Finite Group Databasebeta

← #12103006

Subgroups — #12103006

Order 1280.

1526 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#66150C211yesmaximal, characteristic
2C2×C5×#419C211yesmaximal, characteristic, complemented
3C22×#3967C211yesmaximal, characteristic, complemented
4C2×#65422C211yesmaximal, characteristic
5C2×#65411C211yesmaximal, characteristic, complemented
6#10613965C211yesmaximal, characteristic, complemented
7C2×#65291C211yesmaximal, characteristic
8#10601021C212yesmaximal, complemented
9#10601020C212yesmaximal, complemented
10#10601019C212yesmaximal, complemented
11#10601018C212yesmaximal, complemented
12C23×C5×Q8C2211yescharacteristic
13C22×#709C2211yescharacteristic
14C22×C5×M16C2211yescharacteristic
15C2×C5×#117C2211yescharacteristic
16#66150—24no—
17#66150C418yescomplemented
18C2×#4093C2212yes—
19C2×#4092—22no—
20C2×#4090C2212yes—
21C2×#4075—22no—
22C2×#4072C2211yescharacteristic
23C22×C4⋊Dic5C2211yescharacteristic
24C23×#168C2211yescharacteristic
25C5×#419—24no—
26C2×C5×#103C2212yes—
27C2×C5×#103C2212yes—
28C2×#3967—24no—
29C2×#3967C2212yes—
30C2×#3967C2212yes—
31#65422—24no—
32#65411—24no—
33C2×#3953C2212yes—
34C2×#3948C2212yes—
35C2×#3947C2212yes—
36C2×C8×Dic5C2212yes—
37#65291—24no—
38#65266C2212yes—
39#65265C2212yes—
40C2×#3942C2211yescharacteristic
41C2×#3942C2211yescharacteristic
42C2×#3941C2211yescharacteristic
43C2×#3941C2211yescharacteristic
44#65235—22no—
45#65217—22no—
46#21205—51nomaximal, Sylow
47C22×C5×Q8—21no—
48C22×C5×Q8—22no—
49C22×C5×Q8—22no—
50C22×C5×Q8—22no—
51C22×C5×Q8—22no—
52C22×C5×Q8—22no—
53C22×C5×Q8C2×C414yes—
54C22×C5×Q8C2312yes—
55C23×C4×C5—21no—
56C23×C4×C5C2311yescharacteristic
57C22×C22⋊D5—21no—
58C23×Dic5—21no—
59C2×#709—22no—
60C2×#709—22no—
61C2×#709—22no—
62C2×#709—22no—
63C2×#709—22no—
64C2×#709C2×C414yes—
65C22×D20C2311yescharacteristic
66C22×C4×D5—21no—
67C22×Dic10C2311yescharacteristic
68C2×C5×M16—22no—
69C2×C5×M16—22no—
70C2×C5×M16D414yes—
71C22×C5×C8C2312yes—
72C5×#117—22no—
73C5×#117—22no—
74C2×C5×C4⋊C4C2311yescharacteristic
75C2×C5×C4⋊C4C2311yescharacteristic
76C2×C5×C22⋊C4—21no—
77C2×C42×C5—21no—
78C2×C22⋊Dic5D412yes—
79#4093—24no—
80#4093C2×C418yescomplemented
81#4092—216no—
82#4090—24no—
83#4090C2×C418yescomplemented
84#4075—216no—
85C2×#685—24no—
86#4072C2×C414yes—
87#4072D414yes—
88C2×C4⋊Dic5—22no—
89C2×C4⋊Dic5—22no—
90C2×C4⋊Dic5—21no—
91C2×C4⋊Dic5D412yes—
92C2×C4⋊Dic5C2312yes—
93C2×#683—24no—
94C2×#683—24no—
95C2×C4×Dic5C2312yes—
96C22×#168—22no—
97C22×#168—22no—
98C22×#168D414yes—
99C22×#168C2311yescharacteristic
100C22×#168C2311yescharacteristic
101C5×#103—24no—
102C5×#103—24no—
103#3967—24no—
104#3967—24no—
105#3963—24no—
106#3953—24no—
107#3948—24no—
108#3947—24no—
109C8×Dic5—24no—
110#3942—22no—
111#3942—22no—
112#3941—22no—
113#3941—22no—
114C2×#536—51no—
115C2×#417—51no—
116C2×#419D511yescharacteristic, complemented
117C2×#416—51no—
118C22×#103—51no—
119C2×#405—51no—
120#2317—51no—
121#1960—52no—
122#1959—52no—
123#1958—52no—
124#1956—52no—
125C24×C5C2×D411yescharacteristic
126C23×D5—21no—
127C2×C5×Q8—24no—
128C2×C5×Q8—22no—
