Finite Group Databasebeta

← #12117070

Subgroups — #12117070

Order 1280.

810 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10613890C211yesmaximal, characteristic
2C2×#65253C211yesmaximal, characteristic
3#10605612C211yesmaximal, characteristic
4C5×#1824C211yesmaximal, characteristic
5#10600742C211yesmaximal, characteristic
6#10600719C211yesmaximal, characteristic
7#10600306C211yesmaximal, characteristic
8#10599654C211yesmaximal, characteristic, complemented
9#10599651C211yesmaximal, characteristic, complemented
10#10599614C211yesmaximal, characteristic, complemented
11#10599608C211yesmaximal, characteristic, complemented
12#10599582C211yesmaximal, characteristic, complemented
13#10599581C211yesmaximal, characteristic, complemented
14#10599579C211yesmaximal, characteristic, complemented
15#10599578C211yesmaximal, characteristic, complemented
16C22×C5×C4⋊C4C2211yescharacteristic
17C2×#4072—21no—
18C2×#4072C2211yescharacteristic
19C22×#681C2211yescharacteristic
20C2×C5×C22⋊C8C2211yescharacteristic
21C5×#380C2211yescharacteristic
22#65501C2211yescharacteristic
23#65500—21no—
24C4×C22⋊Dic5C2211yescharacteristic
25C2×#3947C2211yescharacteristic
26C2×C8×Dic5C2211yescharacteristic
27#65274C2211yescharacteristic
28#65274C2211yescharacteristic
29#65270—21no—
30#65270—21no—
31C4⋊C4×Dic5C2211yescharacteristic
32C4⋊C4×Dic5C2211yescharacteristic
33C4⋊C4×Dic5C2211yescharacteristic
34C4⋊C4×Dic5C2211yescharacteristic
35#65262C2211yescharacteristic
36#65262C2211yescharacteristic
37#65253C418yescomplemented
38C2×#3940C2211yescharacteristic
39C2×#3940C2211yescharacteristic
40C2×#3940C2211yescharacteristic
41C2×#3939C2211yescharacteristic
42C2×#3939C2211yescharacteristic
43C2×#3939C2211yescharacteristic
44#65234—21no—
45#65234—21no—
46#64939—21no—
47#64937—21no—
48C5×#340C2211yescharacteristic
49C5×#340C2211yescharacteristic
50C5×#340C2211yescharacteristic
51C5×#340C2211yescharacteristic
52#64771C2211yescharacteristic
53#64771C2211yescharacteristic
54#64771C2211yescharacteristic
55#64771C2211yescharacteristic
56#64770C2211yescharacteristic
57#64770C2211yescharacteristic
58#64749C2211yescharacteristic
59#64749C2211yescharacteristic
60#22624—51nomaximal, Sylow
61C23×C4×C5—21no—
62C23×C4×C5C2311yescharacteristic
63C22×C5×C8C2311yescharacteristic
64C22×C5×C8C2311yescharacteristic
65C2×C5×C4⋊C4—21no—
66C2×C5×C4⋊C4—22no—
67C2×C5×C4⋊C4C2×C414yes—
68C2×C5×C4⋊C4C2311yescharacteristic
69C2×C5×C4⋊C4C2311yescharacteristic
70C2×C5×C4⋊C4C2311yescharacteristic
71C2×C5×C4⋊C4C2311yescharacteristic
72C2×C5×C22⋊C4—21no—
73C2×C22⋊Dic5—21no—
74C2×C22⋊Dic5—21no—
75C2×C22⋊Dic5D411yescharacteristic
