Finite Group Databasebeta

← #12022780

Subgroups — #12022780

Order 1280.

2358 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#4033C211yesmaximal, characteristic, complemented
2C22×#4021C211yesmaximal, characteristic
3C2×#65263C211yesmaximal, characteristic, complemented
4C2×#65277C211yesmaximal, characteristic
5C2×#65277C211yesmaximal, characteristic
6C5×C4⋊#95C211yesmaximal, characteristic, complemented
7C4×#3963C211yesmaximal, characteristic, complemented
8#10600221C214yesmaximal, complemented
9#10600201C214yesmaximal, complemented
10C23×D20C2211yescharacteristic
11C23×Dic10C2211yescharacteristic
12C22×C5×C4⋊C4C2211yescharacteristic
13C22×C5×C4⋊C4C2211yescharacteristic
14C22×C42×C5C2211yescharacteristic
15C22×#685C2212yes—
16C22×#685C2214yes—
17C22×#685C2214yes—
18C22×#685C2212yes—
19C22×#683C2212yes—
20C22×C4×Dic5C2212yes—
21C22×C4×Dic5C2211yescharacteristic
22C22×C4×Dic5C2211yescharacteristic
23C2×#4033—24no—
24C2×#4033—24no—
25C2×#4033—24no—
26C2×#4021—24no—
27C2×#4021—24no—
28C2×#4021C418yescomplemented
29C2×C5×#95C2214yes—
30C2×#3963C2212yes—
31C2×#3963C2214yes—
32C2×#3963C2212yes—
33#65277—24no—
34#65277—24no—
35#65263—24no—
36#14026—51nomaximal, Sylow
37C24×D5C2312yes—
38C23×C4×C5C2312yes—
39C23×C4×C5C2312yes—
40C23×C4×C5C2312yes—
41C23×C4×C5C2311yescharacteristic
42C23×Dic5C2311yescharacteristic
43C23×Dic5C2312yes—
44C23×Dic5C2312yes—
45C23×Dic5C2311yescharacteristic
46C22×D20—22no—
47C22×D20—22no—
48C22×D20—22no—
49C22×D20—22no—
50C22×D20—22no—
51C22×D20—22no—
52C22×D20C2×C414yes—
53C22×Dic10—22no—
54C22×Dic10—22no—
55C22×Dic10—22no—
56C22×Dic10—22no—
57C22×Dic10C2×C414yes—
58C22×Dic10D414yes—
59C22×Dic10D414yes—
60C2×C5×C4⋊C4—22no—
61C2×C5×C4⋊C4—22no—
62C2×C5×C4⋊C4—22no—
63C2×C5×C4⋊C4—22no—
64C2×C5×C4⋊C4—22no—
65C2×C5×C4⋊C4D414yes—
66C2×C42×C5—22no—
67C2×C42×C5—22no—
68C2×C42×C5C2×C414yes—
69C2×#685—28no—
70C2×#685—28no—
71C2×#685—28no—
72C2×#685—24no—
73C2×#685—24no—
74C2×#685—28no—
75C2×#685—28no—
76C2×#685—44no—
77C2×#685—44no—
78C2×#685C2×C4116yes—
79C2×#683—24no—
80C2×#683—24no—
81C2×#683—24no—
82C2×C4×Dic5—24no—
83C2×C4×Dic5—22no—
84C2×C4×Dic5—22no—
85C2×C4×Dic5—24no—
86C2×C4×Dic5—24no—
87C2×C4×Dic5—22no—
88C2×C4×Dic5—22no—
89C2×C4×Dic5—22no—
90C2×C4×Dic5D414yes—
91#4033—232no—
92#4021—232no—
93C5×#95—44no—
94#3963—28no—
95#3963—24no—
96#3963—24no—
97C22×#124—51no—
98C22×C4×Q8—51no—
99C2×#391—51no—
100C2×#390—51no—
101C2×#390—51no—
102#1439—54no—
103#1437—54no—
104C4⋊#95D511yescharacteristic, complemented
105C4×#95—51no—
106C24×C5C2411yescharacteristic
107C23×D5—22no—
108C23×D5—22no—
109C23×D5—22no—
110C23×D5—22no—
111C23×D5—22no—
