Finite Group Databasebeta

← #12127991

Subgroups — #12127991

Order 1280.

1178 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#389C211yesmaximal, characteristic, complemented
2C2×#65255C211yesmaximal, characteristic
3C2×#65255C211yesmaximal, characteristic
4C2×#65253C211yesmaximal, characteristic, complemented
5#10613801C211yesmaximal, characteristic
6C2×#65213C211yesmaximal, characteristic, complemented
7#10602351C211yesmaximal, characteristic, complemented
8#10600428C212yesmaximal, complemented
9#10600427C212yesmaximal, complemented
10#10600423C212yesmaximal, complemented
11#10600422C212yesmaximal, complemented
12C22×#709C2211yescharacteristic
13C22×C5×C4⋊C4—21no—
14C22×C5×C4⋊C4C2211yescharacteristic
15C22×C5×C4⋊C4C2211yescharacteristic
16C22×C42×C5C2211yescharacteristic
17C2×#4072C2211yescharacteristic
18C22×#681C2211yescharacteristic
19C22×#681C2211yescharacteristic
20C5×#389—21no—
21C5×#389—21no—
22C5×#389C2212yes—
23C5×#389C2212yes—
24C2×C5×#95—21no—
25#65502—21no—
26#65502—21no—
27C2×#3942C2212yes—
28C2×#3942C2212yes—
29#65255—24no—
30#65255—24no—
31C2×#3941C2212yes—
32C2×#3941C2212yes—
33#65253—24no—
34C2×#3940C2212yes—
35C2×#3939C2212yes—
36#65228C2211yescharacteristic
37#65228C2211yescharacteristic
38C4×#685—21no—
39C4×#685—21no—
40#65218C2211yescharacteristic
41#65218C2211yescharacteristic
42#65213—24no—
43C2×C4⋊#168C2212yes—
44C2×C4⋊#168C2212yes—
45#64749C2212yes—
46#64749C2212yes—
47#23617—51nomaximal, Sylow
48C23×C4×C5—21no—
49C23×C4×C5—21no—
50C23×C4×C5D411yescharacteristic
51C23×C4×C5D411yescharacteristic
52C23×C4×C5C2311yescharacteristic
53C22×C22⋊D5—21no—
54C2×#709—22no—
55C2×#709—22no—
56C2×#709—22no—
57C2×#709—22no—
58C2×#709—22no—
59C2×#709Q814yes—
60C22×D20C2311yescharacteristic
61C22×C4×D5—21no—
62C22×Dic10C2311yescharacteristic
63C2×C5×C4⋊C4—21no—
64C2×C5×C4⋊C4—22no—
65C2×C5×C4⋊C4—22no—
66C2×C5×C4⋊C4—22no—
67C2×C5×C4⋊C4—21no—
68C2×C5×C4⋊C4—22no—
69C2×C5×C4⋊C4—22no—
70C2×C5×C4⋊C4—44no—
71C2×C5×C4⋊C4D414yes—
72C2×C5×C4⋊C4C2312yes—
73C2×C5×C4⋊C4C2312yes—
74C2×C42×C5—22no—
75C2×C42×C5—22no—
76C2×C42×C5D411yescharacteristic
77C2×C42×C5D411yescharacteristic
78C2×C42×C5C2311yescharacteristic
79C2×C42×C5C2311yescharacteristic
80C2×C22⋊Dic5—21no—
81C2×#685—21no—
82C2×#685—21no—
83C2×#685—21no—
84C2×#685—21no—
85#4072—22no—
86#4072—22no—
87C2×C4⋊Dic5C2311yescharacteristic
88C2×C4⋊Dic5C2311yescharacteristic
89C2×C4×Dic5—21no—
90C2×#681—22no—
91C2×#681—22no—
92C2×#681—22no—
93C2×#681—22no—
94C2×#681—22no—
95C2×#681D414yes—
96C22×#168C2312yes—
97C22×#168C2312yes—
98C5×#95—22no—
99C5×#95—22no—
100#3963—21no—
101#3963—21no—
