Finite Group Databasebeta

← #12102011

Subgroups — #12102011

Order 1280.

1198 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#2297C211yesmaximal, characteristic
2C2×#65521C211yesmaximal, characteristic
3C2×#65521C211yesmaximal, characteristic
4C2×#65217C211yesmaximal, characteristic, complemented
5C2×C4×#681C211yesmaximal, characteristic, complemented
6#10602374C211yesmaximal, characteristic, complemented
7#10602374C211yesmaximal, characteristic, complemented
8#10600468C212yesmaximal, complemented
9#10600467C212yesmaximal, complemented
10#10600438C212yesmaximal, complemented
11#10600437C212yesmaximal, complemented
12C22×C5×#41C2211yescharacteristic
13C2×C5×#117C2211yescharacteristic
14C22×C42×C5C2211yescharacteristic
15C22×C4⋊Dic5C2211yescharacteristic
16C22×C4⋊Dic5C2211yescharacteristic
17C22×#683—21no—
18C22×#681C2211yescharacteristic
19C22×#681C2211yescharacteristic
20C5×#391C2211yescharacteristic
21C5×#391C2211yescharacteristic
22C5×#390C2211yescharacteristic
23C5×#390C2211yescharacteristic
24C5×#386—21no—
25C5×#386—21no—
26C4×C5×C22⋊C4—21no—
27C4×C5×C22⋊C4—21no—
28#65521—24no—
29#65521C418yescomplemented
30C2×#3967C2212yes—
31C2×#3967C2212yes—
32C2×#3964C2212yes—
33C2×#3964C2212yes—
34C2×#3963—21no—
35#65217—21no—
36#65217—21no—
37#65217C2212yes—
38#65217C2212yes—
39C4×#681—24no—
40C2×#3935C2212yes—
41C2×C4×#168C2212yes—
42#64749C2212yes—
43#64749C2212yes—
44#64749C2212yes—
45#64749C2212yes—
46#21093—51nomaximal, Sylow
47C2×C5×#41—22no—
48C2×C5×#41—22no—
49C2×C5×#41—22no—
50C2×C5×#41—22no—
51C2×C5×#41—22no—
52C2×C5×#41C2×C414yes—
53C22×C5×Q8C2311yescharacteristic
54C22×C5×D4—21no—
55C22×C5×D4C2311yescharacteristic
56C23×C4×C5—21no—
57C23×C4×C5D411yescharacteristic
58C23×C4×C5D411yescharacteristic
59C23×C4×C5C2311yescharacteristic
60C23×Dic5—21no—
61C23×Dic5—21no—
62C5×#117—22no—
63C5×#117—22no—
64C2×C5×C4⋊C4C2311yescharacteristic
65C2×C5×C4⋊C4C2311yescharacteristic
66C2×C5×C22⋊C4—21no—
67C2×C5×C22⋊C4—21no—
68C2×C5×C22⋊C4—21no—
69C2×C5×C22⋊C4—21no—
70C2×C5×C22⋊C4—21no—
71C2×C42×C5—22no—
72C2×C42×C5—22no—
73C2×C42×C5—21no—
74C2×C42×C5D411yescharacteristic
75C2×C42×C5D411yescharacteristic
76C2×C42×C5C2311yescharacteristic
77C2×C42×C5C2311yescharacteristic
78C2×C4⋊Dic5—21no—
79C2×C4⋊Dic5—22no—
80C2×C4⋊Dic5—22no—
81C2×C4⋊Dic5—22no—
82C2×C4⋊Dic5—21no—
83C2×C4⋊Dic5C2×C414yes—
84C2×C4⋊Dic5C2312yes—
85C2×C4⋊Dic5C2312yes—
86C2×#683—22no—
87C2×#683—22no—
88C2×#683—44no—
89C2×#681—22no—
90C2×#681—22no—
91C2×#681—22no—
92C2×#681—22no—
93C2×#681C2×C414yes—
94C2×#681D414yes—
95C22×#168C2312yes—
96C22×#168C2312yes—
97C5×#95—21no—
98C5×#95—21no—
99C5×#95—21no—
100C5×#95—21no—
101#3967—24no—
102#3967C2×C418yescomplemented
103#3964—24no—
104#3964C2×C418yescomplemented
105#3963—22no—
106#3963—22no—
107#3935—24no—
108C4×#168—24no—
109C2×#417—51no—
110C2×#417—51no—
111C2×C4×M16—51no—
112#2297D511yescharacteristic
113C2×#389—51no—
114#1936—52no—
115#1935—52no—
116#1934—52no—
117#1933—52no—
118#1736—51no—
119#1736—51no—
120C24×C5—21no—
121C24×C5C2×D411yescharacteristic
122C5×#41—28no—
123C5×#41—28no—
124C5×#41—44no—
125C5×#41—44no—
126C2×C5×Q8—21no—
127C2×C5×Q8—21no—
128C2×C5×Q8—21no—
129C2×C5×Q8—21no—
130C2×C5×Q8—21no—
131C2×C5×Q8C22×C412yes—
