Finite Group Databasebeta

← #13034473

Subgroups — #13034473

Order 1280.

1318 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#65857C211yesmaximal, characteristic
2#10617101C212yesmaximal, complemented
3#10604891C212yesmaximal
4#10604307C212yesmaximal, complemented
5#10604217C212yesmaximal, complemented
6#10604216C212yesmaximal, complemented
7#10604167C212yesmaximal, complemented
8C4×#3999C212yesmaximal, complemented
9C22×#709C2211yescharacteristic
10C2×C5×#117C411yescharacteristic, complemented
11C22×C4⋊Dic5C411yescharacteristic, complemented
12#65857—28no—
13C2×#4038C411yescharacteristic, complemented
14C2×#4038C2211yescharacteristic
15C2×D5×C4⋊C4C411yescharacteristic, complemented
16C2×D5×C4⋊C4C2211yescharacteristic
17C2×#4027C411yescharacteristic, complemented
18C2×#4027C411yescharacteristic, complemented
19C2×#4027C2211yescharacteristic
20C2×#4027C2211yescharacteristic
21C2×C4×D20C2211yescharacteristic
22C2×C4×D20C2211yescharacteristic
23C2×C4×Dic10C411yescharacteristic, complemented
24C2×C4×Dic10C411yescharacteristic, complemented
25C2×#4011C2214yes—
26#65765C2214yes—
27#65764C2214yes—
28C4×C22⋊AGL(1,𝔽5)C2214yes—
29C2×#3999C2214yes—
30#65711—28no—
31#64917C2214yes—
32#64917C2214yes—
33#20725—51nomaximal, Sylow
34C23×C4×C5C2×C411yescharacteristic
35C22×C22⋊D5C2×C411yescharacteristic
36C22×C22⋊D5C2311yescharacteristic
37C23×Dic5C2×C411yescharacteristic
38C23×Dic5C2×C411yescharacteristic
39C2×#709—24no—
40C2×#709—24no—
41C2×#709—24no—
42C22×D20C2311yescharacteristic
43C22×C4×D5C2×C411yescharacteristic
44C22×C4×D5C2×C411yescharacteristic
45C22×C4×D5C2×C411yescharacteristic
46C22×C4×D5C2311yescharacteristic
47C22×C4×D5C2311yescharacteristic
48C22×C4×D5C2311yescharacteristic
49C22×Dic10C2×C411yescharacteristic
50C2×C22⋊AGL(1,𝔽5)C2318yes—
51C2×C4⋊AGL(1,𝔽5)—24no—
52C2×C4×AGL(1,𝔽5)—24no—
53C5×#117—24no—
54C2×C5×C4⋊C4C2×C411yescharacteristic
55C2×C5×C4⋊C4C2×C411yescharacteristic
56C2×C5×C22⋊C4C2×C411yescharacteristic
57C2×C5×C22⋊C4C2×C411yescharacteristic
58C2×C42×C5C2×C411yescharacteristic
59C2×C42×C5C2×C411yescharacteristic
60C2×#685C2×C411yescharacteristic
61C2×#685C2×C411yescharacteristic
62C2×#685C2311yescharacteristic
63C2×#685C2311yescharacteristic
64C2×C4⋊Dic5—24no—
65C2×C4⋊Dic5C2×C411yescharacteristic
66C2×C4⋊Dic5C2×C411yescharacteristic
67C2×C4⋊Dic5C2×C411yescharacteristic
68C2×C4⋊Dic5C2×C411yescharacteristic
69C2×#683C2×C411yescharacteristic
70C2×#683C2×C411yescharacteristic
