Finite Group Databasebeta

← #12102999

Subgroups — #12102999

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×#65860C211yesmaximal, characteristic
2C5×#2317C211yesmaximal, characteristic, complemented
3C2×#65293C211yesmaximal, characteristic, complemented
4C2×#65291C211yesmaximal, characteristic
5C2×#65286C211yesmaximal, characteristic, complemented
6C22×#3942C211yesmaximal, characteristic, complemented
7C2×#65255C211yesmaximal, characteristic
8#10600374C212yesmaximal, complemented
9#10600372C212yesmaximal, complemented
10#10600345C212yesmaximal, complemented
11#10600344C212yesmaximal, complemented
12C22×#709C2211yescharacteristic
13C23×Dic10C2211yescharacteristic
14C2×C5×#117C2211yescharacteristic
15C2×C5×#117C2211yescharacteristic
16C22×C5×C4⋊C4C2211yescharacteristic
17C22×#681C2211yescharacteristic
18C23×#168C2211yescharacteristic
19#65860—24no—
20#65860C418yescomplemented
21C2×#4043—22no—
22C2×#4032—22no—
23C2×#4021C2212yes—
24C2×#4015C2212yes—
25C5×#389—22no—
26C5×#385—22no—
27C5×#384C2212yes—
28C5×#382C2212yes—
29#65293—24no—
30#65291—24no—
31#65286—24no—
32C2×#3942—24no—
33C2×#3942C2212yes—
34C2×#3942C2212yes—
35C2×#3942C2212yes—
36C2×#3942C2212yes—
37C2×#3942C2211yescharacteristic
38C2×#3942C2211yescharacteristic
39C2×#3942C2212yes—
40#65255—24no—
41C2×#3941C2212yes—
42C2×#3941C2211yescharacteristic
43C2×#3941C2211yescharacteristic
44C2×#3935C2212yes—
45C2×C4×#168C2212yes—
46#21205—51nomaximal, Sylow
47C23×C4×C5—21no—
48C23×C4×C5C2311yescharacteristic
49C22×C22⋊D5—21no—
50C23×Dic5—21no—
51C2×#709—22no—
52C2×#709—22no—
53C2×#709—22no—
54C2×#709—22no—
55C2×#709—22no—
56C2×#709C2×C414yes—
57C22×D20C2311yescharacteristic
58C22×C4×D5—21no—
59C22×Dic10—22no—
60C22×Dic10—22no—
61C22×Dic10—22no—
62C22×Dic10—22no—
63C22×Dic10—21no—
64C22×Dic10—22no—
65C22×Dic10C2×C414yes—
66C22×Dic10C2311yescharacteristic
67C22×Dic10C2312yes—
68C5×#117—22no—
69C5×#117—22no—
70C5×#117C2×C414yes—
71C5×#117D414yes—
72C2×C5×C4⋊C4—22no—
73C2×C5×C4⋊C4—22no—
74C2×C5×C4⋊C4—21no—
75C2×C5×C4⋊C4—24no—
76C2×C5×C4⋊C4D412yes—
77C2×C5×C4⋊C4C2311yescharacteristic
78C2×C5×C4⋊C4C2312yes—
79C2×C5×C4⋊C4C2311yescharacteristic
80C2×C5×C22⋊C4—21no—
81C2×C5×C22⋊C4D412yes—
82C2×C42×C5—21no—
83C2×C42×C5C2312yes—
84C2×#685—24no—
85C2×C4⋊Dic5—24no—
86C2×#681—22no—
87C2×#681—22no—
88C2×#681D414yes—
89C22×#168—22no—
90C22×#168—22no—
91C22×#168D414yes—
92C22×#168C2311yescharacteristic
93C22×#168C2311yescharacteristic
94C22×#168C2312yes—
95#4043—216no—
96#4032—216no—
97#4021—24no—
98#4021C2×C418yescomplemented
99#4015—24no—
100#4015C2×C418yescomplemented
101C5×#95—24no—
102#3942—22no—
103#3942—22no—
104#3942—24no—
105#3942—24no—
106#3942—24no—
107#3942—24no—
108#3942—24no—
109#3941—22no—
110#3941—22no—
111#3941—24no—
112#3935—24no—
113C4×#168—24no—
114C2×#536—51no—
115C2×#417—51no—
116C2×#419—51no—
117C2×#416—51no—
118C22×#103—51no—
119C2×#405—51no—
