Finite Group Databasebeta

← #12156295

Subgroups — #12156295

Order 1280.

1100 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#557C211yesmaximal, characteristic, complemented
2C2×#65356C211yesmaximal, characteristic
3#10614207C211yesmaximal, characteristic
4C2×#65316C211yesmaximal, characteristic
5#10602585C212yesmaximal, complemented
6#10602536C212yesmaximal, complemented
7#10602531C211yesmaximal, characteristic, complemented
8#10602428C211yesmaximal, characteristic, complemented
9#10602426C212yesmaximal, complemented
10#10602426C212yesmaximal, complemented
11#10602424C211yesmaximal, characteristic, complemented
12C5×#557—24no—
13C5×#557—24no—
14C2×C5×C4⋊Q8—21no—
15C2×C5×C4⋊Q8C2211yescharacteristic
16C2×C5×#125—21no—
17C2×C5×#125—21no—
18C2×C5×#125C2211yescharacteristic
19C2×C5×#125C2211yescharacteristic
20C2×C4×C5×Q8—21no—
21C2×C4×C5×Q8C2211yescharacteristic
22C2×C4×C5×Q8C2211yescharacteristic
23C2×C4×C5×Q8C2211yescharacteristic
24C2×C4×C5×Q8C2211yescharacteristic
25C2×#4093—21no—
26C22×#688C2212yes—
27C2×#4020C2211yescharacteristic
28C2×#3967C2212yes—
29#65356—24no—
30#65316—24no—
31#65280—21no—
32#65280—21no—
33#65277—21no—
34C2×#3941C2212yes—
35C2×#3941C2212yes—
36C2×#3941C2212yes—
37#65228C2212yes—
38#65228C2212yes—
39C4×C4⋊Dic5C2211yescharacteristic
40C2×C4⋊#168C2211yescharacteristic
41C2×#3935C2211yescharacteristic
42#64749C2212yes—
43#64749C2212yes—
44#64749C2212yes—
45#64744C2212yes—
46#64744C2212yes—
47#25944—51nomaximal, Sylow
48C22×C5×Q8—21no—
49C22×C5×Q8D411yescharacteristic
50C22×C5×Q8C2311yescharacteristic
51C22×D20—21no—
52C22×D20—21no—
53C22×D20C2312yes—
54C5×C4⋊Q8—28no—
55C5×C4⋊Q8—22no—
56C5×C4⋊Q8—22no—
57C5×#125—28no—
58C5×#125—28no—
59C5×#125—22no—
60C5×#125—22no—
61C5×#125—22no—
62C5×#125D414yes—
63C4×C5×Q8—28no—
64C4×C5×Q8—22no—
65C4×C5×Q8—22no—
66C4×C5×Q8—22no—
67C4×C5×Q8—22no—
68C4×C5×Q8—22no—
69C4×C5×Q8—22no—
70C4×C5×Q8—22no—
71C4×C5×Q8D414yes—
72C2×C5×C4⋊C4—21no—
73C2×C5×C4⋊C4—21no—
74C2×C5×C4⋊C4—21no—
75C2×C5×C4⋊C4—21no—
76C2×C5×C4⋊C4—21no—
77C2×C5×C4⋊C4—21no—
78C2×C5×C4⋊C4—21no—
79C2×C5×C4⋊C4—21no—
80C2×C5×C4⋊C4D411yescharacteristic
81C2×C5×C4⋊C4D411yescharacteristic
82C2×C5×C4⋊C4D411yescharacteristic
83C2×C5×C4⋊C4C2311yescharacteristic
84C2×C5×C4⋊C4C2311yescharacteristic
85C2×C5×C4⋊C4C2311yescharacteristic
86C2×C42×C5—21no—
87C2×C42×C5—21no—
88C2×C42×C5—21no—
89C2×C42×C5C2311yescharacteristic
90C2×C42×C5C2311yescharacteristic
91C2×C42×C5C2311yescharacteristic
92#4093—44no—
93C2×#688—24no—
94C2×#688—24no—
95C2×#688—24no—
96C2×#685—22no—
97C2×#685—22no—
98C2×#685—22no—
99C2×#685—22no—
100C2×C4⋊Dic5C2312yes—
101C2×C4×Dic5—21no—
102C22×#168C2312yes—
103C22×#168C2312yes—
104#4020—22no—
105#4020—22no—
106#3967—24no—
107#3963—21no—
108#3941—24no—
109#3941—24no—
110#3941—24no—
