Finite Group Databasebeta

← #12127979

Subgroups — #12127979

Order 1280.

1102 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#2317C211yesmaximal, characteristic, complemented
2#10614001C211yesmaximal, characteristic
3C2×#65291C211yesmaximal, characteristic
4C2×#65284C211yesmaximal, characteristic, complemented
5C2×#65288C211yesmaximal, characteristic, complemented
6C2×#65255C211yesmaximal, characteristic
7C2×#64749C211yesmaximal, characteristic, complemented
8#10600428C212yesmaximal, complemented
9#10600427C212yesmaximal, complemented
10#10600423C212yesmaximal, complemented
11#10600422C212yesmaximal, complemented
12C22×#709C2211yescharacteristic
13C2×C5×#117C2211yescharacteristic
14C2×C5×#117C2211yescharacteristic
15C22×C5×C4⋊C4C2211yescharacteristic
16C22×C4⋊Dic5C2211yescharacteristic
17C22×#681C2211yescharacteristic
18C23×#168C2211yescharacteristic
19C5×#389C2212yes—
20C5×#389C2212yes—
21C5×#385—22no—
22C5×#384—21no—
23C5×#382—21no—
24#65291—24no—
25#65288—24no—
26#65284—24no—
27#65276—21no—
28#65276—21no—
29C2×#3942C2212yes—
30C2×#3942C2211yescharacteristic
31C2×#3942C2211yescharacteristic
32#65255—24no—
33C2×#3941C2212yes—
34C2×#3941C2211yescharacteristic
35C2×#3941C2211yescharacteristic
36C2×#3940C2212yes—
37C2×#3939C2212yes—
38#65247—22no—
39#65228C2212yes—
40#65218C2212yes—
41C2×C4⋊#168C2212yes—
42C2×C4⋊#168C2212yes—
43#64749C2212yes—
44#64749C2212yes—
45#23616—51nomaximal, Sylow
46C23×C4×C5—21no—
47C23×C4×C5C2311yescharacteristic
48C22×C22⋊D5—21no—
49C23×Dic5—21no—
50C2×#709—22no—
51C2×#709—22no—
52C2×#709—22no—
53C2×#709—22no—
54C2×#709—22no—
55C2×#709Q814yes—
56C22×D20C2311yescharacteristic
57C22×C4×D5—21no—
58C22×Dic10C2311yescharacteristic
59C5×#117—22no—
60C5×#117—22no—
61C5×#117—22no—
62C5×#117D414yes—
63C2×C5×C4⋊C4—22no—
64C2×C5×C4⋊C4—22no—
65C2×C5×C4⋊C4—21no—
66C2×C5×C4⋊C4—22no—
67C2×C5×C4⋊C4—22no—
68C2×C5×C4⋊C4D411yescharacteristic
69C2×C5×C4⋊C4D411yescharacteristic
70C2×C5×C4⋊C4C2312yes—
71C2×C5×C4⋊C4C2311yescharacteristic
72C2×C5×C4⋊C4C2311yescharacteristic
73C2×C5×C22⋊C4—21no—
74C2×C5×C22⋊C4D412yes—
75C2×C42×C5—21no—
76C2×C42×C5C2312yes—
77C2×#685—22no—
78C2×#685—22no—
79C2×C4⋊Dic5—22no—
80C2×C4⋊Dic5—22no—
81C2×C4⋊Dic5—21no—
82C2×C4⋊Dic5C2312yes—
83C2×#681—22no—
84C2×#681—22no—
85C2×#681D414yes—
86C22×#168—22no—
87C22×#168—22no—
88C22×#168—22no—
89C22×#168C2311yescharacteristic
90C22×#168C2311yescharacteristic
91C22×#168C2312yes—
92C5×#95—22no—
93C5×#95—22no—
94#3963—22no—
95#3963—22no—
96#3942—22no—
97#3942—22no—
98#3942—24no—
99#3941—22no—
100#3941—22no—
101#3941—24no—
102#3940—24no—
103#3939—24no—
104C4⋊#168—24no—
105C4⋊#168—24no—
106C2×#428—51no—
107C2×#424—51no—
108C2×#417—51no—
109C2×#416—51no—
110#2317D511yescharacteristic, complemented
111#2311—51no—
112#2068—52no—
113#2067—52no—
114#2066—52no—
115#2065—52no—
116C2×#340—51no—
117C24×C5C2×D411yescharacteristic
118C23×D5—21no—
119C22×C4×C5—21no—
120C22×C4×C5—21no—
121C22×C4×C5—22no—
122C22×C4×C5—42no—
123C22×C4×C5—81no—
124C22×C4×C5—81no—
125C22×C4×C5C2×D412yes—
126C22×C4×C5C2×D412yes—