129C2×C5×Q8—28no—
130C2×C5×Q8—28no—
131C2×C5×Q8—22no—
132C2×C5×Q8—22no—
133C2×C5×Q8—22no—
134C2×C5×Q8—22no—
135C2×C5×Q8—44no—
136C2×C5×Q8—44no—
137C2×C5×Q8—42no—
138C2×C5×Q8—42no—
139C2×C5×Q8C22×C414yes—
140C22×C4×C5—24no—
141C22×C4×C5—22no—
142C22×C4×C5—21no—
143C22×C4×C5—21no—
144C22×C4×C5—42no—
145C22×C4×C5—42no—
146C22×C4×C5C22×C412yes—
147C22×C4×C5C2×D411yescharacteristic
148C22×C4×C5C2×D412yes—
149C22×C4×C5C2×D412yes—
150C22×C4×C5C2×D412yes—
151C22×C4×C5#4112yes—
152C22×C4×C5#4112yes—
153C22×C4×C5C2411yescharacteristic
154C2×C22⋊D5—24no—
155C2×C22⋊D5—42no—
156C2×C22⋊D5—42no—
157C22×Dic5—21no—
158C22×Dic5—22no—
159C22×Dic5—42no—
160C22×Dic5—82no—
161C22×Dic5C2×D414yes—
162#709—28no—
163#709—28no—
164#709—44no—
165#709—44no—
166C2×D20—21no—
167C2×D20—21no—
168C2×D20—21no—
169C2×D20—21no—
170C2×D20—21no—
171C2×D20C22×C412yes—
172C2×C4×D5—24no—
173C2×C4×D5—42no—
174C2×C4×D5—42no—
175C2×Dic10—21no—
176C2×Dic10—21no—
177C2×Dic10—21no—
178C2×Dic10—21no—
179C2×Dic10—21no—
180C2×Dic10C22×C412yes—
181C5×M16—28no—
182C2×C5×C8—22no—
183C2×C5×C8—22no—
184C2×C5×C8C2×D414yes—
185C5×C4⋊C4—21no—
186C5×C4⋊C4—21no—
187C5×C4⋊C4—21no—
188C5×C4⋊C4—21no—
189C5×C22⋊C4—42no—
190C42×C5—42no—
191C22⋊Dic5C22⋊C418yes—
192#685—216no—
193C4⋊Dic5—22no—
194C4⋊Dic5—22no—
195C4⋊Dic5—42no—
196C4⋊Dic5C22⋊C418yes—
197#683—216no—
198#683—84no—
199C4×Dic5C22×C414yes—
200C4×Dic5C2×D414yes—
201C2×#168—28no—
202C2×#168—21no—
203C2×#168—21no—
204C2×#168—21no—
205C2×#168—21no—
206C2×#168C2×D412yes—
207C2×#168C2×D412yes—
208C22×#41—51no—
209C23×Q8C2×D511yescharacteristic
210C22×M16C2×D511yescharacteristic
211C23×C8—51no—
212#536—58no—
213#536—104no—
214C2×C4⋊Q8—52no—
215C2×#124—52no—
216C2×#123—102no—
217C2×C22⋊Q8—102no—
218C2×#117—51no—
219C2×#117C2×D511yescharacteristic
220C22×C4⋊C4—51no—
221#419—24no—
222#417—104no—
223#416—104no—
224C2×#103—52no—
225C2×#103—52no—
226C2×#103—52no—
227C2×#103—51no—
228C2×#103—51no—
229C2×#103—104no—
230C2×#103C2×D512yes—
231C2×#103C2×D512yes—
232C2×#102—52no—
233C2×#102—51no—
234C2×#102—51no—
235#405—104no—
236C2×#97—52no—
237C2×C4×C8—52no—
238#389—102no—
239#385—102no—
240#384—52no—
241#382—52no—
242C23×C5—22no—
243C23×C5—22no—
244C23×C5C2×C22⋊C412yes—
245C23×C5#12412yes—
246C23×C5#12712yes—
247C23×C5C22×D411yescharacteristic
248C22×D5—22no—
249C22×D5—41no—
250C22×D5—41no—
251C22×D5—42no—
252C22×D5—42no—
253C5×Q8—24no—
254C5×Q8—24no—
255C5×Q8—42no—
256C5×Q8—42no—
257C5×Q8—48no—
258C2×C4×C5—24no—
259C2×C4×C5—42no—
260C2×C4×C5—42no—
261C2×C4×C5—48no—
262C2×C4×C5—41no—
263C2×C4×C5—41no—
264C2×C4×C5—42no—
265C2×C4×C5—42no—
266C2×C4×C5C2×C22⋊C412yes—
267C2×C4×C5C4×D414yes—
268C2×C4×C5C4×D414yes—
269C2×C4×C5#12114yes—
270C2×C4×C5#12114yes—
271C2×C4×C5#12412yes—
272C2×C4×C5#12712yes—
273C2×C4×C5C23×C411yescharacteristic
274C2×C4×C5C22×D411yescharacteristic
275C2×C4×C5C22×D411yescharacteristic
276C2×C4×C5C22×D411yescharacteristic
277C2×C4×C5C2×#4111yescharacteristic
278C2×C4×C5C2×#4111yescharacteristic
279C22⋊D5—48no—
280C2×Dic5—24no—
281C2×Dic5—24no—
282C2×Dic5—22no—
283C2×Dic5—41no—
284C2×Dic5—41no—