132C2×C5×D4—24no—
133C2×C5×D4—21no—
134C2×C5×D4—21no—
135C2×C5×D4—21no—
136C2×C5×D4—21no—
137C2×C5×D4—21no—
138C2×C5×D4—42no—
139C2×C5×D4—42no—
140C2×C5×D4C22×C412yes—
141C22×C4×C5—22no—
142C22×C4×C5—22no—
143C22×C4×C5—21no—
144C22×C4×C5—24no—
145C22×C4×C5—22no—
146C22×C4×C5—22no—
147C22×C4×C5—21no—
148C22×C4×C5—21no—
149C22×C4×C5—41no—
150C22×C4×C5—41no—
151C22×C4×C5—42no—
152C22×C4×C5—42no—
153C22×C4×C5C22×C412yes—
154C22×C4×C5C2×D412yes—
155C22×C4×C5C2×D411yescharacteristic
156C22×C4×C5C2×D411yescharacteristic
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×C5C2411yescharacteristic
165C22×Dic5—24no—
166C22×Dic5—22no—
167C22×Dic5—22no—
168C22×Dic5—42no—
169C22×Dic5—42no—
170C22×Dic5—42no—
171C22×Dic5—42no—
172C22×Dic5—42no—
173C5×C4⋊C4—21no—
174C5×C4⋊C4—21no—
175C5×C4⋊C4—21no—
176C5×C4⋊C4—21no—
177C5×C22⋊C4—42no—
178C5×C22⋊C4—42no—
179C5×C22⋊C4—42no—
180C5×C22⋊C4—42no—
181C5×C22⋊C4—42no—
182C42×C5—22no—
183C42×C5—22no—
184C42×C5—21no—
185C42×C5—21no—
186C42×C5—21no—
187C42×C5—21no—
188C42×C5—42no—
189C4⋊Dic5—24no—
190C4⋊Dic5—22no—
191C4⋊Dic5—22no—
192C4⋊Dic5—22no—
193C4⋊Dic5—42no—
194C4⋊Dic5C22×C414yes—
195#683—44no—
196#683—44no—
197#681—28no—
198#681—28no—
199C2×#168—22no—
200C2×#168—22no—
201C2×#168—22no—
202C2×#168—22no—
203C2×#168C22×C414yes—
204C2×#168C2×D414yes—
205C22×#41C2×D511yescharacteristic
206C22×M16—51no—
207C22×M16—51no—
208C2×#117C2×D511yescharacteristic
209C22×C4⋊C4—51no—
210C22×C4⋊C4—51no—
211C22×C4⋊C4—101no—
212C22×C42C2×D511yescharacteristic
213#417—58no—
214#417—104no—
215C2×#103—52no—
216C2×#103—52no—
217C2×#102—52no—
218C2×#102—52no—
219C4×M16—104no—
220C2×#97—52no—
221C2×C4×C8—52no—
222#391C2×D511yescharacteristic
223#391C2×D511yescharacteristic
224#390C2×D511yescharacteristic
225#390C2×D511yescharacteristic
226#389—52no—
227#389—52no—
228#389—101no—
229#389—101no—
230#386—21no—
231#386—21no—
232C4×C22⋊C4—21no—
233C4×C22⋊C4—21no—
234C2×#95—101no—
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—22no—
244C23×C5—41no—
245C23×C5—41no—
246C23×C5—41no—
247C23×C5—41no—
248C23×C5—41no—
249C23×C5—41no—
250C23×C5C4×D412yes—
251C23×C5#12112yes—
252C23×C5#12412yes—
253C23×C5C22×D411yescharacteristic
254C5×Q8—22no—
255C5×Q8—22no—
256C5×Q8—41no—
257C5×Q8—41no—
258C5×D4—22no—
259C5×D4—22no—
260C5×D4—48no—
261C5×D4—41no—
262C5×D4—41no—
263C2×C4×C5—21no—
264C2×C4×C5—21no—
265C2×C4×C5—21no—
266C2×C4×C5—21no—
267C2×C4×C5—22no—
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—44no—
276C2×C4×C5—41no—
277C2×C4×C5—41no—
278C2×C4×C5—48no—
279C2×C4×C5—41no—
280C2×C4×C5—41no—
281C2×C4×C5—41no—
282C2×C4×C5—41no—
283C2×C4×C5—41no—
284C2×C4×C5—41no—
285C2×C4×C5#11714yes—
286C2×C4×C5C4×D414yes—
287C2×C4×C5C4×D412yes—
288C2×C4×C5#12112yes—
289C2×C4×C5#12314yes—
290C2×C4×C5#12412yes—
291C2×C4×C5#12614yes—
292C2×C4×C5AGL(1,ℤ/8)12yes—
293C2×C4×C5#13712yes—
294C2×C4×C5C23×C411yescharacteristic
295C2×C4×C5C22×D411yescharacteristic
296C2×C4×C5C2×#4111yescharacteristic
297C2×C4×C5C2×#4111yescharacteristic
298C2×C4×C5C2×#4111yescharacteristic
299C2×C4×C5C2×#4111yescharacteristic
300C2×Dic5—24no—
301C2×Dic5—48no—
302C2×Dic5—42no—
303C2×Dic5—42no—
304C2×Dic5—42no—
305C2×Dic5—42no—
306C2×Dic5—42no—
307C2×Dic5—84no—
308#168—24no—
309#168—24no—
