Finite Group Databasebeta

← #13034369

Subgroups — #13034369

Order 1280.

1602 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#65879C211yesmaximal, characteristic
2#10617506C212yesmaximal
3#10604893C212yesmaximal, complemented
4C2×#64917C212yesmaximal, complemented
5#10604307C212yesmaximal, complemented
6#10604218C212yesmaximal, complemented
7#10604214C212yesmaximal, complemented
8#10604168C212yesmaximal, complemented
9C22×D4×D5C2211yescharacteristic
10C22×C22⋊AGL(1,𝔽5)C2214yes—
11C2×C4×C5×D4C411yescharacteristic, complemented
12C2×D4×Dic5C411yescharacteristic, complemented
13C2×C4×C22⋊D5C411yescharacteristic, complemented
14C2×C4×C22⋊D5C411yescharacteristic, complemented
15C22×#685C2211yescharacteristic
16C22×#685C2211yescharacteristic
17#65879—28no—
18C2×#4039C411yescharacteristic, complemented
19C2×#4038C2211yescharacteristic
20C2×#4027C411yescharacteristic, complemented
21C2×#4027C411yescharacteristic, complemented
22C2×D5×C22⋊C4C2211yescharacteristic
23C2×D5×C22⋊C4C2211yescharacteristic
24C2×C4×D20C411yescharacteristic, complemented
25C2×#4018C2211yescharacteristic
26#65800C2214yes—
27#65779—28no—
28#65764C2214yes—
29#64917C2214yes—
30#64917C2214yes—
31#64917C2214yes—
32#64917C2214yes—
33#20689—51nomaximal, Sylow
34C24×D5C2311yescharacteristic
35C24×D5C2311yescharacteristic
36C23×AGL(1,𝔽5)—24no—
37C22×C5×D4C2×C411yescharacteristic
38C23×C4×C5C2×C411yescharacteristic
39C23×C4×C5C2×C411yescharacteristic, complemented
40C22×C22⋊D5C2×C411yescharacteristic
41C22×C22⋊D5C2×C411yescharacteristic
42C23×Dic5C2×C411yescharacteristic
43C23×Dic5C2×C411yescharacteristic, complemented
44C2×D4×D5—28no—
45C2×D4×D5D418yes—
46C22×D20C2×C411yescharacteristic
47C22×C4×D5C2×C411yescharacteristic
48C22×C4×D5C2×C411yescharacteristic
49C22×C4×D5C2×C411yescharacteristic, complemented
50C22×C4×D5C2×C411yescharacteristic, complemented
51C22×C4×D5C2311yescharacteristic
52C2×C22⋊AGL(1,𝔽5)—216no—
53C2×C22⋊AGL(1,𝔽5)—24no—
54C2×C22⋊AGL(1,𝔽5)—24no—
55C2×C22⋊AGL(1,𝔽5)—24no—
56C2×C22⋊AGL(1,𝔽5)C2318yes—
57C4×C5×D4—24no—
58C2×C5×C4⋊C4C2×C411yescharacteristic, complemented
59C2×C5×C22⋊C4C2×C411yescharacteristic
60C2×C5×C22⋊C4C2×C411yescharacteristic, complemented
61C2×C42×C5C2×C411yescharacteristic, complemented
62C2×C22⋊Dic5C2×C411yescharacteristic
63C2×C22⋊Dic5C2×C411yescharacteristic, complemented
64D4×Dic5—24no—
65C4×C22⋊D5—24no—
66C4×C22⋊D5—24no—
67C2×#685—24no—
68C2×#685—24no—
69C2×#685C2×C411yescharacteristic
70C2×#685C2×C411yescharacteristic
71C2×#685C2×C411yescharacteristic, complemented
72C2×#685C2×C411yescharacteristic, complemented
73C2×#685C2311yescharacteristic
74C2×#685C2311yescharacteristic
75C2×#685C2311yescharacteristic
76C2×#685C2311yescharacteristic
