Finite Group Databasebeta

← #12100717

Subgroups — #12100717

Order 1280.

1766 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#4088C211yesmaximal, characteristic
2C2×#65857C211yesmaximal, characteristic
3C5×#2310C211yesmaximal, characteristic, complemented
4C22×#3967C211yesmaximal, characteristic, complemented
5C2×#65291C211yesmaximal, characteristic
6#10602374C211yesmaximal, characteristic, complemented
7#10602372C211yesmaximal, characteristic, complemented
8#10600483C214yesmaximal, complemented
9#10600480C214yesmaximal, complemented
10C23×C5×Q8C2211yescharacteristic
11C22×#709C2211yescharacteristic
12C2×C5×#117C2211yescharacteristic
13C2×C5×#117C2211yescharacteristic
14C22×#689C2212yes—
15C2×#4088—24no—
16C2×#4088—24no—
17C2×#4088C418yescomplemented
18C22×#688C2212yes—
19C22×C4⋊Dic5C2211yescharacteristic
20C22×#681C2211yescharacteristic
21C23×#168C2211yescharacteristic
22#65857—24no—
23#65857—24no—
24C2×#4038—21no—
25C2×D5×C4⋊C4—21no—
26C2×#4027—22no—
27C2×C4×D20C2212yes—
28C2×C4×Dic10C2212yes—
29C5×#391—21no—
30C5×#391—21no—
31C5×#391C2214yes—
32C5×#381—22no—
33C2×#3967—24no—
34C2×#3967C2214yes—
35#65291—24no—
36C2×#3942C2212yes—
37C2×#3941C2212yes—
38#64749C2214yes—
39#64744C2214yes—
40#20931—51nomaximal, Sylow
41C22×C5×Q8—21no—
42C22×C5×Q8—22no—
43C22×C5×Q8—22no—
44C22×C5×Q8—22no—
45C22×C5×Q8—22no—
46C22×C5×Q8—22no—
47C22×C5×Q8C2×C414yescomplemented
48C22×C5×Q8C2312yes—
49C23×C4×C5—21no—
50C23×C4×C5C2311yescharacteristic
51C22×C22⋊D5—21no—
52C23×Dic5—21no—
53C2×#709—22no—
54C2×#709—22no—
55C2×#709—22no—
56C2×#709C2×C414yes—
57C2×#709D414yes—
58C2×#709D414yes—
59C22×D20C2311yescharacteristic
60C22×C4×D5—21no—
61C22×C4×D5—22no—
62C22×Dic10C2311yescharacteristic
63C5×#117—22no—
64C5×#117—22no—
65C5×#117—22no—
66C5×#117—22no—
67C2×C5×C4⋊C4D411yescharacteristic
68C2×C5×C4⋊C4D411yescharacteristic
69C2×C5×C4⋊C4C2312yes—
70C2×C5×C22⋊C4—21no—
71C2×C5×C22⋊C4D412yes—
72C2×C42×C5—21no—
73C2×C42×C5C2312yes—
74C2×#689—24no—
75C2×#689—24no—
76C2×#689C2×C418yescomplemented
77#4088—232no—
78C2×#688—24no—
79C2×#688—24no—
80C2×#688C2×C418yescomplemented
81C2×#685—22no—
82C2×C4⋊Dic5—21no—
83C2×C4⋊Dic5—22no—
84C2×C4⋊Dic5D414yes—
85C2×C4⋊Dic5C2312yes—
86C2×#683—22no—
87C2×C4×Dic5—22no—
88C2×#681—22no—
89C2×#681—22no—
90C2×#681C2×C414yescomplemented
91C22×#168—22no—
92C22×#168—22no—
93C22×#168—22no—
94C22×#168C2312yes—
95C22×#168C2312yes—
96#4038—28no—
97D5×C4⋊C4—28no—
98#4027—216no—
99C4×D20—24no—
100C4×D20—24no—
101C4×Dic10—24no—
102C4×Dic10—24no—
103C5×#95—24no—
104C5×#95—24no—
105#3967—28no—
106#3942—24no—
107#3941—24no—
108C22×#137—51no—
109C2×#520—51no—
110C2×#416—51no—
111C22×#103—51no—
112#2310D511yescharacteristic, complemented
113#1878—54no—
114#1877—54no—
115#1736—51no—
116#1729—51no—
117C24×C5C2×D411yescharacteristic
118C23×D5—21no—
119C2×C5×Q8—24no—
120C2×C5×Q8—22no—
121C2×C5×Q8—28no—
122C2×C5×Q8—28no—
123C2×C5×Q8—22no—
124C2×C5×Q8—22no—
125C2×C5×Q8—22no—
126C2×C5×Q8—22no—
127C2×C5×Q8—42no—
128C2×C5×Q8—44no—
129C2×C5×Q8—42no—
130C2×C5×Q8—44no—
