Finite Group Databasebeta

← #12103641

Subgroups — #12103641

Order 1280.

874 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#2295C211yesmaximal, characteristic, complemented
2C2×#65315C211yesmaximal, characteristic
3#10605030C211yesmaximal, characteristic
4#10602424C211yesmaximal, characteristic, complemented
5#10600472C211yesmaximal, characteristic
6#10600469C211yesmaximal, characteristic
7#10600468C211yesmaximal, characteristic, complemented
8#10600467C211yesmaximal, characteristic, complemented
9#10600413C211yesmaximal, characteristic, complemented
10#10600412C211yesmaximal, characteristic, complemented
11#10600410C211yesmaximal, characteristic
12#10600408C211yesmaximal, characteristic
13#10600274C211yesmaximal, characteristic, complemented
14#10600267C211yesmaximal, characteristic
15C4×#3935C211yesmaximal, characteristic, complemented
16C2×C4×C5×Q8C2211yescharacteristic
17C22×#688C2211yescharacteristic
18C2×C4×D20C2211yescharacteristic
19C2×#4018—21no—
20C5×#391C2211yescharacteristic
21C5×#391C2211yescharacteristic
22C5×#389C2211yescharacteristic
23C5×#389C2211yescharacteristic
24C5×#387—21no—
25C5×#387—21no—
26C4×C5×C4⋊C4—21no—
27C4×C5×C4⋊C4C2211yescharacteristic
28C5×#376—21no—
29C43×C5C2211yescharacteristic
30C2×#3967C2211yescharacteristic
31C2×#3967C2211yescharacteristic
32C2×#3967C2211yescharacteristic
33#65315C418yescomplemented
34C2×#3941C2211yescharacteristic
35C2×#3941C2211yescharacteristic
36C2×#3941C2211yescharacteristic
37C2×#3940C2211yescharacteristic
38C2×#3939C2211yescharacteristic
39C2×#3939C2211yescharacteristic
40#65229—21no—
41#65229—21no—
42#65228C2211yescharacteristic
43#65228C2211yescharacteristic
44C4×#685—21no—
45C4×C4⋊Dic5C2211yescharacteristic
46#65217C2211yescharacteristic
47#65217C2211yescharacteristic
48C2×#3935C2211yescharacteristic
49C2×#3935C2211yescharacteristic
50C2×#3935C2211yescharacteristic
51C2×#3935C2211yescharacteristic
52C2×C4×#168C2211yescharacteristic
53C2×C4×#168C2211yescharacteristic
54#64749C2211yescharacteristic
55#64749C2211yescharacteristic
56#64749C2211yescharacteristic
57#64749C2211yescharacteristic
58#64749C2211yescharacteristic
59#64749C2211yescharacteristic
60#21288—51nomaximal, Sylow
61C22×C5×Q8C2311yescharacteristic
62C22×D20C2311yescharacteristic
63C22×C4×D5—21no—
64C4×C5×Q8—22no—
65C4×C5×Q8C2×C414yes—
66C2×C5×C4⋊C4—21no—
67C2×C5×C4⋊C4—21no—
68C2×C5×C4⋊C4—21no—
69C2×C5×C4⋊C4C2311yescharacteristic
70C2×C5×C4⋊C4C2311yescharacteristic
71C2×C5×C4⋊C4C2311yescharacteristic
72C2×C42×C5—21no—
73C2×C42×C5—21no—
74C2×C42×C5D411yescharacteristic
75C2×C42×C5D411yescharacteristic
76C2×C42×C5D411yescharacteristic
77C2×C42×C5D411yescharacteristic
78C2×C42×C5C2311yescharacteristic
79C2×C42×C5C2311yescharacteristic
