Finite Group Databasebeta

← #12021276

Subgroups — #12021276

Order 1280.

2006 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#4072C211yesmaximal, characteristic
2C2×C4×C5×C22⋊C4C211yesmaximal, characteristic
3C2×#65501C211yesmaximal, characteristic
4C2×#65224C211yesmaximal, characteristic, complemented
5C2×#65225C211yesmaximal, characteristic, complemented
6#10602339C214yesmaximal
7#10602328C211yesmaximal, characteristic, complemented
8#10602328C211yesmaximal, characteristic, complemented
9#10600200C214yesmaximal, complemented
10C24×C4×C5C2211yescharacteristic
11C22×C5×C22⋊C4C2212yes—
12C22×C42×C5C2211yescharacteristic
13C22×C42×C5C2211yescharacteristic
14C22×C22⋊Dic5C2212yes—
15C22×C22⋊Dic5C2212yes—
16C2×#4072—24no—
17C2×#4072—24no—
18C2×#4072—24no—
19C22×C4⋊Dic5C2211yescharacteristic
20C22×C4⋊Dic5C2211yescharacteristic
21C22×C4⋊Dic5C2214yes—
22C22×C4⋊Dic5C2211yescharacteristic
23C22×C4⋊Dic5C2211yescharacteristic
24C22×C4×Dic5C2212yes—
25C4×C5×C22⋊C4C418yescomplemented
26C2×C5×#95C2212yes—
27#65501—24no—
28C2×#3963C2212yes—
29C2×#3963C2214yes—
30C2×#3963C2214yes—
31C2×#3963C2214yes—
32#65225—24no—
33#65224—24no—
34#13879—51nomaximal, Sylow
35C25×C5C2311yescharacteristic
36C23×C4×C5—22no—
37C23×C4×C5—22no—
38C23×C4×C5—22no—
39C23×C4×C5—22no—
40C23×C4×C5C2×C414yes—
41C23×C4×C5D414yes—
42C23×C4×C5Q814yes—
43C23×C4×C5C2312yes—
44C23×C4×C5C2311yescharacteristic
45C23×C4×C5C2312yes—
46C23×C4×C5C2311yescharacteristic
47C23×Dic5C2312yes—
48C23×Dic5C2312yes—
49C23×Dic5C2312yes—
50C23×Dic5C2312yes—
51C2×C5×C22⋊C4—24no—
52C2×C5×C22⋊C4—24no—
53C2×C5×C22⋊C4C2×C418yes—
54C2×C42×C5—22no—
55C2×C42×C5—22no—
56C2×C42×C5—22no—
57C2×C42×C5—22no—
58C2×C42×C5C2×C414yes—
59C2×C42×C5C2×C414yes—
60C2×C22⋊Dic5—24no—
61C2×C22⋊Dic5—24no—
62C2×C22⋊Dic5—24no—
63C2×C22⋊Dic5—24no—
64C2×C22⋊Dic5—44no—
65#4072—232no—
66C2×C4⋊Dic5—22no—
67C2×C4⋊Dic5—22no—
68C2×C4⋊Dic5—22no—
69C2×C4⋊Dic5—22no—
70C2×C4⋊Dic5—22no—
71C2×C4⋊Dic5—22no—
72C2×C4⋊Dic5—22no—
73C2×C4⋊Dic5—22no—
74C2×C4⋊Dic5—28no—
75C2×C4⋊Dic5—22no—
76C2×C4⋊Dic5—22no—
77C2×C4⋊Dic5—28no—
78C2×C4⋊Dic5—44no—
79C2×C4⋊Dic5D414yes—
80C2×C4⋊Dic5D414yes—
81C2×C4×Dic5—24no—
82C2×C4×Dic5—24no—
83C2×C4×Dic5—24no—
84C5×#95C2×C418yes—
85#3963—24no—
86#3963—28no—
87#3963—28no—
88#3963—44no—
89C22×#117—51no—
90C2×#384—51no—
91C2×#382—51no—
92C2×#380—51no—
93C2×C4×C22⋊C4D511yescharacteristic
94#1439—54no—
95#1436—54no—
96C4⋊#95—51no—
97C4⋊#95—51no—
98C24×C5—21no—
99C24×C5—21no—
100C24×C5—22no—
101C24×C5—22no—
102C24×C5—21no—
103C24×C5—21no—
104C24×C5—22no—
105C24×C5—22no—
106C24×C5C22×C412yes—
