Subgroups — #14283049

Order 1920.

544 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	#11756236	C₂	1	1	yes	maximal, characteristic
2	C₅×#4926	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#65221	C₂	1	1	yes	maximal, characteristic, complemented
4	#11745958	C₂	1	4	yes	maximal, complemented
5	#10601625	—	3	1	no	maximal, Hall
6	C₂×#6817	—	2	1	no	—
7	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
8	#91819	—	2	1	no	—
9	#91819	C₂²	1	2	yes	—
10	#91764	—	2	1	no	—
11	#91763	C₄	1	1	yes	characteristic, complemented
12	C₄×D₆₀	—	2	1	no	—
13	C₄×D₆₀	C₄	1	2	yes	complemented
14	#91758	C₄	1	1	yes	characteristic, complemented
15	C₂×C₃×C₄×Dic₅	—	2	1	no	—
16	C₅×#830	C₂²	1	2	yes	—
17	C₅×C₄⋊(C₃⋊C₈)	—	2	1	no	—
18	C₅×#803	—	2	1	no	—
19	C₃×#3963	—	2	1	no	—
20	C₃×#3963	C₂²	1	2	yes	—
21	#69432	—	5	1	no	maximal, Hall
22	#65811	—	3	1	no	—
23	C₅×#432	—	3	1	no	—
24	#65221	D₃	1	1	yes	characteristic, complemented
25	#64690	—	3	4	no	—
26	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
27	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
28	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
29	C₂×#1220	—	2	2	no	—
30	#6817	—	4	4	no	—
31	C₂×D₆₀	—	2	1	no	—
32	C₂×D₆₀	—	2	1	no	—
33	C₂×D₆₀	C₂×C₄	1	2	yes	—
34	C₂×C₄×D₁₅	—	2	2	no	—
35	C₂×Dic₃₀	—	2	1	no	—
36	C₂²×C₃×Dic₅	—	2	2	no	—
37	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
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	#6714	—	2	2	no	—
42	#6714	—	2	2	no	—
43	C₄⋊Dic₁₅	—	2	1	no	—
44	C₄⋊Dic₁₅	C₂×C₄	1	2	yes	—
45	C₂×C₅×C₃⋊C₈	—	2	2	no	—
46	C₃×C₄×Dic₅	—	2	4	no	—
47	#5477	—	5	1	no	—
48	C₃×#376	—	5	1	no	—
49	#4926	D₅	1	1	yes	characteristic, complemented
50	#4451	—	5	4	no	—
51	C₂×#709	—	6	1	no	—
52	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
53	#4076	—	3	2	no	—
54	#4076	—	6	1	no	—
55	C₂×C₄×Dic₅	—	2	1	no	—
56	#4021	—	6	1	no	—
57	#4020	—	3	1	no	—
58	C₄×D₂₀	—	3	2	no	—
59	C₄×D₂₀	—	6	1	no	—
60	#4015	—	3	1	no	—
61	C₅×C₄⋊C₈	—	6	1	no	—
62	C₅×C₂²⋊C₈	—	3	2	no	—
63	C₅×#97	—	6	1	no	—
64	#3963	—	2	1	no	—
65	#3963	C₂×D₃	1	2	yes	—
66	#1505	—	15	1	no	Sylow
67	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
68	C₂²×D₁₅	—	2	2	no	—
69	C₂×C₃×C₄×C₅	—	2	2	no	—
70	C₂×C₃×C₄×C₅	—	2	2	no	—
71	C₂×C₃×C₄×C₅	—	4	1	no	—
72	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
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	C₂×D₄	1	1	yes	characteristic
78	#1220	—	4	4	no	—
79	C₂×Dic₁₅	—	2	2	no	—
80	D₆₀	—	4	2	no	—
81	D₆₀	—	4	2	no	—
82	D₆₀	—	4	2	no	—
83	C₄×D₁₅	—	4	4	no	—
84	Dic₃₀	—	4	2	no	—
85	C₂×C₃×Dic₅	—	2	4	no	—
86	C₂×C₃×Dic₅	—	4	2	no	—
87	C₂×C₃×Dic₅	—	4	2	no	—
