Finite Group Databasebeta

← #28616119

Subgroups — #28616119

Order 15104.

1148 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21437501C211yesmaximal, characteristic, complemented
2C2×#16837419C211yesmaximal, characteristic
3C2×#16837376C211yesmaximal, characteristic
4#21425266C218yesmaximal, complemented
5C2×#16837173C211yesmaximal, characteristic, complemented
6#21423807C211yesmaximal, characteristic, complemented
7C2×#16837158C211yesmaximal, characteristic, complemented
8C2×C59×#421C211yesmaximal, characteristic
9C22×#11745664C2211yescharacteristic
10#16838033—22no—
11#16838030—22no—
12#16838030—22no—
13#16838028—22no—
14#16837959—22no—
15#16837955—22no—
16#16837955C2214yescomplemented
17C2×#14273686C2211yescharacteristic
18C2×#14273686C2211yescharacteristic
19C2×#14273686C2211yescharacteristic
20#16837419—22no—
21#16837419—22no—
22C2×#14273654C2214yes—
23#16837376—22no—
24#16837376C2214yes—
25C2×#14273642—22no—
26#16837173C2214yescomplemented
27#16837163C2214yescomplemented
28#16837158C2214yescomplemented
29C22×#41×C59C2211yescharacteristic
30C22×M16×C59C2211yescharacteristic
31C2×#117×C59C2211yescharacteristic
32C59×#421—22no—
33C59×#421C2214yes—
34C2×#104×C59—22no—
35C2×#11745664—21no—
36C2×#11745664—21no—
37C2×#11745664—21no—
38C2×#11745664—21no—
39C2×#11745664—21no—
40C2×#11745664C2312yes—
41C22×C59⋊C8—21no—
42#14273727—24no—
43#14273726—24no—
44#14273726—24no—
45#14273719—24no—
46#14273718—24no—
47#14273718—24no—
48#14273706—28no—
49#14273704—28no—
50#14273704—28no—
51C4×Dic118—28no—
52#14273686—21no—
53#14273686—21no—
54#14273686—21no—
55#14273686—21no—
56#14273686—21no—
57#14273686—21no—
58#14273686—21no—
59#14273686D412yes—
60#14273686D412yes—
61#14273686D412yes—
62#14273686C2312yes—
63#14273686C2312yes—
64C2×C4⋊Dic59—21no—
65C2×C4⋊Dic59—21no—
66C2×C4⋊Dic59—21no—
67C2×C22⋊Dic59—21no—
68C2×C22⋊Dic59—21no—
69C2×C22⋊Dic59D412yes—
70C2×C4×Dic59—21no—
71C2×C4×Dic59—21no—
72C2×C4×Dic59C2312yes—
73#14273654—28no—
74#14273642—28no—
75C2×#41×C59—21no—
76C2×#41×C59—21no—
77C2×#41×C59—21no—
78C2×#41×C59—21no—
79C2×#41×C59—21no—
80C2×#41×C59—21no—
81C2×#41×C59—21no—
82C2×#41×C59—21no—
83C2×#41×C59—21no—
84C2×#41×C59—21no—
85C2×#41×C59Q812yescomplemented
86C2×#41×C59C2312yes—
87C22×Q8×C59Q811yescharacteristic, complemented
88C22×D4×C59—21no—
89C22×D4×C59Q811yescharacteristic, complemented
90C23×C4×C59—21no—
91C23×C4×C59C2311yescharacteristic
92C2×M16×C59—21no—
93C2×M16×C59—21no—
94C2×M16×C59—21no—
95C2×M16×C59—21no—
96C2×M16×C59—21no—
97C2×M16×C59C2312yes—
98C22×C8×C59—21no—
99#117×C59—21no—
100#117×C59—21no—
101#117×C59D412yes—
102#117×C59C2312yes—
103C2×C4⋊C4×C59—21no—
104C2×C22⋊C4×C59—21no—
105C2×C42×C59D412yes—
106#104×C59—28no—
107C24×C59—21no—
108C24×C59C2×D411yescharacteristic
109#41×C59—22no—
110#41×C59—22no—
111#41×C59—22no—
112#41×C59—22no—
113#41×C59—22no—
114#41×C59—22no—
115#41×C59—22no—
116#41×C59—22no—
117#41×C59—41no—
118#41×C59—41no—
119#41×C59—41no—
120#41×C59—41no—
121#41×C59—41no—
122#41×C59—41no—
123#41×C59—41no—
124#41×C59—41no—
125C2×Q8×C59—21no—
126C2×Q8×C59—21no—
