Subgroups — #11987474

Order 1280.

428 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10605031	C₂	1	1	yes	maximal, characteristic
2	C₅×#1831	C₂	1	1	yes	maximal, characteristic
3	#10600184	C₂	1	1	yes	maximal, characteristic
4	#10597971	C₂	1	2	yes	maximal
5	#10597969	C₂	1	2	yes	maximal
6	C₂×C₄×Dic₁₀	C₄	1	1	yes	characteristic
7	C₂×C₅×C₄⋊C₈	—	2	1	no	—
8	C₄³×C₅	C₂²	1	1	yes	characteristic
9	C₄×C₄⋊Dic₅	C₄	1	1	yes	characteristic
10	#65222	—	2	1	no	—
11	#65222	—	2	1	no	—
12	#65221	—	2	1	no	—
13	C₄×#683	—	2	1	no	—
14	#65218	C₄	1	1	yes	characteristic
15	#65218	C₂²	1	1	yes	characteristic
16	#65217	C₄	1	1	yes	characteristic
17	#65217	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊#168	—	2	1	no	—
19	C₂×#3935	—	2	1	no	—
20	C₅×#336	—	2	1	no	—
21	C₅×#336	C₂²	1	2	yes	—
22	#64744	C₂²	1	2	yes	—
23	#7615	—	5	1	no	maximal, Sylow
24	C₂²×Dic₁₀	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₅×C₈	—	2	2	no	—
26	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
33	C₂×C₄⋊Dic₅	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄⋊Dic₅	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄⋊Dic₅	C₂×C₄	1	1	yes	characteristic
36	C₂×#683	—	2	1	no	—
37	C₂×#683	—	2	1	no	—
38	C₂×#683	—	2	1	no	—
39	C₂×C₄×Dic₅	—	2	1	no	—
40	C₂×C₄×Dic₅	—	2	1	no	—
41	C₂²×#168	—	2	2	no	—
42	C₄×Dic₁₀	—	2	2	no	—
43	C₄×Dic₁₀	—	2	2	no	—
44	C₅×C₄⋊C₈	—	4	2	no	—
45	#3963	—	2	1	no	—
46	#3963	—	2	1	no	—
47	#3963	—	2	1	no	—
48	#3963	—	2	1	no	—
49	#3963	—	2	1	no	—
50	#3963	—	2	1	no	—
51	#3963	—	2	1	no	—
52	C₄⋊#168	—	2	4	no	—
53	#3935	—	4	2	no	—
54	#2295	—	5	1	no	—
55	#1831	D₅	1	1	yes	characteristic
56	#1767	—	5	1	no	—
57	#1272	—	5	2	no	—
58	#1271	—	5	2	no	—
59	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
66	C₂²×Dic₅	—	2	1	no	—
67	C₂²×Dic₅	—	2	1	no	—
68	C₂²×Dic₅	—	2	1	no	—
69	C₂²×Dic₅	—	2	1	no	—
70	C₂×Dic₁₀	—	2	1	no	—
71	C₂×Dic₁₀	—	4	1	no	—
72	C₂×Dic₁₀	—	4	1	no	—
73	C₂×Dic₁₀	M₁₆	1	2	yes	—
74	C₂×C₅×C₈	—	4	2	no	—
75	C₂×C₅×C₈	—	4	2	no	—
76	C₂×C₅×C₈	—	4	2	no	—
77	C₄²×C₅	—	2	2	no	—
78	C₄²×C₅	—	2	1	no	—
79	C₄²×C₅	—	2	1	no	—
80	C₄²×C₅	—	2	1	no	—
81	C₄²×C₅	—	2	1	no	—
82	C₄²×C₅	—	4	1	no	—
83	C₄²×C₅	—	4	1	no	—
84	C₄²×C₅	—	4	1	no	—
85	C₄²×C₅	SD₁₆	1	2	yes	—
86	C₄²×C₅	Q₁₆	1	2	yes	—
87	C₄⋊Dic₅	—	2	1	no	—
88	C₄⋊Dic₅	—	4	1	no	—
89	C₄⋊Dic₅	—	4	1	no	—
90	C₄⋊Dic₅	M₁₆	1	2	yes	—
91	#683	—	2	4	no	—
92	#683	—	4	2	no	—
93	#683	—	4	1	no	—
94	#683	—	4	1	no	—
95	C₄×Dic₅	—	2	4	no	—
96	C₄×Dic₅	—	4	2	no	—
97	C₂×#168	—	2	4	no	—
98	C₂×#168	—	4	2	no	—
99	C₂×#168	—	4	2	no	—
100	C₂×C₄×Q₈	—	5	1	no	—
101	C₂×C₄⋊C₈	—	2	1	no	—
102	C₂×C₄⋊C₈	—	10	1	no	—
103	C₂×#97	—	10	1	no	—
104	#391	—	5	1	no	—
105	#391	—	5	1	no	—
106	#389	—	5	1	no	—
107	#389	—	5	1	no	—
108	#387	—	10	1	no	—
109	#387	—	10	1	no	—
110	C₄×C₄⋊C₄	—	5	1	no	—
111	C₄×C₄⋊C₄	—	10	1	no	—
112	#376	—	10	1	no	—
113	C₄³	C₂×D₅	1	1	yes	characteristic
114	#336	—	2	1	no	—
115	#336	—	5	2	no	—
116	#336	C₂×D₅	1	2	yes	—
117	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₅	—	2	1	no	—
119	C₂×C₄×C₅	—	2	1	no	—
120	C₂×C₄×C₅	—	2	1	no	—
