Finite Group Databasebeta

← #12100656

Subgroups — #12100656

Order 1280.

2234 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#4095C211yesmaximal, characteristic
2C22×#4039C211yesmaximal, characteristic, complemented
3C2×C5×#417C211yesmaximal, characteristic, complemented
4#10615817C211yesmaximal, characteristic, complemented
5C2×#65420C211yesmaximal, characteristic, complemented
6#10602772C211yesmaximal, characteristic, complemented
7#10602351C211yesmaximal, characteristic, complemented
8#10601069C214yesmaximal, complemented
9#10601001C214yesmaximal, complemented
10C22×C5×#41C2211yescharacteristic
11C23×D20C2211yescharacteristic
12C23×C4×D5—21no—
13C22×C5×M16C2211yescharacteristic
14C22×C5×C4⋊C4C2211yescharacteristic
15C2×#4095—24no—
16C2×#4095—24no—
17C2×#4095C418yescomplemented
18C22×#688C2212yes—
19C22×#686C2212yes—
20C22×#685—21no—
21C2×#4072C2211yescharacteristic
22C22×C4×Dic5C2211yescharacteristic
23C22×#681C2211yescharacteristic
24C2×#4039—22no—
25C2×#4039—24no—
26C2×#4039—24no—
27C2×#4039C2214yes—
28C5×#417—24no—
29C2×C5×#103C2212yescomplemented
30C2×C5×#102C2212yescomplemented
31#65519C2212yescomplemented
32#65510C2212yescomplemented
33#65509—21no—
34#65509—21no—
35C4×C22⋊Dic5—22no—
36#65420—24no—
37C2×#3953C2214yes—
38#64770C2214yescomplemented
39#64749C2214yescomplemented
40#20925—51nomaximal, Sylow
41C24×D5—21no—
42C2×C5×#41—22no—
43C2×C5×#41—22no—
44C2×C5×#41—22no—
45C2×C5×#41—22no—
46C2×C5×#41—22no—
47C2×C5×#41C2×C414yescomplemented
48C22×C5×Q8C2311yescharacteristic
49C22×C5×D4—21no—
50C22×C5×D4C2311yescharacteristic
51C23×C4×C5—21no—
52C23×C4×C5—21no—
53C23×C4×C5C2311yescharacteristic
54C23×Dic5D411yescharacteristic
55C23×Dic5D411yescharacteristic
56C22×D20—21no—
57C22×D20—22no—
58C22×D20—22no—
59C22×D20—22no—
60C22×D20C2×C414yes—
61C22×D20D414yes—
62C22×D20D414yes—
63C22×D20C2312yes—
64C22×C4×D5—28no—
65C22×C4×D5—24no—
66C22×C4×D5—48no—
67C2×C5×M16—22no—
68C2×C5×M16—22no—
69C2×C5×M16—22no—
70C22×C5×C8C2312yes—
71C2×C5×C4⋊C4—21no—
72C2×C5×C4⋊C4—22no—
73C2×C5×C4⋊C4D414yes—
74C2×C5×C4⋊C4C2312yes—
75C2×C22⋊Dic5—21no—
76C2×C22⋊Dic5—22no—
77C2×C22⋊Dic5—22no—
78#4095—232no—
79C2×#688—24no—
80C2×#688—24no—
81C2×#688C2×C418yescomplemented
82C2×#686—24no—
83C2×#686—24no—
84C2×#686C2×C418yescomplemented
85C2×#685—28no—
86C2×#685—24no—
87#4072—22no—
88#4072—22no—
89C2×C4⋊Dic5C2312yes—
90C2×C4×Dic5—22no—
91C2×C4×Dic5—22no—
92C2×C4×Dic5—21no—
93C2×C4×Dic5D412yes—
94C2×C4×Dic5C2312yes—
95C2×#681—22no—
96C2×#681—22no—
97C2×#681C2×C414yescomplemented
98C22×#168C2312yes—
99#4039—216no—
100#4039—28no—
101#4039—28no—
102C5×#103—24no—
103C5×#102—24no—
104#3963—22no—
105#3963—22no—
106#3953—28no—
107C22×AGL(1,ℤ/8)—51no—
108C22×C4×D4—51no—
109C2×#417D511yescharacteristic, complemented
110C2×#418—51no—
111#2297—51no—
112#1879—54no—
113#1877—54no—
114#1736—51no—
115#1728—51no—
116C24×C5—21no—
117C24×C5C2×D411yescharacteristic
118C23×D5—24no—
119C23×D5—24no—
120C23×D5—24no—
121C23×D5—22no—
122C23×D5—44no—
123C23×D5—44no—
124C5×#41—28no—
125C5×#41—28no—
126C5×#41—44no—
127C5×#41—44no—
128C2×C5×Q8—21no—
129C2×C5×Q8—21no—
