Finite Group Databasebeta

← #12133133

Subgroups — #12133133

Order 1280.

1088 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#552C211yesmaximal, characteristic
2C2×#65333C211yesmaximal, characteristic, complemented
3#10614104C211yesmaximal, characteristic
4C2×#65284C211yesmaximal, characteristic, complemented
5#10602532C212yesmaximal, complemented
6#10602490C211yesmaximal, characteristic, complemented
7#10602487C212yesmaximal, complemented
8#10602421C212yesmaximal, complemented
9#10602404C211yesmaximal, characteristic
10#10602368C211yesmaximal, characteristic, complemented
11#10602348C212yesmaximal, complemented
12C5×#552—24no—
13C5×#552—24no—
14C2×C5×#126—21no—
15C2×C5×#125—21no—
16C2×C5×#125C2211yescharacteristic
17C2×C5×#123—21no—
18C2×C5×C22⋊Q8C2211yescharacteristic
19C2×C5×C22⋊Q8C2211yescharacteristic
20C2×C4×C5×Q8C2211yescharacteristic
21C2×C4×C5×D4C2211yescharacteristic
22C2×C5×#117—21no—
23C2×C5×#117C2211yescharacteristic
24C22×C5×C4⋊C4C2211yescharacteristic
25C2×#4081—21no—
26C22×C4⋊Dic5C2211yescharacteristic
27C22×#681C2211yescharacteristic
28C2×#3967C2212yes—
29C2×#3967C2212yes—
30#65501C2212yes—
31#65501C2211yescharacteristic
32C2×C22⋊#168C2212yes—
33C2×C22⋊#168C2211yescharacteristic
34#65333—24no—
35#65284—24no—
36#65272—21no—
37C2×#3940C2212yes—
38C2×#3939C2212yes—
39C2×#3939C2212yes—
40#65242—21no—
41#65232—21no—
42#65217C2212yes—
43#64749C2212yes—
44#64749C2212yes—
45#64749C2212yes—
46#64744C2212yes—
47#24110—51nomaximal, Sylow
48C22×C5×Q8C2311yescharacteristic
49C22×C5×D4D411yescharacteristic
50C23×C4×C5—21no—
51C23×C4×C5C2311yescharacteristic
52C23×C4×C5C2311yescharacteristic
53C23×Dic5—21no—
54C5×#126—28no—
55C5×#125—28no—
56C5×#125—22no—
57C5×#125—22no—
58C5×#123—28no—
59C5×C22⋊Q8—22no—
60C5×C22⋊Q8—22no—
61C5×C22⋊Q8—22no—
62C5×C22⋊Q8D414yes—
63C4×C5×Q8—22no—
64C4×C5×Q8—22no—
65C4×C5×D4—22no—
66C4×C5×D4—22no—
67C5×#117—28no—
68C5×#117—22no—
69C5×#117D414yes—
70C2×C5×C4⋊C4—21no—
71C2×C5×C4⋊C4—21no—
72C2×C5×C4⋊C4—21no—
73C2×C5×C4⋊C4—21no—
74C2×C5×C4⋊C4—21no—
75C2×C5×C4⋊C4—21no—
76C2×C5×C4⋊C4—22no—
77C2×C5×C4⋊C4—22no—
78C2×C5×C4⋊C4D411yescharacteristic
79C2×C5×C4⋊C4C2311yescharacteristic
80C2×C5×C4⋊C4C2311yescharacteristic
81C2×C5×C4⋊C4C2311yescharacteristic
82C2×C5×C22⋊C4—21no—
83C2×C5×C22⋊C4—21no—
84C2×C5×C22⋊C4—21no—
85C2×C5×C22⋊C4D411yescharacteristic
86C2×C5×C22⋊C4D411yescharacteristic
87C2×C42×C5—21no—
88C2×C42×C5—21no—
89C2×C42×C5C2311yescharacteristic
90C2×C22⋊Dic5—21no—
91C2×C22⋊Dic5—22no—
92#4081—44no—
93C2×C4⋊Dic5—22no—
94C2×C4⋊Dic5—22no—
95C2×C4⋊Dic5—21no—
96C2×C4⋊Dic5C2312yes—
97C2×C4⋊Dic5C2312yes—
98C2×#683—22no—
99C2×#681—22no—
100C2×#681—24no—
101C22×#168C2312yes—
102C22×#168C2312yes—
103#3967—24no—
104#3967—24no—
105#3963—22no—
106#3963—22no—
107#3963—21no—
108C22⋊#168—22no—
109C22⋊#168—24no—
110#3940—24no—
111#3939—24no—
112#3939—24no—
