Finite Group Databasebeta

← #12107178

Subgroups — #12107178

Order 1280.

884 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#65117C211yesmaximal, characteristic, complemented
2C2×#65125C211yesmaximal, characteristic
3C4×#4021C211yesmaximal, characteristic
4C5×#1831C211yesmaximal, characteristic, complemented
5#10601266C211yesmaximal, characteristic, complemented
6#10601264C211yesmaximal, characteristic
7C4×#3935C211yesmaximal, characteristic, complemented
8#10599451C211yesmaximal, characteristic, complemented
9#10599451C211yesmaximal, characteristic, complemented
10#10599450C211yesmaximal, characteristic, complemented
11#10599450C211yesmaximal, characteristic, complemented
12#10599443C211yesmaximal, characteristic, complemented
13#10599442C211yesmaximal, characteristic, complemented
14#10599441C211yesmaximal, characteristic, complemented
15#10599440C211yesmaximal, characteristic, complemented
16C22×#676C2211yescharacteristic
17C22×C8×D5C2211yescharacteristic
18C2×#4021C411yescharacteristic
19C2×C4×D20C2211yescharacteristic
20C2×C4×Dic10C2211yescharacteristic
21C2×C5×C4⋊C8C2211yescharacteristic
22C2×C5×C4⋊C8C2211yescharacteristic
23C43×C5C2211yescharacteristic
24C2×#3952C2211yescharacteristic
25C2×#3952C2211yescharacteristic
26C2×#3952C2211yescharacteristic
27C2×#3952C2211yescharacteristic
28C2×#3952C2211yescharacteristic
29C2×#3952C2211yescharacteristic
30C2×#3947C2211yescharacteristic
31C2×#3946C2211yescharacteristic
32C2×#3946C2211yescharacteristic
33C2×C8×Dic5C2211yescharacteristic
34#65229C411yescharacteristic
35#65229C411yescharacteristic
36#65229C411yescharacteristic
37#65229C411yescharacteristic
38#65228C411yescharacteristic
39C4×#685C2211yescharacteristic
40C4×#685C2211yescharacteristic
41C4×#685C2211yescharacteristic
42C4×#685C2211yescharacteristic
43#65225C411yescharacteristic
44#65218C411yescharacteristic
45C2×#3935C2211yescharacteristic
46C2×#3935C2211yescharacteristic
47C2×#3935C2211yescharacteristic
48C2×#3935C2211yescharacteristic
49C2×C4×#168C2211yescharacteristic
50C2×C4×#168C2211yescharacteristic
51#65125—24no—
52#65117—24no—
53C5×#336C2211yescharacteristic
54C5×#336C2211yescharacteristic
55C5×#336C2211yescharacteristic
56C5×#336C2211yescharacteristic
57#64770C2211yescharacteristic
58#64770C2211yescharacteristic
59#64770C2211yescharacteristic
60#64770C2211yescharacteristic
61#21632—51nomaximal, Sylow
62C22×D20C2×C411yescharacteristic
63C22×C4×D5C2311yescharacteristic
64C22×C4×D5C2311yescharacteristic
65C22×Dic10C2×C411yescharacteristic
66C22×C5×C8C2311yescharacteristic
67C22×C5×C8C2311yescharacteristic
