Finite Group Databasebeta

← #12100303

Subgroups — #12100303

Order 1280.

1458 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#65857C211yesmaximal, characteristic
2C5×#2316C211yesmaximal, characteristic, complemented
3C2×#65291C211yesmaximal, characteristic
4C2×#65284C211yesmaximal, characteristic, complemented
5C2×#65255C211yesmaximal, characteristic
6#10602372C211yesmaximal, characteristic, complemented
7C2×#64749C211yesmaximal, characteristic, complemented
8#10600425C212yesmaximal, complemented
9#10600424C212yesmaximal, complemented
10#10600419C212yesmaximal, complemented
11#10600418C212yesmaximal, complemented
12C22×#709C2211yescharacteristic
13C2×C5×#117C2211yescharacteristic
14C2×C5×#117C2211yescharacteristic
15C22×C5×C4⋊C4C2211yescharacteristic
16C22×C4⋊Dic5C2211yescharacteristic
17C22×#681C2211yescharacteristic
18C23×#168C2211yescharacteristic
19#65857—24no—
20#65857—24no—
21C2×#4038—21no—
22C2×D5×C4⋊C4—21no—
23C2×#4027—22no—
24C2×C4×D20C2212yes—
25C2×C4×Dic10C2212yes—
26C5×#389—21no—
27C5×#389—21no—
28C5×#389C2212yes—
29C5×#389C2212yes—
30C5×#380—22no—
31#65291—24no—
32#65284—24no—
33C2×#3942C2211yescharacteristic
34C2×#3942C2211yescharacteristic
35C2×#3942C2212yes—
36#65255C418yescomplemented
37C2×#3941C2211yescharacteristic
38C2×#3941C2211yescharacteristic
39C2×#3941C2212yes—
40C2×#3940C2212yes—
41C2×#3939C2212yes—
42#64749C2212yes—
43#64749C2212yes—
44#64744C2212yes—
45#64744C2212yes—
46#20878—51nomaximal, Sylow
47C23×C4×C5—21no—
48C23×C4×C5C2311yescharacteristic
49C22×C22⋊D5—21no—
50C23×Dic5—21no—
51C2×#709—22no—
52C2×#709—22no—
53C2×#709—22no—
54C2×#709C2×C414yes—
55C2×#709D414yes—
56C2×#709Q814yes—
57C22×D20C2311yescharacteristic
58C22×C4×D5—21no—
59C22×C4×D5—22no—
60C22×Dic10C2311yescharacteristic
61C5×#117—22no—
62C5×#117—22no—
63C5×#117—22no—
64C5×#117—22no—
65C2×C5×C4⋊C4—21no—
66C2×C5×C4⋊C4—22no—
67C2×C5×C4⋊C4—22no—
68C2×C5×C4⋊C4—22no—
69C2×C5×C4⋊C4C2×C414yescomplemented
70C2×C5×C4⋊C4D411yescharacteristic
71C2×C5×C4⋊C4D411yescharacteristic
72C2×C5×C4⋊C4C2311yescharacteristic
73C2×C5×C4⋊C4C2311yescharacteristic
74C2×C5×C4⋊C4C2312yes—
75C2×C5×C22⋊C4—21no—
76C2×C5×C22⋊C4D412yes—
77C2×C42×C5—21no—
78C2×C42×C5C2312yes—
79C2×#685—22no—
80C2×C4⋊Dic5—21no—
81C2×C4⋊Dic5—22no—
82C2×C4⋊Dic5D414yes—
83C2×C4⋊Dic5C2312yes—
84C2×#683—22no—
85C2×C4×Dic5—22no—
86C2×#681—22no—
87C2×#681—22no—
88C2×#681C2×C414yescomplemented
89C22×#168—22no—
90C22×#168—22no—
91C22×#168—22no—
92C22×#168C2311yescharacteristic
93C22×#168C2311yescharacteristic
94C22×#168C2312yes—
95#4038—28no—
96D5×C4⋊C4—28no—
97#4027—216no—
98C4×D20—24no—
99C4×D20—24no—
