Finite Group Databasebeta

← #12156306

Subgroups — #12156306

Order 1280.

1072 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#554C211yesmaximal, characteristic
2C2×#65358C211yesmaximal, characteristic, complemented
3#10614210C211yesmaximal, characteristic, complemented
4C2×#65308C211yesmaximal, characteristic
5#10602587C212yesmaximal, complemented
6#10602537C212yesmaximal, complemented
7#10602531C211yesmaximal, characteristic, complemented
8#10602428C211yesmaximal, characteristic, complemented
9#10602405C212yesmaximal
10#10602405C212yesmaximal
11#10602402C211yesmaximal, characteristic
12C5×#554—24no—
13C5×#554—24no—
14C2×C5×C4⋊Q8C2211yescharacteristic
15C2×C5×#125C2211yescharacteristic
16C2×C5×#125C2211yescharacteristic
17C2×C5×C22⋊Q8—21no—
18C2×C5×C22⋊Q8—21no—
19C2×C5×C22⋊Q8—21no—
20C2×C4×C5×Q8C2211yescharacteristic
21C2×C4×C5×D4C2211yescharacteristic
22C2×C4×C5×D4C2211yescharacteristic
23C2×C4×C5×D4C2211yescharacteristic
24C22×C5×C4⋊C4—21no—
25C22×#687C2212yes—
26C2×#4015C2211yescharacteristic
27#65519—21no—
28C2×#3964C2212yes—
29#65358—24no—
30#65308—24no—
31#65275—21no—
32#65275—21no—
33#65266—21no—
34C2×#3942C2212yes—
35C2×#3942C2212yes—
36C2×#3942C2212yes—
37C4×C4⋊Dic5C2211yescharacteristic
38#65218C2212yes—
39#65218C2212yes—
40C2×C4⋊#168C2211yescharacteristic
41C2×#3935C2211yescharacteristic
42#64749C2212yes—
43#64749C2212yes—
44#64749C2212yes—
45#64744C2212yes—
46#64744C2212yes—
47#25945—51nomaximal, Sylow
48C22×C5×Q8—21no—
49C22×C5×Q8D411yescharacteristic
50C22×C5×D4C2311yescharacteristic
51C23×C4×C5—21no—
52C23×C4×C5—21no—
53C23×C4×C5—21no—
54C22×Dic10C2312yes—
55C5×C4⋊Q8—22no—
56C5×C4⋊Q8—22no—
57C5×#125—22no—
58C5×#125—22no—
59C5×#125—22no—
60C5×#125D414yes—
61C5×C22⋊Q8—28no—
62C5×C22⋊Q8—28no—
63C5×C22⋊Q8—28no—
64C4×C5×Q8—22no—
65C4×C5×Q8—22no—
66C4×C5×D4—22no—
67C4×C5×D4—22no—
68C4×C5×D4—22no—
69C4×C5×D4—22no—
70C4×C5×D4—22no—
71C4×C5×D4D414yes—
72C2×C5×C4⋊C4—28no—
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⋊C4D411yescharacteristic
79C2×C5×C4⋊C4D411yescharacteristic
80C2×C5×C4⋊C4D411yescharacteristic
81C2×C5×C4⋊C4C2311yescharacteristic
82C2×C5×C4⋊C4C2311yescharacteristic
83C2×C5×C4⋊C4C2311yescharacteristic
84C2×C5×C22⋊C4—21no—
85C2×C5×C22⋊C4—21no—
86C2×C5×C22⋊C4—21no—
87C2×C42×C5C2311yescharacteristic
88C2×C42×C5C2311yescharacteristic
89C2×C42×C5C2311yescharacteristic
90C2×#687—24no—
91C2×#687—28no—
92C2×C4⋊Dic5—21no—
93C2×C4⋊Dic5—21no—
94C2×C4⋊Dic5C2312yes—
95C2×C4×Dic5—21no—
96C22×#168C2312yes—
97C22×#168C2312yes—
98#4015—22no—
99#4015—22no—
100#3964—24no—
101#3963—22no—
102#3963—22no—
103#3963—22no—
104#3963—22no—
105#3963—21no—
106#3942—24no—
107#3942—24no—
108#3942—24no—
109C4⋊#168—22no—
110#3935—22no—
111C2×#554D511yescharacteristic
