Finite Group Databasebeta

← #13029306

Subgroups — #13029306

Order 1280.

1346 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#65855C211yesmaximal, characteristic
2#10617092C212yesmaximal, complemented
3#10604877C212yesmaximal, complemented
4#10604290C212yesmaximal, complemented
5#10604202C214yesmaximal, complemented
6#10604163C214yesmaximal, complemented
7C23×D20C411yescharacteristic
8C2×C5×#117C411yescharacteristic
9C2×#4072C2211yescharacteristic
10#65855—28no—
11C2×#4039C411yescharacteristic
12C2×#4039C411yescharacteristic
13C2×#4039C2211yescharacteristic
14C2×#4039C2211yescharacteristic
15C2×D5×C22⋊C4C412yes—
16C2×D5×C22⋊C4C2212yes—
17C2×C4×D20C412yes—
18C2×C4×D20C2212yes—
19#65756C2218yescomplemented
20#65748C2214yescomplemented
21#65705—28no—
22#64933C2214yes—
23#64915C2218yescomplemented
24#64913C2214yescomplemented
25#17369—51nomaximal, Sylow
26C24×D5C2×C412yes—
27C23×C4×C5C2×C411yescharacteristic
28C22×D20—24no—
29C22×D20—24no—
30C22×D20—24no—
31C22×D20C2×C412yes—
32C22×D20C2×C411yescharacteristic
33C22×D20C2×C411yescharacteristic
34C22×C4×D5C2×C412yes—
35C22×C4×D5C2×C412yes—
36C22×C4×D5C2×C412yes—
37C22×C4×D5C2312yes—
38C2×#700—28no—
39C5×#117—24no—
40C2×C5×C4⋊C4C2×C411yescharacteristic
41C2×C5×C4⋊C4C2×C411yescharacteristic
42C2×C5×C22⋊C4C2×C412yes—
43C2×C42×C5C2×C412yes—
44C2×C22⋊Dic5C2×C411yescharacteristic
45C2×C22⋊Dic5C2311yescharacteristic
46C2×#685C2×C412yes—
47C2×#685C2×C412yes—
48C2×#685C2×C412yes—
49C2×#685C2312yes—
50#4072—24no—
51C2×C4⋊Dic5C2×C411yescharacteristic
52C2×C4⋊Dic5C2311yescharacteristic
53C2×C4×Dic5C2×C411yescharacteristic
54C2×C4×Dic5C2311yescharacteristic
55#4039—24no—
56#4039—24no—
57#4039—24no—
58#4039D418yes—
59D5×C22⋊C4—28no—
60D5×C22⋊C4—28no—
61C4×D20—28no—
62C4×D20—28no—
63C22⋊#170C2318yes—
64#4003—28no—
65C4⋊#170—24no—
66C4×#170—24no—
67C2×#518—51no—
68#2975—52no—
69#1465—54no—
70#1462—54no—
71#1460—52no—
72#1458—52no—
73C24×C5C22×C411yescharacteristic
74C23×D5—24no—
75C23×D5—24no—
76C23×D5—24no—
77C23×D5—82no—
78C23×D5—82no—
79C23×D5C22×C412yes—
80C23×D5C22×C412yes—
81C22×C4×C5—22no—
82C22×C4×C5—22no—
83C22×C4×C5M1614yescomplemented
84C22×C4×C5C22×C411yescharacteristic
85C22×C4×C5C22×C412yes—
86C22×C4×C5C22×C412yes—
87C22×C4×C5C22×C411yescharacteristic
88C22×Dic5C22×C411yescharacteristic
89C22×Dic5C22×C411yescharacteristic
90C22×Dic5C22×C411yescharacteristic
91C22×Dic5C2411yescharacteristic
92C2×D20—232no—
93C2×D20—24no—
94C2×D20—24no—
95C2×D20—22no—
96C2×D20—24no—
97C2×D20—22no—
98C2×D20—22no—
99C2×D20—22no—
100C2×D20C22⋊C414yes—
101C2×D20M1614yescomplemented
102C2×C4×D5—24no—
103C2×C4×D5—24no—
104C2×C4×D5—24no—
105C2×C4×D5—82no—
106C2×C4×D5—82no—
107C2×C4×D5—82no—
108C2×C4×D5—82no—
109C2×C4×D5C2×D418yes—
110C2×#170—28no—
111#700—88no—
112C5×C4⋊C4—22no—
113C5×C4⋊C4C22⋊C414yes—
114C5×C22⋊C4—24no—
115C42×C5—24no—
116C22⋊Dic5—22no—
117C22⋊Dic5—22no—
118#685—24no—
119#685—24no—
120#685—24no—
121#685C22⋊C418yes—
122C4⋊Dic5—22no—
123C4⋊Dic5—22no—
124C4×Dic5—22no—
