Finite Group Databasebeta

← #12104549

Subgroups — #12104549

Order 1280.

1466 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#65811C211yesmaximal, characteristic
2C5×#2285C211yesmaximal, characteristic, complemented
3C2×C4×#681C211yesmaximal, characteristic, complemented
4#10600375C212yesmaximal, complemented
5#10600373C212yesmaximal, complemented
6#10600310C214yesmaximal, complemented
7#10600267C214yesmaximal, complemented
8C22×#709—21no—
9C2×C5×#117—21no—
10C22×C42×C5C2211yescharacteristic
11C2×#4076—21no—
12C2×#4076C2212yes—
13C22×#681C2212yes—
14#65811—24no—
15#65811C418yescomplemented
16C2×#4021—21no—
17C2×#4020C2211yescharacteristic
18C2×C4×D20—21no—
19C2×C4×D20C2212yes—
20C2×#4015C2211yescharacteristic
21C5×#384C2212yes—
22C5×#382—21no—
23C5×#380—21no—
24C5×#380C2212yes—
25C4×C5×C4⋊C4—21no—
26C4×C5×C4⋊C4C2212yes—
27C2×#3941C2218yes—
28C4×#681—24no—
29C2×#3935C2212yes—
30C2×C4×#168C2212yes—
31#64749C2218yes—
32#21410—51nomaximal, Sylow
33C23×C4×C5D411yescharacteristic
34C23×C4×C5D411yescharacteristic
35C23×C4×C5C2311yescharacteristic
36C22×C22⋊D5—22no—
37C2×#709—28no—
38C2×#709—48no—
39C22×D20—21no—
40C22×D20—21no—
41C22×D20C2312yes—
42C22×C4×D5—22no—
43C22×Dic10—21no—
44C5×#117—44no—
45C2×C5×C4⋊C4—22no—
46C2×C5×C4⋊C4—22no—
47C2×C5×C4⋊C4C2314yes—
48C2×C5×C22⋊C4—22no—
49C2×C42×C5—22no—
50C2×C42×C5C2×C414yes—
51C2×C42×C5D414yes—
52C2×C42×C5D411yescharacteristic
53C2×C42×C5D411yescharacteristic
54C2×C42×C5C2311yescharacteristic
55C2×C42×C5C2311yescharacteristic
56#4076—28no—
57#4076—24no—
58#4076C2×C418yes—
59C2×#685—22no—
60C2×#685—22no—
61C2×C4⋊Dic5—21no—
62C2×C4⋊Dic5C2312yes—
63C2×#681—24no—
64C2×#681—28no—
65C22×#168C2314yes—
66#4021—28no—
67#4020—22no—
68#4020C2×C414yes—
69C4×D20—28no—
70C4×D20—24no—
71C4×D20C2×C418yes—
72#4015—22no—
73#4015C2×C414yes—
74C5×#95—22no—
75C5×#95—22no—
76#3941—216no—
77#3935—24no—
78C4×#168—24no—
79C2×#532—51no—
80C2×C4×M16—51no—
81#2285D511yescharacteristic, complemented
82#1990—52no—
83#1957—52no—
84#1895—54no—
85#1762—54no—
86C24×C5C2×D411yescharacteristic
87C23×D5—22no—
88C22×C4×C5—22no—
89C22×C4×C5—24no—
90C22×C4×C5—22no—
91C22×C4×C5—24no—
92C22×C4×C5—24no—
93C22×C4×C5C22⋊C414yes—
94C22×C4×C5C22⋊C414yes—
95C22×C4×C5C22×C412yes—
96C22×C4×C5C2×D412yes—
97C22×C4×C5C2×D411yescharacteristic
98C22×C4×C5C2×D411yescharacteristic
99C22×C4×C5C2×D411yescharacteristic
100C22×C4×C5C2×D411yescharacteristic
101C22×C4×C5C2×D411yescharacteristic
102C22×C4×C5C2411yescharacteristic
103C2×C22⋊D5—28no—
104C2×C22⋊D5—48no—
105C22×Dic5—22no—
106#709—432no—
107C2×D20—28no—
108C2×D20—24no—
109C2×D20—24no—
110C2×D20—22no—
111C2×D20—42no—
112C2×D20—44no—
113C2×D20—42no—
114C2×D20C22×C414yes—
115C2×C4×D5—28no—
116C2×C4×D5—48no—
117C2×Dic10—24no—
118C2×Dic10—44no—
119C5×C4⋊C4—28no—
120C5×C4⋊C4—44no—
121C5×C4⋊C4—44no—
122C5×C22⋊C4—44no—
123C42×C5—44no—
124C42×C5C22⋊C414yes—
125C42×C5C22⋊C414yes—
126C42×C5C22×C412yes—
127C42×C5C22×C412yes—
128C42×C5C2×D412yes—
129C42×C5C2×D412yes—
130#685—28no—
131#685—28no—
132C4⋊Dic5—24no—
133C4⋊Dic5—22no—
134C4⋊Dic5C22×C414yes—