127C2×Q8×C59—21no—
128C2×Q8×C59—21no—
129C2×Q8×C59—21no—
130C2×Q8×C59C2×Q812yes—
131C2×D4×C59—22no—
132C2×D4×C59—22no—
133C2×D4×C59—21no—
134C2×D4×C59—21no—
135C2×D4×C59—21no—
136C2×D4×C59—21no—
137C2×D4×C59—21no—
138C2×D4×C59—41no—
139C2×D4×C59—41no—
140C2×D4×C59—41no—
141C2×D4×C59—41no—
142C2×D4×C59C2×Q812yes—
143C22×C4×C59—21no—
144C22×C4×C59—22no—
145C22×C4×C59—22no—
146C22×C4×C59—22no—
147C22×C4×C59—41no—
148C22×C4×C59—41no—
149C22×C4×C59—41no—
150C22×C4×C59—41no—
151C22×C4×C59C2×D411yescharacteristic
152C22×C4×C59C2×D411yescharacteristic
153C22×C4×C59C2×D411yescharacteristic
154C22×C4×C59C2×D411yescharacteristic
155C22×C4×C59C2×D411yescharacteristic
156C22×C4×C59C2×Q811yescharacteristic
157C22×C4×C59C2×Q811yescharacteristic
158C22×C4×C59#4111yescharacteristic
159C22×C4×C59#4111yescharacteristic
160C22×C4×C59#4111yescharacteristic
161C22×C4×C59#4111yescharacteristic
162C22×C4×C59#4111yescharacteristic
163C22×C4×C59#4111yescharacteristic
164C22×C4×C59C2411yescharacteristic
165C22×Dic59—22no—
166C22×Dic59—22no—
167C22×Dic59—41no—
168C2×Dic118—28no—
169M16×C59—22no—
170M16×C59—22no—
171M16×C59—41no—
172M16×C59—41no—
173C2×C8×C59—22no—
174C2×C8×C59—41no—
175C2×C8×C59—41no—
176C4⋊C4×C59—22no—
177C4⋊C4×C59—22no—
178C22⋊C4×C59—22no—
179C22⋊C4×C59—22no—
180C42×C59—22no—
181C42×C59C2×D414yes—
182C22⋊Dic59—22no—
183C22⋊Dic59—22no—
184C22⋊Dic59—22no—
185C22⋊Dic59—22no—
186C22⋊Dic59—41no—
187C22⋊Dic59C2×D414yes—
188C4⋊Dic59—28no—
189C4⋊Dic59—22no—
190C4⋊Dic59—22no—
191C4⋊Dic59—22no—
192C4⋊Dic59—22no—
193C4⋊Dic59—22no—
194C4⋊Dic59—41no—
195Dic59⋊C4—28no—
196Dic59⋊C4—28no—
197C4×Dic59—22no—
198C4×Dic59—22no—
199C4×Dic59—22no—
200C4×Dic59—22no—
201C4×Dic59—41no—
202C4×Dic59C2×D414yes—
203#11745664—22no—
204#11745664—22no—
205#11745664—41no—
206#11745664—41no—
207C2×C59⋊C8—22no—
208C2×C59⋊C8—41no—
209C2×C59⋊C8—41no—
210C23×C59—22no—
211C23×C59—41no—
212C23×C59—41no—
213C23×C59—81no—
214C23×C59—82no—
215C23×C59#12011yescharacteristic
216C23×C59#12011yescharacteristic
217C23×C59C22⋊Q811yescharacteristic
218C23×C59C22⋊Q811yescharacteristic
219C23×C59#12411yescharacteristic
220C23×C59#12411yescharacteristic
221C23×C59C22×D411yescharacteristic
222Q8×C59—22no—
223Q8×C59—22no—
224Q8×C59—41no—
225Q8×C59—41no—
226D4×C59—22no—
227D4×C59—22no—
228D4×C59—42no—
229D4×C59—42no—
230D4×C59—42no—
231D4×C59—42no—
232D4×C59—41no—
233D4×C59—41no—
234C2×C4×C59—24no—
235C2×C4×C59—22no—
236C2×C4×C59—41no—
237C2×C4×C59—41no—
238C2×C4×C59—42no—
239C2×C4×C59—42no—
240C2×C4×C59—42no—
241C2×C4×C59—42no—
242C2×C4×C59—82no—
243C2×C4×C59#12011yescharacteristic
244C2×C4×C59#12011yescharacteristic
245C2×C4×C59C22⋊Q811yescharacteristic
246C2×C4×C59C22⋊Q811yescharacteristic
247C2×C4×C59C22⋊Q811yescharacteristic
248C2×C4×C59C22⋊Q811yescharacteristic
249C2×C4×C59C22⋊Q811yescharacteristic
250C2×C4×C59C22⋊Q811yescharacteristic
251C2×C4×C59C22⋊Q811yescharacteristic
252C2×C4×C59C22⋊Q811yescharacteristic
253C2×C4×C59C22⋊Q811yescharacteristic
254C2×C4×C59C22⋊Q811yescharacteristic
255C2×C4×C59#12411yescharacteristic
256C2×C4×C59#12411yescharacteristic