71C2×#683C2×C411yescharacteristic
72C2×#683C2×C411yescharacteristic
73C2×C4×Dic5C2×C411yescharacteristic
74C2×C4×Dic5C2×C411yescharacteristic
75C2×C4×Dic5C2×C411yescharacteristic
76C2×C4×Dic5C2×C411yescharacteristic
77#4038—24no—
78#4038—24no—
79D5×C4⋊C4—24no—
80D5×C4⋊C4D418yes—
81#4027—24no—
82#4027—24no—
83#4027—24no—
84#4027—24no—
85C4×D20—24no—
86C4×D20—24no—
87C4×Dic10—24no—
88C4×Dic10—24no—
89#4011—28no—
90(C2×AGL(1,𝔽5))⋊C4—24no—
91(C2×AGL(1,𝔽5))⋊C4—28no—
92(C2×AGL(1,𝔽5))⋊C4—24no—
93#3999—28no—
94C2×#520—51no—
95#2313—52no—
96#1820—52no—
97#1799—52no—
98#1799—52no—
99#1782—52no—
100C4×#99—52no—
101C4×#99—52no—
102C24×C5C22×C411yescharacteristic
103C23×D5C22×C411yescharacteristic
104C23×D5C2411yescharacteristic
105C22×AGL(1,𝔽5)—28no—
106C22×C4×C5—22no—
107C22×C4×C5—22no—
108C22×C4×C5—22no—
109C22×C4×C5C22×C411yescharacteristic
110C22×C4×C5C22×C411yescharacteristic
111C22×C4×C5C22×C411yescharacteristic
112C22×C4×C5C22×C411yescharacteristic
113C22×C4×C5C22×C411yescharacteristic
114C22×C4×C5C22×C411yescharacteristic
115C2×C22⋊D5—22no—
116C2×C22⋊D5—22no—
117C2×C22⋊D5—22no—
118C2×C22⋊D5—22no—
119C2×C22⋊D5—22no—
120C2×C22⋊D5#4114yes—
121C22×Dic5—22no—
122C22×Dic5—22no—
123C22×Dic5—81no—
124C22×Dic5—81no—
125C22×Dic5C22×C411yescharacteristic
126C22×Dic5C22×C411yescharacteristic
127C22×Dic5C22×C411yescharacteristic
128C22×Dic5C22×C411yescharacteristic
129C22×Dic5C22×C411yescharacteristic
130C22×Dic5C22×C411yescharacteristic
131#709—232no—
132C2×D20—22no—
133C2×D20—22no—
134C2×D20#4114yes—
135C2×C4×D5—22no—
136C2×C4×D5—22no—
137C2×C4×D5—22no—
138C2×C4×D5—22no—
139C2×C4×D5—22no—
140C2×C4×D5—22no—
141C2×C4×D5—22no—
142C2×C4×D5—81no—
143C2×C4×D5—81no—
144C2×C4×D5—81no—
145C2×C4×D5—81no—
146C2×C4×D5C2×D414yes—
147C2×C4×D5C2×D414yes—
148C2×C4×D5C2×D414yes—
149C2×Dic10—22no—
150C2×Dic10—22no—
151C2×Dic10—22no—
152C22⋊AGL(1,𝔽5)—28no—
153C22⋊AGL(1,𝔽5)—28no—
154C4⋊AGL(1,𝔽5)—84no—
155C4×AGL(1,𝔽5)—84no—
156C5×C4⋊C4—22no—
157C5×C4⋊C4C22⋊C414yes—
158C5×C22⋊C4—22no—
159C5×C22⋊C4—22no—
160C42×C5—22no—
161C42×C5—22no—
162#685—22no—
163#685—22no—
164#685—22no—
165#685—22no—
166C4⋊Dic5—22no—
167C4⋊Dic5—22no—
168C4⋊Dic5—22no—
169C4⋊Dic5C22⋊C414yes—
170#683—22no—
171#683—22no—
172#683C22⋊C414yes—
173#683C22⋊C414yes—
174C4×Dic5—22no—
175C4×Dic5—22no—
176C4×Dic5—22no—
177C4×Dic5—22no—
178C22×#41—51no—
179#520—108no—
180C2×C4×Q8—51no—
181C2×C4×Q8—51no—
182C2×C4×D4—51no—
183C2×C4×D4—51no—
