Finite Group Databasebeta

← #12128175

Subgroups — #12128175

Order 1280.

1166 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#2280C211yesmaximal, characteristic
2C2×#65501C211yesmaximal, characteristic
3C2×#65496C211yesmaximal, characteristic
4C2×#65284C211yesmaximal, characteristic, complemented
5#10602374C211yesmaximal, characteristic, complemented
6#10602374C211yesmaximal, characteristic, complemented
7#10602374C211yesmaximal, characteristic, complemented
8#10600313C212yesmaximal, complemented
9#10600312C212yesmaximal, complemented
10#10600311C212yesmaximal, complemented
11#10600300C212yesmaximal, complemented
12C24×C4×C5C2211yescharacteristic
13C2×C5×#117C2211yescharacteristic
14C2×C5×#117C2211yescharacteristic
15C22×C22⋊Dic5—21no—
16C22×C4⋊Dic5C2211yescharacteristic
17C22×C4⋊Dic5C2211yescharacteristic
18C22×#681C2211yescharacteristic
19C22×#681C2211yescharacteristic
20C5×#381—21no—
21C5×#380C2211yescharacteristic
22C5×#380C2211yescharacteristic
23C5×#380C2211yescharacteristic
24C5×#380C2211yescharacteristic
25C5×#379—21no—
26C4×C5×C22⋊C4—21no—
27C4×C5×C22⋊C4—21no—
28#65501—21no—
29#65501—21no—
30#65501C2212yes—
31#65501C2212yes—
32C2×#3963—21no—
33#65496—24no—
34C2×C22⋊#168C2212yes—
35C2×C22⋊#168C2212yes—
36#65284—24no—
37C2×#3940C2212yescomplemented
38C2×#3939C2212yescomplemented
39#64749C2212yescomplemented
40#64749C2212yescomplemented
41#64749C2212yescomplemented
42#64749C2212yescomplemented
43#64749C2212yescomplemented
44#64749C2212yescomplemented
45#23642—51nomaximal, Sylow
46C25×C5D411yescharacteristic
47C23×C4×C5—22no—
48C23×C4×C5—22no—
49C23×C4×C5—22no—
50C23×C4×C5—22no—
51C23×C4×C5—22no—
52C23×C4×C5—22no—
53C23×C4×C5D411yescharacteristic
54C23×C4×C5D411yescharacteristic
55C23×C4×C5D411yescharacteristic
56C23×C4×C5C2311yescharacteristic
57C23×C4×C5C2311yescharacteristic
58C23×C4×C5C2311yescharacteristic
59C23×Dic5—21no—
60C23×Dic5—21no—
61C5×#117—22no—
62C5×#117—22no—
63C5×#117—22no—
64C5×#117—22no—
65C2×C5×C4⋊C4C2311yescharacteristic
66C2×C5×C4⋊C4C2311yescharacteristic
67C2×C5×C4⋊C4C2311yescharacteristic
68C2×C5×C4⋊C4C2311yescharacteristic
69C2×C5×C22⋊C4—21no—
70C2×C5×C22⋊C4—21no—
71C2×C5×C22⋊C4—21no—
72C2×C5×C22⋊C4—21no—
73C2×C5×C22⋊C4—21no—
74C2×C5×C22⋊C4—21no—
75C2×C42×C5—21no—
76C2×C42×C5—21no—
77C2×C22⋊Dic5—22no—
78C2×C22⋊Dic5—22no—
79C2×C22⋊Dic5—44no—
80C2×C4⋊Dic5—22no—
81C2×C4⋊Dic5—22no—
82C2×C4⋊Dic5—22no—
83C2×C4⋊Dic5—22no—
84C2×C4⋊Dic5—21no—
85C2×C4⋊Dic5—21no—
86C2×C4⋊Dic5C2312yes—
87C2×C4⋊Dic5C2312yes—
88C2×#681—22no—
89C2×#681—22no—
90C2×#681—22no—
91C2×#681—22no—
92C2×#681—22no—
93C2×#681D414yes—
94C22×#168C2312yes—
95C22×#168C2312yes—
96C5×#95—21no—
97C5×#95—21no—