77C2×C4⋊Dic5C2×C411yescharacteristic, complemented
78C2×#683C2×C411yescharacteristic, complemented
79C2×#683C2×C411yescharacteristic, complemented
80C2×C4×Dic5C2×C411yescharacteristic, complemented
81C2×C4×Dic5C2×C411yescharacteristic, complemented
82C2×C4×Dic5C2×C411yescharacteristic, complemented
83#4039—24no—
84#4038—24no—
85#4027—24no—
86#4027—24no—
87D5×C22⋊C4—24no—
88D5×C22⋊C4—24no—
89C4×D20—24no—
90#4018—24no—
91(C2×AGL(1,𝔽5))⋊C4—28no—
92C2×#518—51no—
93#2308—52no—
94#1820—52no—
95#1820—52no—
96#1819—52no—
97#1819—52no—
98C2×#342—52no—
99C2×#342—52no—
100C24×C5C22×C411yescharacteristic
101C24×C5C22×C411yescharacteristic
102C23×D5—24no—
103C23×D5—81no—
104C23×D5—81no—
105C23×D5—81no—
106C23×D5—81no—
107C23×D5C22×C411yescharacteristic
108C23×D5C22×C411yescharacteristic
109C23×D5C2×D414yes—
110C23×D5C2×D414yes—
111C23×D5#4118yes—
112C23×D5C2411yescharacteristic
113C22×AGL(1,𝔽5)—28no—
114C22×AGL(1,𝔽5)—48no—
115C22×AGL(1,𝔽5)—84no—
116C22×AGL(1,𝔽5)—84no—
117C2×C5×D4—24no—
118C2×C5×D4C22⋊C414yes—
119C22×C4×C5—22no—
120C22×C4×C5—22no—
121C22×C4×C5—81no—
122C22×C4×C5—81no—
123C22×C4×C5C22×C411yescharacteristic
124C22×C4×C5C22×C411yescharacteristic
125C22×C4×C5C22×C411yescharacteristic
126C22×C4×C5C22×C411yescharacteristic
127C22×C4×C5C22×C411yescharacteristic
128C2×C22⋊D5—24no—
129C2×C22⋊D5—24no—
130C2×C22⋊D5C22⋊C414yes—
131C2×C22⋊D5C22⋊C414yes—
132C22×Dic5—22no—
133C22×Dic5—22no—
134C22×Dic5—81no—
135C22×Dic5—81no—
136C22×Dic5C22×C411yescharacteristic
137C22×Dic5C22×C411yescharacteristic
138C22×Dic5C22×C411yescharacteristic
139C22×Dic5C22×C411yescharacteristic
140C22×Dic5C22×C411yescharacteristic
141D4×D5—216no—
142D4×D5—216no—
143C2×D20—24no—
144C2×D20C22⋊C414yes—
145C2×C4×D5—24no—
146C2×C4×D5—22no—
147C2×C4×D5—22no—
148C2×C4×D5—22no—
149C2×C4×D5—22no—
150C2×C4×D5—81no—
151C2×C4×D5—81no—
152C2×C4×D5—81no—
153C2×C4×D5—81no—
154C2×C4×D5C2×D414yes—
155C22⋊AGL(1,𝔽5)—216no—
156C22⋊AGL(1,𝔽5)—48no—
157C22⋊AGL(1,𝔽5)—84no—
158C22⋊AGL(1,𝔽5)—84no—
159C5×C4⋊C4—22no—
160C5×C22⋊C4—22no—
161C5×C22⋊C4—22no—
162C42×C5—22no—
163C22⋊Dic5—22no—
164C22⋊Dic5—22no—
165#685—22no—
166#685—22no—
167#685—22no—
168#685—22no—
169#685—22no—
170#685—22no—
171#685—22no—
172#685—22no—
173C4⋊Dic5—22no—
174#683—22no—
175#683—22no—
176C4×Dic5—22no—
177C4×Dic5—22no—
178C4×Dic5—22no—
179C23×D4—51no—
180#518—108no—
181C2×C4×D4—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×C4×D4AGL(1,𝔽5)11yescharacteristic, complemented
189C2×#117—51no—
190C2×#117—51no—
191C22×C22⋊C4—51no—
192C22×C22⋊C4—51no—
193C22×C22⋊C4—51no—
194C22×C22⋊C4—51no—
195C22×C22⋊C4—54no—
196#386—108no—
197#381—54no—
198#379—54no—
199#342—54no—
200#342—54no—
201#342—54no—
202#342—54no—
