅×C₈	#67	1	1	yes	characteristic
119	C₅×C₄⋊C₄	C₄×D₃	1	1	yes	characteristic
120	C₅×C₂²⋊C₄	—	2	1	no	—
121	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
122	#686	—	6	2	no	—
123	#685	—	6	1	no	—
124	C₄⋊Dic₅	—	3	1	no	—
125	C₂×#168	—	3	1	no	—
126	C₂×#168	—	3	1	no	—
127	C₂×#168	—	6	1	no	—
128	C₂×#168	—	6	1	no	—
129	C₈×D₅	—	12	2	no	—
130	C₄×D₈	—	15	1	no	—
131	#436	D₁₅	1	1	yes	characteristic, complemented
132	C₈×D₄	—	15	1	no	—
133	#335	—	15	1	no	—
134	D₄⋊C₈	—	15	1	no	—
135	D₄⋊C₈	—	15	1	no	—
136	#322	—	15	1	no	—
137	C₂²×C₃×C₅	—	4	1	no	—
138	C₂²×C₃×C₅	—	4	1	no	—
139	C₂²×C₃×C₅	—	4	1	no	—
140	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
141	C₂×D₁₅	—	2	2	no	—
142	C₂×D₁₅	—	4	1	no	—
143	C₂×D₁₅	—	4	1	no	—
144	C₃×C₄×C₅	—	2	1	no	—
145	C₃×C₄×C₅	—	4	1	no	—
146	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
147	C₃×C₄×C₅	#131	1	1	yes	characteristic
148	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
149	C₃×C₄×C₅	#135	1	1	yes	characteristic
150	Dic₁₅	—	4	1	no	—
151	C₂×C₃×D₄	C₄×D₅	1	1	yes	characteristic
152	C₂²×C₃×C₄	—	2	1	no	—
153	C₂×D₁₂	—	5	1	no	—
154	C₂×C₄×D₃	—	10	1	no	—
155	C₃×M₁₆	—	4	2	no	—
156	C₂×C₃×C₈	—	2	1	no	—
157	C₂×C₃×C₈	C₂²⋊D₅	1	1	yes	characteristic
158	C₂×C₃×C₈	C₂²⋊D₅	1	1	yes	characteristic
159	C₃×C₄⋊C₄	C₄×D₅	1	1	yes	characteristic
160	C₃×C₂²⋊C₄	—	2	1	no	—
161	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
162	#213	—	10	2	no	—
163	(C₂×D₃)⋊C₄	—	10	1	no	—
164	C₄⋊Dic₃	—	5	1	no	—
165	C₂×C₃⋊C₈	—	5	1	no	—
166	C₂×C₃⋊C₈	—	5	1	no	—
167	C₂×C₃⋊C₈	—	10	1	no	—
168	C₂×C₃⋊C₈	—	10	1	no	—
169	D₃×C₈	—	20	2	no	—
170	C₂³×C₅	—	2	1	no	—
171	C₂²×D₅	—	6	1	no	—
172	C₅×D₄	—	2	1	no	—
173	C₅×D₄	—	2	1	no	—
174	C₂×C₄×C₅	—	2	1	no	—
175	C₂×C₄×C₅	—	4	1	no	—
176	C₂×C₄×C₅	—	4	1	no	—
177	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
178	C₂×C₄×C₅	#235	1	1	yes	characteristic
179	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
180	C₂×Dic₅	—	6	1	no	—
181	D₂₀	—	3	2	no	—
182	D₂₀	—	6	1	no	—
183	C₄×D₅	—	12	1	no	—
184	C₄×D₅	—	12	1	no	—
185	C₅×C₈	—	2	1	no	—
186	C₅×C₈	—	4	1	no	—
187	C₅×C₈	#213	1	2	yes	complemented
188	#168	—	6	1	no	—
189	#168	—	6	1	no	—
190	#168	—	12	1	no	—
191	#168	—	12	1	no	—
192	C₂×D₈	—	15	1	no	—
193	C₂×M₁₆	—	2	1	no	—
194	C₂²×C₈	—	30	1	no	—
195	C₄×D₄	—	15	1	no	—
196	C₄×D₄	C₂×D₁₅	1	1	yes	characteristic
197	#107	—	15	1	no	—
198	C₄⋊C₈	—	15	1	no	—
199	C₄⋊C₈	C₂×D₁₅	1	1	yes	characteristic
