Finite Group Databasebeta

← #12100191

Subgroups — #12100191

Order 1280.

1394 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10619161C211yesmaximal, characteristic
2C2×C5×#421C211yesmaximal, characteristic
3C2×#65521C211yesmaximal, characteristic
4C2×#65410C211yesmaximal, characteristic, complemented
5#10613964C211yesmaximal, characteristic, complemented
6#10602780C211yesmaximal, characteristic, complemented
7#10602371C211yesmaximal, characteristic, complemented
8#10601157C214yesmaximal, complemented
9#10601156C214yesmaximal, complemented
10C22×C5×#41C2211yescharacteristic
11C22×C5×M16C2211yescharacteristic
12C2×C5×#117C2211yescharacteristic
13#66161—21no—
14#66161—21no—
15#66161C2212yes—
16#66161C2212yes—
17#41×Dic5—22no—
18#41×Dic5—22no—
19#66146—21no—
20#66132—21no—
21#66132—22no—
22C2×#4072—22no—
23C2×#4072C2211yescharacteristic
24C2×#4072C2211yescharacteristic
25C22×C4⋊Dic5C2211yescharacteristic
26C22×#681C2211yescharacteristic
27C5×#421C418yescomplemented
28C2×C5×#104C2214yes—
29#65521—22no—
30#65521C2214yes—
31C2×#3967—21no—
32C2×#3964—21no—
33#65410—22no—
34#65410C2214yescomplemented
35C2×#3950—21no—
36C2×#3949—21no—
37#65274—22no—
38#65239—22no—
39#65225C2212yescomplemented
40#65224C2212yescomplemented
41#64774C2212yescomplemented
42#64774C2212yescomplemented
43#64743C2214yescomplemented
44#20865—51nomaximal, Sylow
45C2×C5×#41—21no—
46C2×C5×#41—21no—
47C2×C5×#41—22no—
48C2×C5×#41—22no—
49C2×C5×#41C2×C414yes—
50C2×C5×#41D412yes—
51C2×C5×#41Q814yes—
52C2×C5×#41C2312yes—
53C22×C5×Q8D411yescharacteristic
54C22×C5×D4—21no—
55C22×C5×D4D411yescharacteristic
56C23×C4×C5—21no—
57C23×C4×C5C2311yescharacteristic
58C23×Dic5—21no—
59C2×C5×M16—21no—
60C2×C5×M16—22no—
61C2×C5×M16C2×C414yescomplemented
62C2×C5×M16C2312yes—
63C22×C5×C8—21no—
64C5×#117—22no—
65C5×#117C2×C414yescomplemented
66C2×C5×C4⋊C4—21no—
67C2×C5×C22⋊C4—21no—
68C2×C42×C5C2312yes—
69C2×C22⋊Dic5—24no—
70C2×C22⋊Dic5—21no—
71C2×C22⋊Dic5D412yes—
72Q8×Dic5—22no—
73Q8×Dic5—22no—
74Q8×Dic5—22no—
75Q8×Dic5—22no—
76D4×Dic5—24no—
77D4×Dic5—24no—
78D4×Dic5—22no—
79D4×Dic5—22no—
80D4×Dic5—22no—
81D4×Dic5—22no—
82D4×Dic5—24no—
83D4×Dic5—24no—
84#4072—24no—
85#4072—24no—
86#4072—21no—
87#4072—21no—
88#4072—21no—
89#4072—21no—
90#4072—21no—
91#4072—21no—
92#4072—22no—
93#4072—22no—
94#4072D412yes—
95#4072C2312yes—
96C2×C4⋊Dic5—22no—
97C2×C4⋊Dic5—22no—
98C2×C4⋊Dic5—21no—
99C2×C4⋊Dic5—21no—
100C2×C4⋊Dic5—21no—
101C2×C4⋊Dic5—21no—
102C2×C4⋊Dic5—21no—
103C2×C4⋊Dic5—24no—
104C2×C4⋊Dic5D411yescharacteristic
105C2×C4⋊Dic5D411yescharacteristic
106C2×C4⋊Dic5C2312yes—
107C2×C4⋊Dic5C2311yescharacteristic
108C2×C4⋊Dic5C2311yescharacteristic
109C2×C4×Dic5—24no—
110C2×C4×Dic5—21no—
111C2×C4×Dic5D412yes—
112C2×#681—21no—
113C2×#681—22no—
114C2×#681—22no—
115C2×#681C2312yes—
116C22×#168—21no—
117C5×#104C2×C4116yescomplemented
118#3967—24no—
119#3964—24no—
120#3963—24no—
121#3950—24no—
122#3949—24no—
123#3428—51no—
124C2×#428—51no—
125C2×#421D511yescharacteristic
126C2×#417—51no—
127#2317—51no—
128#1860—54no—
129#1859—54no—
130#1742—51no—
131#1733—51no—
132C24×C5—21no—
133C24×C5C2×D411yescharacteristic
134C5×#41—24no—
135C5×#41—24no—
136C5×#41—24no—
137C5×#41—24no—
138C5×#41—24no—
139C5×#41—22no—
140C5×#41—22no—