76C2×C22⋊Dic5D411yescharacteristic
77#4072—22no—
78#4072—44no—
79#4072C2×C414yes—
80C2×C4⋊Dic5—21no—
81C2×C4⋊Dic5—21no—
82C2×C4⋊Dic5C2311yescharacteristic
83C2×C4⋊Dic5C2311yescharacteristic
84C2×C4⋊Dic5C2311yescharacteristic
85C2×C4⋊Dic5C2311yescharacteristic
86C2×#683—21no—
87C2×#683—21no—
88C2×C4×Dic5—21no—
89C2×C4×Dic5—21no—
90C2×C4×Dic5—21no—
91C2×C4×Dic5—21no—
92C2×C4×Dic5—21no—
93C2×C4×Dic5C2311yescharacteristic
94C2×C4×Dic5C2311yescharacteristic
95C2×#681—22no—
96C2×#681—22no—
97C2×#681C2×C414yes—
98C22×#168C2311yescharacteristic
99C22×#168C2311yescharacteristic
100C5×C22⋊C8—22no—
101C5×#95—21no—
102#3963—21no—
103#3963—21no—
104#3963—21no—
105#3963—21no—
106#3963—21no—
107#3963—21no—
108#3963—21no—
109#3963—21no—
110#3963—21no—
111#3963D411yescharacteristic
112#3963D411yescharacteristic
113#3947—22no—
114C8×Dic5—22no—
115#3940—22no—
116#3940C2×C414yes—
117#3940C2×C414yes—
118#3939—22no—
119#3939C2×C414yes—
120#3939C2×C414yes—
121C2×#428—51no—
122#2346—51no—
123#2344—51no—
124#1944—51no—
125#1941—51no—
126#1921—51no—
127#1920—51no—
128#1919—51no—
129#1918—51no—
130#1825—51no—
131#1824D511yescharacteristic
132#1809—51no—
133#1776—51no—
134C4×#107—51no—
135#1753—51no—
136C24×C5C2×D411yescharacteristic
137C22×C4×C5—21no—
138C22×C4×C5—21no—
139C22×C4×C5—21no—
140C22×C4×C5—21no—
141C22×C4×C5—24no—
142C22×C4×C5—21no—
143C22×C4×C5—21no—
144C22×C4×C5—21no—
145C22×C4×C5—21no—
146C22×C4×C5—42no—
147C22×C4×C5—42no—
148C22×C4×C5C22×C412yes—
149C22×C4×C5C2×D411yescharacteristic
150C22×C4×C5#4112yes—
151C22×C4×C5C2411yescharacteristic
152C22×Dic5—21no—
153C22×Dic5—21no—
154C22×Dic5—21no—
155C22×Dic5—21no—
156C22×Dic5C2×D411yescharacteristic
157C22×Dic5C2×D411yescharacteristic
158C22×Dic5C2×D411yescharacteristic
159C22×Dic5C2×D411yescharacteristic
160C2×C5×C8—21no—
161C2×C5×C8—21no—
162C2×C5×C8—21no—
163C2×C5×C8—21no—
164C2×C5×C8—21no—
165C2×C5×C8—21no—
166C5×C4⋊C4—24no—
167C5×C4⋊C4—21no—
168C5×C4⋊C4—21no—
169C5×C4⋊C4—21no—
170C5×C4⋊C4—21no—
171C5×C4⋊C4—21no—
172C5×C4⋊C4—21no—
173C5×C4⋊C4C22×C412yes—
174C5×C4⋊C4C22×C412yes—
175C5×C22⋊C4—81no—
176C22⋊Dic5—24no—
177C22⋊Dic5—22no—
178C22⋊Dic5—41no—
179C22⋊Dic5—42no—
180C4⋊Dic5—21no—
181C4⋊Dic5—21no—
182C4⋊Dic5—21no—
183C4⋊Dic5—21no—
184C4⋊Dic5—21no—
185C4⋊Dic5—21no—
186C4⋊Dic5—42no—
187C4⋊Dic5—42no—
188C4⋊Dic5C22×C412yes—
189C4⋊Dic5C22×C412yes—
190#683—81no—
191#683—81no—
192C4×Dic5—24no—
193C4×Dic5—21no—
194C4×Dic5—21no—
195C4×Dic5—21no—
196C4×Dic5—21no—
197C4×Dic5—42no—
198C4×Dic5—42no—
199C4×Dic5—42no—
200C4×Dic5—42no—
201#681—28no—
202C2×#168—21no—
203C2×#168—21no—
204C2×#168C22×C412yes—
205C2×#168C22×C412yes—
206C2×#168#4112yes—