107C24×C5C2×D412yes—
108C24×C5C2×Q812yes—
109C24×C5C2411yescharacteristic
110C22×C4×C5—22no—
111C22×C4×C5—21no—
112C22×C4×C5—21no—
113C22×C4×C5—22no—
114C22×C4×C5—21no—
115C22×C4×C5—21no—
116C22×C4×C5—22no—
117C22×C4×C5—22no—
118C22×C4×C5—28no—
119C22×C4×C5—28no—
120C22×C4×C5—28no—
121C22×C4×C5—28no—
122C22×C4×C5—28no—
123C22×C4×C5—28no—
124C22×C4×C5—22no—
125C22×C4×C5—22no—
126C22×C4×C5—22no—
127C22×C4×C5—22no—
128C22×C4×C5—22no—
129C22×C4×C5—22no—
130C22×C4×C5C4⋊C4116yescomplemented
131C22×C4×C5C22×C414yes—
132C22×C4×C5C22×C412yes—
133C22×C4×C5C22×C414yes—
134C22×C4×C5C22×C412yes—
135C22×C4×C5C2×D412yes—
136C22×C4×C5C2×D412yes—
137C22×C4×C5C2×D412yes—
138C22×C4×C5C2×D412yes—
139C22×C4×C5C2×Q812yes—
140C22×C4×C5C2×Q812yes—
141C22×C4×C5#4114yes—
142C22×C4×C5#4114yes—
143C22×C4×C5#4112yes—
144C22×C4×C5#4112yes—
145C22×Dic5—22no—
146C22×Dic5—22no—
147C22×Dic5—22no—
148C22×Dic5—22no—
149C22×Dic5—24no—
150C22×Dic5—24no—
151C22×Dic5—24no—
152C22×Dic5—24no—
153C22×Dic5—22no—
154C22×Dic5—22no—
155C22×Dic5—22no—
156C22×Dic5—22no—
157C22×Dic5—24no—
158C22×Dic5—24no—
159C22×Dic5—22no—
160C22×Dic5—22no—
161C22×Dic5—24no—
162C22×Dic5—24no—
163C22×Dic5C2×D414yes—
164C22×Dic5C2×D414yes—
165C5×C22⋊C4—216no—
166C42×C5—28no—
167C42×C5—28no—
168C22⋊Dic5—216no—
169C22⋊Dic5—48no—
170C4⋊Dic5—28no—
171C4⋊Dic5—28no—
172C4⋊Dic5—416no—
173C4⋊Dic5—44no—
174C4⋊Dic5—44no—
175C4×Dic5—216no—
176C24×C4C2×D511yescharacteristic
177C2×#117—104no—
178C2×#117—104no—
179C2×#117—104no—
180C22×C4⋊C4—51no—
181C22×C4⋊C4—51no—
182C22×C4⋊C4—51no—
183C22×C4⋊C4—51no—
184C22×C4⋊C4—54no—
185C22×C22⋊C4—52no—
186C22×C22⋊C4—52no—
187C22×C22⋊C4C2×D512yes—
188C22×C42—52no—
189C22×C42C2×D511yescharacteristic
190C22×C42C2×D511yescharacteristic
191#384—104no—
192#382—104no—
193#380—104no—
194C4×C22⋊C4Dic518yescomplemented
195C2×#95—54no—
196C2×#95—52no—
197C2×#95—54no—
198C2×#95—54no—
199C2×#95C2×D512yes—
200C23×C5—24no—
201C23×C5—22no—
202C23×C5—22no—
203C23×C5—24no—
204C23×C5—22no—
205C23×C5—22no—
206C23×C5—22no—
207C23×C5—22no—
208C23×C5—22no—
209C23×C5—22no—
210C23×C5—24no—
211C23×C5—22no—
212C23×C5—22no—
213C23×C5—24no—
214C23×C5—42no—
215C23×C5—42no—
216C23×C5—42no—
217C23×C5—42no—
218C23×C5—42no—
219C23×C5—42no—
220C23×C5—42no—
221C23×C5—42no—
222C23×C5C2×C4⋊C414yes—
223C23×C5C23×C411yescharacteristic
224C23×C5C22×D411yescharacteristic
225C23×C5C22×D411yescharacteristic
226C23×C5C22×D411yescharacteristic
227C23×C5C22×Q811yescharacteristic
228C23×C5C2×#4112yes—
229C23×C5C2×#4112yes—
230C23×C5C2×#4111yescharacteristic
231C23×C5C2×#4111yescharacteristic
