Subgroups — #11992604

Order 1280.

384 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2966	C₂	1	1	yes	maximal, characteristic
2	#10602548	C₂	1	2	yes	maximal
3	C₄×#3949	C₂	1	1	yes	maximal, characteristic, complemented
4	#10600525	C₂	1	1	yes	maximal, characteristic, complemented
5	#10598241	C₂	1	2	yes	maximal, complemented
6	C₅×#529	C₂²	1	1	yes	characteristic
7	C₅×#446	C₄	1	1	yes	characteristic, complemented
8	C₅×C₈×Q₈	C₄	1	1	yes	characteristic, complemented
9	C₅×#436	C₄	1	1	yes	characteristic, complemented
10	C₅×#435	—	2	1	no	—
11	C₅×C₈×D₄	—	2	1	no	—
12	C₅×C₈×D₄	C₄	1	1	yes	characteristic, complemented
13	C₅×#433	—	2	1	no	—
14	C₅×#432	C₂²	1	1	yes	characteristic
15	C₅×#405	—	2	1	no	—
16	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
17	C₂×#3949	—	2	1	no	—
18	C₄×C₄⋊Dic₅	C₂²	1	2	yes	—
19	#65215	C₂²	1	2	yes	—
20	#64771	—	2	1	no	—
21	#64758	—	2	2	no	—
22	#64757	—	2	2	no	—
23	#64678	—	2	1	no	—
24	#64677	—	2	1	no	—
25	#8694	—	5	1	no	maximal, Sylow
26	C₂×C₅×M₁₆	—	2	1	no	—
27	C₂²×C₅×C₈	—	2	1	no	—
28	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
29	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
30	C₅×#126	—	2	1	no	—
31	C₅×#125	Q₈	1	1	yes	characteristic
32	C₅×#124	Q₈	1	1	yes	characteristic
33	C₅×#123	—	2	1	no	—
34	C₅×C₂²⋊Q₈	D₄	1	1	yes	characteristic
35	C₅×#121	D₄	1	1	yes	characteristic
36	C₄×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
37	C₄×C₅×D₄	—	2	1	no	—
38	C₄×C₅×D₄	C₂×C₄	1	1	yes	characteristic
39	C₅×#117	—	2	1	no	—
40	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
41	C₂×C₄⋊Dic₅	—	2	2	no	—
42	C₂×C₄×Dic₅	—	2	2	no	—
43	C₅×C₄⋊C₈	—	2	1	no	—
44	C₅×C₄⋊C₈	—	2	1	no	—
45	C₅×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
46	C₅×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
47	C₅×C₂²⋊C₈	—	2	1	no	—
48	C₅×C₂²⋊C₈	—	2	1	no	—
49	C₅×C₂²⋊C₈	—	2	1	no	—
50	C₅×#97	—	2	1	no	—
51	C₅×#97	—	2	1	no	—
52	C₄×C₅×C₈	—	2	1	no	—
53	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
54	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
55	C₄×C₅×C₈	D₄	1	1	yes	characteristic
56	C₄×C₅×C₈	D₄	1	1	yes	characteristic
57	#3963	—	2	2	no	—
58	C₂²⋊#168	—	2	2	no	—
59	#3949	—	2	4	no	—
60	C₄⋊#168	—	2	2	no	—
61	#3935	—	2	2	no	—
62	#2966	D₅	1	1	yes	characteristic
63	C₄×#106	—	5	1	no	—
64	#1568	—	5	1	no	—
65	#1509	—	5	2	no	—
66	#1285	—	5	2	no	—
67	C₂×C₅×Q₈	C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₅×D₄	—	2	1	no	—
69	C₂×C₅×D₄	C₄⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₅	—	2	1	no	—