184C2×C4×D4—51no—
185C2×C4×D4—51no—
186C2×C4×D4—51no—
187C2×C4×D4—51no—
188C2×#117—51no—
189C2×#117—51no—
190C2×#117AGL(1,𝔽5)11yescharacteristic, complemented
191C22×C4⋊C4—51no—
192C22×C4⋊C4—51no—
193C22×C4⋊C4—51no—
194C2×#99—54no—
195C2×#99—54no—
196#387—108no—
197#381—54no—
198#380—54no—
199C4×C22⋊C4—54no—
200#342—54no—
201#342—54no—
202C23×C5—21no—
203C23×C5—21no—
204C23×C5—41no—
205C23×C5—41no—
206C23×C5#11712yes—
207C23×C5C23×C411yescharacteristic
208C22×D5—21no—
209C22×D5—21no—
210C22×D5—41no—
211C22×D5—41no—
212C22×D5—41no—
213C22×D5—41no—
214C22×D5#11712yes—
215C22×D5C22×D412yes—
216C22×D5C2×#4112yes—
217C22×D5C2×#4112yes—
218C2×AGL(1,𝔽5)—48no—
219C2×AGL(1,𝔽5)—48no—
220C2×AGL(1,𝔽5)—48no—
221C2×C4×C5—28no—
222C2×C4×C5—21no—
223C2×C4×C5—21no—
224C2×C4×C5—21no—
225C2×C4×C5—21no—
226C2×C4×C5—21no—
227C2×C4×C5—21no—
228C2×C4×C5—41no—
229C2×C4×C5—41no—
230C2×C4×C5—41no—
231C2×C4×C5—41no—
232C2×C4×C5C2×C22⋊C412yes—
233C2×C4×C5C2×C22⋊C412yes—
234C2×C4×C5C2×C22⋊C412yes—
235C2×C4×C5#11712yes—
236C22⋊D5—28no—
237C22⋊D5—28no—
238C2×Dic5—21no—
239C2×Dic5—21no—
240C2×Dic5—21no—
241C2×Dic5—21no—
242C2×Dic5—21no—
243C2×Dic5—21no—
244C2×Dic5—41no—
245C2×Dic5—41no—
246C2×Dic5—41no—
247C2×Dic5—41no—
248C2×Dic5—82no—
249C2×Dic5—82no—
250C2×Dic5C2×C22⋊C412yes—
251C2×Dic5C2×C22⋊C412yes—
252C2×Dic5C2×C22⋊C412yes—
253C2×Dic5#11712yes—
254D20—28no—
255C4×D5—28no—
256C4×D5—28no—
257C4×D5—44no—
258C4×D5—44no—
259C4×D5—82no—
260C4×D5—82no—
261Dic10—28no—
262C2×#41—104no—
263C2×#41—104no—
264C2×#41—104no—
265C22×Q8—51no—
266C22×D4—51no—
267C22×D4—51no—
268C22×D4—51no—
269C23×C4—51no—
270C23×C4—51no—
271C23×C4—51no—
272C23×C4—51no—
273C23×C4—51no—
274C23×C4—51no—
275C23×C4—51no—
276C23×C4—51no—
277C23×C4C2×AGL(1,𝔽5)11yescharacteristic
278C4×Q8—104no—
279C4×Q8—104no—
280C4×D4—104no—
281C4×D4—104no—
282C4×D4—104no—
283C4×D4—104no—
284C4×D4—104no—
285C4×D4—104no—
286#117—24no—
287#117—104no—
288#117—104no—
289C2×C4⋊C4—51no—
290C2×C4⋊C4—51no—
291C2×C4⋊C4—51no—
292C2×C4⋊C4—58no—
293C2×C4⋊C4—51no—
294C2×C4⋊C4—51no—
295C2×C4⋊C4—51no—
296C2×C4⋊C4—51no—
297C2×C4⋊C4—51no—
298C2×C4⋊C4—104no—
299C2×C4⋊C4—104no—
300C2×C4⋊C4—104no—
301C2×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
302C2×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
303C2×C22⋊C4—51no—
304C2×C22⋊C4—51no—
305C2×C22⋊C4—51no—
306C2×C22⋊C4—51no—
307C2×C22⋊C4—58no—
308C2×C22⋊C4C2×AGL(1,𝔽5)11yescharacteristic
309C2×C22⋊C4C2×AGL(1,𝔽5)11yescharacteristic
