Finite Group Databasebeta

← #12021378

Subgroups — #12021378

Order 1280.

1442 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C5×C4⋊C4C211yesmaximal, characteristic
2C2×#65270C212yesmaximal
3C2×#65225C211yesmaximal, characteristic
4C2×C4×C4⋊Dic5C211yesmaximal, characteristic
5#10602358C212yesmaximal
6#10602328C211yesmaximal, characteristic
7#10602327C211yesmaximal, characteristic
8#10602327C211yesmaximal, characteristic
9#10602327C211yesmaximal, characteristic
10#10600212C212yesmaximal
11#10600200C212yesmaximal
12C22×C5×C4⋊C4C2211yescharacteristic
13C22×C5×C4⋊C4C2211yescharacteristic
14C22×C42×C5C2211yescharacteristic
15C22×C42×C5C2211yescharacteristic
16C22×C42×C5C2211yescharacteristic
17C22×C4⋊Dic5C2212yes—
18C22×C4⋊Dic5C2212yes—
19C22×C4⋊Dic5C2212yes—
20C22×C4×Dic5C2212yes—
21C4×C5×C4⋊C4C418yescomplemented
22C2×C5×#95C2211yescharacteristic
23C2×C5×#95C2211yescharacteristic
24C2×#3963C2212yes—
25C2×#3963C2212yes—
26C2×#3963C2212yes—
27C2×#3963C2212yes—
28C2×#3963C2212yes—
29C2×#3963C2212yes—
30C2×#3963C2211yescharacteristic
31C2×#3963C2211yescharacteristic
32C2×#3963C2211yescharacteristic
33C2×#3963C2212yes—
34C2×#3963C2212yes—
35C2×#3963C2211yescharacteristic
36#65270—28no—
37#65225—24no—
38C4×C4⋊Dic5—24no—
39#13889—51nomaximal, Sylow
40C23×C4×C5C2311yescharacteristic
41C23×C4×C5C2311yescharacteristic
42C23×C4×C5C2311yescharacteristic
43C23×C4×C5C2311yescharacteristic
44C23×C4×C5C2311yescharacteristic
45C23×C4×C5C2311yescharacteristic
46C23×C4×C5C2311yescharacteristic
47C23×Dic5C2312yes—
48C23×Dic5C2312yes—
49C23×Dic5C2312yes—
50C23×Dic5C2312yes—
51C2×C5×C4⋊C4—24no—
52C2×C5×C4⋊C4C2×C414yes—
53C2×C5×C4⋊C4C2×C414yes—
54C2×C5×C4⋊C4D414yes—
55C2×C5×C4⋊C4Q814yes—
56C2×C42×C5—22no—
57C2×C42×C5—22no—
58C2×C42×C5—24no—
59C2×C42×C5—22no—
60C2×C42×C5—22no—
61C2×C42×C5C2×C414yes—
62C2×C42×C5C2×C414yes—
63C2×C42×C5C2×C414yes—
64C2×C4⋊Dic5—24no—
65C2×C4⋊Dic5—24no—
66C2×C4⋊Dic5—24no—
67C2×C4⋊Dic5—24no—
68C2×C4⋊Dic5—24no—
69C2×C4⋊Dic5—24no—
70C2×C4⋊Dic5—24no—
71C2×C4⋊Dic5—42no—
72C2×C4⋊Dic5—42no—
73C2×C4×Dic5—24no—
74C2×C4×Dic5—24no—
75C2×C4×Dic5—42no—
76C5×#95C2×C414yes—
77C5×#95C2×C414yes—
78#3963—22no—
79#3963—22no—
80#3963—24no—
81#3963—24no—
82#3963—24no—
83#3963—24no—
84#3963—24no—
85#3963—24no—
86#3963—22no—
87#3963—24no—
88#3963—24no—
89#3963D414yes—
90C2×#387—52no—
91C2×#382—51no—
92C2×C4×C4⋊C4—51no—
93C2×C4×C4⋊C4D511yescharacteristic
94#1444—52no—
95#1441—52no—
96#1436—52no—
