Finite Group Databasebeta

← #28596543

Subgroups — #28596543

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#21433886C211yesmaximal, characteristic
2#21433036C211yesmaximal, characteristic, complemented
3#21432521C211yesmaximal, characteristic
4#21425039C211yesmaximal, characteristic, complemented
5#21424872C211yesmaximal, characteristic, complemented
6#21424696C211yesmaximal, characteristic, complemented
7#21424696C211yesmaximal, characteristic, complemented
8#21424374C211yesmaximal, characteristic
9#21424374C211yesmaximal, characteristic
10#21424117C211yesmaximal, characteristic
11#21423980C211yesmaximal, characteristic, complemented
12#21423975C211yesmaximal, characteristic, complemented
13#21423940C211yesmaximal, characteristic
14#21423933C211yesmaximal, characteristic
15C59×#1771C211yesmaximal, characteristic, complemented
16C2×#14273733C2211yescharacteristic
17C2×#14273733C2211yescharacteristic
18#16837959C2211yescharacteristic
19C4×#11745685C2211yescharacteristic
20#16837710C411yescharacteristic, complemented
21#16837710C411yescharacteristic, complemented
22#16837703C411yescharacteristic, complemented
23#16837703C411yescharacteristic, complemented
24#16837698C411yescharacteristic, complemented
25#16837693C411yescharacteristic, complemented
26#16837688C411yescharacteristic, complemented
27#16837683C411yescharacteristic, complemented
28#16837530—21no—
29#16837530—21no—
30#16837530—21no—
31#16837530—21no—
32C2×#14273663C2211yescharacteristic
33C2×#14273663C2211yescharacteristic
34#16837440C2211yescharacteristic
35#16837440C2211yescharacteristic
36#16837428C2211yescharacteristic
37#16837428C2211yescharacteristic
38#16837426C2211yescharacteristic
39C2×#14273655C2211yescharacteristic
40C2×#14273655C2211yescharacteristic
41#16837399C2211yescharacteristic
42#16837399C2211yescharacteristic
43#16837359—21no—
44#16837359—21no—
45#16837357C2211yescharacteristic
46#16837357C2211yescharacteristic
47#16837357C2211yescharacteristic
48#16837357C2211yescharacteristic
49C4×#11745664C2211yescharacteristic
50#16837354—21no—
51#97×D59—21no—
52#16837167C2211yescharacteristic
53#16837167C2211yescharacteristic
54#16837162C2211yescharacteristic
55#16837162C2211yescharacteristic
56#16837162C2211yescharacteristic
57#16837162C2211yescharacteristic
58#16837162C2211yescharacteristic
59#16837162C2211yescharacteristic
60C2×C4⋊C8×C59C2211yescharacteristic
61C2×#97×C59C2211yescharacteristic
62C2×#97×C59C2211yescharacteristic
63C2×C4×C8×C59C2211yescharacteristic
64C4×C4⋊C4×C59C2211yescharacteristic
65C59×#336C2211yescharacteristic
66C59×#336C2211yescharacteristic
67C2×#11745685C2311yescharacteristic
68#14273733—22no—
69#14273733—22no—
70#14273733—22no—
71#14273733Q814yescomplemented
72C2×#11745664C2311yescharacteristic
73C2×#11745664C2311yescharacteristic
74C2×#11745664C2311yescharacteristic
75C2×#11745664C2311yescharacteristic
76C2×#11745659—21no—
77C2×#11745659—21no—
