Finite Group Databasebeta

← #12107154

Subgroups — #12107154

Order 1280.

730 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10610756C211yesmaximal, characteristic
2#10610217C211yesmaximal, characteristic
3#10610211C211yesmaximal, characteristic, complemented
4C5×#1771C211yesmaximal, characteristic, complemented
5#10601267C211yesmaximal, characteristic, complemented
6#10600632C211yesmaximal, characteristic
7#10600622C211yesmaximal, characteristic, complemented
8#10600582C211yesmaximal, characteristic
9#10600572C211yesmaximal, characteristic, complemented
10#10600287C211yesmaximal, characteristic
11#10600286C211yesmaximal, characteristic, complemented
12#10599487C211yesmaximal, characteristic
13#10599487C211yesmaximal, characteristic
14#10599438C211yesmaximal, characteristic, complemented
15#10599438C211yesmaximal, characteristic, complemented
16C2×#4047C2211yescharacteristic
17C2×#4047C2211yescharacteristic
18C4×#709C2211yescharacteristic
19#65804C2211yescharacteristic
20C2×C5×C4⋊C8C2211yescharacteristic
21C2×C5×#97C2211yescharacteristic
22C2×C5×#97C2211yescharacteristic
23C2×C4×C5×C8C2211yescharacteristic
24C4×C5×C4⋊C4C2211yescharacteristic
25#65402C2211yescharacteristic
26#65402C2211yescharacteristic
27C2×#3951C2211yescharacteristic
28C2×#3951C2211yescharacteristic
29#65395C2211yescharacteristic
30#65395C2211yescharacteristic
31#65391C2211yescharacteristic
32#65391C2211yescharacteristic
33#65390C2211yescharacteristic
34#65390C2211yescharacteristic
35#65390C2211yescharacteristic
36#65390C2211yescharacteristic
37C2×#3937C2211yescharacteristic
38C2×#3937C2211yescharacteristic
39#65213C2211yescharacteristic
40C4×#681C2211yescharacteristic
41#65140C411yescharacteristic, complemented
42#65139C411yescharacteristic, complemented
43#65136C411yescharacteristic, complemented
44#65136C411yescharacteristic, complemented
45#65135C411yescharacteristic, complemented
46#65131C411yescharacteristic, complemented
47#65118C411yescharacteristic, complemented
48#65118C411yescharacteristic, complemented
49#64997—21no—
50#64997—21no—
51#64996—21no—
52D5×#97—21no—
53#64990—21no—
54#64990—21no—
55#64990—21no—
56#64990—21no—
57C5×#336C2211yescharacteristic
58C5×#336C2211yescharacteristic
59#64772C2211yescharacteristic
60#64772C2211yescharacteristic
61#64743C2211yescharacteristic
62#64743C2211yescharacteristic
63#64743C2211yescharacteristic
64#64743C2211yescharacteristic
65#64743C2211yescharacteristic
66#64743C2211yescharacteristic
67#21629—51nomaximal, Sylow
68C2×#709C2311yescharacteristic
69C22×C5×C8C2311yescharacteristic
70C22×C5×C8C2311yescharacteristic
71C22×C5×C8C2311yescharacteristic
72C22×C5×C8C2311yescharacteristic
73C2×C5×C4⋊C4—21no—
74C2×C42×C5C2311yescharacteristic
75C2×C42×C5C2311yescharacteristic
76C2×C42×C5C2311yescharacteristic
77C4×C22⋊D5—21no—
78C4×C22⋊D5—21no—
79#4072C2311yescharacteristic
80#4072C2311yescharacteristic
81#4072C2311yescharacteristic
82#4070—21no—
83#4070—21no—
84C2×#681C2311yescharacteristic
85C2×#681C2311yescharacteristic
