Finite Group Databasebeta

← #12108402

Subgroups — #12108402

Order 1280.

1326 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#66159C211yesmaximal, characteristic
2C5×#2311C211yesmaximal, characteristic
3C2×#65523C211yesmaximal, characteristic
4#10602378C212yesmaximal, complemented
5#10602352C212yesmaximal, complemented
6#10600318C212yesmaximal, complemented
7#10600309C212yesmaximal, complemented
8#10600307C212yesmaximal, complemented
9#10600296C212yesmaximal, complemented
10C22×C5×#41C2211yescharacteristic
11C23×Dic10C2211yescharacteristic
12C2×C5×#117C2211yescharacteristic
13C22×C5×C4⋊C4C2211yescharacteristic
14#66159—24no—
15#66159C418yescomplemented
16C2×#4090—21no—
17C2×#4082—21no—
18C2×#4081—21no—
19C2×#4081—21no—
20C2×#4072C2211yescharacteristic
21C2×#4070C2212yes—
22C2×#4070C2212yes—
23C22×#681C2211yescharacteristic
24C23×#168C2211yescharacteristic
25C5×#391—21no—
26C5×#390—21no—
27C5×#386—21no—
28C5×#386—21no—
29C5×#380C2212yes—
30C5×#380C2212yes—
31#65523—24no—
32C2×C22⋊#168C2212yes—
33C2×C22⋊#168C2212yes—
34C2×#3942C2212yes—
35C2×#3942C2212yes—
36C2×#3942C2212yes—
37C2×#3942C2212yes—
38#64749C2212yes—
39#64749C2212yes—
40#64749C2212yes—
41#64749C2212yes—
42#21771—51nomaximal, Sylow
43C2×C5×#41—41no—
44C2×C5×#41—41no—
45C2×C5×#41—41no—
46C2×C5×#41—41no—
47C2×C5×#41C2×C414yes—
48C2×C5×#41D414yes—
49C22×C5×Q8D411yescharacteristic
50C22×C5×D4D411yescharacteristic
51C22×C5×D4D411yescharacteristic
52C22×C5×D4D411yescharacteristic
53C23×C4×C5—21no—
54C23×C4×C5C2311yescharacteristic
55C23×C4×C5C2311yescharacteristic
56C23×C4×C5C2311yescharacteristic
57C23×Dic5—21no—
58C22×Dic10—21no—
59C22×Dic10—22no—
60C22×Dic10—41no—
61C22×Dic10—41no—
62C22×Dic10—41no—
63C22×Dic10—41no—
64C22×Dic10C2×C414yescomplemented
65C22×Dic10C2312yes—
66C5×#117—41no—
67C5×#117—41no—
68C2×C5×C4⋊C4—21no—
69C2×C5×C4⋊C4—41no—
70C2×C5×C4⋊C4—41no—
71C2×C5×C4⋊C4C2312yes—
72C2×C5×C4⋊C4C2312yes—
73C2×C5×C22⋊C4—21no—
74C2×C5×C22⋊C4—22no—
75C2×C5×C22⋊C4—22no—
76C2×C42×C5—21no—
77C2×C22⋊Dic5—22no—
78C2×C22⋊Dic5—22no—
79C2×C22⋊Dic5—21no—
80#4090—28no—
81#4082—28no—
82#4081—28no—
83#4081—28no—
84#4072—22no—
85#4072C2×C414yescomplemented
86C2×C4⋊Dic5C2312yes—
87#4070—24no—
88#4070—24no—
89#4070C2×C418yes—
90#4070C2×C418yescomplemented
91C2×#683—22no—
92C2×#683—22no—
93C2×C4×Dic5—21no—
94C2×#681—22no—
95C2×#681—41no—
96C2×#681—41no—
97C22×#168—22no—
98C22×#168—41no—
99C22×#168—41no—
100C22×#168C2312yes—
101C22×#168C2312yes—
