Finite Group Databasebeta

← #12101737

Subgroups — #12101737

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
1C5×#2274C211yesmaximal, characteristic, complemented
2#10614001C211yesmaximal, characteristic
3C2×#65291C211yesmaximal, characteristic
4C2×#65284C211yesmaximal, characteristic, complemented
5C2×#65255C211yesmaximal, characteristic
6#10602372C211yesmaximal, characteristic, complemented
7C2×#64749C211yesmaximal, characteristic, complemented
8#10600258C212yesmaximal, complemented
9#10600257C212yesmaximal, complemented
10#10600256C212yesmaximal, complemented
11#10600255C212yesmaximal, complemented
12C22×#709C2211yescharacteristic
13C2×C5×#117C2211yescharacteristic
14C2×C5×#117C2211yescharacteristic
15C22×C5×C4⋊C4C2211yescharacteristic
16C22×C4⋊Dic5C2211yescharacteristic
17C22×#681C2211yescharacteristic
18C23×#168C2211yescharacteristic
19C4×C5×C4⋊C4—21no—
20C4×C5×C4⋊C4—21no—
21C4×C5×C4⋊C4C2212yes—
22C4×C5×C4⋊C4C2212yes—
23C4×C5×C22⋊C4—22no—
24#65291—24no—
25#65284—24no—
26#65276—21no—
27#65276—21no—
28C2×#3942C2211yescharacteristic
29C2×#3942C2211yescharacteristic
30C2×#3942C2212yes—
31#65255C418yescomplemented
32C2×#3941C2211yescharacteristic
33C2×#3941C2211yescharacteristic
34C2×#3941C2212yes—
35C2×#3940C2212yes—
36C2×#3939C2212yes—
37#65247—22no—
38#65228C2212yes—
39#65218C2212yes—
40#64749C2212yes—
41#64749C2212yes—
42#64744C2212yes—
43#64744C2212yes—
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
58C5×#117—22no—
59C5×#117—22no—
60C5×#117—22no—
61C5×#117D414yes—
62C2×C5×C4⋊C4—21no—
63C2×C5×C4⋊C4—22no—
64C2×C5×C4⋊C4C2×C414yes—
65C2×C5×C4⋊C4D411yescharacteristic
66C2×C5×C4⋊C4D411yescharacteristic
67C2×C5×C4⋊C4C2312yes—
68C2×C5×C4⋊C4C2311yescharacteristic
69C2×C5×C4⋊C4C2311yescharacteristic
70C2×C5×C22⋊C4—21no—
71C2×C5×C22⋊C4D412yes—
72C2×C42×C5—21no—
73C2×C42×C5—22no—
74C2×C42×C5—22no—
75C2×C42×C5C2312yes—
76C2×#685—22no—
77C2×#685—22no—
78C2×C4⋊Dic5—22no—
79C2×C4⋊Dic5—22no—
80C2×C4⋊Dic5—21no—
81C2×C4⋊Dic5C2312yes—
82C2×#681—22no—
83C2×#681—22no—
84C2×#681C2×C414yes—
85C22×#168—22no—
86C22×#168—22no—
87C22×#168—22no—
88C22×#168C2311yescharacteristic
89C22×#168C2311yescharacteristic
90C22×#168C2312yes—
91C5×#95—22no—
92C5×#95—22no—
93#3963—22no—
94#3963—22no—
95#3942—22no—
96#3942—22no—
97#3942C2×C418yescomplemented
98#3941—22no—
99#3941—22no—
100#3941C2×C418yescomplemented
101#3940—24no—
102#3939—24no—
103C2×#428—51no—
104C2×#417—51no—
105C2×#416—51no—
106#2311—51no—
107#2274D511yescharacteristic, complemented
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—24no—
120C22×C4×C5—81no—
121C22×C4×C5—81no—
122C22×C4×C5C22×C412yes—
123C22×C4×C5C2×D412yes—
124C22×C4×C5C2×D412yes—
125C22×C4×C5C2×D412yes—
126C22×C4×C5C2×D411yescharacteristic
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—22no—
137C22×Dic5—42no—
138C22×Dic5—81no—
139C22×Dic5—81no—
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×C4⋊C4—24no—
160C5×C4⋊C4—22no—
161C5×C4⋊C4—22no—
162C5×C4⋊C4—22no—
163C5×C4⋊C4—21no—
164C5×C4⋊C4—21no—
165C5×C4⋊C4C22×C414yes—
166C5×C4⋊C4C2×D412yes—
167C5×C4⋊C4C2×D412yes—
168C5×C22⋊C4—24no—
169C5×C22⋊C4—24no—
170C42×C5—24no—
