Finite Group Databasebeta

← #12102002

Subgroups — #12102002

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
1C2×C5×#417C211yesmaximal, characteristic
2#10615817C211yesmaximal, characteristic
3C2×#65521C211yesmaximal, characteristic
4C2×M16×Dic5C211yesmaximal, characteristic, complemented
5C2×#65262C211yesmaximal, characteristic, complemented
6#10602778C211yesmaximal, characteristic, complemented
7#10602351C211yesmaximal, characteristic, complemented
8#10601049C212yesmaximal, complemented
9#10601047C212yesmaximal, complemented
10#10600981C212yesmaximal, complemented
11#10600979C212yesmaximal, complemented
12C22×C5×#41C2211yescharacteristic
13C22×C5×M16C2211yescharacteristic
14C22×C5×C4⋊C4C2211yescharacteristic
15C2×#4072C2211yescharacteristic
16C22×C4⋊Dic5C2211yescharacteristic
17C22×#683—21no—
18C22×C4×Dic5C2211yescharacteristic
19C22×#681C2211yescharacteristic
20C5×#417—24no—
21C2×C5×#103C2212yes—
22C2×C5×#102C2212yes—
23#65521C418yescomplemented
24#65519C2211yescharacteristic
25#65519C2211yescharacteristic
26C2×#3967C2212yes—
27#65510C2211yescharacteristic
28#65510C2211yescharacteristic
29#65509—21no—
30#65509—21no—
31C2×#3964C2212yes—
32C4×C22⋊Dic5—21no—
33C4×C22⋊Dic5—21no—
34C2×#3963—21no—
35M16×Dic5—24no—
36C2×#3947C2212yes—
37C2×C8×Dic5C2212yes—
38#65262—21no—
39#65262—21no—
40#65262C2212yes—
41#65262C2212yes—
42#64771C2212yes—
43#64771C2212yes—
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×C5—21no—
58C23×C4×C5C2311yescharacteristic
59C23×Dic5—21no—
60C23×Dic5D411yescharacteristic
61C23×Dic5D411yescharacteristic
62C2×C5×M16—22no—
63C2×C5×M16—22no—
64C2×C5×M16D414yes—
65C22×C5×C8C2312yes—
66C2×C5×C4⋊C4—22no—
67C2×C5×C4⋊C4—22no—
68C2×C5×C4⋊C4—21no—
69C2×C5×C4⋊C4C2312yes—
70C2×C22⋊Dic5—21no—
71C2×C22⋊Dic5—21no—
72C2×C22⋊Dic5—21no—
73C2×C22⋊Dic5—21no—
74C2×C22⋊Dic5—21no—
75#4072—22no—
76#4072—22no—
77C2×C4⋊Dic5—21no—
78C2×C4⋊Dic5—22no—
79C2×C4⋊Dic5C2×C414yes—
80C2×C4⋊Dic5C2311yescharacteristic
81C2×C4⋊Dic5C2311yescharacteristic
82C2×C4⋊Dic5C2312yes—
83C2×#683—22no—
84C2×#683—22no—
85C2×#683—44no—
86C2×C4×Dic5—22no—
87C2×C4×Dic5—22no—
88C2×C4×Dic5—21no—
89C2×C4×Dic5D411yescharacteristic
90C2×C4×Dic5D411yescharacteristic
91C2×C4×Dic5C2311yescharacteristic
92C2×C4×Dic5C2311yescharacteristic
93C2×#681—22no—
94C2×#681—22no—
95C2×#681C2×C414yes—
96C22×#168C2312yes—
97C5×#103—24no—
98C5×#102—24no—
99#3967C2×C418yescomplemented
100#3964C2×C418yescomplemented
101#3963—22no—
102#3963—22no—
103#3963—21no—
104#3963—21no—
105#3963—21no—
106#3963—21no—
107#3947—24no—
108C8×Dic5—24no—
109C2×#417—51no—
110C2×#417D511yescharacteristic
111C2×C4×M16—51no—
112#2297—51no—
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—21no—