102#3963—21no—
103#3963—21no—
104#3942—24no—
105#3942—24no—
106#3941—24no—
107#3941—24no—
108#3940—24no—
109#3939—24no—
110C4⋊#168—24no—
111C4⋊#168—24no—
112C2×#428—51no—
113C2×#423—51no—
114C2×#417—51no—
115C2×#417—51no—
116#2297—51no—
117C2×#389D511yescharacteristic, complemented
118#2068—52no—
119#2067—52no—
120#2066—52no—
121#2065—52no—
122#1736—51no—
123C24×C5C2×D411yescharacteristic
124C23×D5—21no—
125C22×C4×C5—22no—
126C22×C4×C5—22no—
127C22×C4×C5—24no—
128C22×C4×C5—22no—
129C22×C4×C5—22no—
130C22×C4×C5—22no—
131C22×C4×C5—22no—
132C22×C4×C5—41no—
133C22×C4×C5—41no—
134C22×C4×C5—42no—
135C22×C4×C5—42no—
136C22×C4×C5—42no—
137C22×C4×C5—42no—
138C22×C4×C5—42no—
139C22×C4×C5C2×D412yes—
140C22×C4×C5C2×D412yes—
141C22×C4×C5C2×D411yescharacteristic
142C22×C4×C5C2×D411yescharacteristic
143C22×C4×C5C2×D411yescharacteristic
144C22×C4×C5C2×D411yescharacteristic
145C22×C4×C5C2×D411yescharacteristic
146C22×C4×C5C2×Q812yes—
147C22×C4×C5#4112yes—
148C22×C4×C5#4112yes—
149C22×C4×C5#4112yes—
150C22×C4×C5C2411yescharacteristic
151C2×C22⋊D5—24no—
152C2×C22⋊D5—42no—
153C2×C22⋊D5—42no—
154C22×Dic5—21no—
155C22×Dic5—21no—
156C22×Dic5—21no—
157#709—28no—
158#709—28no—
159#709—44no—
160#709—44no—
161C2×D20—21no—
162C2×D20—21no—
163C2×D20—21no—
164C2×D20—21no—
165C2×D20—21no—
166C2×D20C2×Q812yes—
167C2×C4×D5—24no—
168C2×C4×D5—42no—
169C2×C4×D5—42no—
170C2×Dic10—21no—
171C2×Dic10—21no—
172C2×Dic10—21no—
173C2×Dic10—21no—
174C2×Dic10—21no—
175C2×Dic10C2×Q812yes—
176C5×C4⋊C4—24no—
177C5×C4⋊C4—22no—
178C5×C4⋊C4—22no—
179C5×C4⋊C4—22no—
180C5×C4⋊C4—44no—
181C5×C4⋊C4—42no—
182C5×C4⋊C4—44no—
183C5×C4⋊C4C2×D414yes—
184C42×C5—22no—
185C42×C5—22no—
186C42×C5—21no—
187C42×C5—21no—
188C42×C5—21no—
189C42×C5—21no—
190C22⋊Dic5—42no—
191#685—42no—
192#685—42no—
193#685—42no—
194#685—42no—
195C4⋊Dic5—21no—
196C4⋊Dic5—21no—
197C4⋊Dic5—21no—
198C4⋊Dic5—21no—
199C4×Dic5—42no—
200#681—28no—
201#681—44no—
202C2×#168—22no—
203C2×#168—22no—
204C2×#168—22no—
205C2×#168—22no—
206C2×#168—22no—
207C2×#168C2×D414yes—
208C22×#41—51no—
209C22×M16—51no—
210C22×M16—51no—
211C2×#117—51no—
212C22×C4⋊C4—21no—
213C22×C4⋊C4C2×D511yescharacteristic
214C22×C4⋊C4C2×D511yescharacteristic
215C22×C42C2×D511yescharacteristic
216#428—104no—
217C2×#107—52no—
218C2×#106—52no—
219#423—104no—
220C2×C4⋊C8—52no—
221C2×C4⋊C8—52no—
222#417—104no—
223#417—104no—
224C2×#103—52no—
225C2×#103—52no—
226C2×#102—52no—
227C2×#102—52no—
228#391—51no—
229#391—51no—
230#390—51no—
231#390—51no—
232#389—21no—
233#389—21no—
234#389C2×D512yes—
235#389C2×D512yes—
236#386—101no—
237#386—101no—
238C4×C22⋊C4—101no—
239C4×C22⋊C4—101no—