71	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₅	#41	1	1	yes	characteristic
74	C₂²×Dic₅	—	4	2	no	—
75	C₅×M₁₆	—	4	2	no	—
76	C₂×C₅×C₈	—	2	2	no	—
77	C₂×C₅×C₈	—	2	1	no	—
78	C₂×C₅×C₈	—	2	1	no	—
79	C₂×C₅×C₈	—	4	1	no	—
80	C₂×C₅×C₈	—	4	2	no	—
81	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
85	C₅×C₄⋊C₄	—	2	1	no	—
86	C₅×C₄⋊C₄	—	2	1	no	—
87	C₅×C₄⋊C₄	—	2	1	no	—
88	C₅×C₄⋊C₄	—	2	1	no	—
89	C₅×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
90	C₅×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
91	C₅×C₂²⋊C₄	—	2	1	no	—
92	C₅×C₂²⋊C₄	—	2	1	no	—
93	C₅×C₂²⋊C₄	—	2	1	no	—
94	C₅×C₂²⋊C₄	—	2	1	no	—
95	C₅×C₂²⋊C₄	—	2	1	no	—
96	C₄²×C₅	—	2	1	no	—
97	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
98	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
99	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
100	C₄²×C₅	#41	1	1	yes	characteristic
101	C₄⋊Dic₅	—	2	4	no	—
102	C₄⋊Dic₅	—	4	2	no	—
103	C₄×Dic₅	—	8	2	no	—
104	C₂×#168	—	4	2	no	—
105	#529	C₂×D₅	1	1	yes	characteristic
106	#446	Dic₅	1	1	yes	characteristic, complemented
107	C₈×Q₈	Dic₅	1	1	yes	characteristic, complemented
108	#436	Dic₅	1	1	yes	characteristic, complemented
109	#435	—	2	1	no	—
110	C₈×D₄	—	2	1	no	—
111	C₈×D₄	Dic₅	1	1	yes	characteristic, complemented
112	#433	—	2	1	no	—
113	#432	—	5	2	no	—
114	#432	C₂×D₅	1	1	yes	characteristic
115	C₂×#106	—	10	1	no	—
116	#405	—	2	1	no	—
117	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
118	C₄×C₄⋊C₄	—	5	2	no	—
119	#340	—	10	1	no	—
120	#335	—	10	1	no	—
121	#334	—	10	1	no	—
122	#328	—	10	2	no	—
123	#327	—	10	2	no	—
124	C₂³×C₅	—	2	1	no	—
125	C₂³×C₅	#121	1	1	yes	characteristic
126	C₅×Q₈	—	4	1	no	—
127	C₅×D₄	—	4	1	no	—
128	C₅×D₄	—	4	2	no	—
129	C₂×C₄×C₅	—	2	2	no	—
130	C₂×C₄×C₅	—	2	1	no	—
131	C₂×C₄×C₅	—	2	1	no	—
132	C₂×C₄×C₅	—	2	1	no	—
133	C₂×C₄×C₅	—	2	1	no	—
134	C₂×C₄×C₅	—	2	1	no	—
135	C₂×C₄×C₅	—	4	1	no	—
136	C₂×C₄×C₅	—	4	1	no	—
137	C₂×C₄×C₅	—	4	1	no	—
138	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
139	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
140	C₂×C₄×C₅	#117	1	1	yes	characteristic
141	C₂×C₄×C₅	#121	1	1	yes	characteristic
142	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
143	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
144	C₂×Dic₅	—	4	4	no	—
145	C₂×Dic₅	—	8	2	no	—
146	C₂×Dic₅	—	8	2	no	—
147	C₅×C₈	—	2	1	no	—
148	C₅×C₈	—	2	1	no	—
149	C₅×C₈	—	2	1	no	—
150	C₅×C₈	—	4	1	no	—
151	C₅×C₈	—	4	1	no	—
152	C₅×C₈	#104	1	2	yes	—
153	#168	—	8	2	no	—
154	C₂×M₁₆	—	2	1	no	—
155	C₂²×C₈	—	2	1	no	—
156	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