102C5×#95—22no—
103C5×#95—22no—
104C22⋊#168—24no—
105C22⋊#168—24no—
106#3942—42no—
107#3942—42no—
108#3942—42no—
109#3942—42no—
110C2×#536—51no—
111C2×#408—51no—
112#2311D511yescharacteristic
113#1894—52no—
114#1892—52no—
115#1873—52no—
116#1872—52no—
117#1825—52no—
118#1787—52no—
119C24×C5C2×D411yescharacteristic
120C24×C5C2×D411yescharacteristic
121C24×C5C2×D411yescharacteristic
122C5×#41—44no—
123C5×#41—44no—
124C5×#41—44no—
125C5×#41—44no—
126C2×C5×Q8—41no—
127C2×C5×Q8—41no—
128C2×C5×Q8C22⋊C414yes—
129C2×C5×D4—41no—
130C2×C5×D4—41no—
131C2×C5×D4—41no—
132C2×C5×D4—41no—
133C2×C5×D4—41no—
134C2×C5×D4—41no—
135C2×C5×D4C22⋊C414yes—
136C2×C5×D4C22⋊C414yes—
137C2×C5×D4C22⋊C414yes—
138C22×C4×C5—21no—
139C22×C4×C5—21no—
140C22×C4×C5—21no—
141C22×C4×C5—21no—
142C22×C4×C5—21no—
143C22×C4×C5—21no—
144C22×C4×C5—21no—
145C22×C4×C5—21no—
146C22×C4×C5—21no—
147C22×C4×C5—21no—
148C22×C4×C5—21no—
149C22×C4×C5—21no—
150C22×C4×C5—22no—
151C22×C4×C5—22no—
152C22×C4×C5—42no—
153C22×C4×C5—82no—
154C22×C4×C5C22×C412yes—
155C22×C4×C5C22×C412yes—
156C22×C4×C5C22×C412yes—
157C22×C4×C5C2×D412yes—
158C22×C4×C5C2×D412yes—
159C22×C4×C5C2×D412yes—
160C22×C4×C5C2×D411yescharacteristic
161C22×C4×C5C2×D411yescharacteristic
162C22×C4×C5C2×D411yescharacteristic
163C22×C4×C5C2411yescharacteristic
164C22×Dic5—24no—
165C22×Dic5—22no—
166C22×Dic5—22no—
167C22×Dic5—42no—
168C22×Dic5—42no—
169C2×Dic10—22no—
170C2×Dic10—22no—
171C2×Dic10—22no—
172C2×Dic10—22no—
173C2×Dic10—24no—
174C2×Dic10—22no—
175C2×Dic10—44no—
176C2×Dic10—44no—
177C2×Dic10—42no—
178C2×Dic10—42no—
179C2×Dic10—44no—
180C2×Dic10—44no—
181C2×Dic10C22×C414yes—
182C5×C4⋊C4—22no—
183C5×C4⋊C4—22no—
184C5×C4⋊C4—22no—
185C5×C4⋊C4—22no—
186C5×C4⋊C4—42no—
187C5×C22⋊C4—42no—
188C5×C22⋊C4—82no—
189C5×C22⋊C4—82no—
190C42×C5—42no—
191C22⋊Dic5—28no—
192C22⋊Dic5—28no—
193C22⋊Dic5—24no—
194C4⋊Dic5—22no—
195C4⋊Dic5C22×C414yes—
196#683—28no—
197#683—28no—
198C4×Dic5—24no—
199#681—44no—
200C2×#168—22no—
201C2×#168—22no—
202C2×#168—22no—
203C2×#168—22no—
204C2×#168—22no—
205C2×#168—22no—
206C2×#168—44no—
207C22×#41C2×D511yescharacteristic
208C23×Q8—51no—
209C22×M16—51no—
210C23×C8—51no—
211#536—58no—
212#536—104no—
213C2×C4⋊Q8—101no—
214C2×#124—101no—
215C2×#123—101no—
216C2×#123—101no—
217C2×C22⋊Q8—52no—
218C2×C22⋊Q8—52no—
219C2×#117—51no—
220C2×#117C2×D511yescharacteristic
221C22×C4⋊C4C2×D511yescharacteristic
222C2×#103—52no—
223C2×#103—52no—
224C2×#103—52no—
225C2×#103—52no—