203C23×C5—22no—
204C23×C5—41no—
205C23×C5—41no—
206C23×C5—41no—
207C23×C5—41no—
208C23×C5C2×C22⋊C412yes—
209C23×C5C2×C22⋊C412yes—
210C23×C5#11714yes—
211C23×C5C23×C411yescharacteristic
212C22×D5—22no—
213C22×D5—28no—
214C22×D5—41no—
215C22×D5—41no—
216C22×D5—41no—
217C22×D5—41no—
218C22×D5—44no—
219C22×D5—44no—
220C22×D5—41no—
221C22×D5—41no—
222C22×D5—44no—
223C22×D5—44no—
224C22×D5—88no—
225C22×D5—88no—
226C22×D5C2×C22⋊C412yes—
227C22×D5C2×C22⋊C412yes—
228C22×D5#11714yes—
229C22×D5#12318yes—
230C22×D5#12318yes—
231C22×D5C22×D412yes—
232C22×D5C2×#4114yes—
233C2×AGL(1,𝔽5)—48no—
234C2×AGL(1,𝔽5)—48no—
235C2×AGL(1,𝔽5)—48no—
236C2×AGL(1,𝔽5)—816no—
237C5×D4—28no—
238C2×C4×C5—22no—
239C2×C4×C5—21no—
240C2×C4×C5—21no—
241C2×C4×C5—21no—
242C2×C4×C5—21no—
243C2×C4×C5—41no—
244C2×C4×C5—41no—
245C2×C4×C5—41no—
246C2×C4×C5—41no—
247C2×C4×C5—82no—
248C2×C4×C5—82no—
249C2×C4×C5C2×C22⋊C412yes—
250C22⋊D5—28no—
251C22⋊D5—28no—
252C2×Dic5—22no—
253C2×Dic5—21no—
254C2×Dic5—21no—
255C2×Dic5—21no—
256C2×Dic5—21no—
257C2×Dic5—41no—
258C2×Dic5—41no—
259C2×Dic5—41no—
260C2×Dic5—41no—
261C2×Dic5—82no—
262C2×Dic5—82no—
263C2×Dic5C2×C22⋊C412yes—
264D20—28no—
265C4×D5—28no—
266C4×D5—82no—
267C4×D5—82no—
268C4×D5—82no—
269C4×D5—82no—
270C25—51no—
271C25—51no—
272C22×D4—51no—
273C22×D4—51no—
274C22×D4—58no—
275C22×D4—51no—
276C22×D4—108no—
277C22×D4C2×AGL(1,𝔽5)11yescharacteristic
278C23×C4—51no—
279C23×C4—51no—
280C23×C4—51no—
281C23×C4—51no—
282C23×C4—51no—
283C23×C4—51no—
284C23×C4—51no—
285C23×C4—104no—
286C23×C4C2×AGL(1,𝔽5)11yescharacteristic
287C23×C4C2×AGL(1,𝔽5)11yescharacteristic, complemented
288C4×D4—24no—
289C4×D4—104no—
290C4×D4—104no—
291C4×D4—104no—
292C4×D4—104no—
293C4×D4—104no—
294C4×D4—104no—
295C4×D4—104no—
296#117—104no—
297#117—104no—
298C2×C4⋊C4—51no—
299C2×C4⋊C4—51no—
300C2×C4⋊C4—51no—
301C2×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic, complemented
302C2×C22⋊C4—51no—
303C2×C22⋊C4—51no—
304C2×C22⋊C4—51no—
305C2×C22⋊C4—51no—
306C2×C22⋊C4—51no—
307C2×C22⋊C4—51no—
308C2×C22⋊C4—51no—
309C2×C22⋊C4—51no—
310C2×C22⋊C4—51no—
311C2×C22⋊C4—51no—
312C2×C22⋊C4—58no—
313C2×C22⋊C4—104no—
314C2×C22⋊C4—104no—
315C2×C22⋊C4—104no—
316C2×C22⋊C4—104no—
317C2×C22⋊C4—1016no—
318C2×C22⋊C4—104no—
319C2×C22⋊C4—104no—
320C2×C22⋊C4—104no—
321C2×C22⋊C4C2×AGL(1,𝔽5)11yescharacteristic
322C2×C22⋊C4C2×AGL(1,𝔽5)11yescharacteristic, complemented
323C2×C42—51no—
324C2×C42—51no—
325C2×C42—51no—
326C2×C42C2×AGL(1,𝔽5)11yescharacteristic, complemented
327#95—108no—
328C22×C5—21no—
329C22×C5—21no—
330C22×C5—22no—
331C22×C5—22no—
332C22×C5—22no—
333C22×C5—22no—
334C22×C5—22no—
335C22×C5—44no—
336C22×C5—44no—
337C22×C5#38114yes—
338C22×C5C22×C22⋊C411yescharacteristic
