Finite Group Databasebeta

← #12963641

Subgroups — #12963641

Order 1280.

2604 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10619198C211yesmaximal, characteristic
2#10619197C211yesmaximal, characteristic, complemented
3#10619067C211yesmaximal, characteristic, complemented
4C5×#2901C211yesmaximal, characteristic, complemented
5#10615541C211yesmaximal, characteristic, complemented
6#10614267C211yesmaximal, characteristic, complemented
7#10614242C211yesmaximal, characteristic, complemented
8#10610721C216yesmaximal, complemented
9#10610720C216yesmaximal, complemented
10#10610711C216yesmaximal, complemented
11C2×#65112C216yesmaximal, complemented
12C5×#585C2211yescharacteristic
13#66283C2211yescharacteristic
14C5×#568C2211yescharacteristic
15C4×C5×#41C2211yescharacteristic
16#66172C414yescomplemented
17#66171C414yescomplemented
18#66170C414yescomplemented
19#66169C414yescomplemented
20#66156C2216yes—
21C2×#4096C2216yes—
22#66153—24no—
23#66153C2211yescharacteristic
24#66153C2216yescomplemented
25C2×#4094—24no—
26C2×#4094C2211yescharacteristic
27C2×#4094C2211yescharacteristic
28C2×#4094C2216yescomplemented
29#66084—24no—
30D5×#131—24no—
31#66079C2216yescomplemented
32C2×#4053C2216yescomplemented
33C5×#421C2216yescomplemented
34C2×C5×#104C2216yescomplemented
35#65413C2216yescomplemented
36C2×#3954C2216yescomplemented
37#65294C2216yescomplemented
38#65286C2213yescomplemented
39#65286C2213yescomplemented
40C2×#3937C2216yescomplemented
41C4×#681C2213yescomplemented
42C2×#3935C2213yescomplemented
43#65113C22124yescomplemented
44#65112C2214yescomplemented
45#65112C22112yescomplemented
46#65112C22112yescomplemented
47#65112C22112yescomplemented
48#60184—51nomaximal, Sylow
49C5×2−1+4—21no—
50C5×2−1+4C2×C414yes—
51C5×H(5,𝔽2)—23no—
52C5×H(5,𝔽2)C2×C414yes—
53C2×C5×#41—24no—
54C2×C5×#41C2316yes—
55C2×C5×#41C2311yescharacteristic
56#4150C2×C414yes—
57#4149C2×C414yes—
58D5×#41—24no—
59#4147—21no—
60#4144—23no—
61C2×#709C2316yes—
62C5×#131—24no—
63C2×C5×M16C2316yes—
64C4×C5×Q8—21no—
65C4×C5×D4—23no—
66C5×#117C2313yes—
67C2×C42×C5C2313yes—
68#4097C2×C4112yescomplemented
69#4096C2×C4112yescomplemented
70#4096C2×C4112yescomplemented
71#4096C2314yes—
72#4096C23112yes—
73#4095C2×C4112yescomplemented
74#4094—24no—
75#4094—212no—
76#4094—26no—
77#4094—26no—
78#4094C2×C414yes—
79#4094C2×C414yes—
80#4094D414yes—
81#4094D414yes—
82#4094C2314yes—
83#4094C23112yes—
84C2×#689C2×C416yescomplemented
85#4088C2×C4112yescomplemented
86C2×#688C2×C416yescomplemented
87C2×#687C2×C416yescomplemented
88#4078C2×C4112yescomplemented
89C2×#686C2×C416yescomplemented
90C2×#681—212no—
91C2×#681—26no—
92C2×#681C2316yes—
93C2×#681C2313yes—
94C2×#681C2313yes—
95C22×#168—26no—
96C22×#168C2313yes—
97C22×#168C2313yes—
98#4053—26no—
99#4053—26no—
100#4053C2314yes—
101#4053C23112yes—
102D5×M16—212no—
103#4047—212no—
104C2×#676—26no—
105C2×C8×D5—26no—
106C5×#104C2314yes—
107C5×#104C23112yes—
108#3954C2314yes—
109#3954C23112yes—
110#3937C23112yes—
111#3937C2314yes—
112#3935C2313yes—
113#3935C2313yes—
114#3935C2311yescharacteristic
115#3935C2313yes—
116C4×#168C2316yes—
117#3584—51no—
118#3573—51no—
119#3573—51no—
120#2957—56no—
121#2956—56no—
122#2955—56no—
123C2×#444—56no—
124#2916—51no—
125#2901—51no—
126#2901D511yescharacteristic, complemented
127#2875—51no—