68C22×C5×C8C2311yescharacteristic
69C22×C5×C8C2311yescharacteristic
70C2×C42×C5C2×C411yescharacteristic
71C2×C42×C5C2×C411yescharacteristic
72C2×C42×C5C2×C411yescharacteristic
73C2×C42×C5C2×C411yescharacteristic
74C2×C42×C5C2311yescharacteristic
75C2×C42×C5C2311yescharacteristic
76C2×C42×C5C2311yescharacteristic
77C2×#685C2×C411yescharacteristic
78C2×#685C2×C411yescharacteristic
79C2×#685C2×C411yescharacteristic
80C2×#685C2×C411yescharacteristic
81C2×#685C2×C411yescharacteristic
82C2×#685C2×C411yescharacteristic
83C2×#685C2×C411yescharacteristic
84C2×#685C2×C411yescharacteristic
85C2×#685C2×C411yescharacteristic
86C2×#685C2×C411yescharacteristic
87C2×C4⋊Dic5C2×C411yescharacteristic
88C2×C4⋊Dic5C2×C411yescharacteristic
89C2×#683C2×C411yescharacteristic
90C2×#683C2×C411yescharacteristic
91C2×C4×Dic5C2311yescharacteristic
92C2×C4×Dic5C2311yescharacteristic
93C22×#168C2311yescharacteristic
94C22×#168C2311yescharacteristic
95C22×#168C2311yescharacteristic
96C22×#168C2311yescharacteristic
97C2×#676—22no—
98C2×#676—22no—
99C2×#676—22no—
100C2×C8×D5—22no—
101C2×C8×D5—22no—
102C2×C8×D5—22no—
103#4021—24no—
104C4×D20—22no—
105C4×D20—22no—
106C4×Dic10—22no—
107C4×Dic10—22no—
108C5×C4⋊C8—22no—
109C5×C4⋊C8—22no—
110#3963C2×C411yescharacteristic
111#3963C2×C411yescharacteristic
112#3963C2×C411yescharacteristic
113#3963C2×C411yescharacteristic
114#3963C2×C411yescharacteristic
115#3963C2×C411yescharacteristic
116#3963C2×C411yescharacteristic
117#3963C2×C411yescharacteristic
118#3952—22no—
119#3952—22no—
120#3952—22no—
121#3952—22no—
122#3952—22no—
123#3952—22no—
124#3947—22no—
125#3946—22no—
126#3946—22no—
127C8×Dic5—22no—
128#3935—22no—
129#3935—22no—
130#3935—22no—
131#3935D414yes—
132C4×#168—22no—
133C4×#168—22no—
134C2×#446—51no—
135C2×#435—51no—
136C4×#124—51no—
137#1985—51no—
138#1955—51no—
139#1924—51no—
140#1924—51no—
141#1923—51no—
142#1923—51no—
143#1831D511yescharacteristic, complemented
144#1753—51no—
145#1753—51no—
146#1752—51no—
147C8×C22⋊C4—51no—
148C4×#97—51no—
149C23×D5C22×C411yescharacteristic
150C23×D5C22×C411yescharacteristic
151C22×C4×C5C22×C411yescharacteristic
152C22×C4×C5C22×C411yescharacteristic
153C22×C4×C5C22×C411yescharacteristic
154C22×C4×C5C22×C411yescharacteristic
155C22×C4×C5C22×C411yescharacteristic
156C22×C4×C5C22×C411yescharacteristic
157C22×C4×C5C2411yescharacteristic
158C22×Dic5C22×C411yescharacteristic
159C22×Dic5C22×C411yescharacteristic
160C22×Dic5C22×C411yescharacteristic
161C22×Dic5C22×C411yescharacteristic
162C22×Dic5C22×C411yescharacteristic
163C22×Dic5C22×C411yescharacteristic
164C2×D20—22no—
165C2×D20—22no—
