Finite Group Databasebeta

← #12133253

Subgroups — #12133253

Order 1280.

1134 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×#554C211yesmaximal, characteristic
2C2×#65380C211yesmaximal, characteristic, complemented
3C2×C4×#689C211yesmaximal, characteristic, complemented
4#10614107C211yesmaximal, characteristic
5#10602580C211yesmaximal, characteristic, complemented
6#10602493C212yesmaximal, complemented
7#10602488C212yesmaximal, complemented
8#10602419C211yesmaximal, characteristic, complemented
9#10602407C211yesmaximal, characteristic
10#10602352C212yesmaximal, complemented
11#10602348C212yesmaximal, complemented
12C5×#554—24no—
13C5×#554—24no—
14C2×C5×C4⋊Q8C2211yescharacteristic
15C2×C5×#125—21no—
16C2×C5×C22⋊Q8—22no—
17C2×C5×C22⋊Q8C2212yes—
18C2×C4×C5×Q8C2211yescharacteristic
19C2×C4×C5×D4—21no—
20C2×C4×C5×D4C2211yescharacteristic
21C22×C5×C4⋊C4C2212yes—
22C22×#689C2211yescharacteristic
23C2×#4070C2212yes—
24C2×C4×Dic10C2211yescharacteristic
25C2×#3967C2211yescharacteristic
26C2×#3967C2212yes—
27#65510—21no—
28#65501C2212yes—
29C2×C22⋊#168C2212yes—
30C2×C22⋊#168C2212yes—
31#65380—24no—
32C4×#689—24no—
33#65262—21no—
34C2×#3942C2211yescharacteristic
35C2×#3942C2212yes—
36C2×#3939C2211yescharacteristic
37C2×#3939C2212yes—
38#65237—22no—
39#65224C2211yescharacteristic
40C2×C4⋊#168C2211yescharacteristic
41C2×C4×#168C2211yescharacteristic
42#64749C2212yes—
43#64749C2212yes—
44#64749C2212yes—
45#24122—51nomaximal, Sylow
46C22×C5×Q8—21no—
47C22×C5×Q8C2311yescharacteristic
48C22×C5×D4D411yescharacteristic
49C23×C4×C5—22no—
50C23×C4×C5C2312yes—
51C22×Dic10C2311yescharacteristic
52C5×C4⋊Q8—22no—
53C5×C4⋊Q8D414yes—
54C5×#125—28no—
55C5×C22⋊Q8—216no—
56C5×C22⋊Q8—24no—
57C5×C22⋊Q8—24no—
58C4×C5×Q8—22no—
59C4×C5×Q8D414yes—
60C4×C5×D4—28no—
61C4×C5×D4—22no—
62C4×C5×D4—22no—
63C2×C5×C4⋊C4—22no—
64C2×C5×C4⋊C4—21no—
65C2×C5×C4⋊C4—21no—
66C2×C5×C4⋊C4—22no—
67C2×C5×C4⋊C4—24no—
68C2×C5×C4⋊C4—24no—
69C2×C5×C4⋊C4D411yescharacteristic
70C2×C5×C4⋊C4C2311yescharacteristic
71C2×C5×C4⋊C4C2312yes—
72C2×C5×C22⋊C4—22no—
73C2×C5×C22⋊C4D412yes—
74C2×C42×C5—21no—
75C2×C42×C5C2311yescharacteristic
76C2×C22⋊Dic5—22no—
77C2×C22⋊Dic5—22no—
78C2×#689—22no—
79C2×#689—22no—
80C2×#689—22no—
81C2×C4⋊Dic5—21no—
82C2×C4⋊Dic5—21no—
83C2×C4⋊Dic5C2311yescharacteristic
84C2×C4⋊Dic5C2312yes—
85#4070—24no—
86#4070—42no—
87C2×#683—21no—
88C2×#683—22no—
89C2×C4×Dic5—21no—
90C22×#168C2311yescharacteristic
91C22×#168C2312yes—
92C22×#168C2311yescharacteristic
93C4×Dic10—22no—
94C4×Dic10—22no—
95#3967—22no—
96#3967—42no—
97#3963—22no—
98C22⋊#168—24no—
99C22⋊#168—42no—
100#3942—22no—
101#3942—24no—
102#3939—22no—
103#3939—24no—
104C4⋊#168—22no—
105C4×#168—22no—
106C2×#554D511yescharacteristic
107C2×#477—51no—
108C2×C4×Q16—51no—
109#2538—51no—
110#2036—52no—
111#2001—52no—
112#1990—51no—
113#1954—51no—
114#1872—52no—
115#1852—52no—
116#1803—51no—
117C24×C5C2×D412yes—
118C2×C5×Q8—24no—
119C2×C5×Q8—21no—
120C2×C5×Q8—21no—
121C2×C5×Q8—21no—