157	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
158	#126	—	2	1	no	—
159	#125	Dic₁₀	1	1	yes	characteristic
160	#124	Dic₁₀	1	1	yes	characteristic
161	#123	—	2	1	no	—
162	C₂²⋊Q₈	D₂₀	1	1	yes	characteristic
163	#121	D₂₀	1	1	yes	characteristic
164	C₄×Q₈	C₂×Dic₅	1	1	yes	characteristic
165	C₄×D₄	—	2	1	no	—
166	C₄×D₄	C₂×Dic₅	1	1	yes	characteristic
167	#117	—	2	1	no	—
168	C₂×C₄⋊C₄	—	10	2	no	—
169	C₂×C₄²	—	10	2	no	—
170	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
171	#106	—	10	4	no	—
172	C₄⋊C₈	—	2	1	no	—
173	C₄⋊C₈	—	2	1	no	—
174	C₄⋊C₈	—	10	2	no	—
175	C₄⋊C₈	C₂×Dic₅	1	1	yes	characteristic
176	C₄⋊C₈	C₂×Dic₅	1	1	yes	characteristic
177	C₂²⋊C₈	—	2	1	no	—
178	C₂²⋊C₈	—	2	1	no	—
179	C₂²⋊C₈	—	2	1	no	—
180	C₂²⋊C₈	—	10	2	no	—
181	#97	—	2	1	no	—
182	#97	—	2	1	no	—
183	#97	—	10	2	no	—
184	C₄×C₈	—	2	1	no	—
185	C₄×C₈	C₂×Dic₅	1	1	yes	characteristic
186	C₄×C₈	C₂×Dic₅	1	1	yes	characteristic
187	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
188	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
189	#95	—	10	2	no	—
190	C₂²×C₅	—	2	1	no	—
191	C₂²×C₅	—	2	1	no	—
192	C₂²×C₅	—	4	1	no	—
193	C₂²×C₅	—	4	1	no	—
194	C₂²×C₅	—	4	1	no	—
195	C₂²×C₅	#380	1	1	yes	characteristic
196	C₂²×C₅	#471	1	2	yes	—
197	C₄×C₅	—	2	1	no	—
198	C₄×C₅	—	2	1	no	—
199	C₄×C₅	—	2	1	no	—
200	C₄×C₅	—	2	1	no	—
201	C₄×C₅	—	2	1	no	—
202	C₄×C₅	—	4	1	no	—
203	C₄×C₅	—	4	1	no	—
204	C₄×C₅	—	4	1	no	—
205	C₄×C₅	C₂×#104	1	1	yes	characteristic
206	C₄×C₅	#421	1	1	yes	characteristic
207	Dic₅	—	8	4	no	—
208	C₂×Q₈	C₄⋊Dic₅	1	1	yes	characteristic
209	C₂×D₄	—	2	1	no	—
210	C₂×D₄	C₄⋊Dic₅	1	1	yes	characteristic
211	C₂²×C₄	—	2	1	no	—
212	C₂²×C₄	—	20	2	no	—
213	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
214	C₂²×C₄	#709	1	1	yes	characteristic
215	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
216	M₁₆	—	4	2	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	2	1	no	—
219	C₂×C₈	—	2	1	no	—
220	C₂×C₈	—	4	1	no	—
221	C₂×C₈	—	4	2	no	—
222	C₂×C₈	—	20	2	no	—
223	C₂×C₈	C₂²⋊Dic₅	1	1	yes	characteristic
224	C₂×C₈	C₂²⋊Dic₅	1	1	yes	characteristic
225	C₂×C₈	C₂²⋊Dic₅	1	1	yes	characteristic
226	C₂×C₈	C₂²⋊Dic₅	1	1	yes	characteristic
227	C₄⋊C₄	—	2	1	no	—
228	C₄⋊C₄	—	2	1	no	—
229	C₄⋊C₄	—	2	1	no	—
230	C₄⋊C₄	—	2	1	no	—
231	C₄⋊C₄	—	10	4	no	—
232	C₄⋊C₄	—	20	2	no	—
233	C₄⋊C₄	C₄⋊Dic₅	1	1	yes	characteristic
234	C₄⋊C₄	C₄⋊Dic₅	1	1	yes	characteristic
235	C₂²⋊C₄	—	2	1	no	—
236	C₂²⋊C₄	—	2	1	no	—
237	C₂²⋊C₄	—	2	1	no	—
238	C₂²⋊C₄	—	2	1	no	—
239	C₂²⋊C₄	—	2	1	no	—
240	C₄²	—	2	1	no	—
241	C₄²	—	40	2	no	—