86C2×#681C2311yescharacteristic
87C2×#681C2311yescharacteristic
88#4047—22no—
89#4047—22no—
90#4047—22no—
91#4047Q814yescomplemented
92C2×#676—21no—
93C2×#676—21no—
94C2×C8×D5—21no—
95C2×C8×D5—21no—
96C4×D20C2×C411yescharacteristic
97C4×D20C2×C411yescharacteristic
98#4018—21no—
99C42×D5—21no—
100#4016C2×C411yescharacteristic
101#4016C2×C411yescharacteristic
102#4015C2×C411yescharacteristic
103#4015C2×C411yescharacteristic
104C4×Dic10—21no—
105C4×Dic10—21no—
106C4×Dic10C2×C411yescharacteristic
107C4×Dic10C2×C411yescharacteristic
108C5×C4⋊C8C2×C411yescharacteristic
109C5×C4⋊C8C2×C411yescharacteristic
110C5×C4⋊C8C2×C411yescharacteristic
111C5×C4⋊C8C2×C411yescharacteristic
112C5×#97—22no—
113C5×#97D411yescharacteristic
114C5×#97D411yescharacteristic
115C5×#97D411yescharacteristic
116C5×#97D411yescharacteristic
117C4×C5×C8—22no—
118C5×#95—21no—
119#3953C2×C411yescharacteristic
120#3953C2×C411yescharacteristic
121#3953C2×C411yescharacteristic
122#3953C2×C411yescharacteristic
123#3951—22no—
124#3951—22no—
125#3950C2×C411yescharacteristic
126#3950C2×C411yescharacteristic
127#3950C2×C411yescharacteristic
128#3950C2×C411yescharacteristic
129#3949C2×C411yescharacteristic
130#3949C2×C411yescharacteristic
131#3949C2×C411yescharacteristic
132#3949C2×C411yescharacteristic
133#3948C2×C411yescharacteristic
134#3948C2×C411yescharacteristic
135#3948C2×C411yescharacteristic
136#3948C2×C411yescharacteristic
137#3947—21no—
138#3947—21no—
139#3947—21no—
140#3947—21no—
141#3946—21no—
142#3946—21no—
143#3946—21no—
144#3946—21no—
145C8×Dic5—21no—
146C8×Dic5—21no—
147C8×Dic5—21no—
148C8×Dic5—21no—
149#3937—22no—
150#3937—22no—
151C4⋊#168—21no—
152C4⋊#168—21no—
153#3935—21no—
154C4×#168—21no—
155#3076—51no—
156#2961—51no—
157#2875—51no—
158#1986—51no—
159#1950—51no—
160#1922—51no—
161#1922—51no—
162#1866—51no—
163#1866—51no—
164#1806—51no—
165#1773—51no—
166#1772—51no—
167#1771D511yescharacteristic, complemented
168#1765—51no—
169#1764—51no—
170C22×C4×C5—21no—
171C22×C4×C5—21no—
172C22×C4×C5#4111yescharacteristic
173C22×C4×C5#4111yescharacteristic
174C22×C4×C5C2411yescharacteristic
175C2×C22⋊D5—21no—
176#709—21no—
177#709—21no—
178#709C2×Q812yes—
179#709#4112yes—
180C2×D20C22×C411yescharacteristic
181C2×C4×D5—21no—
182C2×C4×D5—21no—
183C2×C4×D5—21no—
184C2×Dic10—21no—
185C2×Dic10—21no—
186C2×Dic10C22×C411yescharacteristic
187C2×C5×C8—21no—
188C2×C5×C8—21no—
189C2×C5×C8—21no—
190C2×C5×C8—21no—
191C2×C5×C8—21no—
192C2×C5×C8—21no—
193C2×C5×C8C22×C411yescharacteristic
194C2×C5×C8C22×C411yescharacteristic
195C2×C5×C8C22×C411yescharacteristic
196C2×C5×C8C22×C411yescharacteristic
197C2×C5×C8C2×D411yescharacteristic
198C2×C5×C8C2×D411yescharacteristic
199C2×C5×C8C2×D411yescharacteristic
200C2×C5×C8C2×D411yescharacteristic
201C2×C5×C8C2×Q812yes—
202C2×C5×C8#4112yes—
203C5×C4⋊C4—42no—
204C42×C5—21no—
205C42×C5—21no—
206C42×C5—21no—
207C42×C5—21no—
208C42×C5C22×C411yescharacteristic
209C42×C5C22×C411yescharacteristic
210C42×C5C2×D411yescharacteristic
211C42×C5C2×D411yescharacteristic
