Finite Group Databasebeta

← #28495822

Subgroups — #28495822

Order 15104.

990 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14273653C211yesmaximal, characteristic
2C2×#16837401C211yesmaximal, characteristic
3#21426684C211yesmaximal, characteristic
4C2×#16837267C211yesmaximal, characteristic, complemented
5C2×#16837170C211yesmaximal, characteristic, complemented
6C2×#16837161C211yesmaximal, characteristic, complemented
7C59×#1736C211yesmaximal, characteristic, complemented
8C22×#11745685C2211yescharacteristic
9C22×#11745664C2211yescharacteristic
10C22×C4⋊Dic59C2211yescharacteristic
11C22×C4×Dic59C2211yescharacteristic
12C2×#14273653—22no—
13C2×#14273653—22no—
14C2×#14273653C414yescomplemented
15#16837401—22no—
16#16837401—22no—
17C2×#14273650—21no—
18C2×#14273647—21no—
19#16837308—21no—
20#16837304—21no—
21#16837279—22no—
22#16837267—22no—
23#16837267—22no—
24#16837259—22no—
25C2×C59⋊#95—22no—
26#16837170—22no—
27#16837161—22no—
28C22×M16×C59C2211yescharacteristic
29C2×#117×C59C2211yescharacteristic
30C22×C4⋊C4×C59C2211yescharacteristic
31C59×#340—22no—
32C2×#11745685—21no—
33C2×#11745685—21no—
34C2×#11745685—21no—
35C2×#11745685—21no—
36C2×#11745685—21no—
37C2×#11745685—21no—
38C2×#11745685—21no—
39C2×#11745685—21no—
40C2×#11745685—21no—
41C2×#11745685—21no—
42C2×#11745685C2×C412yes—
43C2×#11745685Q812yes—
44C22×Dic118D411yescharacteristic
45C22×D236D411yescharacteristic
46C22×C22⋊D59—21no—
47C23×Dic59—21no—
48C23×Dic59D412yes—
49C22×C4×D59—21no—
50C2×#11745664—21no—
51C2×#11745664—21no—
52C2×#11745664—21no—
53C2×#11745664—21no—
54C2×#11745664—21no—
55C2×#11745664D412yes—
56C22×C59⋊C8—21no—
57C2×C4⋊Dic59—21no—
58C2×C4⋊Dic59—21no—
59C2×C4⋊Dic59—21no—
60C2×C4⋊Dic59—21no—
61C2×C4⋊Dic59—21no—
62C2×C4⋊Dic59—21no—
63C2×(C2×D59)⋊C4—24no—
64C2×C4×Dic59—21no—
65C2×C4×Dic59—21no—
66C2×C4×Dic59—21no—
67C2×C4×Dic59C2×C412yes—
68C2×C4×Dic59D412yes—
69C2×C4×Dic59D412yes—
70#14273653—28no—
71#14273650—24no—
72#14273647—24no—
73C59⋊#95—28no—
74C59⋊#95—24no—
75C23×C4×C59—21no—
76C23×C4×C59C2311yescharacteristic
77C2×M16×C59—21no—
78C2×M16×C59—21no—
79C2×M16×C59—21no—
80C2×M16×C59—21no—
81C2×M16×C59—21no—
82C2×M16×C59C2×C412yes—
83C22×C8×C59—21no—
84#117×C59—21no—
85#117×C59—21no—
86#117×C59—21no—
87#117×C59—21no—
88C2×C4⋊C4×C59—21no—
89C2×C4⋊C4×C59—21no—
90C2×C4⋊C4×C59—21no—
91C2×C4⋊C4×C59—21no—
92C2×C4⋊C4×C59—21no—
93C2×C4⋊C4×C59D412yes—
94C2×C4⋊C4×C59D412yes—
95C2×C22⋊C4×C59—21no—
96C2×C42×C59—21no—
97C24×C59C2×D411yescharacteristic
98C23×D59—21no—
99C22×C4×C59—22no—
100C22×C4×C59—22no—
101C22×C4×C59—44no—
102C22×C4×C59—41no—
103C22×C4×C59—41no—
104C22×C4×C59C22×C411yescharacteristic
105C22×C4×C59C2×D411yescharacteristic
106C22×C4×C59C2×D411yescharacteristic
107C22×C4×C59C2×D411yescharacteristic
108C22×C4×C59C2×D411yescharacteristic
109C22×C4×C59C2×D411yescharacteristic
110C22×C4×C59C2×D411yescharacteristic
111C22×C4×C59C2×Q811yescharacteristic
112C22×C4×C59#4111yescharacteristic
113C22×C4×C59#4111yescharacteristic
114C22×C4×C59#4111yescharacteristic
115C22×C4×C59#4111yescharacteristic
116C22×C4×C59#4111yescharacteristic
