Finite Group Databasebeta

← #12101733

Subgroups — #12101733

Order 1280.

1126 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#65402C211yesmaximal, characteristic
2#10613917C211yesmaximal, characteristic
3#10613875C211yesmaximal, characteristic, complemented
4C2×#65255C211yesmaximal, characteristic
5C2×#65253C211yesmaximal, characteristic, complemented
6C2×C5×#340C211yesmaximal, characteristic, complemented
7#10602758C211yesmaximal, characteristic, complemented
8#10599587C212yesmaximal, complemented
9#10599586C212yesmaximal, complemented
10#10599585C212yesmaximal, complemented
11#10599584C212yesmaximal, complemented
12C22×#709C2211yescharacteristic
13C23×C5×C8C2211yescharacteristic
14C22×C5×C4⋊C4C2211yescharacteristic
15C22×C5×C4⋊C4C2211yescharacteristic
16C2×#4072C2211yescharacteristic
17C2×#4072C2211yescharacteristic
18C22×#681C2211yescharacteristic
19#65502—22no—
20C4×C22⋊Dic5—22no—
21#65402—24no—
22C2×#3953C2211yescharacteristic
23C2×#3953C2211yescharacteristic
24C2×#3948C2211yescharacteristic
25C2×#3948C2211yescharacteristic
26#65277C2212yes—
27#65276—21no—
28#65276—21no—
29C4⋊C4×Dic5—21no—
30C4⋊C4×Dic5—21no—
31C4⋊C4×Dic5C2212yes—
32C4⋊C4×Dic5C2212yes—
33#65263C2212yes—
34C2×#3942C2212yes—
35#65255C418yescomplemented
36C2×#3941C2212yes—
37#65253—24no—
38C2×#3940C2212yes—
39C2×#3939C2212yes—
40C5×#340C2212yes—
41C5×#340C2212yes—
42#64770C2212yes—
43#64770C2212yes—
44#21062—51nomaximal, Sylow
45C23×C4×C5—21no—
46C23×C4×C5—21no—
47C23×C4×C5C2311yescharacteristic
48C22×C22⋊D5—21no—
49C2×#709—22no—
50C2×#709—22no—
51C2×#709—22no—
52C2×#709—22no—
53C2×#709—22no—
54C2×#709C2×C414yes—
55C22×D20C2311yescharacteristic
56C22×C4×D5—21no—
57C22×Dic10C2311yescharacteristic
58C22×C5×C8—22no—
59C22×C5×C8—22no—
60C22×C5×C8—22no—
61C22×C5×C8C2311yescharacteristic
62C22×C5×C8C2311yescharacteristic
63C2×C5×C4⋊C4—21no—
64C2×C5×C4⋊C4—22no—
65C2×C5×C4⋊C4—22no—
66C2×C5×C4⋊C4—22no—
67C2×C5×C4⋊C4—21no—
68C2×C5×C4⋊C4C2×C414yes—
69C2×C5×C4⋊C4C2312yes—
70C2×C5×C4⋊C4C2312yes—
71C2×C22⋊Dic5—21no—
72C2×C22⋊Dic5D412yes—
73C2×#685—22no—
74C2×#685—22no—
75#4072—22no—
76#4072—22no—
77#4072—22no—
78#4072D414yes—
79C2×C4⋊Dic5D411yescharacteristic
80C2×C4⋊Dic5D411yescharacteristic
81C2×C4⋊Dic5C2311yescharacteristic
82C2×C4⋊Dic5C2311yescharacteristic
83C2×C4×Dic5—21no—
84C2×C4×Dic5—22no—
85C2×C4×Dic5—22no—
86C2×C4×Dic5C2312yes—
87C2×#681—22no—
88C2×#681—22no—
89C2×#681C2×C414yes—
90C22×#168C2312yes—
91#3963—22no—
92#3963—22no—
93#3963—22no—
94#3963—22no—
95#3953—22no—
96#3953—22no—
97#3948—22no—
98#3948—22no—
99#3942C2×C418yescomplemented
100#3941C2×C418yescomplemented
101#3940—24no—
102#3939—24no—
103C2×#428—51no—
104C2×#417—51no—
105C2×#416—51no—
106#2311—51no—
107#2274—51no—
108#1928—52no—
109#1927—52no—
110#1926—52no—
111#1925—52no—
112C2×#340D511yescharacteristic, complemented
113#1729—51no—
114C24×C5C2×D411yescharacteristic
115C23×D5—21no—
116C22×C4×C5—22no—
117C22×C4×C5—24no—
118C22×C4×C5—22no—
119C22×C4×C5—42no—
120C22×C4×C5—81no—
121C22×C4×C5—81no—
122C22×C4×C5—81no—
123C22×C4×C5—81no—
124C22×C4×C5C22×C412yes—
125C22×C4×C5C2×D411yescharacteristic
