Subgroups — #14284060

Order 1920.

528 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#4021	C₂	1	1	yes	maximal, characteristic
2	C₄×#6710	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#4926	C₂	1	1	yes	maximal, characteristic, complemented
4	#11745846	C₂	1	4	yes	maximal, complemented
5	#10601597	—	3	1	no	maximal, Hall
6	C₂²×C₃×D₂₀	C₄	1	1	yes	characteristic
7	C₂²×C₃×Dic₁₀	C₄	1	1	yes	characteristic
8	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
9	C₂×#6710	—	2	1	no	—
10	C₂×C₃×#685	—	2	1	no	—
11	C₂×C₃×#685	—	2	1	no	—
12	C₃×#4021	—	2	1	no	—
13	C₃×#4021	—	2	1	no	—
14	C₃×#4021	C₄	1	2	yes	—
15	C₃×#4021	C₂²	1	2	yes	—
16	#91087	C₂²	1	2	yes	complemented
17	C₄×C₃⋊#168	—	2	1	no	—
18	C₅×#830	—	2	1	no	—
19	C₅×C₄⋊(C₃⋊C₈)	—	2	1	no	—
20	C₅×#803	C₂²	1	2	yes	complemented
21	#71718	—	5	1	no	maximal, Hall
22	C₂×#4021	D₃	1	1	yes	characteristic
23	C₅×#432	—	3	1	no	—
24	C₄×#681	—	3	1	no	—
25	#64684	—	3	4	no	—
26	C₂³×C₃×D₅	—	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×C₃×Dic₅	—	2	1	no	—
31	C₂×C₃×D₂₀	—	2	1	no	—
32	C₂×C₃×D₂₀	—	4	2	no	—
33	C₂×C₃×D₂₀	C₂×C₄	1	2	yes	—
34	C₂×C₃×Dic₁₀	—	2	1	no	—
35	C₂×C₃×Dic₁₀	—	4	2	no	—
36	C₂×C₃×Dic₁₀	C₂×C₄	1	2	yes	—
37	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
38	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
39	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
40	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
41	#6710	—	2	4	no	—
42	C₂×C₃⋊#168	—	2	2	no	—
43	C₂×C₅×C₃⋊C₈	—	2	2	no	—
44	C₃×#685	—	2	2	no	—
45	C₃×#685	—	2	2	no	—
46	C₃×#685	—	2	2	no	—
47	C₃×#685	—	2	2	no	—
48	C₂×C₃×#124	—	5	1	no	—
49	#4926	D₅	1	1	yes	characteristic, complemented
50	C₄×#208	—	5	1	no	—
51	#4540	—	5	4	no	—
52	C₂²×D₂₀	Dic₃	1	1	yes	characteristic
53	C₂²×Dic₁₀	Dic₃	1	1	yes	characteristic
54	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
55	C₂×#685	—	2	1	no	—
56	C₂×#685	—	2	1	no	—
57	C₂×#681	—	6	1	no	—
58	#4021	—	2	1	no	—
59	#4021	—	2	1	no	—
60	#4021	Dic₃	1	2	yes	—
61	#4021	C₂×D₃	1	2	yes	—
62	C₅×C₄⋊C₈	—	6	1	no	—
63	C₅×C₂²⋊C₈	—	6	1	no	—
64	C₅×#97	—	3	2	no	—
65	#3935	—	3	2	no	—
66	C₄×#168	—	6	1	no	—
67	#1527	—	15	1	no	Sylow
68	C₂³×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₃×D₅	—	2	2	no	—
70	C₂²×C₃×D₅	—	4	2	no	—
71	C₂²×C₃×D₅	—	4	2	no	—
72	C₂²×C₃×D₅	—	4	2	no	—
73	C₂×C₃×C₄×C₅	—	2	2	no	—
74	C₂×C₃×C₄×C₅	—	2	2	no	—
75	C₂×C₃×C₄×C₅	—	4	1	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×C₃×Dic₅	—	2	2	no	—
83	C₂×C₃×Dic₅	—	4	2	no	—
84	C₃×D₂₀	—	4	2	no	—
85	C₃×D₂₀	—	4	2	no	—
