Finite Group Databasebeta

← #12110464

Subgroups — #12110464

Order 1280.

1224 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×C5×C22⋊Q8C211yesmaximal, characteristic, complemented
2C2×#65335C211yesmaximal, characteristic
3C2×#65228C211yesmaximal, characteristic
4#10602485C211yesmaximal, characteristic, complemented
5#10602426C212yesmaximal, complemented
6C4×C22⋊#168C211yesmaximal, characteristic, complemented
7#10600469C212yesmaximal, complemented
8#10600408C212yesmaximal, complemented
9#10600314C211yesmaximal, characteristic
10#10600298C211yesmaximal, characteristic, complemented
11#10600256C212yesmaximal, complemented
12C23×D20C2211yescharacteristic
13C2×C5×C22⋊Q8C2211yescharacteristic
14C2×C4×C5×Q8C2211yescharacteristic, complemented
15C22×C42×C5C2211yescharacteristic
16C22×#688C2212yes—
17C22×#685—21no—
18C22×#685—21no—
19C22×C4⋊Dic5C2211yescharacteristic
20C5×#391C2211yescharacteristic, complemented
21C5×#389C2211yescharacteristic, complemented
22C5×#387—21no—
23C5×#385—21no—
24C5×#380C2211yescharacteristic
25C4×C5×C4⋊C4—21no—
26C4×C5×C4⋊C4C2211yescharacteristic, complemented
27C4×C5×C22⋊C4—21no—
28C2×#3967C2212yescomplemented
29C2×C22⋊#168C2211yescharacteristic
30C2×C22⋊#168C2211yescharacteristic
31#65335C418yescomplemented
32C2×#3941C2212yes—
33C2×#3941C2212yes—
34C2×#3941C2212yes—
35C2×#3940C2212yescomplemented
36#65228—21no—
37#65228—21no—
38#65228C2212yes—
39#65228C2212yes—
40C2×C4×#168C2212yescomplemented
41#64749C2212yescomplemented
42#64749C2212yescomplemented
43#64744C2212yescomplemented
44#22043—51nomaximal, Sylow
45C24×D5—21no—
46C22×C5×Q8C2311yescharacteristic
47C23×C4×C5D411yescharacteristic
48C23×C4×C5D411yescharacteristic
49C23×C4×C5C2311yescharacteristic
50C23×Dic5—21no—
51C22×D20—21no—
52C22×D20—22no—
53C22×D20—42no—
54C22×D20—42no—
55C22×D20C2×C414yes—
56C22×D20C2312yes—
57C5×C22⋊Q8—22no—
58C5×C22⋊Q8C2×C414yes—
59C4×C5×Q8—42no—
60C2×C5×C4⋊C4—21no—
61C2×C5×C4⋊C4—21no—
62C2×C5×C4⋊C4—21no—
63C2×C5×C4⋊C4C2311yescharacteristic
64C2×C5×C4⋊C4C2311yescharacteristic
65C2×C5×C4⋊C4C2311yescharacteristic
66C2×C5×C22⋊C4—21no—
67C2×C5×C22⋊C4—21no—
68C2×C42×C5—21no—
69C2×C42×C5—21no—
70C2×C42×C5—42no—
71C2×C42×C5D411yescharacteristic
72C2×C42×C5D411yescharacteristic
73C2×C42×C5C2311yescharacteristic
74C2×C42×C5C2311yescharacteristic
75C2×#688—42no—
76C2×#688—42no—
77C2×#688C2×C418yescomplemented
78C2×#685—22no—
79C2×#685—22no—
80C2×#685—22no—
81C2×#685—22no—
82C2×#685—44no—
83C2×#685—44no—
84C2×C4⋊Dic5—21no—
85C2×C4⋊Dic5—42no—
86C2×C4⋊Dic5C2312yes—
87C22×#168C2312yes—
88C22×#168C2312yes—