98C5×#95—21no—
99C5×#95—21no—
100#3963—22no—
101#3963—22no—
102C22⋊#168—24no—
103C22⋊#168—24no—
104#3940—24no—
105#3939—24no—
106C2×#428—51no—
107C2×#407—51no—
108#2280D511yescharacteristic
109C2×#380—51no—
110#2077—52no—
111#2077—52no—
112#2076—52no—
113#2075—52no—
114#1736—51no—
115#1736—51no—
116#1736—51no—
117C24×C5—22no—
118C24×C5—22no—
119C24×C5—22no—
120C24×C5—22no—
121C24×C5—22no—
122C24×C5—41no—
123C24×C5—41no—
124C24×C5C2×D411yescharacteristic
125C24×C5C2×D411yescharacteristic
126C24×C5C2×D411yescharacteristic
127C22×C4×C5—22no—
128C22×C4×C5—22no—
129C22×C4×C5—22no—
130C22×C4×C5—22no—
131C22×C4×C5—22no—
132C22×C4×C5—22no—
133C22×C4×C5—22no—
134C22×C4×C5—22no—
135C22×C4×C5—22no—
136C22×C4×C5—21no—
137C22×C4×C5—21no—
138C22×C4×C5—21no—
139C22×C4×C5—21no—
140C22×C4×C5—21no—
141C22×C4×C5—21no—
142C22×C4×C5—21no—
143C22×C4×C5—21no—
144C22×C4×C5—21no—
145C22×C4×C5—21no—
146C22×C4×C5—21no—
147C22×C4×C5—21no—
148C22×C4×C5—21no—
149C22×C4×C5—21no—
150C22×C4×C5—21no—
151C22×C4×C5—21no—
152C22×C4×C5—44no—
153C22×C4×C5—44no—
154C22×C4×C5—44no—
155C22×C4×C5—44no—
156C22×C4×C5C2×D412yes—
157C22×C4×C5C2×D411yescharacteristic
158C22×C4×C5C2×D411yescharacteristic
159C22×C4×C5C2×D411yescharacteristic
160C22×C4×C5#4112yes—
161C22×C4×C5#4112yes—
162C22×C4×C5#4112yes—
163C22×C4×C5#4112yes—
164C22×C4×C5#4112yes—
165C22×C4×C5C2411yescharacteristic
166C22×Dic5—22no—
167C22×Dic5—22no—
168C22×Dic5—42no—
169C22×Dic5—42no—
170C22×Dic5—42no—
171C22×Dic5—42no—
172C22×Dic5—42no—
173C22×Dic5—42no—
174C5×C4⋊C4—21no—
175C5×C4⋊C4—21no—
176C5×C4⋊C4—21no—
177C5×C4⋊C4—21no—
178C5×C4⋊C4—21no—
179C5×C4⋊C4—21no—
180C5×C4⋊C4—21no—
181C5×C4⋊C4—21no—
182C5×C22⋊C4—42no—
183C5×C22⋊C4—42no—
184C5×C22⋊C4—42no—
185C5×C22⋊C4—42no—
186C5×C22⋊C4—42no—
187C5×C22⋊C4—42no—
188C42×C5—42no—
189C42×C5—42no—
190C22⋊Dic5—44no—
191C22⋊Dic5—44no—
192C4⋊Dic5—22no—
193C4⋊Dic5—22no—
194C4⋊Dic5—22no—
195C4⋊Dic5—22no—
196C4⋊Dic5—42no—
197C4⋊Dic5—42no—
198#681—28no—
199#681—44no—
200C2×#168—22no—
201C2×#168—22no—
202C2×#168—22no—
203C2×#168—22no—
204C2×#168—22no—
205C2×#168C2×D414yes—
206C24×C4C2×D511yescharacteristic
207C22×M16—51no—
208C22×M16—51no—
209C2×#117C2×D511yescharacteristic
210C2×#117C2×D511yescharacteristic
211C22×C4⋊C4—51no—
212C22×C4⋊C4—51no—
213C22×C22⋊C4—101no—
214#428—104no—
215C2×#107—52no—
216C2×#106—52no—
217#407—104no—
218C2×C22⋊C8—52no—
219C2×C22⋊C8—52no—
220#381—21no—
221#380—52no—
222#380—52no—
223#380—101no—
224#380—101no—
225#380C2×D511yescharacteristic
226#380C2×D511yescharacteristic
227#380C2×D511yescharacteristic
228#380C2×D511yescharacteristic
229#379—21no—
230C4×C22⋊C4—21no—
231C4×C22⋊C4—21no—
232C2×#95—101no—
