Finite Group Databasebeta

← #12162230

Subgroups — #12162230

Order 1280.

804 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#2362C211yesmaximal, characteristic
2C2×#65311C211yesmaximal, characteristic, complemented
3C2×#65284C211yesmaximal, characteristic
4#10605226C211yesmaximal, characteristic
5#10602490C211yesmaximal, characteristic
6#10602485C211yesmaximal, characteristic
7#10602422C211yesmaximal, characteristic, complemented
8#10602369C211yesmaximal, characteristic
9#10600479C211yesmaximal, characteristic, complemented
10#10600476C211yesmaximal, characteristic, complemented
11#10600416C211yesmaximal, characteristic, complemented
12#10600399C211yesmaximal, characteristic, complemented
13#10600261C211yesmaximal, characteristic, complemented
14#10600252C211yesmaximal, characteristic, complemented
15#10600243C211yesmaximal, characteristic
16C2×C5×C22⋊Q8C2211yescharacteristic
17C2×C4×C5×Q8C2211yescharacteristic
18C2×C5×#117C2211yescharacteristic
19C22×C4⋊Dic5C2211yescharacteristic
20C22×#681C2211yescharacteristic
21C2×#4016C2211yescharacteristic
22C5×#391C2211yescharacteristic
23C5×#389C2211yescharacteristic
24C5×#388—21no—
25C5×#388—21no—
26C5×#387—21no—
27C4×C5×C4⋊C4—21no—
28C4×C5×C4⋊C4C2211yescharacteristic
29C4×C5×C22⋊C4C2211yescharacteristic
30C2×#3967C2211yescharacteristic
31C2×#3967C2211yescharacteristic
32C2×#3967C2211yescharacteristic
33C2×#3967C2211yescharacteristic
34#65501C2211yescharacteristic
35C2×C22⋊#168C2211yescharacteristic
36#65311—24no—
37#65284—24no—
38C2×#3940C2211yescharacteristic
39C2×#3940C2211yescharacteristic
40C2×#3940C2211yescharacteristic
41C2×#3939C2211yescharacteristic
42C2×#3939C2211yescharacteristic
43C2×#3939C2211yescharacteristic
44#65235—21no—
45#65231—21no—
46#65225C2211yescharacteristic
47#65217C2211yescharacteristic
48#65217C2211yescharacteristic
49C2×C4⋊#168C2211yescharacteristic
50C2×C4⋊#168C2211yescharacteristic
51#64949—21no—
52#64946—21no—
53#64749C2211yescharacteristic
54#64749C2211yescharacteristic
55#64749C2211yescharacteristic
56#64749C2211yescharacteristic
57#64749C2211yescharacteristic
58#64749C2211yescharacteristic
59#64744C2211yescharacteristic
60#64744C2211yescharacteristic
61#26367—51nomaximal, Sylow
62C22×C5×Q8C2311yescharacteristic
63C23×C4×C5C2311yescharacteristic
64C23×Dic5—21no—
65C5×C22⋊Q8—22no—
66C5×C22⋊Q8Q814yes—
67C4×C5×Q8—22no—
68C4×C5×Q8—41no—
69C5×#117—22no—
70C5×#117—41no—
71C2×C5×C4⋊C4—21no—
72C2×C5×C4⋊C4—21no—
73C2×C5×C4⋊C4—21no—
74C2×C5×C4⋊C4C2311yescharacteristic
75C2×C5×C4⋊C4C2311yescharacteristic
76C2×C5×C4⋊C4C2311yescharacteristic
77C2×C5×C22⋊C4—21no—
78C2×C5×C22⋊C4—21no—
79C2×C5×C22⋊C4D411yescharacteristic
80C2×C5×C22⋊C4D411yescharacteristic
81C2×C42×C5—21no—