126C22×C4×C5C2×D412yes—
127C22×C4×C5#4112yes—
128C22×C4×C5#4112yes—
129C22×C4×C5#4112yes—
130C22×C4×C5#4112yes—
131C22×C4×C5C2411yescharacteristic
132C2×C22⋊D5—24no—
133C2×C22⋊D5—42no—
134C2×C22⋊D5—42no—
135C22×Dic5—21no—
136C22×Dic5—21no—
137C22×Dic5—21no—
138C22×Dic5C2×D412yes—
139C22×Dic5C2×D412yes—
140#709—28no—
141#709—28no—
142#709—44no—
143#709—44no—
144C2×D20—21no—
145C2×D20—21no—
146C2×D20—21no—
147C2×D20—21no—
148C2×D20—21no—
149C2×D20C22×C412yes—
150C2×C4×D5—24no—
151C2×C4×D5—42no—
152C2×C4×D5—42no—
153C2×Dic10—21no—
154C2×Dic10—21no—
155C2×Dic10—21no—
156C2×Dic10—21no—
157C2×Dic10—21no—
158C2×Dic10C22×C412yes—
159C2×C5×C8—21no—
160C2×C5×C8—21no—
161C2×C5×C8—21no—
162C2×C5×C8—21no—
163C2×C5×C8—21no—
164C2×C5×C8—21no—
165C2×C5×C8—44no—
166C5×C4⋊C4—24no—
167C5×C4⋊C4—22no—
168C5×C4⋊C4—22no—
169C5×C4⋊C4—22no—
170C5×C4⋊C4—42no—
171C5×C4⋊C4C22×C414yes—
172C22⋊Dic5—24no—
173C22⋊Dic5—24no—
174#685—82no—
175#685—82no—
176C4⋊Dic5—22no—
177C4⋊Dic5—22no—
178C4⋊Dic5—21no—
179C4⋊Dic5—21no—
180C4⋊Dic5C2×D412yes—
181C4⋊Dic5C2×D412yes—
182C4×Dic5—24no—
183C4×Dic5—22no—
184C4×Dic5—22no—
185C4×Dic5—82no—
186C4×Dic5—82no—
187#681—28no—
188C2×#168—22no—
189C2×#168—22no—
190C2×#168C22×C414yes—
191C22×#41—51no—
192C22×M16—51no—
193C23×C8C2×D511yescharacteristic
194C2×#117—51no—
195C2×#117—51no—
196C22×C4⋊C4C2×D511yescharacteristic
197C22×C4⋊C4C2×D511yescharacteristic
198#428—104no—
199C2×#107—52no—
200C2×#106—52no—
201#417—58no—
202#416—104no—
203C2×#103—52no—
204C2×#103—51no—
205C2×#103—51no—
206C2×#102—52no—
207C2×#102—51no—
208C2×#102—51no—
209#391—52no—
210#390—52no—
211#386—102no—
212#380—101no—
213#380—101no—
214C4×C4⋊C4—52no—
215C4×C4⋊C4—52no—
216C4×C4⋊C4—101no—
217C4×C4⋊C4—101no—
218C4×C22⋊C4—102no—
219#340C2×D512yes—
220#340C2×D512yes—
221#336—52no—
222#336—52no—
223C23×C5—21no—
224C23×C5—21no—
225C23×C5—21no—
226C23×C5—21no—
227C23×C5C4×D412yes—
228C23×C5#12112yes—
229C23×C5#12312yes—
230C23×C5C22×D411yescharacteristic
231C22×D5—22no—
232C22×D5—41no—
233C22×D5—41no—
234C22×D5—41no—
235C22×D5—41no—
236C22×D5—41no—
237C22×D5—41no—
238C2×C4×C5—24no—
239C2×C4×C5—42no—
240C2×C4×C5—42no—
241C2×C4×C5—42no—
242C2×C4×C5—42no—
243C2×C4×C5—42no—
244C2×C4×C5—84no—
245C2×C4×C5—84no—
246C2×C4×C5#11714yes—
247C2×C4×C5C4×D412yes—
248C2×C4×C5C4×D414yes—
249C2×C4×C5#12112yes—
250C2×C4×C5#12312yes—
251C2×C4×C5#12414yes—
252C2×C4×C5#12614yes—
253C2×C4×C5C23×C411yescharacteristic
254C2×C4×C5C22×D411yescharacteristic
255C2×C4×C5C2×#4111yescharacteristic
256C2×C4×C5C2×#4111yescharacteristic
257C2×C4×C5C2×#4111yescharacteristic
258C2×C4×C5C2×#4111yescharacteristic
259C22⋊D5—48no—
260C2×Dic5—22no—
261C2×Dic5—22no—
262C2×Dic5—22no—
263C2×Dic5—22no—
264C2×Dic5—22no—
265C2×Dic5—22no—
266C2×Dic5—22no—
267C2×Dic5—22no—
268C2×Dic5—22no—
269C2×Dic5—41no—
270C2×Dic5—41no—
271C2×Dic5—41no—
272C2×Dic5—41no—
273C2×Dic5—41no—
274C2×Dic5—41no—
275D20—22no—
276D20—22no—
277D20—41no—
278D20—41no—
279C4×D5—48no—
280Dic10—22no—
281Dic10—22no—
282Dic10—41no—
283Dic10—41no—
284C5×C8—41no—
285C5×C8—41no—
286#168—24no—
287C2×#41—54no—