141C5×#41C4⋊C418yescomplemented
142C5×#41C22×C414yes—
143C5×#41C2×Q814yes—
144C2×C5×Q8—21no—
145C2×C5×Q8—22no—
146C2×C5×Q8C4⋊C414yescomplemented
147C2×C5×Q8C2×D412yes—
148C2×C5×D4—24no—
149C2×C5×D4—22no—
150C2×C5×D4—24no—
151C2×C5×D4—22no—
152C2×C5×D4—21no—
153C2×C5×D4—22no—
154C2×C5×D4C4⋊C414yescomplemented
155C2×C5×D4C2×D412yes—
156C22×C4×C5—21no—
157C22×C4×C5—24no—
158C22×C4×C5—22no—
159C22×C4×C5—24no—
160C22×C4×C5—22no—
161C22×C4×C5—22no—
162C22×C4×C5C22×C412yes—
163C22×C4×C5C2×D411yescharacteristic
164C22×C4×C5C2×D411yescharacteristic
165C22×C4×C5C2×D411yescharacteristic
166C22×C4×C5C2×D411yescharacteristic
167C22×C4×C5C2×D411yescharacteristic
168C22×C4×C5C2×Q812yes—
169C22×C4×C5#4112yes—
170C22×C4×C5#4112yes—
171C22×C4×C5C2411yescharacteristic
172C22×Dic5—24no—
173C22×Dic5—24no—
174C22×Dic5—21no—
175C22×Dic5—21no—
176C22×Dic5—22no—
177C22×Dic5—41no—
178C22×Dic5—41no—
179C22×Dic5—82no—
180C22×Dic5C2×D412yes—
181C22×Dic5C2×D412yes—
182C5×M16—24no—
183C5×M16—22no—
184C5×M16C22×C414yes—
185C2×C5×C8—24no—
186C2×C5×C8—22no—
187C5×C4⋊C4—24no—
188C5×C22⋊C4—24no—
189C42×C5—22no—
190C42×C5C22×C414yes—
191C22⋊Dic5—24no—
192C22⋊Dic5—24no—
193C22⋊Dic5—22no—
194C22⋊Dic5—22no—
195C22⋊Dic5—22no—
196C22⋊Dic5—22no—
197C4⋊Dic5—24no—
198C4⋊Dic5—24no—
199C4⋊Dic5—21no—
200C4⋊Dic5—21no—
201C4⋊Dic5—21no—
202C4⋊Dic5—21no—
203C4⋊Dic5—21no—
204C4⋊Dic5—21no—
205C4⋊Dic5—22no—
206C4⋊Dic5—22no—
207C4⋊Dic5—41no—
208C4⋊Dic5—41no—
209C4⋊Dic5—84no—
210C4⋊Dic5C2×D412yes—
211C4⋊Dic5C2×D412yes—
212C4×Dic5—24no—
213C4×Dic5—24no—
214C4×Dic5—22no—
215C4×Dic5—22no—
216C4×Dic5—22no—
217C4×Dic5—22no—
218#681—22no—
219#681—22no—
220#681—42no—
221C2×#168—22no—
222C2×#168—42no—
223C22×#41C2×D511yescharacteristic
224C22×M16—51no—
225C22×M16C2×D511yescharacteristic
226#520—52no—
227#520—52no—
228#520—101no—
229#520—101no—
230#519—101no—
231#518—102no—
232#518—101no—
233C4×#41—102no—
234C4×#41—102no—
235C2×#117—51no—
236C2×#117—51no—
237C2×#117—102no—
238C2×#117C2×D511yescharacteristic
239C22×C4⋊C4—51no—
240#428—54no—
241#428—102no—
242C2×#107—101no—
243C2×#106—101no—
244#421Dic518yescomplemented
245C2×#104C2×D514yes—
246#417—54no—
247#417—102no—
248C2×#103—101no—
249C2×#102—101no—
250#389—102no—
251#385—102no—
252#384—52no—
253#382—52no—
254#343—52no—
255#343—52no—
256#339—54no—
257C23×C5—22no—
258C23×C5—24no—
259C23×C5—41no—
260C23×C5—41no—
261C23×C5—81no—
262C23×C5—81no—
263C23×C5C4×D412yes—
264C23×C5#12011yescharacteristic
265C23×C5#12011yescharacteristic
266C23×C5C22⋊Q812yes—
267C23×C5C22×D411yescharacteristic
268C5×Q8—22no—
269C5×Q8—24no—
270C5×Q8C2×C4⋊C414yes—
271C5×D4—24no—
272C5×D4—24no—
273C5×D4—28no—
274C5×D4—22no—
275C5×D4—24no—
276C5×D4C2×C4⋊C414yes—
277C2×C4×C5—22no—
278C2×C4×C5—24no—
279C2×C4×C5—24no—
280C2×C4×C5—24no—
281C2×C4×C5—28no—
282C2×C4×C5—24no—
283C2×C4×C5—42no—
284C2×C4×C5—42no—
285C2×C4×C5C2×C4⋊C412yes—
286C2×C4×C5C2×C4⋊C412yes—
287C2×C4×C5C4×D412yes—
288C2×C4×C5C4×D412yes—
289C2×C4×C5C4×D412yes—
290C2×C4×C5#12011yescharacteristic
291C2×C4×C5#12011yescharacteristic
292C2×C4×C5C22⋊Q812yes—
293C2×C4×C5C22⋊Q812yes—
294C2×C4×C5C22⋊Q812yes—
295C2×C4×C5#12312yes—
296C2×C4×C5#12312yes—
297C2×C4×C5C23×C411yescharacteristic
298C2×C4×C5C22×D411yescharacteristic
299C2×C4×C5C22×D411yescharacteristic