112C2×#491—51no—
113C2×#461—51no—
114#2568—51no—
115#2026—52no—
116#2024—52no—
117#1989—51no—
118#1985—51no—
119#1877—52no—
120#1877—52no—
121#1762—51no—
122C24×C5—21no—
123C2×C5×Q8—24no—
124C2×C5×Q8—22no—
125C2×C5×Q8—22no—
126C2×C5×Q8—41no—
127C2×C5×Q8—82no—
128C2×C5×D4—21no—
129C2×C5×D4—22no—
130C2×C5×D4C2×D412yes—
131C2×C5×D4#4114yes—
132C22×C4×C5—24no—
133C22×C4×C5—21no—
134C22×C4×C5—21no—
135C22×C4×C5—21no—
136C22×C4×C5—24no—
137C22×C4×C5—24no—
138C22×C4×C5—42no—
139C22×C4×C5—42no—
140C22×C4×C5—82no—
141C22×C4×C5—82no—
142C22×C4×C5C2×D411yescharacteristic
143C22×C4×C5C2×D411yescharacteristic
144C22×C4×C5C2×D411yescharacteristic
145C22×C4×C5C2×D411yescharacteristic
146C22×C4×C5C2×D411yescharacteristic
147C22×C4×C5C2×D411yescharacteristic
148C22×C4×C5C2411yescharacteristic
149C22×Dic5—22no—
150C22×Dic5—22no—
151C2×Dic10—22no—
152C2×Dic10—22no—
153C2×Dic10—24no—
154C2×Dic10—24no—
155C5×C4⋊C4—24no—
156C5×C4⋊C4—24no—
157C5×C4⋊C4—24no—
158C5×C4⋊C4—22no—
159C5×C4⋊C4—24no—
160C5×C4⋊C4—24no—
161C5×C4⋊C4—22no—
162C5×C4⋊C4—22no—
163C5×C4⋊C4—21no—
164C5×C4⋊C4—21no—
165C5×C4⋊C4—21no—
166C5×C4⋊C4C2×D412yes—
167C5×C4⋊C4C2×D412yes—
168C5×C4⋊C4C2×D412yes—
169C5×C22⋊C4—24no—
170C5×C22⋊C4—24no—
171C5×C22⋊C4—24no—
172C42×C5—21no—
173C42×C5—21no—
174C42×C5—21no—
175C42×C5—21no—
176C42×C5C2×D412yes—
177C42×C5C2×D412yes—
178#687—48no—
179C4⋊Dic5—22no—
180C4⋊Dic5—22no—
181C4⋊Dic5—42no—
182C4⋊Dic5—42no—
183C4×Dic5—42no—
184C2×#168—22no—
185C2×#168—22no—
186C2×#168—22no—
187C2×#168—24no—
188C2×#168—22no—
189C22×SD16—52no—
190#554—24no—
191#554—24no—
192C2×C4⋊Q8—51no—
193C2×C4⋊Q8C2×D511yescharacteristic
194C2×#125C2×D511yescharacteristic
195C2×#125C2×D511yescharacteristic
196C2×C22⋊Q8—21no—
197C2×C22⋊Q8—21no—
198C2×C22⋊Q8—21no—
199C2×C4×Q8C2×D511yescharacteristic
200C2×C4×D4C2×D511yescharacteristic
201C2×C4×D4C2×D511yescharacteristic
202C2×C4×D4C2×D511yescharacteristic
203C22×C4⋊C4—21no—
204#491—104no—
205#461—104no—
206C2×C4⋊C8—51no—
207C2×#103—52no—
208C2×#103—52no—
209C2×#103—52no—
210C2×#102—52no—
211C2×#97—51no—
212#400—101no—
213#400—101no—
214#391—52no—
215#391—52no—
216#391—101no—
217#382—101no—
218C4×C4⋊C4—51no—
219#340—52no—
220#340—52no—
221#340—52no—
222#336—52no—
223#336—52no—
224C23×C5—22no—
225C23×C5—24no—
226C23×C5—41no—
227C23×C5—41no—
228C23×C5—42no—
229C23×C5C22×D411yescharacteristic
230C5×Q8—44no—
231C5×Q8—84no—
232C5×D4—22no—
233C5×D4—24no—
234C5×D4#12414yes—
235C2×C4×C5—21no—
236C2×C4×C5—21no—
237C2×C4×C5—21no—
238C2×C4×C5—21no—
239C2×C4×C5—22no—
240C2×C4×C5—21no—
241C2×C4×C5—21no—
242C2×C4×C5—21no—
243C2×C4×C5—22no—
244C2×C4×C5—22no—
245C2×C4×C5—22no—
246C2×C4×C5—21no—
247C2×C4×C5—21no—
248C2×C4×C5—22no—
249C2×C4×C5—22no—
250C2×C4×C5—48no—
251C2×C4×C5—41no—
252C2×C4×C5—41no—