122C2×C5×Q8—21no—
123C2×C5×Q8—21no—
124C2×C5×Q8—81no—
125C2×C5×Q8—81no—
126C2×C5×Q8C2×D412yes—
127C2×C5×D4—22no—
128C2×C5×D4—22no—
129C2×C5×D4Q1614yescomplemented
130C22×C4×C5—21no—
131C22×C4×C5—21no—
132C22×C4×C5—22no—
133C22×C4×C5—28no—
134C22×C4×C5—22no—
135C22×C4×C5—22no—
136C22×C4×C5—22no—
137C22×C4×C5—22no—
138C22×C4×C5—22no—
139C22×C4×C5—22no—
140C22×C4×C5—82no—
141C22×C4×C5—82no—
142C22×C4×C5C2×D412yes—
143C22×C4×C5C2×D411yescharacteristic
144C22×C4×C5C2×D411yescharacteristic
145C22×C4×C5C2411yescharacteristic
146C22×Dic5—21no—
147C22×Dic5—21no—
148C22×Dic5—22no—
149C2×Dic10—21no—
150C2×Dic10—21no—
151C2×Dic10—21no—
152C2×Dic10—21no—
153C2×Dic10—21no—
154C2×Dic10#4112yes—
155C5×C4⋊C4—28no—
156C5×C4⋊C4—28no—
157C5×C4⋊C4—24no—
158C5×C4⋊C4—24no—
159C5×C4⋊C4—22no—
160C5×C4⋊C4—21no—
161C5×C4⋊C4—22no—
162C5×C4⋊C4C2×D412yes—
163C5×C4⋊C4C2×D414yes—
164C5×C22⋊C4—28no—
165C5×C22⋊C4—24no—
166C42×C5—24no—
167C42×C5C2×D412yes—
168C42×C5C2×D412yes—
169C22⋊Dic5—44no—
170C22⋊Dic5—82no—
171#689—28no—
172C4⋊Dic5—21no—
173C4⋊Dic5—21no—
174C4⋊Dic5—22no—
175C4⋊Dic5—22no—
176C4⋊Dic5—42no—
177C4⋊Dic5—81no—
178#683—44no—
179#683—42no—
180C4×Dic5—42no—
181C2×#168—21no—
182C2×#168—21no—
183C2×#168—21no—
184C2×#168—22no—
185C2×#168—22no—
186C2×#168—21no—
187C2×#168—22no—
188C2×#168—21no—
189C2×#168#4112yes—
190C22×Q16—51no—
191#554—24no—
192#554—24no—
193C2×C4⋊Q8C2×D511yescharacteristic
194C2×#125—21no—
195C2×C22⋊Q8—22no—
196C2×C22⋊Q8—52no—
197C2×C22⋊Q8C2×D512yes—
198C2×C4×Q8—51no—
199C2×C4×Q8C2×D511yescharacteristic
200C2×C4×D4—21no—
201C2×C4×D4C2×D511yescharacteristic
202C22×C4⋊C4C2×D512yes—
203#477—104no—
204C4×Q16—104no—
205C2×#107—52no—
206C2×#107—51no—
207C2×C4⋊C8—51no—
208C2×#103—51no—
209C2×#103—52no—
210C2×#103—51no—
211C2×#103—52no—
212C2×C22⋊C8—52no—
213C2×C22⋊C8—52no—
214C2×C4×C8—51no—
215#396—102no—
216#390—101no—
217#389—101no—
218#384—51no—
219#380—52no—
220#340—52no—
221#340—52no—
222#340—52no—
223C23×C5—22no—
224C23×C5—22no—
225C23×C5—22no—
226C23×C5—22no—
227C23×C5—22no—
228C23×C5—22no—
229C23×C5C2×Q1614yes—
230C23×C5C22×D411yescharacteristic
231C5×Q8—22no—
232C5×Q8—22no—
233C5×Q8—41no—
234C5×Q8—41no—
235C5×Q8—84no—
236C5×D4—28no—
237C2×C4×C5—21no—
238C2×C4×C5—21no—
239C2×C4×C5—22no—
240C2×C4×C5—22no—
241C2×C4×C5—22no—
242C2×C4×C5—24no—
243C2×C4×C5—22no—
244C2×C4×C5—22no—
245C2×C4×C5—41no—
246C2×C4×C5—41no—
247C2×C4×C5—41no—
248C2×C4×C5—42no—
249C2×C4×C5—41no—
250C2×C4×C5—42no—
251C2×C4×C5—42no—
252C2×C4×C5—42no—
253C2×C4×C5—42no—
254C2×C4×C5—42no—
255C2×C4×C5—88no—
256C2×C4×C5#12012yes—
257C2×C4×C5#12012yes—
258C2×C4×C5C2×Q1612yes—
259C2×C4×C5#13712yes—
260C2×C4×C5C22×D411yescharacteristic
261C2×C4×C5C22×D411yescharacteristic
262C2×C4×C5C2×#4111yescharacteristic
263C2×C4×C5C2×#4111yescharacteristic
264C2×C4×C5H(5,𝔽2)11yescharacteristic
265C2×C4×C52−1+411yescharacteristic
266C2×Dic5—22no—
267C2×Dic5—41no—
268C2×Dic5—41no—
269C2×Dic5—41no—
270C2×Dic5—42no—
271C2×Dic5—42no—
272C2×Dic5—42no—