125C4×Dic5C2×D414yes—
126C23×D4—51no—
127#518—108no—
128C2×C4×D4—51no—
129C2×C4×D4—51no—
130C2×C4×D4—51no—
131C2×C4×D4—51no—
132C2×C4×D4—52no—
133C2×C4×D4—52no—
134C2×#117—51no—
135C2×#117AGL(1,𝔽5)11yescharacteristic
136C22×C22⋊C4—52no—
137C22×C22⋊C4—52no—
138#436—108no—
139#431—54no—
140#407—58no—
141#324—54no—
142#323—58no—
143#323—54no—
144C23×C5—21no—
145C23×C5—21no—
146C23×C5—41no—
147C23×C5—41no—
148C23×C5C2×M1612yes—
149C23×C5C23×C411yescharacteristic
150C22×D5—22no—
151C22×D5—22no—
152C22×D5—22no—
153C22×D5—22no—
154C22×D5—216no—
155C22×D5—42no—
156C22×D5—42no—
157C22×D5—42no—
158C22×D5—42no—
159C22×D5—84no—
160C22×D5—84no—
161C22×D5—88no—
162C22×D5—88no—
163C22×D5C2×C22⋊C414yes—
164C22×D5C2×M1614yes—
165C2×C4×C5—21no—
166C2×C4×C5—21no—
167C2×C4×C5—28no—
168C2×C4×C5—22no—
169C2×C4×C5—42no—
170C2×C4×C5—42no—
171C2×C4×C5C2×C22⋊C414yes—
172C2×C4×C5C2×C22⋊C412yes—
173C2×C4×C5C2×M1612yes—
174C2×C4×C5C2×M1612yes—
175C2×C4×C5C2×M1612yes—
176C2×Dic5—21no—
177C2×Dic5—21no—
178C2×Dic5—41no—
179C2×Dic5—41no—
180C2×Dic5—41no—
181C2×Dic5—41no—
182C2×Dic5C2×C22⋊C412yes—
183C2×Dic5C22×D412yes—
184D20—216no—
185D20—28no—
186D20—28no—
187C4×D5—48no—
188C4×D5—84no—
189C4×D5—84no—
190C4×D5—84no—
191#170—48no—
192C25—52no—
193C22×D4—51no—
194C22×D4—51no—
195C22×D4—52no—
196C22×D4—104no—
197C22×D4—104no—
198C22×D4—104no—
199C23×C4—52no—
200C23×C4—52no—
201C23×C4—52no—
202C23×C4—52no—
203C23×C4C2×AGL(1,𝔽5)11yescharacteristic
204C2×M16—108no—
205C4×D4—58no—
206C4×D4—108no—
207C4×D4—108no—
208C4×D4—104no—
209C4×D4—104no—
210C4×D4—104no—
211#117—24no—
212#117—104no—
213C2×C4⋊C4—51no—
214C2×C4⋊C4—51no—
215C2×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
216C2×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
217C2×C22⋊C4—52no—
218C2×C22⋊C4—51no—
219C2×C22⋊C4—51no—
220C2×C22⋊C4—52no—
221C2×C22⋊C4—52no—
222C2×C22⋊C4—52no—
223C2×C22⋊C4—108no—
224C2×C22⋊C4—108no—
225C2×C22⋊C4C2×AGL(1,𝔽5)12yes—
226C2×C42—51no—
227C2×C42—51no—
228C2×C42C2×AGL(1,𝔽5)12yes—
229C4⋊C8—104no—
230C22⋊C8—58no—
231C22⋊C8—108no—
232C4×C8—104no—
233C22×C5—21no—
234C22×C5—21no—
235C22×C5—22no—
236C22×C5—42no—
237C22×C5—42no—
238C22×C5—41no—
239C22×C5—41no—
240C22×C5C22×C22⋊C411yescharacteristic
241C22×C5C22×M1611yescharacteristic
242C22×C5C22×M1611yescharacteristic
243C2×D5—24no—
244C2×D5—24no—
245C2×D5—24no—
246C2×D5—24no—
247C2×D5—44no—
248C2×D5—44no—
249C2×D5—44no—
250C2×D5—42no—
251C2×D5—42no—
252C2×D5—44no—
253C2×D5—816no—
254C4×C5—22no—
255C4×C5—24no—
256C4×C5—42no—
257C4×C5#40714yes—
258Dic5—22no—
259Dic5—22no—
260Dic5—41no—
261Dic5—41no—
262C24—52no—
263C24—52no—
264C24—104no—
265C24—104no—
266C24—104no—
267C24—402no—
268C24—402no—
269C24C22×AGL(1,𝔽5)11yescharacteristic
270C2×D4—54no—
271C2×D4—54no—
272C2×D4—104no—
273C2×D4—104no—
274C2×D4—1032no—
275C2×D4—104no—
276C2×D4—102no—
277C2×D4—102no—
278C2×D4—102no—
279C2×D4—102no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—51no—
283C22×C4—51no—
284C22×C4—58no—
285C22×C4—51no—
286C22×C4—51no—