288C2×#41—102no—
289C2×#41—102no—
290C2×#41—102no—
291C2×#41—102no—
292C2×#41—102no—
293C22×Q8—51no—
294C22×D4—51no—
295C22×D4—101no—
296C23×C4—21no—
297C23×C4—21no—
298C23×C4—101no—
299C23×C4C22×D511yescharacteristic
300C2×M16—54no—
301C2×M16—102no—
302C2×M16—102no—
303C22×C8—22no—
304C22×C8—22no—
305C22×C8—22no—
306C22×C8—52no—
307C22×C8C22×D511yescharacteristic
308C22×C8C22×D511yescharacteristic
309#117—54no—
310#117—102no—
311#117—102no—
312#117—102no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—51no—
319C2×C4⋊C4—51no—
320C2×C4⋊C4—51no—
321C2×C4⋊C4—51no—
322C2×C4⋊C4C4×D514yes—
323C2×C4⋊C4C22×D512yes—
324C2×C4⋊C4C22×D512yes—
325C2×C22⋊C4—52no—
326C2×C22⋊C4—101no—
327C2×C22⋊C4—102no—
328C2×C22⋊C4—102no—
329C2×C42—52no—
330C2×C42—101no—
331C2×C42—102no—
332C2×C42—102no—
333#107—104no—
334#106—104no—
335#103—58no—
336#103—102no—
337#103—102no—
338#102—58no—
339#102—102no—
340#102—102no—
341#95—102no—
342#95—102no—
343#95—102no—
344#95—102no—
345C22×C5—22no—
346C22×C5—22no—
347C22×C5—22no—
348C22×C5—22no—
349C22×C5—22no—
350C22×C5—22no—
351C22×C5#38814yes—
352C22×C5C2×C4×D411yescharacteristic
353C22×C5C2×#12111yescharacteristic
354C22×C5C2×#12311yescharacteristic
355C22×C5#57611yescharacteristic
356C22×C5#57711yescharacteristic
357C22×C5#57711yescharacteristic
358C22×C5#57811yescharacteristic
359C2×D5—42no—
360C2×D5—42no—
361C2×D5—42no—
362C2×D5—81no—
363C2×D5—81no—
364C2×D5—81no—
365C2×D5—81no—
366C4×C5—44no—
367C4×C5—82no—
368C4×C5#38814yes—
369C4×C5C2×#11711yescharacteristic
370C4×C5C2×C4×D411yescharacteristic
371C4×C5C2×#12411yescharacteristic
372C4×C5C2×#12611yescharacteristic
373Dic5—42no—
374Dic5—42no—
375Dic5—42no—
376Dic5—42no—
377Dic5—42no—
378C24—101no—
379C24D4×D511yescharacteristic
380#41—108no—
381#41—108no—
382#41—204no—
383#41—204no—
384C2×Q8—52no—
385C2×Q8—101no—
386C2×Q8—101no—
387C2×Q8—101no—
388C2×Q8—101no—
389C2×Q8—101no—
390C2×D4—52no—
391C2×D4—104no—
392C2×D4—101no—
393C2×D4—101no—
394C2×D4—101no—
395C2×D4—101no—
396C2×D4—101no—
397C2×D4—202no—
398C2×D4—202no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—42no—
403C22×C4—52no—
404C22×C4—52no—
405C22×C4—81no—
406C22×C4—81no—
407C22×C4—81no—
408C22×C4—81no—
409C22×C4—101no—
410C22×C4—101no—
411C22×C4—101no—
412C22×C4—104no—
413C22×C4—202no—
414C22×C4—202no—
415C22×C4C2×C4×D512yes—
416C22×C4#70912yes—
417C22×C4#70912yes—
418C22×C4D4×D512yes—
419C22×C4D4×D511yescharacteristic
420C22×C4#71112yes—
421C22×C4#71312yes—
422C22×C4C23×D511yescharacteristic
423M16—108no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—44no—
431C2×C8—54no—
432C2×C8—102no—
433C2×C8—102no—
434C4⋊C4—22no—
435C4⋊C4—24no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—42no—
439C4⋊C4—52no—
440C4⋊C4—52no—
441C4⋊C4—101no—
442C4⋊C4—101no—
443C4⋊C4—102no—
444C4⋊C4—102no—
445C4⋊C4C2×C4×D514yes—
446C22⋊C4—104no—
447C22⋊C4—104no—
448C22⋊C4—402no—
449C22⋊C4—402no—
450C42—102no—
451C42—102no—
452C42—104no—
453C42—402no—
454C42—402no—
455C2×C5—24no—
456C2×C5C2×#38811yescharacteristic
457C2×C5#294611yescharacteristic
458C2×C5#294711yescharacteristic
459C2×C5#304511yescharacteristic