233#340—52no—
234#340—52no—
235#340—52no—
236#340—52no—
237#340—52no—
238#340—52no—
239C23×C5—21no—
240C23×C5—21no—
241C23×C5—21no—
242C23×C5—21no—
243C23×C5—28no—
244C23×C5—21no—
245C23×C5—21no—
246C23×C5—21no—
247C23×C5—21no—
248C23×C5—21no—
249C23×C5—21no—
250C23×C5—21no—
251C23×C5—21no—
252C23×C5—21no—
253C23×C5—21no—
254C23×C5—21no—
255C23×C5—21no—
256C23×C5—44no—
257C23×C5—44no—
258C23×C5—44no—
259C23×C5—44no—
260C23×C5—44no—
261C23×C5—44no—
262C23×C5#12112yes—
263C23×C5#12312yes—
264C23×C5#12312yes—
265C23×C5AGL(1,ℤ/8)12yes—
266C23×C5#13712yes—
267C23×C5C22×D411yescharacteristic
268C2×C4×C5—21no—
269C2×C4×C5—21no—
270C2×C4×C5—21no—
271C2×C4×C5—21no—
272C2×C4×C5—21no—
273C2×C4×C5—21no—
274C2×C4×C5—44no—
275C2×C4×C5—41no—
276C2×C4×C5—41no—
277C2×C4×C5—41no—
278C2×C4×C5—41no—
279C2×C4×C5—41no—
280C2×C4×C5—41no—
281C2×C4×C5—41no—
282C2×C4×C5—41no—
283C2×C4×C5—44no—
284C2×C4×C5—44no—
285C2×C4×C5—41no—
286C2×C4×C5—41no—
287C2×C4×C5—41no—
288C2×C4×C5—41no—
289C2×C4×C5—41no—
290C2×C4×C5—41no—
291C2×C4×C5—41no—
292C2×C4×C5—41no—
293C2×C4×C5—41no—
294C2×C4×C5—41no—
295C2×C4×C5—41no—
296C2×C4×C5—41no—
297C2×C4×C5#12112yes—
298C2×C4×C5#12314yes—
299C2×C4×C5#12312yes—
300C2×C4×C5#12312yes—
301C2×C4×C5#12414yes—
302C2×C4×C5#12614yes—
303C2×C4×C5#12614yes—
304C2×C4×C5C22×D411yescharacteristic
305C2×C4×C5C2×#4111yescharacteristic
306C2×C4×C5C2×#4111yescharacteristic
307C2×C4×C5C2×#4111yescharacteristic
308C2×C4×C5C2×#4111yescharacteristic
309C2×C4×C5C2×#4111yescharacteristic
310C2×Dic5—42no—
311C2×Dic5—42no—
312C2×Dic5—42no—
313C2×Dic5—42no—
314C2×Dic5—42no—
315C2×Dic5—42no—
316C2×Dic5—84no—
317C2×Dic5—84no—
318#168—24no—
319#168—42no—
320C25C22⋊D511yescharacteristic
321C23×C4—22no—
322C23×C4—22no—
323C23×C4—22no—
324C23×C4—22no—
325C23×C4—22no—
326C23×C4—22no—
327C23×C4—101no—
328C23×C4—101no—
329C23×C4C22⋊D511yescharacteristic
330C23×C4C22⋊D511yescharacteristic
331C23×C4C22⋊D511yescharacteristic
332C23×C4C22×D511yescharacteristic
333C23×C4C22×D511yescharacteristic
334C23×C4C22×D511yescharacteristic
335C2×M16—54no—
336C2×M16—102no—
337C2×M16—102no—
338C2×M16—102no—
339C2×M16—102no—
340C2×M16—102no—
341C22×C8—52no—
342C22×C8—52no—
343#117—22no—
344#117—22no—
345#117—22no—
346#117—22no—
347C2×C4⋊C4—52no—
348C2×C4⋊C4—52no—
349C2×C4⋊C4—102no—
350C2×C4⋊C4—102no—
351C2×C4⋊C4—102no—
352C2×C4⋊C4—102no—
353C2×C4⋊C4—101no—
354C2×C4⋊C4—101no—
355C2×C4⋊C4C22×D511yescharacteristic
356C2×C4⋊C4C22×D511yescharacteristic
357C2×C4⋊C4C22×D511yescharacteristic
358C2×C4⋊C4C22×D511yescharacteristic
359C2×C22⋊C4—21no—
360C2×C22⋊C4—21no—
361C2×C22⋊C4—21no—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—21no—
364C2×C22⋊C4—21no—