78C2×C8×D59—21no—
79C2×C8×D59—21no—
80#14273727C2×C411yescharacteristic
81#14273727C2×C411yescharacteristic
82#14273719C2×C411yescharacteristic
83#14273719C2×C411yescharacteristic
84#14273706—21no—
85#14273706—21no—
86C4×Dic118—21no—
87C4×Dic118—21no—
88C4×Dic118C2×C411yescharacteristic
89C4×Dic118C2×C411yescharacteristic
90C4×D236C2×C411yescharacteristic
91C4×D236C2×C411yescharacteristic
92C4×C22⋊D59—21no—
93C4×C22⋊D59—21no—
94#14273686C2311yescharacteristic
95#14273686C2311yescharacteristic
96#14273686C2311yescharacteristic
97#14273684—21no—
98C42×D59—21no—
99#14273663—22no—
100#14273663—22no—
101#14273661C2×C411yescharacteristic
102#14273661C2×C411yescharacteristic
103#14273661C2×C411yescharacteristic
104#14273661C2×C411yescharacteristic
105#14273659C2×C411yescharacteristic
106#14273659C2×C411yescharacteristic
107#14273659C2×C411yescharacteristic
108#14273659C2×C411yescharacteristic
109C4⋊(C59⋊C8)—21no—
110C4⋊(C59⋊C8)—21no—
111Dic59⋊C8—21no—
112Dic59⋊C8—21no—
113Dic59⋊C8—21no—
114Dic59⋊C8—21no—
115#14273655—22no—
116#14273655—22no—
117#14273652C2×C411yescharacteristic
118#14273652C2×C411yescharacteristic
119#14273652C2×C411yescharacteristic
120#14273652C2×C411yescharacteristic
121#14273649C2×C411yescharacteristic
122#14273649C2×C411yescharacteristic
123#14273649C2×C411yescharacteristic
124#14273649C2×C411yescharacteristic
125#14273638—21no—
126#14273637—21no—
127#14273637—21no—
128#14273637—21no—
129#14273637—21no—
130C4×C59⋊C8—21no—
131C8×Dic59—21no—
132C8×Dic59—21no—
133C8×Dic59—21no—
134C8×Dic59—21no—
135C22×C8×C59C2311yescharacteristic
136C22×C8×C59C2311yescharacteristic
137C22×C8×C59C2311yescharacteristic
138C22×C8×C59C2311yescharacteristic
139C2×C4⋊C4×C59—21no—
140C2×C42×C59C2311yescharacteristic
141C2×C42×C59C2311yescharacteristic
142C2×C42×C59C2311yescharacteristic
143C4⋊C8×C59C2×C411yescharacteristic
144C4⋊C8×C59C2×C411yescharacteristic
145C4⋊C8×C59C2×C411yescharacteristic
146C4⋊C8×C59C2×C411yescharacteristic
147#97×C59—22no—
148#97×C59D411yescharacteristic
149#97×C59D411yescharacteristic
150#97×C59D411yescharacteristic
151#97×C59D411yescharacteristic
152C4×C8×C59—22no—
153#95×C59—21no—
154C22×C4×C59—21no—
155C22×C4×C59—21no—
156C22×C4×C59#4111yescharacteristic
157C22×C4×C59#4111yescharacteristic
158C22×C4×C59C2411yescharacteristic
159C2×C22⋊D59—21no—
160#11745685—21no—
161#11745685—21no—
162#11745685C2×Q812yes—
163#11745685#4112yes—
164C2×D236C22×C411yescharacteristic
165C2×C4×D59—21no—
166C2×C4×D59—21no—
167C2×C4×D59—21no—
168C2×Dic118—21no—
169C2×Dic118—21no—
170C2×Dic118C22×C411yescharacteristic
171C2×C8×C59—21no—
172C2×C8×C59—21no—
173C2×C8×C59—21no—
174C2×C8×C59—21no—
175C2×C8×C59—21no—
176C2×C8×C59—21no—
177C2×C8×C59C22×C411yescharacteristic
178C2×C8×C59C22×C411yescharacteristic
179C2×C8×C59C22×C411yescharacteristic
180C2×C8×C59C22×C411yescharacteristic
181C2×C8×C59C2×D411yescharacteristic
182C2×C8×C59C2×D411yescharacteristic