300C2×C4×C5C22×Q811yescharacteristic
301C2×C4×C5C2×#4111yescharacteristic
302C2×C4×C5C2×#4111yescharacteristic
303C2×Dic5—22no—
304C2×Dic5—22no—
305C2×Dic5—22no—
306C2×Dic5—22no—
307C2×Dic5—42no—
308C2×Dic5—42no—
309C2×Dic5—44no—
310C2×Dic5—42no—
311C2×Dic5—81no—
312C2×Dic5—81no—
313C2×Dic5—84no—
314C2×Dic5—84no—
315C2×Dic5—82no—
316C2×Dic5—82no—
317C5×C8—24no—
318#168—42no—
319C2×#41—21no—
320C2×#41—21no—
321C2×#41—22no—
322C2×#41—22no—
323C2×#41Dic1014yes—
324C2×#41D2012yes—
325C2×#41C2×Dic514yes—
326C2×#41C22×D512yes—
327C22×Q8D2011yescharacteristic
328C22×D4—21no—
329C22×D4D2011yescharacteristic
330C23×C4—21no—
331C23×C4—101no—
332C23×C4C22×D511yescharacteristic
333C2×M16—21no—
334C2×M16—22no—
335C2×M16—52no—
336C2×M16—101no—
337C2×M16—102no—
338C2×M16—102no—
339C2×M16C2×Dic514yescomplemented
340C2×M16C22×D512yes—
341C22×C8—21no—
342C22×C8—101no—
343C4×Q8—102no—
344C4×Q8—102no—
345C4×Q8—102no—
346C4×Q8—102no—
347C4×D4—104no—
348C4×D4—104no—
349C4×D4—104no—
350C4×D4—104no—
351C4×D4—102no—
352C4×D4—102no—
353C4×D4—102no—
354C4×D4—102no—
355#117—22no—
356#117—52no—
357#117—52no—
358#117—102no—
359#117—102no—
360#117—104no—
361#117—104no—
362#117—101no—
363#117—101no—
364#117—101no—
365#117—101no—
366#117—101no—
367#117—101no—
368#117C2×Dic514yescomplemented
369C2×C4⋊C4—21no—
370C2×C4⋊C4—51no—
371C2×C4⋊C4—51no—
372C2×C4⋊C4—51no—
373C2×C4⋊C4—51no—
374C2×C4⋊C4—52no—
375C2×C4⋊C4—102no—
376C2×C4⋊C4—102no—
377C2×C4⋊C4—101no—
378C2×C4⋊C4—101no—
379C2×C4⋊C4—101no—
380C2×C4⋊C4—101no—
381C2×C4⋊C4—101no—
382C2×C4⋊C4—104no—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—52no—
385C2×C22⋊C4—104no—
386C2×C22⋊C4—101no—
387C2×C42—52no—
388C2×C42—104no—
389C2×C42—101no—
390C2×C42C22×D512yes—
391#107—104no—
392#106—104no—
393#104C2×Dic5116yescomplemented
394#103—104no—
395#102—104no—
396#95—104no—
397C22×C5—21no—
398C22×C5—21no—
399C22×C5—21no—
400C22×C5—21no—
401C22×C5—21no—
402C22×C5—21no—
403C22×C5—24no—
404C22×C5—42no—
405C22×C5—42no—
406C22×C5—42no—
407C22×C5—42no—
408C22×C5—84no—
409C22×C5#38512yes—
410C22×C5#38512yes—
411C22×C5C2×C4×D411yescharacteristic
412C22×C5C2×#12011yescharacteristic
413C22×C5C2×C22⋊Q811yescharacteristic
414C4×C5—24no—
415C4×C5—44no—
416C4×C5#38512yes—
417C4×C5#38512yes—
418C4×C5C22×C4⋊C411yescharacteristic
419C4×C5C2×C4×D411yescharacteristic
420C4×C5C2×C22⋊Q811yescharacteristic
421C4×C5C2×#12311yescharacteristic
422Dic5—42no—
423Dic5—42no—
424Dic5—42no—
425Dic5—42no—
426Dic5—82no—
427C24—21no—
428C24D4×D511yescharacteristic
429#41—22no—
430#41—22no—
431#41—24no—
432#41—24no—
433#41—24no—
434#41—24no—
435#41—24no—
436#41C4⋊Dic518yescomplemented
437#41C2×Dic1014yes—
438#41C22×Dic514yes—
439C2×Q8—21no—
440C2×Q8—22no—
441C2×Q8C4⋊Dic514yescomplemented
442C2×Q8C2×D2012yes—
443C2×D4—22no—
444C2×D4—24no—
445C2×D4—22no—
446C2×D4—24no—
447C2×D4—21no—
448C2×D4—22no—
449C2×D4C4⋊Dic514yescomplemented
450C2×D4C2×D2012yes—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—52no—
458C22×C4—52no—
459C22×C4—104no—
460C22×C4—104no—
461C22×C4—101no—
462C22×C4—101no—
463C22×C4—102no—
464C22×C4—201no—
465C22×C4—201no—
466C22×C4—402no—
467C22×C4C2×Dic1012yes—
468C22×C4C2×D2011yescharacteristic
469C22×C4C2×D2011yescharacteristic
470C22×C4D4×D511yescharacteristic