120#2317D511yescharacteristic, complemented
121#1960—52no—
122#1959—52no—
123#1958—52no—
124#1956—52no—
125C24×C5C2×D411yescharacteristic
126C23×D5—21no—
127C22×C4×C5—21no—
128C22×C4×C5—21no—
129C22×C4×C5—22no—
130C22×C4×C5—42no—
131C22×C4×C5—82no—
132C22×C4×C5C22×C412yes—
133C22×C4×C5C2×D412yes—
134C22×C4×C5C2×D412yes—
135C22×C4×C5C2×D412yes—
136C22×C4×C5C2×D414yes—
137C22×C4×C5C2×D411yescharacteristic
138C22×C4×C5#4112yes—
139C22×C4×C5#4112yes—
140C22×C4×C5C2411yescharacteristic
141C2×C22⋊D5—24no—
142C2×C22⋊D5—42no—
143C2×C22⋊D5—42no—
144C22×Dic5—24no—
145C22×Dic5—22no—
146C22×Dic5—21no—
147C22×Dic5—42no—
148C22×Dic5—42no—
149#709—28no—
150#709—28no—
151#709—44no—
152#709—44no—
153C2×D20—21no—
154C2×D20—21no—
155C2×D20—21no—
156C2×D20—21no—
157C2×D20—21no—
158C2×D20C22×C412yes—
159C2×C4×D5—24no—
160C2×C4×D5—42no—
161C2×C4×D5—42no—
162C2×Dic10—28no—
163C2×Dic10—28no—
164C2×Dic10—22no—
165C2×Dic10—22no—
166C2×Dic10—21no—
167C2×Dic10—21no—
168C2×Dic10—22no—
169C2×Dic10—22no—
170C2×Dic10—21no—
171C2×Dic10—21no—
172C2×Dic10—24no—
173C2×Dic10—22no—
174C2×Dic10—21no—
175C2×Dic10—44no—
176C2×Dic10—44no—
177C2×Dic10—42no—
178C2×Dic10—42no—
179C2×Dic10C22×C414yes—
180C2×Dic10C22×C412yes—
181C5×C4⋊C4—22no—
182C5×C4⋊C4—22no—
183C5×C4⋊C4—21no—
184C5×C4⋊C4—21no—
185C5×C4⋊C4—21no—
186C5×C4⋊C4—21no—
187C5×C4⋊C4—42no—
188C5×C4⋊C4—84no—
189C5×C4⋊C4C22⋊C418yes—
190C5×C22⋊C4—42no—
191C5×C22⋊C4C22⋊C418yes—
192C42×C5—42no—
193C42×C5C22×C414yes—
194C42×C5C2×D414yes—
195#685—216no—
196C4⋊Dic5—216no—
197#681—28no—
198C2×#168—28no—
199C2×#168—21no—
200C2×#168—22no—
201C2×#168—21no—
202C2×#168—21no—
203C2×#168—21no—
204C2×#168—22no—
205C2×#168C2×D412yes—
206C2×#168C2×D412yes—
207C2×#168C2×D414yes—
208C22×#41—51no—
209C23×Q8—51no—
210C22×M16—51no—
211C23×C8—51no—
212#536—58no—
213#536—104no—
214C2×C4⋊Q8—52no—
215C2×#124—52no—
216C2×#123—102no—
217C2×C22⋊Q8—102no—
218C2×#117C2×D511yescharacteristic
219C2×#117C2×D511yescharacteristic
220C22×C4⋊C4C2×D511yescharacteristic
221#419—104no—
222#417—104no—
223#416—104no—
224C2×#103—52no—
225C2×#103—52no—
226C2×#103—51no—
227C2×#103—51no—
228C2×#103—52no—
229C2×#103—52no—
230C2×#103—52no—
231C2×#103—104no—
232C2×#102—51no—
233C2×#102—51no—
234C2×#102—52no—
235#405—104no—
236C2×#97—52no—
237C2×C4×C8—52no—
238#389—22no—
239#385—22no—
240#384C2×D512yes—
241#382C2×D512yes—
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—
253C2×C4×C5—24no—
254C2×C4×C5—24no—
255C2×C4×C5—41no—
256C2×C4×C5—41no—
257C2×C4×C5—42no—
258C2×C4×C5—42no—
259C2×C4×C5—44no—
260C2×C4×C5—42no—
261C2×C4×C5—84no—
262C2×C4×C5C2×C22⋊C418yes—
263C2×C4×C5C2×C22⋊C412yes—
264C2×C4×C5C4×D414yes—
265C2×C4×C5C4×D414yes—
266C2×C4×C5#12114yes—
267C2×C4×C5#12114yes—
268C2×C4×C5#12412yes—
269C2×C4×C5#12712yes—
270C2×C4×C5C23×C411yescharacteristic
271C2×C4×C5C22×D411yescharacteristic
272C2×C4×C5C22×D411yescharacteristic
273C2×C4×C5C22×D411yescharacteristic
