Finite Group Databasebeta

← #12098167

Subgroups — #12098167

Order 1280.

870 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#65391C211yesmaximal, characteristic
2C2×#65009C211yesmaximal, characteristic
3#10605171C211yesmaximal, characteristic
4C5×#1752C211yesmaximal, characteristic
5C8×C22⋊Dic5C211yesmaximal, characteristic
6#10600742C211yesmaximal, characteristic
7#10600626C211yesmaximal, characteristic, complemented
8#10600625C211yesmaximal, characteristic, complemented
9#10600604C211yesmaximal, characteristic, complemented
10#10600602C211yesmaximal, characteristic, complemented
11#10600241C211yesmaximal, characteristic
12#10599447C211yesmaximal, characteristic, complemented
13#10599444C211yesmaximal, characteristic, complemented
14#10599397C211yesmaximal, characteristic, complemented
15#10599395C211yesmaximal, characteristic, complemented
16C23×C5×C8C2211yescharacteristic
17C2×#4072C2211yescharacteristic
18C22×#681C2211yescharacteristic
19C2×C5×C22⋊C8C2211yescharacteristic
20C2×C5×#97C2211yescharacteristic
21C2×C5×#97C2211yescharacteristic
22C4×C5×C22⋊C4C2211yescharacteristic
23C4×C22⋊Dic5C2211yescharacteristic
24C2×C22⋊#168C2211yescharacteristic
25#65391C418yescomplemented
26C2×#3947C2211yescharacteristic
27C2×#3947C2211yescharacteristic
28C2×#3946C2211yescharacteristic
29C2×#3946C2211yescharacteristic
30C2×#3946C2211yescharacteristic
31C2×#3946C2211yescharacteristic
32C2×#3946C2211yescharacteristic
33C2×#3946C2211yescharacteristic
34C2×C8×Dic5C2211yescharacteristic
35C2×C8×Dic5C2211yescharacteristic
36C2×C8×Dic5C2211yescharacteristic
37C2×C8×Dic5C2211yescharacteristic
38#65234C412yes—
39#65233C412yes—
40#65221C2211yescharacteristic
41C4×#683C2211yescharacteristic
42C4×#683C2211yescharacteristic
43C42×Dic5C2211yescharacteristic
44C2×#3935C2211yescharacteristic
45C2×C4×#168C2211yescharacteristic
46#65009C418yescomplemented
47#64943C412yes—
48#64942C412yes—
49C5×#336C2211yescharacteristic
50C5×#336C2211yescharacteristic
51#64770C2211yescharacteristic
52#64770C2211yescharacteristic
53#64770C2211yescharacteristic
54#64770C2211yescharacteristic
55#64744C2211yescharacteristic
56#64744C2211yescharacteristic
57#20553—51nomaximal, Sylow
58C23×C4×C5C2311yescharacteristic
59C22×C5×C8—41no—
60C22×C5×C8—41no—
61C22×C5×C8C2×C414yes—
62C22×C5×C8C2311yescharacteristic
63C22×C5×C8C2311yescharacteristic
64C22×C5×C8C2311yescharacteristic
65C22×C5×C8C2311yescharacteristic
66C2×C5×C22⋊C4C2×C412yes—
67C2×C42×C5C2311yescharacteristic
68C2×C42×C5C2311yescharacteristic
69C2×C22⋊Dic5C2×C412yes—
70C2×C22⋊Dic5C2×C411yescharacteristic
71C2×C22⋊Dic5C2×C411yescharacteristic
72#4072C2×C414yes—
73#4072C2×C414yes—
74C2×C4⋊Dic5C2×C411yescharacteristic
75C2×C4⋊Dic5C2×C411yescharacteristic
76C2×#683C2×C412yes—
77C2×#683C2×C412yes—
78C2×C4×Dic5C2×C412yes—
79C2×C4×Dic5C2×C412yes—
80C2×C4×Dic5C2311yescharacteristic
81C2×C4×Dic5C2311yescharacteristic
82C2×C4×Dic5C2311yescharacteristic
83C2×C4×Dic5C2311yescharacteristic
84C2×#681—41no—
85C2×#681—41no—
86C2×#681C2×C414yes—