88	C₅×C₃⋊C₈	—	4	2	no	—
89	C₂×#235	—	10	1	no	—
90	C₂×C₃×C₄²	—	10	1	no	—
91	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
92	#930	—	5	2	no	—
93	#930	—	10	1	no	—
94	#875	—	10	1	no	—
95	#874	—	5	1	no	—
96	C₄×D₁₂	—	5	2	no	—
97	C₄×D₁₂	—	10	1	no	—
98	#869	—	5	1	no	—
99	C₃×#95	—	5	2	no	—
100	C₃×#95	—	10	1	no	—
101	#830	C₂×D₅	1	2	yes	—
102	C₄⋊(C₃⋊C₈)	—	2	1	no	—
103	#803	—	2	1	no	—
104	C₂²×C₄×C₅	#67	1	1	yes	characteristic
105	C₂²×C₄×C₅	#67	1	1	yes	characteristic
106	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
107	C₂×C₂²⋊D₅	—	6	2	no	—
108	C₂²×Dic₅	—	2	2	no	—
109	#709	—	12	4	no	—
110	C₂×D₂₀	—	3	2	no	—
111	C₂×D₂₀	—	6	1	no	—
112	C₂×D₂₀	—	6	1	no	—
113	C₂×C₄×D₅	—	6	2	no	—
114	C₂×Dic₁₀	—	6	1	no	—
115	C₂×C₅×C₈	—	6	2	no	—
116	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
117	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
118	C₄²×C₅	D₁₂	1	1	yes	characteristic
119	C₄²×C₅	D₁₂	1	1	yes	characteristic
120	#685	—	6	2	no	—
121	#685	—	6	2	no	—
122	C₄⋊Dic₅	—	3	2	no	—
123	C₄⋊Dic₅	—	6	1	no	—
124	C₄×Dic₅	—	2	4	no	—
125	#532	—	15	1	no	—
126	#432	—	3	1	no	—
127	#376	—	5	1	no	—
128	#327	—	15	4	no	—
129	C₂²×C₃×C₅	—	2	1	no	—
130	C₂²×C₃×C₅	—	2	1	no	—
131	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
132	C₂²×C₃×C₅	AGL(1,ℤ/8)	1	2	yes	—
133	C₂×D₁₅	—	4	2	no	—
134	C₂×D₁₅	—	4	2	no	—
135	C₂×D₁₅	—	4	2	no	—
136	C₃×C₄×C₅	—	2	1	no	—
137	C₃×C₄×C₅	—	2	1	no	—
138	C₃×C₄×C₅	—	2	1	no	—
139	C₃×C₄×C₅	—	2	1	no	—
140	C₃×C₄×C₅	—	2	1	no	—
141	C₃×C₄×C₅	#104	1	2	yes	—
142	Dic₁₅	—	4	2	no	—
143	C₃×Dic₅	—	4	4	no	—
144	C₂²×C₃×C₄	—	10	2	no	—
145	C₂²×C₃×C₄	D₂₀	1	1	yes	characteristic
146	C₂²×C₃×C₄	D₂₀	1	1	yes	characteristic
147	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
148	C₂×#67	—	10	2	no	—
149	#235	—	20	4	no	—
150	C₂×D₁₂	—	5	2	no	—
151	C₂×D₁₂	—	10	1	no	—
152	C₂×D₁₂	—	10	1	no	—
153	C₂×C₄×D₃	—	10	2	no	—
154	C₂×Dic₆	—	10	1	no	—
155	C₃×C₄²	—	10	4	no	—
156	C₃×C₄²	C₄×D₅	1	1	yes	characteristic
157	C₃×C₄²	C₄×D₅	1	1	yes	characteristic
158	C₃×C₄²	C₂²⋊D₅	1	1	yes	characteristic
159	C₃×C₄²	C₂²⋊D₅	1	1	yes	characteristic
160	(C₂×D₃)⋊C₄	—	10	2	no	—
161	(C₂×D₃)⋊C₄	—	10	2	no	—
162	C₄⋊Dic₃	—	5	2	no	—
163	C₄⋊Dic₃	—	10	1	no	—
164	C₂×C₃⋊C₈	—	2	2	no	—
165	C₂³×C₅	C₂×#67	1	1	yes	characteristic
166	C₂²×D₅	—	6	2	no	—
167	C₂×C₄×C₅	—	2	2	no	—
168	C₂×C₄×C₅	—	2	2	no	—
169	C₂×C₄×C₅	—	4	1	no	—
170	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
171	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
172	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
173	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