89C5×#95—21no—
90C5×#95—21no—
91C5×#95—21no—
92C5×#95—21no—
93C5×#95—21no—
94#3967—42no—
95C22⋊#168—22no—
96C22⋊#168C2×C414yes—
97#3941—42no—
98#3941—42no—
99#3941C2×C418yescomplemented
100#3940—42no—
101C4×#168—42no—
102C2×#450—51no—
103C4×C22⋊Q8D511yescharacteristic, complemented
104C2×#390—51no—
105#1968—52no—
106#1934—52no—
107#1926—52no—
108#1880—52no—
109#1872—51no—
110#1853—51no—
111#1786—51no—
112C4×C22⋊C8—51no—
113C24×C5C2×D411yescharacteristic
114C23×D5—24no—
115C23×D5—22no—
116C23×D5—42no—
117C23×D5—42no—
118C23×D5—42no—
119C23×D5—42no—
120C23×D5—42no—
121C23×D5—42no—
122C2×C5×Q8—21no—
123C2×C5×Q8—22no—
124C2×C5×Q8—22no—
125C2×C5×Q8C22×C412yes—
126C22×C4×C5—22no—
127C22×C4×C5—22no—
128C22×C4×C5—22no—
129C22×C4×C5—21no—
130C22×C4×C5—21no—
131C22×C4×C5—22no—
132C22×C4×C5—21no—
133C22×C4×C5—21no—
134C22×C4×C5—41no—
135C22×C4×C5—41no—
136C22×C4×C5—41no—
137C22×C4×C5SD1614yescomplemented
138C22×C4×C5C22×C412yes—
139C22×C4×C5C2×D411yescharacteristic
140C22×C4×C5C2×D411yescharacteristic
141C22×C4×C5C2×D411yescharacteristic
142C22×C4×C5C2×D411yescharacteristic
143C22×C4×C5C2×D411yescharacteristic
144C22×C4×C5#4112yes—
145C22×C4×C5C2411yescharacteristic
146C22×Dic5—22no—
147C22×Dic5—42no—
148C22×Dic5—42no—
149C22×Dic5—42no—
150C2×D20—22no—
151C2×D20—22no—
152C2×D20—22no—
153C2×D20—22no—
154C2×D20—24no—
155C2×D20—22no—
156C2×D20—48no—
157C2×D20—42no—
158C2×D20—42no—
159C2×D20—48no—
160C2×D20C22×C414yes—
161C5×C4⋊C4—24no—
162C5×C4⋊C4—21no—
163C5×C4⋊C4—22no—
164C5×C4⋊C4—22no—
165C5×C4⋊C4—42no—
166C5×C4⋊C4—42no—
167C5×C4⋊C4C22×C412yes—
168C5×C22⋊C4—24no—
169C5×C22⋊C4—42no—
170C42×C5—22no—
171C42×C5—22no—
172C42×C5—41no—
173C42×C5—41no—
174C42×C5—42no—
175C42×C5—42no—
176#688—48no—
177#685—44no—
178#685—44no—
179#685—44no—
180#685—44no—
181C4⋊Dic5—22no—
182C4⋊Dic5—22no—
183C4⋊Dic5—42no—
184C2×#168—22no—
185C2×#168—22no—
186C2×#168—22no—
187C2×#168—22no—
188C2×#168—22no—
189C2×#168C22×C414yes—
190C23×D4—51no—
191C22×SD16—52no—
192C2×C22⋊Q8C2×D511yescharacteristic
193C2×C4×Q8C2×D511yescharacteristic, complemented
194C22×C4⋊C4—51no—
195C22×C22⋊C4—101no—
196C22×C22⋊C4—101no—
197C22×C42C2×D511yescharacteristic
198#450—58no—
199C2×#106—52no—
200C2×#103—52no—
201C2×#102—52no—
202C2×#102—52no—
203C2×#102—52no—
204C2×C22⋊C8—51no—
205C2×C22⋊C8—51no—
206C2×C4×C8—52no—
207#391C2×D511yescharacteristic, complemented
208#390—52no—
209#390—52no—
210#390—101no—
211#390—101no—
212#389C2×D511yescharacteristic, complemented
213#387—21no—
214#385—21no—
215#380C2×D511yescharacteristic
216C4×C4⋊C4—21no—
217C4×C4⋊C4C2×D511yescharacteristic, complemented