226#408—104no—
227C2×C22⋊C8—52no—
228C2×C22⋊C8—52no—
229#391—21no—
230#390—21no—
231#386—21no—
232#386—21no—
233#380C2×D512yes—
234#380C2×D512yes—
235#340—52no—
236#340—52no—
237#340—52no—
238#340—52no—
239C23×C5—21no—
240C23×C5—21no—
241C23×C5—21no—
242C23×C5—21no—
243C23×C5—21no—
244C23×C5—21no—
245C23×C5—22no—
246C23×C5—22no—
247C23×C5—22no—
248C23×C5—22no—
249C23×C5—22no—
250C23×C5—22no—
251C23×C5C2×C22⋊C412yes—
252C23×C5C2×C22⋊C412yes—
253C23×C5C2×C22⋊C412yes—
254C23×C5C22×D411yescharacteristic
255C5×Q8—44no—
256C5×D4—44no—
257C5×D4—44no—
258C5×D4—44no—
259C2×C4×C5—21no—
260C2×C4×C5—21no—
261C2×C4×C5—21no—
262C2×C4×C5—21no—
263C2×C4×C5—21no—
264C2×C4×C5—21no—
265C2×C4×C5—22no—
266C2×C4×C5—22no—
267C2×C4×C5—22no—
268C2×C4×C5—22no—
269C2×C4×C5—22no—
270C2×C4×C5—22no—
271C2×C4×C5—22no—
272C2×C4×C5—22no—
273C2×C4×C5—22no—
274C2×C4×C5—22no—
275C2×C4×C5—22no—
276C2×C4×C5—22no—
277C2×C4×C5—42no—
278C2×C4×C5—42no—
279C2×C4×C5—42no—
280C2×C4×C5—42no—
281C2×C4×C5—42no—
282C2×C4×C5—42no—
283C2×C4×C5—84no—
284C2×C4×C5C2×C22⋊C412yes—
285C2×C4×C5C2×C22⋊C412yes—
286C2×C4×C5C2×C22⋊C412yes—
287C2×C4×C5C23×C411yescharacteristic
288C2×C4×C5C22×D411yescharacteristic
289C2×C4×C5H(5,𝔽2)11yescharacteristic
290C2×C4×C5H(5,𝔽2)11yescharacteristic
291C2×C4×C5H(5,𝔽2)11yescharacteristic
292C2×C4×C5H(5,𝔽2)11yescharacteristic
293C2×Dic5—24no—
294C2×Dic5—24no—
295C2×Dic5—48no—
296C2×Dic5—42no—
297C2×Dic5—42no—
298C2×Dic5—42no—
299C2×Dic5—42no—
300Dic10—24no—
301Dic10—24no—
302Dic10—48no—
303Dic10—42no—
304Dic10—42no—
305#168—42no—
306#168—42no—
307C2×#41—41no—
308C2×#41—41no—
309C2×#41—41no—
310C2×#41—41no—
311C2×#41C4×D514yes—
312C2×#41D2014yes—
313C22×Q8—54no—
314C22×Q8—52no—
315C22×Q8—102no—
316C22×Q8—101no—
317C22×Q8—201no—
318C22×Q8—201no—
319C22×Q8—201no—
320C22×Q8—201no—
321C22×Q8C22⋊D511yescharacteristic
322C22×D4C22⋊D511yescharacteristic
323C22×D4C22⋊D511yescharacteristic
324C22×D4C22⋊D511yescharacteristic
325C23×C4—21no—
326C23×C4—101no—
327C23×C4C22×D511yescharacteristic
328C23×C4C22×D511yescharacteristic
329C23×C4C22×D511yescharacteristic
330C2×M16—102no—
331C2×M16—201no—
332C2×M16—201no—
333C22×C8—52no—
334C22×C8—52no—
335C22×C8—102no—
336C22×C8—201no—
337C22×C8—201no—
338C4⋊Q8—108no—
339#124—108no—
340#123—108no—
341#123—108no—
342C22⋊Q8—58no—
343C22⋊Q8—58no—
344C22⋊Q8—104no—
345C22⋊Q8—104no—
346#117—41no—
347#117—41no—
348#117—54no—
349#117—102no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—41no—