112C23×D5—22no—
113C23×D5—24no—
114C23×D5—24no—
115C23×D5—24no—
116C23×D5—24no—
117C23×D5C22×C414yes—
118C22×C4×C5—22no—
119C22×C4×C5—22no—
120C22×C4×C5—22no—
121C22×C4×C5—22no—
122C22×C4×C5—22no—
123C22×C4×C5—22no—
124C22×C4×C5—24no—
125C22×C4×C5—22no—
126C22×C4×C5—24no—
127C22×C4×C5—22no—
128C22×C4×C5—22no—
129C22×C4×C5—22no—
130C22×C4×C5—22no—
131C22×C4×C5—24no—
132C22×C4×C5—24no—
133C22×C4×C5C22×C414yes—
134C22×C4×C5C22×C412yes—
135C22×C4×C5C2×D414yes—
136C22×C4×C5C2×D412yes—
137C22×C4×C5C2×D412yes—
138C22×C4×C5C2×D412yes—
139C22×C4×C5C2×D412yes—
140C22×C4×C5#4112yes—
141C22×C4×C5#4112yes—
142C22×Dic5—22no—
143C22×Dic5—22no—
144C22×Dic5—21no—
145C22×Dic5—21no—
146C22×Dic5—22no—
147C22×Dic5—22no—
148C22×Dic5—22no—
149C22×Dic5—22no—
150C22×Dic5—21no—
151C22×Dic5—21no—
152C22×Dic5—22no—
153C22×Dic5—22no—
154C22×Dic5—22no—
155C22×Dic5—22no—
156C22×Dic5—22no—
157C22×Dic5—22no—
158C22×Dic5—22no—
159C22×Dic5C22×C414yes—
160C22×Dic5C2×D412yes—
161C22×Dic5C2×D414yes—
162C22×Dic5C2×D414yes—
163C22×Dic5C2×D412yes—
164C22×Dic5#4114yes—
165C22×Dic5#4114yes—
166C2×D20—28no—
167C2×D20—28no—
168C2×D20—28no—
169C2×D20—44no—
170C2×D20—44no—
171C2×D20—44no—
172C2×D20—44no—
173C2×Dic10—28no—
174C2×Dic10—28no—
175C2×Dic10—28no—
176C2×Dic10—28no—
177C2×Dic10—28no—
178C2×Dic10—28no—
179C2×Dic10C22⋊C4116yescomplemented
180C5×C4⋊C4—28no—
181C5×C4⋊C4—44no—
182C42×C5—28no—
183#685—232no—
184#685—416no—
185#685—48no—
186#685—48no—
187#683—216no—
188C4×Dic5—216no—
189C4×Dic5—28no—
190C4×Dic5—44no—
191C23×Q8—51no—
192C23×D4—51no—
193C2×#124—58no—
194C2×#124—104no—
195C2×#124—104no—
196C2×C4×Q8—104no—
197C2×C4×Q8—104no—
198C2×C4×Q8—104no—
199C22×C4⋊C4—52no—
200C22×C4⋊C4C2×D511yescharacteristic
201C22×C4⋊C4C2×D511yescharacteristic
202C22×C22⋊C4—54no—
203C22×C22⋊C4—52no—
204C22×C22⋊C4—54no—
205C22×C22⋊C4—52no—
206C22×C42—51no—
207C22×C42—52no—
208C22×C42—51no—
209C22×C42C2×D511yescharacteristic
210#391—104no—
211#390—104no—
212#390—104no—
213C2×#95—52no—
214C2×#95—52no—
215C2×#95—54no—
216C2×#95C2×D514yes—
217C23×C5C23×C411yescharacteristic
218C23×C5C22×D411yescharacteristic
219C23×C5C22×D411yescharacteristic
220C23×C5C22×D411yescharacteristic
221C23×C5C22×D411yescharacteristic
222C23×C5C2×#4111yescharacteristic
223C23×C5C2×#4111yescharacteristic
224C23×C5C2×#4112yes—
225C23×C5C2×#4112yes—
226C23×C5H(5,𝔽2)12yes—
227C23×C52−1+412yes—
228C22×D5—24no—
229C22×D5—24no—
230C22×D5—24no—
231C22×D5—28no—
232C22×D5—28no—
233C22×D5—42no—
234C22×D5—42no—
235C22×D5—42no—
236C22×D5—42no—
237C22×D5—42no—
238C22×D5—44no—
239C22×D5—42no—
240C22×D5—42no—
241C22×D5—42no—
242C22×D5—44no—
243C22×D5—44no—
244C22×D5—44no—