218C4×C22⋊C4—21no—
219#340—52no—
220#340—52no—
221#336—52no—
222C23×C5—21no—
223C23×C5—21no—
224C23×C5—21no—
225C23×C5—21no—
226C23×C5C4×D412yes—
227C23×C5C2×SD1612yes—
228C23×C5#13512yes—
229C23×C5C22×D411yescharacteristic
230C22×D5—24no—
231C22×D5—48no—
232C22×D5—48no—
233C22×D5—42no—
234C22×D5—42no—
235C22×D5—42no—
236C22×D5—42no—
237C22×D5—42no—
238C22×D5—42no—
239C22×D5—48no—
240C22×D5—84no—
241C22×D5—84no—
242C22×D5—84no—
243C22×D5—84no—
244C5×Q8—24no—
245C5×Q8—42no—
246C2×C4×C5—21no—
247C2×C4×C5—21no—
248C2×C4×C5—21no—
249C2×C4×C5—21no—
250C2×C4×C5—21no—
251C2×C4×C5—21no—
252C2×C4×C5—21no—
253C2×C4×C5—21no—
254C2×C4×C5—24no—
255C2×C4×C5—24no—
256C2×C4×C5—22no—
257C2×C4×C5—22no—
258C2×C4×C5—21no—
259C2×C4×C5—21no—
260C2×C4×C5—21no—
261C2×C4×C5—44no—
262C2×C4×C5—44no—
263C2×C4×C5—41no—
264C2×C4×C5—41no—
265C2×C4×C5—41no—
266C2×C4×C5—41no—
267C2×C4×C5—41no—
268C2×C4×C5—41no—
269C2×C4×C5—41no—
270C2×C4×C5—41no—
271C2×C4×C5—41no—
272C2×C4×C5—41no—
273C2×C4×C5C4×D412yes—
274C2×C4×C5C2×SD1612yes—
275C2×C4×C5C2×SD1612yes—
276C2×C4×C5AGL(1,ℤ/8)12yes—
277C2×C4×C5C23×C411yescharacteristic
278C2×C4×C5C2×#4111yescharacteristic
279C2×C4×C5H(5,𝔽2)12yes—
280C2×C4×C5H(5,𝔽2)12yes—
281C2×Dic5—42no—
282C2×Dic5—42no—
283C2×Dic5—42no—
284C2×Dic5—84no—
285D20—24no—
286D20—24no—
287D20—48no—
288D20—42no—
289D20—42no—
290#168—24no—
291#168—42no—
292C25—101no—
293C22×Q8C22×D511yescharacteristic
294C22×D4—54no—
295C22×D4—52no—
296C22×D4—102no—
297C22×D4—101no—
298C22×D4—202no—
299C22×D4—202no—
300C23×C4—101no—
301C23×C4C22⋊D511yescharacteristic
302C23×C4C22⋊D511yescharacteristic
303C23×C4C22×D511yescharacteristic
304C2×SD16—58no—
305C2×SD16—202no—
306C2×SD16—202no—
307C22×C8—52no—
308C22×C8—52no—
309C22⋊Q8—22no—
310C22⋊Q8C4×D514yes—
311C4×Q8—42no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—52no—
316C2×C4⋊C4—101no—
317C2×C4⋊C4—202no—
318C2×C4⋊C4C22×D511yescharacteristic
319C2×C4⋊C4C22×D511yescharacteristic
320C2×C4⋊C4C22×D511yescharacteristic
321C2×C22⋊C4—21no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—102no—
324C2×C22⋊C4—102no—
325C2×C22⋊C4—102no—
326C2×C22⋊C4—102no—
327C2×C22⋊C4—204no—
328C2×C22⋊C4—204no—
329C2×C42—21no—
330C2×C42—21no—
331C2×C42—42no—
332C2×C42C22⋊D511yescharacteristic
333C2×C42C22⋊D511yescharacteristic
334C2×C42C22×D511yescharacteristic
335C2×C42C22×D511yescharacteristic
336#106—202no—
337#103—202no—
338#102—58no—
339#102—202no—
340#102—202no—
341C22⋊C8—54no—
342C22⋊C8—102no—
343C4×C8—202no—
344#95—21no—
345#95—21no—
346#95—21no—
347#95—21no—
348#95—21no—
