Finite Group Databasebeta

← #12098898

Subgroups — #12098898

Order 1280.

1438 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#4053C211yesmaximal, characteristic
2C2×D5×#107C211yesmaximal, characteristic, complemented
3C2×#65149C211yesmaximal, characteristic, complemented
4#10605734C212yesmaximal
5C5×#1809C211yesmaximal, characteristic, complemented
6#10602781C211yesmaximal, characteristic, complemented
7#10601424C211yesmaximal, characteristic, complemented
8#10601371C211yesmaximal, characteristic, complemented
9#10600305C212yesmaximal, complemented
10#10599662C212yesmaximal, complemented
11#10599660C212yesmaximal, complemented
12C22×#709C2211yescharacteristic
13C22×C5×M16C2211yescharacteristic
14C23×#168C2211yescharacteristic
15C2×#4053—24no—
16C2×#4053—24no—
17C2×#4053C418yescomplemented
18C22×#676C2211yescharacteristic
19C22×#676C2211yescharacteristic
20C22×C8×D5C2211yescharacteristic
21C22×C8×D5C2211yescharacteristic
22C2×#4038C2212yes—
23C2×D5×C4⋊C4C2212yes—
24C2×C5×#107C2211yescharacteristic
25C2×C5×#106C2211yescharacteristic
26C5×#380C2212yes—
27#65501C2212yes—
28C2×#3950C2211yescharacteristic
29C2×#3949C2211yescharacteristic
30#65276C2212yes—
31#65276C2212yes—
32C2×#3939C2212yes—
33C2×#3939C2212yes—
34#65246—22no—
35#65245—22no—
36D5×#107—24no—
37#65149—24no—
38#64954—22no—
39#64935—22no—
40C5×#340C2212yes—
41#64771C2212yes—
42#64749C2212yes—
43#64749C2212yes—
44#20663—51nomaximal, Sylow
45C23×C4×C5C2311yescharacteristic
46C22×C22⋊D5D411yescharacteristic
47C22×C22⋊D5D411yescharacteristic
48C2×#709—22no—
49C2×#709—22no—
50C2×#709—22no—
51C2×#709—22no—
52C2×#709C2×C414yes—
53C2×#709Q814yes—
54C22×D20D411yescharacteristic
55C22×C4×D5—22no—
56C22×C4×D5C2311yescharacteristic
57C22×C4×D5C2311yescharacteristic
58C22×Dic10D411yescharacteristic
59C2×C5×M16—22no—
60C2×C5×M16—22no—
61C2×C5×M16C2×C414yescomplemented
62C22×C5×C8C2311yescharacteristic
63C22×C5×C8C2311yescharacteristic
64C2×C5×C4⋊C4C2312yes—
65C2×C5×C4⋊C4C2312yes—
66C2×C5×C22⋊C4—22no—
67C2×C22⋊Dic5—22no—
68C2×#685—22no—
69C2×#685—22no—
70C2×#685—22no—
71C2×C4⋊Dic5C2312yes—
72C2×C4⋊Dic5C2312yes—
73C2×#683—22no—
74C2×C4×Dic5—22no—
75C22×#168—22no—
76C22×#168—22no—
77C22×#168C2×C414yescomplemented
78C22×#168C2311yescharacteristic
79C22×#168C2311yescharacteristic
80#4053—232no—
81C2×#676—22no—
82C2×#676—22no—
83C2×#676—22no—
84C2×#676—22no—
85C2×#676C2×C414yes—
86C2×#676C2×C414yescomplemented
87C2×C8×D5—22no—
88C2×C8×D5—22no—
89C2×C8×D5C2×C414yes—
90C2×C8×D5C2×C414yescomplemented
91C2×C8×D5D414yes—
