Finite Group Databasebeta

← #28596544

Subgroups — #28596544

Order 15104.

730 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21433878C211yesmaximal, characteristic
2#21433037C211yesmaximal, characteristic, complemented
3#21432525C211yesmaximal, characteristic
4#21424871C211yesmaximal, characteristic, complemented
5#21424698C211yesmaximal, characteristic, complemented
6#21424698C211yesmaximal, characteristic, complemented
7#21424372C211yesmaximal, characteristic
8#21424372C211yesmaximal, characteristic
9#21424117C211yesmaximal, characteristic
10#21423979C211yesmaximal, characteristic, complemented
11#21423974C211yesmaximal, characteristic, complemented
12#21423971C211yesmaximal, characteristic, complemented
13#21423937C211yesmaximal, characteristic
14#21423930C211yesmaximal, characteristic
15C59×#1986C211yesmaximal, characteristic, complemented
16C2×#14273735C2211yescharacteristic
17C2×#14273735C2211yescharacteristic
18C4×#11745685C2211yescharacteristic
19#16837706C411yescharacteristic, complemented
20#16837706C411yescharacteristic, complemented
21#16837699C411yescharacteristic, complemented
22#16837699C411yescharacteristic, complemented
23#16837694C411yescharacteristic, complemented
24#16837689C411yescharacteristic, complemented
25#16837684C411yescharacteristic, complemented
26#16837679C411yescharacteristic, complemented
27#16837533—21no—
28#16837533—21no—
29#16837533—21no—
30#16837533—21no—
31C2×#14273662C2211yescharacteristic
32C2×#14273662C2211yescharacteristic
33#16837438C2211yescharacteristic
34#16837438C2211yescharacteristic
35#16837429C2211yescharacteristic
36#16837429C2211yescharacteristic
37#16837426C2211yescharacteristic
38C2×C4⋊(C59⋊C8)C2211yescharacteristic
39C2×#14273655C2211yescharacteristic
40C2×#14273655C2211yescharacteristic
41#16837396C2211yescharacteristic
42#16837396C2211yescharacteristic
43#16837360C2211yescharacteristic
44#16837360C2211yescharacteristic
45#16837358C2211yescharacteristic
46#16837358C2211yescharacteristic
47#16837356—21no—
48#16837356—21no—
49#16837354—21no—
50C2×#14273638C2211yescharacteristic
51C2×#14273638C2211yescharacteristic
52#97×D59—21no—
53C2×C4×C59⋊C8C2211yescharacteristic
54#16837166C2211yescharacteristic
55#16837166C2211yescharacteristic
56#16837162C2211yescharacteristic
57#16837162C2211yescharacteristic
58#16837157C2211yescharacteristic
59#16837157C2211yescharacteristic
60C59×#533C2211yescharacteristic
61C4×M16×C59C2211yescharacteristic
62C4×C4⋊C4×C59C2211yescharacteristic
63C59×#339C2211yescharacteristic
64C59×#339C2211yescharacteristic
65C59×#339C2211yescharacteristic
66C59×#339C2211yescharacteristic
67C2×#11745685C2311yescharacteristic
68#14273735—22no—
69#14273735—22no—
70#14273735—22no—
71#14273735Q814yescomplemented
72C2×#11745664C2311yescharacteristic
73C2×#11745664C2311yescharacteristic
74C2×#11745659—21no—
75C2×#11745659—21no—
76C22×C59⋊C8C2311yescharacteristic
77C22×C59⋊C8C2311yescharacteristic
78C22×C59⋊C8C2311yescharacteristic
79C22×C59⋊C8C2311yescharacteristic
80C2×C8×D59—21no—
81C2×C8×D59—21no—
82C4×Dic118C2×C411yescharacteristic
83C4×Dic118C2×C411yescharacteristic
84C4×D236C2×C411yescharacteristic
85C4×D236C2×C411yescharacteristic