349C22×C5—22no—
350C22×C5—22no—
351C22×C5—22no—
352C22×C5—22no—
353C22×C5—22no—
354C22×C5—22no—
355C22×C5C4×SD1614yes—
356C22×C5C2×C4×D411yescharacteristic
357C22×C5#53411yescharacteristic
358C22×C5#53511yescharacteristic
359C22×C5C22×SD1611yescharacteristic
360C22×C5C2×#13511yescharacteristic
361C22×C5C2×AGL(1,ℤ/8)11yescharacteristic
362C22×C5#57511yescharacteristic
363C2×D5—44no—
364C2×D5—44no—
365C2×D5—44no—
366C2×D5—816no—
367C2×D5—82no—
368C2×D5—82no—
369C2×D5—82no—
370C2×D5—82no—
371C4×C5—22no—
372C4×C5—22no—
373C4×C5—22no—
374C4×C5—22no—
375C4×C5—42no—
376C4×C5—41no—
377C4×C5—42no—
378C4×C5—81no—
379C4×C5—81no—
380C4×C5#51812yes—
381C4×C5#53812yes—
382Dic5—82no—
383C24—102no—
384C24—104no—
385C24—202no—
386C24—202no—
387C24—202no—
388C24—202no—
389C24—202no—
390C24—202no—
391C24C2×C22⋊D511yescharacteristic
392C2×Q8—21no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8C2×C4×D512yes—
396C2×D4—54no—
397C2×D4—102no—
398C2×D4—102no—
399C2×D4—102no—
400C2×D4—102no—
401C2×D4—102no—
402C2×D4—104no—
403C2×D4—208no—
404C2×D4—202no—
405C2×D4—202no—
406C2×D4—208no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—102no—
419C22×C4—202no—
420C22×C4—202no—
421C22×C4—202no—
422C22×C4#68814yescomplemented
423C22×C4C2×C4×D512yes—
424C22×C4#70912yes—
425C22×C4C2×C22⋊D511yescharacteristic
426C22×C4C2×C22⋊D511yescharacteristic
427C22×C4C2×C22⋊D511yescharacteristic
428C22×C4C2×C22⋊D511yescharacteristic
429C22×C4C2×C22⋊D511yescharacteristic
430C22×C4C23×D511yescharacteristic
431SD16—208no—
432C2×C8—54no—
433C2×C8—102no—
434C2×C8—102no—
435C2×C8—102no—
436C2×C8—102no—
437C2×C8—102no—
438C4⋊C4—21no—
439C4⋊C4—24no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—102no—
445C4⋊C4—102no—
446C4⋊C4—202no—
447C4⋊C4C2×C4×D512yes—
448C22⋊C4—24no—
449C22⋊C4—42no—
450C22⋊C4—204no—
451C22⋊C4—204no—
452C22⋊C4—204no—
453C22⋊C4—204no—
454C42—22no—
455C42—22no—
456C42—41no—
457C42—41no—
458C42—42no—
459C42—42no—
460C2×C5—24no—
461C2×C5#232111yescharacteristic
462C2×C5C2×C4×SD1611yescharacteristic
463C2×C5C4×AGL(1,ℤ/8)11yescharacteristic
464C2×C5#318711yescharacteristic
465C2×C5#318811yescharacteristic
466C2×C5#320811yescharacteristic
467C2×C5#321111yescharacteristic
468D5—84no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—104no—
474C23—208no—
475C23—208no—
476C23—208no—
477C23—202no—
478C23—202no—
479C23—202no—
480C23—202no—
481C23—202no—
482C23—202no—
483C23—404no—
484C23—404no—
485C23—404no—
486C23—404no—
487C23C4×C22⋊D512yes—
488C23C2×#68812yes—
489C23#409612yes—
490C23C22×C22⋊D511yescharacteristic
491Q8—24no—
492Q8—42no—