80C2×C42×C5C2311yescharacteristic
81C2×#688—22no—
82C2×#688—22no—
83C2×#688C2×C414yes—
84C2×#685—21no—
85C2×#685—21no—
86C2×#685—21no—
87C2×#685—21no—
88C2×#685—21no—
89C2×C4⋊Dic5C2311yescharacteristic
90C2×C4⋊Dic5C2311yescharacteristic
91C2×C4⋊Dic5C2311yescharacteristic
92C2×#683—21no—
93C2×#683—21no—
94C2×C4×Dic5—21no—
95C22×#168C2311yescharacteristic
96C22×#168C2311yescharacteristic
97C22×#168C2311yescharacteristic
98C22×#168C2311yescharacteristic
99C4×D20—22no—
100C4×D20C2×C414yes—
101#4018—44no—
102C5×#95—21no—
103C5×#95—21no—
104C5×#95—21no—
105C5×#95—21no—
106C5×#95—21no—
107C5×#95—21no—
108C5×#95—21no—
109#3967—22no—
110#3967—22no—
111#3967C2×C414yes—
112#3963—21no—
113#3963—21no—
114#3963—21no—
115#3941—22no—
116#3941—22no—
117#3941C2×C414yes—
118#3940—22no—
119#3939—22no—
120#3939C2×C414yes—
121#3935—22no—
122#3935—22no—
123#3935—22no—
124#3935C2×C414yes—
125C4×#168—22no—
126C4×#168—22no—
127C2×#440—51no—
128#2295D511yescharacteristic, complemented
129#2293—51no—
130#1973—51no—
131#1968—51no—
132#1943—51no—
133#1942—51no—
134#1936—51no—
135#1935—51no—
136#1934—51no—
137#1933—51no—
138#1776—51no—
139#1763—51no—
140#1762—51no—
141C4×#97—51no—
142C23×D5—21no—
143C2×C5×Q8—21no—
144C2×C5×Q8—21no—
145C2×C5×Q8—21no—
146C2×C5×Q8—21no—
147C2×C5×Q8—21no—
148C2×C5×Q8C22×C412yes—
149C22×C4×C5—21no—
150C22×C4×C5—21no—
151C22×C4×C5—21no—
152C22×C4×C5—21no—
153C22×C4×C5C2×D411yescharacteristic
154C22×C4×C5C2×D411yescharacteristic
155C22×C4×C5C2×D411yescharacteristic
156C22×C4×C5C2×D411yescharacteristic
157C22×C4×C5C2×D411yescharacteristic
158C22×C4×C5C2×D411yescharacteristic
159C22×C4×C5C2411yescharacteristic
160C22×Dic5—21no—
161C22×Dic5—21no—
162C22×Dic5—21no—
163C2×D20—21no—
164C2×D20—21no—
165C2×D20—21no—
166C2×D20—21no—
167C2×D20—21no—
168C2×D20C22×C412yes—
169C2×C4×D5—24no—
170C2×C4×D5—42no—
171C2×C4×D5—42no—
172C5×C4⋊C4—24no—
173C5×C4⋊C4—21no—
174C5×C4⋊C4—21no—
175C5×C4⋊C4—21no—
176C5×C4⋊C4—21no—
177C5×C4⋊C4—21no—
178C5×C4⋊C4—42no—
179C5×C4⋊C4—42no—
180C5×C4⋊C4C22×C412yes—
181C42×C5—24no—
182C42×C5—22no—
183C42×C5—22no—
184C42×C5—21no—
185C42×C5—21no—
186C42×C5—21no—
187C42×C5—21no—
188C42×C5—41no—
189C42×C5—41no—
190C42×C5—42no—
191C42×C5C22×C412yes—
192C42×C5#4112yes—
193#688—28no—
194#685—24no—
195#685—42no—
196#685—42no—
197#685—42no—
198#685—42no—
199C4⋊Dic5—21no—
200C4⋊Dic5—21no—
201C4⋊Dic5—21no—
202C4⋊Dic5—21no—
203C4⋊Dic5—21no—
204C4⋊Dic5C22×C412yes—
205#683—42no—
206#683—42no—
207C4×Dic5—42no—
208C2×#168—21no—
209C2×#168—21no—
210C2×#168—21no—
211C2×#168—21no—
212C2×#168—21no—
213C2×#168—21no—
214C2×#168C22×C412yes—
215C2×#168C22×C412yes—
216C2×#168#4112yes—
217C2×#168#4112yes—
218C2×#168#4112yes—
219C2×#168#4112yes—
220C22×SD16—51no—
221C2×C4×Q8C2×D511yescharacteristic
