Finite Group Databasebeta

← #12108927

Subgroups — #12108927

Order 1280.

1102 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#65872C211yesmaximal, characteristic
2C2×#65407C211yesmaximal, characteristic, complemented
3C2×#65008C211yesmaximal, characteristic, complemented
4C5×#1870C211yesmaximal, characteristic, complemented
5#10602789C211yesmaximal, characteristic, complemented
6#10602404C211yesmaximal, characteristic, complemented
7#10600746C211yesmaximal, characteristic, complemented
8#10600710C212yesmaximal, complemented
9#10600702C212yesmaximal, complemented
10#10599456C212yesmaximal, complemented
11#10599413C212yesmaximal, complemented
12C22×C5×M16C2211yescharacteristic
13C2×C4×C5×D4C2211yescharacteristic
14C2×#4085C411yescharacteristic
15C2×#4081C411yescharacteristic
16C2×C4×C22⋊D5C2211yescharacteristic
17C2×C4×C22⋊D5C2211yescharacteristic
18C2×#4072C2211yescharacteristic
19C22×C4×Dic5C2211yescharacteristic
20C22×#681C2211yescharacteristic
21#65872—24no—
22#65872—24no—
23C2×#4044C411yescharacteristic
24C2×#4035C411yescharacteristic
25C2×#4032C411yescharacteristic
26C2×#4031C411yescharacteristic
27C2×#4025C411yescharacteristic
28C2×#4024C411yescharacteristic
29C2×#4018C2211yescharacteristic
30C2×C4×Dic10C2211yescharacteristic
31C2×C5×C22⋊C8C2211yescharacteristic
32C2×C5×C22⋊C8C2212yes—
33C2×C22⋊#168C2211yescharacteristic
34C2×C22⋊#168C2212yes—
35#65407—24no—
36C2×#3952C2212yes—
37C2×#3952C2212yes—
38C2×#3947C2212yes—
39C2×#3946C2212yes—
40C2×#3946C2212yes—
41C2×#3946C2212yes—
42#65008—24no—
43C5×#336C2212yes—
44#64770C2212yes—
45#64770C2212yes—
46#64744C2212yes—
47#21854—51nomaximal, Sylow
48C22×C5×D4C2×C411yescharacteristic
49C23×C4×C5C2311yescharacteristic
50C23×C4×C5C2311yescharacteristic
51C22×C22⋊D5C2×C411yescharacteristic
52C22×C22⋊D5C2×C411yescharacteristic
53C23×Dic5C2×C411yescharacteristic
54C23×Dic5C2×C411yescharacteristic
55C22×C4×D5C2311yescharacteristic
56C22×Dic10C2×C411yescharacteristic
57C2×C5×M16—22no—
58C2×C5×M16—42no—
59C22×C5×C8C2312yes—
60C22×C5×C8C2312yes—
61C4×C5×D4—22no—
62C4×C5×D4—22no—
63C2×C5×C4⋊C4C2×C411yescharacteristic
64C2×C5×C22⋊C4C2×C411yescharacteristic
65C2×C5×C22⋊C4C2×C411yescharacteristic
66C2×C42×C5C2311yescharacteristic
67C2×C22⋊Dic5C2×C411yescharacteristic
68C2×C22⋊Dic5C2×C411yescharacteristic
69C2×C22⋊Dic5C2×C411yescharacteristic
70C2×C22⋊Dic5C2×C411yescharacteristic
71#4085—24no—
72#4081—24no—
73C4×C22⋊D5—22no—
74C4×C22⋊D5—22no—
75C4×C22⋊D5—22no—
76C4×C22⋊D5—22no—
77C2×#685C2×C411yescharacteristic
78C2×#685C2×C411yescharacteristic
79C2×#685C2×C411yescharacteristic
80C2×#685C2×C411yescharacteristic
81#4072—22no—
82#4072—22no—
83C2×C4⋊Dic5C2×C411yescharacteristic
84C2×C4⋊Dic5C2×C411yescharacteristic