86	C₃×Dic₁₀	—	4	2	no	—
87	C₃×Dic₁₀	—	4	2	no	—
88	C₃⋊#168	—	2	4	no	—
89	C₅×C₃⋊C₈	—	4	2	no	—
90	C₂²×C₃×Q₈	—	5	1	no	—
91	C₂²×C₃×D₄	—	5	1	no	—
92	C₃×#124	—	5	2	no	—
93	C₃×#124	—	5	2	no	—
94	C₃×#124	—	10	1	no	—
95	C₃×#124	—	10	1	no	—
96	C₂×C₃×C₂²⋊C₄	—	10	1	no	—
97	C₂×C₃×C₂²⋊C₄	—	10	1	no	—
98	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
99	C₂×#208	—	10	1	no	—
100	#830	—	2	1	no	—
101	C₄⋊(C₃⋊C₈)	—	2	1	no	—
102	#803	—	5	2	no	—
103	#803	C₂×D₅	1	2	yes	complemented
104	C₄×C₃⋊C₈	—	10	1	no	—
105	C₂³×D₅	—	2	1	no	—
106	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
107	C₂²×C₄×C₅	#67	1	1	yes	characteristic
108	C₂²×C₄×C₅	#67	1	1	yes	characteristic
109	C₂²×Dic₅	—	2	1	no	—
110	C₂×D₂₀	—	2	1	no	—
111	C₂×D₂₀	—	4	2	no	—
112	C₂×D₂₀	C₂×Dic₃	1	2	yes	—
113	C₂×Dic₁₀	—	2	1	no	—
114	C₂×Dic₁₀	—	4	2	no	—
115	C₂×Dic₁₀	C₂×Dic₃	1	2	yes	—
116	C₂×C₅×C₈	—	6	2	no	—
117	C₄²×C₅	C₂×Dic₃	1	1	yes	characteristic
118	C₄²×C₅	#67	1	1	yes	characteristic
119	C₄²×C₅	#67	1	1	yes	characteristic
120	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
121	#685	—	2	2	no	—
122	#685	—	2	2	no	—
123	#685	—	2	2	no	—
124	#685	—	2	2	no	—
125	#681	—	6	4	no	—
126	C₂×#168	—	6	2	no	—
127	C₂×#124	—	5	1	no	—
128	#432	—	3	1	no	—
129	C₄×M₁₆	—	15	1	no	—
130	#329	—	15	4	no	—
131	C₂²×C₃×C₅	—	2	1	no	—
132	C₂²×C₃×C₅	—	2	1	no	—
133	C₂²×C₃×C₅	#104	1	2	yes	—
134	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
135	C₂×C₃×D₅	—	4	2	no	—
136	C₂×C₃×D₅	—	4	2	no	—
137	C₂×C₃×D₅	—	4	2	no	—
138	C₂×C₃×D₅	—	8	4	no	—
139	C₃×C₄×C₅	—	2	1	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	#100	1	2	yes	—
145	C₃×Dic₅	—	4	2	no	—
146	C₂⁴×C₃	—	10	1	no	—
147	C₂×C₃×Q₈	—	5	2	no	—
148	C₂×C₃×Q₈	—	10	1	no	—
149	C₂×C₃×Q₈	—	20	2	no	—
150	C₂×C₃×D₄	—	5	2	no	—
151	C₂×C₃×D₄	—	10	1	no	—
152	C₂×C₃×D₄	—	20	2	no	—
153	C₂²×C₃×C₄	—	10	1	no	—
154	C₂²×C₃×C₄	C₄×D₅	1	1	yes	characteristic
155	C₂²×C₃×C₄	D₂₀	1	1	yes	characteristic
156	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
157	C₃×C₂²⋊C₄	—	10	2	no	—
158	C₃×C₂²⋊C₄	—	10	2	no	—
159	C₃×C₂²⋊C₄	—	10	2	no	—
160	C₃×C₂²⋊C₄	—	10	2	no	—
161	C₃×C₄²	C₄×D₅	1	1	yes	characteristic
162	C₃×C₄²	D₂₀	1	1	yes	characteristic
163	C₃×C₄²	C₂²⋊D₅	1	1	yes	characteristic
164	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
165	#208	—	10	4	no	—
166	C₂×C₃⋊C₈	—	2	2	no	—
167	C₂×C₃⋊C₈	—	10	2	no	—
168	C₂³×C₅	#217	1	1	yes	characteristic
169	C₂²×D₅	—	2	2	no	—
170	C₂²×D₅	—	4	2	no	—
171	C₂²×D₅	—	4	2	no	—
172	C₂²×D₅	—	4	2	no	—
173	C₂×C₄×C₅	—	2	2	no	—
174	C₂×C₄×C₅	—	2	2	no	—
175	C₂×C₄×C₅	—	4	1	no	—
