Finite Group Databasebeta

← #12140669

Subgroups — #12140669

Order 1280.

746 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#2628C211yesmaximal, characteristic
2C2×#65380C211yesmaximal, characteristic, complemented
3#10613966C211yesmaximal, characteristic
4C2×#65289C211yesmaximal, characteristic
5#10602573C211yesmaximal, characteristic, complemented
6#10600414C211yesmaximal, characteristic, complemented
7#10600412C211yesmaximal, characteristic, complemented
8#10600409C211yesmaximal, characteristic, complemented
9#10600407C211yesmaximal, characteristic, complemented
10#10600352C211yesmaximal, characteristic, complemented
11#10600333C211yesmaximal, characteristic, complemented
12#10600320C211yesmaximal, characteristic
13#10600319C211yesmaximal, characteristic
14#10600312C211yesmaximal, characteristic
15#10600311C211yesmaximal, characteristic
16C2×C5×C4⋊Q8C2211yescharacteristic
17C2×C5×#117C2211yescharacteristic
18C2×#4070C2211yescharacteristic
19C2×C4×Dic10C2211yescharacteristic
20C5×#400—21no—
21C5×#399—21no—
22C5×#395—21no—
23C5×#393—21no—
24C5×#389C2211yescharacteristic
25C5×#389C2211yescharacteristic
26C5×#382C2211yescharacteristic
27C5×#380C2211yescharacteristic
28C5×#380C2211yescharacteristic
29#65501C2211yescharacteristic
30C2×C22⋊#168C2211yescharacteristic
31C2×C22⋊#168C2211yescharacteristic
32#65380—24no—
33#65289—24no—
34#65267—21no—
35#65261—21no—
36C2×#3942C2211yescharacteristic
37C2×#3942C2211yescharacteristic
38C2×#3942C2211yescharacteristic
39C2×#3942C2211yescharacteristic
40C2×#3940C2211yescharacteristic
41C2×#3939C2211yescharacteristic
42C2×#3939C2211yescharacteristic
43C2×#3939C2211yescharacteristic
44#65234—21no—
45#65233—21no—
46C4×C4⋊Dic5C2211yescharacteristic
47#65218C2211yescharacteristic
48#65217C2211yescharacteristic
49C2×C4⋊#168C2211yescharacteristic
50C2×C4⋊#168C2211yescharacteristic
51C2×#3935C2211yescharacteristic
52C2×C4×#168C2211yescharacteristic
53#64749C2211yescharacteristic
54#64749C2211yescharacteristic
55#64749C2211yescharacteristic
56#64749C2211yescharacteristic
57#64749C2211yescharacteristic
58#64749C2211yescharacteristic
59#64749C2211yescharacteristic
60#64749C2211yescharacteristic
61#24714—51nomaximal, Sylow
62C22×C5×Q8—21no—
63C23×C4×C5C2311yescharacteristic
64C22×Dic10C2311yescharacteristic
65C5×C4⋊Q8—22no—
66C5×C4⋊Q8—22no—
67C5×#117—22no—
68C5×#117—22no—
69C2×C5×C4⋊C4—21no—
70C2×C5×C4⋊C4—21no—
71C2×C5×C4⋊C4—21no—
72C2×C5×C4⋊C4D411yescharacteristic
73C2×C5×C4⋊C4D411yescharacteristic
74C2×C5×C4⋊C4C2311yescharacteristic
75C2×C5×C4⋊C4C2311yescharacteristic
76C2×C5×C4⋊C4C2311yescharacteristic