85C2×C4⋊Dic5C2×C411yescharacteristic
86C2×#683C2×C411yescharacteristic
87C2×#683C2×C411yescharacteristic
88C2×#683C2×C411yescharacteristic
89C2×#683C2×C411yescharacteristic
90C2×#683C2×C411yescharacteristic
91C2×#683C2×C411yescharacteristic
92C2×C4×Dic5C2×C411yescharacteristic
93C2×C4×Dic5C2×C411yescharacteristic
94C2×C4×Dic5D414yes—
95C2×C4×Dic5D414yes—
96C2×C4×Dic5C2311yescharacteristic
97C2×C4×Dic5C2311yescharacteristic
98C2×C4×Dic5C2311yescharacteristic
99C2×#681—22no—
100C2×#681—42no—
101C22×#168C2312yes—
102C22×#168C2312yes—
103#4044—24no—
104#4035—24no—
105#4032—24no—
106#4031—24no—
107#4025—24no—
108#4024—24no—
109#4018—22no—
110#4018—22no—
111C4×Dic10—22no—
112C4×Dic10—22no—
113C5×C22⋊C8—22no—
114C5×C22⋊C8—42no—
115C22⋊#168—22no—
116C22⋊#168—42no—
117#3952—42no—
118#3952—42no—
119#3947—24no—
120#3946—24no—
121#3946—24no—
122#3946—24no—
123C2×#529—51no—
124C2×#432—51no—
125C2×#423—51no—
126#1985—52no—
127#1955—52no—
128#1870—51no—
129#1870—51no—
130#1870—51no—
131#1870D511yescharacteristic, complemented
132#1798—52no—
133#1798—52no—
134C24×C5C22×C411yescharacteristic
135C24×C5C22×C411yescharacteristic
136C23×D5C22×C411yescharacteristic
137C2×C5×D4—22no—
138C2×C5×D4—41no—
139C2×C5×D4—41no—
140C22×C4×C5—22no—
141C22×C4×C5—22no—
142C22×C4×C5—21no—
143C22×C4×C5—21no—
144C22×C4×C5—22no—
145C22×C4×C5—22no—
146C22×C4×C5C22×C411yescharacteristic
147C22×C4×C5C22×C411yescharacteristic
148C22×C4×C5C22×C411yescharacteristic
149C22×C4×C5C22×C411yescharacteristic
150C22×C4×C5C2×D412yes—
151C22×C4×C5C2×D412yes—
152C22×C4×C5C2411yescharacteristic
153C2×C22⋊D5—22no—
154C2×C22⋊D5—22no—
155C2×C22⋊D5—42no—
156C2×C22⋊D5—41no—
157C2×C22⋊D5—41no—
158C22×Dic5—42no—
159C22×Dic5—41no—
160C22×Dic5—41no—
161C22×Dic5C22⋊C414yes—
162C22×Dic5C22⋊C414yes—
163C22×Dic5C22×C411yescharacteristic
164C22×Dic5C22×C411yescharacteristic
165C22×Dic5C22×C411yescharacteristic
166C22×Dic5C22×C411yescharacteristic
167C22×Dic5C22×C411yescharacteristic
168C22×Dic5C22×C411yescharacteristic
169C22×Dic5C22×C411yescharacteristic
170C2×C4×D5—21no—
171C2×C4×D5—21no—
172C2×C4×D5—22no—
173C2×C4×D5—22no—
174C2×Dic10—22no—
175C2×Dic10—42no—
176C5×M16—44no—
177C2×C5×C8—22no—
178C2×C5×C8—22no—
179C2×C5×C8—22no—
180C2×C5×C8—22no—
181C2×C5×C8—22no—
182C2×C5×C8—22no—
183C5×C4⋊C4—22no—
184C5×C22⋊C4—22no—
185C5×C22⋊C4—22no—
186C42×C5—21no—
187C42×C5—21no—
188C22⋊Dic5—22no—
189C22⋊Dic5—22no—
190C22⋊Dic5—22no—
191C22⋊Dic5—22no—
192#685—22no—
193#685—22no—
194#685—22no—
195#685—22no—
196C4⋊Dic5—22no—
197C4⋊Dic5—22no—
198C4⋊Dic5—22no—
199#683—22no—
200#683—22no—
201#683—22no—
202#683—22no—
203#683—22no—
204#683—22no—
205C4×Dic5—21no—
206C4×Dic5—21no—
207C4×Dic5C22⋊C414yes—
208C4×Dic5C22⋊C414yes—
209C4×Dic5C2×D412yes—
210C4×Dic5C2×D412yes—