111C4⋊#168—22no—
112#3935—22no—
113C2×#557D511yescharacteristic, complemented
114C2×#490—51no—
115C2×#460—51no—
116#2548—51no—
117#2025—52no—
118#2023—52no—
119#1989—51no—
120#1985—51no—
121#1880—52no—
122#1880—52no—
123#1763—51no—
124C23×D5—22no—
125C2×C5×Q8—24no—
126C2×C5×Q8—22no—
127C2×C5×Q8—22no—
128C2×C5×Q8—21no—
129C2×C5×Q8—22no—
130C2×C5×Q8—41no—
131C2×C5×Q8—42no—
132C2×C5×Q8—42no—
133C2×C5×Q8C2×D412yes—
134C2×C5×Q8#4114yes—
135C22×C4×C5—21no—
136C22×C4×C5—21no—
137C22×C4×C5—21no—
138C22×C4×C5—21no—
139C22×C4×C5C2×D411yescharacteristic
140C22×C4×C5C2×D411yescharacteristic
141C22×C4×C5C2×D411yescharacteristic
142C22×C4×C5C2×D411yescharacteristic
143C22×C4×C5C2×D411yescharacteristic
144C22×C4×C5C2×D411yescharacteristic
145C22×C4×C5C2411yescharacteristic
146C22×Dic5—22no—
147C2×D20—22no—
148C2×D20—22no—
149C2×D20—22no—
150C2×D20—22no—
151C2×D20—22no—
152C2×D20—22no—
153C2×D20—42no—
154C2×D20—42no—
155C2×D20—42no—
156C2×D20—42no—
157C2×D20—82no—
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—24no—
165C5×C4⋊C4—24no—
166C5×C4⋊C4—22no—
167C5×C4⋊C4—22no—
168C5×C4⋊C4—22no—
169C5×C4⋊C4—21no—
170C5×C4⋊C4—21no—
171C5×C4⋊C4—21no—
172C5×C4⋊C4C2×D412yes—
173C5×C4⋊C4C2×D412yes—
174C5×C4⋊C4C2×D412yes—
175C42×C5—24no—
176C42×C5—24no—
177C42×C5—24no—
178C42×C5—21no—
179C42×C5—21no—
180C42×C5—21no—
181C42×C5—21no—
182C42×C5C2×D412yes—
183C42×C5C2×D412yes—
184#688—48no—
185#685—44no—
186#685—44no—
187#685—82no—
188#685—82no—
189C4⋊Dic5—22no—
190C4⋊Dic5—22no—
191C4×Dic5—42no—
192C2×#168—22no—
193C2×#168—22no—
194C2×#168—22no—
195C2×#168—22no—
196C2×#168—22no—
197C2×#168—22no—
198C22×SD16—52no—
199#557—24no—
200#557—24no—
201C2×C4⋊Q8—21no—
202C2×C4⋊Q8C2×D511yescharacteristic
203C2×#127—51no—
204C2×#125—21no—
205C2×#125—21no—
206C2×#125C2×D511yescharacteristic
207C2×#125C2×D511yescharacteristic
208C2×#124—101no—
209C2×C4×Q8—21no—
210C2×C4×Q8C2×D511yescharacteristic
211C2×C4×Q8C2×D511yescharacteristic
212C2×C4×Q8C2×D511yescharacteristic
213C2×C4×Q8C2×D511yescharacteristic
214#490—104no—
215#460—104no—
216C2×C4⋊C8—51no—
217C2×#103—52no—
218C2×#102—52no—
219C2×#102—52no—
220C2×#102—52no—
221C2×#97—51no—
222#394—101no—
223#394—101no—
224#390—52no—
225#390—52no—
226#390—101no—
227C4×C4⋊C4—51no—
228#340—52no—
229#340—52no—
230#340—52no—
231#336—52no—
232#336—52no—
233C23×C5C22×D411yescharacteristic
234C22×D5—42no—
235C22×D5—42no—
236C22×D5—42no—
237C22×D5—42no—
238C22×D5—42no—
239C22×D5—42no—
240C22×D5—42no—
241C5×Q8—22no—
242C5×Q8—24no—
243C5×Q8—44no—
244C5×Q8—48no—
245C5×Q8#12414yescomplemented
246C2×C4×C5—22no—
247C2×C4×C5—21no—
248C2×C4×C5—24no—
249C2×C4×C5—21no—
250C2×C4×C5—21no—
251C2×C4×C5—21no—
252C2×C4×C5—22no—
253C2×C4×C5—21no—
254C2×C4×C5—21no—
255C2×C4×C5—22no—
256C2×C4×C5—21no—
257C2×C4×C5—22no—
258C2×C4×C5—22no—
259C2×C4×C5—21no—