142C22×C4×C5—24no—
143C22×C4×C5—22no—
144C22×C4×C5—42no—
145C22×C4×C5—42no—
146C22×C4×C5—42no—
147C22×C4×C5—42no—
148C22×C4×C5—42no—
149C22×C4×C5C22×C412yes—
150C22×C4×C5C2×D411yescharacteristic
151C22×C4×C5C2×D412yes—
152C22×C4×C5#4112yes—
153C22×C4×C5#4112yes—
154C22×C4×C5#4112yes—
155C22×C4×C5#4112yes—
156C22×C4×C5C2411yescharacteristic
157C22×Dic5—24no—
158C22×Dic5—22no—
159C22×Dic5—22no—
160C22×Dic5—22no—
161C22×Dic5—22no—
162C22×Dic5—22no—
163C22×Dic5—21no—
164C22×Dic5—21no—
165C22×Dic5—41no—
166C22×Dic5—41no—
167C22×Dic5—42no—
168C22×Dic5—42no—
169C22×Dic5C2×D411yescharacteristic
170C22×Dic5C2×D411yescharacteristic
171C22×Dic5C2×D411yescharacteristic
172C22×Dic5C2×D411yescharacteristic
173C5×M16—28no—
174C2×C5×C8—22no—
175C2×C5×C8—22no—
176C2×C5×C8C2×D414yes—
177C5×C4⋊C4—22no—
178C5×C4⋊C4—22no—
179C5×C4⋊C4—42no—
180C22⋊Dic5—42no—
181C22⋊Dic5—42no—
182C22⋊Dic5—42no—
183C22⋊Dic5—42no—
184C22⋊Dic5—42no—
185C4⋊Dic5—24no—
186C4⋊Dic5—22no—
187C4⋊Dic5—21no—
188C4⋊Dic5—21no—
189C4⋊Dic5—21no—
190C4⋊Dic5—21no—
191C4⋊Dic5C22×C414yes—
192#683—44no—
193#683—44no—
194C4×Dic5—22no—
195C4×Dic5—22no—
196C4×Dic5—21no—
197C4×Dic5—21no—
198C4×Dic5—21no—
199C4×Dic5—21no—
200C4×Dic5—42no—
201#681—28no—
202C2×#168—22no—
203C2×#168—22no—
204C2×#168C22×C414yes—
205C22×#41C2×D511yescharacteristic
206C22×M16—51no—
207C22×M16C2×D511yescharacteristic
208C2×#117—51no—
209C22×C4⋊C4—51no—
210C22×C4⋊C4—101no—
211C22×C4⋊C4C2×D511yescharacteristic
212C22×C42—51no—
213#417—24no—
214#417—58no—
215C2×#103—52no—
216C2×#103C2×D512yes—
217C2×#102—52no—
218C2×#102C2×D512yes—
219C4×M16—104no—
220C2×#97—52no—
221C2×C4×C8—52no—
222#391—51no—
223#391—51no—
224#390—51no—
225#390—51no—
226#389—52no—
227#389—52no—
228#389—101no—
229#389—101no—
230#386—101no—
231#386—101no—
232C4×C22⋊C4—101no—
233C4×C22⋊C4—101no—
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—22no—
264C2×C4×C5—41no—
265C2×C4×C5—41no—
266C2×C4×C5—48no—
267C2×C4×C5—42no—
268C2×C4×C5—42no—
269C2×C4×C5—42no—
270C2×C4×C5—84no—
271C2×C4×C5#11714yes—
272C2×C4×C5C4×D412yes—
273C2×C4×C5C4×D414yes—
274C2×C4×C5#12112yes—
275C2×C4×C5#12314yes—
276C2×C4×C5#12412yes—
277C2×C4×C5#12614yes—
278C2×C4×C5C23×C411yescharacteristic
279C2×C4×C5C22×D411yescharacteristic
280C2×C4×C5C2×#4111yescharacteristic
281C2×C4×C5C2×#4111yescharacteristic
282C2×C4×C5C2×#4111yescharacteristic
283C2×C4×C5C2×#4111yescharacteristic
284C2×Dic5—21no—
285C2×Dic5—21no—
286C2×Dic5—21no—
287C2×Dic5—21no—
288C2×Dic5—24no—
289C2×Dic5—21no—
290C2×Dic5—21no—
291C2×Dic5—21no—
292C2×Dic5—21no—
293C2×Dic5—21no—
294C2×Dic5—21no—
295C2×Dic5—44no—
296C2×Dic5—44no—
297C2×Dic5—48no—
298C2×Dic5—42no—
299C2×Dic5—42no—