171C42×C5—22no—
172C42×C5—22no—
173C42×C5—82no—
174C42×C5—82no—
175#685—82no—
176#685—82no—
177C4⋊Dic5—22no—
178C4⋊Dic5—22no—
179C4⋊Dic5—42no—
180#681—28no—
181C2×#168—22no—
182C2×#168—22no—
183C2×#168—21no—
184C2×#168—21no—
185C2×#168—21no—
186C2×#168—21no—
187C2×#168—21no—
188C2×#168—21no—
189C2×#168—44no—
190C2×#168C22×C414yes—
191C22×#41—51no—
192C22×M16—51no—
193C23×C8—51no—
194C2×#117C2×D511yescharacteristic
195C2×#117C2×D511yescharacteristic
196C22×C4⋊C4—51no—
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—21no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4C2×D512yes—
217C4×C4⋊C4C2×D512yes—
218C4×C22⋊C4—22no—
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—24no—
239C2×C4×C5—22no—
240C2×C4×C5—22no—
241C2×C4×C5—22no—
242C2×C4×C5—22no—
243C2×C4×C5—22no—
244C2×C4×C5—22no—
245C2×C4×C5—22no—
246C2×C4×C5—22no—
247C2×C4×C5—41no—
248C2×C4×C5—42no—
249C2×C4×C5—41no—
250C2×C4×C5—42no—
251C2×C4×C5—41no—
252C2×C4×C5—41no—
253C2×C4×C5—84no—
254C2×C4×C5#11714yes—
255C2×C4×C5C4×D414yes—
256C2×C4×C5C4×D412yes—
257C2×C4×C5#12112yes—
258C2×C4×C5#12312yes—
259C2×C4×C5#12414yes—
260C2×C4×C5#12614yes—
261C2×C4×C5C23×C411yescharacteristic
262C2×C4×C5C22×D411yescharacteristic
263C2×C4×C5C2×#4111yescharacteristic
264C2×C4×C5C2×#4111yescharacteristic
265C2×C4×C5C2×#4111yescharacteristic
266C2×C4×C5C2×#4111yescharacteristic
267C22⋊D5—48no—
268C2×Dic5—22no—
269C2×Dic5—41no—
270C2×Dic5—41no—
271C2×Dic5—42no—
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—
284#168—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—54no—
301C2×M16—102no—
302C2×M16—102no—
303C22×C8—52no—
304C22×C8—51no—
305C22×C8—51no—
306C22×C8—102no—
307C22×C8—102no—
308C22×C8—102no—
309#117—22no—
310#117—22no—
311#117—22no—
312#117D2014yes—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—52no—
316C2×C4⋊C4—102no—
317C2×C4⋊C4—102no—
318C2×C4⋊C4—101no—
319C2×C4⋊C4C4×D514yes—
320C2×C4⋊C4C22⋊D511yescharacteristic
321C2×C4⋊C4C22⋊D511yescharacteristic
322C2×C4⋊C4C22×D512yes—
323C2×C4⋊C4C22×D511yescharacteristic
324C2×C4⋊C4C22×D511yescharacteristic
325C2×C22⋊C4—21no—
326C2×C22⋊C4—102no—
327C2×C22⋊C4—102no—
328C2×C22⋊C4C22⋊D512yes—
329C2×C42—21no—
330C2×C42—22no—
331C2×C42—22no—
332C2×C42C22×D512yes—
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—22no—
342#95—22no—
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—44no—
369C4×C5—42no—
370C4×C5—42no—
371C4×C5#38814yes—
372C4×C5C2×#11711yescharacteristic
373C4×C5C2×C4×D411yescharacteristic
374C4×C5C2×#12411yescharacteristic
375C4×C5C2×#12611yescharacteristic
376Dic5—42no—
377Dic5—82no—
378C24—101no—
379C24C2×C22⋊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—24no—
403C22×C4—81no—
404C22×C4—81no—
405C22×C4—101no—
406C22×C4—104no—
407C22×C4—102no—
408C22×C4—202no—
409C22×C4—202no—
410C22×C4—202no—
411C22×C4—401no—
412C22×C4—401no—
413C22×C4C2×C4×D512yes—
414C22×C4C2×D2012yes—
415C22×C4#70912yes—
416C22×C4#70912yes—
417C22×C4#70912yes—
418C22×C4#70912yes—
419C22×C4C2×C22⋊D512yes—
420C22×C4C2×C22⋊D512yes—
421C22×C4C2×C22⋊D511yescharacteristic
422C22×C4C23×D511yescharacteristic
423M16—108no—
424C2×C8—54no—