287C22×C4—104no—
288C22×C4—104no—
289C22×C4—104no—
290C22×C4—402no—
291C22×C4—402no—
292C22×C4—402no—
293C22×C4—402no—
294C22×C4#70014yescomplemented
295C22×C4C22×AGL(1,𝔽5)11yescharacteristic
296C22×C4C22×AGL(1,𝔽5)12yes—
297C22×C4C22×AGL(1,𝔽5)12yes—
298C22×C4C22×AGL(1,𝔽5)11yescharacteristic
299M16—408no—
300C2×C8—108no—
301C4⋊C4—22no—
302C4⋊C4—102no—
303C4⋊C4—102no—
304C4⋊C4C22⋊AGL(1,𝔽5)14yes—
305C22⋊C4—24no—
306C22⋊C4—58no—
307C22⋊C4—104no—
308C22⋊C4—104no—
309C22⋊C4—102no—
310C22⋊C4—102no—
311C22⋊C4—104no—
312C42—24no—
313C42—54no—
314C42—102no—
315C2×C5—21no—
316C2×C5—21no—
317C2×C5—42no—
318C2×C5C2×#40711yescharacteristic
319C2×C5#288311yescharacteristic
320C2×C5#288811yescharacteristic
321D5—44no—
322D5—44no—
323C23—21no—
324C23—21no—
325C23—41no—
326C23—41no—
327C23—54no—
328C23—54no—
329C23—1016no—
330C23—102no—
331C23—102no—
332C23—102no—
333C23—102no—
334C23—202no—
335C23—202no—
336C23—202no—
337C23—202no—
338C23—404no—
339C23—404no—
340C23—408no—
341C23—408no—
342C23C2×#70012yes—
343C23C23×AGL(1,𝔽5)11yescharacteristic
344D4—1016no—
345D4—108no—
346D4—108no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—28no—
350C2×C4—22no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—52no—
354C2×C4—52no—
355C2×C4—101no—
356C2×C4—101no—
357C2×C4—208no—
358C2×C4—201no—
359C2×C4—201no—
360C2×C4—201no—
361C2×C4—201no—
362C2×C4—404no—
363C2×C4—404no—
364C2×C4—404no—
365C2×C4C2×#70012yes—
366C2×C4C2×#70012yes—
367C2×C4D5⋊M1612yes—
368C2×C4C2×C22⋊AGL(1,𝔽5)14yes—
369C2×C4C2×C22⋊AGL(1,𝔽5)12yes—
370C8—208no—
371C5#1736911yescharacteristic, complemented, Sylow
372C22—21no—
373C22—21no—
374C22—22no—
375C22—42no—
376C22—42no—
377C22—41no—
378C22—41no—
379C22—104no—
380C22—104no—
381C22—104no—
382C22—104no—
383C22—204no—
384C22—204no—
385C22—204no—
386C22—204no—
387C22—202no—
388C22—202no—
389C22—4016no—
390C22C22×#70011yescharacteristic
391C22C2×D5⋊M1611yescharacteristic
392C22C22×C22⋊AGL(1,𝔽5)11yescharacteristic
393C4—22no—
394C4—24no—
395C4—42no—
396C4—102no—
397C4—102no—
398C4—201no—
399C4—201no—
400C4#6575614yes—
401C2—21no—
402C2—21no—
403C2—42no—
404C2—204no—
405C2—204no—
406C2C2×#6575611yescharacteristic
407C2#1061753811yescharacteristic
408C2#1061755311yescharacteristic
subgroupG/H
centerC22C22×C22⋊AGL(1,𝔽5)
commutatorC23×C5C23×C4
FrattiniC23C23×AGL(1,𝔽5)
FittingC2×C5×#117C4
socleC22×C5C22×C22⋊C4
prime radicalC23×D20C4
cosocleC22×Dic5C24
hypercenterC2×#117AGL(1,𝔽5)
hypocenterC5#17369
LCSG ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. #10617092C2semidirect
3. #10604877C2semidirect
4. #10604290C2semidirect
5. #10604202C2semidirect
6. #10604163C2semidirect
19. #65756C22semidirect
20. #65748C22semidirect
23. #64915C22semidirect
24. #64913C22semidirect
83. C22×C4×C5M16semidirect
101. C2×D20M16semidirect
294. C22×C4#700semidirect
371. C5#17369semidirect

Maximal subgroups

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

subgroupindex
C2×#658552
#106170922
#106048772
#106042902
#106042022
#106041632
#173695