97C4⋊#95—51no—
98#1434—51no—
99#1434—51no—
100#1434—51no—
101C24×C5C2411yescharacteristic
102C22×C4×C5—21no—
103C22×C4×C5—21no—
104C22×C4×C5—21no—
105C22×C4×C5—22no—
106C22×C4×C5—22no—
107C22×C4×C5—21no—
108C22×C4×C5—21no—
109C22×C4×C5—21no—
110C22×C4×C5—22no—
111C22×C4×C5—22no—
112C22×C4×C5—22no—
113C22×C4×C5—21no—
114C22×C4×C5—21no—
115C22×C4×C5—21no—
116C22×C4×C5—22no—
117C22×C4×C5—21no—
118C22×C4×C5—21no—
119C22×C4×C5—21no—
120C22×C4×C5—21no—
121C22×C4×C5—21no—
122C22×C4×C5—22no—
123C22×C4×C5—22no—
124C22×C4×C5C22×C412yes—
125C22×C4×C5C22×C412yes—
126C22×C4×C5C22×C412yes—
127C22×C4×C5C22×C412yes—
128C22×C4×C5C22×C412yes—
129C22×C4×C5C22×C412yes—
130C22×C4×C5C22×C412yes—
131C22×C4×C5C2×D412yes—
132C22×C4×C5C2×D412yes—
133C22×C4×C5C2×D412yes—
134C22×C4×C5C2×D412yes—
135C22×C4×C5C2×Q812yes—
136C22×C4×C5C2×Q812yes—
137C22×C4×C5C2×Q812yes—
138C22×C4×C5#4114yes—
139C22×C4×C5#4114yes—
140C22×C4×C5#4112yes—
141C22×Dic5—22no—
142C22×Dic5—22no—
143C22×Dic5—22no—
144C22×Dic5—22no—
145C22×Dic5—22no—
146C22×Dic5—22no—
147C22×Dic5—22no—
148C22×Dic5—22no—
149C22×Dic5—22no—
150C22×Dic5—22no—
151C22×Dic5—22no—
152C22×Dic5—22no—
153C22×Dic5—22no—
154C22×Dic5—22no—
155C22×Dic5—22no—
156C22×Dic5—22no—
157C22×Dic5—22no—
158C22×Dic5—22no—
159C22×Dic5—22no—
160C22×Dic5—22no—
161C22×Dic5—22no—
162C22×Dic5—22no—
163C22×Dic5—22no—
164C22×Dic5—22no—
165C22×Dic5—22no—
166C22×Dic5—22no—
167C22×Dic5C2×D414yes—
168C22×Dic5#4114yes—
169C5×C4⋊C4—28no—
170C5×C4⋊C4C4⋊C4116yescomplemented
171C42×C5—28no—
172C42×C5—28no—
173C42×C5—28no—
174C4⋊Dic5—216no—
175C4⋊Dic5—48no—
176C4⋊Dic5—48no—
177C4×Dic5—48no—
178C22×C4⋊C4—52no—
179C22×C4⋊C4—52no—
180C22×C4⋊C4—52no—
181C22×C4⋊C4C2×D511yescharacteristic
182C22×C4⋊C4C2×D511yescharacteristic
183C22×C42—52no—
184C22×C42C2×D511yescharacteristic
185C22×C42C2×D511yescharacteristic
186C22×C42C2×D511yescharacteristic
187#387—108no—
188#382—104no—
189C4×C4⋊C4—104no—
190C4×C4⋊C4Dic518yescomplemented
191C2×#95—51no—
192C2×#95—51no—
193C2×#95—51no—
194C2×#95—51no—
195C2×#95—52no—
196C2×#95—52no—
197C2×#95—52no—
198C2×#95—52no—
199C2×#95—52no—
200C2×#95—52no—
201C2×#95—52no—
202C2×#95—52no—
203C2×#95C2×D511yescharacteristic
204C2×#95C2×D511yescharacteristic
205C23×C5C23×C411yescharacteristic
206C23×C5C22×D411yescharacteristic
207C23×C5C22×D411yescharacteristic
208C23×C5C22×Q811yescharacteristic
209C23×C5C2×#4112yes—
210C23×C5C2×#4112yes—
211C23×C5C2×#4111yescharacteristic