274C2×C4×C5C2×#4111yescharacteristic
275C2×C4×C5C2×#4111yescharacteristic
276C22⋊D5—48no—
277C2×Dic5—22no—
278C2×Dic5—24no—
279C2×Dic5—42no—
280C2×Dic5—41no—
281C2×Dic5—42no—
282C2×Dic5—41no—
283C2×Dic5—48no—
284D20—22no—
285D20—22no—
286D20—41no—
287D20—41no—
288C4×D5—48no—
289Dic10—24no—
290Dic10—24no—
291Dic10—22no—
292Dic10—22no—
293Dic10—42no—
294Dic10—42no—
295Dic10—41no—
296Dic10—41no—
297Dic10—48no—
298#168—22no—
299#168—22no—
300#168—24no—
301C2×#41—54no—
302C2×#41—102no—
303C2×#41—102no—
304C2×#41—102no—
305C2×#41—102no—
306C2×#41—102no—
307C22×Q8—54no—
308C22×Q8—52no—
309C22×Q8—51no—
310C22×Q8—102no—
311C22×Q8—101no—
312C22×Q8—102no—
313C22×Q8—102no—
314C22×Q8—102no—
315C22×Q8—102no—
316C22×D4—51no—
317C22×D4—101no—
318C23×C4—21no—
319C23×C4—101no—
320C23×C4—101no—
321C23×C4C22×D511yescharacteristic
322C2×M16—54no—
323C2×M16—102no—
324C2×M16—102no—
325C22×C8—54no—
326C22×C8—51no—
327C22×C8—51no—
328C22×C8—52no—
329C22×C8—102no—
330C22×C8—102no—
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#117C4×D514yes—
340#117D2014yes—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—24no—
345C2×C4⋊C4—104no—
346C2×C4⋊C4C22⋊D512yes—
347C2×C4⋊C4C22×D512yes—
348C2×C4⋊C4C22×D511yescharacteristic
349C2×C4⋊C4C22×D511yescharacteristic
350C2×C22⋊C4—21no—
351C2×C22⋊C4—104no—
352C2×C22⋊C4C22⋊D512yes—
353C2×C42—21no—
354C2×C42C22×D512yes—
355#103—104no—
356#103—104no—
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—24no—
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—28no—
384C4×C5—81no—
385C4×C5—82no—
386C4×C5—81no—
387C4×C5#39014yes—
388C4×C5C2×C4×D411yescharacteristic
389C4×C5C2×C4×D411yescharacteristic
390C4×C5C2×#12111yescharacteristic
391C4×C5C2×#12111yescharacteristic
392Dic5—44no—
393Dic5—42no—
394C24—101no—
395C24C2×C22⋊D511yescharacteristic
396#41—108no—
397#41—108no—
398#41—204no—
399#41—204no—
400C2×Q8—54no—
401C2×Q8—52no—
402C2×Q8—102no—
403C2×Q8—101no—
404C2×Q8—104no—
405C2×Q8—102no—
406C2×Q8—102no—
407C2×Q8—101no—
408C2×Q8—101no—
409C2×Q8—102no—
410C2×Q8—102no—
411C2×Q8—101no—
412C2×Q8—101no—
413C2×Q8—108no—
414C2×Q8—108no—
415C2×Q8—204no—
416C2×Q8—204no—
417C2×Q8—202no—
418C2×Q8—202no—
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—21no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—42no—
432C22×C4—82no—
433C22×C4—102no—
434C22×C4—101no—
435C22×C4—104no—
436C22×C4—104no—
437C22×C4—202no—
438C22×C4—202no—
439C22×C4—202no—
440C22×C4—202no—
441C22×C4C2×C4×D512yes—
442C22×C4C2×D2012yes—
443C22×C4D4×D512yes—
444C22×C4D4×D512yes—
445C22×C4#71312yes—
446C22×C4#71312yes—
447C22×C4C2×C22⋊D514yes—
448C22×C4C2×C22⋊D511yescharacteristic
449C22×C4C23×D511yescharacteristic
450M16—108no—
451C2×C8—52no—
452C2×C8—52no—
453C2×C8—54no—
454C2×C8—101no—
455C2×C8—102no—
456C2×C8—101no—
457C2×C8—101no—
458C2×C8—101no—
459C2×C8—102no—
460C2×C8—108no—