87C22×#168C2311yescharacteristic
88C22×#168C2311yescharacteristic
89C22×#168C2311yescharacteristic
90C22×#168C2311yescharacteristic
91C5×C22⋊C8C2×C414yes—
92C5×#97—41no—
93C5×#97—41no—
94C5×#95C2×C412yes—
95#3963C2×C412yes—
96#3963C2×C412yes—
97#3963C2×C412yes—
98#3963C2×C412yes—
99#3963C2×C412yes—
100C22⋊#168C2×C414yes—
101#3947—41no—
102#3947C2×C414yes—
103#3946C2×C414yes—
104#3946C2×C414yes—
105#3946C2×C414yes—
106#3946C2×C414yes—
107#3946C2×C414yes—
108#3946C2×C414yes—
109C8×Dic5—41no—
110C8×Dic5—41no—
111C8×Dic5—41no—
112C8×Dic5C2×C414yes—
113#3935—41no—
114C4×#168—41no—
115C2×#433—51no—
116C2×#424—51no—
117#2298—51no—
118#1771—51no—
119#1771—51no—
120#1770—51no—
121C8×C4⋊C4—51no—
122#1768—51no—
123#1768—51no—
124#1767—51no—
125C4×C4⋊C8—51no—
126#1753—51no—
127#1753—51no—
128#1752D511yescharacteristic
129C8×C22⋊C4—51no—
130C24×C5C22×C411yescharacteristic
131C22×C4×C5—21no—
132C22×C4×C5—21no—
133C22×C4×C5—21no—
134C22×C4×C5—21no—
135C22×C4×C5C22×C412yes—
136C22×C4×C5C22×C412yes—
137C22×C4×C5C22×C411yescharacteristic
138C22×C4×C5C22×C412yes—
139C22×C4×C5C22×C412yes—
140C22×C4×C5C2411yescharacteristic
141C22×Dic5C22×C412yes—
142C22×Dic5C22×C411yescharacteristic
143C22×Dic5C22×C411yescharacteristic
144C22×Dic5C22×C411yescharacteristic
145C22×Dic5C22×C411yescharacteristic
146C22×Dic5C22×C412yes—
147C2×C5×C8—21no—
148C2×C5×C8—21no—
149C2×C5×C8—21no—
150C2×C5×C8—21no—
151C2×C5×C8—21no—
152C2×C5×C8—21no—
153C2×C5×C8—21no—
154C2×C5×C8—21no—
155C2×C5×C8—44no—
156C2×C5×C8C22×C412yes—
157C2×C5×C8C22×C412yes—
158C2×C5×C8C22×C412yes—
159C2×C5×C8C22×C412yes—
160C5×C22⋊C4—42no—
161C42×C5—21no—
162C42×C5—21no—
163C42×C5—21no—
164C42×C5—21no—
165C22⋊Dic5—24no—
166C22⋊Dic5C4214yes—
167C22⋊Dic5C4214yes—
168C4⋊Dic5C4214yes—
169C4⋊Dic5C4214yes—
170#683—24no—
171#683—42no—
172C4×Dic5—21no—
173C4×Dic5—21no—
174C4×Dic5—21no—
175C4×Dic5—21no—
176C4×Dic5—24no—
177C4×Dic5—42no—
178C4×Dic5C22×C412yes—
179C4×Dic5C22×C412yes—
180C4×Dic5C22×C412yes—
181C4×Dic5C22×C412yes—
182#681—44no—
183C2×#168—21no—
184C2×#168—21no—
185C2×#168—21no—
186C2×#168—21no—
187C2×#168—21no—
188C2×#168—21no—
189C2×#168—21no—
190C2×#168—21no—
191C2×#168C22×C412yes—
192C2×#168C22×C412yes—
193C2×#168C22×C412yes—
194C2×#168C22×C412yes—
195C22×M16—51no—
196C23×C8C2×D511yescharacteristic
197C2×#117—51no—
198#433—58no—
199#424—58no—
200C2×C4⋊C8—51no—
201C2×C4⋊C8—51no—
202C2×C4⋊C8—51no—
203C2×C4⋊C8—51no—
204C2×C4⋊C8—51no—
205C2×C4⋊C8—51no—
206C2×C22⋊C8—51no—
207C2×C22⋊C8C2×D511yescharacteristic
208C2×#97—51no—
209C2×#97—51no—
210C2×#97—51no—
211C2×#97C2×D511yescharacteristic
212C2×#97C2×D511yescharacteristic
213C2×C4×C8—51no—
214C2×C4×C8—51no—
215C2×C4×C8—51no—
216C2×C4×C8—51no—
217C2×C4×C8—51no—
218#388—52no—
219#388—52no—
220#387—52no—
221#387—52no—
222C4×C4⋊C4—51no—
223C4×C4⋊C4—51no—
224C4×C22⋊C4—51no—
225C4×C22⋊C4C2×D511yescharacteristic
226#376—51no—
227C43—51no—
228#336—51no—
229#336—51no—
230#336—51no—
231#336—51no—
232#336—51no—
233#336—51no—
234#336C2×D511yescharacteristic