365C2×C22⋊C4—102no—
366C2×C22⋊C4—102no—
367C2×C22⋊C4—204no—
368C2×C42—21no—
369C2×C42—21no—
370#107—104no—
371#106—104no—
372C22⋊C8—104no—
373C22⋊C8—104no—
374#95—21no—
375#95—21no—
376#95—21no—
377#95—21no—
378#95—102no—
379#95—102no—
380C22×C5—22no—
381C22×C5—22no—
382C22×C5—22no—
383C22×C5—22no—
384C22×C5—22no—
385C22×C5—22no—
386C22×C5—22no—
387C22×C5—22no—
388C22×C5—22no—
389C22×C5—22no—
390C22×C5—22no—
391C22×C5—22no—
392C22×C5—416no—
393C22×C5—41no—
394C22×C5—41no—
395C22×C5—41no—
396C22×C5—41no—
397C22×C5—41no—
398C22×C5—41no—
399C22×C5—41no—
400C22×C5—41no—
401C22×C5#39714yes—
402C22×C5C2×#12111yescharacteristic
403C22×C5C2×#12311yescharacteristic
404C22×C5C2×#12311yescharacteristic
405C22×C5C2×AGL(1,ℤ/8)11yescharacteristic
406C22×C5C2×#13711yescharacteristic
407C22×C5#57511yescharacteristic
408C22×C5#57511yescharacteristic
409C4×C5—41no—
410C4×C5—41no—
411C4×C5—81no—
412C4×C5—81no—
413C4×C5—81no—
414C4×C5—81no—
415C4×C5#39714yes—
416C4×C5C2×#12311yescharacteristic
417C4×C5C2×#12411yescharacteristic
418C4×C5C2×#12611yescharacteristic
419C4×C5C2×#12611yescharacteristic
420Dic5—82no—
421Dic5—82no—
422C24—22no—
423C24—22no—
424C24—22no—
425C24—22no—
426C24—22no—
427C24—41no—
428C24—41no—
429C24C2×C22⋊D511yescharacteristic
430C24C2×C22⋊D511yescharacteristic
431C24C2×C22⋊D511yescharacteristic
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—102no—
462C22×C4—102no—
463C22×C4—202no—
464C22×C4—202no—
465C22×C4—202no—
466C22×C4—202no—
467C22×C4—202no—
468C22×C4—202no—
469C22×C4#70912yes—
470C22×C4#70912yes—
471C22×C4D4×D512yes—
472C22×C4#71112yes—
473C22×C4#71112yes—
474C22×C4#71112yes—
475C22×C4C2×C22⋊D511yescharacteristic
476C22×C4C2×C22⋊D511yescharacteristic
477C22×C4C2×C22⋊D511yescharacteristic
478C22×C4C23×D511yescharacteristic
479M16—108no—
480M16—204no—
481C2×C8—54no—
482C2×C8—102no—
483C2×C8—102no—
484C2×C8—102no—
485C2×C8—102no—
486C2×C8—102no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—102no—
496C4⋊C4—102no—
497C4⋊C4—102no—
498C4⋊C4—102no—
499C4⋊C4—202no—
500C4⋊C4—202no—
501C22⋊C4—42no—
502C22⋊C4—42no—
503C22⋊C4—42no—
504C22⋊C4—42no—
505C22⋊C4—42no—
506C22⋊C4—42no—
507C22⋊C4—204no—
508C22⋊C4—204no—
509C42—42no—
510C42—42no—
511C2×C5—24no—
512C2×C5—42no—
513C2×C5—42no—
514C2×C5C2×#39711yescharacteristic
515C2×C5#305511yescharacteristic
516C2×C5#305611yescharacteristic
517C2×C5#309211yescharacteristic
518C2×C5#309311yescharacteristic
519C2×C5#309411yescharacteristic
520C2×C5#309511yescharacteristic
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—28no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23#407512yes—
545C23#408112yes—
546C23#408512yes—
547C23#409512yes—