310C2×C42—51no—
311C2×C42—51no—
312C2×C42—51no—
313C2×C42—51no—
314C2×C42—104no—
315C2×C42C2×AGL(1,𝔽5)11yescharacteristic
316C2×C42C2×AGL(1,𝔽5)11yescharacteristic
317#99—108no—
318#99—108no—
319#95—104no—
320#95—104no—
321#95—108no—
322C22×C5—21no—
323C22×C5—21no—
324C22×C5—22no—
325C22×C5—42no—
326C22×C5—42no—
327C22×C5—41no—
328C22×C5—41no—
329C22×C5C22×C22⋊C411yescharacteristic
330C22×C5C2×#11711yescharacteristic
331C22×C5C2×#11711yescharacteristic
332C2×D5—22no—
333C2×D5—22no—
334C2×D5—22no—
335C2×D5—42no—
336C2×D5—42no—
337C2×D5—42no—
338C2×D5—41no—
339C2×D5—41no—
340C2×D5—42no—
341C2×D5#52514yes—
342AGL(1,𝔽5)—88no—
343C4×C5—22no—
344C4×C5—22no—
345C4×C5—22no—
346C4×C5—22no—
347C4×C5—41no—
348C4×C5—41no—
349Dic5—22no—
350Dic5—22no—
351Dic5—22no—
352Dic5—22no—
353Dic5—41no—
354Dic5—41no—
355C24—51no—
356C24—51no—
357C24C22×AGL(1,𝔽5)11yescharacteristic
358#41—1032no—
359C2×Q8—102no—
360C2×Q8—102no—
361C2×Q8—102no—
362C2×D4—54no—
363C2×D4—54no—
364C2×D4—102no—
365C2×D4—102no—
366C2×D4—102no—
367C2×D4—102no—
368C2×D4—102no—
369C2×D4—102no—
370C2×D4—102no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—51no—
375C22×C4—51no—
376C22×C4—51no—
377C22×C4—51no—
378C22×C4—54no—
379C22×C4—54no—
380C22×C4—54no—
381C22×C4—51no—
382C22×C4—51no—
383C22×C4—102no—
384C22×C4—102no—
385C22×C4—102no—
386C22×C4—102no—
387C22×C4—102no—
388C22×C4—102no—
389C22×C4—102no—
390C22×C4—102no—
391C22×C4—102no—
392C22×C4—108no—
393C22×C4—401no—
394C22×C4—401no—
395C22×C4—401no—
396C22×C4—401no—
397C22×C4—401no—
398C22×C4—401no—
399C22×C4C22×AGL(1,𝔽5)11yescharacteristic
400C22×C4C22×AGL(1,𝔽5)11yescharacteristic
401C22×C4C22×AGL(1,𝔽5)11yescharacteristic
402C22×C4C22×AGL(1,𝔽5)11yescharacteristic
403C22×C4C22×AGL(1,𝔽5)11yescharacteristic
404C22×C4C22×AGL(1,𝔽5)11yescharacteristic
405C4⋊C4—22no—
406C4⋊C4—54no—
407C4⋊C4—54no—
408C4⋊C4—54no—
409C4⋊C4—102no—
410C4⋊C4—102no—
411C4⋊C4—102no—
412C4⋊C4—102no—
413C4⋊C4—102no—
414C4⋊C4—404no—
415C4⋊C4C22⋊AGL(1,𝔽5)14yes—
416C22⋊C4—22no—
417C22⋊C4—22no—
418C22⋊C4—102no—
419C22⋊C4—102no—
420C22⋊C4—102no—
421C22⋊C4—102no—
422C22⋊C4—108no—
423C22⋊C4—108no—
424C42—22no—
425C42—22no—
426C42—102no—
427C42—102no—
428C42—102no—
429C42—102no—
430C42—404no—
431C2×C5—21no—
432C2×C5—21no—
433C2×C5—42no—
434C2×C5#228011yescharacteristic
435C2×C5#288311yescharacteristic
436C2×C5#288411yescharacteristic
437D5—24no—
438D5—42no—
439C23—21no—
440C23—21no—
441C23—41no—