86C4×C22⋊D59—21no—
87C4×C22⋊D59—21no—
88#14273686C2311yescharacteristic
89#14273684—21no—
90C42×D59—21no—
91#14273662—22no—
92#14273662—22no—
93#14273660C2×C411yescharacteristic
94#14273660C2×C411yescharacteristic
95#14273660C2×C411yescharacteristic
96#14273660C2×C411yescharacteristic
97#14273658C2×C411yescharacteristic
98#14273658C2×C411yescharacteristic
99#14273658C2×C411yescharacteristic
100#14273658C2×C411yescharacteristic
101C4⋊(C59⋊C8)—21no—
102C4⋊(C59⋊C8)—21no—
103C4⋊(C59⋊C8)—21no—
104C4⋊(C59⋊C8)—21no—
105C4⋊(C59⋊C8)—21no—
106C4⋊(C59⋊C8)—21no—
107C4⋊(C59⋊C8)C2×C411yescharacteristic
108C4⋊(C59⋊C8)C2×C411yescharacteristic
109C4⋊(C59⋊C8)C2×C411yescharacteristic
110C4⋊(C59⋊C8)C2×C411yescharacteristic
111#14273655—22no—
112#14273655—22no—
113#14273650C2×C411yescharacteristic
114#14273650C2×C411yescharacteristic
115#14273650C2×C411yescharacteristic
116#14273650C2×C411yescharacteristic
117#14273647C2×C411yescharacteristic
118#14273647C2×C411yescharacteristic
119#14273647C2×C411yescharacteristic
120#14273647C2×C411yescharacteristic
121#14273638—21no—
122#14273638—21no—
123#14273638—22no—
124#14273638D411yescharacteristic
125#14273638D411yescharacteristic
126#14273638D411yescharacteristic
127#14273638D411yescharacteristic
128#14273637—21no—
129#14273637—21no—
130C4×C59⋊C8—21no—
131C4×C59⋊C8—21no—
132C4×C59⋊C8—22no—
133C8×Dic59—21no—
134C8×Dic59—21no—
135C2×M16×C59C2311yescharacteristic
136C2×M16×C59C2311yescharacteristic
137C4⋊Q8×C59C2×C411yescharacteristic
138C4⋊Q8×C59C2×C411yescharacteristic
139#125×C59C2×C411yescharacteristic
140#125×C59C2×C411yescharacteristic
141C22⋊Q8×C59—21no—
142C22⋊Q8×C59—21no—
143C4×Q8×C59—21no—
144C4×Q8×C59—21no—
145#117×C59C2311yescharacteristic
146#117×C59C2311yescharacteristic
147C2×C4⋊C4×C59—21no—
148C2×C42×C59C2311yescharacteristic
149C2×C42×C59C2311yescharacteristic
150C2×C42×C59C2311yescharacteristic
151#97×C59—21no—
152C4×C8×C59—21no—
153#95×C59—21no—
154C2×Q8×C59—21no—
155C2×Q8×C59—21no—
156C22×C4×C59—21no—
157C22×C4×C59—21no—
158C22×C4×C59#4111yescharacteristic
159C22×C4×C59#4111yescharacteristic
160C22×C4×C59C2411yescharacteristic
161C2×C22⋊D59—21no—
162#11745685—21no—
163#11745685—21no—
164#11745685C2×Q812yes—
165#11745685#4112yes—
166C2×D236C22×C411yescharacteristic
167C2×C4×D59—21no—
168C2×C4×D59—21no—
169C2×C4×D59—21no—
170C2×Dic118C22×C411yescharacteristic
171M16×C59—21no—
172M16×C59—21no—
173M16×C59C2×Q812yes—
174M16×C59#4112yes—
175C2×C8×C59—21no—
176C2×C8×C59—21no—
177C4⋊C4×C59—21no—
178C4⋊C4×C59—21no—
179C4⋊C4×C59—21no—
180C4⋊C4×C59—21no—
181C4⋊C4×C59—21no—
182C4⋊C4×C59—21no—
183C4⋊C4×C59—42no—
184C4⋊C4×C59C22×C411yescharacteristic
185C4⋊C4×C59C22×C411yescharacteristic
186C4⋊C4×C59C22×C411yescharacteristic
187C4⋊C4×C59C22×C411yescharacteristic
188C22⋊C4×C59—21no—
189C22⋊C4×C59—21no—
190C42×C59—21no—
191C42×C59—21no—
192C42×C59—21no—
193C42×C59—21no—
194C42×C59—21no—
195C42×C59—21no—
196C42×C59C22×C411yescharacteristic
197C42×C59C22×C411yescharacteristic
198C42×C59C2×D411yescharacteristic
199C42×C59C2×D411yescharacteristic
200C22⋊Dic59—21no—
201(C2×D59)⋊C4—21no—
202(C2×D59)⋊C4—21no—
203C4⋊Dic59C22×C411yescharacteristic