257C2×C4×C59#12411yescharacteristic
258C2×C4×C59#12411yescharacteristic
259C2×C4×C59#12411yescharacteristic
260C2×C4×C59#12411yescharacteristic
261C2×C4×C59#12411yescharacteristic
262C2×C4×C59#12411yescharacteristic
263C2×C4×C59#12411yescharacteristic
264C2×C4×C59#12411yescharacteristic
265C2×C4×C59C22×D411yescharacteristic
266C2×C4×C59C22×D411yescharacteristic
267C2×C4×C59C22×Q811yescharacteristic
268C2×C4×C59C2×#4111yescharacteristic
269C2×C4×C59C2×#4111yescharacteristic
270C2×C4×C59C2×#4111yescharacteristic
271C2×Dic59—24no—
272C2×Dic59—24no—
273C2×Dic59—42no—
274C2×Dic59—82no—
275C2×Dic59—82no—
276C2×Dic59—82no—
277Dic118—88no—
278C8×C59—42no—
279C59⋊C8—42no—
280#23472—591nomaximal, Sylow
281C22×C59—41no—
282C22×C59—41no—
283C22×C59—41no—
284C22×C59—41no—
285C22×C59—41no—
286C22×C59—41no—
287C22×C59—41no—
288C22×C59—42no—
289C22×C59—82no—
290C22×C59—82no—
291C22×C59—82no—
292C22×C59#39311yescharacteristic
293C22×C59#39311yescharacteristic
294C22×C59#39311yescharacteristic
295C22×C59#39311yescharacteristic
296C22×C59C2×#12011yescharacteristic
297C22×C59C2×C22⋊Q811yescharacteristic
298C22×C59C2×#12411yescharacteristic
299C4×C59—44no—
300C4×C59—42no—
301C4×C59#39311yescharacteristic
302C4×C59#39311yescharacteristic
303C4×C59#39311yescharacteristic
304C4×C59#39311yescharacteristic
305C4×C59C2×C22⋊Q811yescharacteristic
306C4×C59C2×C22⋊Q811yescharacteristic
307C4×C59C2×#12411yescharacteristic
308C4×C59C2×#12411yescharacteristic
309Dic59—44no—
310Dic59—44no—
311Dic59—82no—
312#3458—591no—
313C2×#421—591no—
314C2×#421D5911yescharacteristic
315C2×#410—591no—
316#2031—598no—
317C2×#344—591no—
318#1733—591no—
319C2×H(3,ℤ/4)—591no—
320C2×C59—21no—
321C2×C59—21no—
322C2×C59—21no—
323C2×C59—21no—
324C2×C59—21no—
325C2×C59—21no—
326C2×C59—21no—
327C2×C59—82no—
328C2×C59C2×#39311yescharacteristic
329C22×#41C2×D5911yescharacteristic
330C22×M16—591no—
331C22×M16C2×D5911yescharacteristic
332#544—1182no—
333#544—1182no—
334#544—1182no—
335#543—1182no—
336#533—594no—
337#533—1182no—
338#533—1182no—
339C2×#117—591no—
340C2×#117—591no—
341C2×#117—591no—
342C2×#117C2×D5911yescharacteristic
343#421—22no—
344#421—1182no—
345#421—1182no—
346#421C2×D5914yes—
347C2×#104—22no—
348C2×#104—594no—
349#410—594no—
350#410—1182no—
351C2×#99—1182no—
352#344—594no—
353#339—594no—
354H(3,ℤ/4)—594no—
355C59#2347211yescharacteristic, complemented, Sylow
356C2×#41—21no—
357C2×#41—21no—
358C2×#41—21no—
359C2×#41—21no—
360C2×#41—21no—
361C2×#41—21no—
362C2×#41—21no—
363C2×#41—21no—
364C2×#41—21no—
365C2×#41—21no—
366C2×#41Dic11812yescomplemented
367C2×#41C22×D5912yes—
368C22×Q8Dic11811yescharacteristic, complemented
369C22×D4—21no—
370C22×D4Dic11811yescharacteristic, complemented
371C23×C4—21no—
372C23×C4C22×D5911yescharacteristic
373C2×M16—21no—
374C2×M16—21no—
375C2×M16—21no—
376C2×M16—21no—
377C2×M16—21no—
378C2×M16—592no—
379C2×M16—1181no—
380C2×M16—1181no—
381C2×M16—1181no—
382C2×M16—1181no—
383C2×M16—1181no—
384C2×M16C22×D5912yes—
385C22×C8—21no—
386C22×C8—1181no—
387C4⋊Q8—1184no—
388C4⋊Q8—1184no—
389C4⋊Q8—1184no—