222C2×C4×D4—51no—
223C2×#117—101no—
224#440—58no—
225C2×#107—51no—
226C2×#107—51no—
227C2×#106—51no—
228C2×#103—51no—
229C2×#103—51no—
230C2×#103—51no—
231C2×#102—51no—
232C2×#102—51no—
233C2×#102—51no—
234C2×#97—51no—
235C2×#97—51no—
236C2×#97—51no—
237C2×#97—51no—
238C2×C4×C8—51no—
239C2×C4×C8—51no—
240#391C2×D511yescharacteristic
241#391C2×D511yescharacteristic
242#390—51no—
243#390—51no—
244#389—51no—
245#389—51no—
246#389C2×D511yescharacteristic
247#389C2×D511yescharacteristic
248#388—101no—
249#388—101no—
250#387—21no—
251#387—21no—
252C4×C4⋊C4—21no—
253C4×C4⋊C4—51no—
254C4×C4⋊C4C2×D511yescharacteristic
255C4×C22⋊C4—101no—
256#376—21no—
257C43C2×D511yescharacteristic
258#340—51no—
259#340—51no—
260#340—51no—
261#340—51no—
262#340—51no—
263#340—51no—
264C23×C5C22×D411yescharacteristic
265C22×D5—22no—
266C22×D5—41no—
267C22×D5—41no—
268C22×D5—41no—
269C22×D5—41no—
270C22×D5—41no—
271C22×D5—41no—
272C5×Q8—22no—
273C5×Q8—22no—
274C5×Q8—41no—
275C5×Q8—41no—
276C2×C4×C5—21no—
277C2×C4×C5—21no—
278C2×C4×C5—21no—
279C2×C4×C5—21no—
280C2×C4×C5—21no—
281C2×C4×C5—21no—
282C2×C4×C5—21no—
283C2×C4×C5—21no—
284C2×C4×C5—22no—
285C2×C4×C5—22no—
286C2×C4×C5—21no—
287C2×C4×C5—21no—
288C2×C4×C5—21no—
289C2×C4×C5—21no—
290C2×C4×C5—21no—
291C2×C4×C5—21no—
292C2×C4×C5—41no—
293C2×C4×C5—41no—
294C2×C4×C5—41no—
295C2×C4×C5—41no—
296C2×C4×C5—41no—
297C2×C4×C5—41no—
298C2×C4×C5—41no—
299C2×C4×C5—41no—
300C2×C4×C5—41no—
301C2×C4×C5—41no—
302C2×C4×C5C4×D412yes—
303C2×C4×C5C4×D412yes—
304C2×C4×C5AGL(1,ℤ/8)12yes—
305C2×C4×C5#13712yes—
306C2×C4×C5C23×C411yescharacteristic
307C2×C4×C5C2×#4111yescharacteristic
308C2×C4×C5C2×#4111yescharacteristic
309C2×C4×C5C2×#4111yescharacteristic
310C2×C4×C5C2×#4111yescharacteristic
311C2×C4×C5C2×#4111yescharacteristic
312C2×Dic5—22no—
313C2×Dic5—41no—
314C2×Dic5—41no—
315C2×Dic5—41no—
316C2×Dic5—41no—
317C2×Dic5—41no—
318C2×Dic5—41no—
319C2×Dic5—41no—
320C2×Dic5—41no—
321D20—22no—
322D20—22no—
323D20—41no—
324D20—41no—
325C4×D5—48no—
326#168—22no—
327#168—22no—
328#168—22no—
329#168#11714yescomplemented
330C22×Q8C22×D511yescharacteristic
331C22×D4—51no—
332C23×C4—101no—
333C2×SD16—54no—
334C2×SD16—102no—
335C2×SD16—102no—
336C22×C8—51no—
337C22×C8—51no—
338C22×C8—51no—
339C22×C8—51no—
340C4×Q8—22no—
341C4×Q8C4×D514yes—
342C4×D4—54no—
343C4×D4—102no—
344#117—204no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—51no—
349C2×C4⋊C4—51no—
350C2×C4⋊C4—51no—
351C2×C4⋊C4—101no—
352C2×C4⋊C4—101no—
353C2×C4⋊C4C22×D511yescharacteristic
354C2×C4⋊C4C22×D511yescharacteristic
355C2×C4⋊C4C22×D511yescharacteristic
356C2×C22⋊C4—101no—
357C2×C22⋊C4—101no—
358C2×C22⋊C4—101no—
359C2×C22⋊C4—101no—
360C2×C22⋊C4—101no—
361C2×C42—21no—
362C2×C42—21no—
363C2×C42—101no—