352C2×C4⋊C4—41no—
353C2×C4⋊C4—52no—
354C2×C4⋊C4—102no—
355C2×C4⋊C4—102no—
356C2×C4⋊C4C22×D512yes—
357C2×C4⋊C4C22×D512yes—
358C2×C22⋊C4—21no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—101no—
362C2×C22⋊C4—102no—
363C2×C22⋊C4—102no—
364C2×C42—21no—
365C2×C42—101no—
366#103—202no—
367#103—202no—
368#103—202no—
369#103—202no—
370C22⋊C8—104no—
371C22⋊C8—104no—
372#95—22no—
373#95—22no—
374C22×C5—22no—
375C22×C5—22no—
376C22×C5—22no—
377C22×C5—24no—
378C22×C5—24no—
379C22×C5—24no—
380C22×C5—41no—
381C22×C5—41no—
382C22×C5—41no—
383C22×C5—42no—
384C22×C5—42no—
385C22×C5—42no—
386C22×C5—41no—
387C22×C5—41no—
388C22×C5—41no—
389C22×C5C22×C22⋊C411yescharacteristic
390C22×C5#53411yescharacteristic
391C22×C5#53411yescharacteristic
392C22×C5#57712yes—
393C22×C5#57812yes—
394C4×C5—22no—
395C4×C5—22no—
396C4×C5—22no—
397C4×C5—82no—
398C4×C5—82no—
399C4×C5#51811yescharacteristic
400C4×C5#51811yescharacteristic
401C4×C5#53411yescharacteristic
402C4×C5#53411yescharacteristic
403Dic5—44no—
404Dic5—44no—
405C24C2×C22⋊D511yescharacteristic
406C24C2×C22⋊D511yescharacteristic
407C24C2×C22⋊D511yescharacteristic
408#41—44no—
409#41—44no—
410#41—44no—
411#41—44no—
412C2×Q8—41no—
413C2×Q8—41no—
414C2×Q8—54no—
415C2×Q8—102no—
416C2×Q8—104no—
417C2×Q8—102no—
418C2×Q8—102no—
419C2×Q8—102no—
420C2×Q8—102no—
421C2×Q8—204no—
422C2×Q8—204no—
423C2×Q8—202no—
424C2×Q8—202no—
425C2×Q8—204no—
426C2×Q8—204no—
427C2×Q8#68514yes—
428C2×D4—41no—
429C2×D4—41no—
430C2×D4—41no—
431C2×D4—41no—
432C2×D4—41no—
433C2×D4—41no—
434C2×D4#68514yes—
435C2×D4#68514yes—
436C2×D4#68514yes—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—42no—
452C22×C4—82no—
453C22×C4—102no—
454C22×C4—102no—
455C22×C4—104no—
456C22×C4—202no—
457C22×C4—202no—
458C22×C4C2×C4×D512yes—
459C22×C4C2×C4×D512yes—
460C22×C4C2×C4×D512yes—
461C22×C4C2×D2012yes—
462C22×C4C2×D2012yes—
463C22×C4C2×D2012yes—
464C22×C4C2×C22⋊D511yescharacteristic
465C22×C4C2×C22⋊D511yescharacteristic
466C22×C4C2×C22⋊D511yescharacteristic
467C22×C4C23×D511yescharacteristic
468M16—204no—
469C2×C8—102no—
470C2×C8—102no—
471C2×C8—102no—
472C2×C8—102no—
473C2×C8—102no—
474C2×C8—102no—
475C2×C8—204no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—42no—
481C4⋊C4—54no—
482C4⋊C4—102no—
483C4⋊C4—108no—
484C4⋊C4—108no—
485C22⋊C4—42no—
486C22⋊C4—82no—
487C22⋊C4—82no—
488C22⋊C4—104no—
489C22⋊C4—108no—
490C22⋊C4—108no—
491C42—42no—
492C42—104no—
493C2×C5—42no—
494C2×C5—42no—