232C23×C5H(5,𝔽2)12yes—
233C23×C52−1+412yes—
234C2×C4×C5—24no—
235C2×C4×C5—22no—
236C2×C4×C5—22no—
237C2×C4×C5—24no—
238C2×C4×C5—22no—
239C2×C4×C5—22no—
240C2×C4×C5—22no—
241C2×C4×C5—22no—
242C2×C4×C5—24no—
243C2×C4×C5—24no—
244C2×C4×C5—24no—
245C2×C4×C5—24no—
246C2×C4×C5—232no—
247C2×C4×C5—24no—
248C2×C4×C5—24no—
249C2×C4×C5—24no—
250C2×C4×C5—42no—
251C2×C4×C5—42no—
252C2×C4×C5—42no—
253C2×C4×C5—42no—
254C2×C4×C5C2×C4⋊C414yes—
255C2×C4×C5C2×C4⋊C414yes—
256C2×C4×C5#11718yescomplemented
257C2×C4×C5C4×D414yes—
258C2×C4×C5C4×D414yes—
259C2×C4×C5#12014yes—
260C2×C4×C5C22⋊Q814yes—
261C2×C4×C5C22⋊Q814yes—
262C2×C4×C5#12314yes—
263C2×Dic5—24no—
264C2×Dic5—24no—
265C2×Dic5—28no—
266C2×Dic5—28no—
267C2×Dic5—42no—
268C2×Dic5—42no—
269C2×Dic5—42no—
270C2×Dic5—42no—
271C2×Dic5—44no—
272C2×Dic5—44no—
273C2×Dic5—42no—
274C2×Dic5—42no—
275C2×Dic5—42no—
276C2×Dic5—42no—
277C2×Dic5—42no—
278C2×Dic5—42no—
279C2×Dic5—44no—
280C2×Dic5—44no—
281C2×Dic5—44no—
282C2×Dic5—44no—
283C25C22×D511yescharacteristic
284C23×C4—22no—
285C23×C4—22no—
286C23×C4—22no—
287C23×C4—22no—
288C23×C4—52no—
289C23×C4—52no—
290C23×C4—52no—
291C23×C4—52no—
292C23×C4Dic1014yes—
293C23×C4D2014yes—
294C23×C4C2×Dic514yes—
295C23×C4C22×D512yes—
296C23×C4C22×D511yescharacteristic
297C23×C4C22×D512yes—
298C23×C4C22×D511yescharacteristic
299#117—1032no—
300C2×C4⋊C4—54no—
301C2×C4⋊C4—54no—
302C2×C4⋊C4—102no—
303C2×C4⋊C4—102no—
304C2×C4⋊C4—102no—
305C2×C4⋊C4—102no—
306C2×C4⋊C4—108no—
307C2×C4⋊C4—102no—
308C2×C4⋊C4—102no—
309C2×C4⋊C4—108no—
310C2×C4⋊C4—102no—
311C2×C4⋊C4—102no—
312C2×C4⋊C4—102no—
313C2×C4⋊C4—102no—
314C2×C4⋊C4—204no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—104no—
318C2×C22⋊C4—104no—
319C2×C22⋊C4—104no—
320C2×C22⋊C4—104no—
321C2×C22⋊C4—204no—
322C2×C22⋊C4C2×Dic518yes—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—104no—
328C2×C42—104no—
329C2×C42—104no—
330C2×C42C2×Dic514yes—
331C2×C42C2×Dic514yes—
332#95—108no—
333#95—104no—
334#95—108no—
335#95—204no—
336#95C2×Dic518yes—
337C22×C5—24no—
338C22×C5—24no—
339C22×C5—24no—
340C22×C5—44no—
341C22×C5—44no—
342C22×C5—42no—
343C22×C5—42no—
344C22×C5—44no—
345C22×C5—44no—
346C22×C5—42no—
347C22×C5—42no—
348C22×C5C22×C4⋊C411yescharacteristic
349C22×C5C2×#11712yes—
350C22×C5C2×C4×D411yescharacteristic
351C22×C5C2×C4×D411yescharacteristic
352C22×C5#52012yes—
353C22×C5C2×#12011yescharacteristic
354C22×C5C2×C22⋊Q811yescharacteristic
355C22×C5C2×C22⋊Q811yescharacteristic
356C22×C5C2×#12311yescharacteristic
357C22×C5#53211yescharacteristic
358C22×C5#53211yescharacteristic
359C22×C5#53312yes—