100C4×Dic10—24no—
101C4×Dic10—24no—
102C5×#95—22no—
103C5×#95—22no—
104#3942—22no—
105#3942—22no—
106#3942C2×C418yescomplemented
107#3941—22no—
108#3941—22no—
109#3941C2×C418yescomplemented
110#3940—24no—
111#3939—24no—
112C2×#520—51no—
113C2×#428—51no—
114C2×#417—51no—
115C2×#416—51no—
116#2316D511yescharacteristic, complemented
117#1865—52no—
118#1864—52no—
119#1863—52no—
120#1862—52no—
121C2×#340—51no—
122#1729—51no—
123C24×C5C2×D411yescharacteristic
124C23×D5—21no—
125C22×C4×C5—22no—
126C22×C4×C5—24no—
127C22×C4×C5—21no—
128C22×C4×C5—21no—
129C22×C4×C5—81no—
130C22×C4×C5—81no—
131C22×C4×C5C22×C412yes—
132C22×C4×C5C2×D412yes—
133C22×C4×C5C2×D412yes—
134C22×C4×C5C2×D412yes—
135C22×C4×C5C2×D412yes—
136C22×C4×C5C2×D411yescharacteristic
137C22×C4×C5C2×Q812yes—
138C22×C4×C5#4112yes—
139C22×C4×C5#4112yes—
140C22×C4×C5C2411yescharacteristic
141C2×C22⋊D5—24no—
142C2×C22⋊D5—24no—
143C2×C22⋊D5—24no—
144C22×Dic5—21no—
145C22×Dic5—22no—
146C22×Dic5—24no—
147C22×Dic5—81no—
148C22×Dic5—81no—
149#709—28no—
150#709—28no—
151#709—28no—
152#709C4⋊C4116yescomplemented
153C2×D20—21no—
154C2×D20—21no—
155C2×D20—21no—
156C2×D20C22×C412yes—
157C2×D20C2×D412yes—
158C2×D20C2×Q812yes—
159C2×C4×D5—24no—
160C2×C4×D5—24no—
161C2×C4×D5—24no—
162C2×C4×D5—28no—
163C2×C4×D5—82no—
164C2×C4×D5—82no—
165C2×Dic10—21no—
166C2×Dic10—21no—
167C2×Dic10—21no—
168C2×Dic10C22×C412yes—
169C2×Dic10C2×D412yes—
170C2×Dic10C2×Q812yes—
171C5×C4⋊C4—24no—
172C5×C4⋊C4—22no—
173C5×C4⋊C4—22no—
174C5×C4⋊C4—22no—
175C5×C4⋊C4—21no—
176C5×C4⋊C4—21no—
177C5×C4⋊C4—21no—
178C5×C4⋊C4—21no—
179C5×C4⋊C4—82no—
180C5×C4⋊C4—82no—
181C5×C4⋊C4C22×C414yes—
182C5×C22⋊C4—24no—
183C5×C22⋊C4—42no—
184C42×C5—22no—
185C42×C5—22no—
186C42×C5—42no—
187#685—28no—
188C4⋊Dic5—24no—
189C4⋊Dic5—22no—
190C4⋊Dic5C2×D414yes—
191#683—28no—
192C4×Dic5—28no—
193#681—28no—
194C2×#168—22no—
195C2×#168—22no—
196C2×#168—21no—
197C2×#168—21no—
198C2×#168—21no—
199C2×#168—21no—
200C2×#168—21no—
201C2×#168—21no—
202C2×#168—44no—
203C2×#168C22×C414yes—
204C22×#41—51no—
205C22×M16—51no—
206C23×C8—51no—
207#520—104no—
208#520—104no—
209C2×C4×Q8—52no—
210C2×C4×D4—52no—
211C2×C4×D4—102no—
212C2×#117—101no—
213C2×#117C2×D511yescharacteristic
214C2×#117C2×D511yescharacteristic
215C22×C4⋊C4—51no—
216C22×C4⋊C4—101no—
217C22×C4⋊C4C2×D511yescharacteristic
218#428—104no—
219C2×#107—52no—
220C2×#106—52no—
221#417—58no—
222#416—104no—
223C2×#103—52no—
224C2×#103—51no—
225C2×#103—51no—
226C2×#102—52no—
227C2×#102—51no—
228C2×#102—51no—
229#389—21no—
230#389—21no—
231#389C2×D512yes—
232#389C2×D512yes—
233#380—22no—
234#340—52no—
235#340—52no—