207C2×#168#4112yes—
208C22×M16—51no—
209C2×#117—51no—
210C2×#117—101no—
211C22×C4⋊C4C2×D511yescharacteristic
212#428—58no—
213C2×#107—51no—
214C2×#107—51no—
215C2×#107—51no—
216C2×#106—51no—
217C2×#106—51no—
218C2×#106—51no—
219C2×C22⋊C8C2×D511yescharacteristic
220C2×#97—51no—
221C2×C4×C8—51no—
222#389—51no—
223#389—51no—
224#389—51no—
225#389—51no—
226#388—101no—
227#388—101no—
228#387—101no—
229#387—101no—
230#382—101no—
231#381—101no—
232#380—51no—
233#380C2×D511yescharacteristic
234C4×C4⋊C4—51no—
235C4×C4⋊C4—51no—
236C4×C4⋊C4—51no—
237C4×C4⋊C4—51no—
238C4×C22⋊C4—51no—
239#376—101no—
240#340—51no—
241#340—51no—
242#340—51no—
243#340—51no—
244#340—51no—
245#340—51no—
246#340C2×D511yescharacteristic
247#340C2×D511yescharacteristic
248#340C2×D511yescharacteristic
249#340C2×D511yescharacteristic
250#336—51no—
251#336—51no—
252C23×C5—21no—
253C23×C5—21no—
254C23×C5—21no—
255C23×C5—21no—
256C23×C5—21no—
257C23×C5—21no—
258C23×C5C4×D412yes—
259C23×C5C22×D411yescharacteristic
260C2×C4×C5—21no—
261C2×C4×C5—21no—
262C2×C4×C5—22no—
263C2×C4×C5—22no—
264C2×C4×C5—22no—
265C2×C4×C5—22no—
266C2×C4×C5—22no—
267C2×C4×C5—22no—
268C2×C4×C5—48no—
269C2×C4×C5—41no—
270C2×C4×C5—41no—
271C2×C4×C5—41no—
272C2×C4×C5—41no—
273C2×C4×C5—41no—
274C2×C4×C5—41no—
275C2×C4×C5—41no—
276C2×C4×C5—41no—
277C2×C4×C5—41no—
278C2×C4×C5—41no—
279C2×C4×C5C4×D412yes—
280C2×C4×C5C23×C411yescharacteristic
281C2×C4×C5C2×#4111yescharacteristic
282C2×C4×C5C2×#4111yescharacteristic
283C2×C4×C5C2×#4111yescharacteristic
284C2×C4×C52−1+411yescharacteristic
285C2×C4×C52−1+411yescharacteristic
286C2×Dic5—21no—
287C2×Dic5—21no—
288C2×Dic5—22no—
289C2×Dic5—22no—
290C2×Dic5—21no—
291C2×Dic5—21no—
292C2×Dic5—21no—
293C2×Dic5—21no—
294C2×Dic5—21no—
295C2×Dic5—21no—
296C2×Dic5—21no—
297C2×Dic5—21no—
298C2×Dic5—21no—
299C2×Dic5—41no—
300C2×Dic5—41no—
301C2×Dic5—41no—
302C2×Dic5—41no—
303C2×Dic5—41no—
304C2×Dic5—41no—
305C2×Dic5—41no—
306C2×Dic5—41no—
307C2×Dic5—41no—
308C2×Dic5—41no—
309C2×Dic5AGL(1,ℤ/8)12yes—
310C5×C8—41no—
311C5×C8—41no—
312#168—22no—
313#168#11714yescomplemented
314C23×C4—21no—
315C23×C4C22×D511yescharacteristic
316C2×M16—54no—
317C2×M16—102no—
318C2×M16—102no—
319C22×C8—51no—
320C22×C8—51no—
321C22×C8C22×D511yescharacteristic
322C22×C8C22×D511yescharacteristic
323#117—54no—
324#117—102no—
325#117—204no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—51no—
329C2×C4⋊C4—51no—
330C2×C4⋊C4—51no—
331C2×C4⋊C4—51no—
332C2×C4⋊C4—101no—
333C2×C4⋊C4—101no—
334C2×C4⋊C4—101no—
335C2×C4⋊C4—101no—
336C2×C4⋊C4C4×D514yes—
337C2×C4⋊C4C22×D511yescharacteristic
338C2×C4⋊C4C22×D511yescharacteristic