253C2×C4×C5—42no—
254C2×C4×C5—42no—
255C2×C4×C5—84no—
256C2×C4×C5—84no—
257C2×C4×C5#12012yes—
258C2×C4×C5#12012yes—
259C2×C4×C5#13712yes—
260C2×C4×C5C22×D411yescharacteristic
261C2×C4×C5C22×D411yescharacteristic
262C2×C4×C5C2×#4112yes—
263C2×C4×C52−1+412yes—
264C2×Dic5—42no—
265C2×Dic5—42no—
266C2×Dic5—44no—
267C2×Dic5—42no—
268C2×Dic5—42no—
269Dic10—42no—
270Dic10—42no—
271Dic10—44no—
272#168—42no—
273#168—42no—
274C22×Q8—21no—
275C22×Q8—52no—
276C22×Q8C22⋊D511yescharacteristic
277C22×D4C22×D511yescharacteristic
278C23×C4—21no—
279C23×C4—21no—
280C23×C4—21no—
281C2×SD16—108no—
282C2×SD16—104no—
283C22×C8—52no—
284C22×C8—52no—
285C4⋊Q8—22no—
286C4⋊Q8—22no—
287C4⋊Q8—102no—
288C4⋊Q8—102no—
289#125—22no—
290#125—22no—
291#125—22no—
292#125C22⋊D514yes—
293C22⋊Q8—28no—
294C22⋊Q8—28no—
295C22⋊Q8—28no—
296C4×Q8—22no—
297C4×Q8—22no—
298C4×D4—22no—
299C4×D4—22no—
300C4×D4—22no—
301C4×D4—22no—
302C4×D4—22no—
303C4×D4C22⋊D514yes—
304C2×C4⋊C4—28no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—52no—
311C2×C4⋊C4—101no—
312C2×C4⋊C4—101no—
313C2×C4⋊C4C22⋊D511yescharacteristic
314C2×C4⋊C4C22⋊D511yescharacteristic
315C2×C4⋊C4C22⋊D511yescharacteristic
316C2×C4⋊C4C22×D511yescharacteristic
317C2×C4⋊C4C22×D511yescharacteristic
318C2×C4⋊C4C22×D511yescharacteristic
319C2×C22⋊C4—21no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—21no—
322C2×C42—101no—
323C2×C42C22×D511yescharacteristic
324C2×C42C22×D511yescharacteristic
325C2×C42C22×D511yescharacteristic
326C4⋊C8—102no—
327#103—104no—
328#103—104no—
329#103—104no—
330#102—104no—
331#97—102no—
332#95—101no—
333#95—102no—
334#95—102no—
335#95—102no—
336#95—102no—
337C22×C5—24no—
338C22×C5—44no—
339C22×C5—44no—
340C22×C5—42no—
341C22×C5—42no—
342C22×C5C2×#12011yescharacteristic
343C22×C5#54812yes—
344C22×C5C2×#13711yescharacteristic
345C22×C5#57511yescharacteristic
346C22×C5#57712yes—
347C4×C5—22no—
348C4×C5—22no—
349C4×C5—22no—
350C4×C5—41no—
351C4×C5—42no—
352C4×C5—41no—
353C4×C5—41no—
354C4×C5—42no—
355C4×C5—42no—
356C4×C5C2×#12411yescharacteristic
357C4×C5#53611yescharacteristic
358C4×C5#54812yes—
359Dic5—82no—
360Dic5—82no—
361C24—21no—
362C2×Q8—24no—
363C2×Q8—22no—
364C2×Q8—22no—
365C2×Q8—41no—
366C2×Q8—82no—
367C2×Q8—104no—
368C2×Q8—104no—
369C2×Q8—102no—
370C2×Q8—102no—
371C2×D4—22no—
372C2×D4—21no—
373C2×D4#71314yes—
374C2×D4C2×C22⋊D512yes—
375C22×C4—24no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—82no—
384C22×C4—82no—
385C22×C4—102no—
386C22×C4—102no—
387C22×C4C2×C22⋊D511yescharacteristic
388C22×C4C2×C22⋊D511yescharacteristic
389C22×C4C2×C22⋊D511yescharacteristic
390C22×C4C2×C22⋊D511yescharacteristic
391C22×C4C2×C22⋊D511yescharacteristic
392C22×C4C2×C22⋊D511yescharacteristic
393C22×C4C23×D511yescharacteristic