166C2×D20—41no—
167C2×C4×D5—21no—
168C2×C4×D5—21no—
169C2×C4×D5—21no—
170C2×C4×D5—21no—
171C2×C4×D5—21no—
172C2×C4×D5—21no—
173C2×C4×D5—21no—
174C2×C4×D5—21no—
175C2×C4×D5—21no—
176C2×C4×D5—21no—
177C2×C4×D5—21no—
178C2×C4×D5—21no—
179C2×Dic10—22no—
180C2×Dic10—22no—
181C2×Dic10—41no—
182C2×C5×C8—21no—
183C2×C5×C8—21no—
184C2×C5×C8—21no—
185C2×C5×C8—21no—
186C2×C5×C8—21no—
187C2×C5×C8—21no—
188C2×C5×C8—21no—
189C2×C5×C8—21no—
190C2×C5×C8—21no—
191C2×C5×C8—21no—
192C2×C5×C8—21no—
193C2×C5×C8—21no—
194C42×C5—22no—
195C42×C5—22no—
196C42×C5—21no—
197C42×C5—21no—
198C42×C5—21no—
199C42×C5—21no—
200C42×C5—41no—
201C42×C5—41no—
202C42×C5C2×D412yes—
203C42×C5#4112yes—
204#685—22no—
205#685—22no—
206#685—22no—
207#685—22no—
208#685—22no—
209#685—22no—
210#685—22no—
211#685—41no—
212#685—41no—
213#685M1614yescomplemented
214C4⋊Dic5—22no—
215C4⋊Dic5—22no—
216#683—22no—
217#683—22no—
218C4×Dic5—21no—
219C4×Dic5—21no—
220C4×Dic5—21no—
221C4×Dic5—21no—
222C2×#168—21no—
223C2×#168—21no—
224C2×#168—21no—
225C2×#168—21no—
226C2×#168—21no—
227C2×#168—21no—
228C2×#168C2×D412yes—
229C2×#168C2×D412yes—
230C2×#168#4112yes—
231C2×#168#4112yes—
232C2×#168#4112yes—
233C2×#168#4112yes—
234#676—44no—
235C8×D5—44no—
236C22×M16—51no—
237C23×C8—51no—
238C2×#124—51no—
239C2×C4×Q8—51no—
240C2×C4×D4—51no—
241#446—104no—
242#435—104no—
243C2×C4⋊C8—51no—
244C2×C4⋊C8—51no—
245C2×C4⋊C8C2×D511yescharacteristic
246C2×C4⋊C8C2×D511yescharacteristic
247C2×C22⋊C8—51no—
248C2×C22⋊C8—51no—
249C2×C22⋊C8—51no—
250C2×C22⋊C8—51no—
251C2×C22⋊C8—51no—
252C2×C22⋊C8—51no—
253C2×#97—51no—
254C2×#97—51no—
255C2×#97—51no—
256C2×#97—51no—
257C2×#97—51no—
258C2×C4×C8—51no—
259C2×C4×C8—51no—
260C2×C4×C8—51no—
261#391—51no—
262#390—51no—
263#388—51no—
264#388—51no—
265#388—51no—
266#388—51no—
267#382—51no—
268C4×C22⋊C4—51no—
269C4×C22⋊C4—51no—
270C4×C22⋊C4—51no—
271C4×C22⋊C4—51no—
272C43C2×D511yescharacteristic
273#336—51no—
274#336—51no—
275#336—51no—
276#336—51no—
277#336C2×D511yescharacteristic
278#336C2×D511yescharacteristic
279#336C2×D511yescharacteristic
280#336C2×D511yescharacteristic
281C23×C5C23×C411yescharacteristic
282C22×D5—21no—
283C22×D5—21no—
284C22×D5—21no—
285C22×D5—21no—
286C22×D5—21no—
287C22×D5—21no—
288C22×D5—21no—
289C22×D5—21no—
290C22×D5—21no—
291C22×D5—21no—
292C22×D5—21no—
293C22×D5—21no—
294C22×D5—21no—
295C22×D5C2×M1612yes—
296C2×C4×C5—21no—
297C2×C4×C5—21no—
298C2×C4×C5—21no—
299C2×C4×C5—21no—
300C2×C4×C5—21no—
301C2×C4×C5—21no—
302C2×C4×C5—21no—
303C2×C4×C5—21no—
304C2×C4×C5—21no—
305C2×C4×C5—21no—
306C2×C4×C5—21no—
307C2×C4×C5—21no—
308C2×C4×C5—21no—
