Finite Group Databasebeta

← #12027331

Subgroups — #12027331

Order 1280.

1606 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#385C211yesmaximal, characteristic
2C2×#65501C212yesmaximal
3C2×#65270C212yesmaximal, complemented
4#10602358C212yesmaximal, complemented
5#10602341C212yesmaximal
6#10602338C211yesmaximal, characteristic
7#10602338C211yesmaximal, characteristic
8#10600205C212yesmaximal, complemented
9#10600200C212yesmaximal, complemented
10C24×C4×C5C2211yescharacteristic
11C22×C5×C4⋊C4C2212yes—
12C22×C5×C22⋊C4C2211yescharacteristic
13C22×C5×C22⋊C4C2211yescharacteristic
14C22×C22⋊Dic5C2212yes—
15C22×C22⋊Dic5C2212yes—
16C22×C4⋊Dic5C2212yes—
17C22×C4⋊Dic5C2212yes—
18C22×C4×Dic5C2211yescharacteristic
19C22×C4×Dic5C2211yescharacteristic
20C5×#385C418yescomplemented
21C2×C5×#95C2212yes—
22#65501—28no—
23C2×#3963C2212yes—
24C2×#3963C2212yes—
25C2×#3963C2212yes—
26C2×#3963C2212yes—
27C2×#3963C2212yes—
28C2×#3963C2212yes—
29C2×#3963C2211yescharacteristic
30C2×#3963C2212yes—
31C2×#3963C2211yescharacteristic
32C2×#3963C2212yes—
33#65270—28no—
34#14409—51nomaximal, Sylow
35C25×C5C2311yescharacteristic
36C23×C4×C5—24no—
37C23×C4×C5—24no—
38C23×C4×C5—41no—
39C23×C4×C5—41no—
40C23×C4×C5C2×C414yes—
41C23×C4×C5C2311yescharacteristic
42C23×C4×C5C2312yes—
43C23×C4×C5C2311yescharacteristic
44C23×C4×C5C2312yes—
45C23×Dic5C2312yes—
46C23×Dic5C2312yes—
47C23×Dic5C2312yes—
48C23×Dic5C2312yes—
49C2×C5×C4⋊C4—24no—
50C2×C5×C4⋊C4—24no—
51C2×C5×C4⋊C4C2×C418yes—
52C2×C5×C22⋊C4—24no—
53C2×C5×C22⋊C4—41no—
54C2×C5×C22⋊C4—41no—
55C2×C5×C22⋊C4C2×C414yes—
56C2×C5×C22⋊C4C2×C414yes—
57C2×C22⋊Dic5—24no—
58C2×C22⋊Dic5—24no—
59C2×C22⋊Dic5—24no—
60C2×C22⋊Dic5—24no—
61C2×C22⋊Dic5—42no—
62C2×C22⋊Dic5—42no—
63C2×C4⋊Dic5—24no—
64C2×C4⋊Dic5—24no—
65C2×C4⋊Dic5—24no—
66C2×C4⋊Dic5—24no—
67C2×C4⋊Dic5—42no—
68C2×C4⋊Dic5—42no—
69C2×C4×Dic5—24no—
70C2×C4×Dic5—41no—
71C2×C4×Dic5—42no—
72C2×C4×Dic5—41no—
73C5×#95C2×C418yes—
74#3963—24no—
75#3963—24no—
76#3963—24no—
77#3963—24no—
78#3963—42no—
79#3963—42no—
80#3963—42no—
81#3963—41no—
82#3963—41no—
83#3963D418yes—
84C2×#387—52no—
85C2×#385D511yescharacteristic
86C2×#380—52no—
87#1442—52no—
88#1441—52no—
89#1440—52no—
90#1437—51no—
91#1437—51no—
92#1436—52no—
93C24×C5—22no—
94C24×C5—22no—
95C24×C5—22no—
96C24×C5—21no—
97C24×C5—21no—
98C24×C5—22no—
99C24×C5—22no—
100C24×C5C22×C412yes—
101C24×C5#4114yes—
102C24×C5C2411yescharacteristic
103C22×C4×C5—22no—
104C22×C4×C5—22no—
105C22×C4×C5—22no—
106C22×C4×C5—22no—
107C22×C4×C5—22no—
108C22×C4×C5—22no—
109C22×C4×C5—22no—
110C22×C4×C5—22no—
111C22×C4×C5—22no—
112C22×C4×C5—22no—
113C22×C4×C5—21no—
114C22×C4×C5—21no—
115C22×C4×C5—21no—
116C22×C4×C5—21no—
117C22×C4×C5—22no—
118C22×C4×C5—22no—
119C22×C4×C5—22no—
120C22×C4×C5—22no—
121C22×C4×C5—216no—
122C22×C4×C5—44no—
123C22×C4×C5—44no—
124C22×C4×C5—48no—
125C22×C4×C5—44no—