285C2×Dic5—42no—
286C2×Dic5—44no—
287C2×Dic5—84no—
288C2×Dic5C2×C22⋊C418yes—
289D20—22no—
290D20—22no—
291D20—41no—
292D20—41no—
293C4×D5—48no—
294Dic10—22no—
295Dic10—22no—
296Dic10—41no—
297Dic10—41no—
298C5×C8—24no—
299#168—22no—
300#168—22no—
301C2×#41—54no—
302C2×#41—102no—
303C2×#41—102no—
304C2×#41—102no—
305C2×#41—102no—
306C2×#41—102no—
307C22×Q8—21no—
308C22×Q8—22no—
309C22×Q8—22no—
310C22×Q8—22no—
311C22×Q8—22no—
312C22×Q8—22no—
313C22×Q8—51no—
314C22×Q8C4×D514yes—
315C22×Q8C22×D512yes—
316C22×D4—51no—
317C22×D4—101no—
318C23×C4—21no—
319C23×C4—101no—
320C23×C4—101no—
321C23×C4C22×D511yescharacteristic
322C2×M16—22no—
323C2×M16—22no—
324C2×M16C22⋊D514yes—
325C22×C8—54no—
326C22×C8—51no—
327C22×C8—51no—
328C22×C8—102no—
329C22×C8—102no—
330C22×C8C22×D512yes—
331C4⋊Q8—58no—
332C4⋊Q8—104no—
333#124—58no—
334#124—104no—
335#123—1016no—
336C22⋊Q8—1016no—
337#117—22no—
338#117—22no—
339#117—54no—
340#117—54no—
341C2×C4⋊C4—52no—
342C2×C4⋊C4—52no—
343C2×C4⋊C4—104no—
344C2×C4⋊C4—102no—
345C2×C4⋊C4—102no—
346C2×C4⋊C4—101no—
347C2×C4⋊C4—104no—
348C2×C4⋊C4C22×D511yescharacteristic
349C2×C4⋊C4C22×D511yescharacteristic
350C2×C22⋊C4—21no—
351C2×C22⋊C4—52no—
352C2×C22⋊C4—104no—
353C2×C42—21no—
354C2×C42—52no—
355#103—24no—
356#103—24no—
357#103—102no—
358#103—102no—
359#103—104no—
360#103—104no—
361#103—104no—
362#102—102no—
363#102—102no—
364#102—104no—
365#97—104no—
366C4×C8—104no—
367#95—104no—
368C22×C5—24no—
369C22×C5—24no—
370C22×C5—22no—
371C22×C5—22no—
372C22×C5#39014yes—
373C22×C5C22×C22⋊C411yescharacteristic
374C22×C5C2×#12411yescharacteristic
375C22×C5C2×#12711yescharacteristic
376C22×C5#57712yes—
377C22×C5#57812yes—
378C2×D5—44no—
379C2×D5—42no—
380C2×D5—82no—
381C2×D5—81no—
382C2×D5—81no—
383C4×C5—44no—
384C4×C5—81no—
385C4×C5—81no—
386C4×C5#39014yes—
387C4×C5C2×C4×D411yescharacteristic
388C4×C5C2×C4×D411yescharacteristic
389C4×C5C2×#12111yescharacteristic
390C4×C5C2×#12111yescharacteristic
391Dic5—28no—
392Dic5—42no—
393Dic5—82no—
394C24—101no—
395C24D4×D511yescharacteristic
396#41—108no—
397#41—108no—
398#41—204no—
399#41—204no—
400C2×Q8—22no—
401C2×Q8—24no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—28no—
407C2×Q8—28no—
408C2×Q8—44no—
409C2×Q8—42no—
410C2×Q8—44no—
411C2×Q8—42no—
412C2×Q8—52no—
413C2×Q8—101no—
414C2×Q8—101no—
415C2×Q8—101no—
416C2×Q8—101no—
417C2×Q8—101no—
418C2×Q8C2×C4×D514yes—
419C2×D4—52no—
420C2×D4—104no—
421C2×D4—101no—
422C2×D4—101no—
423C2×D4—101no—
424C2×D4—101no—
425C2×D4—101no—
426C2×D4—202no—
427C2×D4—202no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—54no—
435C22×C4—101no—
436C22×C4—104no—
437C22×C4—102no—
438C22×C4—202no—
439C22×C4—202no—
440C22×C4—202no—
441C22×C4—402no—
442C22×C4C2×C4×D512yes—
443C22×C4#70912yes—
444C22×C4D4×D512yes—
445C22×C4D4×D512yes—
446C22×C4D4×D511yescharacteristic
447C22×C4#71112yes—
448C22×C4C2×C22⋊D512yes—
449C22×C4C23×D511yescharacteristic
450M16—28no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—52no—
454C2×C8—52no—
455C2×C8—101no—
456C2×C8—101no—
457C2×C8—101no—
458C2×C8—101no—
459C2×C8—108no—