548C23#409712yes—
549C23C22×C22⋊D511yescharacteristic
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—44no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—202no—
580C2×C4—202no—
581C2×C4—202no—
582C2×C4—202no—
583C2×C4—202no—
584C2×C4—202no—
585C2×C4—404no—
586C2×C4—404no—
587C2×C4#402514yes—
588C2×C4#402514yes—
589C2×C4#402914yes—
590C2×C4#403114yes—
591C2×C4#407512yes—
592C2×C4#408112yes—
593C2×C4#408512yes—
594C2×C4C2×#70911yescharacteristic
595C2×C4C2×#70911yescharacteristic
596C2×C4C2×D4×D511yescharacteristic
597C2×C4C2×#71111yescharacteristic
598C2×C4C2×#71111yescharacteristic
599C2×C4C2×#71111yescharacteristic
600C8—104no—
601C8—202no—
602C5#2364211yescharacteristic, complemented, Sylow
603C22—22no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22—22no—
615C22—416no—
616C22—41no—
617C22—41no—
618C22—41no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22—41no—
624C22#6524114yes—
625C22C2×#407511yescharacteristic
626C22C2×#408111yescharacteristic
627C22C2×#408511yescharacteristic
628C22C2×#409511yescharacteristic
629C22#6615611yescharacteristic
630C22#6615611yescharacteristic
631C22C2×#409711yescharacteristic
632C4—41no—
633C4—41no—
634C4—81no—
635C4—81no—
636C4—81no—
637C4—81no—
638C4—402no—
639C4—402no—
640C4#6524114yes—
641C4C2×#402511yescharacteristic
642C4C2×#402511yescharacteristic
643C4C2×#402911yescharacteristic
644C4C2×#403111yescharacteristic
645C2—24no—
646C2—42no—
647C2—42no—
648C2C2×#6524111yescharacteristic
649C2#1061394811yescharacteristic
650C2#1061395111yescharacteristic
651C2#1061464511yescharacteristic
652C2#1061464611yescharacteristic
653C2#1061465511yescharacteristic
654C2#1061465611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D5
commutator, cosocleC22×C4×C5C24
FrattiniC22×C4C23×D5
FittingC5×#2280C2
socleC23×C5C22×D4
prime radicalC25×C5D4
hypercenter#2280D5
hypocenterC5#23642
LCSG ▷ C22×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C24×C4 ◁ #2280

Complements

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

NH ≅ G/Ntype
4. C2×#65284C2semidirect
5. #10602374C2semidirect
6. #10602374C2semidirect
7. #10602374C2semidirect
8. #10600313C2semidirect
9. #10600312C2semidirect
10. #10600311C2semidirect
11. #10600300C2semidirect
37. C2×#3940C22semidirect
38. C2×#3939C22semidirect
39. #64749C22semidirect
40. #64749C22semidirect
41. #64749C22semidirect
42. #64749C22semidirect
43. #64749C22semidirect
44. #64749C22semidirect
602. C5#23642semidirect

Maximal subgroups

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

subgroupindex
C5×#22802
C2×#655012
C2×#654962
C2×#652842
#106023742
#106003132
#106003122
#106003112
#106003002
#236425