121	C₂×C₄×C₅	—	2	1	no	—
122	C₂×C₄×C₅	—	2	1	no	—
123	C₂×C₄×C₅	—	2	1	no	—
124	C₂×C₄×C₅	—	2	1	no	—
125	C₂×C₄×C₅	—	2	1	no	—
126	C₂×C₄×C₅	—	2	1	no	—
127	C₂×C₄×C₅	—	2	1	no	—
128	C₂×C₄×C₅	—	2	1	no	—
129	C₂×C₄×C₅	—	2	1	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	#103	1	2	yes	—
134	C₂×C₄×C₅	#103	1	2	yes	—
135	C₂×C₄×C₅	#104	1	2	yes	—
136	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
137	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
138	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
139	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
140	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
141	C₂×C₄×C₅	#137	1	1	yes	characteristic
142	C₂×Dic₅	—	2	2	no	—
143	C₂×Dic₅	—	2	2	no	—
144	C₂×Dic₅	—	4	1	no	—
145	C₂×Dic₅	—	4	1	no	—
146	C₂×Dic₅	—	4	1	no	—
147	C₂×Dic₅	—	4	1	no	—
148	C₂×Dic₅	—	4	1	no	—
149	C₂×Dic₅	—	4	1	no	—
150	C₂×Dic₅	—	4	1	no	—
151	C₂×Dic₅	—	4	1	no	—
152	C₂×Dic₅	—	4	1	no	—
153	C₂×Dic₅	—	4	1	no	—
154	Dic₁₀	—	4	2	no	—
155	Dic₁₀	—	4	2	no	—
156	C₅×C₈	—	8	2	no	—
157	#168	—	4	4	no	—
158	C₂²×Q₈	—	5	1	no	—
159	C₂²×C₈	—	2	2	no	—
160	C₂²×C₈	—	10	2	no	—
161	C₄×Q₈	—	10	2	no	—
162	C₄×Q₈	—	10	2	no	—
163	C₂×C₄⋊C₄	—	5	1	no	—
164	C₂×C₄⋊C₄	—	5	1	no	—
165	C₂×C₄⋊C₄	—	5	1	no	—
166	C₂×C₄⋊C₄	—	10	1	no	—
167	C₂×C₄⋊C₄	—	10	1	no	—
168	C₂×C₄⋊C₄	—	10	1	no	—
169	C₂×C₄²	—	10	1	no	—
170	C₂×C₄²	—	10	1	no	—
171	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
172	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
173	C₂×C₄²	D₂₀	1	1	yes	characteristic
174	C₂×C₄²	D₂₀	1	1	yes	characteristic
175	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
176	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
177	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
178	C₄⋊C₈	—	4	2	no	—
179	C₄⋊C₈	—	10	4	no	—
180	#97	—	20	2	no	—
181	#95	—	10	1	no	—
182	#95	—	10	1	no	—
183	#95	—	10	1	no	—
184	#95	—	10	1	no	—
185	#95	—	10	1	no	—
186	#95	—	10	1	no	—
187	#95	—	10	1	no	—
188	C₂²×C₅	#407	1	1	yes	characteristic
189	C₂²×C₅	#410	1	1	yes	characteristic
190	C₂²×C₅	#413	1	1	yes	characteristic
191	C₂²×C₅	C₂×#103	1	1	yes	characteristic
192	C₂²×C₅	#417	1	1	yes	characteristic
193	C₂²×C₅	C₂×#104	1	1	yes	characteristic
194	C₂²×C₅	#421	1	1	yes	characteristic
195	C₄×C₅	—	2	2	no	—
196	C₄×C₅	—	2	1	no	—
197	C₄×C₅	—	2	1	no	—
198	C₄×C₅	—	2	2	no	—
199	C₄×C₅	—	2	1	no	—
200	C₄×C₅	—	2	1	no	—
201	C₄×C₅	—	2	1	no	—
202	C₄×C₅	—	2	1	no	—
203	C₄×C₅	—	2	2	no	—
204	C₄×C₅	—	4	1	no	—
205	Dic₅	—	4	2	no	—
206	Dic₅	—	4	2	no	—
207	Dic₅	—	8	1	no	—
208	Dic₅	—	8	1	no	—
209	C₂×Q₈	—	5	2	no	—
210	C₂×Q₈	—	10	1	no	—
211	C₂×Q₈	—	20	1	no	—
212	C₂×Q₈	—	20	1	no	—
213	C₂²×C₄	—	10	1	no	—
214	C₂²×C₄	—	10	1	no	—
215	C₂²×C₄	—	10	1	no	—
216	C₂²×C₄	—	10	1	no	—
217	C₂²×C₄	#685	1	1	yes	characteristic
218	C₂²×C₄	#685	1	1	yes	characteristic
219	C₂²×C₄	#685	1	1	yes	characteristic
220	C₂²×C₄	#685	1	1	yes	characteristic
221	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
222	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
223	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