493D4—104no—
494D4—104no—
495D4—208no—
496D4—202no—
497D4—202no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—202no—
526C2×C4—202no—
527C2×C4—202no—
528C2×C4—404no—
529C2×C4C4×C22⋊D512yes—
530C2×C4C2×#68812yes—
531C2×C4C2×#68812yes—
532C2×C4#409512yes—
533C2×C4C22×C4×D511yescharacteristic
534C2×C4C2×#70911yescharacteristic
535C2×C4#414412yes—
536C2×C4#414912yes—
537C8—104no—
538C8—202no—
539C5#2204311yescharacteristic, complemented, Sylow
540C22—22no—
541C22—22no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—204no—
547C22—204no—
548C22—204no—
549C22—4016no—
550C22—402no—
551C22—402no—
552C22—402no—
553C22—402no—
554C22C4×#68814yes—
555C22C2×C4×C22⋊D511yescharacteristic
556C22#6613811yescharacteristic
557C22C22×#68811yescharacteristic
558C22C2×#409511yescharacteristic
559C22C2×#409611yescharacteristic
560C22#6615611yescharacteristic
561C22#6616411yescharacteristic
562C4—22no—
563C4—22no—
564C4—22no—
565C4—22no—
566C4—42no—
567C4—41no—
568C4—42no—
569C4—81no—
570C4—81no—
571C4—402no—
572C4#6587912yes—
573C4#6589412yes—
574C2—24no—
575C2—404no—
576C2#1061412611yescharacteristic
577C2C2×C4×#68811yescharacteristic
578C2C4×#409511yescharacteristic
579C2#1061446711yescharacteristic
580C2#1061466011yescharacteristic
581C2#1061487511yescharacteristic
582C2#1061508611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D5
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC4×C5×C22⋊Q8C2
socleC23×C5C22×D4
prime radicalC23×D20C22
cosocleC22×C4×C5C24
hypercenterC4×C22⋊Q8D5
hypocenterC5#22043
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C4×C5×C22⋊Q8C2semidirect
4. #10602485C2semidirect
5. #10602426C2semidirect
6. C4×C22⋊#168C2semidirect
7. #10600469C2semidirect
8. #10600408C2semidirect
10. #10600298C2semidirect
11. #10600256C2semidirect
14. C2×C4×C5×Q8C22semidirect
20. C5×#391C22semidirect
21. C5×#389C22semidirect
26. C4×C5×C4⋊C4C22semidirect
28. C2×#3967C22semidirect
31. #65335C4semidirect
35. C2×#3940C22semidirect
40. C2×C4×#168C22semidirect
41. #64749C22semidirect
42. #64749C22semidirect
43. #64744C22semidirect
77. C2×#688C2×C4semidirect
99. #3941C2×C4semidirect
103. C4×C22⋊Q8D5semidirect
137. C22×C4×C5SD16semidirect
193. C2×C4×Q8C2×D5semidirect
207. #391C2×D5semidirect
212. #389C2×D5semidirect
217. C4×C4⋊C4C2×D5semidirect
422. C22×C4#688semidirect
539. C5#22043semidirect

Maximal subgroups

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

subgroupindex
C4×C5×C22⋊Q82
C2×#653352
C2×#652282
#106024852
#106024262
C4×C22⋊#1682
#106004692
#106004082
#106003142
#106002982
#106002562
#220435