460C2×C5#304611yescharacteristic
461C2×C5#309811yescharacteristic
462C2×C5#309911yescharacteristic
463D5—82no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—102no—
469C23—201no—
470C23—201no—
471C23—201no—
472C23—201no—
473C23—201no—
474C23—201no—
475C23#402712yes—
476C23#403012yes—
477C23#404012yes—
478C23C2×D4×D511yescharacteristic
479Q8—102no—
480Q8—102no—
481Q8—201no—
482Q8—201no—
483D4—102no—
484D4—102no—
485D4—208no—
486D4—201no—
487D4—201no—
488C2×C4—24no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—84no—
495C2×C4—84no—
496C2×C4—102no—
497C2×C4—102no—
498C2×C4—102no—
499C2×C4—102no—
500C2×C4—102no—
501C2×C4—102no—
502C2×C4—102no—
503C2×C4—102no—
504C2×C4—102no—
505C2×C4—201no—
506C2×C4—201no—
507C2×C4—201no—
508C2×C4—201no—
509C2×C4—201no—
510C2×C4—201no—
511C2×C4—208no—
512C2×C4#401814yes—
513C2×C4#402712yes—
514C2×C4#403012yes—
515C2×C4#403114yes—
516C2×C4#403914yes—
517C2×C4#404012yes—
518C2×C4#404414yes—
519C2×C4C22×C4×D511yescharacteristic
520C2×C4C2×#70911yescharacteristic
521C2×C4C2×#70911yescharacteristic
522C2×C4C2×D4×D511yescharacteristic
523C2×C4C2×#71111yescharacteristic
524C2×C4C2×#71311yescharacteristic
525C8—41no—
526C8—41no—
527C8—104no—
528C5#2106211yescharacteristic, complemented, Sylow
529C22—22no—
530C22—22no—
531C22—22no—
532C22—22no—
533C22—22no—
534C22—22no—
535C22—202no—
536C22—202no—
537C22—202no—
538C22—401no—
539C22—401no—
540C22—401no—
541C22—401no—
542C22#6495814yes—
543C22C2×#402711yescharacteristic
544C22C2×#403011yescharacteristic
545C22C2×#404011yescharacteristic
546C22#6609111yescharacteristic
547C22#6609611yescharacteristic
548C22#6609611yescharacteristic
549C22#6610011yescharacteristic
550C4—44no—
551C4—82no—
552C4—202no—
553C4—202no—
554C4—202no—
555C4—202no—
556C4—202no—
557C4#6495814yes—
558C4C2×#401811yescharacteristic
559C4C2×#403111yescharacteristic
560C4C2×#403911yescharacteristic
561C4C2×#404411yescharacteristic
562C2—24no—
563C2—402no—
564C2C2×#6495811yescharacteristic
565C2#1061045711yescharacteristic
566C2#1061049811yescharacteristic
567C2#1061050511yescharacteristic
568C2#1061054611yescharacteristic
569C2#1061084311yescharacteristic
570C2#1061087211yescharacteristic
subgroupG/H
centerC23C2×D4×D5
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC2×C5×#340C2
socleC23×C5C22×D4
prime radicalC22×#709C22
cosocleC22×C4×C5C24
hypercenterC2×#340D5
hypocenterC5#21062
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#340

Complements

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

NH ≅ G/Ntype
3. #10613875C2semidirect
5. C2×#65253C2semidirect
6. C2×C5×#340C2semidirect
7. #10602758C2semidirect
8. #10599587C2semidirect
9. #10599586C2semidirect
10. #10599585C2semidirect
11. #10599584C2semidirect
35. #65255C4semidirect
99. #3942C2×C4semidirect
100. #3941C2×C4semidirect
112. C2×#340D5semidirect
528. C5#21062semidirect

Maximal subgroups

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

subgroupindex
C2×#654022
#106139172
#106138752
C2×#652552
C2×#652532
C2×C5×#3402
#106027582
#105995872
#105995862
#105995852
#105995842
#210625