135#681—216no—
136C2×#168—24no—
137C2×#168—24no—
138C2×#168#4118yescomplemented
139C22×#41—101no—
140C22×M16—52no—
141#532—58no—
142#532—104no—
143C2×C4⋊Q8—51no—
144C2×#127—51no—
145C2×#124—101no—
146C2×#121—52no—
147C2×#121—101no—
148C2×C4×D4—52no—
149C2×C4×D4—101no—
150C2×#117—21no—
151C22×C42C2×D511yescharacteristic
152C2×#102—58no—
153C4×M16—104no—
154C2×#97—52no—
155C2×C4×C8—52no—
156#384C2×D512yes—
157#382—21no—
158#380—21no—
159#380C2×D512yes—
160C4×C4⋊C4—21no—
161C4×C4⋊C4C2×D512yes—
162#340—58no—
163C23×C5—22no—
164C23×C5—21no—
165C23×C5—22no—
166C23×C5—21no—
167C23×C5C2×C22⋊C412yes—
168C23×C5C22×D411yescharacteristic
169C22×D5—24no—
170C22×D5—44no—
171C22×D5—44no—
172C22×D5—42no—
173C22×D5—42no—
174C2×C4×C5—22no—
175C2×C4×C5—21no—
176C2×C4×C5—22no—
177C2×C4×C5—21no—
178C2×C4×C5—24no—
179C2×C4×C5—24no—
180C2×C4×C5—28no—
181C2×C4×C5—28no—
182C2×C4×C5—44no—
183C2×C4×C5—44no—
184C2×C4×C5—44no—
185C2×C4×C5C2×C22⋊C412yes—
186C2×C4×C5C2×C22⋊C412yes—
187C2×C4×C5C2×C22⋊C412yes—
188C2×C4×C5C2×C22⋊C412yes—
189C2×C4×C5C2×C22⋊C412yes—
190C2×C4×C5AGL(1,ℤ/8)12yes—
191C2×C4×C5AGL(1,ℤ/8)14yes—
192C2×C4×C5#13712yes—
193C2×C4×C5C23×C411yescharacteristic
194C2×C4×C5C22×D411yescharacteristic
195C2×C4×C5C2×#4114yes—
196C22⋊D5—416no—
197C2×Dic5—24no—
198C2×Dic5—44no—
199D20—28no—
200D20—44no—
201D20—48no—
202D20—48no—
203C4×D5—416no—
204Dic10—48no—
205#168—28no—
206C2×#41—108no—
207C2×#41—208no—
208C22×Q8—101no—
209C22×D4—52no—
210C22×D4—102no—
211C22×D4—101no—
212C22×D4—101no—
213C23×C4—102no—
214C23×C4C22⋊D511yescharacteristic
215C23×C4C22⋊D511yescharacteristic
216C23×C4C22×D511yescharacteristic
217C2×M16—108no—
218C2×M16—104no—
219C22×C8—54no—
220C4⋊Q8—54no—
221C4⋊Q8—102no—
222#127—54no—
223#127—102no—
224#124—108no—
225#121—58no—
226#121—108no—
227#121—104no—
228C4×D4—58no—
229C4×D4—108no—
230C4×D4—104no—
231#117—44no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—52no—
235C2×C4⋊C4—101no—
236C2×C4⋊C4C22×D514yes—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—102no—
239C2×C22⋊C4—102no—
240C2×C42—22no—
241C2×C42C4×D514yes—
242C2×C42D2014yes—
243C2×C42C22⋊D511yescharacteristic
244C2×C42C22⋊D511yescharacteristic
245C2×C42C22×D511yescharacteristic
246C2×C42C22×D511yescharacteristic
247#102—1016no—
248#97—104no—
249C4×C8—104no—
250#95—22no—
251#95—22no—
252C22×C5—22no—
253C22×C5—22no—
254C22×C5—22no—
255C22×C5—42no—
256C22×C5—42no—
257C22×C5C22×C22⋊C411yescharacteristic
258C22×C5#53212yes—
259C22×C5C2×AGL(1,ℤ/8)11yescharacteristic
260C22×C5C2×AGL(1,ℤ/8)12yes—
261C22×C5C2×#13711yescharacteristic
262C2×D5—44no—
263C2×D5—44no—
264C2×D5—44no—
265C2×D5—84no—
266C2×D5—84no—
267C4×C5—22no—
268C4×C5—22no—
269C4×C5—22no—
270C4×C5—84no—
271C4×C5#41714yes—
272C4×C5#41814yes—
273C4×C5C4×#4112yes—
274C4×C5#52512yes—
275Dic5—44no—
276C24—102no—
277C24C2×C22⋊D511yescharacteristic
278#41—2032no—
279C2×Q8—104no—
280C2×Q8—204no—
281C2×D4—54no—
282C2×D4—108no—
283C2×D4—102no—
284C2×D4—108no—
285C2×D4—104no—
286C2×D4—104no—
287C2×D4—208no—
288C2×D4—202no—
289C2×D4—202no—
290C2×D4—204no—