364C2×C42C22⋊D511yescharacteristic
365C2×C42C22⋊D511yescharacteristic
366C2×C42C22⋊D511yescharacteristic
367C2×C42C22⋊D511yescharacteristic
368C2×C42C22×D511yescharacteristic
369C2×C42C22×D511yescharacteristic
370C2×C42C22×D511yescharacteristic
371#107—54no—
372#107—102no—
373#106—102no—
374#103—54no—
375#103—102no—
376#103—102no—
377#102—54no—
378#102—102no—
379#102—102no—
380#97—54no—
381#97—102no—
382#97—102no—
383#97—102no—
384C4×C8—102no—
385C4×C8—102no—
386#95—21no—
387#95—21no—
388#95—21no—
389#95—21no—
390#95—21no—
391#95—21no—
392#95—21no—
393#95—101no—
394#95—101no—
395#95—101no—
396C22×C5C2×C4×D411yescharacteristic
397C22×C5#53211yescharacteristic
398C22×C5#53211yescharacteristic
399C22×C5C2×AGL(1,ℤ/8)11yescharacteristic
400C22×C5C2×#13711yescharacteristic
401C22×C5#57511yescharacteristic
402C22×C5#57511yescharacteristic
403C2×D5—42no—
404C2×D5—42no—
405C2×D5—42no—
406C2×D5—81no—
407C2×D5—81no—
408C2×D5—81no—
409C2×D5—81no—
410C4×C5—22no—
411C4×C5—22no—
412C4×C5—22no—
413C4×C5—22no—
414C4×C5—41no—
415C4×C5—41no—
416C4×C5—42no—
417C4×C5—42no—
418C4×C5—81no—
419C4×C5—81no—
420C4×C5C2×#11711yescharacteristic
421C4×C5C4×#4111yescharacteristic
422C4×C5#52911yescharacteristic
423C4×C5#52911yescharacteristic
424Dic5—42no—
425Dic5—81no—
426Dic5—81no—
427C24—101no—
428C2×Q8—21no—
429C2×Q8—21no—
430C2×Q8—21no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8C2×C4×D512yes—
434C2×D4—52no—
435C2×D4—101no—
436C2×D4—101no—
437C2×D4—101no—
438C2×D4—101no—
439C2×D4—101no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—101no—
445C22×C4—104no—
446C22×C4—101no—
447C22×C4—101no—
448C22×C4—202no—
449C22×C4—202no—
450C22×C4C2×C22⋊D511yescharacteristic
451C22×C4C2×C22⋊D511yescharacteristic
452C22×C4C2×C22⋊D511yescharacteristic
453C22×C4C2×C22⋊D511yescharacteristic
454C22×C4C2×C22⋊D511yescharacteristic
455C22×C4C2×C22⋊D511yescharacteristic
456C22×C4C23×D511yescharacteristic
457SD16—108no—
458C2×C8—52no—
459C2×C8—52no—
460C2×C8—52no—
461C2×C8—52no—
462C2×C8—52no—
463C2×C8—52no—
464C2×C8—101no—
465C2×C8—101no—
466C2×C8—101no—
467C2×C8—101no—
468C2×C8—101no—
469C2×C8—101no—
470C4⋊C4—21no—
471C4⋊C4—24no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—52no—
479C4⋊C4—101no—
480C4⋊C4—101no—
481C4⋊C4—101no—
482C4⋊C4—101no—
483C4⋊C4—101no—
484C4⋊C4—202no—
485C4⋊C4—202no—
486C4⋊C4C2×C4×D512yes—
487C22⋊C4—104no—
488C22⋊C4—202no—
489C22⋊C4—202no—
490C22⋊C4—202no—
491C22⋊C4—202no—
492C42—21no—
493C42—21no—
494C42—21no—
495C42—21no—
496C42—24no—
497C42—22no—
498C42—22no—
499C42—41no—
500C42—41no—
501C42—42no—
502C42—202no—
503C42C2×C4×D512yes—
504C42#70912yes—
505C2×C5#229311yescharacteristic
506C2×C5C4×AGL(1,ℤ/8)11yescharacteristic
507C2×C5C4×#13711yescharacteristic
508C2×C5#318011yescharacteristic