212C23×C5C2×#4112yes—
213C23×C5H(5,𝔽2)11yescharacteristic
214C23×C52−1+412yes—
215C23×C52−1+411yescharacteristic
216C2×C4×C5—22no—
217C2×C4×C5—22no—
218C2×C4×C5—22no—
219C2×C4×C5—22no—
220C2×C4×C5—22no—
221C2×C4×C5—22no—
222C2×C4×C5—22no—
223C2×C4×C5—22no—
224C2×C4×C5—22no—
225C2×C4×C5—22no—
226C2×C4×C5—22no—
227C2×C4×C5—22no—
228C2×C4×C5—24no—
229C2×C4×C5—24no—
230C2×C4×C5—24no—
231C2×C4×C5—24no—
232C2×C4×C5—22no—
233C2×C4×C5—22no—
234C2×C4×C5—24no—
235C2×C4×C5—22no—
236C2×C4×C5—22no—
237C2×C4×C5—22no—
238C2×C4×C5—22no—
239C2×C4×C5—24no—
240C2×C4×C5—24no—
241C2×C4×C5—24no—
242C2×C4×C5—22no—
243C2×C4×C5—22no—
244C2×C4×C5—22no—
245C2×C4×C5—22no—
246C2×C4×C5—22no—
247C2×C4×C5—42no—
248C2×C4×C5—42no—
249C2×C4×C5C2×C4⋊C414yes—
250C2×C4×C5C2×C4⋊C414yes—
251C2×C4×C5C2×C4⋊C414yes—
252C2×C4×C5#11714yes—
253C2×C4×C5#11714yes—
254C2×C4×C5C4×D414yes—
255C2×Dic5—24no—
256C2×Dic5—24no—
257C2×Dic5—24no—
258C2×Dic5—24no—
259C2×Dic5—24no—
260C2×Dic5—24no—
261C2×Dic5—42no—
262C2×Dic5—42no—
263C2×Dic5—42no—
264C2×Dic5—42no—
265C2×Dic5—42no—
266C2×Dic5—42no—
267C2×Dic5—42no—
268C2×Dic5—42no—
269C2×Dic5—42no—
270C2×Dic5—42no—
271C2×Dic5—42no—
272C2×Dic5—42no—
273C2×Dic5—42no—
274C2×Dic5—42no—
275C2×Dic5—42no—
276C2×Dic5—42no—
277C2×Dic5—42no—
278C2×Dic5—42no—
279C2×Dic5—42no—
280C2×Dic5—42no—
281C2×Dic5—42no—
282C2×Dic5—42no—
283C23×C4—52no—
284C23×C4—52no—
285C23×C4—52no—
286C23×C4—52no—
287C23×C4C22×D511yescharacteristic
288C23×C4C22×D511yescharacteristic
289C23×C4C22×D511yescharacteristic
290C23×C4C22×D511yescharacteristic
291C23×C4C22×D511yescharacteristic
292C23×C4C22×D511yescharacteristic
293C23×C4C22×D511yescharacteristic
294C2×C4⋊C4—24no—
295C2×C4⋊C4—104no—
296C2×C4⋊C4—104no—
297C2×C4⋊C4—104no—
298C2×C4⋊C4—104no—
299C2×C4⋊C4—104no—
300C2×C4⋊C4—104no—
301C2×C4⋊C4—104no—
302C2×C4⋊C4—202no—
303C2×C4⋊C4—202no—
304C2×C4⋊C4Dic1014yes—
305C2×C4⋊C4D2014yes—
306C2×C4⋊C4C2×Dic514yes—
307C2×C4⋊C4C2×Dic514yes—
308C2×C42—24no—
309C2×C42—22no—
310C2×C42—22no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—104no—
314C2×C42—104no—
315C2×C42—202no—
316C2×C42C2×Dic514yes—
317C2×C42C2×Dic514yes—
318C2×C42C2×Dic514yes—
319#95—54no—
320#95—104no—
321#95—104no—
322#95—104no—
323#95—104no—
324#95—104no—
325#95—104no—
326#95—104no—
327#95—104no—
328#95—102no—
329#95—102no—
330#95—102no—
331#95C2×Dic514yes—
332#95C2×Dic514yes—
333C22×C5C22×C4⋊C411yescharacteristic
334C22×C5C2×#11711yescharacteristic