183C2×C8×C59C2×D411yescharacteristic
184C2×C8×C59C2×D411yescharacteristic
185C2×C8×C59C2×Q812yes—
186C2×C8×C59#4112yes—
187C4⋊C4×C59—42no—
188C42×C59—21no—
189C42×C59—21no—
190C42×C59—21no—
191C42×C59—21no—
192C42×C59C22×C411yescharacteristic
193C42×C59C22×C411yescharacteristic
194C42×C59C2×D411yescharacteristic
195C42×C59C2×D411yescharacteristic
196C22⋊Dic59—21no—
197C22⋊Dic59—21no—
198C22⋊Dic59—21no—
199(C2×D59)⋊C4—21no—
200(C2×D59)⋊C4—21no—
201C4⋊Dic59—21no—
202C4⋊Dic59—21no—
203C4⋊Dic59C22×C411yescharacteristic
204C4⋊Dic59C22×C411yescharacteristic
205C4⋊Dic59C22×C411yescharacteristic
206C4⋊Dic59C22×C411yescharacteristic
207C4⋊Dic59C22×C411yescharacteristic
208C4⋊Dic59C22×C411yescharacteristic
209Dic59⋊C4—21no—
210Dic59⋊C4—21no—
211Dic59⋊C4—21no—
212Dic59⋊C4—21no—
213Dic59⋊C4—21no—
214Dic59⋊C4—21no—
215C4×Dic59—21no—
216C4×Dic59—21no—
217C4×Dic59—21no—
218C4×Dic59—21no—
219C4×Dic59—21no—
220#11745664—21no—
221#11745664—21no—
222#11745664—21no—
223#11745664—21no—
224#11745664—21no—
225#11745664—21no—
226#11745664C2×Q812yes—
227#11745664#4112yes—
228C2×C59⋊C8—21no—
229C2×C59⋊C8—21no—
230C2×C59⋊C8—21no—
231C2×C59⋊C8—21no—
232#11745659—24no—
233#11745659—24no—
234C8×D59—24no—
235C8×D59—24no—
236C23×C59C2×#4111yescharacteristic
237C22×D59—21no—
238C2×C4×C59—22no—
239C2×C4×C59—41no—
240C2×C4×C59—41no—
241C2×C4×C59—41no—
242C2×C4×C59—41no—
243C2×C4×C59—41no—
244C2×C4×C59—41no—
245C2×C4×C59—41no—
246C2×C4×C59C4×D411yescharacteristic
247C2×C4×C59C4×D411yescharacteristic
248C2×C4×C59C4×D411yescharacteristic
249C2×C4×C59C4×D411yescharacteristic
250C2×C4×C59C23×C411yescharacteristic
251C2×C4×C59C22×D411yescharacteristic
252C2×C4×C59C22×Q811yescharacteristic
253C2×C4×C59C2×#4111yescharacteristic
254C2×C4×C59C2×#4111yescharacteristic
255C2×C4×C59C2×#4111yescharacteristic
256C22⋊D59—22no—
257C22⋊D59—22no—
258C2×Dic59—21no—
259C2×Dic59—21no—
260C2×Dic59—21no—
261C2×Dic59—21no—
262C2×Dic59—21no—
263C2×Dic59—21no—
264C2×Dic59—21no—
265D236—21no—
266D236C4×Q812yes—
267C4×D59—22no—
268C4×D59—22no—
269C4×D59—81no—
270C4×D59—81no—
271Dic118—21no—
272Dic118—81no—
273Dic118—81no—
274Dic118C4×Q812yes—
275C8×C59—21no—
276C8×C59—21no—
277C8×C59—21no—
278C8×C59—21no—
279C8×C59—21no—
280C8×C59—21no—
281C8×C59C22⋊Q812yes—
282C8×C59C22⋊Q812yes—
283C59⋊C8—22no—
284C59⋊C8—22no—
285C59⋊C8—41no—
286C59⋊C8—41no—
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—81no—
302C4×C59C2×C4×Q811yescharacteristic
303C4×C59C4×#4111yescharacteristic
304C4×C59C2×C22⋊Q811yescharacteristic
305C4×C59#52511yescharacteristic
306Dic59—22no—
307Dic59—41no—
308Dic59—41no—
309Dic59—41no—
310Dic59—41no—
311Dic59—41no—
312Dic59—41no—
313#3076—591no—
314#2961—591no—
315#2875—591no—
316#1986—591no—
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#1771D5911yescharacteristic, complemented
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#533—591no—
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⋊C8C2×D5911yescharacteristic