509C2×C5#318111yescharacteristic
510C2×C5#318211yescharacteristic
511C2×C5#318311yescharacteristic
512D5—82no—
513C23—102no—
514C23—201no—
515C23—201no—
516C23—201no—
517C23—201no—
518C23—201no—
519C23—201no—
520C23C22×C22⋊D511yescharacteristic
521Q8—22no—
522Q8—22no—
523Q8—41no—
524Q8—41no—
525D4—102no—
526D4—102no—
527D4—201no—
528D4—201no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—102no—
556C2×C4—208no—
557C2×C4—201no—
558C2×C4—201no—
559C2×C4—201no—
560C2×C4—201no—
561C2×C4—201no—
562C2×C4—201no—
563C2×C4—201no—
564C2×C4—201no—
565C2×C4C4×C22⋊D512yes—
566C2×C4C4×C22⋊D512yes—
567C2×C4#408812yes—
568C2×C4#409512yes—
569C2×C4C22×C4×D511yescharacteristic
570C2×C4C2×#70911yescharacteristic
571C2×C4D5×#4111yescharacteristic
572C2×C4D5×#4111yescharacteristic
573C2×C4D5×#4111yescharacteristic
574C2×C4D5×#4111yescharacteristic
575C8—54no—
576C8—102no—
577C8—102no—
578C8—102no—
579C5#2128811yescharacteristic, complemented, Sylow
580C22—202no—
581C22—202no—
582C22—202no—
583C22—401no—
584C22—401no—
585C22—401no—
586C22—401no—
587C22C2×C4×C22⋊D511yescharacteristic
588C22C2×#408811yescharacteristic
589C22C2×#409511yescharacteristic
590C22#6615611yescharacteristic
591C22#6615611yescharacteristic
592C22#6616611yescharacteristic
593C22#6616611yescharacteristic
594C4—22no—
595C4—22no—
596C4—22no—
597C4—22no—
598C4—41no—
599C4—41no—
600C4—42no—
601C4—42no—
602C4—81no—
603C4—81no—
604C4—202no—
605C4—401no—
606C4—401no—
607C4D5×#11711yescharacteristic
608C4#6585611yescharacteristic
609C4#6586211yescharacteristic
610C4#6586211yescharacteristic
611C2—402no—
612C2#1061395611yescharacteristic
613C2C4×#408811yescharacteristic
614C2C4×#409511yescharacteristic
615C2#1061486811yescharacteristic
616C2#1061487011yescharacteristic
617C2#1061507711yescharacteristic
618C2#1061507811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D5
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC5×#2295C2
socleC23×C5C22×D4
prime radicalC22×D20C23
cosocleC22×C4×C5C24
hypercenter#2295D5
hypocenterC5#21288
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C43 ◁ #2295

Complements

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

NH ≅ G/Ntype
1. C5×#2295C2semidirect
4. #10602424C2semidirect
7. #10600468C2semidirect
8. #10600467C2semidirect
9. #10600413C2semidirect
10. #10600412C2semidirect
13. #10600274C2semidirect
15. C4×#3935C2semidirect
33. #65315C4semidirect
128. #2295D5semidirect
329. #168#117semidirect
579. C5#21288semidirect

Maximal subgroups

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

subgroupindex
C5×#22952
C2×#653152
#106050302
#106024242
#106004722
#106004692
#106004682
#106004672
#106004132
#106004122
#106004102
#106004082
#106002742
#106002672
C4×#39352
#212885