339C22×C5C2×#11712yes—
340C2×D5—22no—
341C2×D5—22no—
342C2×D5—22no—
343C2×D5—22no—
344C2×D5—22no—
345C2×D5—22no—
346C2×D5—22no—
347C2×D5—48no—
348C2×D5—44no—
349C2×D5—44no—
350C2×D5—48no—
351C2×D5—44no—
352C2×D5—88no—
353C2×D5#38114yes—
354C2×D5C2×#12314yes—
355AGL(1,𝔽5)—88no—
356C4×C5—22no—
357C4×C5—41no—
358C4×C5—41no—
359C4×C5—41no—
360C4×C5—41no—
361Dic5—22no—
362Dic5—41no—
363Dic5—41no—
364Dic5—41no—
365Dic5—41no—
366C24—54no—
367C24—51no—
368C24—51no—
369C24—58no—
370C24—54no—
371C24—51no—
372C24—104no—
373C24—401no—
374C24—401no—
375C24—401no—
376C24—401no—
377C24C22×AGL(1,𝔽5)11yescharacteristic
378C24C22×AGL(1,𝔽5)11yescharacteristic
379C2×D4—24no—
380C2×D4—54no—
381C2×D4—54no—
382C2×D4—54no—
383C2×D4—104no—
384C2×D4—1016no—
385C2×D4—1016no—
386C2×D4—104no—
387C2×D4—104no—
388C2×D4C22⋊AGL(1,𝔽5)14yes—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—51no—
392C22×C4—54no—
393C22×C4—51no—
394C22×C4—51no—
395C22×C4—51no—
396C22×C4—51no—
397C22×C4—81no—
398C22×C4—81no—
399C22×C4—104no—
400C22×C4—102no—
401C22×C4—102no—
402C22×C4—102no—
403C22×C4—102no—
404C22×C4—102no—
405C22×C4—102no—
406C22×C4—108no—
407C22×C4—208no—
408C22×C4—401no—
409C22×C4—401no—
410C22×C4—401no—
411C22×C4—401no—
412C22×C4—404no—
413C22×C4—404no—
414C22×C4—401no—
415C22×C4—401no—
416C22×C4C22×AGL(1,𝔽5)11yescharacteristic
417C22×C4C22×AGL(1,𝔽5)11yescharacteristic
418C22×C4C22×AGL(1,𝔽5)11yescharacteristic
419C22×C4C22×AGL(1,𝔽5)11yescharacteristic
420C22×C4C22×AGL(1,𝔽5)11yescharacteristic
421C4⋊C4—22no—
422C4⋊C4—102no—
423C4⋊C4—102no—
424C4⋊C4—102no—
425C22⋊C4—22no—
426C22⋊C4—22no—
427C22⋊C4—102no—
428C22⋊C4—102no—
429C22⋊C4—102no—
430C22⋊C4—102no—
431C22⋊C4—102no—
432C22⋊C4—102no—
433C22⋊C4—102no—
434C22⋊C4—102no—
435C22⋊C4—102no—
436C22⋊C4—102no—
437C22⋊C4—1016no—
438C22⋊C4—208no—
439C22⋊C4—404no—
440C22⋊C4—404no—
441C42—22no—
442C42—102no—
443C42—102no—
444C42—102no—
445C2×C5—22no—
446C2×C5—24no—
447C2×C5—42no—
448C2×C5C2×#38111yescharacteristic
449C2×C5#288311yescharacteristic
450C2×C5#288311yescharacteristic
451D5—24no—
452D5—24no—
453D5—42no—
454C23—22no—
455C23—41no—
456C23—41no—
457C23—41no—
458C23—41no—
459C23—52no—
460C23—52no—
461C23—54no—
462C23—54no—
463C23—58no—
464C23—58no—
465C23—52no—
466C23—108no—
467C23—102no—
468C23—201no—
469C23—201no—
470C23—204no—
471C23—201no—
472C23—201no—
473C23—204no—
474C23—201no—
475C23—201no—
476C23—204no—
477C23—204no—
478C23—408no—
479C23—408no—
480C23#413014yes—
481C23C2×C22⋊AGL(1,𝔽5)12yes—
482C23C2×C22⋊AGL(1,𝔽5)12yes—
483C23C23×AGL(1,𝔽5)11yescharacteristic
484D4—28no—
485D4—108no—
486D4—108no—
487D4—108no—
488C2×C4—22no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—52no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—108no—