240C2×#95—21no—
241#340—52no—
242#340—52no—
243C23×C5—21no—
244C23×C5—21no—
245C23×C5—21no—
246C23×C5—21no—
247C23×C5#12112yes—
248C23×C5#12112yes—
249C23×C5C22⋊Q812yes—
250C23×C5C22×D411yescharacteristic
251C22×D5—22no—
252C22×D5—41no—
253C22×D5—41no—
254C22×D5—41no—
255C22×D5—41no—
256C22×D5—41no—
257C22×D5—41no—
258C2×C4×C5—21no—
259C2×C4×C5—21no—
260C2×C4×C5—21no—
261C2×C4×C5—21no—
262C2×C4×C5—21no—
263C2×C4×C5—21no—
264C2×C4×C5—24no—
265C2×C4×C5—21no—
266C2×C4×C5—21no—
267C2×C4×C5—21no—
268C2×C4×C5—21no—
269C2×C4×C5—44no—
270C2×C4×C5—44no—
271C2×C4×C5—48no—
272C2×C4×C5—42no—
273C2×C4×C5—42no—
274C2×C4×C5—42no—
275C2×C4×C5—42no—
276C2×C4×C5—42no—
277C2×C4×C5—84no—
278C2×C4×C5#12114yes—
279C2×C4×C5#12112yes—
280C2×C4×C5#12112yes—
281C2×C4×C5C22⋊Q814yes—
282C2×C4×C5C22⋊Q814yes—
283C2×C4×C5C22⋊Q812yes—
284C2×C4×C5#12614yes—
285C2×C4×C5AGL(1,ℤ/8)12yes—
286C2×C4×C5#13712yes—
287C2×C4×C5C22×D411yescharacteristic
288C2×C4×C5C22×D411yescharacteristic
289C2×C4×C5C22×Q811yescharacteristic
290C2×C4×C5C2×#4111yescharacteristic
291C2×C4×C5C2×#4111yescharacteristic
292C2×C4×C5C2×#4111yescharacteristic
293C22⋊D5—48no—
294C2×Dic5—22no—
295C2×Dic5—41no—
296C2×Dic5—41no—
297C2×Dic5—41no—
298C2×Dic5—41no—
299C2×Dic5—41no—
300C2×Dic5—41no—
301C2×Dic5—41no—
302C2×Dic5—41no—
303D20—22no—
304D20—22no—
305D20—41no—
306D20—41no—
307C4×D5—48no—
308Dic10—22no—
309Dic10—22no—
310Dic10—41no—
311Dic10—41no—
312#168—24no—
313#168—42no—
314C2×#41—54no—
315C2×#41—102no—
316C2×#41—102no—
317C2×#41—102no—
318C2×#41—102no—
319C2×#41—102no—
320C22×Q8—51no—
321C22×D4—51no—
322C22×D4—101no—
323C23×C4—21no—
324C23×C4—21no—
325C23×C4—101no—
326C23×C4C22⋊D511yescharacteristic
327C23×C4C22⋊D511yescharacteristic
328C23×C4C22×D511yescharacteristic
329C2×M16—54no—
330C2×M16—102no—
331C2×M16—102no—
332C2×M16—102no—
333C2×M16—102no—
334C2×M16—102no—
335C22×C8—52no—
336C22×C8—52no—
337#117—102no—
338#117—102no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—44no—
347C2×C4⋊C4—51no—
348C2×C4⋊C4—51no—
349C2×C4⋊C4D2014yes—
350C2×C4⋊C4C22×D512yes—
351C2×C4⋊C4C22×D512yes—
352C2×C22⋊C4—101no—
353C2×C22⋊C4—101no—
354C2×C22⋊C4—101no—
355C2×C22⋊C4—101no—
356C2×C22⋊C4—101no—
357C2×C42—22no—
358C2×C42—22no—
359C2×C42—101no—
360C2×C42C22⋊D511yescharacteristic
361C2×C42C22⋊D511yescharacteristic
362C2×C42C22×D511yescharacteristic
363C2×C42C22×D511yescharacteristic
364#107—104no—
365#106—104no—
366C4⋊C8—104no—
367C4⋊C8—104no—
368#103—104no—
369#103—104no—
370#102—104no—
371#102—104no—
372#95—22no—
373#95—22no—
374#95—101no—
375#95—101no—
376#95—101no—
377#95—101no—
378C22×C5—22no—
379C22×C5—22no—
380C22×C5—22no—