113C2×#552D511yescharacteristic
114C2×#484—51no—
115C2×#428—51no—
116#2486—51no—
117#2059—52no—
118#2035—52no—
119#1894—51no—
120#1870—51no—
121#1863—52no—
122#1852—52no—
123#1825—51no—
124C24×C5C2×D411yescharacteristic
125C24×C5C2×D411yescharacteristic
126C2×C5×Q8—21no—
127C2×C5×Q8—22no—
128C2×C5×Q8—22no—
129C2×C5×Q8C2×D412yes—
130C2×C5×D4—22no—
131C2×C5×D4—22no—
132C2×C5×D4—41no—
133C22×C4×C5—22no—
134C22×C4×C5—22no—
135C22×C4×C5—24no—
136C22×C4×C5—21no—
137C22×C4×C5—21no—
138C22×C4×C5—21no—
139C22×C4×C5—21no—
140C22×C4×C5—21no—
141C22×C4×C5—21no—
142C22×C4×C5—22no—
143C22×C4×C5—22no—
144C22×C4×C5—42no—
145C22×C4×C5—42no—
146C22×C4×C5C2×D411yescharacteristic
147C22×C4×C5C2×D411yescharacteristic
148C22×C4×C5C2×D411yescharacteristic
149C22×C4×C5C2×D411yescharacteristic
150C22×C4×C5C2×D412yes—
151C22×C4×C5C2×D412yes—
152C22×C4×C5C2411yescharacteristic
153C22×Dic5—22no—
154C22×Dic5—22no—
155C22×Dic5—42no—
156C22×Dic5—42no—
157C22×Dic5—42no—
158C5×C4⋊C4—24no—
159C5×C4⋊C4—24no—
160C5×C4⋊C4—24no—
161C5×C4⋊C4—24no—
162C5×C4⋊C4—24no—
163C5×C4⋊C4—24no—
164C5×C4⋊C4—22no—
165C5×C4⋊C4—21no—
166C5×C4⋊C4—21no—
167C5×C4⋊C4—21no—
168C5×C4⋊C4C2×D412yes—
169C5×C4⋊C4C2×D412yes—
170C5×C4⋊C4C2×D412yes—
171C5×C22⋊C4—24no—
172C5×C22⋊C4—24no—
173C5×C22⋊C4—24no—
174C5×C22⋊C4—22no—
175C5×C22⋊C4—22no—
176C42×C5—24no—
177C42×C5—24no—
178C42×C5—21no—
179C42×C5—21no—
180C22⋊Dic5—42no—
181C22⋊Dic5—44no—
182C4⋊Dic5—22no—
183C4⋊Dic5—22no—
184C4⋊Dic5—22no—
185C4⋊Dic5—22no—
186C4⋊Dic5—42no—
187#683—44no—
188#681—28no—
189C2×#168—22no—
190C2×#168—22no—
191C2×#168—22no—
192C2×#168—22no—
193C2×#168—22no—
194C2×#168#4114yes—
195C22×M16—51no—
196#552—24no—
197#552—24no—
198C2×#126—21no—
199C2×#125—21no—
200C2×#125C2×D511yescharacteristic
201C2×#123—21no—
202C2×#123—101no—
203C2×C22⋊Q8C2×D511yescharacteristic
204C2×C22⋊Q8C2×D511yescharacteristic
205C2×C4×Q8C2×D511yescharacteristic
206C2×C4×D4C2×D511yescharacteristic
207C2×#117—21no—
208C2×#117C2×D511yescharacteristic
209C22×C4⋊C4—51no—
210C22×C4⋊C4C2×D511yescharacteristic
211#484—104no—
212#428—104no—
213C2×#107—52no—
214C2×#107—52no—
215C2×#106—52no—
216C2×#103—52no—
217C2×#103—52no—
218C2×C22⋊C8—51no—
219C2×C22⋊C8—52no—
220#398—101no—
221#397—101no—
222#389—52no—
223#381—101no—
224#380—51no—
225#380—52no—
226#340—52no—
227#340—52no—
228#340—52no—
229#336—52no—
230C23×C5—21no—
231C23×C5—22no—
232C23×C5—22no—
233C23×C5—21no—
234C23×C5—21no—
235C23×C5—21no—
236C23×C5—22no—
237C23×C5—22no—
238C23×C5#12012yes—
239C23×C5AGL(1,ℤ/8)12yes—
240C23×C5C22×D411yescharacteristic
241C5×Q8—24no—
242C5×Q8—41no—
243C5×Q8—41no—
244C5×D4—44no—
245C2×C4×C5—22no—
246C2×C4×C5—22no—
247C2×C4×C5—22no—
248C2×C4×C5—24no—
249C2×C4×C5—21no—
250C2×C4×C5—22no—
251C2×C4×C5—22no—
252C2×C4×C5—22no—
253C2×C4×C5—21no—
254C2×C4×C5—21no—
255C2×C4×C5—22no—
256C2×C4×C5—22no—
257C2×C4×C5—21no—
258C2×C4×C5—21no—