310C2×#41—22no—
311C2×#41—22no—
312C2×#41—22no—
313C2×#41—22no—
314C2×#41—22no—
315C2×#41C4×D514yes—
316C22×Q8C22×D511yescharacteristic
317C22×D4—21no—
318C22×D4C22×D511yescharacteristic
319C23×C4—21no—
320C23×C4—101no—
321C23×C4—101no—
322C23×C4C22⋊D511yescharacteristic
323C23×C4C22⋊D511yescharacteristic
324C23×C4C22×D511yescharacteristic
325C2×M16—54no—
326C2×M16—54no—
327C2×M16—102no—
328C2×M16—102no—
329C2×M16—102no—
330C2×M16—102no—
331C22×C8—52no—
332C22×C8—52no—
333#117—22no—
334#117—22no—
335C2×C4⋊C4—54no—
336C2×C4⋊C4—52no—
337C2×C4⋊C4—52no—
338C2×C4⋊C4—102no—
339C2×C4⋊C4—101no—
340C2×C4⋊C4—102no—
341C2×C4⋊C4—102no—
342C2×C4⋊C4—101no—
343C2×C4⋊C4—102no—
344C2×C4⋊C4—102no—
345C2×C4⋊C4—204no—
346C2×C4⋊C4C22×D511yescharacteristic
347C2×C4⋊C4C22×D511yescharacteristic
348C2×C22⋊C4—21no—
349C2×C22⋊C4—21no—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—21no—
353C2×C42—22no—
354C2×C42—22no—
355C2×C42—21no—
356C2×C42C22⋊D511yescharacteristic
357C2×C42C22⋊D511yescharacteristic
358C2×C42C22×D511yescharacteristic
359C2×C42C22×D511yescharacteristic
360#103—58no—
361#103—104no—
362#102—58no—
363#102—104no—
364#97—104no—
365C4×C8—104no—
366#95—21no—
367#95—21no—
368#95—21no—
369#95—21no—
370#95—102no—
371#95—102no—
372C22×C5—22no—
373C22×C5—22no—
374C22×C5—22no—
375C22×C5—22no—
376C22×C5—22no—
377C22×C5—22no—
378C22×C5—42no—
379C22×C5—42no—
380C22×C5—42no—
381C22×C5—81no—
382C22×C5—81no—
383C22×C5—81no—
384C22×C5—81no—
385C22×C5#38814yes—
386C22×C5C2×C4×D411yescharacteristic
387C22×C5C2×#12111yescharacteristic
388C22×C5C2×#12411yescharacteristic
389C22×C5C2×AGL(1,ℤ/8)11yescharacteristic
390C22×C5C2×#13711yescharacteristic
391C22×C5#57511yescharacteristic
392C22×C5#57511yescharacteristic
393C4×C5—22no—
394C4×C5—22no—
395C4×C5—41no—
396C4×C5—41no—
397C4×C5—42no—
398C4×C5—81no—
399C4×C5—81no—
400C4×C5#38814yes—
401C4×C5C2×#11711yescharacteristic
402C4×C5C2×C4×D411yescharacteristic
403C4×C5C2×#12311yescharacteristic
404C4×C5C2×#12611yescharacteristic
405Dic5—44no—
406Dic5—82no—
407C24—21no—
408C24C2×C22⋊D511yescharacteristic
409#41—28no—
410#41—28no—
411#41—44no—
412#41—44no—
413C2×Q8—21no—
414C2×Q8—21no—
415C2×Q8—21no—
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8C2×C4×D512yes—
419C2×D4—24no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4—21no—
425C2×D4—42no—
426C2×D4—42no—
427C2×D4C2×C4×D512yes—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—41no—
437C22×C4—41no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—102no—
441C22×C4—104no—
442C22×C4—102no—
443C22×C4—202no—
444C22×C4—202no—
445C22×C4—202no—
446C22×C4—202no—
447C22×C4—202no—
448C22×C4C2×C4×D512yes—
449C22×C4#70912yes—
450C22×C4D4×D512yes—
451C22×C4#71112yes—
452C22×C4#71112yes—
453C22×C4#71112yes—
454C22×C4C2×C22⋊D511yescharacteristic
455C22×C4C2×C22⋊D511yescharacteristic
456C22×C4C2×C22⋊D511yescharacteristic
457C22×C4C2×C22⋊D511yescharacteristic
458C22×C4C2×C22⋊D511yescharacteristic
459C22×C4C23×D511yescharacteristic
460M16—108no—
461M16—108no—
462C2×C8—54no—
463C2×C8—54no—
464C2×C8—102no—
465C2×C8—102no—
466C2×C8—102no—
467C2×C8—102no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—54no—
473C4⋊C4—102no—
474C4⋊C4—104no—
475C4⋊C4—102no—
476C4⋊C4—102no—