131C2×C5×Q8C22×C414yes—
132C22×C4×C5—24no—
133C22×C4×C5—22no—
134C22×C4×C5—22no—
135C22×C4×C5—42no—
136C22×C4×C5—42no—
137C22×C4×C5C22×C412yes—
138C22×C4×C5C2×D412yes—
139C22×C4×C5C2×D412yes—
140C22×C4×C5C2×D412yes—
141C22×C4×C5C2×D412yes—
142C22×C4×C5C2×D412yes—
143C22×C4×C5C2×D411yescharacteristic
144C22×C4×C5#4112yes—
145C22×C4×C5#4112yes—
146C22×C4×C5C2411yescharacteristic
147C2×C22⋊D5—24no—
148C2×C22⋊D5—24no—
149C2×C22⋊D5—24no—
150C22×Dic5—21no—
151C22×Dic5—22no—
152C22×Dic5—24no—
153C22×Dic5—82no—
154#709—28no—
155#709—28no—
156#709—28no—
157#709C22⋊C4116yes—
158C2×D20—21no—
159C2×D20—21no—
160C2×D20—21no—
161C2×D20C22×C412yes—
162C2×D20C2×D412yes—
163C2×D20C2×D412yes—
164C2×C4×D5—24no—
165C2×C4×D5—24no—
166C2×C4×D5—24no—
167C2×C4×D5—28no—
168C2×C4×D5—84no—
169C2×Dic10—21no—
170C2×Dic10—21no—
171C2×Dic10—21no—
172C2×Dic10C22×C412yes—
173C2×Dic10C2×D412yes—
174C2×Dic10C2×D412yes—
175C5×C4⋊C4—22no—
176C5×C4⋊C4—22no—
177C5×C4⋊C4—22no—
178C5×C4⋊C4—22no—
179C5×C22⋊C4—24no—
180C5×C22⋊C4—42no—
181C42×C5—22no—
182C42×C5—22no—
183C42×C5—42no—
184#689—216no—
185#688—216no—
186#685—28no—
187C4⋊Dic5—24no—
188C4⋊Dic5—22no—
189C4⋊Dic5C2×D414yes—
190#683—28no—
191C4×Dic5—28no—
192#681—28no—
193C2×#168—22no—
194C2×#168—22no—
195C2×#168—22no—
196C2×#168—22no—
197C2×#168—22no—
198C2×#168—44no—
199C2×#168C22×C414yes—
200C22×#41—51no—
201C23×Q8C2×D511yescharacteristic
202C2×#137—58no—
203C2×#137—104no—
204C2×#137—104no—
205C22×Q16—52no—
206C22×SD16—52no—
207C22×M16—51no—
208C23×C8—51no—
209#520—104no—
210#520—104no—
211C2×C4×Q8—52no—
212C2×C4×D4—52no—
213C2×C4×D4—102no—
214C2×#117—101no—
215C2×#117C2×D511yescharacteristic
216C2×#117C2×D511yescharacteristic
217C22×C4⋊C4—51no—
218C22×C4⋊C4—101no—
219#416—104no—
220C2×#103—54no—
221C2×#103—52no—
222C2×#103—104no—
223C2×#102—52no—
224#391—21no—
225#391—21no—
226#391C2×D514yes—
227#381—22no—
228#340—54no—
229#336—54no—
230C23×C5—22no—
231C23×C5—22no—
232C23×C5C4×D412yes—
233C23×C5#12012yes—
234C23×C5#12112yes—
235C23×C5C22×D411yescharacteristic
236C22×D5—22no—
237C22×D5—22no—
238C22×D5—22no—
239C22×D5—42no—
240C22×D5—42no—
241C5×Q8—24no—
242C5×Q8—24no—
243C5×Q8—42no—
244C5×Q8—42no—
245C5×Q8—48no—
246C2×C4×C5—24no—
247C2×C4×C5—22no—
248C2×C4×C5—22no—
249C2×C4×C5—24no—
250C2×C4×C5—24no—
251C2×C4×C5—42no—
252C2×C4×C5—42no—
253C2×C4×C5—48no—
254C2×C4×C5—42no—
255C2×C4×C5—42no—
256C2×C4×C5—42no—
257C2×C4×C5C2×C22⋊C414yes—
258C2×C4×C5C4×D414yes—
259C2×C4×C5C4×D412yes—
260C2×C4×C5#12012yes—
261C2×C4×C5#12114yes—
262C2×C4×C5#12112yes—
263C2×C4×C5#12314yes—
264C2×C4×C5C23×C411yescharacteristic
265C2×C4×C5C22×D411yescharacteristic
266C2×C4×C5C22×D411yescharacteristic
267C2×C4×C5C22×D411yescharacteristic
268C2×C4×C5C2×#4111yescharacteristic
269C2×C4×C5C2×#4111yescharacteristic
270C22⋊D5—216no—
271C2×Dic5—22no—
272C2×Dic5—22no—
273C2×Dic5—22no—
274C2×Dic5—24no—
275C2×Dic5—44no—
276C2×Dic5—84no—
277D20—22no—
278D20—22no—
279D20—22no—
280D20C2×C22⋊C414yes—
281C4×D5—216no—
282C4×D5—88no—
283Dic10—22no—
284Dic10—22no—