127C22×C4×C5C2×D412yes—
128C22×C4×C5C2×D412yes—
129C22×C4×C5C2×D411yescharacteristic
130C22×C4×C5C2×Q812yes—
131C22×C4×C5#4112yes—
132C22×C4×C5#4112yes—
133C22×C4×C5#4112yes—
134C22×C4×C5C2411yescharacteristic
135C2×C22⋊D5—24no—
136C2×C22⋊D5—42no—
137C2×C22⋊D5—42no—
138C22×Dic5—21no—
139C22×Dic5—22no—
140C22×Dic5—42no—
141C22×Dic5—81no—
142C22×Dic5—81no—
143#709—28no—
144#709—28no—
145#709—44no—
146#709—44no—
147C2×D20—21no—
148C2×D20—21no—
149C2×D20—21no—
150C2×D20—21no—
151C2×D20—21no—
152C2×D20C2×Q812yes—
153C2×C4×D5—24no—
154C2×C4×D5—42no—
155C2×C4×D5—42no—
156C2×Dic10—21no—
157C2×Dic10—21no—
158C2×Dic10—21no—
159C2×Dic10—21no—
160C2×Dic10—21no—
161C2×Dic10C2×Q812yes—
162C5×C4⋊C4—22no—
163C5×C4⋊C4—22no—
164C5×C4⋊C4—21no—
165C5×C4⋊C4—21no—
166C5×C4⋊C4—22no—
167C5×C4⋊C4—22no—
168C5×C4⋊C4—42no—
169C5×C4⋊C4—82no—
170C5×C4⋊C4—82no—
171C5×C4⋊C4C2×D412yes—
172C5×C4⋊C4C2×D412yes—
173C5×C22⋊C4—24no—
174C5×C22⋊C4—24no—
175C42×C5—24no—
176C42×C5—22no—
177C42×C5—22no—
178#685—82no—
179#685—82no—
180C4⋊Dic5—22no—
181C4⋊Dic5—22no—
182C4⋊Dic5—42no—
183#681—28no—
184C2×#168—22no—
185C2×#168—22no—
186C2×#168—21no—
187C2×#168—21no—
188C2×#168—21no—
189C2×#168—21no—
190C2×#168—21no—
191C2×#168—21no—
192C2×#168—44no—
193C2×#168C2×D414yes—
194C22×#41—51no—
195C22×M16—51no—
196C23×C8—51no—
197C2×#117C2×D511yescharacteristic
198C2×#117C2×D511yescharacteristic
199C22×C4⋊C4—51no—
200C22×C4⋊C4C2×D511yescharacteristic
201#428—104no—
202C2×#107—52no—
203C2×#106—52no—
204#424—104no—
205C2×C4⋊C8—52no—
206C2×C4⋊C8—52no—
207#417—104no—
208#416—104no—
209C2×#103—51no—
210C2×#103—52no—
211C2×#103—51no—
212C2×#102—51no—
213C2×#102—52no—
214C2×#102—51no—
215#391—52no—
216#390—52no—
217#389C2×D512yes—
218#389C2×D512yes—
219#386—102no—
220#385—22no—
221#384—21no—
222#382—21no—
223#380—101no—
224#380—101no—
225#340—52no—
226#340—52no—
227C23×C5—21no—
228C23×C5—21no—
229C23×C5—21no—
230C23×C5—21no—
231C23×C5#12112yes—
232C23×C5#12112yes—
233C23×C5C22⋊Q812yes—
234C23×C5C22×D411yescharacteristic
235C22×D5—22no—
236C22×D5—41no—
237C22×D5—41no—
238C22×D5—41no—
239C22×D5—41no—
240C22×D5—41no—
241C22×D5—41no—
242C2×C4×C5—22no—
243C2×C4×C5—22no—
244C2×C4×C5—22no—
245C2×C4×C5—22no—
246C2×C4×C5—22no—
247C2×C4×C5—22no—
248C2×C4×C5—22no—
249C2×C4×C5—22no—
250C2×C4×C5—41no—
251C2×C4×C5—41no—
252C2×C4×C5—41no—
253C2×C4×C5—41no—
254C2×C4×C5—42no—
255C2×C4×C5—42no—
256C2×C4×C5—42no—
257C2×C4×C5—84no—
258C2×C4×C5#12114yes—
259C2×C4×C5#12112yes—
260C2×C4×C5#12112yes—
261C2×C4×C5C22⋊Q814yes—
262C2×C4×C5C22⋊Q814yes—
263C2×C4×C5C22⋊Q812yes—
264C2×C4×C5#12614yes—
265C2×C4×C5C22×D411yescharacteristic
266C2×C4×C5C22×D411yescharacteristic
267C2×C4×C5C22×Q811yescharacteristic
268C2×C4×C5C2×#4111yescharacteristic
269C2×C4×C5C2×#4111yescharacteristic
270C2×C4×C5C2×#4111yescharacteristic
271C22⋊D5—48no—
272C2×Dic5—22no—
273C2×Dic5—41no—
274C2×Dic5—41no—
275C2×Dic5—42no—
276C2×Dic5—42no—
277C2×Dic5—42no—
278C2×Dic5—84no—
279D20—22no—
280D20—22no—
281D20—41no—
282D20—41no—
283C4×D5—48no—
284Dic10—22no—
285Dic10—22no—
286Dic10—41no—
287Dic10—41no—