117C22×C4×C59#4111yescharacteristic
118C2×C22⋊D59—22no—
119C2×C22⋊D59—22no—
120C2×C22⋊D59—41no—
121C2×C22⋊D59—41no—
122C2×C22⋊D59—41no—
123C2×C22⋊D59—41no—
124C22×Dic59—22no—
125C22×Dic59—22no—
126C22×Dic59—21no—
127C22×Dic59—22no—
128C22×Dic59—22no—
129C22×Dic59—22no—
130C22×Dic59—22no—
131C22×Dic59—42no—
132C22×Dic59—42no—
133C22×Dic59—44no—
134#11745685—22no—
135#11745685—22no—
136#11745685—22no—
137#11745685—22no—
138#11745685—22no—
139#11745685—22no—
140#11745685—22no—
141#11745685—22no—
142#11745685—41no—
143#11745685—41no—
144#11745685—41no—
145#11745685—41no—
146#11745685—41no—
147#11745685—41no—
148#11745685—41no—
149#11745685—41no—
150C2×D236—21no—
151C2×D236—21no—
152C2×D236—21no—
153C2×D236—21no—
154C2×D236—21no—
155C2×D236C4⋊C412yes—
156C2×C4×D59—22no—
157C2×C4×D59—22no—
158C2×C4×D59—41no—
159C2×C4×D59—41no—
160C2×C4×D59—41no—
161C2×C4×D59—41no—
162C2×Dic118—21no—
163C2×Dic118—21no—
164C2×Dic118—21no—
165C2×Dic118—21no—
166C2×Dic118—21no—
167C2×Dic118C4⋊C412yes—
168M16×C59—22no—
169M16×C59—22no—
170M16×C59—41no—
171M16×C59—41no—
172C2×C8×C59—22no—
173C2×C8×C59—41no—
174C2×C8×C59—41no—
175C4⋊C4×C59—22no—
176C4⋊C4×C59—22no—
177C4⋊C4×C59—22no—
178C4⋊C4×C59—22no—
179C4⋊C4×C59—41no—
180C4⋊C4×C59—41no—
181C22⋊C4×C59—41no—
182C22⋊C4×C59—41no—
183C42×C59—41no—
184C42×C59—41no—
185(C2×D59)⋊C4—84no—
186C4⋊Dic59—41no—
187C4⋊Dic59—41no—
188C4⋊Dic59—41no—
189C4⋊Dic59—41no—
190C4×Dic59—22no—
191C4×Dic59—22no—
192C4×Dic59—22no—
193C4×Dic59C22⋊C414yescomplemented
194#11745664—22no—
195#11745664—22no—
196#11745664—41no—
197#11745664—41no—
198C2×C59⋊C8—22no—
199C2×C59⋊C8—41no—
200C2×C59⋊C8—41no—
201C23×C59—22no—
202C23×C59—22no—
203C23×C59C4×D411yescharacteristic
204C23×C59#12011yescharacteristic
205C23×C59#12011yescharacteristic
206C23×C59#12111yescharacteristic
207C23×C59#12111yescharacteristic
208C23×C59C22⋊Q811yescharacteristic
209C23×C59#12411yescharacteristic
210C22×D59—22no—
211C22×D59—41no—
212C22×D59—41no—
213C22×D59—42no—
214C22×D59—42no—
215C2×C4×C59—44no—
216C2×C4×C59—42no—
217C2×C4×C59—42no—
218C2×C4×C59—42no—
219C2×C4×C59—42no—
220C2×C4×C59—82no—
221C2×C4×C59—82no—
222C2×C4×C59C2×C22⋊C411yescharacteristic
223C2×C4×C59C2×C4⋊C411yescharacteristic
224C2×C4×C59#11711yescharacteristic
225C2×C4×C59C4×D411yescharacteristic
226C2×C4×C59C4×D411yescharacteristic
227C2×C4×C59C4×D411yescharacteristic
228C2×C4×C59#12011yescharacteristic
229C2×C4×C59#12111yescharacteristic
230C2×C4×C59#12111yescharacteristic
231C2×C4×C59#12111yescharacteristic
232C2×C4×C59#12111yescharacteristic
233C2×C4×C59#12111yescharacteristic
234C2×C4×C59#12111yescharacteristic
235C2×C4×C59C22⋊Q811yescharacteristic
236C2×C4×C59C22⋊Q811yescharacteristic
237C2×C4×C59C22⋊Q811yescharacteristic
238C2×C4×C59C22⋊Q811yescharacteristic
239C2×C4×C59C22⋊Q811yescharacteristic
240C2×C4×C59#12311yescharacteristic
241C2×C4×C59#12311yescharacteristic
242C2×C4×C59#12311yescharacteristic
243C2×C4×C59#12311yescharacteristic
244C2×C4×C59#12311yescharacteristic
245C2×C4×C59#12411yescharacteristic
246C2×C4×C59#12411yescharacteristic