204C4⋊Dic59C22×C411yescharacteristic
205Dic59⋊C4—21no—
206Dic59⋊C4—21no—
207C4×Dic59—21no—
208C4×Dic59—21no—
209C4×Dic59—21no—
210#11745664—21no—
211#11745664—21no—
212#11745664—21no—
213#11745664—21no—
214C2×C59⋊C8—21no—
215C2×C59⋊C8—21no—
216C2×C59⋊C8—21no—
217C2×C59⋊C8—21no—
218C2×C59⋊C8—21no—
219C2×C59⋊C8—21no—
220C2×C59⋊C8—21no—
221C2×C59⋊C8—21no—
222C2×C59⋊C8C22×C411yescharacteristic
223C2×C59⋊C8C22×C411yescharacteristic
224C2×C59⋊C8C22×C411yescharacteristic
225C2×C59⋊C8C22×C411yescharacteristic
226C2×C59⋊C8C2×D411yescharacteristic
227C2×C59⋊C8C2×D411yescharacteristic
228C2×C59⋊C8C2×D411yescharacteristic
229C2×C59⋊C8C2×D411yescharacteristic
230C2×C59⋊C8C2×Q812yes—
231C2×C59⋊C8#4112yes—
232#11745659—24no—
233#11745659—24no—
234C8×D59—24no—
235C8×D59—24no—
236C23×C59C2×#4111yescharacteristic
237C22×D59—21no—
238Q8×C59—81no—
239Q8×C59—81no—
240C2×C4×C59—22no—
241C2×C4×C59—21no—
242C2×C4×C59—21no—
243C2×C4×C59—21no—
244C2×C4×C59—21no—
245C2×C4×C59—41no—
246C2×C4×C59—41no—
247C2×C4×C59—41no—
248C2×C4×C59—41no—
249C2×C4×C59—41no—
250C2×C4×C59—41no—
251C2×C4×C59—41no—
252C2×C4×C59C4×D411yescharacteristic
253C2×C4×C59C4×D411yescharacteristic
254C2×C4×C59C4×D411yescharacteristic
255C2×C4×C59C4×D411yescharacteristic
256C2×C4×C59C23×C411yescharacteristic
257C2×C4×C59C22×D411yescharacteristic
258C2×C4×C59C22×Q811yescharacteristic
259C2×C4×C59C2×#4111yescharacteristic
260C2×C4×C59C2×#4111yescharacteristic
261C2×C4×C59C2×#4111yescharacteristic
262C22⋊D59—22no—
263C22⋊D59—22no—
264C2×Dic59—21no—
265C2×Dic59—21no—
266C2×Dic59—21no—
267D236—21no—
268D236C4×Q812yes—
269C4×D59—22no—
270C4×D59—22no—
271C4×D59—81no—
272C4×D59—81no—
273Dic118—21no—
274Dic118C4×Q812yes—
275C8×C59—22no—
276C8×C59—22no—
277C59⋊C8—21no—
278C59⋊C8—21no—
279C59⋊C8—21no—
280C59⋊C8—21no—
281C59⋊C8—21no—
282C59⋊C8—21no—
283C59⋊C8—41no—
284C59⋊C8—41no—
285C59⋊C8C22⋊Q812yes—
286C59⋊C8C22⋊Q812yes—
287#21629—591nomaximal, Sylow
288C22×C59—21no—
289C22×C59—21no—
290C22×C59C2×C4×D411yescharacteristic
291C22×C59#52911yescharacteristic
292C22×C59#53311yescharacteristic
293C2×D59—22no—
294C2×D59—41no—
295C2×D59—41no—
296C4×C59—21no—
297C4×C59—21no—
298C4×C59—21no—
299C4×C59—21no—
300C4×C59—42no—
301C4×C59—41no—
302C4×C59—41no—
303C4×C59—41no—
304C4×C59—41no—
305C4×C59—81no—
306C4×C59C2×C4×Q811yescharacteristic
307C4×C59C4×#4111yescharacteristic
308C4×C59C2×C22⋊Q811yescharacteristic
309C4×C59#52511yescharacteristic
310Dic59—22no—
311Dic59—41no—
312Dic59—41no—
313#3076—591no—
314#2961—591no—
315#2875—591no—
316#1986D5911yescharacteristic, complemented
317#1950—591no—
318#1922—591no—
319#1922—591no—
320#1866—591no—
321#1866—591no—
322#1806—591no—
323#1773—591no—
324#1772—591no—
325#1771—591no—
326#1765—591no—
327#1764—591no—
328C2×C59—21no—
329C2×C59—21no—
330C2×C59C4×C22⋊Q811yescharacteristic
331C2×C59#295611yescharacteristic
332C2×C59#295711yescharacteristic
333D59—42no—
334C2×#131—591no—
335C2×#131—591no—
336#533C2×D5911yescharacteristic
337C4×#41—591no—
338#479—591no—
339#479—591no—
340#478—591no—
341#478—591no—
342#477—591no—
343#476—591no—
344#475—591no—
345#474—591no—
346#446—1181no—
347#446—1181no—
348#446—1181no—
349#446—1181no—
350C2×#108—591no—