285Dic10—22no—
286Dic10C2×C22⋊C414yes—
287#168—24no—
288#168—42no—
289C2×#41—54no—
290C2×#41—54no—
291C2×#41—54no—
292C2×#41—102no—
293C2×#41—102no—
294C2×#41—102no—
295C22×Q8—22no—
296C22×Q8—21no—
297C22×Q8—22no—
298C22×Q8—22no—
299C22×Q8—22no—
300C22×Q8—22no—
301C22×Q8—51no—
302C22×Q8C4×D514yescomplemented
303C22×Q8C22×D512yes—
304C22×D4—51no—
305C22×D4—101no—
306C23×C4—21no—
307C23×C4—101no—
308C23×C4—101no—
309C23×C4—102no—
310C23×C4C22×D511yescharacteristic
311#137—1032no—
312C2×Q16—58no—
313C2×Q16—104no—
314C2×Q16—104no—
315C2×SD16—58no—
316C2×SD16—104no—
317C2×SD16—104no—
318C2×M16—54no—
319C2×M16—102no—
320C2×M16—102no—
321C22×C8—52no—
322C22×C8—52no—
323C22×C8—102no—
324C22×C8—102no—
325C22×C8—102no—
326C4×Q8—104no—
327C4×Q8—104no—
328C4×D4—104no—
329C4×D4—104no—
330C4×D4—1016no—
331#117—22no—
332#117—22no—
333#117—22no—
334#117—22no—
335#117—108no—
336C2×C4⋊C4—54no—
337C2×C4⋊C4—52no—
338C2×C4⋊C4—108no—
339C2×C4⋊C4—102no—
340C2×C4⋊C4—101no—
341C2×C4⋊C4—102no—
342C2×C4⋊C4C22⋊D511yescharacteristic
343C2×C4⋊C4C22⋊D511yescharacteristic
344C2×C4⋊C4C22×D512yes—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—102no—
347C2×C22⋊C4C22⋊D512yes—
348C2×C42—21no—
349C2×C42—102no—
350C2×C42C22×D512yes—
351#103—104no—
352#103—108no—
353#102—104no—
354#95—24no—
355#95—24no—
356C22×C5—24no—
357C22×C5—22no—
358C22×C5—22no—
359C22×C5—24no—
360C22×C5#38614yes—
361C22×C5C2×C4×D411yescharacteristic
362C22×C5C2×#12011yescharacteristic
363C22×C5C2×#12111yescharacteristic
364C22×C5#57712yes—
365C22×C5#57812yes—
366C2×D5—24no—
367C2×D5—44no—
368C2×D5—42no—
369C2×D5—42no—
370C2×D5—44no—
371C4×C5—42no—
372C4×C5—42no—
373C4×C5—44no—
374C4×C5—82no—
375C4×C5#38614yes—
376C4×C5C22×C22⋊C411yescharacteristic
377C4×C5C2×C4×D411yescharacteristic
378C4×C5C2×#12111yescharacteristic
379C4×C5C2×#12311yescharacteristic
380Dic5—24no—
381Dic5—44no—
382C24—101no—
383C24C2×C22⋊D511yescharacteristic
384#41—516no—
385#41—108no—
386#41—108no—
387#41—108no—
388C2×Q8—22no—
389C2×Q8—24no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—28no—
395C2×Q8—28no—
396C2×Q8—44no—
397C2×Q8—44no—
398C2×Q8—42no—
399C2×Q8—42no—
400C2×Q8—52no—
401C2×Q8—52no—
402C2×Q8—52no—
403C2×Q8—101no—
404C2×Q8—101no—
405C2×Q8—101no—
406C2×Q8C2×C4×D514yes—
407C2×D4—52no—
408C2×D4—52no—
409C2×D4—52no—
410C2×D4—104no—
411C2×D4—104no—
412C2×D4—104no—
413C2×D4—101no—
414C2×D4—101no—
415C2×D4—101no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—108no—
422C22×C4—101no—
423C22×C4—104no—
424C22×C4—104no—
425C22×C4—104no—
426C22×C4—102no—
427C22×C4—104no—
428C22×C4—404no—
429C22×C4—402no—
430C22×C4C2×C4×D512yes—
431C22×C4#70912yes—
432C22×C4D4×D512yes—
433C22×C4D4×D512yes—
434C22×C4D4×D512yes—
435C22×C4#71112yes—
436C22×C4C2×C22⋊D512yes—
437C22×C4C2×C22⋊D512yes—
438C22×C4C2×C22⋊D511yescharacteristic
439C22×C4C23×D511yescharacteristic
440Q16—1016no—
441SD16—1016no—
442M16—108no—
443C2×C8—54no—
444C2×C8—102no—
445C2×C8—102no—
446C2×C8—102no—
447C2×C8—102no—
448C2×C8—102no—