245C22×D5—44no—
246C22×D5—44no—
247C22×D5—44no—
248C22×D5—44no—
249C22×D5—44no—
250C22×D5—44no—
251C2×C4×C5—24no—
252C2×C4×C5—24no—
253C2×C4×C5—24no—
254C2×C4×C5—24no—
255C2×C4×C5—28no—
256C2×C4×C5—42no—
257C2×C4×C5—42no—
258C2×C4×C5—42no—
259C2×C4×C5—42no—
260C2×C4×C5—42no—
261C2×C4×C5—42no—
262C2×C4×C5—44no—
263C2×C4×C5—42no—
264C2×C4×C5—42no—
265C2×C4×C5—42no—
266C2×C4×C5—44no—
267C2×C4×C5—44no—
268C2×C4×C5C2×C22⋊C414yes—
269C2×C4×C5C4×D414yes—
270C2×C4×C5C4×D414yes—
271C2×C4×C5#12014yes—
272C2×C4×C5#12114yes—
273C2×C4×C5#12114yes—
274C2×C4×C5#12314yes—
275C2×Dic5—24no—
276C2×Dic5—24no—
277C2×Dic5—22no—
278C2×Dic5—24no—
279C2×Dic5—24no—
280C2×Dic5—24no—
281C2×Dic5—24no—
282C2×Dic5—22no—
283C2×Dic5—24no—
284C2×Dic5—24no—
285C2×Dic5—22no—
286C2×Dic5—24no—
287C2×Dic5—22no—
288C2×Dic5—22no—
289C2×Dic5—24no—
290C2×Dic5—24no—
291C2×Dic5—42no—
292C2×Dic5—42no—
293C2×Dic5—42no—
294C2×Dic5—42no—
295C2×Dic5—42no—
296C2×Dic5—42no—
297C2×Dic5C2×C22⋊C418yes—
298C2×Dic5#12414yescomplemented
299D20—416no—
300Dic10—232no—
301C25—52no—
302C22×Q8—54no—
303C22×Q8—54no—
304C22×Q8—54no—
305C22×Q8—102no—
306C22×Q8—102no—
307C22×Q8—102no—
308C22×Q8—102no—
309C22×D4—54no—
310C22×D4—102no—
311C22×D4—102no—
312C22×D4—102no—
313C22×D4—102no—
314C22×D4—102no—
315C22×D4—102no—
316C23×C4—52no—
317C23×C4—51no—
318C23×C4—51no—
319C23×C4—52no—
320C23×C4C22×D512yes—
321C23×C4C22×D512yes—
322C23×C4C22×D512yes—
323C23×C4C22×D511yescharacteristic
324#124—1032no—
325C4×Q8—1032no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—104no—
332C2×C4⋊C4—104no—
333C2×C4⋊C4—104no—
334C2×C4⋊C4D2014yes—
335C2×C22⋊C4—516no—
336C2×C22⋊C4—108no—
337C2×C22⋊C4—108no—
338C2×C22⋊C4—104no—
339C2×C22⋊C4—104no—
340C2×C22⋊C4—108no—
341C2×C22⋊C4—108no—
342C2×C22⋊C4—108no—
343C2×C22⋊C4—204no—
344C2×C22⋊C4—204no—
345C2×C42—22no—
346C2×C42—22no—
347C2×C42—54no—
348C2×C42—104no—
349C2×C42—104no—
350C2×C42—104no—
351C2×C42—102no—
352C2×C42—102no—
353C2×C42—102no—
354C2×C42—102no—
355C2×C42—102no—
356C2×C42C4×D514yes—
357#95—44no—
358#95—104no—
359#95—104no—
360#95—108no—
361C22×C5C22×C22⋊C411yescharacteristic
362C22×C5C2×C4×D411yescharacteristic
363C22×C5C2×C4×D411yescharacteristic
364C22×C5C4×#4112yes—
365C22×C5#52012yes—
366C22×C5C2×#12011yescharacteristic
367C22×C5C2×#12111yescharacteristic
368C22×C5C2×#12111yescharacteristic
369C22×C5C2×#12311yescharacteristic
370C22×C5C2×#12411yescharacteristic
371C22×C5#52912yes—
372C22×C5#53211yescharacteristic
373C22×C5#53511yescharacteristic
374C22×C5#53611yescharacteristic
375C22×C5#54212yes—
376C22×C5#54612yes—
377C22×C5#54812yes—
378C22×C5#54812yes—
379C22×C5#54812yes—