212C22⋊Dic5—21no—
213C22⋊Dic5—21no—
214C22⋊Dic5—21no—
215#685—21no—
216#685—21no—
217C4⋊Dic5—21no—
218C4⋊Dic5—21no—
219C4⋊Dic5C22×C411yescharacteristic
220C4⋊Dic5C22×C411yescharacteristic
221C4⋊Dic5C22×C411yescharacteristic
222C4⋊Dic5C22×C411yescharacteristic
223C4⋊Dic5C22×C411yescharacteristic
224C4⋊Dic5C22×C411yescharacteristic
225#683—21no—
226#683—21no—
227#683—21no—
228#683—21no—
229#683—21no—
230#683—21no—
231C4×Dic5—21no—
232C4×Dic5—21no—
233C4×Dic5—21no—
234C4×Dic5—21no—
235C4×Dic5—21no—
236#681—21no—
237#681—21no—
238#681—21no—
239#681—21no—
240#681—21no—
241#681—21no—
242#681C2×Q812yes—
243#681#4112yes—
244C2×#168—21no—
245C2×#168—21no—
246C2×#168—21no—
247C2×#168—21no—
248#676—24no—
249#676—24no—
250C8×D5—24no—
251C8×D5—24no—
252C2×#131—51no—
253C2×#131—51no—
254#533—51no—
255C4×#41—51no—
256#479—51no—
257#479—51no—
258#478—51no—
259#478—51no—
260#477—51no—
261#476—51no—
262#475—51no—
263#474—51no—
264#446—101no—
265#446—101no—
266#446—101no—
267#446—101no—
268C2×#108—51no—
269C2×#108—51no—
270#427—51no—
271#427—51no—
272#424—51no—
273#424—51no—
274#423—51no—
275C2×C4⋊C8C2×D511yescharacteristic
276C2×#104—51no—
277C2×#104—51no—
278#416—51no—
279#416—51no—
280#405—51no—
281#405—51no—
282#405—51no—
283#405—51no—
284#405—101no—
285#405—101no—
286C4×M16—51no—
287C4×M16—101no—
288C2×#97—101no—
289C2×#97C2×D511yescharacteristic
290C2×#97C2×D511yescharacteristic
291C2×C4×C8C2×D511yescharacteristic
292C4×C4⋊C4C2×D511yescharacteristic
293#341—51no—
294#341—51no—
295#339—51no—
296#339—51no—
297#339—51no—
298#339—51no—
299#339—51no—
300#339—51no—
301#336C2×D511yescharacteristic
302#336C2×D511yescharacteristic
303C23×C5C2×#4111yescharacteristic
304C22×D5—21no—
305C2×C4×C5—22no—
306C2×C4×C5—41no—
307C2×C4×C5—41no—
308C2×C4×C5—41no—
309C2×C4×C5—41no—
310C2×C4×C5—41no—
311C2×C4×C5—41no—
312C2×C4×C5—41no—
313C2×C4×C5C4×D411yescharacteristic
314C2×C4×C5C4×D411yescharacteristic
315C2×C4×C5C4×D411yescharacteristic
316C2×C4×C5C4×D411yescharacteristic
317C2×C4×C5C23×C411yescharacteristic
318C2×C4×C5C22×D411yescharacteristic
319C2×C4×C5C22×Q811yescharacteristic
320C2×C4×C5C2×#4111yescharacteristic
321C2×C4×C5C2×#4111yescharacteristic
322C2×C4×C5C2×#4111yescharacteristic
323C22⋊D5—22no—
324C22⋊D5—22no—
325C2×Dic5—21no—
326C2×Dic5—21no—
327C2×Dic5—21no—
328C2×Dic5—21no—
329C2×Dic5—21no—
330C2×Dic5—21no—
331C2×Dic5—21no—
332D20—21no—
333D20C4×Q812yes—
334C4×D5—22no—
335C4×D5—22no—
336C4×D5—81no—
337C4×D5—81no—
338Dic10—21no—
339Dic10—81no—
340Dic10—81no—
341Dic10C4×Q812yes—
342C5×C8—21no—
343C5×C8—21no—
344C5×C8—21no—
345C5×C8—21no—
346C5×C8—21no—
347C5×C8—21no—
348C5×C8C22⋊Q812yes—
349C5×C8C22⋊Q812yes—
350#168—22no—
351#168—22no—
352#168—41no—
353#168—41no—
354C2×#41—51no—
355#131—54no—
356#131—102no—
357#131—102no—
358#131—102no—
359C2×M16—51no—
360C2×M16—51no—
361C2×M16—51no—
362C2×M16—51no—
363C2×M16—101no—
364C2×M16—101no—
365C22×C8—101no—
366C22×C8—101no—
367C22×C8C22×D511yescharacteristic
368C22×C8C22×D511yescharacteristic
369C22×C8C22×D511yescharacteristic
