Finite Group Databasebeta

← #12106250

Subgroups — #12106250

Order 1280.

868 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#2875C211yesmaximal, characteristic, complemented
2#10615536C211yesmaximal, characteristic
3#10615523C212yesmaximal, complemented
4C2×#65006C211yesmaximal, characteristic, complemented
5#10602806C211yesmaximal, characteristic, complemented
6#10602438C212yesmaximal, complemented
7#10600694C211yesmaximal, characteristic, complemented
8#10600691C212yesmaximal, complemented
9#10600690C211yesmaximal, characteristic, complemented
10#10600667C212yesmaximal, complemented
11#10600622C211yesmaximal, characteristic, complemented
12C2×C5×#131C2211yescharacteristic
13C2×C5×#131C2211yescharacteristic
14C4×C5×#41C2211yescharacteristic
15#66156C2212yes—
16C2×#4050C2211yescharacteristic
17C2×#4047C2212yes—
18C5×#405—21no—
19C5×#405—21no—
20C5×#405C2211yescharacteristic
21C5×#405C2211yescharacteristic
22C4×C5×M16—21no—
23C4×C5×M16C2211yescharacteristic
24C2×C5×#97—21no—
25C2×C5×#97C2211yescharacteristic
26#65525C2212yes—
27#65434—24no—
28#65433—24no—
29#65432C412yescomplemented
30#65431C412yescomplemented
31#65430C414yescomplemented
32#65428C2212yes—
33#65421C2212yes—
34#65413C2212yes—
35C2×#3951C2211yescharacteristic
36C2×#3951C2211yescharacteristic
37#65395C2211yescharacteristic
38#65390C2212yes—
39#65295C2212yes—
40C2×#3937C2212yes—
41#65006—21no—
42#65006—21no—
43#65006—21no—
44#65006—21no—
45#64779C2212yes—
46#64753C2212yes—
47#64743C2212yes—
48#21559—51nomaximal, Sylow
49C2×C5×#41C2311yescharacteristic
50C2×#709C2312yes—
51C5×#131—21no—
52C5×#131—21no—
53C5×#131—22no—
54C5×#131C2×C412yes—
55C5×#131C2×C412yes—
56C5×#131D414yes—
57C2×C5×M16—21no—
58C2×C5×M16—21no—
59C2×C5×M16C2311yescharacteristic
60C2×C5×M16C2311yescharacteristic
61C22×C5×C8—21no—
62C22×C5×C8—21no—
63C22×C5×C8C2311yescharacteristic
64C22×C5×C8C2311yescharacteristic
65C4×C5×Q8—21no—
66C4×C5×D4—21no—
67C4×C5×D4—22no—
68C5×#117—21no—
69C5×#117C2311yescharacteristic
70C2×C42×C5—21no—
71C2×C42×C5C2311yescharacteristic
72#4097C2×C414yes—
73#4096—24no—
74#4095C2×C414yes—
75#4088—22no—
76#4078—22no—
77#4072C2312yes—
78C2×#681C2312yes—
79C2×#681C2312yes—
80C2×Dic20C2×C411yescharacteristic, complemented
81#4050—21no—
82#4050—21no—
83#4050—21no—
84#4050—21no—
85C2×D40C2×C411yescharacteristic, complemented
86C2×#677C2×C412yescomplemented
87#4047—22no—
88#4047—22no—
89#4047—22no—
90#4047—22no—
91C2×#676—22no—
92C2×C8×D5—22no—
93C5×#97—21no—
94C5×#97—21no—
95C5×#97—21no—
96C5×#97D411yescharacteristic
97C5×#97D411yescharacteristic
98C5×#97D411yescharacteristic
99C5×#97D411yescharacteristic
100C4×C5×C8—21no—
101C4×C5×C8—21no—
102C4×C5×C8—21no—
103#3969—24no—
104#3957—24no—
105#3956C2×C414yescomplemented
106#3955C2×C414yescomplemented
107#3954—24no—
108#3951—21no—
109#3951—21no—
110#3951C2×C412yescomplemented
111#3951C2×C412yescomplemented
112#3950—21no—
113#3949—21no—
114#3947—22no—
115C8×Dic5—22no—
116#3938—24no—
117#3937—22no—
118#3937—22no—