247C2×C4×C59#12611yescharacteristic
248C2×C4×C59#12611yescharacteristic
249C2×C4×C59#12711yescharacteristic
250C22⋊D59—42no—
251C22⋊D59—42no—
252C22⋊D59—42no—
253C22⋊D59—42no—
254C2×Dic59—22no—
255C2×Dic59—24no—
256C2×Dic59—44no—
257C2×Dic59—42no—
258C2×Dic59—42no—
259C2×Dic59—44no—
260C2×Dic59—41no—
261C2×Dic59—41no—
262C2×Dic59—44no—
263C2×Dic59—42no—
264C2×Dic59—44no—
265C2×Dic59—82no—
266D236—22no—
267D236—22no—
268D236—41no—
269D236—41no—
270C4×D59—42no—
271C4×D59—42no—
272C4×D59—42no—
273C4×D59—42no—
274Dic118—22no—
275Dic118—22no—
276Dic118—41no—
277Dic118—41no—
278C8×C59—42no—
279C59⋊C8—42no—
280#11079—591nomaximal, Sylow
281C22×C59—22no—
282C22×C59—22no—
283C22×C59—21no—
284C22×C59—21no—
285C22×C59—21no—
286C22×C59—21no—
287C22×C59—22no—
288C22×C59—22no—
289C22×C59#38511yescharacteristic
290C22×C59#38611yescharacteristic
291C22×C59#38811yescharacteristic
292C22×C59#39211yescharacteristic
293C22×C59#39311yescharacteristic
294C22×C59#39411yescharacteristic
295C22×C59#39611yescharacteristic
296C22×C59#45412yes—
297C22×C59#45512yes—
298C2×D59—44no—
299C2×D59—42no—
300C2×D59—81no—
301C2×D59—81no—
302C2×D59—82no—
303C4×C59—82no—
304C4×C59—82no—
305C4×C59#38011yescharacteristic
306C4×C59#38511yescharacteristic
307C4×C59#38811yescharacteristic
308C4×C59#39011yescharacteristic
309C4×C59#39411yescharacteristic
310C4×C59#39611yescharacteristic
311C4×C59#39711yescharacteristic
312C4×C59#39911yescharacteristic
313Dic59—44no—
314Dic59—42no—
315Dic59—82no—
316C22×#104—591no—
317C2×#417—591no—
318#2311—591no—
319C2×#382—591no—
320C2×#343—591no—
321#1736D5911yescharacteristic, complemented
322C2×#339—591no—
323C2×C59—21no—
324C2×C59—21no—
325C2×C59—21no—
326C2×C59—21no—
327C2×C59#143911yescharacteristic
328C2×C59#186012yes—
329C2×C59#200311yescharacteristic
330C2×C59#200411yescharacteristic
331C2×C59#204012yes—
332D59—82no—
333C22×#41—591no—
334C22×M16—591no—
335C22×M16C2×D5911yescharacteristic
336C2×#117C2×D5911yescharacteristic
337C22×C4⋊C4—591no—
338C22×C4⋊C4C2×D5911yescharacteristic
339C22×C42—591no—
340C2×#104—594no—
341C2×#104—1182no—
342C2×#104—1182no—
343#417—1182no—
344#417—1182no—
345C2×#103—1181no—
346C2×#102—1181no—
347#391—1181no—
348#390—1181no—
349#386—1182no—
350#382—1182no—
351#382—1182no—
352#380—1182no—
353C2×#95—1182no—
354#343—1182no—
355#340—22no—
356#339—1182no—
357C59#1107911yescharacteristic, complemented, Sylow
358C2×#41—592no—
359C2×#41—592no—
360C2×#41—1181no—
361C2×#41—1181no—
362C2×#41—1181no—
363C2×#41—1181no—
364C2×#41—1181no—
365C2×#41—1181no—
366C2×#41—1181no—
367C2×#41—1181no—
368C2×#41—1181no—
369C2×#41—1181no—
370C22×Q8—591no—
371C22×D4—591no—
372C22×D4—1181no—
373C23×C4—21no—
374C23×C4—592no—
375C23×C4—1181no—
376C23×C4—1181no—
377C23×C4C22×D5911yescharacteristic
378C2×M16—21no—
379C2×M16—21no—
380C2×M16—21no—
381C2×M16—21no—
382C2×M16—21no—
383C2×M16—592no—
384C2×M16—1181no—
385C2×M16—1181no—
386C2×M16—1181no—
387C2×M16—1181no—
388C2×M16—1181no—
389C2×M16C4×D5912yes—
390C22×C8—21no—
391C22×C8—1181no—
392#117—21no—
393#117—21no—
394#117—21no—