242	C₄²	C₂×Dic₁₀	1	1	yes	characteristic
243	C₄²	#709	1	1	yes	characteristic
244	C₄²	C₂²×Dic₅	1	1	yes	characteristic
245	C₄²	C₂×C₂²⋊D₅	1	1	yes	characteristic
246	C₂×C₅	—	2	2	no	—
247	C₂×C₅	—	8	1	no	—
248	C₂×C₅	#1806	1	1	yes	characteristic
249	C₂×C₅	#1812	1	1	yes	characteristic
250	C₂×C₅	#1815	1	1	yes	characteristic
251	C₂³	—	2	1	no	—
252	C₂³	#4076	1	1	yes	characteristic
253	Q₈	—	4	1	no	—
254	D₄	—	4	1	no	—
255	D₄	—	4	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	4	1	no	—
263	C₂×C₄	—	4	1	no	—
264	C₂×C₄	—	4	1	no	—
265	C₂×C₄	—	20	4	no	—
266	C₂×C₄	—	40	2	no	—
267	C₂×C₄	—	40	2	no	—
268	C₂×C₄	#4070	1	1	yes	characteristic
269	C₂×C₄	#4070	1	1	yes	characteristic
270	C₂×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
271	C₂×C₄	#4072	1	1	yes	characteristic
272	C₂×C₄	#4076	1	1	yes	characteristic
273	C₂×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
274	C₈	—	2	1	no	—
275	C₈	—	2	1	no	—
276	C₈	—	2	1	no	—
277	C₈	—	4	1	no	—
278	C₈	—	4	1	no	—
279	C₈	—	40	2	no	—
280	C₈	#3969	1	2	yes	—
281	C₅	#8694	1	1	yes	characteristic, complemented, Sylow
282	C₂²	—	2	1	no	—
283	C₂²	—	2	1	no	—
284	C₂²	—	4	1	no	—
285	C₂²	—	4	1	no	—
286	C₂²	—	4	1	no	—
287	C₂²	#65430	1	2	yes	—
288	C₂²	#65501	1	1	yes	characteristic
289	C₄	—	2	1	no	—
290	C₄	—	2	1	no	—
291	C₄	—	2	1	no	—
292	C₄	—	2	1	no	—
293	C₄	—	2	1	no	—
294	C₄	—	4	1	no	—
295	C₄	—	4	1	no	—
296	C₄	—	4	1	no	—
297	C₄	—	40	4	no	—
298	C₄	C₂×#3969	1	1	yes	characteristic
299	C₄	#65525	1	1	yes	characteristic
300	C₂	—	2	2	no	—
301	C₂	—	8	1	no	—
302	C₂	#10602439	1	1	yes	characteristic
303	C₂	#10602805	1	1	yes	characteristic
304	C₂	#10602808	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#65501
commutator	C₄²×C₅	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₅×#2966	C₂
socle	C₂²×C₅	#380
prime radical	C₅×#121	D₄
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	#2966	D₅
hypocenter	C₅	#8694
LCS	G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2966

Complements

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

N	H ≅ G/N	type
3. C₄×#3949	C₂	semidirect
4. #10600525	C₂	semidirect
5. #10598241	C₂	semidirect
7. C₅×#446	C₄	semidirect
8. C₅×C₈×Q₈	C₄	semidirect
9. C₅×#436	C₄	semidirect
12. C₅×C₈×D₄	C₄	semidirect
106. #446	Dic₅	semidirect
107. C₈×Q₈	Dic₅	semidirect
108. #436	Dic₅	semidirect
111. C₈×D₄	Dic₅	semidirect
281. C₅	#8694	semidirect

Maximal subgroups

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

subgroup	index
C₅×#2966	2
#10602548	2
C₄×#3949	2
#10600525	2
#10598241	2
#8694	5