471C22×C4D4×D511yescharacteristic
472C22×C4D4×D511yescharacteristic
473C22×C4#71112yes—
474C22×C4#71112yes—
475C22×C4C22×Dic512yes—
476C22×C4C23×D511yescharacteristic
477M16—22no—
478M16—24no—
479M16—102no—
480M16—102no—
481M16—202no—
482M16C22×Dic514yes—
483C2×C8—22no—
484C2×C8—24no—
485C2×C8—102no—
486C2×C8—202no—
487C4⋊C4—24no—
488C4⋊C4—52no—
489C4⋊C4—52no—
490C4⋊C4—104no—
491C4⋊C4—104no—
492C4⋊C4—101no—
493C4⋊C4—101no—
494C4⋊C4—101no—
495C4⋊C4—101no—
496C4⋊C4—101no—
497C4⋊C4—101no—
498C4⋊C4—102no—
499C4⋊C4—102no—
500C4⋊C4—201no—
501C4⋊C4—201no—
502C4⋊C4—404no—
503C22⋊C4—24no—
504C22⋊C4—104no—
505C22⋊C4—104no—
506C22⋊C4—102no—
507C22⋊C4—102no—
508C22⋊C4—102no—
509C22⋊C4—102no—
510C42—22no—
511C42—104no—
512C42—104no—
513C42—102no—
514C42—102no—
515C42—102no—
516C42—102no—
517C42C22×Dic514yes—
518C2×C5—21no—
519C2×C5—22no—
520C2×C5—21no—
521C2×C5—22no—
522C2×C5—44no—
523C2×C5C2×#38511yescharacteristic
524C2×C5#302111yescharacteristic
525C2×C5#302211yescharacteristic
526C23—22no—
527C23—24no—
528C23—41no—
529C23—41no—
530C23—81no—
531C23—81no—
532C23#402412yes—
533C23#402811yescharacteristic
534C23#402811yescharacteristic
535C23D4×Dic512yes—
536C23C2×D4×D511yescharacteristic
537Q8—22no—
538Q8—24no—
539Q8C2×C4⋊Dic514yes—
540D4—24no—
541D4—24no—
542D4—28no—
543D4—22no—
544D4—24no—
545D4C2×C4⋊Dic514yes—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—28no—
551C2×C4—24no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—102no—
555C2×C4—102no—
556C2×C4—102no—
557C2×C4—102no—
558C2×C4—202no—
559C2×C4—202no—
560C2×C4—204no—
561C2×C4—202no—
562C2×C4—401no—
563C2×C4—401no—
564C2×C4—404no—
565C2×C4—404no—
566C2×C4—402no—
567C2×C4—402no—
568C2×C4#402412yes—
569C2×C4#402412yes—
570C2×C4#402412yes—
571C2×C4#402811yescharacteristic
572C2×C4#402811yescharacteristic
573C2×C4#403212yes—
574C2×C4#403212yes—
575C2×C4C2×C4⋊Dic512yes—
576C2×C4C2×C4⋊Dic512yes—
577C2×C4D4×Dic512yes—
578C2×C4D4×Dic512yes—
579C2×C4D4×Dic512yes—
580C2×C4C22×Dic1011yescharacteristic
581C2×C4C22×D2011yescharacteristic
582C2×C4C2×D4×D511yescharacteristic
583C2×C4C2×#71111yescharacteristic
584C2×C4C2×#71111yescharacteristic
585C2×C4C23×Dic511yescharacteristic
586C8—24no—
587C8—202no—
588C5#2086511yescharacteristic, complemented, Sylow
589C22—21no—
590C22—21no—
591C22—21no—
592C22—21no—
593C22—21no—
594C22—21no—
595C22—24no—
596C22—42no—
597C22—42no—
598C22—42no—
599C22—42no—
600C22—84no—
601C22#6523912yes—
602C22#6523912yes—
603C22C2×#402411yescharacteristic
604C22C2×#402811yescharacteristic
605C22C2×D4×Dic511yescharacteristic
606C4—24no—
607C4—44no—
608C4—202no—
609C4—202no—
610C4—202no—
611C4—202no—
612C4—402no—
613C4#6523912yes—
614C4#6523912yes—
615C4C2×#402411yescharacteristic
616C4C2×#403211yescharacteristic
617C4C22×C4⋊Dic511yescharacteristic
618C4C2×D4×Dic511yescharacteristic
619C2—21no—
620C2—22no—
621C2—21no—
622C2—22no—
623C2—44no—
624C2#1061072511yescharacteristic
625C2#1061072611yescharacteristic
626C2C2×#6523911yescharacteristic
subgroupG/H
centerC22C2×#4028
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC2×C5×#421C2
socleC22×C5C2×#120
prime radicalC22×C5×#41C22
cosocleC22×C4×C5C24
hypercenterC2×#421D5
hypocenterC5#20865
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#421