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—591no—
369C4×M16—1181no—
370C2×#97—1181no—
371C2×#97C2×D5911yescharacteristic
372C2×#97C2×D5911yescharacteristic
373C2×C4×C8C2×D5911yescharacteristic
374C4×C4⋊C4C2×D5911yescharacteristic
375#341—591no—
376#341—591no—
377#339—591no—
378#339—591no—
379#339—591no—
380#339—591no—
381#339—591no—
382#339—591no—
383#336C2×D5911yescharacteristic
384#336C2×D5911yescharacteristic
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—591no—
394C2×M16—591no—
395C2×M16—1181no—
396C2×M16—1181no—
397C22×C8—1181no—
398C22×C8—1181no—
399C22×C8C22×D5911yescharacteristic
400C22×C8C22×D5911yescharacteristic
401C22×C8C22×D5911yescharacteristic
402C22×C8C22×D5911yescharacteristic
403C4⋊Q8—591no—
404C4⋊Q8—591no—
405#125—591no—
406#125—591no—
407C22⋊Q8—1181no—
408C22⋊Q8—1181no—
409C4×Q8—591no—
410C4×Q8—591no—
411C4×Q8—1181no—
412C4×Q8—1181no—
413C4×D4—591no—
414C4×D4—591no—
415C4×D4—1181no—
416C4×D4—1181no—
417#117—591no—
418#117—591no—
419#117—591no—
420#117—1181no—
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—1181no—
437C4⋊C8—1181no—
438C4⋊C8—1181no—
439C4⋊C8—1181no—
440C4⋊C8—1181no—
441C4⋊C8—1181no—
442C4⋊C8C4×D5911yescharacteristic
443C4⋊C8C4×D5911yescharacteristic
444C4⋊C8C4×D5911yescharacteristic
445C4⋊C8C4×D5911yescharacteristic
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—22no—
457#97—1181no—
458#97—1181no—
459#97—1181no—
460#97—1181no—
461#97—1181no—
462#97D23611yescharacteristic
463#97D23611yescharacteristic
464#97D23611yescharacteristic
465#97D23611yescharacteristic
466C4×C8—22no—
467C4×C8—1181no—
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—591no—
478C2×Q8—1181no—
479C2×Q8—1181no—
480C2×D4—591no—
481C2×D4—1181no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—1181no—
485C22×C4—1181no—
486C22×C4—1181no—
487C22×C4#1174568511yescharacteristic
488C22×C4#1174568511yescharacteristic
489C22×C4C23×D5911yescharacteristic
490M16—592no—
491M16—592no—
492M16—1184no—
493M16—1181no—
494M16—1181no—
495M16—1184no—
496M16—1181no—
497M16—1181no—
498M16—1181no—
499M16—1181no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—1184no—
507C2×C8—1181no—
508C2×C8—1181no—
509C2×C8—1184no—
510C2×C8—1181no—
511C2×C8—1181no—
512C2×C8C2×C4×D5911yescharacteristic
513C2×C8C2×C4×D5911yescharacteristic
514C2×C8C2×C4×D5911yescharacteristic
515C2×C8C2×C4×D5911yescharacteristic
516C2×C8C2×D23611yescharacteristic
517C2×C8C2×D23611yescharacteristic
518C2×C8C2×D23611yescharacteristic
519C2×C8C2×D23611yescharacteristic
520C2×C8#1174568712yes—
521C2×C8Q8×D5912yes—
522C4⋊C4—42no—
523C4⋊C4—591no—
524C4⋊C4—591no—
525C4⋊C4—591no—
526C4⋊C4—591no—
527C4⋊C4—591no—
528C4⋊C4—591no—
529C4⋊C4—1181no—
530C4⋊C4—1181no—
531C4⋊C4—1181no—
532C4⋊C4—1181no—
533C4⋊C4—1181no—
534C4⋊C4—1181no—
535C4⋊C4—1181no—
536C4⋊C4—1181no—
537C22⋊C4—1181no—
538C22⋊C4—1181no—
539C22⋊C4—1181no—
540C22⋊C4—1181no—
541C22⋊C4—1181no—