370C22×C8C22×D511yescharacteristic
371C4⋊Q8—51no—
372C4⋊Q8—51no—
373#125—51no—
374#125—51no—
375C22⋊Q8—101no—
376C22⋊Q8—101no—
377C4×Q8—51no—
378C4×Q8—51no—
379C4×Q8—101no—
380C4×Q8—101no—
381C4×D4—51no—
382C4×D4—51no—
383C4×D4—101no—
384C4×D4—101no—
385#117—51no—
386#117—51no—
387#117—51no—
388#117—101no—
389C2×C4⋊C4—21no—
390C2×C42—101no—
391C2×C42C22×D511yescharacteristic
392C2×C42C22×D511yescharacteristic
393C2×C42C22×D511yescharacteristic
394#108—102no—
395#108—102no—
396#107—51no—
397#107—51no—
398#107—51no—
399#107—51no—
400#106—51no—
401#106—51no—
402#106—51no—
403#106—51no—
404C4⋊C8—101no—
405C4⋊C8—101no—
406C4⋊C8—101no—
407C4⋊C8—101no—
408C4⋊C8—101no—
409C4⋊C8—101no—
410C4⋊C8C4×D511yescharacteristic
411C4⋊C8C4×D511yescharacteristic
412C4⋊C8C4×D511yescharacteristic
413C4⋊C8C4×D511yescharacteristic
414#104—102no—
415#104—102no—
416#103—51no—
417#103—51no—
418#103—51no—
419#103—51no—
420#102—51no—
421#102—51no—
422#102—51no—
423#102—51no—
424#97—22no—
425#97—101no—
426#97—101no—
427#97—101no—
428#97—101no—
429#97—101no—
430#97D2011yescharacteristic
431#97D2011yescharacteristic
432#97D2011yescharacteristic
433#97D2011yescharacteristic
434C4×C8—22no—
435C4×C8—101no—
436C4×C8—101no—
437C4×C8—101no—
438C4×C8—101no—
439C4×C8—101no—
440#95—21no—
441C22×C5—21no—
442C22×C5—21no—
443C22×C5C2×C4×D411yescharacteristic
444C22×C5#52911yescharacteristic
445C22×C5#53311yescharacteristic
446C2×D5—22no—
447C2×D5—41no—
448C2×D5—41no—
449C4×C5—21no—
450C4×C5—21no—
451C4×C5—21no—
452C4×C5—21no—
453C4×C5—42no—
454C4×C5—81no—
455C4×C5C2×C4×Q811yescharacteristic
456C4×C5C4×#4111yescharacteristic
457C4×C5C2×C22⋊Q811yescharacteristic
458C4×C5#52511yescharacteristic
459Dic5—22no—
460Dic5—41no—
461Dic5—41no—
462Dic5—41no—
463Dic5—41no—
464Dic5—41no—
465Dic5—41no—
466#41—52no—
467#41—52no—
468#41—101no—
469#41—101no—
470C2×Q8—51no—
471C2×Q8—101no—
472C2×Q8—101no—
473C2×D4—51no—
474C2×D4—101no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—101no—
478C22×C4—101no—
479C22×C4—101no—
480C22×C4#70911yescharacteristic
481C22×C4#70911yescharacteristic
482C22×C4C23×D511yescharacteristic
483M16—52no—
484M16—52no—
485M16—104no—
486M16—101no—
487M16—101no—
488M16—104no—
489M16—101no—
490M16—101no—
491M16—101no—
492M16—101no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—104no—
500C2×C8—101no—
501C2×C8—101no—
502C2×C8—104no—
503C2×C8—101no—
504C2×C8—101no—
505C2×C8C2×C4×D511yescharacteristic
506C2×C8C2×C4×D511yescharacteristic
507C2×C8C2×C4×D511yescharacteristic
508C2×C8C2×C4×D511yescharacteristic
509C2×C8C2×D2011yescharacteristic
510C2×C8C2×D2011yescharacteristic
511C2×C8C2×D2011yescharacteristic
512C2×C8C2×D2011yescharacteristic
513C2×C8#71112yes—
514C2×C8Q8×D512yes—
515C4⋊C4—42no—
516C4⋊C4—51no—
517C4⋊C4—51no—
518C4⋊C4—51no—
519C4⋊C4—51no—
520C4⋊C4—51no—
521C4⋊C4—51no—
522C4⋊C4—101no—
523C4⋊C4—101no—
524C4⋊C4—101no—
525C4⋊C4—101no—
526C4⋊C4—101no—
527C4⋊C4—101no—
528C4⋊C4—101no—
529C4⋊C4—101no—
530C22⋊C4—101no—
531C22⋊C4—101no—
532C22⋊C4—101no—
533C22⋊C4—101no—
