Finite Group Databasebeta

← #12101735

Subgroups — #12101735

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×C5×#428C211yesmaximal, characteristic, complemented
2C2×#65422C211yesmaximal, characteristic
3#10613954C211yesmaximal, characteristic, complemented
4C2×#65291C211yesmaximal, characteristic
5#10613917C211yesmaximal, characteristic
6#10602773C211yesmaximal, characteristic, complemented
7C2×#64749C211yesmaximal, characteristic, complemented
8#10601417C212yesmaximal, complemented
9#10601416C212yesmaximal, complemented
10#10601363C212yesmaximal, complemented
11#10601362C212yesmaximal, complemented
12C22×#709C2211yescharacteristic
13C22×C5×M16C2211yescharacteristic
14C2×C5×#117C2211yescharacteristic
15C22×C5×C4⋊C4C2211yescharacteristic
16C2×#4072C2211yescharacteristic
17C22×C4⋊Dic5C2211yescharacteristic
18C23×#168C2211yescharacteristic
19C5×#428—24no—
20C2×C5×#107C2212yes—
21C2×C5×#106C2212yes—
22#65502—22no—
23#65422C418yescomplemented
24C2×#3953C2212yes—
25C2×#3948C2212yes—
26#65291—24no—
27#65277C2212yes—
28#65276—21no—
29#65276—21no—
30C4⋊C4×Dic5C2212yes—
31C4⋊C4×Dic5C2212yes—
32#65263C2212yes—
33C2×#3942C2211yescharacteristic
34C2×#3942C2211yescharacteristic
35C2×#3941C2211yescharacteristic
36C2×#3941C2211yescharacteristic
37C22⋊C4×Dic5—22no—
38C4×C4⋊Dic5—21no—
39C4×C4⋊Dic5—21no—
40#64770C2212yes—
41#64770C2212yes—
42#64749C2212yes—
43#64749C2212yes—
44#21062—51nomaximal, Sylow
45C23×C4×C5—21no—
46C23×C4×C5C2311yescharacteristic
47C22×C22⋊D5—21no—
48C23×Dic5—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
58C2×C5×M16—22no—
59C2×C5×M16—22no—
60C2×C5×M16C2×C414yes—
61C22×C5×C8C2312yes—
62C5×#117—22no—
63C5×#117D414yes—
64C2×C5×C4⋊C4—22no—
65C2×C5×C4⋊C4—22no—
66C2×C5×C4⋊C4—21no—
67C2×C5×C4⋊C4C2312yes—
68C2×C5×C4⋊C4C2311yescharacteristic
69C2×C5×C4⋊C4C2311yescharacteristic
70C2×C5×C22⋊C4—21no—
71C2×C42×C5—21no—
72C2×C22⋊Dic5D412yes—
73C2×#685—22no—
74C2×#685—22no—
75#4072—22no—
76#4072—22no—
77C2×C4⋊Dic5—21no—
78C2×C4⋊Dic5—22no—
79C2×C4⋊Dic5C2×C414yes—
80C2×C4⋊Dic5D411yescharacteristic
81C2×C4⋊Dic5D411yescharacteristic
82C2×C4⋊Dic5C2312yes—
83C2×C4×Dic5—22no—
84C2×C4×Dic5—22no—
85C2×C4×Dic5C2312yes—
86C22×#168—22no—
87C22×#168—22no—
88C22×#168—22no—
89C22×#168C2311yescharacteristic
90C22×#168C2311yescharacteristic
91C5×#107—24no—
92C5×#106—24no—
93#3963—22no—
94#3963—22no—
95#3963—22no—
96#3963—22no—
97#3953C2×C418yescomplemented
98#3948C2×C418yescomplemented
99#3942—22no—
100#3942—22no—
101#3941—22no—
102#3941—22no—
103C2×#428D511yescharacteristic, complemented
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×#340—51no—
113#1729—51no—
114C24×C5C2×D411yescharacteristic
115C23×D5—21no—
116C22×C4×C5—21no—
117C22×C4×C5—21no—
118C22×C4×C5—22no—
119C22×C4×C5—42no—
120C22×C4×C5—81no—
121C22×C4×C5—81no—
122C22×C4×C5C22×C412yes—
123C22×C4×C5C2×D411yescharacteristic
124C22×C4×C5C2×D412yes—
125C22×C4×C5#4112yes—
126C22×C4×C5#4112yes—
127C22×C4×C5#4112yes—
128C22×C4×C5#4112yes—
129C22×C4×C5C2411yescharacteristic
130C2×C22⋊D5—24no—
131C2×C22⋊D5—42no—
132C2×C22⋊D5—42no—
133C22×Dic5—21no—
134C22×Dic5—22no—
135C22×Dic5—24no—
136C22×Dic5—81no—
137C22×Dic5—81no—
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—
159C5×M16—28no—