211C4×Dic5C2×D412yes—
212C4×Dic5C2×D412yes—
213#681—44no—
214C2×#168—22no—
215C2×#168—22no—
216C2×#168—22no—
217C2×#168—22no—
218C2×#168—22no—
219C2×#168—22no—
220C22×M16—51no—
221C22×M16C2×D511yescharacteristic
222#529—104no—
223#529—104no—
224C2×#126—51no—
225C2×#126—51no—
226C2×#125—51no—
227C2×#124—51no—
228C2×#123—51no—
229C2×#123—51no—
230C2×C22⋊Q8—51no—
231C2×#121—51no—
232C2×C4×Q8—51no—
233C2×C4×D4—51no—
234C2×C4×D4—51no—
235C2×C4×D4C2×D511yescharacteristic
236C2×#117—51no—
237C2×#117—51no—
238C22×C42—51no—
239#432—104no—
240#423—104no—
241C2×C4⋊C8—52no—
242C2×C4⋊C8—52no—
243C2×C4⋊C8—52no—
244C2×C22⋊C8—51no—
245C2×C22⋊C8—52no—
246C2×C22⋊C8—52no—
247C2×C22⋊C8—52no—
248C2×C22⋊C8C2×D511yescharacteristic
249C2×C22⋊C8C2×D512yes—
250C2×#97—52no—
251#336—52no—
252#336—52no—
253#336—52no—
254#336C2×D512yes—
255C23×C5—22no—
256C23×C5—21no—
257C23×C5—21no—
258C23×C5—22no—
259C23×C5—21no—
260C23×C5—22no—
261C23×C5—22no—
262C23×C5—21no—
263C23×C5—21no—
264C23×C5C2×C22⋊C412yes—
265C23×C5C23×C411yescharacteristic
266C22×D5—21no—
267C22×D5—22no—
268C22×D5—21no—
269C22×D5—21no—
270C22×D5—22no—
271C5×D4—44no—
272C2×C4×C5—22no—
273C2×C4×C5—22no—
274C2×C4×C5—22no—
275C2×C4×C5—24no—
276C2×C4×C5—21no—
277C2×C4×C5—21no—
278C2×C4×C5—21no—
279C2×C4×C5—22no—
280C2×C4×C5—22no—
281C2×C4×C5—21no—
282C2×C4×C5—21no—
283C2×C4×C5—41no—
284C2×C4×C5—41no—
285C2×C4×C5—42no—
286C2×C4×C5C2×C22⋊C412yes—
287C2×C4×C5C22×D411yescharacteristic
288C2×C4×C5C22×D411yescharacteristic
289C2×C4×C5H(5,𝔽2)12yes—
290C2×C4×C52−1+412yes—
291C22⋊D5—44no—
292C22⋊D5—44no—
293C2×Dic5—22no—
294C2×Dic5—21no—
295C2×Dic5—21no—
296C2×Dic5—21no—
297C2×Dic5—21no—
298C2×Dic5—22no—
299C2×Dic5—21no—
300C2×Dic5—21no—
301C2×Dic5—21no—
302C2×Dic5—22no—
303C2×Dic5—22no—
304C2×Dic5—21no—
305C2×Dic5—21no—
306C2×Dic5—44no—
307C2×Dic5—44no—
308C2×Dic5C2×C22⋊C412yes—
309C2×Dic5C2×C22⋊C412yes—
310C2×Dic5C2×C22⋊C412yes—
311C2×Dic5C2×C22⋊C412yes—
312C4×D5—41no—
313C4×D5—41no—
314C4×D5—42no—
315Dic10—44no—
316C5×C8—42no—
317C5×C8—42no—
318#168—42no—
319#168—42no—
320C22×Q8—51no—
321C22×D4—51no—
322C22×D4—51no—
323C22×D4C4×D511yescharacteristic
324C23×C4—51no—
325C23×C4—51no—
326C23×C4—51no—
327C23×C4C22×D511yescharacteristic
328C23×C4C22×D511yescharacteristic
329C2×M16—22no—
330C2×M16—42no—
331C2×M16—102no—
332C2×M16—202no—
333C22×C8—52no—
334C22×C8—52no—
335C22×C8C22×D512yes—
336C22×C8C22×D512yes—
337#126—104no—
338#126—104no—
339#125—104no—
340#124—104no—
341#123—104no—
342#123—104no—
343C22⋊Q8—104no—
344#121—104no—
345C4×Q8—102no—
346C4×Q8—102no—
347C4×D4—22no—
348C4×D4—22no—
349C4×D4—102no—
350C4×D4—102no—
351C4×D4—102no—
352C4×D4—102no—
353#117—102no—
354#117—102no—
355#117—102no—
356#117—102no—
357C2×C4⋊C4—51no—