309C2×C4×C5—21no—
310C2×C4×C5C4×D412yes—
311C2×C4×C5C4×D412yes—
312C2×C4×C5C2×M1612yes—
313C2×C4×C5#13112yes—
314C2×C4×C5C22×D411yescharacteristic
315C2×C4×C5C2×#4111yescharacteristic
316C2×C4×C5C2×#4111yescharacteristic
317C2×C4×C5C2×#4111yescharacteristic
318C2×C4×C5C2×#4111yescharacteristic
319C2×C4×C5C2×#4111yescharacteristic
320C2×Dic5—21no—
321C2×Dic5—21no—
322C2×Dic5—21no—
323C2×Dic5—21no—
324C2×Dic5—21no—
325C2×Dic5—21no—
326C2×Dic5—21no—
327C2×Dic5—21no—
328C2×Dic5—21no—
329C2×Dic5—21no—
330C2×Dic5—21no—
331C2×Dic5—21no—
332C2×Dic5—21no—
333C2×Dic5—21no—
334C2×Dic5—21no—
335C2×Dic5—21no—
336C2×Dic5—21no—
337C2×Dic5C2×M1612yes—
338D20—44no—
339C4×D5—41no—
340C4×D5—41no—
341C4×D5—41no—
342C4×D5—41no—
343C4×D5—41no—
344C4×D5—41no—
345C4×D5—41no—
346C4×D5—41no—
347Dic10—44no—
348C5×C8—41no—
349C5×C8—41no—
350C5×C8—41no—
351C5×C8—41no—
352#168—22no—
353#168—22no—
354#168—22no—
355#168#12414yescomplemented
356C22×Q8—51no—
357C22×D4—51no—
358C23×C4—51no—
359C23×C4—51no—
360C2×M16—102no—
361C2×M16—102no—
362C2×M16—102no—
363C22×C8—51no—
364C22×C8—51no—
365C22×C8—51no—
366C22×C8—51no—
367C22×C8—102no—
368C22×C8—102no—
369C22×C8—102no—
370C22×C8C22×D511yescharacteristic
371C22×C8C22×D511yescharacteristic
372C22×C8C22×D511yescharacteristic
373C22×C8C22×D511yescharacteristic
374#124—104no—
375C4×Q8—102no—
376C4×Q8—102no—
377C4×D4—102no—
378C4×D4—102no—
379C2×C4⋊C4—51no—
380C2×C4⋊C4—51no—
381C2×C4⋊C4—51no—
382C2×C4⋊C4—51no—
383C2×C22⋊C4—51no—
384C2×C22⋊C4—51no—
385C2×C22⋊C4—51no—
386C2×C22⋊C4—51no—
387C2×C22⋊C4—51no—
388C2×C22⋊C4—51no—
389C2×C22⋊C4—51no—
390C2×C22⋊C4—51no—
391C2×C22⋊C4—51no—
392C2×C22⋊C4—51no—
393C2×C42—51no—
394C2×C42—51no—
395C2×C42C4×D511yescharacteristic
396C2×C42C4×D511yescharacteristic
397C2×C42C4×D511yescharacteristic
398C2×C42C4×D511yescharacteristic
399C2×C42C22×D511yescharacteristic
400C2×C42C22×D511yescharacteristic
401C2×C42C22×D511yescharacteristic
402C4⋊C8—22no—
403C4⋊C8—22no—
404C4⋊C8—102no—
405C4⋊C8—102no—
406C22⋊C8—102no—
407C22⋊C8—102no—
408C22⋊C8—102no—
409C22⋊C8—102no—
410C22⋊C8—102no—
411C22⋊C8—102no—
412#97—54no—
413#97—102no—
414#97—102no—
415#97—102no—
416#97—102no—
417C4×C8—102no—
418C4×C8—102no—
419C4×C8—102no—
420#95—51no—
421#95—51no—
422#95—51no—
423#95—51no—
424#95—51no—
425#95—51no—
426#95—51no—
427#95—51no—
428C22×C5C2×C4×D411yescharacteristic
429C22×C5C4×#4111yescharacteristic
430C22×C5C4×#4111yescharacteristic
431C22×C5C22×M1611yescharacteristic
432C22×C5C2×#13111yescharacteristic
433C22×C5#56811yescharacteristic
434C22×C5#56811yescharacteristic
435C2×D5—22no—
436C2×D5—22no—