380C22×C5#55012yes—
381C22×C5#55012yes—
382C22×C5#55612yes—
383C22×C5#55912yes—
384C2×D5—44no—
385C2×D5—48no—
386C2×D5—48no—
387C2×D5—44no—
388C2×D5—44no—
389C2×D5—84no—
390C2×D5—82no—
391C2×D5—82no—
392C2×D5—84no—
393C2×D5—82no—
394C2×D5—82no—
395C4×C5—44no—
396C4×C5—44no—
397C4×C5—82no—
398C4×C5#38618yes—
399Dic5—24no—
400Dic5—28no—
401Dic5—44no—
402Dic5—42no—
403C24—54no—
404C24—102no—
405C24—102no—
406C24—102no—
407C24—102no—
408C24—102no—
409C24—104no—
410C24—102no—
411C24—104no—
412C24—104no—
413C24—104no—
414C24C23×D511yescharacteristic
415C2×Q8—516no—
416C2×Q8—108no—
417C2×Q8—108no—
418C2×Q8—108no—
419C2×Q8—108no—
420C2×Q8—108no—
421C2×Q8—108no—
422C2×D4—108no—
423C2×D4—108no—
424C2×D4—108no—
425C2×D4—204no—
426C2×D4—204no—
427C2×D4—204no—
428C2×D4—204no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—54no—
445C22×C4—54no—
446C22×C4—52no—
447C22×C4—54no—
448C22×C4—54no—
449C22×C4—52no—
450C22×C4—54no—
451C22×C4—101no—
452C22×C4—102no—
453C22×C4—102no—
454C22×C4—101no—
455C22×C4—102no—
456C22×C4—102no—
457C22×C4—102no—
458C22×C4—102no—
459C22×C4—101no—
460C22×C4—102no—
461C22×C4—101no—
462C22×C4—102no—
463C22×C4—102no—
464C22×C4—102no—
465C22×C4—102no—
466C22×C4—102no—
467C22×C4—102no—
468C22×C4C2×C4×D514yes—
469C22×C4C2×C4×D512yes—
470C22×C4C2×D2014yes—
471C22×C4C2×D2012yes—
472C22×C4#70912yes—
473C22×C4D4×D512yes—
474C22×C4D4×D512yes—
475C22×C4D4×D512yes—
476C22×C4#71312yes—
477C4⋊C4—28no—
478C4⋊C4—44no—
479C4⋊C4—1016no—
480C22⋊C4—1032no—
481C22⋊C4—2016no—
482C22⋊C4—208no—
483C22⋊C4—208no—
484C42—28no—
485C42—1016no—
486C42—108no—
487C42—204no—
488C2×C5C2×#38611yescharacteristic
489C2×C5C4×#12411yescharacteristic
490C2×C5#232611yescharacteristic
491C2×C5#234112yes—
492C2×C5#236712yes—
493C2×C5#242612yes—
494C2×C5#243111yescharacteristic
495C2×C5#243611yescharacteristic
496C2×C5#244812yes—
497C2×C5#254511yescharacteristic
498C2×C5#254811yescharacteristic
499D5—84no—
500C23—104no—
501C23—104no—
502C23—104no—
503C23—108no—
504C23—108no—
505C23—202no—
506C23—202no—
507C23—204no—
508C23—202no—
509C23—202no—
510C23—202no—
511C23—202no—
512C23—204no—
513C23—204no—
514C23—204no—
515C23—204no—
516C23—204no—
517C23—204no—
518C23—204no—
519C23—204no—
520C23—202no—
521C23—202no—
522C23—204no—
523C23C22×C4×D511yescharacteristic
524C23C22×D2011yescharacteristic
525C23C2×#70911yescharacteristic
526C23C2×D4×D511yescharacteristic
527C23C2×D4×D511yescharacteristic
528C23C2×D4×D511yescharacteristic
529C23#414412yes—
530C23C2×#71311yescharacteristic
531C23#414712yes—
532C23D5×#4112yes—
533C23D5×#4112yes—
534Q8—1032no—