358C2×C4⋊C4—51no—
359C2×C4⋊C4—51no—
360C2×C4⋊C4—51no—
361C2×C4⋊C4—51no—
362C2×C4⋊C4—51no—
363C2×C4⋊C4—51no—
364C2×C4⋊C4—51no—
365C2×C4⋊C4—51no—
366C2×C4⋊C4C4×D511yescharacteristic
367C2×C22⋊C4—51no—
368C2×C22⋊C4—51no—
369C2×C22⋊C4—51no—
370C2×C22⋊C4—51no—
371C2×C22⋊C4—51no—
372C2×C22⋊C4—51no—
373C2×C22⋊C4—51no—
374C2×C22⋊C4—51no—
375C2×C22⋊C4C4×D511yescharacteristic
376C2×C22⋊C4C4×D511yescharacteristic
377C2×C42—51no—
378C2×C42—51no—
379C2×C42—54no—
380C2×C42—54no—
381C2×C42—51no—
382C2×C42—51no—
383C2×C42—51no—
384C2×C42C22×D511yescharacteristic
385C4⋊C8—104no—
386C4⋊C8—104no—
387C4⋊C8—104no—
388C22⋊C8—22no—
389C22⋊C8—42no—
390C22⋊C8—102no—
391C22⋊C8—202no—
392C22⋊C8—202no—
393C22⋊C8—202no—
394#97—104no—
395C22×C5—22no—
396C22×C5—22no—
397C22×C5—22no—
398C22×C5—42no—
399C22×C5—42no—
400C22×C5—41no—
401C22×C5—41no—
402C22×C5—41no—
403C22×C5—42no—
404C22×C5—42no—
405C22×C5C22×C22⋊C411yescharacteristic
406C22×C5#52012yes—
407C22×C5#56811yescharacteristic
408C22×C5#56811yescharacteristic
409C22×C5#56812yes—
410C2×D5—41no—
411C2×D5—41no—
412C2×D5—41no—
413C2×D5—42no—
414C2×D5—42no—
415C4×C5—22no—
416C4×C5—41no—
417C4×C5—41no—
418C4×C5—41no—
419C4×C5#53511yescharacteristic
420C4×C5#53611yescharacteristic
421C4×C5#53712yes—
422Dic5—22no—
423Dic5—22no—
424Dic5—22no—
425Dic5—22no—
426Dic5—41no—
427Dic5—41no—
428Dic5—41no—
429C24—51no—
430C24C2×C4×D511yescharacteristic
431C24C2×C4×D511yescharacteristic
432C2×Q8—102no—
433C2×Q8—202no—
434C2×D4—22no—
435C2×D4—41no—
436C2×D4—41no—
437C2×D4—102no—
438C2×D4—102no—
439C2×D4—202no—
440C2×D4—201no—
441C2×D4—201no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—51no—
449C22×C4—51no—
450C22×C4—51no—
451C22×C4—51no—
452C22×C4—54no—
453C22×C4—54no—
454C22×C4—51no—
455C22×C4—51no—
456C22×C4—51no—
457C22×C4—101no—
458C22×C4—101no—
459C22×C4—102no—
460C22×C4—102no—
461C22×C4—201no—
462C22×C4—201no—
463C22×C4—202no—
464C22×C4C2×C4×D511yescharacteristic
465C22×C4C2×C4×D511yescharacteristic
466C22×C4C2×C4×D511yescharacteristic
467C22×C4C2×C4×D511yescharacteristic
468C22×C4D4×D512yes—
469C22×C4D4×D512yes—
470C22×C4C23×D511yescharacteristic
471M16—44no—
472M16—204no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—102no—
480C2×C8—102no—
481C2×C8—102no—
482C2×C8—102no—
483C2×C8—102no—
484C2×C8—102no—
485C4⋊C4—22no—
486C4⋊C4—102no—
487C4⋊C4—102no—
488C4⋊C4—102no—
489C4⋊C4—102no—
490C4⋊C4—102no—
491C4⋊C4—102no—
492C4⋊C4—102no—
493C4⋊C4—102no—
494C4⋊C4—102no—
495C22⋊C4—22no—
496C22⋊C4—22no—
497C22⋊C4—102no—
498C22⋊C4—102no—
499C22⋊C4—102no—
500C22⋊C4—102no—
501C22⋊C4—102no—
502C22⋊C4—102no—
503C22⋊C4—102no—
504C22⋊C4—102no—
505C42—21no—
506C42—21no—
507C42—52no—
508C42—52no—
509C42—52no—
510C42—52no—
511C42—54no—
512C42—54no—
513C42—101no—