77C2×C5×C4⋊C4C2311yescharacteristic
78C2×C5×C22⋊C4—21no—
79C2×C5×C22⋊C4—21no—
80C2×C5×C22⋊C4D411yescharacteristic
81C2×C5×C22⋊C4D411yescharacteristic
82C2×C42×C5C2311yescharacteristic
83C2×C42×C5C2311yescharacteristic
84C2×C22⋊Dic5—21no—
85C2×C22⋊Dic5—21no—
86C2×C4⋊Dic5C2311yescharacteristic
87C2×C4⋊Dic5C2311yescharacteristic
88C2×C4⋊Dic5C2311yescharacteristic
89#4070—22no—
90#4070—22no—
91C2×#683—21no—
92C2×#683—21no—
93C2×#683—21no—
94C2×C4×Dic5—21no—
95C2×C4×Dic5—21no—
96C22×#168C2311yescharacteristic
97C22×#168C2311yescharacteristic
98C22×#168C2311yescharacteristic
99C22×#168C2311yescharacteristic
100C4×Dic10—22no—
101C4×Dic10—22no—
102C5×#95—21no—
103C5×#95—21no—
104C5×#95—21no—
105C5×#95—21no—
106C5×#95—21no—
107C5×#95—21no—
108#3963—21no—
109#3963—21no—
110#3963—21no—
111#3963—21no—
112#3963—21no—
113C22⋊#168—22no—
114C22⋊#168—22no—
115#3942—22no—
116#3942—22no—
117#3942—22no—
118#3942—22no—
119#3940—22no—
120#3939—22no—
121#3939—22no—
122#3939—22no—
123C4⋊#168—22no—
124C4⋊#168D414yes—
125#3935—22no—
126C4×#168—22no—
127C2×#477—51no—
128C2×#433—51no—
129#2628D511yescharacteristic
130#2323—51no—
131#2076—51no—
132#2075—51no—
133#2036—51no—
134#2035—51no—
135#1988—51no—
136#1949—51no—
137#1944—51no—
138#1942—51no—
139#1935—51no—
140#1933—51no—
141#1805—51no—
142C24×C5C2×D411yescharacteristic
143C2×C5×Q8—42no—
144C2×C5×Q8—81no—
145C2×C5×Q8—81no—
146C22×C4×C5—21no—
147C22×C4×C5—21no—
148C22×C4×C5—21no—
149C22×C4×C5—21no—
150C22×C4×C5—21no—
151C22×C4×C5—21no—
152C22×C4×C5—21no—
153C22×C4×C5—21no—
154C22×C4×C5—21no—
155C22×C4×C5—21no—
156C22×C4×C5C2×D411yescharacteristic
157C22×C4×C5C2×D411yescharacteristic
158C22×C4×C5C2×D411yescharacteristic
159C22×C4×C5C2×D411yescharacteristic
160C22×C4×C5C2×D411yescharacteristic
161C22×C4×C5C2411yescharacteristic
162C22×Dic5—21no—
163C22×Dic5—21no—
164C22×Dic5—21no—
165C22×Dic5—21no—
166C2×Dic10—21no—
167C2×Dic10—21no—
168C2×Dic10—21no—
169C2×Dic10—21no—
170C2×Dic10—21no—
171C2×Dic10#4112yes—
172C5×C4⋊C4—22no—
173C5×C4⋊C4—22no—
174C5×C4⋊C4—21no—
175C5×C4⋊C4—21no—
176C5×C4⋊C4—21no—
177C5×C4⋊C4—21no—
178C5×C4⋊C4—21no—
179C5×C4⋊C4—21no—
180C5×C4⋊C4—21no—
181C5×C4⋊C4—21no—
182C5×C4⋊C4—42no—
183C5×C4⋊C4—81no—
184C5×C4⋊C4—81no—
185C5×C22⋊C4—22no—
186C5×C22⋊C4—22no—
187C5×C22⋊C4—81no—
188C5×C22⋊C4—81no—
189C42×C5—21no—
190C42×C5—21no—
191C42×C5C2×D412yes—
192C42×C5#4112yes—
193C22⋊Dic5—24no—
194C22⋊Dic5—81no—
195C4⋊Dic5—21no—
196C4⋊Dic5—21no—
197C4⋊Dic5—21no—
198C4⋊Dic5—21no—
199C4⋊Dic5—21no—
200C4⋊Dic5#4112yes—
201#683—24no—
202#683—42no—
203#683—81no—
204C4×Dic5—42no—
205C4×Dic5—81no—
206C2×#168—21no—
207C2×#168—21no—