335C22×C5C2×#11711yescharacteristic
336C22×C5C2×C4×D411yescharacteristic
337C22×C5C4×#4111yescharacteristic
338C22×C5#51911yescharacteristic
339C22×C5#52011yescharacteristic
340C22×C5#52512yes—
341C22×C5#52912yes—
342C22×C5#53212yes—
343C22×C5#53312yes—
344C22×C5#54112yes—
345C22×C5#54212yes—
346C22×C5D4211yescharacteristic
347C22×C5#54712yes—
348C22×C5#54711yescharacteristic
349C22×C5#54812yes—
350C22×C5D4×Q811yescharacteristic
351C22×C5#55112yes—
352C22×C5#55212yes—
353C22×C5#55212yes—
354C22×C5#55411yescharacteristic
355C22×C5#55712yes—
356C4×C5—24no—
357C4×C5—24no—
358C4×C5—24no—
359C4×C5—24no—
360C4×C5—24no—
361C4×C5—24no—
362C4×C5—42no—
363Dic5—44no—
364Dic5—44no—
365Dic5—82no—
366Dic5—82no—
367C24C23×D511yescharacteristic
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—54no—
391C22×C4—54no—
392C22×C4—102no—
393C22×C4—102no—
394C22×C4—102no—
395C22×C4—102no—
396C22×C4—102no—
397C22×C4—102no—
398C22×C4—102no—
399C22×C4—102no—
400C22×C4—102no—
401C22×C4—102no—
402C22×C4—102no—
403C22×C4—102no—
404C22×C4—102no—
405C22×C4—102no—
406C22×C4—102no—
407C22×C4—102no—
408C22×C4—102no—
409C22×C4—102no—
410C22×C4—102no—
411C22×C4—102no—
412C22×C4—102no—
413C22×C4—102no—
414C22×C4—102no—
415C22×C4—102no—
416C22×C4—102no—
417C22×C4—102no—
418C22×C4C2×Dic1012yes—
419C22×C4C2×Dic1012yes—
420C22×C4C2×Dic1012yes—
421C22×C4C2×D2012yes—
422C22×C4C2×D2012yes—
423C22×C4C2×D2012yes—
424C22×C4#70914yes—
425C22×C4#70914yes—
426C22×C4D4×D512yes—
427C22×C4#71112yes—
428C22×C4C22×Dic512yes—
429C22×C4C22×Dic512yes—
430C22×C4C22×Dic512yes—
431C22×C4C22×Dic512yes—
432C22×C4C22×Dic512yes—
433C22×C4C22×Dic512yes—
434C22×C4C22×Dic512yes—
435C4⋊C4—28no—
436C4⋊C4—1016no—
437C4⋊C4—208no—
438C4⋊C4—208no—
439C4⋊C4C4⋊Dic5116yescomplemented
440C42—28no—
441C42—28no—
442C42—28no—
443C42—208no—
444C2×C5#228511yescharacteristic
445C2×C5#229011yescharacteristic
446C2×C5#232011yescharacteristic
447C2×C5#234311yescharacteristic
448C2×C5#234411yescharacteristic
449C2×C5D4×C4⋊C411yescharacteristic
450C2×C5#235111yescharacteristic
451C2×C5#246112yes—
452C2×C5#246212yes—
453C2×C5#251912yes—
454C2×C5#252012yes—
455C23C22×Dic1011yescharacteristic
456C23C22×D2011yescharacteristic
457C23C2×#70912yes—
458C23C2×#70912yes—
459C23C2×D4×D511yescharacteristic
460C23C2×#71111yescharacteristic
461C23#414712yes—
462C23D5×#4112yes—
463C23#414911yescharacteristic
464C23#415011yescharacteristic
465C23C23×Dic511yescharacteristic
