Finite Group Databasebeta

← #12100889

Subgroups — #12100889

Order 1280.

1694 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#66159C211yesmaximal, characteristic
2C2×#65850C211yesmaximal, characteristic, complemented
3C5×#2896C211yesmaximal, characteristic
4#10615803C211yesmaximal, characteristic
5C22×#3956C211yesmaximal, characteristic, complemented
6#10602775C211yesmaximal, characteristic, complemented
7#10602367C211yesmaximal, characteristic, complemented
8#10601138C214yesmaximal, complemented
9#10601118C214yesmaximal, complemented
10C22×C5×#41C2211yescharacteristic
11C23×Dic10C2211yescharacteristic
12C22×C5×M16C2211yescharacteristic
13C2×C5×#117C2211yescharacteristic
14#66159—24no—
15#66159C418yescomplemented
16C2×#4090C2212yes—
17C2×#4082C2212yes—
18C2×#4081—22no—
19C2×#4072C2211yescharacteristic
20C2×#4072C2211yescharacteristic
21C2×#4070—22no—
22C22×#681C2211yescharacteristic
23#65850—22no—
24#65850—22no—
25#65850—22no—
26#65850C2214yes—
27C2×#4033—22no—
28C2×#4023—22no—
29C2×C4×Dic10—22no—
30C5×#419C2212yes—
31C5×#418C2212yes—
32C5×#417—22no—
33C5×#416—22no—
34#65522—22no—
35#65521—22no—
36#65514C2212yes—
37#65504C2212yes—
38C2×#3956—24no—
39C2×#3956C2214yes—
40#64775C2214yes—
41#64752C2214yes—
42#20954—51nomaximal, Sylow
43C2×C5×#41—22no—
44C2×C5×#41—21no—
45C2×C5×#41—21no—
46C2×C5×#41—21no—
47C2×C5×#41—21no—
48C2×C5×#41—22no—
49C2×C5×#41—22no—
50C2×C5×#41C2×C414yes—
51C22×C5×Q8C2311yescharacteristic
52C22×C5×D4—21no—
53C22×C5×D4C2311yescharacteristic
54C23×C4×C5—21no—
55C23×C4×C5C2311yescharacteristic
56C23×Dic5D412yes—
57C22×Dic10—21no—
58C22×Dic10—21no—
59C22×Dic10—21no—
60C22×Dic10—21no—
61C22×Dic10—22no—
62C22×Dic10—22no—
63C22×Dic10C2×C414yes—
64C22×Dic10D412yes—
65C22×Dic10D414yes—
66C22×Dic10C2312yes—
67C2×C5×M16—21no—
68C2×C5×M16—22no—
69C2×C5×M16—22no—
70C2×C5×M16C2312yes—
71C22×C5×C8—21no—
72C5×#117—21no—
73C5×#117—21no—
74C5×#117D412yes—
75C5×#117C2312yes—
76C2×C5×C4⋊C4D412yes—
77C2×C5×C22⋊C4—21no—
78C2×C42×C5—21no—
79C2×C22⋊Dic5—21no—
80C2×C22⋊Dic5—21no—
81C2×C22⋊Dic5—24no—
82#4090—24no—
83#4090C2×C418yescomplemented
84#4082—24no—
85#4082C2×C418yescomplemented
86#4081—216no—
87#4072—21no—
88#4072—21no—
89#4072—22no—
90#4072C2×C414yes—
91#4072D412yes—
92#4072C2312yes—
93C2×C4⋊Dic5—21no—
94C2×C4⋊Dic5D412yes—
95#4070—216no—
96C2×#683—24no—
97C2×#683—24no—
98C2×C4×Dic5—24no—
99C2×C4×Dic5—21no—
100C2×C4×Dic5C2312yes—
101C2×#681—21no—
102C2×#681—22no—
103C2×#681—22no—
104C2×#681C2312yes—
105C22×#168—21no—
106#4033—28no—
107#4033—28no—
108#4023—28no—
109#4023—28no—
110C4×Dic10—28no—
111C4×Dic10—28no—