300C2×Dic5—41no—
301C2×Dic5—41no—
302C2×Dic5—41no—
303C2×Dic5—41no—
304C2×Dic5—41no—
305C2×Dic5—41no—
306C2×Dic5AGL(1,ℤ/8)12yes—
307C2×Dic5#13712yes—
308C5×C8—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—21no—
321C23×C4—51no—
322C23×C4—51no—
323C23×C4—101no—
324C23×C4C22×D511yescharacteristic
325C2×M16—22no—
326C2×M16—22no—
327C2×M16—54no—
328C2×M16—102no—
329C2×M16—102no—
330C2×M16C22⋊D514yes—
331C22×C8—52no—
332C22×C8C22×D512yes—
333#117—102no—
334#117—102no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—54no—
339C2×C4⋊C4—52no—
340C2×C4⋊C4—51no—
341C2×C4⋊C4—51no—
342C2×C4⋊C4—101no—
343C2×C4⋊C4—102no—
344C2×C4⋊C4—102no—
345C2×C4⋊C4—102no—
346C2×C4⋊C4—204no—
347C2×C4⋊C4C22×D512yes—
348C2×C22⋊C4—101no—
349C2×C22⋊C4—101no—
350C2×C22⋊C4—101no—
351C2×C22⋊C4—101no—
352C2×C22⋊C4—101no—
353C2×C42—51no—
354C2×C42—51no—
355C2×C42—51no—
356C2×C42—51no—
357C2×C42—102no—
358C2×C42—102no—
359C2×C42—101no—
360#103—24no—
361#103—58no—
362#102—24no—
363#102—58no—
364#97—104no—
365C4×C8—104no—
366#95—102no—
367#95—102no—
368#95—101no—
369#95—101no—
370#95—101no—
371#95—101no—
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—42no—
394C4×C5—82no—
395C4×C5#38814yes—
396C4×C5C2×#11711yescharacteristic
397C4×C5C2×C4×D411yescharacteristic
398C4×C5C2×#12311yescharacteristic
399C4×C5C2×#12611yescharacteristic
400Dic5—22no—
401Dic5—22no—
402Dic5—44no—
403Dic5—41no—
404Dic5—41no—
405Dic5—81no—
406Dic5—81no—
407C24—21no—
408C24D4×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—24no—
430C22×C4—22no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—51no—
437C22×C4—51no—
438C22×C4—51no—
439C22×C4—51no—
440C22×C4—102no—
441C22×C4—102no—
442C22×C4—102no—
443C22×C4—104no—
444C22×C4—102no—
445C22×C4—102no—
446C22×C4—101no—
447C22×C4—101no—
448C22×C4—201no—
449C22×C4—201no—
450C22×C4—202no—
451C22×C4—202no—
452C22×C4C2×C4×D512yes—
453C22×C4#70912yes—
454C22×C4D4×D511yescharacteristic
455C22×C4#71112yes—
456C22×C4#71112yes—
457C22×C4#71112yes—
458C22×C4C2×C22⋊D512yes—
459C22×C4C23×D511yescharacteristic
460M16—28no—
461M16—108no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—54no—
465C2×C8—102no—
466C2×C8—102no—
467C2×C8C2×C22⋊D514yes—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—42no—
471C4⋊C4—54no—
472C4⋊C4—102no—
473C4⋊C4—104no—
474C4⋊C4—101no—
475C4⋊C4—101no—
476C4⋊C4—101no—
477C4⋊C4—101no—
478C4⋊C4—204no—
479C4⋊C4—204no—
480C22⋊C4—202no—
481C22⋊C4—202no—
482C22⋊C4—202no—
483C22⋊C4—202no—
484C22⋊C4—202no—
485C42—101no—
486C42—101no—
487C42—101no—
488C42—101no—
489C42—102no—
490C42—102no—
491C42—202no—
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—
512C23#402712yes—
513C23#403112yes—