461C4⋊C4—22no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—22no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—42no—
468C4⋊C4—84no—
469C4⋊C4—1016no—
470C4⋊C4#68518yes—
471C22⋊C4—42no—
472C22⋊C4—1016no—
473C22⋊C4#68518yes—
474C42—42no—
475C42C2×C4×D514yes—
476C42C2×D2014yes—
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—201no—
489C23—202no—
490C23—202no—
491C23C2×#68512yes—
492C23#408612yes—
493C23#409312yes—
494C23C22×C22⋊D511yescharacteristic
495Q8—104no—
496Q8—104no—
497Q8—102no—
498Q8—102no—
499Q8—202no—
500Q8—202no—
501Q8—201no—
502Q8—201no—
503Q8—208no—
504D4—102no—
505D4—102no—
506D4—208no—
507D4—201no—
508D4—201no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—44no—
516C2×C4—42no—
517C2×C4—84no—
518C2×C4—104no—
519C2×C4—102no—
520C2×C4—201no—
521C2×C4—202no—
522C2×C4—201no—
523C2×C4—202no—
524C2×C4—208no—
525C2×C4—208no—
526C2×C4#403914yes—
527C2×C4#403914yes—
528C2×C4#404114yes—
529C2×C4#404114yes—
530C2×C4C2×#68518yes—
531C2×C4C2×#68512yes—
532C2×C4#408612yes—
533C2×C4#409312yes—
534C2×C4C22×C4×D511yescharacteristic
535C2×C4C22×D2011yescharacteristic
536C2×C4C2×D4×D511yescharacteristic
537C2×C4C2×D4×D511yescharacteristic
538C2×C4C2×#71311yescharacteristic
539C2×C4C2×#71311yescharacteristic
540C8—102no—
541C8—102no—
542C8—104no—
543C5#2120511yescharacteristic, complemented, Sylow
544C22—24no—
545C22—22no—
546C22—22no—
547C22—24no—
548C22—204no—
549C22—202no—
550C22—401no—
551C22—401no—
552C22—402no—
553C22#6527714yes—
554C22C22×#68511yescharacteristic
555C22C2×#408611yescharacteristic
556C22C2×#409311yescharacteristic
557C22#6617012yes—
558C22#6617212yes—
559C4—28no—
560C4—82no—
561C4—81no—
562C4—81no—
563C4—202no—
564C4—204no—
565C4#6527714yes—
566C4C2×#403911yescharacteristic
567C4C2×#403911yescharacteristic
568C4C2×#404111yescharacteristic
569C4C2×#404111yescharacteristic
570C2—24no—
571C2—402no—
572C2C2×#6527711yescharacteristic
573C2#1061430112yes—
574C2#1061511211yescharacteristic
575C2#1061513411yescharacteristic
576C2#1061520212yes—
subgroupG/H
centerC23C22×C22⋊D5
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC5×#2317C2
socleC23×C5C22×D4
prime radicalC22×#709C22
cosocleC22×C4×C5C24
hypercenter#2317D5
hypocenterC5#21205
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2317

Complements

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

NH ≅ G/Ntype
2. C5×#2317C2semidirect
3. C2×#65293C2semidirect
5. C2×#65286C2semidirect
6. C22×#3942C2semidirect
8. #10600374C2semidirect
9. #10600372C2semidirect
10. #10600345C2semidirect
11. #10600344C2semidirect
20. #65860C4semidirect
98. #4021C2×C4semidirect
100. #4015C2×C4semidirect
120. #2317D5semidirect
543. C5#21205semidirect

Maximal subgroups

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

subgroupindex
C2×#658602
C5×#23172
C2×#652932
C2×#652912
C2×#652862
C22×#39422
C2×#652552
#106003742
#106003722
#106003452
#106003442
#212055