112C5×#103—24no—
113C5×#102—24no—
114#3967—24no—
115#3964—24no—
116#3956—28no—
117C2×#536—51no—
118C2×#519—51no—
119#2896—51no—
120#2896D511yescharacteristic
121C22×#101—51no—
122#1888—54no—
123#1886—54no—
124#1733—51no—
125#1733—51no—
126C24×C5—21no—
127C24×C5C2×D411yescharacteristic
128C5×#41—24no—
129C5×#41—24no—
130C5×#41—24no—
131C5×#41—24no—
132C5×#41—42no—
133C5×#41—42no—
134C5×#41—42no—
135C5×#41—42no—
136C2×C5×Q8—21no—
137C2×C5×Q8—21no—
138C2×C5×Q8—21no—
139C2×C5×Q8—21no—
140C2×C5×Q8—21no—
141C2×C5×Q8C22×C412yes—
142C2×C5×D4—24no—
143C2×C5×D4—21no—
144C2×C5×D4—21no—
145C2×C5×D4—21no—
146C2×C5×D4—21no—
147C2×C5×D4—21no—
148C2×C5×D4—41no—
149C2×C5×D4—41no—
150C2×C5×D4—42no—
151C2×C5×D4C22×C412yes—
152C22×C4×C5—22no—
153C22×C4×C5—21no—
154C22×C4×C5—24no—
155C22×C4×C5—41no—
156C22×C4×C5—41no—
157C22×C4×C5—42no—
158C22×C4×C5C22×C412yes—
159C22×C4×C5C2×D411yescharacteristic
160C22×C4×C5C2×D412yes—
161C22×C4×C5C2×D411yescharacteristic
162C22×C4×C5C2×D411yescharacteristic
163C22×C4×C5C2×D411yescharacteristic
164C22×C4×C5C2×D411yescharacteristic
165C22×C4×C5#4112yes—
166C22×C4×C5#4112yes—
167C22×C4×C5C2411yescharacteristic
168C22×Dic5—22no—
169C22×Dic5—22no—
170C22×Dic5—24no—
171C22×Dic5—22no—
172C22×Dic5—22no—
173C22×Dic5C22⋊C418yes—
174C22×Dic5C2×D414yes—
175C2×Dic10—24no—
176C2×Dic10—24no—
177C2×Dic10—24no—
178C2×Dic10—24no—
179C2×Dic10—22no—
180C2×Dic10—22no—
181C2×Dic10—24no—
182C2×Dic10—24no—
183C2×Dic10—24no—
184C2×Dic10—24no—
185C2×Dic10—24no—
186C2×Dic10—22no—
187C2×Dic10—22no—
188C2×Dic10—22no—
189C2×Dic10—22no—
190C2×Dic10C22⋊C418yes—
191C2×Dic10C22×C414yes—
192C2×Dic10C2×D414yes—
193C5×M16—22no—
194C5×M16—22no—
195C5×M16—42no—
196C2×C5×C8—22no—
197C2×C5×C8—42no—
198C5×C4⋊C4—22no—
199C5×C4⋊C4C2×D414yes—
200C5×C22⋊C4—22no—
201C5×C22⋊C4—22no—
202C42×C5—22no—
203C42×C5—22no—
204C22⋊Dic5—24no—
205C22⋊Dic5—22no—
206C22⋊Dic5—22no—
207C22⋊Dic5—216no—
208C4⋊Dic5—24no—
209C4⋊Dic5—22no—
210C4⋊Dic5C2×D414yes—
211#683—216no—
212#683—216no—
213C4×Dic5—216no—
214C4×Dic5—22no—
215C4×Dic5—22no—
216C4×Dic5—22no—
217C4×Dic5C22×C414yes—
218#681—22no—
219#681—22no—
220#681—42no—
221C2×#168—22no—
222C2×#168—42no—
223C22×#41C2×D511yescharacteristic
224C23×Q8—51no—
225C22×M16—51no—
226C22×M16C2×D511yescharacteristic
227#536—58no—
228#536—104no—
229C2×C4⋊Q8—52no—
230C2×#124—52no—
231C2×#123—102no—
232C2×C22⋊Q8—102no—
233#519—54no—
234#519—102no—
235#519—102no—
236#519—102no—
237C2×C4×Q8—102no—
238C2×C4×Q8—102no—
239C2×#117—51no—
240C2×#117—51no—
241C2×#117—102no—
242C2×#117C2×D511yescharacteristic
243#419—52no—