160C2×C5×C8—22no—
161C2×C5×C8—22no—
162C2×C5×C8C22×C414yes—
163C5×C4⋊C4—21no—
164C5×C4⋊C4—21no—
165C5×C4⋊C4—22no—
166C5×C4⋊C4—22no—
167C5×C4⋊C4—42no—
168C5×C4⋊C4C2×D412yes—
169C5×C4⋊C4C2×D412yes—
170C5×C22⋊C4—24no—
171C42×C5—24no—
172C22⋊Dic5—24no—
173#685—82no—
174#685—82no—
175C4⋊Dic5—24no—
176C4⋊Dic5—22no—
177C4⋊Dic5—22no—
178C4⋊Dic5—22no—
179C4⋊Dic5C22×C414yes—
180C4×Dic5—22no—
181C4×Dic5—22no—
182C4×Dic5—82no—
183C4×Dic5—82no—
184C2×#168—21no—
185C2×#168—21no—
186C2×#168—21no—
187C2×#168—21no—
188C2×#168—21no—
189C2×#168—21no—
190C2×#168—44no—
191C22×#41—51no—
192C22×M16C2×D511yescharacteristic
193C23×C8—51no—
194C2×#117—51no—
195C2×#117C2×D511yescharacteristic
196C22×C4⋊C4—51no—
197C22×C4⋊C4C2×D511yescharacteristic
198#428—24no—
199C2×#107C2×D512yes—
200C2×#106C2×D512yes—
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#340—52no—
220#340—52no—
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—22no—
239C2×C4×C5—22no—
240C2×C4×C5—41no—
241C2×C4×C5—41no—
242C2×C4×C5—41no—
243C2×C4×C5—41no—
244C2×C4×C5—42no—
245C2×C4×C5—42no—
246C2×C4×C5—42no—
247C2×C4×C5—84no—
248C2×C4×C5#11714yes—
249C2×C4×C5C4×D412yes—
250C2×C4×C5C4×D414yes—
251C2×C4×C5#12112yes—
252C2×C4×C5#12312yes—
253C2×C4×C5#12414yes—
254C2×C4×C5#12614yes—
255C2×C4×C5C23×C411yescharacteristic
256C2×C4×C5C22×D411yescharacteristic
257C2×C4×C5C2×#4111yescharacteristic
258C2×C4×C5C2×#4111yescharacteristic
259C2×C4×C5C2×#4111yescharacteristic
260C2×C4×C5C2×#4111yescharacteristic
261C22⋊D5—48no—
262C2×Dic5—24no—
263C2×Dic5—22no—
264C2×Dic5—22no—
265C2×Dic5—22no—
266C2×Dic5—22no—
267C2×Dic5—22no—
268C2×Dic5—22no—
269C2×Dic5—22no—
270C2×Dic5—41no—
271C2×Dic5—41no—
272C2×Dic5—42no—
273C2×Dic5—42no—
274C2×Dic5—84no—
275D20—22no—
276D20—22no—
277D20—41no—
278D20—41no—
279C4×D5—48no—
280Dic10—22no—
281Dic10—22no—
282Dic10—41no—
283Dic10—41no—
284C5×C8—24no—
285#168—41no—
286#168—41no—
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—101no—
298C23×C4—101no—
299C23×C4C22×D511yescharacteristic
300C2×M16—22no—
301C2×M16—22no—
302C2×M16C4×D514yes—
303C22×C8—51no—
304C22×C8—51no—
305C22×C8—102no—
306C22×C8—102no—
307C22×C8—102no—
308C22×C8C22×D512yes—
309#117—22no—
310#117—102no—
311#117—102no—
312#117C22⋊D514yes—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—54no—
317C2×C4⋊C4—52no—
318C2×C4⋊C4—51no—
319C2×C4⋊C4—51no—
320C2×C4⋊C4—101no—
321C2×C4⋊C4—102no—
322C2×C4⋊C4C22×D512yes—
323C2×C4⋊C4C22×D511yescharacteristic
324C2×C4⋊C4C22×D511yescharacteristic
325C2×C22⋊C4—21no—
326C2×C22⋊C4—52no—
327C2×C22⋊C4—102no—
328C2×C22⋊C4—102no—
329C2×C42—21no—
330C2×C42—52no—
331C2×C42—102no—
332C2×C42—102no—
333#107—24no—
334#106—24no—
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—42no—
367C4×C5—42no—
368C4×C5—82no—
369C4×C5#38814yes—
370C4×C5C2×#11711yescharacteristic
371C4×C5C2×C4×D411yescharacteristic
372C4×C5C2×#12411yescharacteristic
373C4×C5C2×#12611yescharacteristic
374Dic5—42no—
375Dic5—42no—
376Dic5—44no—
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—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—42no—
403C22×C4—52no—
404C22×C4—52no—
405C22×C4—81no—
406C22×C4—81no—
407C22×C4—101no—
408C22×C4—102no—
409C22×C4—104no—
410C22×C4—104no—
411C22×C4—202no—
412C22×C4—202no—