390#125—1184no—
391#125—1184no—
392#125—1184no—
393C22⋊Q8—1188no—
394C22⋊Q8—1188no—
395C22⋊Q8—1188no—
396C4×Q8—1188no—
397#117—21no—
398#117—21no—
399#117—592no—
400#117—592no—
401#117—592no—
402#117—592no—
403#117—592no—
404#117—1181no—
405#117—1181no—
406#117—1181no—
407#117—1181no—
408#117—1181no—
409#117—1181no—
410#117—1181no—
411#117C22⋊D5912yes—
412#117C22×D5912yes—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—1181no—
415C2×C4⋊C4—1181no—
416C2×C4⋊C4—1181no—
417C2×C22⋊C4—21no—
418C2×C22⋊C4—592no—
419C2×C22⋊C4—1181no—
420C2×C22⋊C4—1181no—
421C2×C42—592no—
422C2×C42—1181no—
423C2×C42—1181no—
424C2×C42C22⋊D5912yes—
425#104—28no—
426#104—1188no—
427#99—1188no—
428C24—21no—
429C24D4×D5911yescharacteristic
430#41—22no—
431#41—22no—
432#41—22no—
433#41—22no—
434#41—22no—
435#41—22no—
436#41—22no—
437#41—22no—
438#41—41no—
439#41—41no—
440#41—41no—
441#41—41no—
442#41—41no—
443#41—41no—
444#41—41no—
445#41—41no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—1188no—
452C2×Q8C2×Dic11812yes—
453C2×D4—22no—
454C2×D4—22no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—41no—
461C2×D4—41no—
462C2×D4—41no—
463C2×D4—41no—
464C2×D4C2×Dic11812yes—
465C22×C4—21no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—41no—
470C22×C4—41no—
471C22×C4—41no—
472C22×C4—41no—
473C22×C4—1182no—
474C22×C4—1182no—
475C22×C4—2361no—
476C22×C4C2×Dic11811yescharacteristic
477C22×C4C2×Dic11811yescharacteristic
478C22×C4#1174568511yescharacteristic
479C22×C4#1174568511yescharacteristic
480C22×C4D4×D5911yescharacteristic
481C22×C4D4×D5911yescharacteristic
482C22×C4D4×D5911yescharacteristic
483C22×C4#1174568711yescharacteristic
484C22×C4#1174568711yescharacteristic
485C22×C4#1174568711yescharacteristic
486C22×C4#1174568711yescharacteristic
487C22×C4C2×C22⋊D5911yescharacteristic
488C22×C4C2×C22⋊D5911yescharacteristic
489C22×C4C23×D5911yescharacteristic
490M16—22no—
491M16—22no—
492M16—41no—
493M16—41no—
494M16—1182no—
495M16—1182no—
496M16—2361no—
497M16—2361no—
498C2×C8—22no—
499C2×C8—41no—
500C2×C8—41no—
501C2×C8—1182no—
502C2×C8—2361no—
503C2×C8—2361no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—1188no—
507C4⋊C4—1188no—
508C4⋊C4—1188no—
509C4⋊C4—1182no—
510C4⋊C4—1182no—
511C4⋊C4—1182no—
512C4⋊C4—1182no—
513C4⋊C4—1182no—
514C4⋊C4—2361no—
515C22⋊C4—22no—
516C22⋊C4—22no—
517C22⋊C4—594no—
518C22⋊C4—1182no—
519C22⋊C4—1182no—
520C22⋊C4—1182no—
521C22⋊C4—1182no—
522C22⋊C4—2361no—
523C42—22no—
524C42—594no—
525C42—1182no—
526C42—1182no—
527C42—1182no—
528C42—1182no—
529C42—2361no—
530C42C2×C22⋊D5914yes—
531C23—22no—
532C23—41no—
533C23—41no—
534C23—81no—
535C23—82no—
536C23#1427369511yescharacteristic
537C23#1427369511yescharacteristic
538C23#1427370411yescharacteristic
539C23#1427370411yescharacteristic
540C23#1427371311yescharacteristic
541C23#1427371311yescharacteristic
542C23C2×D4×D5911yescharacteristic
543Q8—22no—
544Q8—22no—
545Q8—41no—
546Q8—41no—
547Q8—4728no—
548D4—22no—
549D4—22no—
550D4—42no—
551D4—42no—
552D4—42no—
553D4—42no—
554D4—41no—
555D4—41no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—82no—