235#336C2×D511yescharacteristic
236C23×C5—21no—
237C23×C5—21no—
238C23×C5—22no—
239C23×C5—22no—
240C23×C5C2×C4212yes—
241C23×C5C23×C411yescharacteristic
242C2×C4×C5—21no—
243C2×C4×C5—21no—
244C2×C4×C5—22no—
245C2×C4×C5—22no—
246C2×C4×C5—22no—
247C2×C4×C5—22no—
248C2×C4×C5—22no—
249C2×C4×C5—22no—
250C2×C4×C5—22no—
251C2×C4×C5—22no—
252C2×C4×C5—22no—
253C2×C4×C5—22no—
254C2×C4×C5C2×C4212yes—
255C2×C4×C5C23×C411yescharacteristic
256C2×C4×C5C23×C411yescharacteristic
257C2×C4×C5H(5,𝔽2)11yescharacteristic
258C2×C4×C5H(5,𝔽2)11yescharacteristic
259C2×C4×C52−1+411yescharacteristic
260C2×C4×C52−1+411yescharacteristic
261C2×Dic5—22no—
262C2×Dic5—22no—
263C2×Dic5—22no—
264C2×Dic5—22no—
265C2×Dic5—22no—
266C2×Dic5—22no—
267C2×Dic5—22no—
268C2×Dic5—22no—
269C2×Dic5—22no—
270C2×Dic5C2×C4212yes—
271C2×Dic5C2×C4212yes—
272C2×Dic5C2×C4212yes—
273C2×Dic5C2×C4212yes—
274C2×Dic5#13114yes—
275C5×C8—22no—
276C5×C8—22no—
277C5×C8—22no—
278C5×C8—22no—
279#168—22no—
280#168—22no—
281#168—22no—
282#168—22no—
283C23×C4C22×D511yescharacteristic
284C2×M16—54no—
285C2×M16—201no—
286C2×M16—201no—
287C22×C8—41no—
288C22×C8—41no—
289C22×C8—51no—
290C22×C8—51no—
291C22×C8—51no—
292C22×C8—51no—
293C22×C8C4×D514yes—
294C22×C8C22×D511yescharacteristic
295C22×C8C22×D511yescharacteristic
296C22×C8C22×D511yescharacteristic
297C22×C8C22×D511yescharacteristic
298#117—54no—
299#117—54no—
300C2×C4⋊C4—51no—
301C2×C4⋊C4—51no—
302C2×C4⋊C4—52no—
303C2×C4⋊C4—52no—
304C2×C22⋊C4—51no—
305C2×C22⋊C4—51no—
306C2×C22⋊C4—52no—
307C2×C22⋊C4C4×D512yes—
308C2×C42—51no—
309C2×C42—51no—
310C2×C42—52no—
311C2×C42—51no—
312C2×C42—51no—
313C2×C42—52no—
314C2×C42C22×D511yescharacteristic
315C2×C42C22×D511yescharacteristic
316C4⋊C8—54no—
317C4⋊C8—54no—
318C4⋊C8—54no—
319C4⋊C8—54no—
320C4⋊C8—54no—
321C4⋊C8—54no—
322C22⋊C8—54no—
323C22⋊C8C4×D514yes—
324#97—41no—
325#97—41no—
326#97—54no—
327#97—201no—
328#97—201no—
329C4×C8—54no—
330C4×C8—201no—
331C4×C8—201no—
332C4×C8—201no—
333C4×C8—201no—
334#95—52no—
335#95—52no—
336#95—52no—
337#95—52no—
338#95—52no—
339#95C4×D512yes—
340C22×C5—22no—
341C22×C5—22no—
342C22×C5—22no—
343C22×C5—42no—
344C22×C5—42no—
345C22×C5C22×C4211yescharacteristic
346C22×C5#52011yescharacteristic
347C22×C5#52011yescharacteristic
348C22×C5C2×#13112yes—
349C22×C5#56812yes—
350C4×C5—22no—
351C4×C5—42no—
352C4×C5—42no—
353C4×C5#51811yescharacteristic
354C4×C5#51911yescharacteristic
355C4×C5#52011yescharacteristic
356C4×C5#52011yescharacteristic
357Dic5—22no—
358Dic5—22no—
359Dic5—22no—
360Dic5—22no—
361Dic5—24no—
362Dic5—42no—
363C24C2×C4×D511yescharacteristic
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—51no—
369C22×C4—51no—
370C22×C4—52no—
371C22×C4—51no—
372C22×C4—51no—
373C22×C4—52no—
374C22×C4C2×C4×D512yes—
375C22×C4C2×C4×D512yes—
376C22×C4C2×C4×D512yes—
377C22×C4C2×C4×D512yes—
378C22×C4C2×C4×D511yescharacteristic
379C22×C4C23×D511yescharacteristic
380M16—204no—
381C2×C8—21no—
382C2×C8—21no—
383C2×C8—21no—
384C2×C8—21no—
385C2×C8—21no—
386C2×C8—21no—
387C2×C8—21no—
388C2×C8—21no—