92C2×C8×D5D414yes—
93#4038—24no—
94#4038—24no—
95D5×C4⋊C4—24no—
96D5×C4⋊C4—24no—
97C5×#107—22no—
98C5×#106—22no—
99C5×#95—22no—
100#3963—22no—
101#3963—22no—
102#3963—22no—
103#3950—22no—
104#3949—22no—
105#3939—24no—
106#3939—24no—
107C22×#131—51no—
108C2×#427—51no—
109C22×#107—51no—
110#2311—52no—
111#1809—51no—
112#1809—51no—
113#1809—51no—
114#1809D511yescharacteristic, complemented
115#1808—52no—
116#1808—52no—
117#1736—52no—
118C24×C5C2×D411yescharacteristic
119C23×D5C2×D411yescharacteristic
120C23×D5C2×D411yescharacteristic
121C22×C4×C5—21no—
122C22×C4×C5—21no—
123C22×C4×C5—21no—
124C22×C4×C5—21no—
125C22×C4×C5—22no—
126C22×C4×C5—22no—
127C22×C4×C5C22×C412yes—
128C22×C4×C5C2×D411yescharacteristic
129C22×C4×C5C2×Q812yes—
130C22×C4×C5C2411yescharacteristic
131C2×C22⋊D5—22no—
132C2×C22⋊D5—22no—
133C2×C22⋊D5—22no—
134C2×C22⋊D5—22no—
135C2×C22⋊D5C4⋊C414yes—
136C2×C22⋊D5C4⋊C414yes—
137C22×Dic5—22no—
138C22×Dic5—22no—
139C22×Dic5C2×D411yescharacteristic
140C22×Dic5C2×D411yescharacteristic
141#709—28no—
142#709—28no—
143#709—28no—
144#709—28no—
145C2×D20—22no—
146C2×D20—22no—
147C2×D20C4⋊C414yes—
148C2×C4×D5—21no—
149C2×C4×D5—21no—
150C2×C4×D5—21no—
151C2×C4×D5—21no—
152C2×C4×D5—44no—
153C2×C4×D5—82no—
154C2×C4×D5—82no—
155C2×C4×D5C22×C412yes—
156C2×C4×D5C22×C412yes—
157C2×C4×D5C2×D412yes—
158C2×C4×D5C2×D412yes—
159C2×C4×D5C2×Q812yes—
160C2×C4×D5C2×Q812yes—
161C2×C4×D5#4112yes—
162C2×C4×D5#4112yes—
163C2×Dic10—22no—
164C2×Dic10—22no—
165C2×Dic10C4⋊C414yes—
166C5×M16—28no—
167C2×C5×C8—21no—
168C2×C5×C8—21no—
169C2×C5×C8C22×C412yes—
170C2×C5×C8C22×C412yes—
171C2×C5×C8C2×D412yes—
172C2×C5×C8C2×D412yes—
173C5×C4⋊C4—22no—
174C5×C4⋊C4—22no—
175C5×C4⋊C4—22no—
176C5×C4⋊C4—22no—
177C5×C22⋊C4—82no—
178C22⋊Dic5—82no—
179#685—44no—
180#685—82no—
181#685—82no—
182C4⋊Dic5—22no—
183C4⋊Dic5—22no—
184C4⋊Dic5—22no—
185C4⋊Dic5—22no—
186#683—44no—
187C4×Dic5—44no—
188C2×#168—21no—
189C2×#168—21no—
190C2×#168—28no—
191C2×#168C22×C412yes—
192C2×#168C22×C412yes—
193C2×#168C2×D412yes—
194C2×#168C2×D412yes—
195#676—28no—
196#676—28no—
197C8×D5—28no—
198C8×D5C22⋊C4116yescomplemented
199C22×#41—51no—
200C2×#131—58no—
201C2×#131—104no—
202C2×#131—104no—
203C22×M16—51no—
204C22×M16—51no—
205C22×M16C2×D511yescharacteristic
206C23×C8—51no—
207C23×C8—51no—
208C23×C8—51no—
209C2×#117—52no—
210C22×C4⋊C4—52no—
211#427—104no—
212C2×#107—51no—
213C2×#107—52no—
214C2×#107—52no—
215C2×#107—104no—
216C2×#107C2×D511yescharacteristic
217C2×#106—51no—
218C2×#106C2×D511yescharacteristic