128C5×#41—24no—
129C5×#41—212no—
130C5×#41—26no—
131C5×#41—26no—
132C5×#41C22×C4112yes—
133C5×#41C22×C414yes—
134C5×#41C2×D414yes—
135C5×#41C2414yes—
136C2×C5×Q8—24no—
137C2×C5×Q8C22×C416yes—
138C2×C5×Q8#4111yescharacteristic
139C2×C5×D4—26no—
140C2×C5×D4—212no—
141C2×C5×D4C22×C416yes—
142C2×C5×D4#4113yes—
143C22×C4×C5—26no—
144C22×C4×C5—23no—
145C22×C4×C5C2413yes—
146C2×C22⋊D5—26no—
147#713—24no—
148Q8×D5—24no—
149#711—212no—
150D4×D5—212no—
151#709—26no—
152#709—26no—
153#709C22×C4112yes—
154#709C2414yes—
155C2×D20C22×C416yes—
156C2×C4×D5—26no—
157C2×Dic10C22×C416yes—
158C5×M16—26no—
159C5×M16C2414yes—
160C2×C5×C8—26no—
161C5×C4⋊C4—23no—
162C5×C22⋊C4—23no—
163C42×C5—23no—
164C42×C5C2411yescharacteristic
165C42×C5C2413yes—
166#689C22×C414yes—
167#689C22×C4112yes—
168#688C22×C414yes—
169#688C22×C4112yes—
170#687C22×C414yes—
171#687C22×C4112yes—
172#686C22×C414yes—
173#686C22×C4112yes—
174#681—26no—
175#681—24no—
176#681—212no—
177#681C22×C4112yes—
178#681C2×D4112yes—
179#681C2414yes—
180C2×#168—26no—
181C2×#168—24no—
182C2×#168—212no—
183C2×#168C22×C416yes—
184C2×#168C22×C416yes—
185C2×#168C2×D416yes—
186C2×#168C2×D416yes—
187C2×#168C2413yes—
188C2×#168C2411yescharacteristic
189C2×#168C2413yes—
190C2×#168C2411yescharacteristic
191#676—24no—
192#676—212no—
193C8×D5—24no—
194C8×D5—212no—
195#585—51no—
196#585C2×D511yescharacteristic
197#578—54no—
198#577—54no—
199#577—54no—
200#576—54no—
201#575—56no—
202C2×#135—56no—
203#568—51no—
204#568—56no—
205#568—56no—
206#568—104no—
207#568—104no—
208#568C2×D511yescharacteristic
209C2×#131—51no—
210C2×#131—51no—
211C2×#131—56no—
212C2×#131—56no—
213C2×#131—104no—
214C2×#131—104no—
215C4×#41C2×D511yescharacteristic
216#444—54no—
217#444—512no—
218#444—512no—
219#444—512no—
220#443—524no—
221#430—56no—
222C2×#108—56no—
223#421—56no—
224#421C2×D516yescomplemented
225C2×#104—56no—
226C2×#104C2×D516yescomplemented
227#405—53no—
228#405—53no—
229C4×M16—53no—
230C2×#97—53no—
231C23×C5—26no—
232C23×C5C2×#4113yes—
233C22×D5—26no—
234C5×Q8—26no—
235C5×Q8C4×D414yes—
236C5×Q8C23×C414yes—
237C5×D4—212no—
238C5×D4—24no—
239C5×D4—26no—
240C5×D4C4×D4112yes—
241C5×D4C23×C414yes—
242C2×C4×C5—26no—
243C2×C4×C5—212no—
244C2×C4×C5—24no—
245C2×C4×C5—21no—
246C2×C4×C5—23no—
247C2×C4×C5—83no—
248C2×C4×C5C23×C416yes—
249C2×C4×C5C22×D416yes—
250C2×C4×C5C2×#4113yes—
251C2×C4×C5C2511yescharacteristic
252C22⋊D5—212no—
253C22⋊D5—24no—
254C2×Dic5—26no—
255D20—26no—
256D20C23×C414yes—
257C4×D5—212no—
258C4×D5—24no—
259Dic10—26no—
260Dic10C23×C414yes—
261C5×C8—24no—
262#168—24no—
263#168C23×C411yescharacteristic
264#168C23×C411yescharacteristic
265#168C23×C416yes—
266#168C22×D411yescharacteristic
267#168C22×D416yes—
268#168C22×D411yescharacteristic
2692−1+4—21no—
2702−1+4—54no—
2712−1+4—101no—
2722−1+4C4×D514yes—
273H(5,𝔽2)—23no—
274H(5,𝔽2)—54no—
275H(5,𝔽2)—103no—
276H(5,𝔽2)C4×D514yes—
277C2×#41—24no—
278C2×#41—56no—
279C2×#41—104no—
280C2×#41C22×D516yes—
281C2×#41C22×D511yescharacteristic
282#137—512no—
283#137—512no—
284AGL(1,ℤ/8)—512no—
285AGL(1,ℤ/8)—512no—
286#135—512no—
287#135—512no—
288#135—54no—
289#135—512no—
290C2×Q16—56no—
291C2×SD16—56no—
292C2×SD16—56no—
293C2×D8—56no—
294#131—24no—
295#131—54no—
296#131—54no—
297#131—54no—
298#131—54no—
299#131—54no—
300#131—512no—
301#131—54no—
302#131—512no—