291C22×C4—24no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—102no—
297C22×C4—108no—
298C22×C4—208no—
299C22×C4#68514yes—
300C22×C4#68514yes—
301C22×C4C2×C4×D512yes—
302C22×C4C2×D2012yes—
303C22×C4C2×C22⋊D511yescharacteristic
304C22×C4C2×C22⋊D511yescharacteristic
305C22×C4C2×C22⋊D511yescharacteristic
306C22×C4C2×C22⋊D511yescharacteristic
307C22×C4C2×C22⋊D511yescharacteristic
308C22×C4C23×D511yescharacteristic
309M16—1016no—
310C2×C8—58no—
311C2×C8—104no—
312C2×C8—104no—
313C4⋊C4—28no—
314C4⋊C4—44no—
315C4⋊C4—44no—
316C4⋊C4—54no—
317C4⋊C4—102no—
318C4⋊C4—104no—
319C22⋊C4—44no—
320C22⋊C4—108no—
321C22⋊C4—108no—
322C42—44no—
323C42#68514yes—
324C42#68514yes—
325C42C2×C4×D512yes—
326C42C2×C4×D512yes—
327C42C2×D2012yes—
328C42C2×D2012yes—
329C2×C5—42no—
330C2×C5#229711yescharacteristic
331C2×C5C2×#41811yescharacteristic
332C2×C5C2×#41711yescharacteristic
333C2×C5#318212yes—
334C2×C5#318412yes—
335D5—84no—
336C23—21no—
337C23—21no—
338C23—22no—
339C23—22no—
340C23—104no—
341C23—202no—
342C23—202no—
343C23—204no—
344C23—204no—
345C23C2×#68512yes—
346C23C22×C22⋊D511yescharacteristic
347Q8—208no—
348D4—108no—
349D4—2016no—
350D4—208no—
351D4—204no—
352D4—208no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—28no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—28no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—104no—
365C2×C4—2016no—
366C2×C4—204no—
367C2×C4C2×#68512yes—
368C2×C4C2×#68512yes—
369C2×C4C2×#68512yes—
370C2×C4C2×#68512yes—
371C2×C4C2×#68512yes—
372C2×C4#407812yes—
373C2×C4#408812yes—
374C2×C4#409514yes—
375C2×C4C22×C4×D511yescharacteristic
376C2×C4C22×D2011yescharacteristic
377C2×C4D5×#4114yes—
378C8—108no—
379C5#2141011yescharacteristic, complemented, Sylow
380C22—22no—
381C22—22no—
382C22—22no—
383C22—42no—
384C22—42no—
385C22—204no—
386C22—204no—
387C22—204no—
388C22—404no—
389C22—404no—
390C22C22×#68511yescharacteristic
391C22C2×#407811yescharacteristic
392C22C2×#408811yescharacteristic
393C22C2×#409512yes—
394C22#6616612yes—
395C4—22no—
396C4—22no—
397C4—22no—
398C4—84no—
399C4—204no—
400C4#6525514yes—
401C4#6529014yes—
402C4#6585612yes—
403C4#6585812yes—
404C2—42no—
405C2—404no—
406C2C2×#6525511yescharacteristic
407C2C2×#6529011yescharacteristic
408C2#1061395911yescharacteristic
409C2#1061502612yes—
410C2#1061507812yes—
subgroupG/H
centerC23C22×C22⋊D5
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC5×#2285C2
socleC23×C5C22×D4
prime radicalC2×#65811C2
cosocleC22×C4×C5C24
hypercenter#2285D5
hypocenterC5#21410
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2285

Complements

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

NH ≅ G/Ntype
2. C5×#2285C2semidirect
3. C2×C4×#681C2semidirect
4. #10600375C2semidirect
5. #10600373C2semidirect
6. #10600310C2semidirect
7. #10600267C2semidirect
15. #65811C4semidirect
81. #2285D5semidirect
138. C2×#168#41semidirect
379. C5#21410semidirect

Maximal subgroups

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

subgroupindex
C2×#658112
C5×#22852
C2×C4×#6812
#106003752
#106003732
#106003102
#106002672
#214105