273C2×Dic5—41no—
274C2×Dic5—41no—
275Dic10—22no—
276Dic10—22no—
277Dic10—41no—
278Dic10—41no—
279#168—22no—
280#168—41no—
281#168—42no—
282C22×Q8—21no—
283C22×Q8—51no—
284C22×Q8C22×D511yescharacteristic
285C22×D4C22⋊D511yescharacteristic
286C23×C4—22no—
287C23×C4C22×D512yes—
288C2×Q16—102no—
289C2×Q16—102no—
290C2×Q16—102no—
291C22×C8—51no—
292C22×C8—51no—
293C22×C8—52no—
294C4⋊Q8—22no—
295C4⋊Q8C22⋊D514yes—
296#125—28no—
297C22⋊Q8—216no—
298C22⋊Q8—24no—
299C22⋊Q8—24no—
300C22⋊Q8—104no—
301C22⋊Q8—202no—
302C4×Q8—22no—
303C4×Q8—102no—
304C4×Q8—102no—
305C4×Q8C22⋊D514yes—
306C4×D4—28no—
307C4×D4—22no—
308C4×D4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—51no—
316C2×C4⋊C4—52no—
317C2×C4⋊C4—101no—
318C2×C4⋊C4—102no—
319C2×C4⋊C4—101no—
320C2×C4⋊C4—101no—
321C2×C4⋊C4C22⋊D511yescharacteristic
322C2×C4⋊C4C22×D511yescharacteristic
323C2×C4⋊C4C22×D512yes—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—102no—
326C2×C22⋊C4—102no—
327C2×C22⋊C4C22⋊D512yes—
328C2×C42—21no—
329C2×C42—101no—
330C2×C42C22×D511yescharacteristic
331#107—102no—
332#107—104no—
333C4⋊C8—102no—
334#103—102no—
335#103—102no—
336#103—104no—
337#103—202no—
338C22⋊C8—104no—
339C22⋊C8—202no—
340C4×C8—102no—
341#95—102no—
342C22×C5—24no—
343C22×C5—24no—
344C22×C5—24no—
345C22×C5—42no—
346C22×C5—42no—
347C22×C5—42no—
348C22×C5—42no—
349C22×C5C2×#12011yescharacteristic
350C22×C5#54611yescharacteristic
351C22×C5#54711yescharacteristic
352C22×C5C22×Q1611yescharacteristic
353C22×C5C2×#13711yescharacteristic
354C22×C5#57611yescharacteristic
355C22×C5#57711yescharacteristic
356C4×C5—22no—
357C4×C5—42no—
358C4×C5—42no—
359C4×C5—42no—
360C4×C5—44no—
361C4×C5#45114yes—
362C4×C5#52511yescharacteristic
363C4×C5#53711yescharacteristic
364C4×C5#54811yescharacteristic
365C4×C5#55011yescharacteristic
366Dic5—42no—
367Dic5—82no—
368Dic5—81no—
369C24C2×C22⋊D512yes—
370C2×Q8—24no—
371C2×Q8—21no—
372C2×Q8—21no—
373C2×Q8—21no—
374C2×Q8—21no—
375C2×Q8—21no—
376C2×Q8—52no—
377C2×Q8—81no—
378C2×Q8—81no—
379C2×Q8—101no—
380C2×Q8—101no—
381C2×Q8—101no—
382C2×Q8—101no—
383C2×Q8—101no—
384C2×Q8C2×C22⋊D512yes—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4#68914yescomplemented
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—28no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—82no—
399C22×C4—82no—
400C22×C4—101no—
401C22×C4—101no—
402C22×C4—102no—
403C22×C4C2×C22⋊D512yes—
404C22×C4C2×C22⋊D511yescharacteristic
405C22×C4C2×C22⋊D511yescharacteristic
406C22×C4C23×D511yescharacteristic
407Q16—108no—
408C2×C8—52no—
409C2×C8—101no—
410C2×C8—101no—
411C2×C8—102no—
412C2×C8—101no—
413C2×C8—102no—
414C2×C8—101no—
415C2×C8—101no—
416C2×C8—102no—
417C4⋊C4—28no—
418C4⋊C4—28no—
419C4⋊C4—24no—
420C4⋊C4—24no—
421C4⋊C4—22no—
422C4⋊C4—21no—
423C4⋊C4—22no—
424C4⋊C4—102no—
425C4⋊C4—102no—
426C4⋊C4—101no—
427C4⋊C4—101no—
428C4⋊C4—204no—
429C4⋊C4—202no—
430C4⋊C4—202no—
431C4⋊C4—401no—
432C4⋊C4C2×C22⋊D512yes—
433C4⋊C4C2×C22⋊D514yes—
434C22⋊C4—28no—