208C2×#168—21no—
209C2×#168—21no—
210C2×#168—21no—
211C2×#168—21no—
212C2×#168—21no—
213C2×#168—21no—
214C2×#168—21no—
215C2×#168—21no—
216C2×#168C2×D412yes—
217C2×#168C2×D412yes—
218C2×C4⋊Q8C2×D511yescharacteristic
219C2×C22⋊Q8—51no—
220C2×C4×Q8—51no—
221C2×#117C2×D511yescharacteristic
222#477—104no—
223#433—104no—
224C2×#107—51no—
225C2×#107—51no—
226C2×#107—51no—
227C2×#106—51no—
228C2×C4⋊C8—51no—
229C2×C4⋊C8—51no—
230C2×#103—51no—
231C2×#103—51no—
232C2×#103—51no—
233C2×#103—51no—
234C2×C22⋊C8—51no—
235C2×C22⋊C8—51no—
236C2×#97—51no—
237C2×C4×C8—51no—
238#400—21no—
239#399—21no—
240#395—21no—
241#393—21no—
242#391—51no—
243#389—51no—
244#389—101no—
245#389C2×D511yescharacteristic
246#389C2×D511yescharacteristic
247#388—101no—
248#388—101no—
249#387—101no—
250#382C2×D511yescharacteristic
251#380—51no—
252#380C2×D511yescharacteristic
253#380C2×D511yescharacteristic
254C4×C4⋊C4—51no—
255#340—51no—
256#340—51no—
257#340—51no—
258#340—51no—
259#340—51no—
260#340—51no—
261#340—51no—
262#340—51no—
263C23×C5—21no—
264C23×C5—21no—
265C23×C5—21no—
266C23×C5—21no—
267C23×C5—21no—
268C23×C5—21no—
269C23×C5—21no—
270C23×C5C22×D411yescharacteristic
271C5×Q8—84no—
272C2×C4×C5—21no—
273C2×C4×C5—21no—
274C2×C4×C5—21no—
275C2×C4×C5—21no—
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—21no—
285C2×C4×C5—41no—
286C2×C4×C5—41no—
287C2×C4×C5—41no—
288C2×C4×C5—41no—
289C2×C4×C5—41no—
290C2×C4×C5—41no—
291C2×C4×C5—41no—
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#12112yes—
302C2×C4×C5#12112yes—
303C2×C4×C5C22×D411yescharacteristic
304C2×C4×C5C2×#4111yescharacteristic
305C2×C4×C5C2×#4111yescharacteristic
306C2×C4×C5C2×#4111yescharacteristic
307C2×C4×C52−1+411yescharacteristic
308C2×C4×C52−1+411yescharacteristic
309C2×Dic5—22no—
310C2×Dic5—22no—
311C2×Dic5—41no—
312C2×Dic5—41no—
313C2×Dic5—41no—
314C2×Dic5—41no—
315C2×Dic5—41no—
316C2×Dic5—41no—
317C2×Dic5—41no—
318C2×Dic5—41no—
319C2×Dic5—41no—
320C2×Dic5—41no—
321Dic10—22no—
322Dic10—22no—
323Dic10—41no—
324Dic10—41no—
325#168—22no—
326#168—22no—
327#168—41no—
328#168—41no—
329C22×Q8—21no—
330C22×Q8—51no—
331C23×C4C22×D511yescharacteristic
332C22×C8—51no—
333C22×C8—51no—
334C22×C8—51no—
335C22×C8—51no—
336C4⋊Q8—22no—
337C4⋊Q8—22no—
338C22⋊Q8—102no—
339C22⋊Q8—102no—
340C4×Q8—102no—
341C4×Q8—102no—
342#117—22no—
343#117—22no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—51no—
348C2×C4⋊C4—51no—
349C2×C4⋊C4—51no—
350C2×C4⋊C4—101no—
351C2×C4⋊C4—101no—
352C2×C4⋊C4—101no—
353C2×C4⋊C4C22⋊D511yescharacteristic
354C2×C4⋊C4C22⋊D511yescharacteristic
355C2×C4⋊C4C22×D511yescharacteristic