219#391—102no—
220#390—102no—
221#386—102no—
222#386—102no—
223#380—52no—
224#380—52no—
225#380—52no—
226#380C2×D512yes—
227#340—52no—
228#340—52no—
229#340—52no—
230#340C2×D512yes—
231C23×C5—21no—
232C23×C5—21no—
233C23×C5—22no—
234C23×C5—22no—
235C23×C5C2×C4⋊C412yes—
236C23×C5C22×D411yescharacteristic
237C22×D5—22no—
238C22×D5—22no—
239C22×D5—21no—
240C22×D5—21no—
241C22×D5—22no—
242C22×D5—22no—
243C22×D5C2×C4⋊C412yes—
244C22×D5C2×C4⋊C412yes—
245C22×D5#12112yes—
246C22×D5#12112yes—
247C2×C4×C5—21no—
248C2×C4×C5—21no—
249C2×C4×C5—22no—
250C2×C4×C5—22no—
251C2×C4×C5—22no—
252C2×C4×C5—22no—
253C2×C4×C5—42no—
254C2×C4×C5—42no—
255C2×C4×C5—42no—
256C2×C4×C5—42no—
257C2×C4×C5—42no—
258C2×C4×C5—42no—
259C2×C4×C5C2×C4⋊C412yes—
260C2×C4×C5C23×C411yescharacteristic
261C2×C4×C5C22×D411yescharacteristic
262C2×C4×C5C22×D411yescharacteristic
263C2×C4×C5C22×Q811yescharacteristic
264C2×C4×C5C2×#4111yescharacteristic
265C2×C4×C5C2×#4111yescharacteristic
266C22⋊D5—28no—
267C22⋊D5—28no—
268C2×Dic5—21no—
269C2×Dic5—21no—
270C2×Dic5—42no—
271C2×Dic5—42no—
272C2×Dic5—42no—
273C2×Dic5—42no—
274C2×Dic5—42no—
275C2×Dic5—42no—
276C2×Dic5C2×C4⋊C412yes—
277C2×Dic5C2×C4⋊C412yes—
278C2×Dic5#12112yes—
279C2×Dic5#12112yes—
280D20—28no—
281C4×D5—22no—
282C4×D5—22no—
283C4×D5—22no—
284C4×D5—22no—
285C4×D5—88no—
286C4×D5C2×C22⋊C414yes—
287C4×D5#11714yes—
288C4×D5C22⋊Q814yes—
289C4×D5C22⋊Q814yes—
290Dic10—28no—
291C5×C8—22no—
292C5×C8C2×C22⋊C414yes—
293#168—22no—
294#168C2×C22⋊C414yes—
295C2×#41—54no—
296C2×#41—54no—
297C2×#41—102no—
298C2×#41—102no—
299C2×#41—102no—
300C2×#41—102no—
301C22×Q8—51no—
302C22×D4—51no—
303C22×D4—51no—
304C22×D4—51no—
305C23×C4—51no—
306C23×C4—51no—
307C23×C4—102no—
308C23×C4C22×D511yescharacteristic
309#131—1032no—
310C2×M16—22no—
311C2×M16—22no—
312C2×M16—54no—
313C2×M16—54no—
314C2×M16—102no—
315C2×M16—102no—
316C2×M16—102no—
317C2×M16—102no—
318C2×M16C4×D514yescomplemented
319C22×C8—51no—
320C22×C8—54no—
321C22×C8—54no—
322C22×C8—54no—
323C22×C8—54no—
324C22×C8—51no—
325C22×C8—54no—
326C22×C8—102no—
327C22×C8—102no—
328C22×C8—102no—
329C22×C8—102no—
330C22×C8C22×D511yescharacteristic
331C22×C8C22×D511yescharacteristic
332#117—104no—
333#117—104no—
334C2×C4⋊C4—52no—
335C2×C4⋊C4—52no—
336C2×C4⋊C4—104no—
337C2×C4⋊C4—104no—
338C2×C4⋊C4—102no—
339C2×C4⋊C4C22×D512yes—
340C2×C4⋊C4C22×D512yes—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—102no—
343C2×C22⋊C4—102no—
344C2×C22⋊C4—102no—
345C2×C22⋊C4—102no—
346C2×C42—102no—
347#107—22no—
348#107—104no—
349#107—104no—