449C2×C8—204no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—54no—
455C4⋊C4—108no—
456C4⋊C4—102no—
457C4⋊C4—104no—
458C22⋊C4—24no—
459C22⋊C4—42no—
460C22⋊C4—108no—
461C42—22no—
462C42—22no—
463C42—42no—
464C42—108no—
465C2×C5—24no—
466C2×C5C2×#38611yescharacteristic
467C2×C5#294712yes—
468C2×C5#300911yescharacteristic
469C2×C5#301111yescharacteristic
470C2×C5#304612yes—
471D5—44no—
472C23—22no—
473C23—22no—
474C23—102no—
475C23—102no—
476C23—102no—
477C23—202no—
478C23—202no—
479C23C4×C22⋊D512yes—
480C23#408312yes—
481C23#408412yes—
482C23C22×C22⋊D511yescharacteristic
483Q8—24no—
484Q8—24no—
485Q8—42no—
486Q8—42no—
487Q8—48no—
488Q8—54no—
489Q8—102no—
490Q8—102no—
491Q8—102no—
492D4—54no—
493D4—1016no—
494D4—102no—
495D4—102no—
496D4—102no—
497C2×C4—24no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—48no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—102no—
509C2×C4—102no—
510C2×C4—102no—
511C2×C4—1016no—
512C2×C4—104no—
513C2×C4—204no—
514C2×C4—408no—
515C2×C4—404no—
516C2×C4D5×C22⋊C414yes—
517C2×C4#402714yes—
518C2×C4#402914yes—
519C2×C4#403014yes—
520C2×C4C4×C22⋊D512yes—
521C2×C4#408312yes—
522C2×C4#408412yes—
523C2×C4C22×C4×D511yescharacteristic
524C2×C4C2×#70911yescharacteristic
525C2×C4C2×D4×D511yescharacteristic
526C2×C4C2×D4×D511yescharacteristic
527C2×C4C2×D4×D511yescharacteristic
528C2×C4C2×#71111yescharacteristic
529C8—104no—
530C8—202no—
531C5#2093111yescharacteristic, complemented, Sylow
532C22—24no—
533C22—22no—
534C22—24no—
535C22—22no—
536C22—104no—
537C22—204no—
538C22—202no—
539C22—204no—
540C22—202no—
541C22#6524514yes—
542C22C2×C4×C22⋊D511yescharacteristic
543C22C2×#408311yescharacteristic
544C22C2×#408411yescharacteristic
545C22#6617112yes—
546C22#6617212yes—
547C4—44no—
548C4—42no—
549C4—42no—
550C4—82no—
551C4—104no—
552C4—204no—
553C4#6524514yes—
554C4C2×D5×C22⋊C411yescharacteristic
555C4C2×#402711yescharacteristic
556C4C2×#402911yescharacteristic
557C4C2×#403011yescharacteristic
558C2—24no—
559C2—204no—
560C2C2×#6524511yescharacteristic
561C2#1061427812yes—
562C2#1061513511yescharacteristic
563C2#1061513712yes—
564C2#1061514211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D5
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC5×#2310C2
socleC23×C5C22×D4
prime radicalC22×#709C22
cosocleC22×C4×C5C24
hypercenter#2310D5
hypocenterC5#20931
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2310

Complements

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

NH ≅ G/Ntype
3. C5×#2310C2semidirect
4. C22×#3967C2semidirect
6. #10602374C2semidirect
7. #10602372C2semidirect
8. #10600483C2semidirect
9. #10600480C2semidirect
17. C2×#4088C4semidirect
47. C22×C5×Q8C2×C4semidirect
76. C2×#689C2×C4semidirect
80. C2×#688C2×C4semidirect
90. C2×#681C2×C4semidirect
112. #2310D5semidirect
302. C22×Q8C4×D5semidirect
531. C5#20931semidirect

Maximal subgroups

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

subgroupindex
C22×#40882
C2×#658572
C5×#23102
C22×#39672
C2×#652912
#106023742
#106023722
#106004832
#106004802
#209315