119#3067—51no—
120#2955—52no—
121C2×#444—51no—
122#2875D511yescharacteristic, complemented
123#1976—51no—
124#1953—51no—
125#1932—52no—
126#1906—52no—
127#1814—51no—
128#1773—51no—
129#1757—52no—
130C5×#41—21no—
131C5×#41—21no—
132C5×#41C22×C412yes—
133C5×#41C2×D412yes—
134C2×C5×Q8#4111yescharacteristic
135C2×C5×D4—21no—
136C2×C5×D4#4111yescharacteristic
137C22×C4×C5—21no—
138C22×C4×C5—22no—
139C22×C4×C5—21no—
140C22×C4×C5C2411yescharacteristic
141C2×C22⋊D5—22no—
142#709—22no—
143#709—22no—
144#709—22no—
145#709—22no—
146C2×D20C22×C412yes—
147C2×C4×D5—22no—
148C2×Dic10C22×C412yes—
149C5×M16—21no—
150C5×M16—21no—
151C5×M16—22no—
152C5×M16—22no—
153C5×M16—24no—
154C5×M16C22×C412yes—
155C5×M16C2×D412yes—
156C2×C5×C8—21no—
157C2×C5×C8—21no—
158C2×C5×C8—22no—
159C2×C5×C8—22no—
160C2×C5×C8—24no—
161C2×C5×C8C22×C411yescharacteristic
162C2×C5×C8C22×C411yescharacteristic
163C2×C5×C8C2×D412yes—
164C2×C5×C8C2×D411yescharacteristic
165C2×C5×C8C2×D411yescharacteristic
166C2×C5×C8C2×D411yescharacteristic
167C2×C5×C8C2×D411yescharacteristic
168C2×C5×C8#4111yescharacteristic
169C2×C5×C8#4111yescharacteristic
170C2×C5×C8#4112yes—
171C5×C4⋊C4—22no—
172C5×C4⋊C4—21no—
173C5×C22⋊C4—22no—
174C5×C22⋊C4#4112yes—
175C42×C5—22no—
176C42×C5—21no—
177C42×C5—21no—
178C42×C5C2×D411yescharacteristic
179C42×C5C2×D411yescharacteristic
180C22⋊Dic5—22no—
181#689—24no—
182#688—24no—
183#687—24no—
184#686—24no—
185C4⋊Dic5—22no—
186C4×Dic5—22no—
187#681—22no—
188#681—22no—
189#681—22no—
190#681C22×C414yes—
191C2×#168—22no—
192C2×#168—22no—
193Dic20—21no—
194Dic20—21no—
195D40—21no—
196D40—21no—
197#677—22no—
198#677—22no—
199#676—42no—
200#676—42no—
201C8×D5—42no—
202C8×D5—42no—
203#575—52no—
204C2×#135—51no—
205C2×#131—52no—
206C2×#131C2×D511yescharacteristic
207C2×#131C2×D511yescharacteristic
208C4×#41C2×D511yescharacteristic
209#473—52no—
210#472—52no—
211#471—54no—
212#444—104no—
213#444—101no—
214#444—101no—
215#444—101no—
216#444—101no—
217#443—104no—
218#430—52no—
219C2×#108—51no—
220C2×#108—51no—
221#427—51no—
222#421—52no—
223#421—52no—
224C2×#104—52no—
225#413—52no—
226#410—52no—
227#405—21no—
228#405—21no—
229#405—52no—
230#405C2×D511yescharacteristic
231#405C2×D511yescharacteristic
232C4×M16—21no—
233C4×M16C2×D511yescharacteristic
234C2×#97—21no—
235C2×#97C2×D511yescharacteristic
236#344—52no—
237#344—52no—
238#339—52no—
239C23×C5—21no—
240C23×C5C2×#4111yescharacteristic
241C22×D5—22no—
242C5×Q8—21no—
243C5×Q8C4×D412yes—
244C5×D4—22no—
245C5×D4—22no—
246C5×D4—21no—
247C5×D4C4×D412yes—
248C2×C4×C5—21no—
249C2×C4×C5—22no—
250C2×C4×C5—22no—
251C2×C4×C5—21no—
252C2×C4×C5—22no—
253C2×C4×C5—21no—
254C2×C4×C5—81no—
255C2×C4×C5—82no—
256C2×C4×C5C23×C411yescharacteristic
257C2×C4×C5C22×D411yescharacteristic
258C2×C4×C5C22×D411yescharacteristic
259C2×C4×C5C2×#4111yescharacteristic
260C2×C4×C5C2×#4111yescharacteristic
261C2×C4×C5C2×#4111yescharacteristic
262C22⋊D5—42no—
263C22⋊D5—42no—
264C2×Dic5—22no—
265C2×Dic5—42no—
266D20—22no—
267D20—22no—
268C4×D5—42no—