130C2×C5×Q8—21no—
131C2×C5×Q8—21no—
132C2×C5×Q8—21no—
133C2×C5×Q8C22×C412yescomplemented
134C2×C5×D4—24no—
135C2×C5×D4—21no—
136C2×C5×D4—21no—
137C2×C5×D4—21no—
138C2×C5×D4—21no—
139C2×C5×D4—21no—
140C2×C5×D4—42no—
141C2×C5×D4—42no—
142C2×C5×D4C22×C412yescomplemented
143C22×C4×C5—24no—
144C22×C4×C5—22no—
145C22×C4×C5—21no—
146C22×C4×C5—24no—
147C22×C4×C5—44no—
148C22×C4×C5—42no—
149C22×C4×C5—42no—
150C22×C4×C5C22×C412yes—
151C22×C4×C5C2×D412yes—
152C22×C4×C5C2×D412yes—
153C22×C4×C5C2×D411yescharacteristic
154C22×C4×C5C2×D412yes—
155C22×C4×C5#4112yes—
156C22×C4×C5#4112yes—
157C22×C4×C5C2411yescharacteristic
158C22×Dic5—22no—
159C22×Dic5—24no—
160C22×Dic5—22no—
161C22×Dic5—22no—
162C22×Dic5—42no—
163C22×Dic5C2×D412yes—
164C22×Dic5C2×D412yes—
165C2×D20—24no—
166C2×D20—28no—
167C2×D20—24no—
168C2×D20—24no—
169C2×D20—28no—
170C2×D20—28no—
171C2×D20—22no—
172C2×D20—22no—
173C2×D20—22no—
174C2×D20C22⋊C4116yescomplemented
175C2×D20C22×C414yes—
176C2×D20C2×D414yes—
177C2×D20C2×D414yes—
178C2×C4×D5—216no—
179C2×C4×D5—432no—
180C2×C4×D5—48no—
181C2×C4×D5—48no—
182C5×M16—44no—
183C2×C5×C8—22no—
184C2×C5×C8—22no—
185C2×C5×C8—22no—
186C5×C4⋊C4—24no—
187C5×C4⋊C4—22no—
188C5×C4⋊C4C2×D414yes—
189C22⋊Dic5—44no—
190C22⋊Dic5—42no—
191C22⋊Dic5—44no—
192#688—216no—
193#686—216no—
194#685—216no—
195C4⋊Dic5—22no—
196C4⋊Dic5—22no—
197C4×Dic5—24no—
198C4×Dic5—22no—
199C4×Dic5—22no—
200C4×Dic5—42no—
201#681—28no—
202C2×#168—22no—
203C2×#168—22no—
204C2×#168C22×C414yescomplemented
205C22×#41C2×D511yescharacteristic
206C23×D4—51no—
207C24×C4—101no—
208C2×AGL(1,ℤ/8)—58no—
209C2×AGL(1,ℤ/8)—104no—
210C2×AGL(1,ℤ/8)—104no—
211C22×SD16—52no—
212C22×D8—52no—
213C22×M16—51no—
214C22×M16C2×D511yescharacteristic
215C2×C4×D4—54no—
216C2×C4×D4—102no—
217C2×C4×D4—104no—
218C2×C4×D4—104no—
219C2×#117—51no—
220C22×C4⋊C4C2×D511yescharacteristic
221C22×C22⋊C4—101no—
222C22×C42—51no—
223#418—104no—
224#417—24no—
225C2×#103C2×D512yescomplemented
226C2×#102—54no—
227C2×#102C2×D512yescomplemented
228#391—52no—
229#390—52no—
230#386—101no—
231#386—101no—
232C4×C22⋊C4—102no—
233#340—54no—
234#336—54no—
235C23×C5—22no—
236C23×C5—22no—
237C23×C5—22no—
238C23×C5—41no—
239C23×C5—41no—
240C23×C5—42no—
241C23×C5—42no—
242C23×C5C4×D412yes—
243C23×C5#12012yes—
244C23×C5#12112yes—
245C23×C5C22×D411yescharacteristic
246C22×D5—24no—
247C22×D5—24no—
248C22×D5—24no—
249C22×D5—216no—
250C22×D5—48no—
251C22×D5—416no—
252C22×D5—48no—
253C22×D5—416no—
254C22×D5—44no—
255C22×D5—44no—
256C5×Q8—22no—
257C5×Q8—22no—
258C5×Q8—41no—
259C5×Q8—41no—
260C5×D4—22no—
261C5×D4—22no—
262C5×D4—48no—
263C5×D4—41no—
264C5×D4—41no—
265C2×C4×C5—22no—
266C2×C4×C5—24no—
267C2×C4×C5—48no—
268C2×C4×C5—41no—
269C2×C4×C5—41no—
270C2×C4×C5—44no—
271C2×C4×C5—48no—
272C2×C4×C5C2×C22⋊C414yes—
273C2×C4×C5C4×D412yes—
274C2×C4×C5C4×D414yes—
275C2×C4×C5#12012yes—
276C2×C4×C5#12112yes—
277C2×C4×C5#12114yes—
278C2×C4×C5#12314yes—
279C2×C4×C5C23×C411yescharacteristic
280C2×C4×C5C22×D411yescharacteristic
281C2×C4×C5C22×D411yescharacteristic
282C2×C4×C5C22×D411yescharacteristic
283C2×C4×C5C2×#4111yescharacteristic