477C4⋊C4—204no—
478C4⋊C4—202no—
479C4⋊C4—204no—
480C22⋊C4—42no—
481C22⋊C4—42no—
482C22⋊C4—42no—
483C22⋊C4—42no—
484C22⋊C4—42no—
485C42—21no—
486C42—21no—
487C42—21no—
488C42—21no—
489C42—22no—
490C42—22no—
491C42—42no—
492C2×C5—24no—
493C2×C5—82no—
494C2×C5C2×#38811yescharacteristic
495C2×C5C4×AGL(1,ℤ/8)11yescharacteristic
496C2×C5C4×#13711yescharacteristic
497C2×C5#305511yescharacteristic
498C2×C5#305611yescharacteristic
499C2×C5#314111yescharacteristic
500C2×C5#314211yescharacteristic
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—22no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23C4×C22⋊D512yes—
513C23#408212yes—
514C23#408512yes—
515C23C22×C22⋊D511yescharacteristic
516Q8—22no—
517Q8—22no—
518Q8—41no—
519Q8—41no—
520D4—22no—
521D4—22no—
522D4—48no—
523D4—41no—
524D4—41no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—48no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—104no—
548C2×C4—208no—
549C2×C4—202no—
550C2×C4—202no—
551C2×C4—202no—
552C2×C4—202no—
553C2×C4—202no—
554C2×C4—404no—
555C2×C4#402314yes—
556C2×C4#402514yes—
557C2×C4#402714yes—
558C2×C4#402914yes—
559C2×C4C4×C22⋊D512yes—
560C2×C4#408212yes—
561C2×C4#408512yes—
562C2×C4#409512yes—
563C2×C4#409712yes—
564C2×C4C22×C4×D511yescharacteristic
565C2×C4C2×#70911yescharacteristic
566C2×C4C2×D4×D511yescharacteristic
567C2×C4C2×#71111yescharacteristic
568C2×C4C2×#71111yescharacteristic
569C2×C4C2×#71111yescharacteristic
570C8—104no—
571C8—104no—
572C5#2109311yescharacteristic, complemented, Sylow
573C22—22no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22—81no—
586C22#6523414yes—
587C22C2×C4×C22⋊D511yescharacteristic
588C22C2×#408211yescharacteristic
589C22C2×#408511yescharacteristic
590C22C2×#409511yescharacteristic
591C22#6615611yescharacteristic
592C22#6615611yescharacteristic
593C22C2×#409711yescharacteristic
594C4—22no—
595C4—22no—
596C4—42no—
597C4—41no—
598C4—41no—
599C4—81no—
600C4—81no—
601C4—204no—
602C4—402no—
603C4#6523414yes—
604C4C2×#402311yescharacteristic
605C4C2×#402511yescharacteristic
606C4C2×#402711yescharacteristic
607C4C2×#402911yescharacteristic
608C2—24no—
609C2—82no—
610C2C2×#6523411yescharacteristic
611C2C4×#409511yescharacteristic
612C2C4×#409711yescharacteristic
613C2#1061465511yescharacteristic
614C2#1061465611yescharacteristic
615C2#1061501411yescharacteristic
616C2#1061501511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D5
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC5×#2297C2
socleC23×C5C22×D4
prime radicalC22×C5×#41C22
cosocleC22×C4×C5C24
hypercenter#2297D5
hypocenterC5#21093
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2297

Complements

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

NH ≅ G/Ntype
4. C2×#65217C2semidirect
5. C2×C4×#681C2semidirect
6. #10602374C2semidirect
7. #10602374C2semidirect
8. #10600468C2semidirect
9. #10600467C2semidirect
10. #10600438C2semidirect
11. #10600437C2semidirect
29. #65521C4semidirect
102. #3967C2×C4semidirect
104. #3964C2×C4semidirect
572. C5#21093semidirect

Maximal subgroups

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

subgroupindex
C5×#22972
C2×#655212
C2×#652172
C2×C4×#6812
#106023742
#106004682
#106004672
#106004382
#106004372
#210935