437C2×D5—22no—
438C2×D5—41no—
439C2×D5—41no—
440C2×D5—41no—
441C2×D5—41no—
442C2×D5—41no—
443C2×D5—41no—
444C2×D5—41no—
445C2×D5—41no—
446C2×D5—41no—
447C2×D5—41no—
448C2×D5—41no—
449C4×C5—22no—
450C4×C5—22no—
451C4×C5—22no—
452C4×C5—22no—
453C4×C5—41no—
454C4×C5—41no—
455C4×C5C2×#12411yescharacteristic
456C4×C5#52911yescharacteristic
457C4×C5#52911yescharacteristic
458C4×C5#53211yescharacteristic
459Dic5—22no—
460Dic5—41no—
461Dic5—41no—
462Dic5—41no—
463Dic5—41no—
464Dic5—41no—
465C24—51no—
466C24—51no—
467C2×Q8—102no—
468C2×Q8—102no—
469C2×Q8—201no—
470C2×D4—102no—
471C2×D4—102no—
472C2×D4—201no—
473C22×C4—51no—
474C22×C4—51no—
475C22×C4—51no—
476C22×C4—51no—
477C22×C4—51no—
478C22×C4—51no—
479C22×C4—101no—
480C22×C4—101no—
481C22×C4—101no—
482C22×C4—101no—
483C22×C4—101no—
484C22×C4—101no—
485C22×C4—101no—
486C22×C4—101no—
487C22×C4—101no—
488C22×C4—101no—
489C22×C4—101no—
490C22×C4—101no—
491C22×C4C2×C4×D511yescharacteristic
492C22×C4C2×C4×D511yescharacteristic
493C22×C4C2×C4×D511yescharacteristic
494C22×C4C2×C4×D511yescharacteristic
495C22×C4C2×C4×D511yescharacteristic
496C22×C4C2×C4×D511yescharacteristic
497C22×C4C23×D511yescharacteristic
498M16—204no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—52no—
512C2×C8—52no—
513C2×C8—52no—
514C2×C8—52no—
515C2×C8—52no—
516C2×C8—52no—
517C2×C8—101no—
518C2×C8—101no—
519C2×C8—101no—
520C2×C8—101no—
521C2×C8—101no—
522C2×C8—101no—
523C2×C8—204no—
524C4⋊C4—102no—
525C4⋊C4—102no—
526C4⋊C4—102no—
527C4⋊C4—102no—
528C22⋊C4—54no—
529C22⋊C4—102no—
530C22⋊C4—102no—
531C22⋊C4—102no—
532C22⋊C4—102no—
533C22⋊C4—102no—
534C22⋊C4—102no—
535C22⋊C4—102no—
536C22⋊C4—201no—
537C22⋊C4—201no—
538C42—21no—
539C42—21no—
540C42—21no—
541C42—21no—
542C42—22no—
543C42—22no—
544C42—41no—
545C42—41no—
546C42—101no—
547C42—101no—
548C42—101no—
549C42—101no—
550C42D4×D512yes—
551C42#71312yes—
552C2×C5C4×#12411yescharacteristic
553C2×C5D4×M1611yescharacteristic
554C2×C5#293511yescharacteristic
555C2×C5#296511yescharacteristic
556C2×C5#296711yescharacteristic
557C2×C5#296811yescharacteristic
558C2×C5#297011yescharacteristic
559D5—42no—
560D5—81no—
561C23—52no—
562C23—101no—
563C23—101no—
564C23—101no—
565C23—101no—
566C23—101no—
567C23—101no—
568C23—101no—
569C23—101no—
570C23—101no—
571C23—101no—
572C23—101no—
573C23—101no—
574C23—101no—
575C23C22×C4×D511yescharacteristic
576Q8—204no—
577D4—204no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—52no—
593C2×C4—101no—
594C2×C4—101no—
595C2×C4—101no—
596C2×C4—101no—
597C2×C4—101no—
598C2×C4—101no—
599C2×C4—101no—
600C2×C4—101no—
601C2×C4—101no—
602C2×C4—101no—