565C2×C4—1184no—
566C2×C4—1184no—
567C2×C4—4722no—
568C2×C4—4722no—
569C2×C4—4722no—
570C2×C4—2362no—
571C2×C4#1427369511yescharacteristic
572C2×C4#1427369511yescharacteristic
573C2×C4#1427370411yescharacteristic
574C2×C4#1427370411yescharacteristic
575C2×C4#1427370411yescharacteristic
576C2×C4#1427370411yescharacteristic
577C2×C4#1427370411yescharacteristic
578C2×C4#1427370411yescharacteristic
579C2×C4#1427370611yescharacteristic
580C2×C4#1427370611yescharacteristic
581C2×C4#1427370611yescharacteristic
582C2×C4#1427370611yescharacteristic
583C2×C4#1427371311yescharacteristic
584C2×C4#1427371311yescharacteristic
585C2×C4#1427371311yescharacteristic
586C2×C4#1427371311yescharacteristic
587C2×C4#1427371311yescharacteristic
588C2×C4#1427371311yescharacteristic
589C2×C4#1427371411yescharacteristic
590C2×C4#1427371411yescharacteristic
591C2×C4#1427371411yescharacteristic
592C2×C4#1427371411yescharacteristic
593C2×C4C2×D4×D5911yescharacteristic
594C2×C4C22×C22⋊D5911yescharacteristic
595C2×C4C22×Dic11811yescharacteristic
596C2×C4C2×#1174568711yescharacteristic
597C2×C4C2×#1174568711yescharacteristic
598C2×C4C2×#1174568511yescharacteristic
599C8—42no—
600C8—2362no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22—41no—
606C22—41no—
607C22—41no—
608C22—42no—
609C22—82no—
610C22—82no—
611C22—82no—
612C22#1683731411yescharacteristic
613C22#1683731411yescharacteristic
614C22#1683731411yescharacteristic
615C22#1683731411yescharacteristic
616C22C2×#1427369511yescharacteristic
617C22C2×#1427370411yescharacteristic
618C22C2×#1427371311yescharacteristic
619C4—44no—
620C4—42no—
621C4—4722no—
622C4—2364no—
623C4—2364no—
624C4#1683731411yescharacteristic
625C4#1683731411yescharacteristic
626C4#1683731411yescharacteristic
627C4#1683731411yescharacteristic
628C4C2×#1427370411yescharacteristic
629C4C2×#1427370611yescharacteristic
630C4C2×#1427371311yescharacteristic
631C4C2×#1427371411yescharacteristic
632C2—21no—
633C2—21no—
634C2—21no—
635C2—21no—
636C2—21no—
637C2—21no—
638C2—21no—
639C2—82no—
640C2C2×#1683731411yescharacteristic
subgroupG/H
centerC4C2×#14273714
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC2×C59×#421C2
socleC2×C59C2×#393
prime radicalC22×#41×C59C22
hypercenterC2×#421D59
hypocenterC59#23472
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C4 ◁ C23×C4 ◁ C2×#421

Complements

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

NH ≅ G/Ntype
1. #21437501C2semidirect
4. #21425266C2semidirect
5. C2×#16837173C2semidirect
6. #21423807C2semidirect
7. C2×#16837158C2semidirect
16. #16837955C22semidirect
26. #16837173C22semidirect
27. #16837163C22semidirect
28. #16837158C22semidirect
85. C2×#41×C59Q8semidirect
87. C22×Q8×C59Q8semidirect
89. C22×D4×C59Q8semidirect
355. C59#23472semidirect
366. C2×#41Dic118semidirect
368. C22×Q8Dic118semidirect
370. C22×D4Dic118semidirect

Maximal subgroups

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

subgroupindex
#214375012
C2×#168374192
C2×#168373762
#214252662
C2×#168371732
#214238072
C2×#168371582
C2×C59×#4212
#2347259