174	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
175	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
176	C₂²⋊D₅	—	12	4	no	—
177	C₂×Dic₅	—	2	4	no	—
178	C₂×Dic₅	—	4	2	no	—
179	C₂×Dic₅	—	4	2	no	—
180	C₂×Dic₅	—	6	2	no	—
181	D₂₀	—	12	2	no	—
182	D₂₀	—	12	2	no	—
183	D₂₀	—	12	2	no	—
184	C₄×D₅	—	12	4	no	—
185	Dic₁₀	—	12	2	no	—
186	C₅×C₈	—	12	2	no	—
187	C₂×#41	—	30	1	no	—
188	C₄⋊Q₈	—	15	1	no	—
189	#127	—	15	1	no	—
190	#124	—	30	1	no	—
191	#121	—	15	2	no	—
192	#121	—	30	1	no	—
193	C₄×D₄	—	15	2	no	—
194	C₄×D₄	—	30	1	no	—
195	C₂×C₄²	—	10	1	no	—
196	C₂×C₄²	D₃×D₅	1	1	yes	characteristic
197	C₄⋊C₈	—	6	1	no	—
198	C₂²⋊C₈	—	3	2	no	—
199	#97	—	6	1	no	—
200	#95	—	5	2	no	—
201	#95	—	10	1	no	—
202	C₂×C₃×C₅	—	2	2	no	—
203	C₂×C₃×C₅	#410	1	1	yes	characteristic
204	C₂×C₃×C₅	#418	1	1	yes	characteristic
205	C₂×C₃×C₅	C₂×#104	1	1	yes	characteristic
206	D₁₅	—	8	2	no	—
207	C₂³×C₃	C₂×D₂₀	1	1	yes	characteristic
208	C₂²×D₃	—	10	2	no	—
209	C₂×C₃×C₄	—	2	2	no	—
210	C₂×C₃×C₄	—	2	2	no	—
211	C₂×C₃×C₄	—	4	1	no	—
212	C₂×C₃×C₄	—	10	4	no	—
213	C₂×C₃×C₄	—	20	2	no	—
214	C₂×C₃×C₄	—	20	2	no	—
215	C₂×C₃×C₄	#685	1	1	yes	characteristic
216	C₂×C₃×C₄	#685	1	1	yes	characteristic
217	C₂×C₃×C₄	#685	1	1	yes	characteristic
218	C₂×C₃×C₄	#685	1	1	yes	characteristic
219	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
220	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
221	#67	—	20	4	no	—
222	C₂×Dic₃	—	10	2	no	—
223	D₁₂	—	20	2	no	—
224	D₁₂	—	20	2	no	—
225	D₁₂	—	20	2	no	—
226	C₄×D₃	—	20	4	no	—
227	Dic₆	—	20	2	no	—
228	C₃⋊C₈	—	4	2	no	—
229	C₂²×C₅	—	2	1	no	—
230	C₂²×C₅	—	2	1	no	—
231	C₂²×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
232	C₂²×C₅	#949	1	2	yes	—
233	C₂×D₅	—	12	2	no	—
234	C₂×D₅	—	12	2	no	—
235	C₂×D₅	—	12	2	no	—
236	C₄×C₅	—	2	1	no	—
237	C₄×C₅	—	2	1	no	—
238	C₄×C₅	—	2	1	no	—
239	C₄×C₅	—	2	1	no	—
240	C₄×C₅	—	2	1	no	—
241	C₄×C₅	#805	1	2	yes	—
242	Dic₅	—	4	4	no	—
243	Dic₅	—	12	2	no	—
244	#41	—	60	4	no	—
245	C₂×Q₈	—	30	1	no	—
246	C₂×D₄	—	15	2	no	—
247	C₂×D₄	—	30	2	no	—
248	C₂×D₄	—	30	1	no	—
249	C₂×D₄	—	30	1	no	—
250	C₂²×C₄	—	10	2	no	—
251	C₂²×C₄	—	30	2	no	—
252	C₂²×C₄	#1202	1	1	yes	characteristic
253	C₂²×C₄	#1202	1	1	yes	characteristic
254	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
255	C₂×C₈	—	6	2	no	—
256	C₄⋊C₄	—	15	2	no	—
257	C₄⋊C₄	—	30	1	no	—
258	C₂²⋊C₄	—	30	2	no	—
259	C₂²⋊C₄	—	30	2	no	—
260	C₄²	—	10	4	no	—
261	C₄²	#1200	1	1	yes	characteristic
262	C₄²	#1200	1	1	yes	characteristic
263	C₄²	#1203	1	1	yes	characteristic
264	C₄²	#1203	1	1	yes	characteristic
265	C₃×C₅	#1505	1	1	yes	characteristic, complemented, Hall
266	C₂²×C₃	—	2	1	no	—