356C2×C4⋊C4C22×D511yescharacteristic
357C2×C4⋊C4C22×D511yescharacteristic
358C2×C4⋊C4C22×D511yescharacteristic
359C2×C22⋊C4—21no—
360C2×C22⋊C4—21no—
361C2×C22⋊C4—101no—
362C2×C22⋊C4—101no—
363C2×C22⋊C4C22⋊D511yescharacteristic
364C2×C22⋊C4C22⋊D511yescharacteristic
365C2×C42—101no—
366C2×C42—101no—
367C2×C42C22×D511yescharacteristic
368C2×C42C22×D511yescharacteristic
369#107—102no—
370#107—102no—
371#107—102no—
372#106—102no—
373C4⋊C8—54no—
374C4⋊C8—102no—
375#103—102no—
376#103—102no—
377#103—102no—
378#103—102no—
379C22⋊C8—102no—
380C22⋊C8—102no—
381#97—102no—
382C4×C8—102no—
383#95—21no—
384#95—21no—
385#95—21no—
386#95—21no—
387#95—21no—
388#95—21no—
389#95—101no—
390#95—101no—
391#95—101no—
392#95—101no—
393#95—101no—
394C22×C5—41no—
395C22×C5—41no—
396C22×C5—41no—
397C22×C5—41no—
398C22×C5—41no—
399C22×C5—41no—
400C22×C5—41no—
401C22×C5C2×#12111yescharacteristic
402C22×C5#52511yescharacteristic
403C22×C5#53611yescharacteristic
404C22×C5#57611yescharacteristic
405C22×C5#57711yescharacteristic
406C22×C5#57811yescharacteristic
407C22×C5#57811yescharacteristic
408C4×C5—22no—
409C4×C5—22no—
410C4×C5—41no—
411C4×C5—41no—
412C4×C5—41no—
413C4×C5—81no—
414C4×C5—81no—
415C4×C5—81no—
416C4×C5—81no—
417C4×C5#54711yescharacteristic
418C4×C5#54811yescharacteristic
419C4×C5#55211yescharacteristic
420C4×C5#55611yescharacteristic
421Dic5—42no—
422Dic5—42no—
423Dic5—81no—
424Dic5—81no—
425C24C2×C22⋊D511yescharacteristic
426C2×Q8—42no—
427C2×Q8—52no—
428C2×Q8—81no—
429C2×Q8—81no—
430C2×Q8—101no—
431C2×Q8—101no—
432C2×Q8—101no—
433C2×Q8—101no—
434C2×Q8—101no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—101no—
446C22×C4—101no—
447C22×C4—101no—
448C22×C4—101no—
449C22×C4C2×C22⋊D511yescharacteristic
450C22×C4C2×C22⋊D511yescharacteristic
451C22×C4C2×C22⋊D511yescharacteristic
452C22×C4C2×C22⋊D511yescharacteristic
453C22×C4C2×C22⋊D511yescharacteristic
454C22×C4C23×D511yescharacteristic
455C2×C8—52no—
456C2×C8—52no—
457C2×C8—101no—
458C2×C8—101no—
459C2×C8—101no—
460C2×C8—101no—
461C2×C8—101no—
462C2×C8—101no—
463C2×C8—101no—
464C2×C8—101no—
465C2×C8—101no—
466C2×C8—101no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—42no—
478C4⋊C4—52no—
479C4⋊C4—81no—
480C4⋊C4—81no—
481C4⋊C4—104no—
482C4⋊C4—101no—
483C4⋊C4—101no—
484C4⋊C4—101no—
485C4⋊C4—101no—
486C4⋊C4—101no—
487C4⋊C4—202no—
488C4⋊C4—401no—
489C22⋊C4—22no—
490C22⋊C4—22no—
491C22⋊C4—81no—
492C22⋊C4—81no—
493C22⋊C4—104no—
494C22⋊C4—401no—
495C42—21no—
496C42—21no—
497C42—202no—
498C42—401no—
499C42D4×D512yes—
500C42#71112yes—
501C2×C5—81no—
502C2×C5#241511yescharacteristic
503C2×C5#305911yescharacteristic