395#117—21no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—21no—
400C2×C4⋊C4—21no—
401C2×C4⋊C4—1181no—
402C2×C4⋊C4—1181no—
403C2×C4⋊C4—1181no—
404C2×C4⋊C4—1181no—
405C2×C4⋊C4—1181no—
406C2×C4⋊C4—1181no—
407C2×C4⋊C4D23612yes—
408C2×C4⋊C4C22⋊D5912yes—
409C2×C22⋊C4—21no—
410C2×C22⋊C4—1184no—
411C2×C42—21no—
412C2×C42—592no—
413C2×C42—592no—
414C2×C42—592no—
415C2×C42—1181no—
416C2×C42—1181no—
417C2×C42—1181no—
418#104—1188no—
419#103—1184no—
420#102—1184no—
421#95—1188no—
422#95—1184no—
423C24—1181no—
424C24D4×D5911yescharacteristic
425#41—1182no—
426#41—1182no—
427#41—1182no—
428#41—1182no—
429#41—1182no—
430#41—1182no—
431#41—1182no—
432#41—1182no—
433#41—2361no—
434#41—2361no—
435#41—2361no—
436#41—2361no—
437#41—2361no—
438#41—2361no—
439#41—2361no—
440#41—2361no—
441C2×Q8—592no—
442C2×Q8—1181no—
443C2×Q8—1181no—
444C2×Q8—1181no—
445C2×Q8—1181no—
446C2×Q8—1181no—
447C2×D4—592no—
448C2×D4—1182no—
449C2×D4—1182no—
450C2×D4—1181no—
451C2×D4—1181no—
452C2×D4—1181no—
453C2×D4—1181no—
454C2×D4—1181no—
455C2×D4—2361no—
456C2×D4—2361no—
457C2×D4—2361no—
458C2×D4—2361no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—44no—
462C22×C4—41no—
463C22×C4—41no—
464C22×C4—1182no—
465C22×C4—1181no—
466C22×C4—1182no—
467C22×C4—1182no—
468C22×C4—1182no—
469C22×C4—1182no—
470C22×C4—1182no—
471C22×C4—1182no—
472C22×C4—1182no—
473C22×C4—2364no—
474C22×C4—2362no—
475C22×C4—2362no—
476C22×C4—2361no—
477C22×C4—2361no—
478C22×C4—2361no—
479C22×C4—2361no—
480C22×C4C2×C4×D5911yescharacteristic
481C22×C4C2×D23611yescharacteristic
482C22×C4#1174568511yescharacteristic
483C22×C4#1174568511yescharacteristic
484C22×C4#1174568511yescharacteristic
485C22×C4#1174568511yescharacteristic
486C22×C4D4×D5911yescharacteristic
487C22×C4D4×D5911yescharacteristic
488C22×C4D4×D5911yescharacteristic
489C22×C4D4×D5911yescharacteristic
490C22×C4#1174568711yescharacteristic
491C22×C4Q8×D5911yescharacteristic
492C22×C4#1174568911yescharacteristic
493C22×C4C2×C22⋊D5911yescharacteristic
494M16—22no—
495M16—22no—
496M16—41no—
497M16—41no—
498M16—1182no—
499M16—1182no—
500M16—2361no—
501M16—2361no—
502C2×C8—22no—
503C2×C8—41no—
504C2×C8—41no—
505C2×C8—1182no—
506C2×C8—2361no—
507C2×C8—2361no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—41no—
513C4⋊C4—41no—
514C4⋊C4—2361no—
515C4⋊C4—2361no—
516C4⋊C4—2361no—
517C4⋊C4—2361no—
518C22⋊C4—41no—
519C22⋊C4—41no—
520C22⋊C4—4724no—
521C42—41no—
522C42—41no—
523C42—594no—
524C42—1182no—
525C42—1182no—
526C42—1182no—
527C23—22no—
528C23—22no—
529C23—1182no—
530C23—2361no—
531C23—2361no—
532C23—2362no—
533C23—2362no—
534C23#1427368911yescharacteristic
535C23#1427369511yescharacteristic
536C23#1427369711yescharacteristic
537C23#1427370011yescharacteristic
538C23#1427370011yescharacteristic
539C23#1427370511yescharacteristic
540C23#1427371311yescharacteristic
541Q8—1182no—
542Q8—1182no—
543Q8—2361no—
544Q8—2361no—
545D4—1182no—
546D4—1182no—
547D4—2362no—
548D4—2362no—
549D4—2362no—
550D4—2362no—
551D4—2361no—
552D4—2361no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—82no—
559C2×C4—82no—
560C2×C4—1182no—
561C2×C4—1184no—