267	C₂²×C₃	—	2	1	no	—
268	C₂²×C₃	#4054	1	2	yes	—
269	C₂²×C₃	C₂×#685	1	1	yes	characteristic
270	C₂×D₃	—	20	2	no	—
271	C₂×D₃	—	20	2	no	—
272	C₂×D₃	—	20	2	no	—
273	C₃×C₄	—	2	1	no	—
274	C₃×C₄	—	2	1	no	—
275	C₃×C₄	—	2	1	no	—
276	C₃×C₄	—	2	1	no	—
277	C₃×C₄	—	2	1	no	—
278	C₃×C₄	—	20	4	no	—
279	C₃×C₄	#3957	1	2	yes	—
280	Dic₃	—	20	2	no	—
281	C₂×C₅	—	2	2	no	—
282	C₂×C₅	C₂×#805	1	1	yes	characteristic
283	C₂×C₅	#5001	1	1	yes	characteristic
284	C₂×C₅	#5006	1	1	yes	characteristic
285	D₅	—	24	2	no	—
286	C₂³	—	30	2	no	—
287	C₂³	C₂×#1202	1	1	yes	characteristic
288	Q₈	—	60	2	no	—
289	D₄	—	60	2	no	—
290	D₄	—	60	2	no	—
291	D₄	—	60	4	no	—
292	D₄	—	60	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	10	4	no	—
297	C₂×C₄	—	20	2	no	—
298	C₂×C₄	—	20	2	no	—
299	C₂×C₄	—	30	2	no	—
300	C₂×C₄	—	60	4	no	—
301	C₂×C₄	#6667	1	1	yes	characteristic
302	C₂×C₄	#6667	1	1	yes	characteristic
303	C₂×C₄	#6667	1	1	yes	characteristic
304	C₂×C₄	#6667	1	1	yes	characteristic
305	C₂×C₄	C₂×#1200	1	1	yes	characteristic
306	C₂×C₄	C₂×#1203	1	1	yes	characteristic
307	C₈	—	12	2	no	—
308	C₂×C₃	—	2	2	no	—
309	C₂×C₃	#65295	1	1	yes	characteristic
310	C₂×C₃	#65420	1	1	yes	characteristic
311	C₂×C₃	C₂×#3957	1	1	yes	characteristic
312	D₃	—	40	2	no	—
313	C₅	#69432	1	1	yes	characteristic, complemented, Sylow
314	C₂²	—	2	1	no	—
315	C₂²	—	2	1	no	—
316	C₂²	—	60	2	no	—
317	C₂²	—	60	2	no	—
318	C₂²	—	60	2	no	—
319	C₂²	#91301	1	2	yes	—
320	C₂²	C₂×#6667	1	1	yes	characteristic
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	20	4	no	—
327	C₄	—	60	2	no	—
328	C₄	#90981	1	2	yes	—
329	C₃	#10601625	1	1	yes	characteristic, complemented, Sylow
330	C₂	—	2	2	no	—
331	C₂	—	120	2	no	—
332	C₂	#11748392	1	1	yes	characteristic
333	C₂	C₂×#90981	1	1	yes	characteristic
334	C₂	#11748626	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#6667
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₂²×C₄	C₂×D₃×D₅
Fitting	C₂×C₃×C₄²×C₅	C₂²
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	#11756236	C₂
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	C₂×C₄²	D₃×D₅
hypocenter	C₃×C₅	#1505
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. C₅×#4926	C₂	semidirect
3. C₃×#65221	C₂	semidirect
4. #11745958	C₂	semidirect
11. #91763	C₄	semidirect
13. C₄×D₆₀	C₄	semidirect
14. #91758	C₄	semidirect
24. #65221	D₃	semidirect
49. #4926	D₅	semidirect
265. C₃×C₅	#1505	semidirect
313. C₅	#69432	semidirect
329. C₃	#10601625	semidirect

Maximal subgroups

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

subgroup	index
#11756236	2
C₅×#4926	2
C₃×#65221	2
#11745958	2
#10601625	3
#69432	5