504C2×C5#306211yescharacteristic
505C2×C5#312111yescharacteristic
506C2×C5#312411yescharacteristic
507C2×C5#324211yescharacteristic
508C2×C5#324311yescharacteristic
509C23—21no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23C22×C22⋊D511yescharacteristic
517Q8—84no—
518Q8—102no—
519Q8—102no—
520Q8—201no—
521Q8—201no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—102no—
552C2×C4—102no—
553C2×C4—201no—
554C2×C4—201no—
555C2×C4—201no—
556C2×C4—201no—
557C2×C4—201no—
558C2×C4—201no—
559C2×C4—201no—
560C2×C4—201no—
561C2×C4—201no—
562C2×C4—201no—
563C2×C4#408512yes—
564C2×C4#408512yes—
565C2×C4C2×D4×D511yescharacteristic
566C2×C4C2×#71111yescharacteristic
567C2×C4#414711yescharacteristic
568C2×C4#414711yescharacteristic
569C2×C4D5×#4111yescharacteristic
570C2×C4D5×#4111yescharacteristic
571C8—102no—
572C8—102no—
573C8—201no—
574C8—201no—
575C5#2471411yescharacteristic, complemented, Sylow
576C22—41no—
577C22—41no—
578C22—41no—
579C22—41no—
580C22—41no—
581C22—41no—
582C22—41no—
583C22C2×#408511yescharacteristic
584C22#6615011yescharacteristic
585C22#6616211yescharacteristic
586C22#6616911yescharacteristic
587C22#6617011yescharacteristic
588C22#6617211yescharacteristic
589C22#6617211yescharacteristic
590C4—22no—
591C4—22no—
592C4—41no—
593C4—41no—
594C4—41no—
595C4—81no—
596C4—81no—
597C4—81no—
598C4—81no—
599C4—202no—
600C4—202no—
601C4—401no—
602C4—401no—
603C4#6596511yescharacteristic
604C4#6596711yescharacteristic
605C4#6597311yescharacteristic
606C4#6597511yescharacteristic
607C2—81no—
608C2#1061454511yescharacteristic
609C2#1061511011yescharacteristic
610C2#1061513011yescharacteristic
611C2#1061519211yescharacteristic
612C2#1061519811yescharacteristic
613C2#1061520911yescharacteristic
614C2#1061521211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D5
commutator, cosocleC22×C4×C5C24
FrattiniC22×C4C23×D5
FittingC5×#2628C2
socleC23×C5C22×D4
prime radicalC24×C5C2×D4
hypercenter#2628D5
hypocenterC5#24714
LCSG ▷ C22×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2628

Complements

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

NH ≅ G/Ntype
2. C2×#65380C2semidirect
5. #10602573C2semidirect
6. #10600414C2semidirect
7. #10600412C2semidirect
8. #10600409C2semidirect
9. #10600407C2semidirect
10. #10600352C2semidirect
11. #10600333C2semidirect
575. C5#24714semidirect

Maximal subgroups

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

subgroupindex
C5×#26282
C2×#653802
#106139662
C2×#652892
#106025732
#106004142
#106004122
#106004092
#106004072
#106003522
#106003332
#106003202
#106003192
#106003122
#106003112
#247145