535D4—2016no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—28no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—58no—
554C2×C4—54no—
555C2×C4—102no—
556C2×C4—104no—
557C2×C4—102no—
558C2×C4—104no—
559C2×C4—104no—
560C2×C4—104no—
561C2×C4—104no—
562C2×C4—104no—
563C2×C4—104no—
564C2×C4—104no—
565C2×C4—102no—
566C2×C4—102no—
567C2×C4—104no—
568C2×C4—104no—
569C2×C4—102no—
570C2×C4—104no—
571C2×C4—202no—
572C2×C4—202no—
573C2×C4—202no—
574C2×C4—202no—
575C2×C4—202no—
576C2×C4—202no—
577C2×C4C4×D2014yes—
578C2×C4D5×C22⋊C414yes—
579C2×C4#402814yes—
580C2×C4#403014yes—
581C2×C4#403914yes—
582C2×C4#404014yes—
583C2×C4#404114yes—
584C5#1402611yescharacteristic, complemented, Sylow
585C22—208no—
586C22—204no—
587C22—204no—
588C22—208no—
589C22—204no—
590C22—404no—
591C22—402no—
592C22—404no—
593C22—402no—
594C22—402no—
595C22—402no—
596C22C2×C4×D2011yescharacteristic
597C22C2×D5×C22⋊C411yescharacteristic
598C22C2×#402811yescharacteristic
599C22C2×#403011yescharacteristic
600C22C2×#403911yescharacteristic
601C22#6583812yes—
602C22C2×#404011yescharacteristic
603C22C2×#404111yescharacteristic
604C22#6585612yes—
605C22#6588812yes—
606C22#6589312yes—
607C22#6590212yes—
608C22#6591012yes—
609C22#6593312yes—
610C22#6593312yes—
611C22#6596712yes—
612C22#6596712yes—
613C22#6600211yescharacteristic
614C22D5×#12411yescharacteristic
615C22#6602512yes—
616C22#6604911yescharacteristic
617C22#6605811yescharacteristic
618C22#6606712yes—
619C4—44no—
620C4—44no—
621C4—82no—
622C4—108no—
623C4—104no—
624C4—204no—
625C4—202no—
626C4#6495418yes—
627C2—404no—
628C2C2×#6495411yescharacteristic
629C2#1060538512yes—
630C2#1060560812yes—
631C2#1060588011yescharacteristic
632C2#1060605211yescharacteristic
633C2#1060773812yes—
634C2#1060794912yes—
635C2#1060804411yescharacteristic
636C2#1060808311yescharacteristic
637C2#1060823111yescharacteristic
638C2#1060826511yescharacteristic
subgroupG/H
center, FrattiniC24C23×D5
commutatorC23×C5C23×C4
FittingC5×C4⋊#95C2
socle, cosocleC24×C5C24
prime radicalC23×D20C22
hypercenterC4⋊#95D5
hypocenterC5#14026
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C24 ◁ C4⋊#95

Complements

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

NH ≅ G/Ntype
1. C22×#4033C2semidirect
3. C2×#65263C2semidirect
6. C5×C4⋊#95C2semidirect
7. C4×#3963C2semidirect
8. #10600221C2semidirect
9. #10600201C2semidirect
28. C2×#4021C4semidirect
104. C4⋊#95D5semidirect
179. C2×Dic10C22⋊C4semidirect
298. C2×Dic5#124semidirect
584. C5#14026semidirect

Maximal subgroups

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

subgroupindex
C22×#40332
C22×#40212
C2×#652632
C2×#652772
C5×C4⋊#952
C4×#39632
#106002212
#106002012
#140265