176	C₂×C₄×C₅	#217	1	1	yes	characteristic
177	C₂×C₄×C₅	#217	1	1	yes	characteristic
178	C₂×C₄×C₅	#217	1	1	yes	characteristic
179	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
180	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
181	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
182	C₂×Dic₅	—	2	2	no	—
183	C₂×Dic₅	—	4	2	no	—
184	D₂₀	—	4	2	no	—
185	D₂₀	—	4	2	no	—
186	Dic₁₀	—	4	2	no	—
187	Dic₁₀	—	4	2	no	—
188	C₅×C₈	—	12	2	no	—
189	#168	—	6	4	no	—
190	C₂²×Q₈	—	5	1	no	—
191	C₂²×D₄	—	5	1	no	—
192	C₂×M₁₆	—	30	1	no	—
193	#124	—	5	2	no	—
194	#124	—	5	2	no	—
195	#124	—	10	1	no	—
196	#124	—	10	1	no	—
197	C₂×C₂²⋊C₄	—	10	1	no	—
198	C₂×C₂²⋊C₄	—	10	1	no	—
199	C₂×C₄²	D₃×D₅	1	1	yes	characteristic
200	C₄⋊C₈	—	6	1	no	—
201	C₂²⋊C₈	—	6	1	no	—
202	#97	—	3	2	no	—
203	#97	—	15	2	no	—
204	C₄×C₈	—	30	1	no	—
205	C₂×C₃×C₅	—	2	2	no	—
206	C₂×C₃×C₅	C₂×#100	1	1	yes	characteristic
207	C₂×C₃×C₅	C₂×#104	1	1	yes	characteristic
208	C₂×C₃×C₅	#421	1	1	yes	characteristic
209	C₃×D₅	—	8	2	no	—
210	C₂³×C₃	—	10	2	no	—
211	C₂³×C₃	—	20	2	no	—
212	C₂³×C₃	—	20	2	no	—
213	C₂³×C₃	—	20	2	no	—
214	C₂³×C₃	#685	1	1	yes	characteristic
215	C₃×Q₈	—	20	2	no	—
216	C₃×Q₈	—	20	2	no	—
217	C₃×D₄	—	20	2	no	—
218	C₃×D₄	—	20	2	no	—
219	C₂×C₃×C₄	—	2	2	no	—
220	C₂×C₃×C₄	—	2	2	no	—
221	C₂×C₃×C₄	—	4	1	no	—
222	C₂×C₃×C₄	—	10	2	no	—
223	C₂×C₃×C₄	—	20	2	no	—
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	C₃⋊C₈	—	4	2	no	—
231	C₃⋊C₈	—	10	4	no	—
232	C₂²×C₅	—	2	1	no	—
233	C₂²×C₅	—	2	1	no	—
234	C₂²×C₅	#837	1	2	yes	—
235	C₂²×C₅	C₂×#217	1	1	yes	characteristic
236	C₂×D₅	—	4	2	no	—
237	C₂×D₅	—	4	2	no	—
238	C₂×D₅	—	4	2	no	—
239	C₂×D₅	—	8	4	no	—
240	C₄×C₅	—	2	1	no	—
241	C₄×C₅	—	2	1	no	—
242	C₄×C₅	—	2	1	no	—
243	C₄×C₅	—	2	1	no	—
244	C₄×C₅	—	2	1	no	—
245	C₄×C₅	#833	1	2	yes	—
246	Dic₅	—	4	2	no	—
247	C₂⁴	—	10	1	no	—
248	C₂×Q₈	—	5	2	no	—
249	C₂×Q₈	—	10	1	no	—
250	C₂×Q₈	—	20	2	no	—
251	C₂×D₄	—	5	2	no	—
252	C₂×D₄	—	10	1	no	—
253	C₂×D₄	—	20	2	no	—
254	C₂²×C₄	—	10	1	no	—
255	C₂²×C₄	D₅×Dic₃	1	1	yes	characteristic
256	C₂²×C₄	#1201	1	1	yes	characteristic
257	C₂²×C₄	#1202	1	1	yes	characteristic
258	M₁₆	—	30	4	no	—
259	C₂×C₈	—	6	2	no	—
260	C₂×C₈	—	30	2	no	—
261	C₂²⋊C₄	—	10	2	no	—
262	C₂²⋊C₄	—	10	2	no	—
263	C₂²⋊C₄	—	10	2	no	—
264	C₂²⋊C₄	—	10	2	no	—
265	C₄²	D₅×Dic₃	1	1	yes	characteristic
266	C₄²	#1201	1	1	yes	characteristic
267	C₄²	#1202	1	1	yes	characteristic
268	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
269	C₃×C₅	#1527	1	1	yes	characteristic, complemented, Hall
270	C₂²×C₃	—	2	1	no	—