435C22⋊C4—24no—
436C22⋊C4—204no—
437C22⋊C4—402no—
438C42—24no—
439C42—202no—
440C42C2×C22⋊D512yes—
441C42C2×C22⋊D512yes—
442C2×C5—44no—
443C2×C5#240711yescharacteristic
444C2×C5C2×#45111yescharacteristic
445C2×C5#301211yescharacteristic
446C2×C5#329911yescharacteristic
447C2×C5#333311yescharacteristic
448C2×C5#333811yescharacteristic
449C2×C5#335611yescharacteristic
450C23—22no—
451C23—22no—
452C23—22no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23C2×#68914yes—
457C23C22×C22⋊D511yescharacteristic
458Q8—22no—
459Q8—22no—
460Q8—41no—
461Q8—41no—
462Q8—84no—
463Q8—102no—
464Q8—102no—
465Q8—201no—
466Q8—201no—
467D4—28no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—24no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—42no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—88no—
487C2×C4—102no—
488C2×C4—201no—
489C2×C4—201no—
490C2×C4—202no—
491C2×C4—202no—
492C2×C4—201no—
493C2×C4—202no—
494C2×C4—201no—
495C2×C4—201no—
496C2×C4#408812yes—
497C2×C4C2×#68912yes—
498C2×C4#409912yes—
499C2×C4#409912yes—
500C2×C4D5×#4111yescharacteristic
501C2×C4D5×#4111yescharacteristic
502C2×C4#414911yescharacteristic
503C2×C4#415011yescharacteristic
504C2×C4C22×C22⋊D511yescharacteristic
505C2×C4C22×C22⋊D511yescharacteristic
506C8—102no—
507C8—201no—
508C8—202no—
509C5#2412211yescharacteristic, complemented, Sylow
510C22—24no—
511C22—24no—
512C22—24no—
513C22—42no—
514C22—42no—
515C22—42no—
516C22—42no—
517C22C2×#408811yescharacteristic
518C22C22×#68911yescharacteristic
519C22#6615811yescharacteristic
520C22#6616311yescharacteristic
521C22#6616911yescharacteristic
522C22#6617011yescharacteristic
523C22C2×#409911yescharacteristic
524C4—22no—
525C4—42no—
526C4—42no—
527C4—42no—
528C4—44no—
529C4—202no—
530C4—401no—
531C4—402no—
532C4#6551714yes—
533C4#6616211yescharacteristic
534C4#6616511yescharacteristic
535C4#6616711yescharacteristic
536C4#6616811yescharacteristic
537C2—44no—
538C2#1061535411yescharacteristic
539C2#1061535711yescharacteristic
540C2#1061537111yescharacteristic
541C2#1061537311yescharacteristic
542C2C2×#6551711yescharacteristic
543C2#1061582811yescharacteristic
544C2#1061583311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D5
commutator, cosocleC22×C4×C5C24
FrattiniC22×C4C23×D5
FittingC2×C5×#554C2
socleC23×C5C22×D4
prime radicalC22×C5×D4D4
hypercenterC2×#554D5
hypocenterC5#24122
LCSG ▷ C22×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#554

Complements

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

NH ≅ G/Ntype
2. C2×#65380C2semidirect
3. C2×C4×#689C2semidirect
5. #10602580C2semidirect
6. #10602493C2semidirect
7. #10602488C2semidirect
8. #10602419C2semidirect
10. #10602352C2semidirect
11. #10602348C2semidirect
129. C2×C5×D4Q16semidirect
387. C2×D4#689semidirect
509. C5#24122semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#5542
C2×#653802
C2×C4×#6892
#106141072
#106025802
#106024932
#106024882
#106024192
#106024072
#106023522
#106023482
#241225