501C2×C4—102no—
502C2×C4—101no—
503C2×C4—101no—
504C2×C4—101no—
505C2×C4—101no—
506C2×C4—201no—
507C2×C4—201no—
508C2×C4—201no—
509C2×C4—201no—
510C2×C4—208no—
511C2×C4—208no—
512C2×C4—208no—
513C2×C4—402no—
514C2×C4—402no—
515C2×C4—402no—
516C2×C4—402no—
517C2×C4—402no—
518C2×C4—402no—
519C2×C4—4016no—
520C2×C4C2×C22⋊AGL(1,𝔽5)12yes—
521C5#2068911yescharacteristic, complemented, Sylow
522C22—21no—
523C22—22no—
524C22—21no—
525C22—22no—
526C22—22no—
527C22—22no—
528C22—22no—
529C22—44no—
530C22—44no—
531C22—54no—
532C22—54no—
533C22—102no—
534C22—102no—
535C22—102no—
536C22—102no—
537C22—102no—
538C22—102no—
539C22—102no—
540C22—208no—
541C22—204no—
542C22—204no—
543C22—208no—
544C22—204no—
545C22—408no—
546C22#6576414yes—
547C22C2×#413012yes—
548C22C22×C22⋊AGL(1,𝔽5)11yescharacteristic
549C4—22no—
550C4—41no—
551C4—41no—
552C4—41no—
553C4—41no—
554C4—102no—
555C4—201no—
556C4—201no—
557C4—201no—
558C4—201no—
559C4—408no—
560C2—22no—
561C2—24no—
562C2—42no—
563C2—104no—
564C2—104no—
565C2—202no—
566C2C2×#6576411yescharacteristic
567C2#1061748711yescharacteristic
568C2#1061753811yescharacteristic
subgroupG/H
centerC22C22×C22⋊AGL(1,𝔽5)
commutatorC23×C5C23×C4
FrattiniC23C23×AGL(1,𝔽5)
FittingC2×C4×C5×D4C4
socleC22×C5C22×C22⋊C4
prime radicalC22×D4×D5C22
cosocleC23×D5C24
hypercenterC2×C4×D4AGL(1,𝔽5)
hypocenterC5#20689
LCSG ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C22×D4 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
3. #10604893C2semidirect
4. C2×#64917C2semidirect
5. #10604307C2semidirect
6. #10604218C2semidirect
7. #10604214C2semidirect
8. #10604168C2semidirect
11. C2×C4×C5×D4C4semidirect
12. C2×D4×Dic5C4semidirect
13. C2×C4×C22⋊D5C4semidirect
14. C2×C4×C22⋊D5C4semidirect
18. C2×#4039C4semidirect
20. C2×#4027C4semidirect
21. C2×#4027C4semidirect
24. C2×C4×D20C4semidirect
39. C23×C4×C5C2×C4semidirect
43. C23×Dic5C2×C4semidirect
49. C22×C4×D5C2×C4semidirect
50. C22×C4×D5C2×C4semidirect
58. C2×C5×C4⋊C4C2×C4semidirect
60. C2×C5×C22⋊C4C2×C4semidirect
61. C2×C42×C5C2×C4semidirect
63. C2×C22⋊Dic5C2×C4semidirect
71. C2×#685C2×C4semidirect
72. C2×#685C2×C4semidirect
77. C2×C4⋊Dic5C2×C4semidirect
78. C2×#683C2×C4semidirect
79. C2×#683C2×C4semidirect
80. C2×C4×Dic5C2×C4semidirect
81. C2×C4×Dic5C2×C4semidirect
82. C2×C4×Dic5C2×C4semidirect
188. C2×C4×D4AGL(1,𝔽5)semidirect
287. C23×C4C2×AGL(1,𝔽5)semidirect
301. C2×C4⋊C4C2×AGL(1,𝔽5)semidirect
322. C2×C22⋊C4C2×AGL(1,𝔽5)semidirect
326. C2×C42C2×AGL(1,𝔽5)semidirect
521. C5#20689semidirect

Maximal subgroups

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

subgroupindex
C2×#658792
#106175062
#106048932
C2×#649172
#106043072
#106042182
#106042142
#106041682
#206895