284C2×C4×C5C2×#4111yescharacteristic
285C2×Dic5—28no—
286C2×Dic5—22no—
287C2×Dic5—22no—
288C2×Dic5—22no—
289C2×Dic5—22no—
290C2×Dic5—22no—
291C2×Dic5—44no—
292C2×Dic5—42no—
293C2×Dic5—42no—
294C2×Dic5—42no—
295C2×Dic5AGL(1,ℤ/8)14yescomplemented
296D20—24no—
297D20—24no—
298D20—24no—
299D20—216no—
300D20C2×C22⋊C418yes—
301C4×D5—432no—
302C5×C8—42no—
303#168—24no—
304C25—101no—
305C2×#41—22no—
306C2×#41—22no—
307C2×#41—22no—
308C2×#41—22no—
309C2×#41—22no—
310C2×#41C4×D514yescomplemented
311C22×Q8C22×D511yescharacteristic
312C22×D4—21no—
313C22×D4—54no—
314C22×D4—54no—
315C22×D4—54no—
316C22×D4—52no—
317C22×D4—102no—
318C22×D4—101no—
319C22×D4—102no—
320C22×D4—102no—
321C22×D4C22×D511yescharacteristic
322C23×C4—21no—
323C23×C4—21no—
324C23×C4—51no—
325C23×C4—51no—
326C23×C4—104no—
327C23×C4—108no—
328C23×C4—208no—
329C23×C4C22×D511yescharacteristic
330AGL(1,ℤ/8)—1032no—
331C2×SD16—58no—
332C2×SD16—104no—
333C2×SD16—104no—
334C2×D8—58no—
335C2×D8—104no—
336C2×D8—104no—
337C2×M16—22no—
338C2×M16—22no—
339C2×M16—22no—
340C2×M16—54no—
341C2×M16—102no—
342C2×M16—102no—
343C22×C8—52no—
344C22×C8C22×D512yes—
345C4×D4—108no—
346C4×D4—1016no—
347C4×D4—108no—
348#117—102no—
349#117—102no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—52no—
353C2×C4⋊C4C22⋊D514yes—
354C2×C4⋊C4C22×D512yes—
355C2×C22⋊C4—104no—
356C2×C22⋊C4—108no—
357C2×C22⋊C4—102no—
358C2×C22⋊C4—102no—
359C2×C22⋊C4—101no—
360C2×C42—52no—
361C2×C42—52no—
362C2×C42—102no—
363C2×C42—102no—
364C2×C42—101no—
365#103—24no—
366#102—24no—
367#102—108no—
368#95—102no—
369#95—102no—
370C22×C5—24no—
371C22×C5—22no—
372C22×C5—22no—
373C22×C5—24no—
374C22×C5—44no—
375C22×C5—42no—
376C22×C5—81no—
377C22×C5—81no—
378C22×C5—82no—
379C22×C5#38614yes—
380C22×C5C2×C4×D411yescharacteristic
381C22×C5C2×#12011yescharacteristic
382C22×C5C2×#12111yescharacteristic
383C22×C5C2×AGL(1,ℤ/8)12yes—
384C22×C5#57512yes—
385C2×D5—28no—
386C2×D5—432no—
387C2×D5—48no—
388C2×D5—44no—
389C2×D5—44no—
390C2×D5—48no—
391C4×C5—44no—
392C4×C5—42no—
393C4×C5#38614yes—
394C4×C5C22×C22⋊C411yescharacteristic
395C4×C5C2×C4×D411yescharacteristic
396C4×C5C2×#12111yescharacteristic
397C4×C5C2×#12311yescharacteristic
398Dic5—24no—
399Dic5—42no—
400Dic5—82no—
401C24—21no—
402C24—102no—
403C24—104no—
404C24—104no—
405C24—104no—
406C24—204no—
407C24—204no—
408C24D4×D511yescharacteristic
409#41—28no—
410#41—28no—
411#41—44no—
412#41—44no—
413C2×Q8—21no—
414C2×Q8—21no—
415C2×Q8—21no—
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8C2×C4×D512yescomplemented
419C2×D4—24no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4—21no—
425C2×D4—42no—
426C2×D4—42no—
427C2×D4—54no—
428C2×D4—54no—
429C2×D4—54no—
430C2×D4—516no—
431C2×D4—102no—
432C2×D4—102no—
433C2×D4—102no—
434C2×D4—104no—
435C2×D4—108no—
436C2×D4—104no—
437C2×D4—108no—
438C2×D4—104no—
439C2×D4—108no—
440C2×D4C2×C4×D512yescomplemented
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—44no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—52no—
449C22×C4—52no—
450C22×C4—1016no—
451C22×C4—104no—
452C22×C4—102no—
453C22×C4—102no—
454C22×C4—102no—
455C22×C4—208no—