82C2×C42×C5—21no—
83C2×C42×C5—21no—
84C2×C42×C5C2311yescharacteristic
85C2×C42×C5C2311yescharacteristic
86C2×C22⋊Dic5—21no—
87C2×C4⋊Dic5—22no—
88C2×C4⋊Dic5—21no—
89C2×C4⋊Dic5—41no—
90C2×C4⋊Dic5C2311yescharacteristic
91C2×C4⋊Dic5C2311yescharacteristic
92C2×C4⋊Dic5C2311yescharacteristic
93C2×C4⋊Dic5C2311yescharacteristic
94C2×#683—21no—
95C2×#683—21no—
96C2×#683—21no—
97C2×#683—21no—
98C2×#681—22no—
99C2×#681—22no—
100C2×#681—22no—
101C22×#168C2311yescharacteristic
102C22×#168C2311yescharacteristic
103C22×#168C2311yescharacteristic
104C22×#168C2311yescharacteristic
105#4016—22no—
106#4016—41no—
107C5×#95—21no—
108C5×#95—21no—
109C5×#95—21no—
110C5×#95—21no—
111C5×#95D411yescharacteristic
112C5×#95D411yescharacteristic
113#3967—22no—
114#3967—22no—
115#3967—22no—
116#3967—41no—
117#3963—21no—
118#3963—21no—
119#3963—21no—
120#3963—21no—
121#3963—21no—
122C22⋊#168—22no—
123#3940—22no—
124#3940—41no—
125#3940D414yes—
126#3939—22no—
127#3939—22no—
128#3939—22no—
129C4⋊#168—22no—
130C4⋊#168—22no—
131C2×#479—51no—
132C2×#428—51no—
133#2442—51no—
134#2362D511yescharacteristic
135#2097—51no—
136#2087—51no—
137#2068—51no—
138#2059—51no—
139#1949—51no—
140#1919—51no—
141#1894—51no—
142#1872—51no—
143#1865—51no—
144#1854—51no—
145#1851—51no—
146C24×C5C2×D411yescharacteristic
147C2×C5×Q8—21no—
148C2×C5×Q8—21no—
149C2×C5×Q8—21no—
150C2×C5×Q8—21no—
151C2×C5×Q8—21no—
152C2×C5×Q8C2×Q812yes—
153C22×C4×C5—21no—
154C22×C4×C5—21no—
155C22×C4×C5—21no—
156C22×C4×C5—21no—
157C22×C4×C5—21no—
158C22×C4×C5—21no—
159C22×C4×C5—21no—
160C22×C4×C5—21no—
161C22×C4×C5C2×D411yescharacteristic
162C22×C4×C5C2×D411yescharacteristic
163C22×C4×C5C2×D411yescharacteristic
164C22×C4×C5C2×D411yescharacteristic
165C22×C4×C5C2×D411yescharacteristic
166C22×C4×C5C2×Q812yes—
167C22×C4×C5#4112yes—
168C22×C4×C5C2411yescharacteristic
169C22×Dic5—21no—
170C22×Dic5—21no—
171C22×Dic5—21no—
172C22×Dic5—21no—
173C22×Dic5—42no—
174C22×Dic5—42no—
175C22×Dic5—81no—
176C5×C4⋊C4—24no—
177C5×C4⋊C4—21no—
178C5×C4⋊C4—21no—
179C5×C4⋊C4—21no—
180C5×C4⋊C4—21no—
181C5×C4⋊C4—42no—
182C5×C4⋊C4—42no—
183C5×C4⋊C4C2×D412yes—
184C5×C4⋊C4C2×Q812yes—
185C5×C22⋊C4—24no—
186C5×C22⋊C4—41no—
187C5×C22⋊C4—41no—
188C5×C22⋊C4—42no—
189C42×C5—21no—
190C42×C5—21no—
191C42×C5—21no—
192C42×C5—21no—
193C42×C5—42no—
194C42×C5—42no—
195C42×C5—42no—
196C22⋊Dic5—42no—
197C4⋊Dic5—21no—
198C4⋊Dic5—21no—
199C4⋊Dic5—21no—
200C4⋊Dic5—21no—
201C4⋊Dic5—21no—
202C4⋊Dic5—21no—
203C4⋊Dic5—21no—
204C4⋊Dic5—42no—
205C4⋊Dic5C2×D412yes—
206#683—42no—
207#683—42no—
208#683—81no—
209#683—81no—
210#681—28no—
211C2×#168—21no—
212C2×#168—21no—
213C2×#168—21no—
214C2×#168—21no—
215C2×#168—21no—
216C2×#168—21no—
217C2×#168—21no—
218C2×#168—21no—
219C2×#168—21no—