200	#102	—	15	1	no	—
201	#102	—	15	1	no	—
202	C₂²⋊C₈	—	2	1	no	—
203	C₂²⋊C₈	—	30	1	no	—
204	C₄×C₈	—	15	1	no	—
205	C₄×C₈	—	15	1	no	—
206	C₄×C₈	C₂×D₁₅	1	1	yes	characteristic
207	C₂×C₃×C₅	—	8	1	no	—
208	C₂×C₃×C₅	#435	1	1	yes	characteristic
209	C₂×C₃×C₅	C₄×D₈	1	1	yes	characteristic
210	C₂×C₃×C₅	#444	1	1	yes	characteristic
211	D₁₅	—	4	2	no	—
212	C₂³×C₃	—	2	1	no	—
213	C₂²×D₃	—	10	1	no	—
214	C₃×D₄	—	2	1	no	—
215	C₃×D₄	—	2	1	no	—
216	C₂×C₃×C₄	—	2	1	no	—
217	C₂×C₃×C₄	—	4	1	no	—
218	C₂×C₃×C₄	—	4	1	no	—
219	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
220	C₂×C₃×C₄	#709	1	1	yes	characteristic
221	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
222	C₂×Dic₃	—	10	1	no	—
223	D₁₂	—	5	2	no	—
224	D₁₂	—	10	1	no	—
225	C₄×D₃	—	20	1	no	—
226	C₄×D₃	—	20	1	no	—
227	C₃×C₈	—	2	1	no	—
228	C₃×C₈	—	4	1	no	—
229	C₃×C₈	#686	1	2	yes	complemented
230	C₃⋊C₈	—	10	1	no	—
231	C₃⋊C₈	—	10	1	no	—
232	C₃⋊C₈	—	20	1	no	—
233	C₃⋊C₈	—	20	1	no	—
234	C₂²×C₅	—	4	1	no	—
235	C₂²×C₅	—	4	1	no	—
236	C₂²×C₅	—	4	1	no	—
237	C₂²×C₅	C₄×#67	1	1	yes	characteristic
238	C₂×D₅	—	6	2	no	—
239	C₂×D₅	—	12	1	no	—
240	C₂×D₅	—	12	1	no	—
241	C₄×C₅	—	2	1	no	—
242	C₄×C₅	—	4	1	no	—
243	C₄×C₅	D₃×M₁₆	1	1	yes	characteristic
244	C₄×C₅	#907	1	1	yes	characteristic
245	C₄×C₅	C₂×#213	1	1	yes	characteristic
246	C₄×C₅	#950	1	1	yes	characteristic
247	Dic₅	—	12	1	no	—
248	C₂×D₄	—	15	1	no	—
249	C₂×D₄	C₄×D₁₅	1	1	yes	characteristic
250	C₂²×C₄	—	2	1	no	—
251	C₂²×C₄	—	30	1	no	—
252	D₈	—	30	2	no	—
253	M₁₆	—	4	2	no	—
254	C₂×C₈	—	2	1	no	—
255	C₂×C₈	—	15	1	no	—
256	C₂×C₈	—	15	1	no	—
257	C₂×C₈	—	30	1	no	—
258	C₂×C₈	—	30	1	no	—
259	C₂×C₈	—	60	2	no	—
260	C₂×C₈	#1220	1	1	yes	characteristic
261	C₂×C₈	#1220	1	1	yes	characteristic
262	C₄⋊C₄	—	15	1	no	—
263	C₄⋊C₄	C₄×D₁₅	1	1	yes	characteristic
264	C₂²⋊C₄	—	2	1	no	—
265	C₂²⋊C₄	—	30	1	no	—
266	C₄²	C₂²×D₁₅	1	1	yes	characteristic
267	C₃×C₅	#1581	1	1	yes	characteristic, complemented, Hall
268	C₂²×C₃	—	4	1	no	—
269	C₂²×C₃	—	4	1	no	—
270	C₂²×C₃	—	4	1	no	—
271	C₂²×C₃	C₄×C₂²⋊D₅	1	1	yes	characteristic
272	C₂×D₃	—	10	2	no	—
273	C₂×D₃	—	20	1	no	—
274	C₂×D₃	—	20	1	no	—
275	C₃×C₄	—	2	1	no	—
276	C₃×C₄	—	4	1	no	—
277	C₃×C₄	D₅×M₁₆	1	1	yes	characteristic
278	C₃×C₄	#4053	1	1	yes	characteristic
279	C₃×C₄	C₂×#686	1	1	yes	characteristic
280	C₃×C₄	#4096	1	1	yes	characteristic
281	Dic₃	—	20	1	no	—
282	C₂×C₅	—	8	1	no	—