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—24no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—24no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—24no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—104no—
500C2×C4—104no—
501C2×C4—104no—
502C2×C4—104no—
503C2×C4—104no—
504C2×C4—104no—
505C2×C4—202no—
506C2×C4—202no—
507C2×C4—202no—
508C2×C4—202no—
509C2×C4—202no—
510C2×C4—202no—
511C2×C4—202no—
512C2×C4—202no—
513C2×C4—202no—
514C2×C4—202no—
515C2×C4—202no—
516C2×C4—202no—
517C2×C4—202no—
518C2×C4—202no—
519C2×C4—202no—
520C2×C4—202no—
521C2×C4—202no—
522C2×C4—202no—
523C2×C4—202no—
524C2×C4—202no—
525C2×C4—202no—
526C2×C4—202no—
527C2×C4C2×C4⋊Dic514yes—
528C2×C4C2×C4⋊Dic514yes—
529C2×C4C2×C4⋊Dic514yes—
530C2×C4#407214yes—
531C2×C4#407214yes—
532C2×C4D4×Dic514yes—
533C5#1388911yescharacteristic, complemented, Sylow
534C22#6580412yes—
535C22#6581112yes—
536C22#6584112yes—
537C22#6584712yes—
538C22#6586112yes—
539C22#6586512yes—
540C22D4×Dic1011yescharacteristic
541C22#6587212yes—
542C22#6587711yescharacteristic
543C22#6588112yes—
544C22D4×D2011yescharacteristic
545C22#6588911yescharacteristic
546C22#6590312yes—
547C22#6591012yes—
548C22#6591612yes—
549C22#6591712yes—
550C22C22×C4⋊Dic511yescharacteristic
551C22C2×#407211yescharacteristic
552C22C2×#407211yescharacteristic
553C22C2×D4×Dic511yescharacteristic
554C22#6614611yescharacteristic
555C22#41×Dic511yescharacteristic
556C22#6616111yescharacteristic
557C4—24no—
558C4—24no—
559C4—24no—
560C4—24no—
561C4—24no—
562C4—24no—
563C4—42no—
564C4—204no—
565C4—204no—
566C4—402no—
567C4—402no—
568C2#1060540912yes—
569C2#1060542912yes—
570C2#1060543112yes—
571C2#1060551312yes—
572C2#1061379311yescharacteristic
573C2#1061389011yescharacteristic
574C2#1061396811yescharacteristic
575C2#1061404911yescharacteristic
576C2D4×C4⋊Dic511yescharacteristic
577C2#1061418511yescharacteristic
578C2#1061419111yescharacteristic
subgroupG/H
center, FrattiniC24C23×D5
commutatorC23×C5C23×C4
FittingC2×C4×C5×C4⋊C4C2
socle, prime radical, cosocleC24×C5C24
hypercenterC2×C4×C4⋊C4D5
hypocenterC5#13889
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
21. C4×C5×C4⋊C4C4semidirect
170. C5×C4⋊C4C4⋊C4semidirect
190. C4×C4⋊C4Dic5semidirect
439. C4⋊C4C4⋊Dic5semidirect
533. C5#13889semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C5×C4⋊C42
C2×#652702
C2×#652252
C2×C4×C4⋊Dic52
#106023582
#106023282
#106023272
#106002122
#106002002
#138895