456C22×C4—2032no—
457C22×C4—208no—
458C22×C4—202no—
459C22×C4C2×C4×D512yes—
460C22×C4#70912yes—
461C22×C4D4×D512yes—
462C22×C4D4×D512yes—
463C22×C4D4×D511yescharacteristic
464C22×C4#71112yes—
465C22×C4C2×C22⋊D512yes—
466C22×C4C23×D511yescharacteristic
467SD16—1016no—
468D8—1016no—
469M16—44no—
470M16—108no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—54no—
475C2×C8—102no—
476C2×C8—102no—
477C4⋊C4—22no—
478C4⋊C4—24no—
479C4⋊C4—102no—
480C4⋊C4—102no—
481C4⋊C4C2×C22⋊D514yes—
482C22⋊C4—1016no—
483C22⋊C4—204no—
484C22⋊C4—204no—
485C22⋊C4—202no—
486C42—102no—
487C42—102no—
488C42—104no—
489C42—202no—
490C2×C5—24no—
491C2×C5—82no—
492C2×C5C2×#38611yescharacteristic
493C2×C5C4×AGL(1,ℤ/8)12yes—
494C2×C5#300311yescharacteristic
495C2×C5#300511yescharacteristic
496C2×C5#303612yes—
497D5—48no—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—41no—
502C23—42no—
503C23—42no—
504C23—41no—
505C23—104no—
506C23—104no—
507C23—104no—
508C23—1016no—
509C23—208no—
510C23—208no—
511C23—2016no—
512C23—2016no—
513C23—204no—
514C23—204no—
515C23#402712yes—
516C23#403012yes—
517C23#408312yes—
518C23C2×D4×D511yescharacteristic
519Q8—22no—
520Q8—22no—
521Q8—41no—
522Q8—41no—
523D4—22no—
524D4—22no—
525D4—48no—
526D4—41no—
527D4—41no—
528D4—58no—
529D4—104no—
530D4—104no—
531D4—104no—
532D4—1016no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—48no—
536C2×C4—41no—
537C2×C4—44no—
538C2×C4—41no—
539C2×C4—48no—
540C2×C4—54no—
541C2×C4—108no—
542C2×C4—102no—
543C2×C4—102no—
544C2×C4—102no—
545C2×C4—102no—
546C2×C4—102no—
547C2×C4—2032no—
548C2×C4—204no—
549C2×C4—202no—
550C2×C4—202no—
551C2×C4—202no—
552C2×C4D5×C22⋊C414yes—
553C2×C4#402712yes—
554C2×C4#402914yes—
555C2×C4#403012yes—
556C2×C4C4×C22⋊D514yes—
557C2×C4#408312yes—
558C2×C4#408414yes—
559C2×C4C22×C4×D511yescharacteristic
560C2×C4C2×#70911yescharacteristic
561C2×C4C2×D4×D511yescharacteristic
562C2×C4C2×D4×D511yescharacteristic
563C2×C4C2×#71111yescharacteristic
564C2×C4C22×C22⋊D511yescharacteristic
565C8—42no—
566C8—104no—
567C5#2092511yescharacteristic, complemented, Sylow
568C22—22no—
569C22—22no—
570C22—24no—
571C22—24no—
572C22—42no—
573C22—44no—
574C22—81no—
575C22—81no—
576C22—82no—
577C22—108no—
578C22—2032no—
579C22—204no—
580C22—204no—
581C22—208no—
582C22—208no—
583C22#6524514yes—
584C22C2×#402711yescharacteristic
585C22C2×#403011yescharacteristic
586C22D5×AGL(1,ℤ/8)12yes—
587C22#6611112yes—
588C22C2×#408311yescharacteristic
589C4—42no—
590C4—44no—
591C4—104no—
592C4—202no—
593C4—402no—
594C4#6524514yes—
595C4C2×D5×C22⋊C411yescharacteristic
596C4C2×#402911yescharacteristic
597C4C2×C4×C22⋊D511yescharacteristic
598C4C2×#408411yescharacteristic
599C2—24no—
600C2—82no—
601C2—208no—
602C2#1061059712yes—
603C2#1061064112yes—
604C2C2×#6524511yescharacteristic
605C2#1061567511yescharacteristic
606C2#1061571111yescharacteristic
subgroupG/H
centerC23C2×D4×D5
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC2×C5×#417C2
socleC23×C5C22×D4
prime radicalC22×#4095C2
cosocleC22×C4×C5C24
hypercenterC2×#417D5
hypocenterC5#20925
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#417