495C2×C5—42no—
496C2×C5#230811yescharacteristic
497C2×C5#289812yes—
498C2×C5#320011yescharacteristic
499C2×C5#320112yes—
500C2×C5#320211yescharacteristic
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23C2×#68512yes—
514C23C2×#68512yes—
515C23C2×#68512yes—
516C23C22×C22⋊D511yescharacteristic
517Q8—44no—
518Q8—104no—
519Q8—104no—
520Q8—208no—
521Q8—202no—
522Q8—202no—
523D4—44no—
524D4—44no—
525D4—44no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—84no—
551C2×C4—104no—
552C2×C4—104no—
553C2×C4—208no—
554C2×C4—202no—
555C2×C4—202no—
556C2×C4—202no—
557C2×C4—202no—
558C2×C4C2×#68512yes—
559C2×C4C2×#68512yes—
560C2×C4C2×#68512yes—
561C2×C4C22×C4×D511yescharacteristic
562C2×C4C22×D2011yescharacteristic
563C2×C4#414411yescharacteristic
564C2×C4#414411yescharacteristic
565C2×C4#414411yescharacteristic
566C2×C4#414411yescharacteristic
567C8—202no—
568C8—202no—
569C5#2177111yescharacteristic, complemented, Sylow
570C22—24no—
571C22—24no—
572C22—24no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—41no—
580C22—41no—
581C22—41no—
582C22—42no—
583C22—42no—
584C22—42no—
585C22C22×#68511yescharacteristic
586C22#6613811yescharacteristic
587C22#6613811yescharacteristic
588C22#6617012yes—
589C22#6617212yes—
590C4—22no—
591C4—22no—
592C4—22no—
593C4—82no—
594C4—82no—
595C4—204no—
596C4—204no—
597C4#6582011yescharacteristic
598C4#6582011yescharacteristic
599C4#6582711yescharacteristic
600C4#6582711yescharacteristic
601C2—42no—
602C2—42no—
603C2—42no—
604C2#1061385211yescharacteristic
605C2#1061430112yes—
606C2#1061512311yescharacteristic
607C2#1061514411yescharacteristic
608C2#1061515912yes—
subgroupG/H
centerC23C22×C22⋊D5
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC5×#2311C2
socleC23×C5C22×D4
prime radicalC22×C5×#41C22
cosocleC22×C4×C5C24
hypercenter#2311D5
hypocenterC5#21771
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C22×#41 ◁ #2311

Complements

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

NH ≅ G/Ntype
4. #10602378C2semidirect
5. #10602352C2semidirect
6. #10600318C2semidirect
7. #10600309C2semidirect
8. #10600307C2semidirect
9. #10600296C2semidirect
15. #66159C4semidirect
64. C22×Dic10C2×C4semidirect
85. #4072C2×C4semidirect
90. #4070C2×C4semidirect
569. C5#21771semidirect

Maximal subgroups

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

subgroupindex
C2×#661592
C5×#23112
C2×#655232
#106023782
#106023522
#106003182
#106003092
#106003072
#106002962
#217715