220C2×#168—21no—
221C2×#168C2×D412yes—
222C2×#168#4112yes—
223C22×M16—51no—
224C2×#125—51no—
225C2×C22⋊Q8C2×D511yescharacteristic
226C2×C4×Q8C2×D511yescharacteristic
227C2×#117C2×D511yescharacteristic
228C22×C4⋊C4—51no—
229#479—104no—
230#428—104no—
231C2×#107—51no—
232C2×#107—51no—
233C2×#107—51no—
234C2×#106—51no—
235C2×#106—51no—
236C2×#106—51no—
237C2×C4⋊C8—51no—
238C2×C4⋊C8—51no—
239C2×#103—51no—
240C2×#103—51no—
241C2×#103—51no—
242C2×#103—51no—
243C2×C22⋊C8—51no—
244#400—101no—
245#398—101no—
246#391C2×D511yescharacteristic
247#389—51no—
248#389—51no—
249#389C2×D511yescharacteristic
250#388—21no—
251#388—21no—
252#387—21no—
253#385—101no—
254#382—51no—
255#380—51no—
256#380—101no—
257C4×C4⋊C4—21no—
258C4×C4⋊C4C2×D511yescharacteristic
259C4×C22⋊C4C2×D511yescharacteristic
260#340—51no—
261#340—51no—
262#340—51no—
263#340—51no—
264#340—51no—
265#340—51no—
266#336—51no—
267#336—51no—
268C23×C5—21no—
269C23×C5—21no—
270C23×C5—21no—
271C23×C5—21no—
272C23×C5—21no—
273C23×C5—21no—
274C23×C5C22⋊Q812yes—
275C23×C5C22×D411yescharacteristic
276C5×Q8—22no—
277C5×Q8—22no—
278C5×Q8—41no—
279C5×Q8—41no—
280C2×C4×C5—21no—
281C2×C4×C5—21no—
282C2×C4×C5—22no—
283C2×C4×C5—22no—
284C2×C4×C5—22no—
285C2×C4×C5—22no—
286C2×C4×C5—22no—
287C2×C4×C5—22no—
288C2×C4×C5—21no—
289C2×C4×C5—21no—
290C2×C4×C5—22no—
291C2×C4×C5—21no—
292C2×C4×C5—21no—
293C2×C4×C5—21no—
294C2×C4×C5—21no—
295C2×C4×C5—21no—
296C2×C4×C5—21no—
297C2×C4×C5—21no—
298C2×C4×C5—21no—
299C2×C4×C5—21no—
300C2×C4×C5—21no—
301C2×C4×C5—41no—
302C2×C4×C5—41no—
303C2×C4×C5—41no—
304C2×C4×C5—41no—
305C2×C4×C5—41no—
306C2×C4×C5—41no—
307C2×C4×C5—41no—
308C2×C4×C5—41no—
309C2×C4×C5—41no—
310C2×C4×C5C22⋊Q812yes—
311C2×C4×C5C22×D411yescharacteristic
312C2×C4×C5C22×Q811yescharacteristic
313C2×C4×C5C2×#4111yescharacteristic
314C2×C4×C5C2×#4111yescharacteristic
315C2×C4×C5H(5,𝔽2)11yescharacteristic
316C2×C4×C52−1+411yescharacteristic
317C2×Dic5—22no—
318C2×Dic5—41no—
319C2×Dic5—41no—
320C2×Dic5—41no—
321C2×Dic5—41no—
322C2×Dic5—41no—
323C2×Dic5—41no—
324C2×Dic5—41no—
325C2×Dic5—41no—
326C2×Dic5—41no—
327C2×Dic5—41no—
328C2×Dic5—41no—
329C2×Dic5—84no—
330#168—22no—
331#168—22no—
332#168—41no—
333#168—41no—
334C22×Q8C22×D511yescharacteristic
335C23×C4—101no—
336C23×C4C22×D511yescharacteristic
337C2×M16—102no—
338C2×M16—102no—
339C2×M16—102no—
340C22×C8—51no—
341C22×C8—51no—
342C22×C8—51no—
343C22×C8—51no—
344#125—102no—
345#125—201no—
346C22⋊Q8—22no—
347C22⋊Q8Dic1014yes—
348C4×Q8—22no—
349C4×Q8—41no—
350#117—22no—
351#117—41no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—51no—
356C2×C4⋊C4—51no—
357C2×C4⋊C4—51no—
358C2×C4⋊C4—51no—
359C2×C4⋊C4—102no—
360C2×C4⋊C4—101no—
361C2×C4⋊C4—101no—
362C2×C4⋊C4—101no—