360C22×C5#53511yescharacteristic
361C22×C5#53611yescharacteristic
362C22×C5#54412yes—
363C22×C5#54612yes—
364C22×C5#54712yes—
365C22×C5#54812yes—
366C22×C5#55012yes—
367C22×C5#55212yes—
368C22×C5#55312yes—
369C22×C5#55512yes—
370C22×C5#55612yes—
371C4×C5—28no—
372C4×C5—24no—
373C4×C5—44no—
374C4×C5#38518yes—
375Dic5—44no—
376Dic5—44no—
377Dic5—82no—
378Dic5—82no—
379C24—22no—
380C24—22no—
381C24—21no—
382C24—21no—
383C24—21no—
384C24—21no—
385C24—22no—
386C24—22no—
387C24C2×Dic1012yes—
388C24C2×D2012yes—
389C24C22×Dic512yes—
390C24C23×D511yescharacteristic
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—28no—
400C22×C4—28no—
401C22×C4—28no—
402C22×C4—28no—
403C22×C4—28no—
404C22×C4—28no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—54no—
412C22×C4—54no—
413C22×C4—102no—
414C22×C4—102no—
415C22×C4—104no—
416C22×C4—104no—
417C22×C4—102no—
418C22×C4—102no—
419C22×C4—104no—
420C22×C4—104no—
421C22×C4—102no—
422C22×C4—102no—
423C22×C4—104no—
424C22×C4—104no—
425C22×C4—102no—
426C22×C4—102no—
427C22×C4—102no—
428C22×C4—102no—
429C22×C4—104no—
430C22×C4—104no—
431C22×C4C4⋊Dic5116yescomplemented
432C22×C4C2×Dic1012yes—
433C22×C4C2×Dic1012yes—
434C22×C4C2×D2012yes—
435C22×C4C2×D2012yes—
436C22×C4#70914yes—
437C22×C4#70914yes—
438C22×C4D4×D512yes—
439C22×C4D4×D512yes—
440C22×C4#71112yes—
441C22×C4#71112yes—
442C22×C4C22×Dic514yes—
443C22×C4C22×Dic512yes—
444C22×C4C22×Dic514yes—
445C22×C4C22×Dic512yes—
446C4⋊C4—108no—
447C4⋊C4—108no—
448C4⋊C4—2016no—
449C4⋊C4—204no—
450C4⋊C4—204no—
451C22⋊C4—216no—
452C22⋊C4—1016no—
453C22⋊C4—208no—
454C42—28no—
455C42—28no—
456C42—1016no—
457C2×C5—44no—
458C2×C5#228511yescharacteristic
459C2×C5C2×#38511yescharacteristic
460C2×C5#231711yescharacteristic
461C2×C5#235212yes—
462C2×C5#235412yes—
463C2×C5#243111yescharacteristic
464C2×C5#243211yescharacteristic
465C2×C5#243912yes—
466C2×C5#244112yes—
467C2×C5#249911yescharacteristic
468C2×C5#250011yescharacteristic
469C23—24no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—24no—
477C23—22no—
478C23—22no—
479C23—24no—
480C23—22no—
481C23—22no—
482C23—24no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23C2×C4⋊Dic514yes—
492C23C22×Dic1011yescharacteristic
493C23C22×D2011yescharacteristic
494C23C2×#70912yes—
495C23C2×#70912yes—
496C23C2×D4×D511yescharacteristic
497C23C2×D4×D511yescharacteristic
498C23C2×#71111yescharacteristic
499C23C2×#71111yescharacteristic
500C23#414912yes—
501C23#415012yes—
502C23C23×Dic511yescharacteristic
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—24no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—232no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—104no—