288#168—24no—
289#168—41no—
290#168—41no—
291C2×#41—54no—
292C2×#41—102no—
293C2×#41—102no—
294C2×#41—102no—
295C2×#41—102no—
296C2×#41—102no—
297C22×Q8—51no—
298C22×D4—51no—
299C22×D4—101no—
300C23×C4—21no—
301C23×C4—101no—
302C23×C4—101no—
303C23×C4C22×D511yescharacteristic
304C2×M16—54no—
305C2×M16—102no—
306C2×M16—102no—
307C22×C8—52no—
308C22×C8—51no—
309C22×C8—51no—
310C22×C8—102no—
311C22×C8—102no—
312C22×C8—102no—
313#117—22no—
314#117—22no—
315#117—22no—
316#117D2014yes—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—52no—
323C2×C4⋊C4—102no—
324C2×C4⋊C4—102no—
325C2×C4⋊C4—101no—
326C2×C4⋊C4C22⋊D511yescharacteristic
327C2×C4⋊C4C22⋊D511yescharacteristic
328C2×C4⋊C4C22×D512yes—
329C2×C4⋊C4C22×D511yescharacteristic
330C2×C4⋊C4C22×D511yescharacteristic
331C2×C22⋊C4—21no—
332C2×C22⋊C4—102no—
333C2×C22⋊C4—102no—
334C2×C22⋊C4C22⋊D512yes—
335C2×C42—21no—
336C2×C42C22×D512yes—
337#107—104no—
338#106—104no—
339C4⋊C8—104no—
340C4⋊C8—104no—
341#103—104no—
342#103—102no—
343#103—102no—
344#102—104no—
345#102—102no—
346#102—102no—
347#95—22no—
348#95—22no—
349#95—102no—
350#95—102no—
351C22×C5—22no—
352C22×C5—22no—
353C22×C5—22no—
354C22×C5—22no—
355C22×C5—22no—
356C22×C5—22no—
357C22×C5#39614yes—
358C22×C5C2×#12111yescharacteristic
359C22×C5C2×#12111yescharacteristic
360C22×C5C2×C22⋊Q811yescharacteristic
361C22×C5#57611yescharacteristic
362C22×C5#57711yescharacteristic
363C22×C5#57711yescharacteristic
364C22×C5#57811yescharacteristic
365C2×D5—42no—
366C2×D5—42no—
367C2×D5—42no—
368C2×D5—81no—
369C2×D5—81no—
370C2×D5—81no—
371C2×D5—81no—
372C4×C5—42no—
373C4×C5—42no—
374C4×C5—42no—
375C4×C5—42no—
376C4×C5—82no—
377C4×C5#39614yes—
378C4×C5C2×#12111yescharacteristic
379C4×C5C2×C22⋊Q811yescharacteristic
380C4×C5C2×C22⋊Q811yescharacteristic
381C4×C5C2×#12611yescharacteristic
382Dic5—42no—
383Dic5—82no—
384C24—101no—
385C24C2×C22⋊D511yescharacteristic
386#41—108no—
387#41—108no—
388#41—204no—
389#41—204no—
390C2×Q8—52no—
391C2×Q8—101no—
392C2×Q8—101no—
393C2×Q8—101no—
394C2×Q8—101no—
395C2×Q8—101no—
396C2×D4—52no—
397C2×D4—104no—
398C2×D4—101no—
399C2×D4—101no—
400C2×D4—101no—
401C2×D4—101no—
402C2×D4—101no—
403C2×D4—202no—
404C2×D4—202no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—42no—
409C22×C4—81no—
410C22×C4—81no—
411C22×C4—101no—
412C22×C4—104no—
413C22×C4—102no—
414C22×C4—202no—
415C22×C4—202no—
416C22×C4—202no—
417C22×C4—401no—
418C22×C4—401no—
419C22×C4C2×D2012yes—
420C22×C4#70912yes—
421C22×C4D4×D512yes—
422C22×C4#71112yes—
423C22×C4Q8×D512yes—
424C22×C4#71312yes—
425C22×C4C2×C22⋊D512yes—
426C22×C4C2×C22⋊D512yes—
427C22×C4C2×C22⋊D511yescharacteristic
428C22×C4C23×D511yescharacteristic
429M16—108no—
430C2×C8—54no—
431C2×C8—102no—
432C2×C8—102no—
433C2×C8—101no—
434C2×C8—101no—
435C2×C8—101no—
436C2×C8—101no—
437C2×C8—101no—
438C2×C8—101no—
439C2×C8—204no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—42no—
447C4⋊C4—82no—
448C4⋊C4—82no—
449C4⋊C4—102no—
450C4⋊C4—102no—
451C4⋊C4—202no—
452C4⋊C4C2×D2012yes—
453C4⋊C4C2×D2012yes—
454C22⋊C4—24no—
455C22⋊C4—24no—