244#419C2×D512yes—
245#418—52no—
246#418C2×D512yes—
247#417—22no—
248#417—102no—
249#416—22no—
250#416—102no—
251C2×#101—54no—
252C2×#101—104no—
253#339—54no—
254#339—54no—
255C23×C5—22no—
256C23×C5—42no—
257C23×C5—44no—
258C23×C5—41no—
259C23×C5—41no—
260C23×C5C2×C22⋊C412yes—
261C23×C5#12011yescharacteristic
262C23×C5#12011yescharacteristic
263C23×C5#12312yes—
264C23×C5C22×D411yescharacteristic
265C5×Q8—22no—
266C5×Q8—22no—
267C5×Q8—41no—
268C5×Q8—41no—
269C5×D4—22no—
270C5×D4—22no—
271C5×D4—44no—
272C5×D4—44no—
273C5×D4—41no—
274C5×D4—41no—
275C2×C4×C5—22no—
276C2×C4×C5—24no—
277C2×C4×C5—44no—
278C2×C4×C5—41no—
279C2×C4×C5—41no—
280C2×C4×C5—44no—
281C2×C4×C5—44no—
282C2×C4×C5C2×C22⋊C412yes—
283C2×C4×C5C4×D412yes—
284C2×C4×C5C4×D412yes—
285C2×C4×C5C4×D412yes—
286C2×C4×C5C4×D412yes—
287C2×C4×C5#12011yescharacteristic
288C2×C4×C5#12011yescharacteristic
289C2×C4×C5#12112yes—
290C2×C4×C5#12112yes—
291C2×C4×C5#12112yes—
292C2×C4×C5#12112yes—
293C2×C4×C5#12312yes—
294C2×C4×C5C23×C411yescharacteristic
295C2×C4×C5C22×D411yescharacteristic
296C2×C4×C5C22×D411yescharacteristic
297C2×C4×C5C22×D411yescharacteristic
298C2×C4×C5C2×#4111yescharacteristic
299C2×C4×C5C2×#4111yescharacteristic
300C2×Dic5—24no—
301C2×Dic5—24no—
302C2×Dic5—28no—
303C2×Dic5—28no—
304C2×Dic5—24no—
305C2×Dic5—24no—
306C2×Dic5—44no—
307C2×Dic5—44no—
308C2×Dic5C2×C22⋊C418yes—
309Dic10—24no—
310Dic10—24no—
311Dic10—24no—
312Dic10—24no—
313Dic10—28no—
314Dic10—28no—
315C5×C8—42no—
316#168—42no—
317C2×#41—22no—
318C2×#41—22no—
319C2×#41—21no—
320C2×#41—21no—
321C2×#41—22no—
322C2×#41—21no—
323C2×#41—21no—
324C2×#41C4×D514yes—
325C22×Q8—54no—
326C22×Q8—52no—
327C22×Q8—54no—
328C22×Q8—52no—
329C22×Q8—102no—
330C22×Q8—101no—
331C22×Q8—101no—
332C22×Q8—102no—
333C22×Q8—101no—
334C22×Q8—101no—
335C22×Q8C22×D511yescharacteristic
336C22×D4—21no—
337C22×D4C22×D511yescharacteristic
338C23×C4—21no—
339C23×C4—52no—
340C23×C4C22×D511yescharacteristic
341C2×M16—21no—
342C2×M16—22no—
343C2×M16—22no—
344C2×M16—52no—
345C2×M16—101no—
346C2×M16—102no—
347C2×M16—102no—
348C2×M16C22×D512yes—
349C22×C8—21no—
350C22×C8—101no—
351C4⋊Q8—58no—
352C4⋊Q8—104no—
353#124—58no—
354#124—104no—
355#123—1016no—
356C22⋊Q8—1016no—
357C4×Q8—108no—
358C4×Q8—108no—
359C4×Q8—108no—
360C4×Q8—108no—
361#117—21no—
362#117—21no—
363#117—54no—
364#117—52no—
365#117—52no—
366#117—108no—
367#117—108no—
368#117—101no—
369#117—102no—
370#117—101no—
371#117C22⋊D512yes—
372#117C22×D512yes—
373C2×C4⋊C4—52no—
374C2×C4⋊C4—104no—
375C2×C4⋊C4—104no—
376C2×C4⋊C4—101no—
377C2×C4⋊C4C22⋊D512yes—
378C2×C22⋊C4—21no—
379C2×C22⋊C4—104no—