303#131—1012no—
304#131—106no—
305#131—106no—
306#131—106no—
307#131—106no—
308#131—1012no—
309#131—104no—
310C2×M16—53no—
311C2×M16—53no—
312C2×M16—56no—
313C2×M16—106no—
314C2×M16—106no—
315C2×M16—1012no—
316C2×M16—1012no—
317C2×M16C22×D516yes—
318C22×C8—53no—
319C22×C8—53no—
320C22×C8—106no—
321C22×C8—106no—
322C4×Q8—21no—
323C4×D4—23no—
324#117C22×D513yes—
325C2×C42C22×D513yes—
326#108—54no—
327#108—512no—
328#104—54no—
329#104—512no—
330#104C22×D514yes—
331#104C22×D5112yes—
332#97—51no—
333#97—53no—
334#97—53no—
335#97—53no—
336C4×C8—56no—
337C22×C5—24no—
338C22×C5—43no—
339C22×C5—86no—
340C22×C5C2×C4×D416yes—
341C22×C5C22×#4111yescharacteristic
342C2×D5—24no—
343C2×D5—86no—
344C4×C5—24no—
345C4×C5—43no—
346C4×C5—41no—
347C4×C5C2×C4×D416yes—
348C4×C5C24×C411yescharacteristic
349C4×C5C23×D411yescharacteristic
350Dic5—24no—
351#41—24no—
352#41—212no—
353#41—26no—
354#41—26no—
355#41—512no—
356#41—54no—
357#41—104no—
358#41—1012no—
359#41—106no—
360#41—106no—
361#41C2×C4×D5112yes—
362#41C2×C4×D514yes—
363#41D4×D514yes—
364#41C23×D514yes—
365C2×Q8—24no—
366C2×Q8—56no—
367C2×Q8—104no—
368C2×Q8C2×C4×D516yes—
369C2×Q8#71111yescharacteristic
370C2×D4—26no—
371C2×D4—212no—
372C2×D4—56no—
373C2×D4—106no—
374C2×D4—1012no—
375C2×D4C2×C4×D516yes—
376C2×D4#71113yes—
377C22×C4—26no—
378C22×C4—23no—
379C22×C4—106no—
380C22×C4C23×D513yes—
381Q16—54no—
382Q16—512no—
383SD16—54no—
384SD16—512no—
385SD16—54no—
386SD16—512no—
387D8—54no—
388D8—512no—
389M16—26no—
390M16—512no—
391M16—512no—
392M16—54no—
393M16—104no—
394M16—1012no—
395M16—104no—
396M16—1012no—
397M16—106no—
398M16C23×D514yes—
399C2×C8—26no—
400C2×C8—56no—
401C2×C8—56no—
402C2×C8—56no—
403C2×C8—56no—
404C2×C8—53no—
405C2×C8—53no—
406C2×C8—51no—
407C2×C8—51no—
408C2×C8—104no—
409C2×C8—1012no—
410C2×C8—104no—
411C2×C8—1012no—
412C2×C8—106no—
413C4⋊C4—23no—
414C22⋊C4—23no—
415C42—23no—
416C42C23×D513yes—
417C42C23×D511yescharacteristic
418C2×C5—21no—
419C2×C5—26no—
420C2×C5—44no—
421C2×C5C22×C4×D411yescharacteristic
422D5—44no—
423C23—26no—
424C23—106no—
425C23C2×#71113yes—
426Q8—26no—
427Q8—54no—
428Q8—106no—
429Q8#402714yes—
430Q8C22×C4×D514yes—
431D4—212no—
432D4—24no—
433D4—26no—
434D4—54no—
435D4—1012no—
436D4—104no—
437D4—106no—
438D4#4027112yes—
439D4C22×C4×D514yes—
440C2×C4—26no—
441C2×C4—212no—
442C2×C4—24no—
443C2×C4—21no—
444C2×C4—23no—
445C2×C4—83no—
446C2×C4—106no—
447C2×C4—1012no—
448C2×C4—104no—
449C2×C4C22×C4×D516yes—
450C2×C4C2×D4×D516yes—
451C2×C4C2×#71113yes—
452C2×C4C24×D511yescharacteristic
453C8—24no—
454C8—51no—
455C8—51no—
456C8—56no—
457C8—51no—
458C8—51no—
459C8—56no—
460C8—104no—
461C5#6018411yescharacteristic, complemented, Sylow
462C22—24no—
463C22—43no—
464C22—86no—
465C22—104no—
466C22—406no—
467C22C2×#402716yes—
468C22C22×#71111yescharacteristic
469C4—24no—
470C4—41no—
471C4—43no—
472C4—104no—
473C4C2×#402716yes—
474C4C23×C4×D511yescharacteristic
475C4C22×D4×D511yescharacteristic
476C2—21no—
477C2—26no—
478C2—44no—
479C2—204no—
480C2C22×#402711yescharacteristic
subgroupG/H
centerC4C22×D4×D5
commutatorC4×C5C24×C4
FrattiniC2×C4C24×D5
FittingC5×#2901C2
socleC2×C5C22×C4×D4
prime radical#10619198C2
cosocleC2×C4×C5C25
hypercenter#2901D5
hypocenterC5#60184
LCSG ▷ C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C4 ◁ C2×#41 ◁ #2901