460C2×C8C2×C22⋊D514yes—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—58no—
466C4⋊C4—1016no—
467C4⋊C4—102no—
468C4⋊C4—102no—
469C4⋊C4—202no—
470C4⋊C4—404no—
471C22⋊C4—42no—
472C22⋊C4—58no—
473C22⋊C4—1016no—
474C42—42no—
475C42—54no—
476C42—54no—
477C2×C5—24no—
478C2×C5C2×#39011yescharacteristic
479C2×C5#289812yes—
480C2×C5#313712yes—
481C2×C5#315411yescharacteristic
482C2×C5#315611yescharacteristic
483D5—82no—
484C23—22no—
485C23—22no—
486C23—102no—
487C23—201no—
488C23—202no—
489C23—202no—
490C23—201no—
491C23D5×C22⋊C412yes—
492C23#403112yes—
493C23#408612yes—
494C23C2×D4×D511yescharacteristic
495Q8—24no—
496Q8—24no—
497Q8—42no—
498Q8—42no—
499Q8—48no—
500Q8—102no—
501Q8—102no—
502Q8—201no—
503Q8—201no—
504D4—102no—
505D4—102no—
506D4—208no—
507D4—201no—
508D4—201no—
509C2×C4—24no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—48no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—58no—
518C2×C4—104no—
519C2×C4—104no—
520C2×C4—102no—
521C2×C4—201no—
522C2×C4—201no—
523C2×C4—208no—
524C2×C4—204no—
525C2×C4—202no—
526C2×C4—404no—
527C2×C4D5×C22⋊C412yes—
528C2×C4#402714yes—
529C2×C4#403014yes—
530C2×C4#403112yes—
531C2×C4C4×C22⋊D514yes—
532C2×C4#408514yes—
533C2×C4#408612yes—
534C2×C4C22×C4×D511yescharacteristic
535C2×C4C2×#70911yescharacteristic
536C2×C4C2×D4×D511yescharacteristic
537C2×C4C2×D4×D511yescharacteristic
538C2×C4C2×#71111yescharacteristic
539C2×C4C22×C22⋊D511yescharacteristic
540C8—24no—
541C8—102no—
542C8—102no—
543C5#2120511yescharacteristic, complemented, Sylow
544C22—24no—
545C22—22no—
546C22—22no—
547C22—24no—
548C22—204no—
549C22—202no—
550C22—401no—
551C22—401no—
552C22—402no—
553C22#6524614yes—
554C22C2×D5×C22⋊C411yescharacteristic
555C22C2×#403111yescharacteristic
556C22#6611212yes—
557C22#6611312yes—
558C22C2×#408611yescharacteristic
559C4—44no—
560C4—81no—
561C4—81no—
562C4—108no—
563C4—202no—
564C4—402no—
565C4#6524614yes—
566C4C2×#402711yescharacteristic
567C4C2×#403011yescharacteristic
568C4C2×C4×C22⋊D511yescharacteristic
569C4C2×#408511yescharacteristic
570C2—24no—
571C2—402no—
572C2#1061068312yes—
573C2#1061068812yes—
574C2C2×#6524611yescharacteristic
575C2#1061570511yescharacteristic
576C2#1061570611yescharacteristic
subgroupG/H
centerC23C2×D4×D5
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC2×C5×#419C2
socleC23×C5C22×D4
prime radicalC22×#709C22
cosocleC22×C4×C5C24
hypercenterC2×#419D5
hypocenterC5#21205
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#419

Complements

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

NH ≅ G/Ntype
2. C2×C5×#419C2semidirect
3. C22×#3967C2semidirect
5. C2×#65411C2semidirect
6. #10613965C2semidirect
8. #10601021C2semidirect
9. #10601020C2semidirect
10. #10601019C2semidirect
11. #10601018C2semidirect
17. #66150C4semidirect
80. #4093C2×C4semidirect
83. #4090C2×C4semidirect
116. C2×#419D5semidirect
543. C5#21205semidirect

Maximal subgroups

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

subgroupindex
C2×#661502
C2×C5×#4192
C22×#39672
C2×#654222
C2×#654112
#106139652
C2×#652912
#106010212
#106010202
#106010192
#106010182
#212055