603C2×C4—101no—
604C2×C4—101no—
605C2×C4—101no—
606C2×C4—101no—
607C2×C4—101no—
608C2×C4—101no—
609C2×C4—101no—
610C2×C4—201no—
611C2×C4—201no—
612C2×C4—201no—
613C2×C4—201no—
614C2×C4—201no—
615C2×C4—201no—
616C2×C4—201no—
617C2×C4—201no—
618C2×C4#403912yes—
619C2×C4#403912yes—
620C2×C4#404712yes—
621C2×C4D5×M1612yes—
622C2×C4C2×D4×D511yescharacteristic
623C2×C4C2×#71311yescharacteristic
624C2×C4D5×#4111yescharacteristic
625C2×C4D5×#4111yescharacteristic
626C2×C4D5×#4111yescharacteristic
627C2×C4D5×#4111yescharacteristic
628C8—41no—
629C8—41no—
630C8—41no—
631C8—41no—
632C8—54no—
633C8—102no—
634C8—102no—
635C8—102no—
636C5#2163211yescharacteristic, complemented, Sylow
637C22—102no—
638C22—102no—
639C22—102no—
640C22—201no—
641C22—201no—
642C22—201no—
643C22—201no—
644C22—201no—
645C22—201no—
646C22—201no—
647C22—201no—
648C22—201no—
649C22—201no—
650C22—201no—
651C22C2×#403911yescharacteristic
652C22#6585611yescharacteristic
653C22#6585611yescharacteristic
654C22C2×#404711yescharacteristic
655C22C2×D5×M1611yescharacteristic
656C22#6607911yescharacteristic
657C22#6607911yescharacteristic
658C4—22no—
659C4—22no—
660C4—22no—
661C4—22no—
662C4—41no—
663C4—41no—
664C4—102no—
665C4—201no—
666C4—201no—
667C4—201no—
668C4—201no—
669C4—201no—
670C4#6600211yescharacteristic
671C4D5×#12411yescharacteristic
672C4#6602511yescharacteristic
673C4#6602511yescharacteristic
674C2—202no—
675C2—401no—
676C2#1060588011yescharacteristic
677C2#1061073511yescharacteristic
678C2#1061077411yescharacteristic
679C2#1061100911yescharacteristic
680C2#1061101011yescharacteristic
681C2#1061105711yescharacteristic
682C2#1061105811yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D5
commutator, socleC23×C5C23×C4
FittingC5×#1831C2
prime radicalC22×D20C2×C4
cosocleC22×C4×C5C24
hypercenter#1831D5
hypocenterC5#21632
LCSG ▷ C23×C5 ▷ C5
UCSC1 ◁ C22×C4 ◁ #1831

Complements

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

NH ≅ G/Ntype
1. C2×#65117C2semidirect
4. C5×#1831C2semidirect
5. #10601266C2semidirect
7. C4×#3935C2semidirect
8. #10599451C2semidirect
9. #10599451C2semidirect
10. #10599450C2semidirect
11. #10599450C2semidirect
12. #10599443C2semidirect
13. #10599442C2semidirect
14. #10599441C2semidirect
15. #10599440C2semidirect
143. #1831D5semidirect
213. #685M16semidirect
355. #168#124semidirect
636. C5#21632semidirect

Maximal subgroups

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

subgroupindex
C2×#651172
C2×#651252
C4×#40212
C5×#18312
#106012662
#106012642
C4×#39352
#105994512
#105994502
#105994432
#105994422
#105994412
#105994402
#216325