126C22×C4×C5C22×C412yes—
127C22×C4×C5C22×C414yes—
128C22×C4×C5C22×C412yes—
129C22×C4×C5C22×C414yes—
130C22×C4×C5C2×D414yes—
131C22×C4×C5#4114yes—
132C22×Dic5—22no—
133C22×Dic5—22no—
134C22×Dic5—22no—
135C22×Dic5—22no—
136C22×Dic5—22no—
137C22×Dic5—22no—
138C22×Dic5—22no—
139C22×Dic5—22no—
140C22×Dic5—22no—
141C22×Dic5—22no—
142C22×Dic5—22no—
143C22×Dic5—22no—
144C22×Dic5—22no—
145C22×Dic5—22no—
146C22×Dic5—22no—
147C22×Dic5—22no—
148C22×Dic5—22no—
149C22×Dic5—22no—
150C22×Dic5—22no—
151C22×Dic5—22no—
152C22×Dic5—22no—
153C22×Dic5—22no—
154C22×Dic5—22no—
155C22×Dic5—22no—
156C22×Dic5—22no—
157C22×Dic5—22no—
158C22×Dic5C2×D414yes—
159C22×Dic5C2×D414yes—
160C5×C4⋊C4—216no—
161C5×C22⋊C4—28no—
162C5×C22⋊C4—44no—
163C22⋊Dic5—48no—
164C22⋊Dic5—48no—
165C4⋊Dic5—48no—
166C4⋊Dic5—48no—
167C4×Dic5—44no—
168C4×Dic5—44no—
169C24×C4C2×D511yescharacteristic
170C22×C4⋊C4—52no—
171C22×C4⋊C4—52no—
172C22×C4⋊C4C2×D512yes—
173C22×C22⋊C4—52no—
174C22×C22⋊C4—52no—
175C22×C22⋊C4C2×D511yescharacteristic
176C22×C22⋊C4C2×D511yescharacteristic
177C22×C42—51no—
178C22×C42—51no—
179#387—108no—
180#385Dic518yescomplemented
181#380—108no—
182C2×#95—52no—
183C2×#95—52no—
184C2×#95—52no—
185C2×#95—52no—
186C2×#95—52no—
187C2×#95—52no—
188C2×#95—52no—
189C2×#95—51no—
190C2×#95—51no—
191C2×#95—52no—
192C2×#95C2×D512yes—
193C23×C5—24no—
194C23×C5—22no—
195C23×C5—22no—
196C23×C5—24no—
197C23×C5—24no—
198C23×C5—24no—
199C23×C5—24no—
200C23×C5—24no—
201C23×C5—24no—
202C23×C5—24no—
203C23×C5—42no—
204C23×C5—42no—
205C23×C5—42no—
206C23×C5—42no—
207C23×C5—41no—
208C23×C5—42no—
209C23×C5—41no—
210C23×C5—41no—
211C23×C5—42no—
212C23×C5—41no—
213C23×C5#11714yes—
214C23×C5C23×C411yescharacteristic
215C23×C5C22×D412yes—
216C23×C5C2×#4112yes—
217C23×C5C2×#4112yes—
218C23×C5H(5,𝔽2)11yescharacteristic
219C23×C5H(5,𝔽2)11yescharacteristic
220C23×C5H(5,𝔽2)12yes—
221C23×C52−1+412yes—
222C23×C52−1+412yes—
223C2×C4×C5—24no—
224C2×C4×C5—24no—
225C2×C4×C5—24no—
226C2×C4×C5—24no—
227C2×C4×C5—24no—
228C2×C4×C5—24no—
229C2×C4×C5—24no—
230C2×C4×C5—24no—
231C2×C4×C5—24no—
232C2×C4×C5—22no—
233C2×C4×C5—22no—
234C2×C4×C5—24no—
235C2×C4×C5—24no—
236C2×C4×C5—416no—
237C2×C4×C5—41no—
238C2×C4×C5—42no—
239C2×C4×C5—42no—
240C2×C4×C5—42no—
241C2×C4×C5—41no—
242C2×C4×C5—42no—
243C2×C4×C5—42no—
244C2×C4×C5—41no—
245C2×C4×C5—42no—
246C2×C4×C5—41no—
247C2×C4×C5C4×D418yes—
248C2×Dic5—24no—
249C2×Dic5—24no—
250C2×Dic5—24no—
251C2×Dic5—24no—
252C2×Dic5—24no—
253C2×Dic5—24no—
254C2×Dic5—42no—
255C2×Dic5—42no—
256C2×Dic5—42no—
257C2×Dic5—42no—
258C2×Dic5—42no—
259C2×Dic5—42no—
260C2×Dic5—42no—
261C2×Dic5—42no—
262C2×Dic5—42no—
263C2×Dic5—42no—
264C2×Dic5—42no—
265C2×Dic5—42no—
266C2×Dic5—42no—
267C2×Dic5—42no—
268C2×Dic5—42no—
269C2×Dic5—42no—
270C2×Dic5—42no—
271C2×Dic5—42no—
272C2×Dic5—42no—
273C2×Dic5—42no—
274C2×Dic5—42no—
275C2×Dic5—42no—
276C25C22×D511yescharacteristic