389C2×C8—44no—
390C2×C8—52no—
391C2×C8—52no—
392C2×C8—52no—
393C2×C8—52no—
394C2×C8—101no—
395C2×C8—101no—
396C2×C8—101no—
397C2×C8—101no—
398C2×C8—101no—
399C2×C8—101no—
400C2×C8—101no—
401C2×C8—101no—
402C2×C8C2×C4×D512yes—
403C2×C8C2×C4×D512yes—
404C2×C8C2×C4×D512yes—
405C2×C8C2×C4×D512yes—
406C4⋊C4—54no—
407C4⋊C4—54no—
408C4⋊C4—104no—
409C4⋊C4—202no—
410C22⋊C4—42no—
411C22⋊C4—54no—
412C22⋊C4—54no—
413C22⋊C4—104no—
414C42—21no—
415C42—21no—
416C42—21no—
417C42—21no—
418C42—52no—
419C42—52no—
420C42—52no—
421C42—52no—
422C42—104no—
423C42—101no—
424C42—101no—
425C42—101no—
426C42—101no—
427C42—202no—
428C2×C5—42no—
429C2×C5#227411yescharacteristic
430C2×C5C4×#13111yescharacteristic
431C2×C5#287511yescharacteristic
432C2×C5#299212yes—
433C2×C5#299312yes—
434C23—22no—
435C23—22no—
436C23—21no—
437C23—21no—
438C23C42×D512yes—
439C23C22×C4×D511yescharacteristic
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—52no—
453C2×C4—52no—
454C2×C4—52no—
455C2×C4—52no—
456C2×C4—54no—
457C2×C4—102no—
458C2×C4—102no—
459C2×C4—102no—
460C2×C4—102no—
461C2×C4—102no—
462C2×C4—102no—
463C2×C4—102no—
464C2×C4—102no—
465C2×C4—102no—
466C2×C4C42×D512yes—
467C2×C4C22×C4×D511yescharacteristic
468C2×C4C22×C4×D511yescharacteristic
469C2×C4#414411yescharacteristic
470C2×C4#414411yescharacteristic
471C2×C4#415011yescharacteristic
472C2×C4#415011yescharacteristic
473C8—22no—
474C8—22no—
475C8—22no—
476C8—22no—
477C8—102no—
478C8—102no—
479C8—102no—
480C8—102no—
481C5#2055311yescharacteristic, complemented, Sylow
482C22—22no—
483C22—22no—
484C22—22no—
485C22—42no—
486C22—42no—
487C22C2×C42×D511yescharacteristic
488C22#6588011yescharacteristic
489C22#6588011yescharacteristic
490C22#6607912yes—
491C22D5×#13112yes—
492C4—22no—
493C4—42no—
494C4—42no—
495C4—102no—
496C4—102no—
497C4—102no—
498C4—102no—
499C4—104no—
500C4—202no—
501C4#6582011yescharacteristic
502C4#6585011yescharacteristic
503C4#6588011yescharacteristic
504C4#6588011yescharacteristic
505C2—42no—
506C2#1060516611yescharacteristic
507C2#1061021711yescharacteristic
508C2#1061028011yescharacteristic
509C2#1061119512yes—
510C2#1061119612yes—
subgroupG/H
center, FrattiniC22×C4C23×D5
commutatorC22×C5C22×C42
FittingC5×#1752C2
socleC23×C5C23×C4
prime radicalC24×C5C22×C4
cosocleC22×C4×C5C24
hypercenter#1752D5
hypocenterC5#20553
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C22×C4 ◁ #1752

Complements

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

NH ≅ G/Ntype
7. #10600626C2semidirect
8. #10600625C2semidirect
9. #10600604C2semidirect
10. #10600602C2semidirect
12. #10599447C2semidirect
13. #10599444C2semidirect
14. #10599397C2semidirect
15. #10599395C2semidirect
25. #65391C4semidirect
46. #65009C4semidirect
481. C5#20553semidirect

Maximal subgroups

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

subgroupindex
C2×#653912
C2×#650092
#106051712
C5×#17522
C8×C22⋊Dic52
#106007422
#106006262
#106006252
#106006042
#106006022
#106002412
#105994472
#105994442
#105993972
#105993952
#205535