260C2×C4×C5—21no—
261C2×C4×C5—22no—
262C2×C4×C5—22no—
263C2×C4×C5—41no—
264C2×C4×C5—41no—
265C2×C4×C5—41no—
266C2×C4×C5—41no—
267C2×C4×C5—42no—
268C2×C4×C5#12012yes—
269C2×C4×C5#12012yes—
270C2×C4×C5AGL(1,ℤ/8)12yes—
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—
278D20—42no—
279D20—42no—
280D20—42no—
281D20—42no—
282D20—84no—
283D20—84no—
284#168—42no—
285#168—42no—
286C22×Q8—21no—
287C22×Q8C22⋊D511yescharacteristic
288C22×Q8C22×D511yescharacteristic
289C22×D4—52no—
290C22×D4—101no—
291C22×D4—101no—
292C2×SD16—104no—
293C2×SD16—104no—
294C2×SD16—104no—
295C22×C8—52no—
296C22×C8—52no—
297C4⋊Q8—28no—
298C4⋊Q8—22no—
299C4⋊Q8—22no—
300#127—102no—
301#127—102no—
302#125—28no—
303#125—28no—
304#125—22no—
305#125—22no—
306#125—22no—
307#125C22⋊D514yes—
308#124—204no—
309C4×Q8—28no—
310C4×Q8—22no—
311C4×Q8—22no—
312C4×Q8—22no—
313C4×Q8—22no—
314C4×Q8—22no—
315C4×Q8—22no—
316C4×Q8—22no—
317C4×Q8C22⋊D514yes—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—52no—
327C2×C4⋊C4C22⋊D511yescharacteristic
328C2×C4⋊C4C22⋊D511yescharacteristic
329C2×C4⋊C4C22⋊D511yescharacteristic
330C2×C4⋊C4C22×D511yescharacteristic
331C2×C4⋊C4C22×D511yescharacteristic
332C2×C4⋊C4C22×D511yescharacteristic
333C2×C22⋊C4—102no—
334C2×C22⋊C4—102no—
335C2×C22⋊C4—102no—
336C2×C22⋊C4—102no—
337C2×C42—21no—
338C2×C42—21no—
339C2×C42—21no—
340C2×C42—101no—
341C2×C42C22×D511yescharacteristic
342C2×C42C22×D511yescharacteristic
343C2×C42C22×D511yescharacteristic
344C4⋊C8—102no—
345#103—104no—
346#102—104no—
347#102—104no—
348#102—104no—
349#97—102no—
350#95—101no—
351C22×C5C2×#12011yescharacteristic
352C22×C5#54812yes—
353C22×C5C2×AGL(1,ℤ/8)11yescharacteristic
354C22×C5#57511yescharacteristic
355C22×C5#57712yes—
356C2×D5—82no—
357C2×D5—82no—
358C2×D5—82no—
359C2×D5—82no—
360C2×D5—82no—
361C2×D5—82no—
362C2×D5—82no—
363C4×C5—24no—
364C4×C5—22no—
365C4×C5—22no—
366C4×C5—22no—
367C4×C5—41no—
368C4×C5—42no—
369C4×C5—42no—
370C4×C5—41no—
371C4×C5—41no—
372C4×C5—42no—
373C4×C5C2×#12411yescharacteristic
374C4×C5#53611yescharacteristic
375C4×C5#54812yes—
376Dic5—82no—
377C24—102no—
378C2×Q8—24no—
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8—21no—
382C2×Q8—22no—
383C2×Q8—41no—
384C2×Q8—42no—
385C2×Q8—42no—
386C2×Q8#71314yes—
387C2×Q8C2×C22⋊D512yes—
388C2×D4—102no—
389C2×D4—102no—
390C2×D4—102no—
391C2×D4—102no—
392C2×D4—102no—
393C2×D4—102no—
394C2×D4—202no—
395C2×D4—202no—
396C2×D4—202no—
397C2×D4—202no—
398C2×D4—402no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—102no—
404C22×C4C2×C22⋊D511yescharacteristic
405C22×C4C2×C22⋊D511yescharacteristic
406C22×C4C2×C22⋊D511yescharacteristic
407C22×C4C2×C22⋊D511yescharacteristic
408C22×C4C2×C22⋊D511yescharacteristic
409C22×C4C2×C22⋊D511yescharacteristic
410C22×C4C23×D511yescharacteristic