363C2×C4⋊C4—101no—
364C2×C4⋊C4—101no—
365C2×C4⋊C4—201no—
366C2×C4⋊C4C22×D511yescharacteristic
367C2×C4⋊C4C22×D511yescharacteristic
368C2×C4⋊C4C22×D511yescharacteristic
369C2×C22⋊C4—21no—
370C2×C22⋊C4—21no—
371C2×C22⋊C4—101no—
372C2×C22⋊C4C22⋊D511yescharacteristic
373C2×C22⋊C4C22⋊D511yescharacteristic
374C2×C42—21no—
375C2×C42—21no—
376C2×C42—21no—
377C2×C42C22×D511yescharacteristic
378C2×C42C22×D511yescharacteristic
379#107—102no—
380#107—102no—
381#107—102no—
382#106—54no—
383#106—102no—
384#106—201no—
385C4⋊C8—102no—
386C4⋊C8—102no—
387#103—102no—
388#103—102no—
389#103—102no—
390#103—201no—
391C22⋊C8—102no—
392#95—21no—
393#95—21no—
394#95—21no—
395#95—21no—
396#95—101no—
397#95—101no—
398#95—101no—
399#95—101no—
400#95—101no—
401#95C22⋊D511yescharacteristic
402#95C22⋊D511yescharacteristic
403C22×C5—22no—
404C22×C5—22no—
405C22×C5—22no—
406C22×C5—41no—
407C22×C5—41no—
408C22×C5—41no—
409C22×C5—41no—
410C22×C5C2×C22⋊Q811yescharacteristic
411C22×C5D4211yescharacteristic
412C22×C5#54711yescharacteristic
413C22×C5#57511yescharacteristic
414C22×C5#57511yescharacteristic
415C22×C5#57711yescharacteristic
416C22×C5#57811yescharacteristic
417C4×C5—22no—
418C4×C5—42no—
419C4×C5—41no—
420C4×C5—41no—
421C4×C5—42no—
422C4×C5—41no—
423C4×C5—42no—
424C4×C5—41no—
425C4×C5—81no—
426C4×C5#52511yescharacteristic
427C4×C5#53911yescharacteristic
428C4×C5D4×Q811yescharacteristic
429C4×C5#55411yescharacteristic
430Dic5—42no—
431Dic5—81no—
432Dic5—81no—
433Dic5—81no—
434C24C2×C22⋊D511yescharacteristic
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8C2×Dic1012yes—
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—101no—
450C22×C4—101no—
451C22×C4—101no—
452C22×C4—101no—
453C22×C4—202no—
454C22×C4—202no—
455C22×C4—401no—
456C22×C4C2×Dic1012yes—
457C22×C4#70912yes—
458C22×C4C2×C22⋊D511yescharacteristic
459C22×C4C2×C22⋊D511yescharacteristic
460C22×C4C2×C22⋊D511yescharacteristic
461C22×C4C2×C22⋊D511yescharacteristic
462C22×C4C2×C22⋊D511yescharacteristic
463C22×C4C23×D511yescharacteristic
464M16—108no—
465C2×C8—52no—
466C2×C8—52no—
467C2×C8—101no—
468C2×C8—101no—
469C2×C8—101no—
470C2×C8—101no—
471C2×C8—101no—
472C2×C8—101no—
473C2×C8—101no—
474C2×C8—101no—
475C2×C8—101no—
476C2×C8—101no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—24no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—52no—
485C4⋊C4—101no—
486C4⋊C4—101no—
487C4⋊C4—101no—
488C4⋊C4—101no—
489C4⋊C4—101no—
490C4⋊C4—101no—
491C4⋊C4—101no—
492C4⋊C4—202no—
493C4⋊C4—202no—
494C4⋊C4—202no—
495C4⋊C4—401no—
496C4⋊C4—401no—
497C4⋊C4C2×Dic1012yes—
498C4⋊C4D4×D512yes—
499C22⋊C4—24no—
500C22⋊C4—41no—
501C22⋊C4—41no—
502C22⋊C4—42no—
503C22⋊C4—202no—
504C42—21no—
505C42—21no—
506C42—21no—
507C42—21no—