236#336—52no—
237#336—52no—
238C23×C5—21no—
239C23×C5—21no—
240C23×C5—21no—
241C23×C5—21no—
242C23×C5C4×D412yes—
243C23×C5#12012yes—
244C23×C5C22⋊Q812yes—
245C23×C5C22×D411yescharacteristic
246C22×D5—22no—
247C22×D5—22no—
248C22×D5—22no—
249C22×D5—41no—
250C22×D5—41no—
251C22×D5—41no—
252C22×D5—41no—
253C2×C4×C5—24no—
254C2×C4×C5—22no—
255C2×C4×C5—22no—
256C2×C4×C5—22no—
257C2×C4×C5—22no—
258C2×C4×C5—22no—
259C2×C4×C5—22no—
260C2×C4×C5—42no—
261C2×C4×C5—42no—
262C2×C4×C5—41no—
263C2×C4×C5—41no—
264C2×C4×C5—41no—
265C2×C4×C5—41no—
266C2×C4×C5—41no—
267C2×C4×C5—41no—
268C2×C4×C5—84no—
269C2×C4×C5C2×C4⋊C414yes—
270C2×C4×C5C4×D414yes—
271C2×C4×C5C4×D412yes—
272C2×C4×C5#12012yes—
273C2×C4×C5C22⋊Q814yes—
274C2×C4×C5C22⋊Q812yes—
275C2×C4×C5#12314yes—
276C2×C4×C5C23×C411yescharacteristic
277C2×C4×C5C22×D411yescharacteristic
278C2×C4×C5C22×D411yescharacteristic
279C2×C4×C5C22×Q811yescharacteristic
280C2×C4×C5C2×#4111yescharacteristic
281C2×C4×C5C2×#4111yescharacteristic
282C22⋊D5—216no—
283C2×Dic5—22no—
284C2×Dic5—22no—
285C2×Dic5—22no—
286C2×Dic5—24no—
287C2×Dic5—42no—
288C2×Dic5—42no—
289C2×Dic5—84no—
290D20—22no—
291D20—22no—
292D20—22no—
293D20C2×C4⋊C414yes—
294C4×D5—216no—
295C4×D5—88no—
296Dic10—22no—
297Dic10—22no—
298Dic10—22no—
299Dic10C2×C4⋊C414yes—
300#168—24no—
301#168—41no—
302#168—41no—
303C2×#41—54no—
304C2×#41—54no—
305C2×#41—54no—
306C2×#41—102no—
307C2×#41—102no—
308C2×#41—102no—
309C22×Q8—51no—
310C22×D4—51no—
311C22×D4—101no—
312C23×C4—21no—
313C23×C4—101no—
314C23×C4—101no—
315C23×C4—102no—
316C23×C4C22×D511yescharacteristic
317C2×M16—54no—
318C2×M16—102no—
319C2×M16—102no—
320C22×C8—52no—
321C22×C8—51no—
322C22×C8—51no—
323C22×C8—102no—
324C22×C8—102no—
325C22×C8—102no—
326C4×Q8—104no—
327C4×Q8—104no—
328C4×D4—104no—
329C4×D4—104no—
330C4×D4—1016no—
331#117—22no—
332#117—22no—
333#117—22no—
334#117—22no—
335#117—108no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—54no—
341C2×C4⋊C4—52no—
342C2×C4⋊C4—108no—
343C2×C4⋊C4—102no—
344C2×C4⋊C4—101no—
345C2×C4⋊C4—102no—
346C2×C4⋊C4C4×D514yescomplemented
347C2×C4⋊C4C22⋊D511yescharacteristic
348C2×C4⋊C4C22⋊D511yescharacteristic
349C2×C4⋊C4C22×D511yescharacteristic
350C2×C4⋊C4C22×D511yescharacteristic
351C2×C4⋊C4C22×D512yes—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—102no—
354C2×C22⋊C4C22⋊D512yes—
355C2×C42—21no—
356C2×C42—102no—
357C2×C42C22×D512yes—
358#107—104no—
359#106—104no—
360#103—58no—
361#103—102no—
362#103—102no—
363#102—58no—
364#102—102no—
365#102—102no—
366#95—22no—
367#95—22no—
368C22×C5—22no—
369C22×C5—22no—
370C22×C5—22no—
371C22×C5—22no—