524C2×C4—104no—
525C2×C4—108no—
526C2×C4—108no—
527C2×C4—202no—
528C2×C4—202no—
529C2×C4—202no—
530C2×C4—202no—
531C2×C4—204no—
532C2×C4—204no—
533C2×C4—202no—
534C2×C4—202no—
535C2×C4—202no—
536C2×C4—202no—
537C2×C4—202no—
538C2×C4—202no—
539C2×C4—204no—
540C2×C4—204no—
541C2×C4—204no—
542C2×C4—204no—
543C2×C4#402414yes—
544C2×C4#402414yes—
545C2×C4#402814yes—
546C2×C4#403214yes—
547C2×C4C2×C4⋊Dic514yes—
548C2×C4C2×C4⋊Dic514yes—
549C2×C4#407218yescomplemented
550C2×C4D4×Dic514yes—
551C2×C4D4×Dic514yes—
552C5#1387911yescharacteristic, complemented, Sylow
553C22—24no—
554C22—24no—
555C22—24no—
556C22—42no—
557C22—44no—
558C22—44no—
559C22—42no—
560C22—44no—
561C22—44no—
562C22—42no—
563C22—42no—
564C22#6580412yes—
565C22#6581111yescharacteristic
566C22#6581111yescharacteristic
567C22C2×#402411yescharacteristic
568C22C2×#402411yescharacteristic
569C22C2×#402811yescharacteristic
570C22C2×#403211yescharacteristic
571C22#6585312yes—
572C22#6585911yescharacteristic
573C22#6586011yescharacteristic
574C22#6587512yes—
575C22#6587612yes—
576C22#6588412yes—
577C22#6588512yes—
578C22#6588612yes—
579C22#6588712yes—
580C22#6589712yes—
581C22#6589912yes—
582C22C22×C4⋊Dic511yescharacteristic
583C22C2×#407212yes—
584C22C2×D4×Dic511yescharacteristic
585C22C2×D4×Dic511yescharacteristic
586C22#6616112yes—
587C4—28no—
588C4—24no—
589C4—44no—
590C4—204no—
591C4—204no—
592C4—402no—
593C4—402no—
594C4#6523918yes—
595C2—44no—
596C2#1060522011yescharacteristic
597C2#1060522312yes—
598C2#1060522512yes—
599C2#1060524511yescharacteristic
600C2#1060524611yescharacteristic
601C2#1060531711yescharacteristic
602C2#1061379311yescharacteristic
603C2C2×#6523911yescharacteristic
604C2#1061396411yescharacteristic
605C2#1061406612yes—
606C2#1061406812yes—
subgroupG/H
center, FrattiniC24C23×D5
commutatorC23×C5C23×C4
FittingC2×C4×C5×C22⋊C4C2
socle, cosocleC24×C5C24
prime radicalC25×C5C23
hypercenterC2×C4×C22⋊C4D5
hypocenterC5#13879
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
4. C2×#65224C2semidirect
5. C2×#65225C2semidirect
7. #10602328C2semidirect
8. #10602328C2semidirect
9. #10600200C2semidirect
25. C4×C5×C22⋊C4C4semidirect
130. C22×C4×C5C4⋊C4semidirect
194. C4×C22⋊C4Dic5semidirect
256. C2×C4×C5#117semidirect
431. C22×C4C4⋊Dic5semidirect
549. C2×C4#4072semidirect
552. C5#13879semidirect

Maximal subgroups

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

subgroupindex
C22×#40722
C2×C4×C5×C22⋊C42
C2×#655012
C2×#652242
C2×#652252
#106023392
#106023282
#106002002
#138795