Complements

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

NH ≅ G/Ntype
2. #10619197C2semidirect
3. #10619067C2semidirect
4. C5×#2901C2semidirect
5. #10615541C2semidirect
6. #10614267C2semidirect
7. #10614242C2semidirect
8. #10610721C2semidirect
9. #10610720C2semidirect
10. #10610711C2semidirect
11. C2×#65112C2semidirect
16. #66172C4semidirect
17. #66171C4semidirect
18. #66170C4semidirect
19. #66169C4semidirect
24. #66153C22semidirect
28. C2×#4094C22semidirect
31. #66079C22semidirect
32. C2×#4053C22semidirect
33. C5×#421C22semidirect
34. C2×C5×#104C22semidirect
35. #65413C22semidirect
36. C2×#3954C22semidirect
37. #65294C22semidirect
38. #65286C22semidirect
39. #65286C22semidirect
40. C2×#3937C22semidirect
41. C4×#681C22semidirect
42. C2×#3935C22semidirect
43. #65113C22semidirect
44. #65112C22semidirect
45. #65112C22semidirect
46. #65112C22semidirect
47. #65112C22semidirect
68. #4097C2×C4semidirect
69. #4096C2×C4semidirect
70. #4096C2×C4semidirect
73. #4095C2×C4semidirect
84. C2×#689C2×C4semidirect
85. #4088C2×C4semidirect
86. C2×#688C2×C4semidirect
87. C2×#687C2×C4semidirect
88. #4078C2×C4semidirect
89. C2×#686C2×C4semidirect
126. #2901D5semidirect
224. #421C2×D5semidirect
226. C2×#104C2×D5semidirect
461. C5#60184semidirect

Maximal subgroups

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

subgroupindex
#106191982
#106191972
#106190672
C5×#29012
#106155412
#106142672
#106142422
#106107212
#106107202
#106107112
C2×#651122
#601845