351C2×#108—591no—
352#427—591no—
353#427—591no—
354#424—591no—
355#424—591no—
356#423—591no—
357C2×C4⋊C8—591no—
358C2×#104—591no—
359C2×#104—591no—
360#416—591no—
361#416—591no—
362#405—591no—
363#405—591no—
364#405—591no—
365#405—591no—
366#405—1181no—
367#405—1181no—
368C4×M16—1181no—
369C4×M16C2×D5911yescharacteristic
370C2×#97—591no—
371C2×#97—591no—
372C2×#97—1181no—
373C2×C4×C8—591no—
374C4×C4⋊C4C2×D5911yescharacteristic
375#341—591no—
376#341—591no—
377#339—591no—
378#339—591no—
379#339C2×D5911yescharacteristic
380#339C2×D5911yescharacteristic
381#339C2×D5911yescharacteristic
382#339C2×D5911yescharacteristic
383#336—591no—
384#336—591no—
385C59#2162911yescharacteristic, complemented, Sylow
386C2×#41—591no—
387#131—594no—
388#131—1182no—
389#131—1182no—
390#131—1182no—
391C2×M16—591no—
392C2×M16—591no—
393C2×M16—1181no—
394C2×M16—1181no—
395C2×M16C22×D5911yescharacteristic
396C2×M16C22×D5911yescharacteristic
397C22×C8—591no—
398C22×C8—591no—
399C22×C8—591no—
400C22×C8—591no—
401C22×C8—1181no—
402C22×C8—1181no—
403C4⋊Q8C4×D5911yescharacteristic
404C4⋊Q8C4×D5911yescharacteristic
405#125C4×D5911yescharacteristic
406#125C4×D5911yescharacteristic
407C22⋊Q8—21no—
408C22⋊Q8—21no—
409C4×Q8—21no—
410C4×Q8—21no—
411C4×Q8—591no—
412C4×Q8—591no—
413C4×D4—591no—
414C4×D4—591no—
415C4×D4—1181no—
416C4×D4—1181no—
417#117—591no—
418#117—1181no—
419#117C22×D5911yescharacteristic
420#117C22×D5911yescharacteristic
421C2×C4⋊C4—21no—
422C2×C42—1181no—
423C2×C42C22×D5911yescharacteristic
424C2×C42C22×D5911yescharacteristic
425C2×C42C22×D5911yescharacteristic
426#108—1182no—
427#108—1182no—
428#107—591no—
429#107—591no—
430#107—591no—
431#107—591no—
432#106—591no—
433#106—591no—
434#106—591no—
435#106—591no—
436C4⋊C8—591no—
437C4⋊C8—591no—
438C4⋊C8—591no—
439C4⋊C8—591no—
440C4⋊C8—1181no—
441C4⋊C8—1181no—
442C4⋊C8—1181no—
443C4⋊C8—1181no—
444C4⋊C8—1181no—
445C4⋊C8—1181no—
446#104—1182no—
447#104—1182no—
448#103—591no—
449#103—591no—
450#103—591no—
451#103—591no—
452#102—591no—
453#102—591no—
454#102—591no—
455#102—591no—
456#97—21no—
457#97—591no—
458#97—591no—
459#97—591no—
460#97—591no—
461#97—1182no—
462#97—1181no—
463#97—1181no—
464#97—1181no—
465#97—1181no—
466C4×C8—21no—
467C4×C8—1182no—
468C4×C8—1181no—
469C4×C8—1181no—
470C4×C8—1181no—
471C4×C8—1181no—
472#95—21no—
473#41—592no—
474#41—592no—
475#41—1181no—
476#41—1181no—
477C2×Q8—21no—
478C2×Q8—21no—
479C2×Q8—591no—
480C2×D4—591no—
481C2×D4—1181no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—1181no—
485C22×C4—1181no—
486C22×C4—1181no—
487C22×C4#1174568711yescharacteristic
488C22×C4#1174568711yescharacteristic
489C22×C4C23×D5911yescharacteristic
490M16—21no—
491M16—21no—
492M16—1184no—
493M16—1184no—
494M16—1181no—
495M16—1181no—
496M16—1181no—
497M16—1181no—
498M16Q8×D5912yes—
499M16#1174568912yes—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—591no—
503C2×C8—591no—
504C2×C8—592no—
505C2×C8—592no—
506C2×C8—591no—
507C2×C8—591no—
508C2×C8—591no—
509C2×C8—591no—
510C2×C8—591no—
511C2×C8—591no—
512C2×C8—1184no—
513C2×C8—1184no—
514C2×C8—1181no—
515C2×C8—1181no—
516C2×C8—1181no—
517C2×C8—1181no—
518C2×C8—1181no—
519C2×C8—1181no—
520C2×C8—1181no—