456C22⋊C4—402no—
457C22⋊C4—402no—
458C42—22no—
459C42—22no—
460C42—24no—
461C2×C5—24no—
462C2×C5C2×#39611yescharacteristic
463C2×C5#304511yescharacteristic
464C2×C5#304611yescharacteristic
465C2×C5#306211yescharacteristic
466C2×C5#306311yescharacteristic
467C2×C5#308311yescharacteristic
468C2×C5#308411yescharacteristic
469D5—82no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—102no—
475C23—201no—
476C23—201no—
477C23—201no—
478C23—201no—
479C23—201no—
480C23—201no—
481C23#407612yes—
482C23#408512yes—
483C23#409212yes—
484C23C22×C22⋊D511yescharacteristic
485Q8—102no—
486Q8—102no—
487Q8—201no—
488Q8—201no—
489D4—102no—
490D4—102no—
491D4—208no—
492D4—201no—
493D4—201no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—84no—
510C2×C4—102no—
511C2×C4—201no—
512C2×C4—201no—
513C2×C4—208no—
514C2×C4—202no—
515C2×C4—202no—
516C2×C4—202no—
517C2×C4—404no—
518C2×C4#404114yes—
519C2×C4#404214yes—
520C2×C4#404314yes—
521C2×C4#404414yes—
522C2×C4#407612yes—
523C2×C4#408512yes—
524C2×C4#409212yes—
525C2×C4C22×D2011yescharacteristic
526C2×C4C2×#70911yescharacteristic
527C2×C4C2×D4×D511yescharacteristic
528C2×C4C2×#71111yescharacteristic
529C2×C4C2×Q8×D511yescharacteristic
530C2×C4C2×#71311yescharacteristic
531C8—104no—
532C8—201no—
533C8—201no—
534C5#2361611yescharacteristic, complemented, Sylow
535C22—22no—
536C22—22no—
537C22—22no—
538C22—22no—
539C22—22no—
540C22—22no—
541C22—202no—
542C22—202no—
543C22—202no—
544C22—401no—
545C22—401no—
546C22—401no—
547C22—401no—
548C22#6528314yes—
549C22C2×#407611yescharacteristic
550C22C2×#408511yescharacteristic
551C22C2×#409211yescharacteristic
552C22#6616911yescharacteristic
553C22#6617011yescharacteristic
554C22#6617111yescharacteristic
555C22#6617211yescharacteristic
556C4—42no—
557C4—42no—
558C4—42no—
559C4—42no—
560C4—82no—
561C4—202no—
562C4—402no—
563C4#6528314yes—
564C4C2×#404111yescharacteristic
565C4C2×#404211yescharacteristic
566C4C2×#404311yescharacteristic
567C4C2×#404411yescharacteristic
568C2—24no—
569C2—402no—
570C2C2×#6528311yescharacteristic
571C2#1061429611yescharacteristic
572C2#1061430311yescharacteristic
573C2#1061511011yescharacteristic
574C2#1061513111yescharacteristic
575C2#1061520611yescharacteristic
576C2#1061520711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D5
commutator, cosocleC22×C4×C5C24
FrattiniC22×C4C23×D5
FittingC5×#2317C2
socleC23×C5C22×D4
prime radicalC22×#709C22
hypercenter#2317D5
hypocenterC5#23616
LCSG ▷ C22×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2317

Complements

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

NH ≅ G/Ntype
1. C5×#2317C2semidirect
4. C2×#65284C2semidirect
5. C2×#65288C2semidirect
7. C2×#64749C2semidirect
8. #10600428C2semidirect
9. #10600427C2semidirect
10. #10600423C2semidirect
11. #10600422C2semidirect
110. #2317D5semidirect
534. C5#23616semidirect

Maximal subgroups

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

subgroupindex
C5×#23172
#106140012
C2×#652912
C2×#652842
C2×#652882
C2×#652552
C2×#647492
#106004282
#106004272
#106004232
#106004222
#236165