542C42—21no—
543C42—21no—
544C42—21no—
545C42—21no—
546C42—1181no—
547C42—1181no—
548C42—1181no—
549C42—1181no—
550C42—1181no—
551C42C2×C4×D5911yescharacteristic
552C42C2×C4×D5911yescharacteristic
553C42C2×D23611yescharacteristic
554C42C2×D23611yescharacteristic
555C23—1181no—
556C23C2×#1174568511yescharacteristic
557Q8—592no—
558Q8—1181no—
559Q8—4721no—
560Q8—4721no—
561D4—592no—
562D4—1182no—
563D4—1182no—
564D4—1181no—
565C2×C4—22no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—1181no—
574C2×C4—1182no—
575C2×C4—1182no—
576C2×C4—1181no—
577C2×C4—1181no—
578C2×C4—1181no—
579C2×C4—1181no—
580C2×C4—1181no—
581C2×C4—1181no—
582C2×C4—4721no—
583C2×C4—4721no—
584C2×C4C4×D23611yescharacteristic
585C2×C4C4×D23611yescharacteristic
586C2×C4C4×D23611yescharacteristic
587C2×C4C4×D23611yescharacteristic
588C2×C4C22×C4×D5911yescharacteristic
589C2×C4C22×D23611yescharacteristic
590C2×C4C2×Q8×D5911yescharacteristic
591C2×C4#41×D5911yescharacteristic
592C2×C4#41×D5911yescharacteristic
593C2×C4C2×#1174568711yescharacteristic
594C8—21no—
595C8—21no—
596C8—21no—
597C8—21no—
598C8—21no—
599C8—21no—
600C8—1182no—
601C8—1182no—
602C8—2361no—
603C8—2361no—
604C8#1427370712yes—
605C8#1427370712yes—
606C22—21no—
607C22—21no—
608C22—1182no—
609C22—2361no—
610C22—2361no—
611C22C2×C4×D23611yescharacteristic
612C22#1683793411yescharacteristic
613C22#1683795611yescharacteristic
614C4—21no—
615C4—21no—
616C4—21no—
617C4—21no—
618C4—42no—
619C4—81no—
620C4—1182no—
621C4—2361no—
622C4—2361no—
623C4—2361no—
624C4—2361no—
625C4—2361no—
626C4—2361no—
627C4C4×Q8×D5911yescharacteristic
628C4C4×#1174568711yescharacteristic
629C4C2×#1427370711yescharacteristic
630C4#1683791411yescharacteristic
631C2—21no—
632C2—21no—
633C2—2362no—
634C2C4×#1427370711yescharacteristic
635C2#2143300711yescharacteristic
636C2#2143301411yescharacteristic
subgroupG/H
centerC2×C4C22×D236
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1771C2
socleC22×C59C2×C4×D4
prime radicalC2×#11745685C23
cosocleC22×C4×C59C24
hypercenter#1771D59
hypocenterC59#21629
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×#97 ◁ #1771

Complements

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

NH ≅ G/Ntype
2. #21433036C2semidirect
4. #21425039C2semidirect
5. #21424872C2semidirect
6. #21424696C2semidirect
7. #21424696C2semidirect
11. #21423980C2semidirect
12. #21423975C2semidirect
15. C59×#1771C2semidirect
20. #16837710C4semidirect
21. #16837710C4semidirect
22. #16837703C4semidirect
23. #16837703C4semidirect
24. #16837698C4semidirect
25. #16837693C4semidirect
26. #16837688C4semidirect
27. #16837683C4semidirect
71. #14273733Q8semidirect
325. #1771D59semidirect
385. C59#21629semidirect

Maximal subgroups

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

subgroupindex
#214338862
#214330362
#214325212
#214250392
#214248722
#214246962
#214243742
#214241172
#214239802
#214239752
#214239402
#214239332
C59×#17712
#2162959