380C2×C22⋊C4—101no—
381C2×C22⋊C4—101no—
382C2×C42—21no—
383C2×C42—52no—
384C2×C42—104no—
385C2×C42—101no—
386#103—24no—
387#103—104no—
388#102—24no—
389#102—104no—
390#101—108no—
391C22×C5—22no—
392C22×C5—22no—
393C22×C5—22no—
394C22×C5—42no—
395C22×C5—42no—
396C22×C5—44no—
397C22×C5—42no—
398C22×C5—82no—
399C22×C5—82no—
400C22×C5#38612yes—
401C22×C5#38612yes—
402C22×C5C22×C22⋊C411yescharacteristic
403C22×C5C2×#12011yescharacteristic
404C22×C5C2×#12311yescharacteristic
405C4×C5—44no—
406C4×C5—42no—
407C4×C5#38612yes—
408C4×C5#38612yes—
409C4×C5C2×C4×D411yescharacteristic
410C4×C5C2×C4×D411yescharacteristic
411C4×C5C2×#12111yescharacteristic
412C4×C5C2×#12111yescharacteristic
413Dic5—28no—
414Dic5—44no—
415Dic5—44no—
416C24—21no—
417C24D4×D511yescharacteristic
418#41—24no—
419#41—24no—
420#41—24no—
421#41—24no—
422#41—42no—
423#41—42no—
424#41—42no—
425#41—42no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8—21no—
430C2×Q8—21no—
431C2×Q8—54no—
432C2×Q8—54no—
433C2×Q8—58no—
434C2×Q8—102no—
435C2×Q8—102no—
436C2×Q8—102no—
437C2×Q8—102no—
438C2×Q8—104no—
439C2×Q8—104no—
440C2×Q8—104no—
441C2×Q8—104no—
442C2×Q8—104no—
443C2×Q8—104no—
444C2×Q8—104no—
445C2×Q8—102no—
446C2×Q8—102no—
447C2×Q8—104no—
448C2×Q8—104no—
449C2×Q8C2×C4×D512yes—
450C2×D4—24no—
451C2×D4—21no—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—42no—
457C2×D4—41no—
458C2×D4—41no—
459C2×D4C2×C4×D512yes—
460C22×C4—21no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—42no—
464C22×C4—41no—
465C22×C4—41no—
466C22×C4—54no—
467C22×C4—58no—
468C22×C4—104no—
469C22×C4—102no—
470C22×C4—102no—
471C22×C4—102no—
472C22×C4—102no—
473C22×C4C2×C4×D512yes—
474C22×C4#70912yes—
475C22×C4D4×D512yes—
476C22×C4D4×D511yescharacteristic
477C22×C4D4×D511yescharacteristic
478C22×C4D4×D511yescharacteristic
479C22×C4#71112yes—
480C22×C4C2×C22⋊D511yescharacteristic
481C22×C4C2×C22⋊D511yescharacteristic
482C22×C4C23×D511yescharacteristic
483M16—22no—
484M16—22no—
485M16—42no—
486M16—102no—
487M16—102no—
488M16—202no—
489C2×C8—22no—
490C2×C8—42no—
491C2×C8—102no—
492C2×C8—202no—
493C4⋊C4—22no—
494C4⋊C4—54no—
495C4⋊C4—1016no—
496C4⋊C4—1016no—
497C4⋊C4—104no—
498C4⋊C4—102no—
499C4⋊C4C2×C22⋊D514yes—
500C22⋊C4—22no—
501C22⋊C4—22no—
502C22⋊C4—1016no—
503C22⋊C4—102no—
504C22⋊C4—104no—
505C22⋊C4—102no—
506C42—22no—
507C42—22no—
508C42—54no—
509C42—1016no—
510C42—102no—
511C42—102no—
512C42—102no—
513C2×C5—21no—
514C2×C5—21no—
515C2×C5—22no—
516C2×C5—22no—
517C2×C5—82no—
518C2×C5C2×#38611yescharacteristic
519C2×C5#302012yes—
520C23—22no—
521C23—42no—
522C23—44no—