277C23×C4—24no—
278C23×C4—24no—
279C23×C4—41no—
280C23×C4—41no—
281C23×C4—52no—
282C23×C4—52no—
283C23×C4—52no—
284C23×C4—52no—
285C23×C4C2×Dic514yes—
286C23×C4C22×D511yescharacteristic
287C23×C4C22×D512yes—
288C23×C4C22×D511yescharacteristic
289C23×C4C22×D512yes—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—104no—
293C2×C4⋊C4—104no—
294C2×C4⋊C4—104no—
295C2×C4⋊C4—104no—
296C2×C4⋊C4—202no—
297C2×C4⋊C4—202no—
298C2×C4⋊C4C2×Dic518yes—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—41no—
301C2×C22⋊C4—41no—
302C2×C22⋊C4—104no—
303C2×C22⋊C4—104no—
304C2×C22⋊C4—104no—
305C2×C22⋊C4—104no—
306C2×C22⋊C4—202no—
307C2×C22⋊C4—202no—
308C2×C22⋊C4C2×Dic514yes—
309C2×C22⋊C4C2×Dic514yes—
310C2×C42—104no—
311C2×C42—201no—
312C2×C42—201no—
313C2×C42—202no—
314#95—58no—
315#95—104no—
316#95—104no—
317#95—104no—
318#95—104no—
319#95—201no—
320#95—201no—
321#95—202no—
322#95—202no—
323#95—202no—
324#95C2×Dic518yes—
325C22×C5—24no—
326C22×C5—28no—
327C22×C5—44no—
328C22×C5—42no—
329C22×C5—42no—
330C22×C5—44no—
331C22×C5—44no—
332C22×C5—44no—
333C22×C5—44no—
334C22×C5C2×#11711yescharacteristic
335C22×C5C2×C4×D412yes—
336C22×C5#51811yescharacteristic
337C22×C5#51911yescharacteristic
338C22×C5#52012yes—
339C22×C5#53612yes—
340C22×C5#53712yes—
341C22×C5#53912yes—
342C22×C5#53912yes—
343C22×C5#54112yes—
344C22×C5#54212yes—
345C22×C5D4211yescharacteristic
346C22×C5#54612yes—
347C22×C5#54612yes—
348C22×C5#54812yes—
349C22×C5#55111yescharacteristic
350C22×C5#55212yes—
351C22×C5#55312yes—
352C22×C5#56011yescharacteristic
353C22×C5#56212yes—
354C22×C5#56311yescharacteristic
355C4×C5—28no—
356C4×C5—44no—
357C4×C5—42no—
358C4×C5—42no—
359Dic5—44no—
360Dic5—44no—
361Dic5—82no—
362Dic5—82no—
363C24—22no—
364C24—22no—
365C24—21no—
366C24—21no—
367C24—22no—
368C24—22no—
369C24—22no—
370C24#70914yes—
371C24C22×Dic512yes—
372C24C23×D511yescharacteristic
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—216no—
392C22×C4—44no—
393C22×C4—44no—
394C22×C4—44no—
395C22×C4—48no—
396C22×C4—54no—
397C22×C4—54no—
398C22×C4—102no—
399C22×C4—102no—
400C22×C4—102no—
401C22×C4—102no—
402C22×C4—102no—
403C22×C4—102no—
404C22×C4—102no—
405C22×C4—102no—
406C22×C4—102no—
407C22×C4—102no—
408C22×C4—102no—
409C22×C4—102no—
410C22×C4—102no—
411C22×C4—102no—
412C22×C4—102no—
413C22×C4—102no—
414C22×C4—102no—
415C22×C4—102no—
416C22×C4—102no—
417C22×C4—102no—
418C22×C4—102no—
419C22×C4—102no—
420C22×C4—102no—
421C22×C4—102no—
422C22×C4—102no—
423C22×C4—102no—
424C22×C4D4×D514yes—
425C22×C4#71114yes—
426C22×C4C22×Dic512yes—
427C22×C4C22×Dic514yes—
428C22×C4C22×Dic512yes—
429C22×C4C22×Dic514yes—
430C4⋊C4—216no—
431C4⋊C4—208no—
432C4⋊C4—208no—
433C22⋊C4—28no—
434C22⋊C4—44no—
435C22⋊C4—208no—
436C22⋊C4—208no—
437C42—204no—
438C42—204no—
439C2×C5—44no—
440C2×C5#231311yescharacteristic
441C2×C5#232312yes—
442C2×C5#235212yes—