381C22×C5—22no—
382C22×C5—22no—
383C22×C5—22no—
384C22×C5#39614yes—
385C22×C5C2×#12111yescharacteristic
386C22×C5C2×#12111yescharacteristic
387C22×C5C2×C22⋊Q811yescharacteristic
388C22×C5C2×AGL(1,ℤ/8)11yescharacteristic
389C22×C5C2×#13711yescharacteristic
390C22×C5#57511yescharacteristic
391C22×C5#57511yescharacteristic
392C2×D5—42no—
393C2×D5—42no—
394C2×D5—42no—
395C2×D5—81no—
396C2×D5—81no—
397C2×D5—81no—
398C2×D5—81no—
399C4×C5—22no—
400C4×C5—22no—
401C4×C5—44no—
402C4×C5—41no—
403C4×C5—41no—
404C4×C5—82no—
405C4×C5#39614yes—
406C4×C5C2×#12111yescharacteristic
407C4×C5C2×C22⋊Q811yescharacteristic
408C4×C5C2×C22⋊Q811yescharacteristic
409C4×C5C2×#12611yescharacteristic
410Dic5—42no—
411Dic5—81no—
412Dic5—81no—
413C24—101no—
414C24C2×C22⋊D511yescharacteristic
415#41—108no—
416#41—108no—
417#41—204no—
418#41—204no—
419C2×Q8—52no—
420C2×Q8—101no—
421C2×Q8—101no—
422C2×Q8—101no—
423C2×Q8—101no—
424C2×Q8—101no—
425C2×D4—52no—
426C2×D4—104no—
427C2×D4—101no—
428C2×D4—101no—
429C2×D4—101no—
430C2×D4—101no—
431C2×D4—101no—
432C2×D4—202no—
433C2×D4—202no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—41no—
442C22×C4—41no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—101no—
449C22×C4—104no—
450C22×C4—101no—
451C22×C4—101no—
452C22×C4—202no—
453C22×C4—202no—
454C22×C4C2×D2012yes—
455C22×C4#70912yes—
456C22×C4D4×D512yes—
457C22×C4#71112yes—
458C22×C4Q8×D512yes—
459C22×C4#71312yes—
460C22×C4C2×C22⋊D511yescharacteristic
461C22×C4C2×C22⋊D511yescharacteristic
462C22×C4C2×C22⋊D511yescharacteristic
463C22×C4C2×C22⋊D511yescharacteristic
464C22×C4C2×C22⋊D511yescharacteristic
465C22×C4C23×D511yescharacteristic
466M16—108no—
467M16—204no—
468C2×C8—54no—
469C2×C8—102no—
470C2×C8—102no—
471C2×C8—102no—
472C2×C8—102no—
473C2×C8—102no—
474C4⋊C4—22no—
475C4⋊C4—24no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—44no—
479C4⋊C4—44no—
480C4⋊C4—42no—
481C4⋊C4—101no—
482C4⋊C4—101no—
483C4⋊C4—101no—
484C4⋊C4—101no—
485C4⋊C4C2×D2014yes—
486C22⋊C4—202no—
487C22⋊C4—202no—
488C22⋊C4—202no—
489C22⋊C4—202no—
490C22⋊C4—202no—
491C42—21no—
492C42—21no—
493C42—21no—
494C42—21no—
495C42—22no—
496C42—22no—
497C42—202no—
498C2×C5—24no—
499C2×C5C2×#39611yescharacteristic
500C2×C5#303611yescharacteristic
501C2×C5#303711yescharacteristic
502C2×C5#305511yescharacteristic
503C2×C5#305611yescharacteristic
504C2×C5#307911yescharacteristic
505C2×C5#308011yescharacteristic
506D5—82no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—102no—
512C23—201no—
513C23—201no—
514C23—201no—
515C23—201no—
516C23—201no—
517C23—201no—
518C23#407612yes—
519C23#408512yes—
520C23#409212yes—