411SD16—208no—
412C2×C8—102no—
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—24no—
425C4⋊C4—24no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—102no—
433C4⋊C4—102no—
434C4⋊C4C2×C22⋊D512yes—
435C4⋊C4C2×C22⋊D512yes—
436C4⋊C4C2×C22⋊D512yes—
437C22⋊C4—204no—
438C22⋊C4—204no—
439C22⋊C4—402no—
440C22⋊C4—402no—
441C42—24no—
442C42—24no—
443C42—24no—
444C42—21no—
445C42—21no—
446C42—21no—
447C42—21no—
448C42—202no—
449C42C2×C22⋊D512yes—
450C42C2×C22⋊D512yes—
451C2×C5#243611yescharacteristic
452C2×C5#300611yescharacteristic
453C2×C5#301011yescharacteristic
454C2×C5#336812yes—
455C2×C5#337112yes—
456D5—162no—
457C23—202no—
458C23—202no—
459C23—202no—
460C23—202no—
461C23—202no—
462C23—202no—
463C23—202no—
464C23C22×C22⋊D511yescharacteristic
465Q8—22no—
466Q8—24no—
467Q8—44no—
468Q8—48no—
469Q8#409314yescomplemented
470D4—202no—
471D4—202no—
472D4—202no—
473D4—202no—
474D4—404no—
475D4—404no—
476C2×C4—24no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—22no—
490C2×C4—21no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—42no—
498C2×C4—202no—
499C2×C4—202no—
500C2×C4—202no—
501C2×C4#409512yes—
502C2×C4#409912yes—
503C2×C4#409912yes—
504C2×C4C2×#71312yes—
505C2×C4#414712yes—
506C2×C4C22×C22⋊D511yescharacteristic
507C2×C4C22×C22⋊D511yescharacteristic
508C8—202no—
509C8—202no—
510C5#2594411yescharacteristic, complemented, Sylow
511C22—402no—
512C22—402no—
513C22—402no—
514C22—402no—
515C22—402no—
516C22—402no—
517C22—402no—
518C22#6615112yes—
519C22C2×#409511yescharacteristic
520C22#6615611yescharacteristic
521C22#6617112yes—
522C22C2×#409911yescharacteristic
523C4—22no—
524C4—22no—
525C4—22no—
526C4—24no—
527C4—41no—
528C4—42no—
529C4—42no—
530C4—41no—
531C4—41no—
532C4—42no—
533C4—402no—
534C4C2×#409311yescharacteristic
535C4#6615011yescharacteristic
536C4#6615112yes—
537C2—802no—
538C2#1061542112yes—
539C2#1061542212yes—
540C2#1061579711yescharacteristic
541C2#1061581311yescharacteristic
542C2#1061584411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D5
commutator, cosocleC22×C4×C5C24
FrattiniC22×C4C23×D5
FittingC2×C5×#557C2
socleC23×C5C22×D4
prime radicalC2×#4020C22
hypercenterC2×#557D5
hypocenterC5#25944
LCSG ▷ C22×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#557

Complements

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

NH ≅ G/Ntype
1. C2×C5×#557C2semidirect
5. #10602585C2semidirect
6. #10602536C2semidirect
7. #10602531C2semidirect
8. #10602428C2semidirect
9. #10602426C2semidirect
10. #10602426C2semidirect
11. #10602424C2semidirect
113. C2×#557D5semidirect
245. C5×Q8#124semidirect
469. Q8#4093semidirect
510. C5#25944semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#5572
C2×#653562
#106142072
C2×#653162
#106025852
#106025362
#106025312
#106024282
#106024262
#106024242
#259445