514C23#408412yes—
515C23C2×D4×D511yescharacteristic
516Q8—22no—
517Q8—22no—
518Q8—41no—
519Q8—41no—
520D4—22no—
521D4—22no—
522D4—48no—
523D4—41no—
524D4—41no—
525C2×C4—22no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—48no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—52no—
533C2×C4—52no—
534C2×C4—84no—
535C2×C4—101no—
536C2×C4—101no—
537C2×C4—101no—
538C2×C4—101no—
539C2×C4—104no—
540C2×C4—101no—
541C2×C4—101no—
542C2×C4—101no—
543C2×C4—101no—
544C2×C4—101no—
545C2×C4—101no—
546C2×C4—204no—
547C2×C4—204no—
548C2×C4—208no—
549C2×C4—202no—
550C2×C4—202no—
551C2×C4—201no—
552C2×C4—201no—
553C2×C4—201no—
554C2×C4—201no—
555C2×C4—201no—
556C2×C4—201no—
557C2×C4#402314yes—
558C2×C4#402514yes—
559C2×C4#402712yes—
560C2×C4#403112yes—
561C2×C4C4×C22⋊D514yes—
562C2×C4#408114yes—
563C2×C4#408412yes—
564C2×C4C22×C4×D511yescharacteristic
565C2×C4C2×#70911yescharacteristic
566C2×C4C2×#71111yescharacteristic
567C2×C4C2×#71111yescharacteristic
568C2×C4C2×#71111yescharacteristic
569C2×C4C22×C22⋊D511yescharacteristic
570C8—24no—
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#6523314yes—
587C22C2×#402711yescharacteristic
588C22C2×#403111yescharacteristic
589C22D5×AGL(1,ℤ/8)11yescharacteristic
590C22#6610711yescharacteristic
591C22D5×#13711yescharacteristic
592C22#6611111yescharacteristic
593C22C2×#408411yescharacteristic
594C4—42no—
595C4—82no—
596C4—102no—
597C4—102no—
598C4—201no—
599C4—201no—
600C4—204no—
601C4—401no—
602C4—401no—
603C4#6523314yes—
604C4C2×#402311yescharacteristic
605C4C2×#402511yescharacteristic
606C4C2×C4×C22⋊D511yescharacteristic
607C4C2×#408111yescharacteristic
608C2—24no—
609C2—82no—
610C2#1061059711yescharacteristic
611C2#1061059811yescharacteristic
612C2#1061060911yescharacteristic
613C2#1061061011yescharacteristic
614C2C2×#6523311yescharacteristic
615C2#1061568111yescharacteristic
616C2#1061571611yescharacteristic
subgroupG/H
centerC23C2×D4×D5
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC2×C5×#417C2
socleC23×C5C22×D4
prime radicalC22×C5×#41C22
cosocleC22×C4×C5C24
hypercenterC2×#417D5
hypocenterC5#21093
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#417

Complements

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

NH ≅ G/Ntype
4. C2×M16×Dic5C2semidirect
5. C2×#65262C2semidirect
6. #10602778C2semidirect
7. #10602351C2semidirect
8. #10601049C2semidirect
9. #10601047C2semidirect
10. #10600981C2semidirect
11. #10600979C2semidirect
23. #65521C4semidirect
99. #3967C2×C4semidirect
100. #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
C2×C5×#4172
#106158172
C2×#655212
C2×M16×Dic52
C2×#652622
#106027782
#106023512
#106010492
#106010472
#106009812
#106009792
#210935