508C42—42no—
509C42—42no—
510C42—42no—
511C2×C5—42no—
512C2×C5#244911yescharacteristic
513C2×C5#308011yescharacteristic
514C2×C5#308211yescharacteristic
515C2×C5#326911yescharacteristic
516C2×C5#327511yescharacteristic
517C2×C5#334911yescharacteristic
518C2×C5#335111yescharacteristic
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23#407012yes—
526C23C22×C22⋊D511yescharacteristic
527Q8—22no—
528Q8—22no—
529Q8—41no—
530Q8—41no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—102no—
562C2×C4—201no—
563C2×C4—201no—
564C2×C4—201no—
565C2×C4—201no—
566C2×C4—201no—
567C2×C4—201no—
568C2×C4—201no—
569C2×C4—201no—
570C2×C4—201no—
571C2×C4—201no—
572C2×C4—201no—
573C2×C4—404no—
574C2×C4#407012yes—
575C2×C4C22×Dic1011yescharacteristic
576C2×C4C2×#70911yescharacteristic
577C2×C4C2×D4×D511yescharacteristic
578C2×C4#414411yescharacteristic
579C2×C4D5×#4111yescharacteristic
580C2×C4#415011yescharacteristic
581C8—102no—
582C8—102no—
583C8—201no—
584C8—201no—
585C5#2636711yescharacteristic, complemented, Sylow
586C22—22no—
587C22—22no—
588C22—22no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22C2×#407011yescharacteristic
594C22D4×C22⋊D511yescharacteristic
595C22#6615611yescharacteristic
596C22#6615611yescharacteristic
597C22#6615811yescharacteristic
598C22#6617111yescharacteristic
599C22#6617211yescharacteristic
600C4—22no—
601C4—41no—
602C4—42no—
603C4—41no—
604C4—41no—
605C4—42no—
606C4—42no—
607C4—41no—
608C4—81no—
609C4—202no—
610C4—401no—
611C4—401no—
612C4—401no—
613C4D4×Dic1011yescharacteristic
614C4#6587311yescharacteristic
615C4#6588111yescharacteristic
616C4#6589811yescharacteristic
617C2—42no—
618C2#1061409111yescharacteristic
619C2#1061424011yescharacteristic
620C2#1061428111yescharacteristic
621C2#1061525911yescharacteristic
622C2#1061527511yescharacteristic
623C2#1061532711yescharacteristic
624C2#1061539111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D5
commutator, cosocleC22×C4×C5C24
FrattiniC22×C4C23×D5
FittingC5×#2362C2
socleC23×C5C22×D4
prime radicalC24×C5C2×D4
hypercenter#2362D5
hypocenterC5#26367
LCSG ▷ C22×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2362

Complements

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

NH ≅ G/Ntype
2. C2×#65311C2semidirect
7. #10602422C2semidirect
9. #10600479C2semidirect
10. #10600476C2semidirect
11. #10600416C2semidirect
12. #10600399C2semidirect
13. #10600261C2semidirect
14. #10600252C2semidirect
585. C5#26367semidirect

Maximal subgroups

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

subgroupindex
C5×#23622
C2×#653112
C2×#652842
#106052262
#106024902
#106024852
#106024222
#106023692
#106004792
#106004762
#106004162
#106003992
#106002612
#106002522
#106002432
#263675