521C23C22×C22⋊D511yescharacteristic
522Q8—102no—
523Q8—102no—
524Q8—201no—
525Q8—201no—
526D4—102no—
527D4—102no—
528D4—208no—
529D4—201no—
530D4—201no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—24no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—48no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—84no—
551C2×C4—102no—
552C2×C4—201no—
553C2×C4—201no—
554C2×C4—208no—
555C2×C4—201no—
556C2×C4—201no—
557C2×C4—201no—
558C2×C4—201no—
559C2×C4—201no—
560C2×C4—201no—
561C2×C4#404114yes—
562C2×C4#404214yes—
563C2×C4#404314yes—
564C2×C4#404414yes—
565C2×C4#407612yes—
566C2×C4#407812yes—
567C2×C4#408512yes—
568C2×C4#408812yes—
569C2×C4#409212yes—
570C2×C4C22×D2011yescharacteristic
571C2×C4C2×#70911yescharacteristic
572C2×C4C2×D4×D511yescharacteristic
573C2×C4C2×#71111yescharacteristic
574C2×C4C2×Q8×D511yescharacteristic
575C2×C4C2×#71311yescharacteristic
576C8—104no—
577C8—202no—
578C5#2361711yescharacteristic, complemented, Sylow
579C22—22no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—22no—
584C22—22no—
585C22—202no—
586C22—202no—
587C22—202no—
588C22—401no—
589C22—401no—
590C22—401no—
591C22—401no—
592C22#6528314yes—
593C22C2×#407611yescharacteristic
594C22C2×#407811yescharacteristic
595C22C2×#408511yescharacteristic
596C22C2×#408811yescharacteristic
597C22C2×#409211yescharacteristic
598C22#6615611yescharacteristic
599C22#6615611yescharacteristic
600C4—22no—
601C4—22no—
602C4—44no—
603C4—41no—
604C4—41no—
605C4—82no—
606C4—202no—
607C4—401no—
608C4—401no—
609C4#6528314yes—
610C4C2×#404111yescharacteristic
611C4C2×#404211yescharacteristic
612C4C2×#404311yescharacteristic
613C4C2×#404411yescharacteristic
614C2—24no—
615C2—402no—
616C2C2×#6528311yescharacteristic
617C2#1061403811yescharacteristic
618C2#1061417211yescharacteristic
619C2#1061435211yescharacteristic
620C2#1061446311yescharacteristic
621C2#1061474811yescharacteristic
622C2#1061494311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D5
commutator, cosocleC22×C4×C5C24
FrattiniC22×C4C23×D5
FittingC2×C5×#389C2
socleC23×C5C22×D4
prime radicalC22×#709C22
hypercenterC2×#389D5
hypocenterC5#23617
LCSG ▷ C22×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
1. C2×C5×#389C2semidirect
4. C2×#65253C2semidirect
6. C2×#65213C2semidirect
7. #10602351C2semidirect
8. #10600428C2semidirect
9. #10600427C2semidirect
10. #10600423C2semidirect
11. #10600422C2semidirect
117. C2×#389D5semidirect
578. C5#23617semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#3892
C2×#652552
C2×#652532
#106138012
C2×#652132
#106023512
#106004282
#106004272
#106004232
#106004222
#236175