534C22⋊C4—101no—
535C42—21no—
536C42—21no—
537C42—21no—
538C42—21no—
539C42—101no—
540C42—101no—
541C42—101no—
542C42—101no—
543C42—101no—
544C42C2×C4×D511yescharacteristic
545C42C2×C4×D511yescharacteristic
546C42C2×D2011yescharacteristic
547C42C2×D2011yescharacteristic
548C2×C5—21no—
549C2×C5—21no—
550C2×C5C4×C22⋊Q811yescharacteristic
551C2×C5#295611yescharacteristic
552C2×C5#295711yescharacteristic
553D5—42no—
554C23—101no—
555C23C2×#70911yescharacteristic
556Q8—52no—
557Q8—101no—
558Q8—401no—
559Q8—401no—
560D4—52no—
561D4—102no—
562D4—102no—
563D4—101no—
564C2×C4—22no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—101no—
573C2×C4—102no—
574C2×C4—102no—
575C2×C4—101no—
576C2×C4—101no—
577C2×C4—101no—
578C2×C4—101no—
579C2×C4—101no—
580C2×C4—101no—
581C2×C4—401no—
582C2×C4—401no—
583C2×C4C4×D2011yescharacteristic
584C2×C4C4×D2011yescharacteristic
585C2×C4C4×D2011yescharacteristic
586C2×C4C4×D2011yescharacteristic
587C2×C4C22×C4×D511yescharacteristic
588C2×C4C22×D2011yescharacteristic
589C2×C4C2×#71111yescharacteristic
590C2×C4C2×Q8×D511yescharacteristic
591C2×C4D5×#4111yescharacteristic
592C2×C4D5×#4111yescharacteristic
593C8—21no—
594C8—21no—
595C8—21no—
596C8—21no—
597C8—21no—
598C8—21no—
599C8—102no—
600C8—102no—
601C8—201no—
602C8—201no—
603C8#404312yes—
604C8#404312yes—
605C5#2162911yescharacteristic, complemented, Sylow
606C22—21no—
607C22—21no—
608C22—102no—
609C22—201no—
610C22—201no—
611C22C2×C4×D2011yescharacteristic
612C22#6589111yescharacteristic
613C22#6591211yescharacteristic
614C4—21no—
615C4—21no—
616C4—21no—
617C4—21no—
618C4—42no—
619C4—81no—
620C4—102no—
621C4—201no—
622C4—201no—
623C4—201no—
624C4—201no—
625C4—201no—
626C4—201no—
627C4C2×#404311yescharacteristic
628C4#6585811yescharacteristic
629C4C4×#71111yescharacteristic
630C4C4×Q8×D511yescharacteristic
631C2—21no—
632C2—21no—
633C2—202no—
634C2C4×#404311yescharacteristic
635C2#1061027211yescharacteristic
636C2#1061031111yescharacteristic
subgroupG/H
centerC2×C4C22×D20
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC5×#1771C2
socleC22×C5C2×C4×D4
prime radicalC2×#709C23
cosocleC22×C4×C5C24
hypercenter#1771D5
hypocenterC5#21629
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C4 ◁ C2×#97 ◁ #1771

Complements

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

NH ≅ G/Ntype
3. #10610211C2semidirect
4. C5×#1771C2semidirect
5. #10601267C2semidirect
7. #10600622C2semidirect
9. #10600572C2semidirect
11. #10600286C2semidirect
14. #10599438C2semidirect
15. #10599438C2semidirect
41. #65140C4semidirect
42. #65139C4semidirect
43. #65136C4semidirect
44. #65136C4semidirect
45. #65135C4semidirect
46. #65131C4semidirect
47. #65118C4semidirect
48. #65118C4semidirect
91. #4047Q8semidirect
167. #1771D5semidirect
605. C5#21629semidirect

Maximal subgroups

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

subgroupindex
#106107562
#106102172
#106102112
C5×#17712
#106012672
#106006322
#106006222
#106005822
#106005722
#106002872
#106002862
#105994872
#105994382
#216295