514C42—101no—
515C2×C5—42no—
516C2×C5—81no—
517C2×C5#231111yescharacteristic
518C2×C5#287911yescharacteristic
519C2×C5#287911yescharacteristic
520C2×C5#299512yes—
521C2×C5#299512yes—
522D5—81no—
523C23—21no—
524C23—21no—
525C23—22no—
526C23—22no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—22no—
531C23—22no—
532C23—101no—
533C23—101no—
534C23—101no—
535C23—102no—
536C23—102no—
537C23D5×C22⋊C412yes—
538C23C22×C4×D511yescharacteristic
539Q8—204no—
540D4—44no—
541D4—204no—
542D4—204no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—41no—
557C2×C4—52no—
558C2×C4—52no—
559C2×C4—52no—
560C2×C4—52no—
561C2×C4—101no—
562C2×C4—101no—
563C2×C4—101no—
564C2×C4—101no—
565C2×C4—102no—
566C2×C4—102no—
567C2×C4—101no—
568C2×C4—101no—
569C2×C4—101no—
570C2×C4—101no—
571C2×C4—101no—
572C2×C4—102no—
573C2×C4—102no—
574C2×C4—204no—
575C2×C4—204no—
576C2×C4—201no—
577C2×C4—202no—
578C2×C4—201no—
579C2×C4D5×C22⋊C412yes—
580C2×C4C2×D4×D511yescharacteristic
581C2×C4C2×D4×D511yescharacteristic
582C2×C4#414412yes—
583C2×C4#415012yes—
584C8—42no—
585C8—42no—
586C8—202no—
587C8—202no—
588C5#2185411yescharacteristic, complemented, Sylow
589C22—22no—
590C22—22no—
591C22—22no—
592C22—42no—
593C22—42no—
594C22—41no—
595C22—42no—
596C22—41no—
597C22—41no—
598C22—42no—
599C22—201no—
600C22—202no—
601C22—201no—
602C22—201no—
603C22—202no—
604C22C2×D5×C22⋊C411yescharacteristic
605C22#6588012yes—
606C22#6607911yescharacteristic
607C22#6607911yescharacteristic
608C22#6608412yes—
609C4—22no—
610C4—41no—
611C4—41no—
612C4—41no—
613C4—102no—
614C4—102no—
615C4—102no—
616C4—102no—
617C4—201no—
618C4—201no—
619C4—201no—
620C4#6592611yescharacteristic
621C4#6593112yes—
622C4#6598211yescharacteristic
623C2—42no—
624C2—81no—
625C2—401no—
626C2#1060639311yescharacteristic
627C2#1061036111yescharacteristic
628C2#1061039311yescharacteristic
629C2#1061119812yes—
630C2#1061125912yes—
subgroupG/H
center, FrattiniC22×C4C23×D5
commutator, socleC23×C5C23×C4
FittingC5×#1870C2
prime radicalC2×#4085C4
cosocleC22×C4×C5C24
hypercenter#1870D5
hypocenterC5#21854
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C22×C4 ◁ #1870

Complements

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

NH ≅ G/Ntype
2. C2×#65407C2semidirect
3. C2×#65008C2semidirect
4. C5×#1870C2semidirect
5. #10602789C2semidirect
6. #10602404C2semidirect
7. #10600746C2semidirect
8. #10600710C2semidirect
9. #10600702C2semidirect
10. #10599456C2semidirect
11. #10599413C2semidirect
131. #1870D5semidirect
588. C5#21854semidirect

Maximal subgroups

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

subgroupindex
C2×#658722
C2×#654072
C2×#650082
C5×#18702
#106027892
#106024042
#106007462
#106007102
#106007022
#105994562
#105994132
#218545