339C2×C4⋊C4C22×D511yescharacteristic
340C2×C4⋊C4C22×D511yescharacteristic
341C2×C22⋊C4—21no—
342C2×C22⋊C4—51no—
343C2×C22⋊C4—51no—
344C2×C22⋊C4—101no—
345C2×C22⋊C4—101no—
346C2×C42—51no—
347C2×C42—51no—
348C2×C42—101no—
349C2×C42—101no—
350C2×C42—101no—
351C2×C42—101no—
352C2×C42—101no—
353#107—54no—
354#107—54no—
355#107—102no—
356#106—54no—
357#106—54no—
358#106—102no—
359C22⋊C8—22no—
360#97—102no—
361C4×C8—102no—
362#95—21no—
363#95—51no—
364#95—51no—
365#95—101no—
366#95—101no—
367#95—101no—
368#95—101no—
369#95—101no—
370#95—101no—
371#95—101no—
372#95—101no—
373#95—101no—
374C22×C5—22no—
375C22×C5—22no—
376C22×C5—22no—
377C22×C5—41no—
378C22×C5—41no—
379C22×C5—41no—
380C22×C5—41no—
381C22×C5C2×C4×D411yescharacteristic
382C22×C5#54711yescharacteristic
383C22×C5#54711yescharacteristic
384C22×C5C2×AGL(1,ℤ/8)11yescharacteristic
385C22×C5#57511yescharacteristic
386C22×C5#57711yescharacteristic
387C22×C5#57811yescharacteristic
388C4×C5—22no—
389C4×C5—42no—
390C4×C5—42no—
391C4×C5—81no—
392C4×C5—81no—
393C4×C5C2×#11711yescharacteristic
394C4×C5#51911yescharacteristic
395C4×C5#55211yescharacteristic
396C4×C5#55211yescharacteristic
397Dic5—22no—
398Dic5—41no—
399Dic5—41no—
400Dic5—41no—
401Dic5—42no—
402Dic5—42no—
403Dic5—81no—
404Dic5—81no—
405C24D4×D511yescharacteristic
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—24no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—51no—
418C22×C4—51no—
419C22×C4—51no—
420C22×C4—51no—
421C22×C4—101no—
422C22×C4—101no—
423C22×C4—101no—
424C22×C4—101no—
425C22×C4C2×C4×D512yes—
426C22×C4D4×D511yescharacteristic
427C22×C4#71112yes—
428C22×C4C23×D511yescharacteristic
429M16—108no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—52no—
437C2×C8—52no—
438C2×C8—52no—
439C2×C8—52no—
440C2×C8—101no—
441C2×C8—101no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—24no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—52no—
450C4⋊C4—52no—
451C4⋊C4—101no—
452C4⋊C4—101no—
453C4⋊C4—101no—
454C4⋊C4—101no—
455C4⋊C4—101no—
456C4⋊C4—101no—
457C4⋊C4—202no—
458C4⋊C4—202no—
459C4⋊C4—401no—
460C4⋊C4—401no—
461C4⋊C4C2×C4×D512yes—
462C4⋊C4C2×C4×D512yes—
463C22⋊C4—81no—
464C22⋊C4—102no—
465C22⋊C4—104no—
466C22⋊C4—201no—
467C22⋊C4—202no—
468C42—101no—
469C42—101no—
470C42—101no—
471C42—101no—
472C42—104no—
473C42—202no—
474C42—202no—
475C42—202no—
476C42—202no—
477C2×C5—42no—
478C2×C5#234411yescharacteristic
479C2×C5C4×AGL(1,ℤ/8)11yescharacteristic
480C2×C5#294711yescharacteristic
481C2×C5#334311yescharacteristic
482C2×C5#334811yescharacteristic
483C2×C5#335211yescharacteristic
484C2×C5#335411yescharacteristic
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23#402712yes—