350#107—102no—
351#106—22no—
352#106—102no—
353#95—22no—
354#95—102no—
355#95—102no—
356#95—102no—
357C22×C5—24no—
358C22×C5—22no—
359C22×C5—41no—
360C22×C5—41no—
361C22×C5—42no—
362C22×C5C22×C4⋊C411yescharacteristic
363C22×C5C2×#12111yescharacteristic
364C22×C5C2×#12111yescharacteristic
365C22×C5#57612yes—
366C22×C5#57812yes—
367C2×D5—24no—
368C2×D5—22no—
369C2×D5—22no—
370C2×D5—24no—
371C2×D5—24no—
372C2×D5—22no—
373C2×D5—41no—
374C2×D5—41no—
375C2×D5—42no—
376C2×D5#38014yes—
377C4×C5—22no—
378C4×C5—82no—
379C4×C5—82no—
380C4×C5C22×C22⋊C411yescharacteristic
381C4×C5C2×#11711yescharacteristic
382C4×C5C2×C22⋊Q811yescharacteristic
383C4×C5C2×C22⋊Q811yescharacteristic
384Dic5—22no—
385Dic5—82no—
386Dic5—82no—
387Dic5#38014yes—
388C24—51no—
389C24—51no—
390C24D4×D511yescharacteristic
391#41—108no—
392#41—108no—
393#41—108no—
394#41—108no—
395C2×Q8—54no—
396C2×Q8—102no—
397C2×Q8—102no—
398C2×D4—54no—
399C2×D4—54no—
400C2×D4—54no—
401C2×D4—102no—
402C2×D4—102no—
403C2×D4—102no—
404C2×D4—102no—
405C2×D4—102no—
406C2×D4—102no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—51no—
414C22×C4—52no—
415C22×C4—52no—
416C22×C4—52no—
417C22×C4—52no—
418C22×C4—52no—
419C22×C4—52no—
420C22×C4—51no—
421C22×C4—52no—
422C22×C4—52no—
423C22×C4—101no—
424C22×C4—101no—
425C22×C4—101no—
426C22×C4—101no—
427C22×C4—102no—
428C22×C4—102no—
429C22×C4—204no—
430C22×C4—402no—
431C22×C4—402no—
432C22×C4C2×C4×D512yes—
433C22×C4D4×D511yescharacteristic
434C22×C4Q8×D512yes—
435C22×C4C23×D511yescharacteristic
436M16—28no—
437M16—108no—
438M16—108no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—52no—
442C2×C8—52no—
443C2×C8—52no—
444C2×C8—516no—
445C2×C8—52no—
446C2×C8—108no—
447C2×C8—108no—
448C2×C8—101no—
449C2×C8—101no—
450C2×C8C2×C4×D512yes—
451C2×C8C2×C4×D512yes—
452C2×C8D4×D512yes—
453C2×C8D4×D512yes—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—102no—
459C4⋊C4—102no—
460C4⋊C4—102no—
461C4⋊C4—102no—
462C4⋊C4—204no—
463C22⋊C4—82no—
464C22⋊C4—204no—
465C22⋊C4—402no—
466C22⋊C4—402no—
467C22⋊C4—402no—
468C42—204no—
469C2×C5—42no—
470C2×C5C2×#38011yescharacteristic
471C2×C5#291312yes—
472C2×C5#305811yescharacteristic
473C2×C5#305911yescharacteristic
474C2×C5#306112yes—
475D5—24no—
476D5—42no—
477C23—22no—
478C23—22no—
479C23—21no—
480C23—21no—
481C23—52no—
482C23—52no—
483C23—52no—
484C23—52no—
485C23—102no—
486C23—102no—
487C23—102no—
488C23—102no—
489C23—101no—
490C23—101no—
491C23D5×C4⋊C412yes—
492C23C2×D4×D511yescharacteristic
493Q8—108no—
494D4—108no—
495D4—108no—