443C2×C5#235811yescharacteristic
444C2×C5#237611yescharacteristic
445C2×C5#268112yes—
446C2×C5#269012yes—
447C2×C5#269112yes—
448C2×C5#274712yes—
449C23—24no—
450C23—24no—
451C23—24no—
452C23—24no—
453C23—22no—
454C23—24no—
455C23—24no—
456C23—22no—
457C23—24no—
458C23—24no—
459C23—41no—
460C23—41no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—41no—
467C23—41no—
468C23—42no—
469C23#407214yes—
470C23C2×#70912yes—
471C23C2×D4×D512yes—
472C23C2×#71112yes—
473C23#414411yescharacteristic
474C23#414411yescharacteristic
475C23#414712yes—
476C23#414912yes—
477C23#415012yes—
478C23C23×Dic511yescharacteristic
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—416no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—42no—
503C2×C4—104no—
504C2×C4—104no—
505C2×C4—104no—
506C2×C4—104no—
507C2×C4—104no—
508C2×C4—104no—
509C2×C4—202no—
510C2×C4—202no—
511C2×C4—202no—
512C2×C4—202no—
513C2×C4—202no—
514C2×C4—202no—
515C2×C4—202no—
516C2×C4—202no—
517C2×C4—202no—
518C2×C4—202no—
519C2×C4—202no—
520C2×C4—202no—
521C2×C4—202no—
522C2×C4—202no—
523C2×C4—202no—
524C2×C4—202no—
525C2×C4—202no—
526C2×C4—202no—
527C2×C4—202no—
528C2×C4—202no—
529C2×C4—202no—
530C2×C4—202no—
531C2×C4D4×Dic518yes—
532C5#1440911yescharacteristic, complemented, Sylow
533C22—28no—
534C22—24no—
535C22—44no—
536C22—42no—
537C22—42no—
538C22—44no—
539C22—44no—
540C22—44no—
541C22—44no—
542C22#6584912yes—
543C22#6584912yes—
544C22#6587512yes—
545C22#6588412yes—
546C22#6588612yes—
547C22#6589912yes—
548C22#6592111yescharacteristic
549C22#6592411yescharacteristic
550C22#6592512yes—
551C22#6595912yes—
552C22#6596212yes—
553C22#6596312yes—
554C22#6597712yes—
555C22#6598511yescharacteristic
556C22#6600011yescharacteristic
557C22#6600112yes—
558C22C2×#407211yescharacteristic
559C22C2×D4×Dic512yes—
560C22#6613211yescharacteristic
561C22#6614611yescharacteristic
562C22#6616112yes—
563C4—28no—
564C4—42no—
565C4—42no—
566C4—44no—
567C4—204no—
568C4—204no—
569C4—402no—
570C4—402no—
571C2—44no—
572C2#1060624512yes—
573C2#1060624612yes—
574C2#1060627812yes—
575C2#1060636312yes—
576C2#1061388611yescharacteristic
577C2#1061406612yes—
578C2#1061431011yescharacteristic
579C2#1061447512yes—
580C2#1061457111yescharacteristic
subgroupG/H
center, FrattiniC24C23×D5
commutatorC23×C5C23×C4
FittingC2×C5×#385C2
socle, cosocleC24×C5C24
prime radicalC25×C5C23
hypercenterC2×#385D5
hypocenterC5#14409
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C24 ◁ C2×#385

Complements

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

NH ≅ G/Ntype
3. C2×#65270C2semidirect
4. #10602358C2semidirect
8. #10600205C2semidirect
9. #10600200C2semidirect
20. C5×#385C4semidirect
180. #385Dic5semidirect
532. C5#14409semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#3852
C2×#655012
C2×#652702
#106023582
#106023412
#106023382
#106002052
#106002002
#144095