492C23C2×D4×D511yescharacteristic
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—48no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—52no—
513C2×C4—101no—
514C2×C4—101no—
515C2×C4—102no—
516C2×C4—102no—
517C2×C4—101no—
518C2×C4—101no—
519C2×C4—101no—
520C2×C4—101no—
521C2×C4—101no—
522C2×C4—101no—
523C2×C4—101no—
524C2×C4—101no—
525C2×C4—101no—
526C2×C4—201no—
527C2×C4—201no—
528C2×C4—201no—
529C2×C4—201no—
530C2×C4—201no—
531C2×C4—201no—
532C2×C4—201no—
533C2×C4—201no—
534C2×C4—201no—
535C2×C4—201no—
536C2×C4#402712yes—
537C2×C4C22×C4×D511yescharacteristic
538C2×C4C2×#71111yescharacteristic
539C2×C4D5×#4111yescharacteristic
540C2×C4D5×#4111yescharacteristic
541C2×C4#415011yescharacteristic
542C2×C4#415011yescharacteristic
543C8—41no—
544C8—41no—
545C8—54no—
546C8—102no—
547C5#2262411yescharacteristic, complemented, Sylow
548C22—22no—
549C22—22no—
550C22—22no—
551C22—41no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22C2×#402711yescharacteristic
556C22#6598911yescharacteristic
557C22#6598911yescharacteristic
558C22#6609611yescharacteristic
559C22#6610011yescharacteristic
560C22D5×AGL(1,ℤ/8)11yescharacteristic
561C22#6610711yescharacteristic
562C4—22no—
563C4—42no—
564C4—42no—
565C4—81no—
566C4—81no—
567C4—102no—
568C4—202no—
569C4—201no—
570C4—201no—
571C4—202no—
572C4—201no—
573C4—401no—
574C4—401no—
575C4#6585011yescharacteristic
576C4D5×#11711yescharacteristic
577C4#6598411yescharacteristic
578C4#6598411yescharacteristic
579C2—42no—
580C2#1060579111yescharacteristic
581C2#1061050511yescharacteristic
582C2#1061064511yescharacteristic
583C2#1061306711yescharacteristic
584C2#1061306911yescharacteristic
585C2#1061320811yescharacteristic
586C2#1061321111yescharacteristic
subgroupG/H
centerC23C2×D4×D5
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC5×#1824C2
socleC23×C5C22×D4
prime radicalC24×C5C2×D4
cosocleC22×C4×C5C24
hypercenter#1824D5
hypocenterC5#22624
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1824

Complements

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

NH ≅ G/Ntype
8. #10599654C2semidirect
9. #10599651C2semidirect
10. #10599614C2semidirect
11. #10599608C2semidirect
12. #10599582C2semidirect
13. #10599581C2semidirect
14. #10599579C2semidirect
15. #10599578C2semidirect
37. #65253C4semidirect
313. #168#117semidirect
547. C5#22624semidirect

Maximal subgroups

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

subgroupindex
#106138902
C2×#652532
#106056122
C5×#18242
#106007422
#106007192
#106003062
#105996542
#105996512
#105996142
#105996082
#105995822
#105995812
#105995792
#105995782
#226245