259C2×C4×C5—22no—
260C2×C4×C5—48no—
261C2×C4×C5—42no—
262C2×C4×C5—41no—
263C2×C4×C5—41no—
264C2×C4×C5—41no—
265C2×C4×C5—41no—
266C2×C4×C5—42no—
267C2×C4×C5—41no—
268C2×C4×C5—41no—
269C2×C4×C5—42no—
270C2×C4×C5#12012yes—
271C2×C4×C5C22×D411yescharacteristic
272C2×C4×C5C22×D411yescharacteristic
273C2×C4×C5C2×#4112yes—
274C2×C4×C52−1+412yes—
275C2×Dic5—42no—
276C2×Dic5—42no—
277C2×Dic5—42no—
278C2×Dic5—42no—
279C2×Dic5—42no—
280C2×Dic5—42no—
281C2×Dic5—84no—
282#168—24no—
283#168—42no—
284C22×Q8C22×D511yescharacteristic
285C22×D4C22⋊D511yescharacteristic
286C23×C4—21no—
287C23×C4—101no—
288C23×C4C22×D511yescharacteristic
289C23×C4C22×D511yescharacteristic
290C2×M16—104no—
291C2×M16—102no—
292C22×C8—52no—
293C22×C8—52no—
294#126—28no—
295#125—28no—
296#125—22no—
297#125—22no—
298#123—28no—
299#123—204no—
300C22⋊Q8—22no—
301C22⋊Q8—22no—
302C22⋊Q8—22no—
303C22⋊Q8C22⋊D514yes—
304C4×Q8—22no—
305C4×Q8—22no—
306C4×D4—22no—
307C4×D4—22no—
308#117—28no—
309#117—22no—
310#117C22⋊D514yes—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—52no—
320C2×C4⋊C4—52no—
321C2×C4⋊C4—102no—
322C2×C4⋊C4—102no—
323C2×C4⋊C4—101no—
324C2×C4⋊C4—102no—
325C2×C4⋊C4C22⋊D511yescharacteristic
326C2×C4⋊C4C22×D511yescharacteristic
327C2×C4⋊C4C22×D511yescharacteristic
328C2×C4⋊C4C22×D511yescharacteristic
329C2×C22⋊C4—21no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—101no—
333C2×C22⋊C4—102no—
334C2×C22⋊C4C22⋊D511yescharacteristic
335C2×C22⋊C4C22⋊D511yescharacteristic
336C2×C42—21no—
337C2×C42—21no—
338C2×C42C22×D511yescharacteristic
339#107—104no—
340#107—104no—
341#106—104no—
342#103—104no—
343#103—104no—
344C22⋊C8—104no—
345C22⋊C8—102no—
346#95—102no—
347#95—101no—
348#95—102no—
349C22×C5—24no—
350C22×C5—22no—
351C22×C5—22no—
352C22×C5—22no—
353C22×C5—22no—
354C22×C5—42no—
355C22×C5—42no—
356C22×C5—42no—
357C22×C5—41no—
358C22×C5—41no—
359C22×C5C2×#12011yescharacteristic
360C22×C5#54712yes—
361C22×C5C2×AGL(1,ℤ/8)11yescharacteristic
362C22×C5#57511yescharacteristic
363C22×C5#57812yes—
364C4×C5—22no—
365C4×C5—41no—
366C4×C5—42no—
367C4×C5—41no—
368C4×C5—42no—
369C4×C5—42no—
370C4×C5—41no—
371C4×C5—42no—
372C4×C5#52511yescharacteristic
373C4×C5#53611yescharacteristic
374C4×C5#54712yes—
375Dic5—82no—
376Dic5—82no—
377C24C2×C22⋊D511yescharacteristic
378C24C2×C22⋊D511yescharacteristic
379C2×Q8—21no—
380C2×Q8—22no—
381C2×Q8—22no—
382C2×Q8C2×C22⋊D512yes—
383C2×D4—22no—
384C2×D4—22no—
385C2×D4—41no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—102no—
400C22×C4—102no—
401C22×C4—202no—
402C22×C4—202no—
403C22×C4—202no—
404C22×C4C2×C22⋊D511yescharacteristic
405C22×C4C2×C22⋊D511yescharacteristic
406C22×C4C2×C22⋊D511yescharacteristic
407C22×C4C2×C22⋊D511yescharacteristic
408C22×C4C2×C22⋊D512yes—
409C22×C4C2×C22⋊D512yes—
410C22×C4C23×D511yescharacteristic
411M16—108no—
412C2×C8—54no—
413C2×C8—102no—
414C2×C8—102no—