394SD16—208no—
395C2×C8—102no—
396C2×C8—104no—
397C2×C8—102no—
398C2×C8—102no—
399C2×C8—102no—
400C4⋊C4—24no—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—22no—
404C4⋊C4—24no—
405C4⋊C4—24no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—21no—
409C4⋊C4—21no—
410C4⋊C4—21no—
411C4⋊C4—102no—
412C4⋊C4—102no—
413C4⋊C4—202no—
414C4⋊C4—202no—
415C4⋊C4C2×C22⋊D512yes—
416C4⋊C4C2×C22⋊D512yes—
417C4⋊C4C2×C22⋊D512yes—
418C22⋊C4—24no—
419C22⋊C4—24no—
420C22⋊C4—24no—
421C42—21no—
422C42—21no—
423C42—21no—
424C42—21no—
425C42—202no—
426C42C2×C22⋊D512yes—
427C42C2×C22⋊D512yes—
428C2×C5—44no—
429C2×C5#243611yescharacteristic
430C2×C5#300711yescharacteristic
431C2×C5#300911yescharacteristic
432C2×C5#337212yes—
433C2×C5#337612yes—
434C23—24no—
435C23—22no—
436C23—41no—
437C23—41no—
438C23—42no—
439C23C22×C22⋊D511yescharacteristic
440Q8—44no—
441Q8—84no—
442Q8—202no—
443Q8—202no—
444Q8—204no—
445D4—22no—
446D4—24no—
447D4#409314yes—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—22no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—48no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—84no—
469C2×C4—84no—
470C2×C4—202no—
471C2×C4—204no—
472C2×C4—202no—
473C2×C4—202no—
474C2×C4—202no—
475C2×C4#409712yes—
476C2×C4#409912yes—
477C2×C4#409912yes—
478C2×C4C2×#71312yes—
479C2×C4#414712yes—
480C2×C4C22×C22⋊D511yescharacteristic
481C2×C4C22×C22⋊D511yescharacteristic
482C8—202no—
483C8—202no—
484C5#2594511yescharacteristic, complemented, Sylow
485C22—24no—
486C22—42no—
487C22—42no—
488C22—44no—
489C22—44no—
490C22#6615112yes—
491C22#6615611yescharacteristic
492C22C2×#409711yescharacteristic
493C22#6617012yes—
494C22C2×#409911yescharacteristic
495C4—22no—
496C4—22no—
497C4—22no—
498C4—41no—
499C4—42no—
500C4—41no—
501C4—41no—
502C4—42no—
503C4—42no—
504C4—402no—
505C4—402no—
506C4C2×#409311yescharacteristic
507C4#6615011yescharacteristic
508C4#6615112yes—
509C2—44no—
510C2#1061537512yes—
511C2#1061537612yes—
512C2#1061579711yescharacteristic
513C2#1061581511yescharacteristic
514C2#1061583611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D5
commutator, cosocleC22×C4×C5C24
FrattiniC22×C4C23×D5
FittingC2×C5×#554C2
socleC23×C5C22×D4
prime radicalC22×C5×D4C23
hypercenterC2×#554D5
hypocenterC5#25945
LCSG ▷ C22×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#554

Complements

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

NH ≅ G/Ntype
2. C2×#65358C2semidirect
3. #10614210C2semidirect
5. #10602587C2semidirect
6. #10602537C2semidirect
7. #10602531C2semidirect
8. #10602428C2semidirect
484. C5#25945semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#5542
C2×#653582
#106142102
C2×#653082
#106025872
#106025372
#106025312
#106024282
#106024052
#106024022
#259455