521C2×C8—1181no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—42no—
529C4⋊C4—591no—
530C4⋊C4—591no—
531C4⋊C4—1181no—
532C4⋊C4—1181no—
533C4⋊C4C2×C4×D5911yescharacteristic
534C4⋊C4C2×C4×D5911yescharacteristic
535C4⋊C4C2×C4×D5911yescharacteristic
536C4⋊C4C2×C4×D5911yescharacteristic
537C22⋊C4—21no—
538C22⋊C4—21no—
539C22⋊C4—1181no—
540C22⋊C4—1181no—
541C22⋊C4—1181no—
542C42—21no—
543C42—21no—
544C42—21no—
545C42—21no—
546C42—21no—
547C42—21no—
548C42—1181no—
549C42—1181no—
550C42—1181no—
551C42C2×C4×D5911yescharacteristic
552C42C2×C4×D5911yescharacteristic
553C42D4×D5911yescharacteristic
554C42D4×D5911yescharacteristic
555C23—1181no—
556C23C2×#1174568711yescharacteristic
557Q8—81no—
558Q8—81no—
559Q8—592no—
560Q8—1181no—
561D4—592no—
562D4—1182no—
563D4—1182no—
564D4—1181no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—1181no—
578C2×C4—1182no—
579C2×C4—1182no—
580C2×C4—1181no—
581C2×C4—1181no—
582C2×C4—4721no—
583C2×C4—4721no—
584C2×C4(C22⋊D59)⋊C411yescharacteristic
585C2×C4(C22⋊D59)⋊C411yescharacteristic
586C2×C4(C22⋊D59)⋊C411yescharacteristic
587C2×C4(C22⋊D59)⋊C411yescharacteristic
588C2×C4C22×C4×D5911yescharacteristic
589C2×C4C2×D4×D5911yescharacteristic
590C2×C4C2×Q8×D5911yescharacteristic
591C2×C4#41×D5911yescharacteristic
592C2×C4#41×D5911yescharacteristic
593C2×C4C2×#1174568911yescharacteristic
594C8—22no—
595C8—22no—
596C8—592no—
597C8—592no—
598C8—1181no—
599C8—1181no—
600C8—1181no—
601C8—1181no—
602C8—1181no—
603C8—1181no—
604C8—2361no—
605C8—2361no—
606C22—21no—
607C22—21no—
608C22—1182no—
609C22—2361no—
610C22—2361no—
611C22C2×(C22⋊D59)⋊C411yescharacteristic
612C22#1683793111yescharacteristic
613C22#1683795711yescharacteristic
614C4—21no—
615C4—21no—
616C4—21no—
617C4—21no—
618C4—42no—
619C4—41no—
620C4—41no—
621C4—41no—
622C4—41no—
623C4—81no—
624C4—1182no—
625C4—2361no—
626C4—2361no—
627C4C4×Q8×D5911yescharacteristic
628C4C4×#1174568911yescharacteristic
629C4C22⋊Q8×D5911yescharacteristic
630C4#1683791511yescharacteristic
631C2—21no—
632C2—21no—
633C2—2362no—
634C2#2142663411yescharacteristic
635C2#2143300211yescharacteristic
636C2#2143300911yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1986C2
socleC22×C59C2×C4×D4
prime radicalC2×#11745685C23
cosocleC22×C4×C59C24
hypercenter#1986D59
hypocenterC59#21629
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1986

Complements

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

NH ≅ G/Ntype
2. #21433037C2semidirect
4. #21424871C2semidirect
5. #21424698C2semidirect
6. #21424698C2semidirect
10. #21423979C2semidirect
11. #21423974C2semidirect
12. #21423971C2semidirect
15. C59×#1986C2semidirect
19. #16837706C4semidirect
20. #16837706C4semidirect
21. #16837699C4semidirect
22. #16837699C4semidirect
23. #16837694C4semidirect
24. #16837689C4semidirect
25. #16837684C4semidirect
26. #16837679C4semidirect
71. #14273735Q8semidirect
316. #1986D59semidirect
385. C59#21629semidirect

Maximal subgroups

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

subgroupindex
#214338782
#214330372
#214325252
#214248712
#214246982
#214243722
#214241172
#214239792
#214239742
#214239712
#214239372
#214239302
C59×#19862
#2162959