415C2×C8—102no—
416C2×C8—102no—
417C2×C8—102no—
418C4⋊C4—24no—
419C4⋊C4—24no—
420C4⋊C4—24no—
421C4⋊C4—24no—
422C4⋊C4—24no—
423C4⋊C4—24no—
424C4⋊C4—22no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—102no—
429C4⋊C4—102no—
430C4⋊C4—102no—
431C4⋊C4—102no—
432C4⋊C4—204no—
433C4⋊C4—202no—
434C4⋊C4C2×C22⋊D512yes—
435C4⋊C4C2×C22⋊D512yes—
436C4⋊C4C2×C22⋊D512yes—
437C22⋊C4—24no—
438C22⋊C4—24no—
439C22⋊C4—24no—
440C22⋊C4—22no—
441C22⋊C4—22no—
442C22⋊C4—202no—
443C22⋊C4—204no—
444C42—24no—
445C42—24no—
446C42—21no—
447C42—21no—
448C2×C5—42no—
449C2×C5—42no—
450C2×C5#240611yescharacteristic
451C2×C5#300611yescharacteristic
452C2×C5#301111yescharacteristic
453C2×C5#334312yes—
454C2×C5#335212yes—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—22no—
459C23—22no—
460C23—22no—
461C23—22no—
462C23—21no—
463C23#409512yes—
464C23#409912yes—
465C23C22×C22⋊D511yescharacteristic
466Q8—24no—
467Q8—41no—
468Q8—41no—
469D4—44no—
470C2×C4—22no—
471C2×C4—24no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—21no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—22no—
485C2×C4—48no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—202no—
496C2×C4—202no—
497C2×C4—202no—
498C2×C4—202no—
499C2×C4—202no—
500C2×C4—202no—
501C2×C4—404no—
502C2×C4#409912yes—
503C2×C4D5×#4112yes—
504C2×C4#415012yes—
505C2×C4C22×C22⋊D511yescharacteristic
506C2×C4C22×C22⋊D511yescharacteristic
507C8—104no—
508C8—202no—
509C5#2411011yescharacteristic, complemented, Sylow
510C22—22no—
511C22—22no—
512C22—22no—
513C22—24no—
514C22—22no—
515C22—41no—
516C22—42no—
517C22—42no—
518C22—42no—
519C22—41no—
520C22C2×#409511yescharacteristic
521C22#6615611yescharacteristic
522C22#6615812yes—
523C22#6617212yes—
524C22C2×#409911yescharacteristic
525C4—22no—
526C4—41no—
527C4—42no—
528C4—42no—
529C4—41no—
530C4—42no—
531C4—41no—
532C4—42no—
533C4—402no—
534C4—402no—
535C4#6615812yes—
536C4#6615911yescharacteristic
537C4#6616211yescharacteristic
538C2—42no—
539C2—42no—
540C2#1061532312yes—
541C2#1061532812yes—
542C2#1061581311yescharacteristic
543C2#1061582111yescharacteristic
544C2#1061584811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D5
commutator, cosocleC22×C4×C5C24
FrattiniC22×C4C23×D5
FittingC2×C5×#552C2
socleC23×C5C22×D4
prime radicalC22×C5×D4D4
hypercenterC2×#552D5
hypocenterC5#24110
LCSG ▷ C22×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#552

Complements

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

NH ≅ G/Ntype
2. C2×#65333C2semidirect
4. C2×#65284C2semidirect
5. #10602532C2semidirect
6. #10602490C2semidirect
7. #10602487C2semidirect
8. #10602421C2semidirect
10. #10602368C2semidirect
11. #10602348C2semidirect
509. C5#24110semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#5522
C2×#653332
#106141042
C2×#652842
#106025322
#106024902
#106024872
#106024212
#106024042
#106023682
#106023482
#241105