372C22×C5—22no—
373C22×C5—22no—
374C22×C5#38514yes—
375C22×C5C2×C4×D411yescharacteristic
376C22×C5C2×#12011yescharacteristic
377C22×C5C2×C22⋊Q811yescharacteristic
378C22×C5#57611yescharacteristic
379C22×C5#57711yescharacteristic
380C22×C5#57711yescharacteristic
381C22×C5#57811yescharacteristic
382C2×D5—24no—
383C2×D5—42no—
384C2×D5—42no—
385C2×D5—42no—
386C2×D5—42no—
387C2×D5—42no—
388C2×D5—42no—
389C4×C5—44no—
390C4×C5—42no—
391C4×C5—42no—
392C4×C5—81no—
393C4×C5—81no—
394C4×C5#38514yes—
395C4×C5C22×C4⋊C411yescharacteristic
396C4×C5C2×C4×D411yescharacteristic
397C4×C5C2×C22⋊Q811yescharacteristic
398C4×C5C2×#12311yescharacteristic
399Dic5—24no—
400Dic5—44no—
401C24—101no—
402C24C2×C22⋊D511yescharacteristic
403#41—516no—
404#41—108no—
405#41—108no—
406#41—108no—
407C2×Q8—52no—
408C2×Q8—52no—
409C2×Q8—52no—
410C2×Q8—101no—
411C2×Q8—101no—
412C2×Q8—101no—
413C2×D4—52no—
414C2×D4—52no—
415C2×D4—52no—
416C2×D4—104no—
417C2×D4—104no—
418C2×D4—104no—
419C2×D4—101no—
420C2×D4—101no—
421C2×D4—101no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—81no—
427C22×C4—81no—
428C22×C4—108no—
429C22×C4—101no—
430C22×C4—104no—
431C22×C4—104no—
432C22×C4—104no—
433C22×C4—102no—
434C22×C4—104no—
435C22×C4—402no—
436C22×C4—402no—
437C22×C4—401no—
438C22×C4—401no—
439C22×C4C2×C4×D512yes—
440C22×C4#70912yes—
441C22×C4D4×D512yes—
442C22×C4D4×D512yes—
443C22×C4Q8×D512yes—
444C22×C4#71312yes—
445C22×C4C2×C22⋊D512yes—
446C22×C4C2×C22⋊D512yes—
447C22×C4C2×C22⋊D511yescharacteristic
448C22×C4C23×D511yescharacteristic
449M16—108no—
450C2×C8—54no—
451C2×C8—102no—
452C2×C8—102no—
453C2×C8—101no—
454C2×C8—101no—
455C2×C8—101no—
456C2×C8—101no—
457C2×C8—101no—
458C2×C8—101no—
459C2×C8—204no—
460C4⋊C4—22no—
461C4⋊C4—24no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—54no—
469C4⋊C4—82no—
470C4⋊C4—82no—
471C4⋊C4—108no—
472C4⋊C4—102no—
473C4⋊C4—104no—
474C4⋊C4C2×C4×D514yes—
475C22⋊C4—24no—
476C22⋊C4—42no—
477C22⋊C4—108no—
478C42—22no—
479C42—22no—
480C42—42no—
481C42—108no—
482C2×C5—24no—
483C2×C5C2×#38511yescharacteristic
484C2×C5#294611yescharacteristic
485C2×C5#294711yescharacteristic
486C2×C5#301211yescharacteristic
487C2×C5#301311yescharacteristic
488C2×C5#308311yescharacteristic
489C2×C5#308411yescharacteristic
490D5—44no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—102no—
496C23—102no—
497C23—102no—
498C23—201no—
499C23—201no—
500C23—201no—
501C23—201no—
502C23C4×C22⋊D512yes—
503C23#408312yes—
504C23#409212yes—
505C23C22×C22⋊D511yescharacteristic
506Q8—54no—
507Q8—102no—
508Q8—102no—
509Q8—102no—
510D4—54no—
511D4—1016no—
512D4—102no—
513D4—102no—
514D4—102no—