224	C₂×C₈	—	4	2	no	—
225	C₂×C₈	—	4	2	no	—
226	C₂×C₈	—	4	2	no	—
227	C₂×C₈	—	10	4	no	—
228	C₂×C₈	—	20	2	no	—
229	C₂×C₈	—	20	2	no	—
230	C₄⋊C₄	—	5	2	no	—
231	C₄⋊C₄	—	10	4	no	—
232	C₄⋊C₄	—	10	1	no	—
233	C₄⋊C₄	—	20	1	no	—
234	C₄⋊C₄	—	20	1	no	—
235	C₄⋊C₄	—	20	2	no	—
236	C₄⋊C₄	—	20	1	no	—
237	C₄⋊C₄	—	20	1	no	—
238	C₄²	—	2	2	no	—
239	C₄²	—	2	1	no	—
240	C₄²	—	2	1	no	—
241	C₄²	—	2	1	no	—
242	C₄²	—	2	1	no	—
243	C₄²	—	4	1	no	—
244	C₄²	—	4	1	no	—
245	C₄²	—	4	1	no	—
246	C₄²	—	10	4	no	—
247	C₄²	—	20	2	no	—
248	C₄²	#677	1	2	yes	—
249	C₄²	Dic₂₀	1	2	yes	—
250	C₂×C₅	#1464	1	1	yes	characteristic
251	C₂×C₅	#1487	1	1	yes	characteristic
252	C₂×C₅	#1489	1	1	yes	characteristic
253	C₂×C₅	#1503	1	1	yes	characteristic
254	C₂×C₅	#1512	1	1	yes	characteristic
255	C₂×C₅	#1532	1	1	yes	characteristic
256	C₂×C₅	#1550	1	1	yes	characteristic
257	C₂³	C₂×#685	1	1	yes	characteristic
258	Q₈	—	20	2	no	—
259	Q₈	—	20	2	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	10	2	no	—
276	C₂×C₄	—	10	2	no	—
277	C₂×C₄	—	20	1	no	—
278	C₂×C₄	—	20	1	no	—
279	C₂×C₄	—	20	1	no	—
280	C₂×C₄	—	20	1	no	—
281	C₂×C₄	—	20	1	no	—
282	C₂×C₄	—	20	1	no	—
283	C₂×C₄	—	20	1	no	—
284	C₂×C₄	—	20	1	no	—
285	C₂×C₄	—	20	1	no	—
286	C₂×C₄	—	20	1	no	—
287	C₂×C₄	#3948	1	2	yes	—
288	C₂×C₄	#3948	1	2	yes	—
289	C₂×C₄	#3957	1	2	yes	—
290	C₂×C₄	C₂×#677	1	1	yes	characteristic
291	C₂×C₄	C₂×Dic₂₀	1	1	yes	characteristic
292	C₂×C₄	D₅×M₁₆	1	1	yes	characteristic
293	C₂×C₄	D₅×M₁₆	1	1	yes	characteristic
294	C₂×C₄	#4078	1	1	yes	characteristic
295	C₂×C₄	#4088	1	1	yes	characteristic
296	C₈	—	8	2	no	—
297	C₈	—	20	4	no	—
298	C₅	#7615	1	1	yes	characteristic, complemented, Sylow
299	C₂²	#65255	1	1	yes	characteristic
300	C₂²	#65294	1	1	yes	characteristic
301	C₂²	#65295	1	1	yes	characteristic
302	C₂²	C₂×#3948	1	1	yes	characteristic
303	C₂²	#65415	1	1	yes	characteristic
304	C₂²	#65421	1	1	yes	characteristic
305	C₂²	C₂×#3957	1	1	yes	characteristic
306	C₄	—	2	2	no	—
307	C₄	—	2	2	no	—
308	C₄	—	2	1	no	—
309	C₄	—	2	1	no	—
310	C₄	—	2	2	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	1	no	—
315	C₄	—	4	1	no	—
316	C₄	—	20	2	no	—
317	C₄	—	20	2	no	—
318	C₄	—	40	1	no	—
319	C₄	—	40	1	no	—
320	C₂	#10600337	1	1	yes	characteristic
321	C₂	#10601231	1	1	yes	characteristic
322	C₂	#10601290	1	1	yes	characteristic
323	C₂	#10601563	1	1	yes	characteristic
324	C₂	#10601585	1	1	yes	characteristic
325	C₂	#10601600	1	1	yes	characteristic
326	C₂	#10601619	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#685
commutator	C₂²×C₄×C₅	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₅×#1831	C₂
socle, prime radical	C₂³×C₅	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	#1831	D₅
hypocenter	C₅	#7615
LCS	G ▷ C₂²×C₄×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1831

Complements

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

N	H ≅ G/N	type
298. C₅	#7615	semidirect

Maximal subgroups

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

subgroup	index
#10605031	2
C₅×#1831	2
#10600184	2
#10597971	2
#10597969	2
#7615	5