496D4—108no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—52no—
510C2×C4—52no—
511C2×C4—52no—
512C2×C4—54no—
513C2×C4—54no—
514C2×C4—54no—
515C2×C4—54no—
516C2×C4—52no—
517C2×C4—101no—
518C2×C4—101no—
519C2×C4—102no—
520C2×C4—102no—
521C2×C4—102no—
522C2×C4—102no—
523C2×C4—202no—
524C2×C4—202no—
525C2×C4—202no—
526C2×C4—202no—
527C2×C4—202no—
528C2×C4—202no—
529C2×C4—408no—
530C2×C4D5×C4⋊C412yes—
531C2×C4C22×C4×D511yescharacteristic
532C2×C4C2×D4×D511yescharacteristic
533C2×C4C2×D4×D511yescharacteristic
534C2×C4C2×Q8×D511yescharacteristic
535C2×C4D5×#4111yescharacteristic
536C2×C4D5×#4111yescharacteristic
537C8—22no—
538C8—54no—
539C8—102no—
540C8D5×C22⋊C414yes—
541C5#2066311yescharacteristic, complemented, Sylow
542C22—22no—
543C22—24no—
544C22—42no—
545C22—41no—
546C22—41no—
547C22—54no—
548C22—104no—
549C22—102no—
550C22—102no—
551C22—104no—
552C22—102no—
553C22—104no—
554C22—202no—
555C22—201no—
556C22—201no—
557C22C2×D5×C4⋊C411yescharacteristic
558C22D5×#12111yescharacteristic
559C22D5×#12111yescharacteristic
560C22#6610812yes—
561C22#6611312yes—
562C4—22no—
563C4—54no—
564C4—82no—
565C4—82no—
566C4—102no—
567C4—402no—
568C4—402no—
569C4C2×D5×C22⋊C411yescharacteristic
570C4D5×#11711yescharacteristic
571C4D5×C22⋊Q811yescharacteristic
572C4D5×C22⋊Q811yescharacteristic
573C2—42no—
574C2—104no—
575C2—202no—
576C2D5×#38011yescharacteristic
577C2#1061094112yes—
578C2#1061253912yes—
579C2#1061259611yescharacteristic
580C2#1061263611yescharacteristic
subgroupG/H
centerC23C2×D4×D5
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC5×#1809C2
socleC23×C5C22×D4
prime radicalC22×#709C22
cosocleC22×C4×C5C24
hypercenter#1809D5
hypocenterC5#20663
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1809

Complements

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

NH ≅ G/Ntype
2. C2×D5×#107C2semidirect
3. C2×#65149C2semidirect
5. C5×#1809C2semidirect
6. #10602781C2semidirect
7. #10601424C2semidirect
8. #10601371C2semidirect
9. #10600305C2semidirect
10. #10599662C2semidirect
11. #10599660C2semidirect
17. C2×#4053C4semidirect
61. C2×C5×M16C2×C4semidirect
77. C22×#168C2×C4semidirect
86. C2×#676C2×C4semidirect
90. C2×C8×D5C2×C4semidirect
114. #1809D5semidirect
198. C8×D5C22⋊C4semidirect
318. C2×M16C4×D5semidirect
541. C5#20663semidirect

Maximal subgroups

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

subgroupindex
C22×#40532
C2×D5×#1072
C2×#651492
#106057342
C5×#18092
#106027812
#106014242
#106013712
#106003052
#105996622
#105996602
#206635