269C4×D5—42no—
270Dic10—22no—
271Dic10—22no—
272C5×C8—22no—
273C5×C8—22no—
274C5×C8C4×D411yescharacteristic
275C5×C8C4×D411yescharacteristic
276C5×C8C4×D411yescharacteristic
277C5×C8C4×D411yescharacteristic
278C5×C8#12112yes—
279C5×C8#12112yes—
280#168—24no—
281#168—42no—
282C2×#41—52no—
283C2×#41C22×D511yescharacteristic
284#137—54no—
285#137—102no—
286AGL(1,ℤ/8)—54no—
287AGL(1,ℤ/8)—102no—
288#135—104no—
289#135—101no—
290#135—101no—
291#135—101no—
292#135—101no—
293C2×Q16—51no—
294C2×SD16—52no—
295C2×D8—51no—
296#131—21no—
297#131—21no—
298#131—22no—
299#131—102no—
300#131—102no—
301#131—102no—
302#131—102no—
303#131C4×D512yes—
304#131C4×D512yes—
305#131D2014yes—
306C2×M16—21no—
307C2×M16—21no—
308C2×M16—52no—
309C2×M16—52no—
310C2×M16—102no—
311C2×M16C22×D511yescharacteristic
312C2×M16C22×D511yescharacteristic
313C22×C8—21no—
314C22×C8—21no—
315C22×C8—102no—
316C22×C8C22×D511yescharacteristic
317C22×C8C22×D511yescharacteristic
318C4×Q8—21no—
319C4×D4—21no—
320C4×D4—22no—
321#117—21no—
322#117—52no—
323#117C22×D511yescharacteristic
324C2×C42—21no—
325C2×C42C22×D511yescharacteristic
326#108—52no—
327#108—52no—
328#108—104no—
329#108—101no—
330#108—101no—
331#107—101no—
332#106—101no—
333#104—104no—
334#104—104no—
335#104—102no—
336#104—102no—
337#101—54no—
338#100—54no—
339#99—104no—
340#97—21no—
341#97—21no—
342#97—21no—
343#97—102no—
344#97C22⋊D511yescharacteristic
345#97C22⋊D511yescharacteristic
346#97C22⋊D511yescharacteristic
347#97C22⋊D511yescharacteristic
348C4×C8—21no—
349C4×C8—21no—
350C4×C8—21no—
351C4×C8—102no—
352C22×C5—22no—
353C22×C5—41no—
354C22×C5—42no—
355C22×C5C4×#4111yescharacteristic
356C22×C5#52511yescharacteristic
357C22×C5#53211yescharacteristic
358C2×D5—42no—
359C2×D5—82no—
360C4×C5—22no—
361C4×C5—42no—
362C4×C5—41no—
363C4×C5—41no—
364C4×C5C2×C4×D411yescharacteristic
365C4×C5C2×C4×D411yescharacteristic
366C4×C5C2×#12111yescharacteristic
367C4×C5#52911yescharacteristic
368Dic5—42no—
369Dic5—82no—
370#41—21no—
371#41—21no—
372#41—102no—
373#41—102no—
374#41—102no—
375#41—102no—
376#41C2×C4×D512yes—
377#41C2×D2012yes—
378C2×Q8—52no—
379C2×Q8#70911yescharacteristic
380C2×D4—21no—
381C2×D4—52no—
382C2×D4—102no—
383C2×D4#70911yescharacteristic
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—21no—
387C22×C4—102no—
388C22×C4C23×D511yescharacteristic
389Q16—104no—
390Q16—101no—
391Q16—101no—
392SD16—104no—
393SD16—104no—
394SD16—102no—
395SD16—102no—
396D8—104no—
397D8—101no—
398D8—101no—
399M16—21no—
400M16—21no—
401M16—22no—
402M16—22no—
403M16—24no—
404M16—54no—
405M16—102no—
406M16—102no—
407M16—102no—
408M16—202no—
409M16—202no—
410M16C2×C4×D512yes—
411M16C2×D2012yes—
412C2×C8—21no—
413C2×C8—21no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—24no—
417C2×C8—102no—
418C2×C8—102no—
419C2×C8—202no—
420C2×C8—202no—
421C2×C8C2×C4×D511yescharacteristic
422C2×C8C2×C4×D511yescharacteristic
423C2×C8C2×D2012yes—
424C2×C8#70911yescharacteristic
425C2×C8#70911yescharacteristic
426C2×C8#70912yes—