Complements

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

NH ≅ G/Ntype
2. C22×#4039C2semidirect
3. C2×C5×#417C2semidirect
4. #10615817C2semidirect
5. C2×#65420C2semidirect
6. #10602772C2semidirect
7. #10602351C2semidirect
8. #10601069C2semidirect
9. #10601001C2semidirect
17. C2×#4095C4semidirect
29. C2×C5×#103C22semidirect
30. C2×C5×#102C22semidirect
31. #65519C22semidirect
32. #65510C22semidirect
38. #64770C22semidirect
39. #64749C22semidirect
47. C2×C5×#41C2×C4semidirect
81. C2×#688C2×C4semidirect
84. C2×#686C2×C4semidirect
97. C2×#681C2×C4semidirect
109. C2×#417D5semidirect
133. C2×C5×Q8C22×C4semidirect
142. C2×C5×D4C22×C4semidirect
174. C2×D20C22⋊C4semidirect
204. C2×#168C22×C4semidirect
225. C2×#103C2×D5semidirect
227. C2×#102C2×D5semidirect
295. C2×Dic5AGL(1,ℤ/8)semidirect
310. C2×#41C4×D5semidirect
418. C2×Q8C2×C4×D5semidirect
440. C2×D4C2×C4×D5semidirect
567. C5#20925semidirect

Maximal subgroups

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

subgroupindex
C22×#40952
C22×#40392
C2×C5×#4172
#106158172
C2×#654202
#106027722
#106023512
#106010692
#106010012
#209255