425C2×C8—102no—
426C2×C8—102no—
427C2×C8—101no—
428C2×C8—101no—
429C2×C8—101no—
430C2×C8—101no—
431C2×C8—101no—
432C2×C8—101no—
433C2×C8—204no—
434C4⋊C4—22no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—24no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—102no—
441C4⋊C4—102no—
442C4⋊C4—202no—
443C4⋊C4C2×C4×D514yes—
444C4⋊C4C2×D2012yes—
445C4⋊C4C2×D2012yes—
446C22⋊C4—24no—
447C22⋊C4—24no—
448C22⋊C4—402no—
449C22⋊C4—402no—
450C42—22no—
451C42—22no—
452C42—24no—
453C42—82no—
454C42—82no—
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—
475C23C4×C22⋊D512yes—
476C23#407512yes—
477C23#407612yes—
478C23C22×C22⋊D511yescharacteristic
479Q8—102no—
480Q8—102no—
481Q8—201no—
482Q8—201no—
483D4—102no—
484D4—102no—
485D4—208no—
486D4—201no—
487D4—201no—
488C2×C4—24no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—41no—
500C2×C4—42no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—84no—
504C2×C4—102no—
505C2×C4—201no—
506C2×C4—201no—
507C2×C4—208no—
508C2×C4—202no—
509C2×C4—202no—
510C2×C4—202no—
511C2×C4—404no—
512C2×C4#401814yes—
513C2×C4C4×D2014yes—
514C2×C4#402114yes—
515C2×C4#402214yes—
516C2×C4C4×C22⋊D512yes—
517C2×C4#407512yes—
518C2×C4#407612yes—
519C2×C4C22×C4×D511yescharacteristic
520C2×C4C22×D2011yescharacteristic
521C2×C4C2×#70911yescharacteristic
522C2×C4C2×#70911yescharacteristic
523C2×C4C2×#70911yescharacteristic
524C2×C4C2×#70911yescharacteristic
525C8—104no—
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#6522914yes—
543C22C2×C4×C22⋊D511yescharacteristic
544C22C2×#407511yescharacteristic
545C22C2×#407611yescharacteristic
546C22#6616911yescharacteristic
547C22#6617011yescharacteristic
548C22#6617111yescharacteristic
549C22#6617211yescharacteristic
550C4—42no—
551C4—42no—
552C4—42no—
553C4—42no—
554C4—44no—
555C4—202no—
556C4—402no—
557C4#6522914yes—
558C4C2×#401811yescharacteristic
559C4C2×C4×D2011yescharacteristic
560C4C2×#402111yescharacteristic
561C4C2×#402211yescharacteristic
562C2—24no—
563C2—402no—
564C2C2×#6522911yescharacteristic
565C2#1061426811yescharacteristic
566C2#1061427811yescharacteristic
567C2#1061429611yescharacteristic
568C2#1061429811yescharacteristic
569C2#1061430311yescharacteristic
570C2#1061430511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D5
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC5×#2274C2
socleC23×C5C22×D4
prime radicalC22×#709C22
cosocleC22×C4×C5C24
hypercenter#2274D5
hypocenterC5#21062
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2274

Complements

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

NH ≅ G/Ntype
1. C5×#2274C2semidirect
4. C2×#65284C2semidirect
6. #10602372C2semidirect
7. C2×#64749C2semidirect
8. #10600258C2semidirect
9. #10600257C2semidirect
10. #10600256C2semidirect
11. #10600255C2semidirect
31. #65255C4semidirect
97. #3942C2×C4semidirect
100. #3941C2×C4semidirect
107. #2274D5semidirect
528. C5#21062semidirect

Maximal subgroups

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

subgroupindex
C5×#22742
#106140012
C2×#652912
C2×#652842
C2×#652552
#106023722
C2×#647492
#106002582
#106002572
#106002562
#106002552
#210625