515C2×C4—24no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—84no—
531C2×C4—102no—
532C2×C4—102no—
533C2×C4—102no—
534C2×C4—1016no—
535C2×C4—104no—
536C2×C4—202no—
537C2×C4—202no—
538C2×C4—408no—
539C2×C4—404no—
540C2×C4D5×C4⋊C414yes—
541C2×C4#403914yes—
542C2×C4#404014yes—
543C2×C4#404214yes—
544C2×C4C4×C22⋊D512yes—
545C2×C4#408312yes—
546C2×C4#409212yes—
547C2×C4C22×C4×D511yescharacteristic
548C2×C4C2×#70911yescharacteristic
549C2×C4C2×D4×D511yescharacteristic
550C2×C4C2×D4×D511yescharacteristic
551C2×C4C2×Q8×D511yescharacteristic
552C2×C4C2×#71311yescharacteristic
553C8—104no—
554C8—201no—
555C8—201no—
556C5#2087811yescharacteristic, complemented, Sylow
557C22—22no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—104no—
564C22—202no—
565C22—202no—
566C22—202no—
567C22—202no—
568C22—202no—
569C22—202no—
570C22#6527814yes—
571C22C2×C4×C22⋊D511yescharacteristic
572C22C2×#408311yescharacteristic
573C22C2×#409211yescharacteristic
574C22#6616911yescharacteristic
575C22#6617011yescharacteristic
576C22#6617111yescharacteristic
577C22#6617211yescharacteristic
578C4—42no—
579C4—42no—
580C4—44no—
581C4—81no—
582C4—81no—
583C4—104no—
584C4—204no—
585C4#6527814yes—
586C4C2×D5×C4⋊C411yescharacteristic
587C4C2×#403911yescharacteristic
588C4C2×#404011yescharacteristic
589C4C2×#404211yescharacteristic
590C2—24no—
591C2—204no—
592C2C2×#6527811yescharacteristic
593C2#1061426811yescharacteristic
594C2#1061427811yescharacteristic
595C2#1061515511yescharacteristic
596C2#1061515811yescharacteristic
597C2#1061520611yescharacteristic
598C2#1061520711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D5
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC5×#2316C2
socleC23×C5C22×D4
prime radicalC22×#709C22
cosocleC22×C4×C5C24
hypercenter#2316D5
hypocenterC5#20878
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2316

Complements

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

NH ≅ G/Ntype
2. C5×#2316C2semidirect
4. C2×#65284C2semidirect
6. #10602372C2semidirect
7. C2×#64749C2semidirect
8. #10600425C2semidirect
9. #10600424C2semidirect
10. #10600419C2semidirect
11. #10600418C2semidirect
36. #65255C4semidirect
69. C2×C5×C4⋊C4C2×C4semidirect
88. C2×#681C2×C4semidirect
106. #3942C2×C4semidirect
109. #3941C2×C4semidirect
116. #2316D5semidirect
152. #709C4⋊C4semidirect
346. C2×C4⋊C4C4×D5semidirect
556. C5#20878semidirect

Maximal subgroups

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

subgroupindex
C2×#658572
C5×#23162
C2×#652912
C2×#652842
C2×#652552
#106023722
C2×#647492
#106004252
#106004242
#106004192
#106004182
#208785