Complements

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

NH ≅ G/Ntype
4. C2×#65410C2semidirect
5. #10613964C2semidirect
6. #10602780C2semidirect
7. #10602371C2semidirect
8. #10601157C2semidirect
9. #10601156C2semidirect
27. C5×#421C4semidirect
34. #65410C22semidirect
39. #65225C22semidirect
40. #65224C22semidirect
41. #64774C22semidirect
42. #64774C22semidirect
43. #64743C22semidirect
61. C2×C5×M16C2×C4semidirect
65. C5×#117C2×C4semidirect
117. C5×#104C2×C4semidirect
141. C5×#41C4⋊C4semidirect
146. C2×C5×Q8C4⋊C4semidirect
154. C2×C5×D4C4⋊C4semidirect
244. #421Dic5semidirect
339. C2×M16C2×Dic5semidirect
368. #117C2×Dic5semidirect
393. #104C2×Dic5semidirect
436. #41C4⋊Dic5semidirect
441. C2×Q8C4⋊Dic5semidirect
449. C2×D4C4⋊Dic5semidirect
588. C5#20865semidirect

Maximal subgroups

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

subgroupindex
#106191612
C2×C5×#4212
C2×#655212
C2×#654102
#106139642
#106027802
#106023712
#106011572
#106011562
#208655