523C23—41no—
524C23—41no—
525C23D5×C22⋊C412yes—
526C23#402912yes—
527C23#408311yescharacteristic
528C23#408311yescharacteristic
529C23C2×D4×D511yescharacteristic
530Q8—22no—
531Q8—22no—
532Q8—41no—
533Q8—41no—
534Q8—104no—
535Q8—104no—
536Q8—104no—
537Q8—104no—
538Q8—108no—
539Q8—108no—
540D4—22no—
541D4—22no—
542D4—44no—
543D4—44no—
544D4—41no—
545D4—41no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—44no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—58no—
554C2×C4—104no—
555C2×C4—104no—
556C2×C4—108no—
557C2×C4—108no—
558C2×C4—104no—
559C2×C4—104no—
560C2×C4—204no—
561C2×C4—204no—
562C2×C4D5×C22⋊C412yes—
563C2×C4#402712yes—
564C2×C4#402712yes—
565C2×C4#402912yes—
566C2×C4#403012yes—
567C2×C4#403012yes—
568C2×C4C4×C22⋊D512yes—
569C2×C4C4×C22⋊D512yes—
570C2×C4#408311yescharacteristic
571C2×C4#408311yescharacteristic
572C2×C4#408412yes—
573C2×C4#408412yes—
574C2×C4C22×C4×D511yescharacteristic
575C2×C4C2×#70911yescharacteristic
576C2×C4C2×D4×D511yescharacteristic
577C2×C4C2×D4×D511yescharacteristic
578C2×C4C2×#71111yescharacteristic
579C2×C4C22×C22⋊D511yescharacteristic
580C8—42no—
581C8—202no—
582C5#2095411yescharacteristic, complemented, Sylow
583C22—22no—
584C22—22no—
585C22—22no—
586C22—42no—
587C22—42no—
588C22—44no—
589C22—42no—
590C22—82no—
591C22—82no—
592C22#6524512yes—
593C22#6524512yes—
594C22C2×D5×C22⋊C411yescharacteristic
595C22C2×#402911yescharacteristic
596C22C2×#408311yescharacteristic
597C4—42no—
598C4—44no—
599C4—108no—
600C4—204no—
601C4—204no—
602C4#6524512yes—
603C4#6524512yes—
604C4C2×#402711yescharacteristic
605C4C2×#403011yescharacteristic
606C4C2×C4×C22⋊D511yescharacteristic
607C4C2×#408411yescharacteristic
608C2—21no—
609C2—21no—
610C2—22no—
611C2—22no—
612C2—82no—
613C2C2×#6524511yescharacteristic
614C2#1061573712yes—
subgroupG/H
centerC22C2×#4083
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC5×#2896C2
socleC22×C5C2×#120
prime radicalC22×C5×#41C22
cosocleC22×C4×C5C24
hypercenter#2896D5
hypocenterC5#20954
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2896

Complements

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

NH ≅ G/Ntype
2. C2×#65850C2semidirect
5. C22×#3956C2semidirect
6. #10602775C2semidirect
7. #10602367C2semidirect
8. #10601138C2semidirect
9. #10601118C2semidirect
15. #66159C4semidirect
83. #4090C2×C4semidirect
85. #4082C2×C4semidirect
582. C5#20954semidirect

Maximal subgroups

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

subgroupindex
C2×#661592
C2×#658502
C5×#28962
#106158032
C22×#39562
#106027752
#106023672
#106011382
#106011182
#209545