427C2×C8C2×C22⋊D511yescharacteristic
428C2×C8C2×C22⋊D511yescharacteristic
429C2×C8C2×C22⋊D511yescharacteristic
430C2×C8C2×C22⋊D511yescharacteristic
431C4⋊C4—22no—
432C4⋊C4—21no—
433C4⋊C4—102no—
434C22⋊C4—22no—
435C22⋊C4—102no—
436C22⋊C4#70912yes—
437C42—22no—
438C42—21no—
439C42—21no—
440C42—102no—
441C42C2×C22⋊D511yescharacteristic
442C42C2×C22⋊D511yescharacteristic
443C2×C5—21no—
444C2×C5—21no—
445C2×C5—21no—
446C2×C5—42no—
447C2×C5C4×#12111yescharacteristic
448D5—82no—
449C23—21no—
450C23—102no—
451C23C2×#70911yescharacteristic
452Q8—21no—
453Q8—102no—
454Q8—102no—
455Q8C4×D2012yes—
456D4—22no—
457D4—22no—
458D4—21no—
459D4—102no—
460D4—102no—
461D4—202no—
462D4—202no—
463D4C4×D2012yes—
464C2×C4—21no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—21no—
468C2×C4—22no—
469C2×C4—21no—
470C2×C4—81no—
471C2×C4—82no—
472C2×C4—102no—
473C2×C4—202no—
474C2×C4—202no—
475C2×C4—202no—
476C2×C4C22×C4×D511yescharacteristic
477C2×C4C22×D2011yescharacteristic
478C2×C4C2×#70911yescharacteristic
479C2×C4C2×#70911yescharacteristic
480C2×C4C2×#70911yescharacteristic
481C2×C4C22×C22⋊D511yescharacteristic
482C8—22no—
483C8—22no—
484C8—104no—
485C8—202no—
486C8C4×C22⋊D511yescharacteristic
487C8C4×C22⋊D511yescharacteristic
488C8C4×C22⋊D511yescharacteristic
489C8C4×C22⋊D511yescharacteristic
490C8#407612yes—
491C8#407612yes—
492C5#2155911yescharacteristic, complemented, Sylow
493C22—22no—
494C22—41no—
495C22—42no—
496C22—202no—
497C22—402no—
498C22C4×#70911yescharacteristic
499C22#6581111yescharacteristic
500C22#6612511yescharacteristic
501C4—22no—
502C4—41no—
503C4—42no—
504C4—41no—
505C4—202no—
506C4—402no—
507C4C2×C4×D2011yescharacteristic
508C4#6581311yescharacteristic
509C4C2×C4×C22⋊D511yescharacteristic
510C4C2×#407611yescharacteristic
511C2—21no—
512C2—21no—
513C2—21no—
514C2—42no—
515C2—402no—
516C2C4×#407611yescharacteristic
subgroupG/H
centerC4C2×#4076
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC5×#2875C2
socleC2×C5C4×#121
prime radical#10615536C2
cosocleC22×C4×C5C24
hypercenter#2875D5
hypocenterC5#21559
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C4 ◁ C22×C8 ◁ #2875

Complements

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

NH ≅ G/Ntype
1. C5×#2875C2semidirect
3. #10615523C2semidirect
4. C2×#65006C2semidirect
5. #10602806C2semidirect
6. #10602438C2semidirect
7. #10600694C2semidirect
8. #10600691C2semidirect
9. #10600690C2semidirect
10. #10600667C2semidirect
11. #10600622C2semidirect
29. #65432C4semidirect
30. #65431C4semidirect
31. #65430C4semidirect
80. C2×Dic20C2×C4semidirect
85. C2×D40C2×C4semidirect
86. C2×#677C2×C4semidirect
105. #3956C2×C4semidirect
106. #3955C2×C4semidirect
110. #3951C2×C4semidirect
111. #3951C2×C4semidirect
122. #2875D5semidirect
492. C5#21559semidirect

Maximal subgroups

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

subgroupindex
C5×#28752
#106155362
#106155232
C2×#650062
#106028062
#106024382
#106006942
#106006912
#106006902
#106006672
#106006222
#215595