562C2×C4—4722no—
563C2×C4—2364no—
564C2×C4—2362no—
565C2×C4—2364no—
566C2×C4—2364no—
567C2×C4—2364no—
568C2×C4—2362no—
569C2×C4—2362no—
570C2×C4—2361no—
571C2×C4—2361no—
572C2×C4—2362no—
573C2×C4—2362no—
574C2×C4—2362no—
575C2×C4—2362no—
576C2×C4C22⋊C4×D5911yescharacteristic
577C2×C4C4⋊C4×D5911yescharacteristic
578C2×C4#1427368411yescharacteristic
579C2×C4(C22⋊D59)⋊C411yescharacteristic
580C2×C4C4×C22⋊D5911yescharacteristic
581C2×C4C4×D23611yescharacteristic
582C2×C4#1427369711yescharacteristic
583C2×C4#1427369911yescharacteristic
584C2×C4#1427370011yescharacteristic
585C2×C4#1427370011yescharacteristic
586C2×C4#1427370011yescharacteristic
587C2×C4#1427370111yescharacteristic
588C2×C4#1427370211yescharacteristic
589C2×C4#1427370311yescharacteristic
590C2×C4#1427370511yescharacteristic
591C2×C4#1427370511yescharacteristic
592C2×C4#1427370511yescharacteristic
593C2×C4#1427370711yescharacteristic
594C2×C4#1427370911yescharacteristic
595C2×C4#1427371011yescharacteristic
596C2×C4#1427371211yescharacteristic
597C2×C4#1427371211yescharacteristic
598C2×C4#1427371211yescharacteristic
599C2×C4#1427371311yescharacteristic
600C2×C4#1427371511yescharacteristic
601C2×C4#1427372111yescharacteristic
602C2×C4#1427372211yescharacteristic
603C2×C4#1427372311yescharacteristic
604C8—42no—
605C8—2362no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—21no—
611C22—21no—
612C22—21no—
613C22—21no—
614C22—4722no—
615C22—4721no—
616C22—4721no—
617C22—2362no—
618C22—2364no—
619C22#1683727311yescharacteristic
620C22#1683728011yescharacteristic
621C22#1683729111yescharacteristic
622C22#1683731111yescharacteristic
623C22#1683731511yescharacteristic
624C22#1683731711yescharacteristic
625C22#1683732611yescharacteristic
626C22#1683757312yes—
627C22#1683757812yes—
628C4—82no—
629C4—82no—
630C4—4722no—
631C4—2364no—
632C4—2362no—
633C4#1683726111yescharacteristic
634C4#1683727611yescharacteristic
635C4#1683729311yescharacteristic
636C4#1683730211yescharacteristic
637C4#1683732011yescharacteristic
638C4#1683732411yescharacteristic
639C4#1683733111yescharacteristic
640C4#1683733711yescharacteristic
641C2—21no—
642C2—21no—
643C2—21no—
644C2—21no—
645C2—4722no—
646C2#2142234011yescharacteristic
647C2#2142432612yes—
648C2#2142511911yescharacteristic
649C2#2142512611yescharacteristic
650C2#2142532312yes—
subgroupG/H
centerC23#14273695
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC59×#1736C2
socleC23×C59#120
prime radicalC22×#11745685C22
cosocleC23×C4×C59C23
hypercenter#1736D59
hypocenterC59#11079
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1736

Complements

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

NH ≅ G/Ntype
4. C2×#16837267C2semidirect
5. C2×#16837170C2semidirect
6. C2×#16837161C2semidirect
7. C59×#1736C2semidirect
14. C2×#14273653C4semidirect
193. C4×Dic59C22⋊C4semidirect
321. #1736D59semidirect
357. C59#11079semidirect

Maximal subgroups

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

subgroupindex
C22×#142736532
C2×#168374012
#214266842
C2×#168372672
C2×#168371702
C2×#168371612
C59×#17362
#1107959