271	C₂²×C₃	—	2	1	no	—
272	C₂²×C₃	—	20	2	no	—
273	C₂²×C₃	—	20	2	no	—
274	C₂²×C₃	—	20	2	no	—
275	C₂²×C₃	—	40	4	no	—
276	C₂²×C₃	#3957	1	2	yes	—
277	C₂²×C₃	C₂×#685	1	1	yes	characteristic
278	C₃×C₄	—	2	1	no	—
279	C₃×C₄	—	2	1	no	—
280	C₃×C₄	—	2	1	no	—
281	C₃×C₄	—	2	1	no	—
282	C₃×C₄	—	2	1	no	—
283	C₃×C₄	—	20	2	no	—
284	C₃×C₄	#3955	1	2	yes	—
285	C₂×C₅	—	2	2	no	—
286	C₂×C₅	C₂×#833	1	1	yes	characteristic
287	C₂×C₅	C₂×#837	1	1	yes	characteristic
288	C₂×C₅	#5236	1	1	yes	characteristic
289	D₅	—	8	2	no	—
290	C₂³	—	10	2	no	—
291	C₂³	—	20	2	no	—
292	C₂³	—	20	2	no	—
293	C₂³	—	20	2	no	—
294	C₂³	#6665	1	1	yes	characteristic
295	Q₈	—	20	2	no	—
296	Q₈	—	20	2	no	—
297	D₄	—	20	2	no	—
298	D₄	—	20	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	4	1	no	—
302	C₂×C₄	—	10	2	no	—
303	C₂×C₄	—	20	2	no	—
304	C₂×C₄	#6665	1	1	yes	characteristic
305	C₂×C₄	#6665	1	1	yes	characteristic
306	C₂×C₄	#6665	1	1	yes	characteristic
307	C₂×C₄	C₂×D₅×Dic₃	1	1	yes	characteristic
308	C₂×C₄	C₂×#1201	1	1	yes	characteristic
309	C₂×C₄	C₂×#1202	1	1	yes	characteristic
310	C₈	—	12	2	no	—
311	C₈	—	30	4	no	—
312	C₂×C₃	—	2	2	no	—
313	C₂×C₃	—	40	2	no	—
314	C₂×C₃	#65294	1	1	yes	characteristic
315	C₂×C₃	C₂×#3955	1	1	yes	characteristic
316	C₂×C₃	C₂×#3957	1	1	yes	characteristic
317	C₅	#71718	1	1	yes	characteristic, complemented, Sylow
318	C₂²	—	2	1	no	—
319	C₂²	—	2	1	no	—
320	C₂²	—	20	2	no	—
321	C₂²	—	20	2	no	—
322	C₂²	—	20	2	no	—
323	C₂²	—	40	4	no	—
324	C₂²	#90976	1	2	yes	—
325	C₂²	C₂×#6665	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	2	1	no	—
331	C₄	—	20	2	no	—
332	C₄	#90958	1	2	yes	—
333	C₃	#10601597	1	1	yes	characteristic, complemented, Sylow
334	C₂	—	2	2	no	—
335	C₂	—	40	2	no	—
336	C₂	C₂×#90958	1	1	yes	characteristic
337	C₂	C₂×#90976	1	1	yes	characteristic
338	C₂	#11748421	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#6665
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₄²	C₂×D₃×D₅
Fitting	C₂×C₃×C₄²×C₅	C₂²
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₂²×C₃×D₂₀	C₄
cosocle	C₃×C₄²×C₅	C₂³
hypercenter	C₂×C₄²	D₃×D₅
hypocenter	C₃×C₅	#1527
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
2. C₄×#6710	C₂	semidirect
3. C₅×#4926	C₂	semidirect
4. #11745846	C₂	semidirect
16. #91087	C₂²	semidirect
20. C₅×#803	C₂²	semidirect
49. #4926	D₅	semidirect
103. #803	C₂×D₅	semidirect
269. C₃×C₅	#1527	semidirect
317. C₅	#71718	semidirect
333. C₃	#10601597	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×#4021	2
C₄×#6710	2
C₅×#4926	2
#11745846	2
#10601597	3
#71718	5