413C22×C4—401no—
414C22×C4—401no—
415C22×C4C2×C4×D512yes—
416C22×C4#70912yes—
417C22×C4D4×D511yescharacteristic
418C22×C4#71112yes—
419C22×C4#71312yes—
420C22×C4#71312yes—
421C22×C4C2×C22⋊D512yes—
422C22×C4C23×D511yescharacteristic
423M16—28no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—101no—
427C2×C8—101no—
428C2×C8—101no—
429C2×C8—101no—
430C2×C8—101no—
431C2×C8—101no—
432C2×C8—204no—
433C2×C8C2×C4×D514yes—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—42no—
439C4⋊C4—54no—
440C4⋊C4—102no—
441C4⋊C4—104no—
442C4⋊C4—102no—
443C4⋊C4—102no—
444C4⋊C4C2×C22⋊D512yes—
445C4⋊C4C2×C22⋊D512yes—
446C22⋊C4—24no—
447C22⋊C4—104no—
448C22⋊C4—402no—
449C22⋊C4—402no—
450C42—24no—
451C42—102no—
452C42—102no—
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#403912yes—
476C23#404012yes—
477C23#408512yes—
478C23C2×D4×D511yescharacteristic
479Q8—102no—
480Q8—102no—
481Q8—201no—
482Q8—201no—
483D4—102no—
484D4—102no—
485D4—208no—
486D4—201no—
487D4—201no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—84no—
498C2×C4—104no—
499C2×C4—102no—
500C2×C4—102no—
501C2×C4—102no—
502C2×C4—102no—
503C2×C4—102no—
504C2×C4—102no—
505C2×C4—102no—
506C2×C4—201no—
507C2×C4—202no—
508C2×C4—202no—
509C2×C4—201no—
510C2×C4—208no—
511C2×C4—404no—
512C2×C4#403814yes—
513C2×C4#403912yes—
514C2×C4#404012yes—
515C2×C4#404414yes—
516C2×C4C4×C22⋊D514yes—
517C2×C4#408512yes—
518C2×C4#409314yes—
519C2×C4C22×C4×D511yescharacteristic
520C2×C4C2×#70911yescharacteristic
521C2×C4C2×#71111yescharacteristic
522C2×C4C2×#71311yescharacteristic
523C2×C4C2×#71311yescharacteristic
524C2×C4C22×C22⋊D511yescharacteristic
525C8—24no—
526C8—201no—
527C8—201no—
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#6527914yes—
543C22C2×#403911yescharacteristic
544C22C2×#404011yescharacteristic
545C22#6610811yescharacteristic
546C22#6610911yescharacteristic
547C22#6611211yescharacteristic
548C22#6611311yescharacteristic
549C22C2×#408511yescharacteristic
550C4—42no—
551C4—42no—
552C4—82no—
553C4—202no—
554C4—202no—
555C4—202no—
556C4—204no—
557C4#6527914yes—
558C4C2×#403811yescharacteristic
559C4C2×#404411yescharacteristic
560C4C2×C4×C22⋊D511yescharacteristic
561C4C2×#409311yescharacteristic
562C2—24no—
563C2—402no—
564C2#1061093211yescharacteristic
565C2#1061093311yescharacteristic
566C2#1061095211yescharacteristic
567C2#1061095311yescharacteristic
568C2C2×#6527911yescharacteristic
569C2#1061566711yescharacteristic
570C2#1061570211yescharacteristic
subgroupG/H
centerC23C2×D4×D5
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC2×C5×#428C2
socleC23×C5C22×D4
prime radicalC22×#709C22
cosocleC22×C4×C5C24
hypercenterC2×#428D5
hypocenterC5#21062
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
1. C2×C5×#428C2semidirect
3. #10613954C2semidirect
6. #10602773C2semidirect
7. C2×#64749C2semidirect
8. #10601417C2semidirect
9. #10601416C2semidirect
10. #10601363C2semidirect
11. #10601362C2semidirect
23. #65422C4semidirect
97. #3953C2×C4semidirect
98. #3948C2×C4semidirect
103. C2×#428D5semidirect
528. C5#21062semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#4282
C2×#654222
#106139542
C2×#652912
#106139172
#106027732
C2×#647492
#106014172
#106014162
#106013632
#106013622
#210625