442C23—41no—
443C23—52no—
444C23—52no—
445C23—52no—
446C23—52no—
447C23—101no—
448C23—101no—
449C23—201no—
450C23—201no—
451C23—201no—
452C23—201no—
453C23#413012yes—
454C23C23×AGL(1,𝔽5)11yescharacteristic
455Q8—108no—
456D4—108no—
457D4—108no—
458D4—108no—
459C2×C4—28no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—52no—
471C2×C4—52no—
472C2×C4—52no—
473C2×C4—52no—
474C2×C4—108no—
475C2×C4—108no—
476C2×C4—101no—
477C2×C4—101no—
478C2×C4—101no—
479C2×C4—101no—
480C2×C4—101no—
481C2×C4—101no—
482C2×C4—204no—
483C2×C4—204no—
484C2×C4—201no—
485C2×C4—201no—
486C2×C4—201no—
487C2×C4—201no—
488C2×C4—208no—
489C2×C4—208no—
490C2×C4—208no—
491C2×C4—402no—
492C2×C4—402no—
493C2×C4—402no—
494C2×C4—402no—
495C2×C4#413012yes—
496C2×C4C2×C22⋊AGL(1,𝔽5)12yes—
497C2×C4C2×C22⋊AGL(1,𝔽5)12yes—
498C2×C4C2×C22⋊AGL(1,𝔽5)12yes—
499C5#2072511yescharacteristic, complemented, Sylow
500C22—21no—
501C22—21no—
502C22—22no—
503C22—42no—
504C22—42no—
505C22—41no—
506C22—41no—
507C22—54no—
508C22—102no—
509C22—102no—
510C22—102no—
511C22—202no—
512C22—202no—
513C22—202no—
514C22—202no—
515C22—201no—
516C22—201no—
517C22C2×#413011yescharacteristic
518C22C2×#413011yescharacteristic
519C22C22×C22⋊AGL(1,𝔽5)11yescharacteristic
520C4—22no—
521C4—22no—
522C4—22no—
523C4—22no—
524C4—41no—
525C4—41no—
526C4—102no—
527C4—102no—
528C4—102no—
529C4—102no—
530C4—201no—
531C4—201no—
532C4—408no—
533C2—21no—
534C2—21no—
535C2—42no—
536C2—104no—
537C2—202no—
538C2#1061748011yescharacteristic
539C2#1061753811yescharacteristic
540C2#1061753911yescharacteristic
subgroupG/H
centerC22C22×C22⋊AGL(1,𝔽5)
commutatorC23×C5C23×C4
FrattiniC23C23×AGL(1,𝔽5)
FittingC2×C5×#117C4
socleC22×C5C22×C22⋊C4
prime radicalC22×#709C22
cosocleC23×D5C24
hypercenterC2×#117AGL(1,𝔽5)
hypocenterC5#20725
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. #10617101C2semidirect
4. #10604307C2semidirect
5. #10604217C2semidirect
6. #10604216C2semidirect
7. #10604167C2semidirect
8. C4×#3999C2semidirect
10. C2×C5×#117C4semidirect
11. C22×C4⋊Dic5C4semidirect
13. C2×#4038C4semidirect
15. C2×D5×C4⋊C4C4semidirect
17. C2×#4027C4semidirect
18. C2×#4027C4semidirect
23. C2×C4×Dic10C4semidirect
24. C2×C4×Dic10C4semidirect
190. C2×#117AGL(1,𝔽5)semidirect
499. C5#20725semidirect

Maximal subgroups

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

subgroupindex
C2×#658572
#106171012
#106048912
#106043072
#106042172
#106042162
#106041672
C4×#39992
#207255