283	C₂×C₅	C₄×#213	1	1	yes	characteristic
284	C₂×C₅	#5127	1	1	yes	characteristic
285	C₂×C₅	#5145	1	1	yes	characteristic
286	D₅	—	12	2	no	—
287	C₂³	—	2	1	no	—
288	C₂³	—	30	1	no	—
289	D₄	—	2	1	no	—
290	D₄	—	2	1	no	—
291	D₄	—	15	2	no	—
292	D₄	—	30	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	4	1	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	30	1	no	—
297	C₂×C₄	—	60	1	no	—
298	C₂×C₄	—	60	1	no	—
299	C₂×C₄	C₂×C₄×D₁₅	1	1	yes	characteristic
300	C₂×C₄	#6817	1	1	yes	characteristic
301	C₂×C₄	C₂×#1220	1	1	yes	characteristic
302	C₈	—	2	1	no	—
303	C₈	—	4	1	no	—
304	C₈	—	30	1	no	—
305	C₈	—	30	1	no	—
306	C₈	—	60	1	no	—
307	C₈	—	60	1	no	—
308	C₈	#6715	1	2	yes	complemented
309	C₂×C₃	—	8	1	no	—
310	C₂×C₃	C₄×#686	1	1	yes	characteristic
311	C₂×C₃	#65416	1	1	yes	characteristic
312	C₂×C₃	#65434	1	1	yes	characteristic
313	D₃	—	20	2	no	—
314	C₅	#70862	1	1	yes	characteristic, complemented, Sylow
315	C₂²	—	4	1	no	—
316	C₂²	—	4	1	no	—
317	C₂²	—	4	1	no	—
318	C₂²	—	30	2	no	—
319	C₂²	—	60	1	no	—
320	C₂²	—	60	1	no	—
321	C₂²	C₄×#1220	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	4	1	no	—
324	C₄	—	60	1	no	—
325	C₄	M₁₆×D₁₅	1	1	yes	characteristic
326	C₄	#91796	1	1	yes	characteristic
327	C₄	C₂×#6715	1	1	yes	characteristic
328	C₄	#91839	1	1	yes	characteristic
329	C₃	#10601655	1	1	yes	characteristic, complemented, Sylow
330	C₂	—	8	1	no	—
331	C₂	—	60	2	no	—
332	C₂	C₄×#6715	1	1	yes	characteristic
333	C₂	#11751228	1	1	yes	characteristic
334	C₂	#11751246	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×#436	C₂
socle	C₂²×C₃×C₅	C₄×D₄
prime radical	C₂×#6715	C₄
cosocle	C₃×C₄²×C₅	C₂³
hypercenter	#436	D₁₅
hypocenter	C₃×C₅	#1581
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #436

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×#436	C₂	semidirect
3. C₈×D₆₀	C₂	semidirect
4. D₄⋊(C₃⋊#168)	C₂	semidirect
5. #11746304	C₂	semidirect
6. #11746300	C₂	semidirect
7. #11746276	C₂	semidirect
25. C₅×#436	D₃	semidirect
51. C₃×#436	D₅	semidirect
86. D₆₀	M₁₆	semidirect
91. C₃×C₅×C₈	D₈	semidirect
131. #436	D₁₅	semidirect
187. C₅×C₈	#213	semidirect
229. C₃×C₈	#686	semidirect
267. C₃×C₅	#1581	semidirect
308. C₈	#6715	semidirect
314. C₅	#70862	semidirect
329. C₃	#10601655	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅×#436	2
C₄×#6715	2
C₈×D₆₀	2
D₄⋊(C₃⋊#168)	2
#11746304	2
#11746300	2
#11746276	2
#10601655	3
#70862	5