Finite Group Databasebeta

← #28607191

Subgroups — #28607191

Order 15104.

850 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837524C211yesmaximal, characteristic
2#21432972C211yesmaximal, characteristic
3#21432694C211yesmaximal, characteristic, complemented
4#21432689C211yesmaximal, characteristic, complemented
5#21424881C211yesmaximal, characteristic, complemented
6#21424874C211yesmaximal, characteristic, complemented
7#21424329C211yesmaximal, characteristic, complemented
8#21424327C211yesmaximal, characteristic, complemented
9#21424322C211yesmaximal, characteristic, complemented
10#21424320C211yesmaximal, characteristic, complemented
11#21424261C211yesmaximal, characteristic, complemented
12#21424158C211yesmaximal, characteristic, complemented
13#21424153C211yesmaximal, characteristic, complemented
14#21423981C211yesmaximal, characteristic, complemented
15C59×#2952C211yesmaximal, characteristic
16C2×#14273734C2211yescharacteristic
17C2×#14273734C2211yescharacteristic
18#16837885C2211yescharacteristic
19#16837885C2211yescharacteristic
20#16837524C418yescomplemented
21#16837503—21no—
22#16837503—21no—
23Q16×Dic59—21no—
24D8×Dic59—21no—
25C2×#14273663C2211yescharacteristic
26C2×#14273662C2211yescharacteristic
27#16837445—21no—
28#16837445—21no—
29#16837445C2211yescharacteristic
30#16837445C2211yescharacteristic
31#16837442—21no—
32#16837442—21no—
33#16837442C2211yescharacteristic
34#16837442C2211yescharacteristic
35#16837438—21no—
36#16837438—21no—
37#16837438C2211yescharacteristic
38C2×#14273661C2211yescharacteristic
39C2×#14273660—21no—
40C2×#14273660C2211yescharacteristic
41C2×#14273658—21no—
42C2×#14273658C2211yescharacteristic
43C2×#14273654C2211yescharacteristic
44C2×#14273654C2211yescharacteristic
45#16837403C2211yescharacteristic
46#16837403C2211yescharacteristic
47#16837357C2211yescharacteristic
48C2×#14273637C2211yescharacteristic
49#16837271C2211yescharacteristic
50#16837267C2211yescharacteristic
51#16837171C2211yescharacteristic
52#16837171C2211yescharacteristic
53#16837170C2211yescharacteristic
54#16837170C2211yescharacteristic
55#16837163C2211yescharacteristic
56#16837161C2211yescharacteristic
57C2×#135×C59C2211yescharacteristic
58C59×#520C2211yescharacteristic
59C59×#520C2211yescharacteristic
60C59×#442—21no—
61C59×#441—21no—
62C4×SD16×C59—21no—
63C4×SD16×C59—21no—
64C2×#106×C59C2211yescharacteristic
65C59×#417C2211yescharacteristic
66C59×#417C2211yescharacteristic
67C59×#405C2211yescharacteristic
68#14273734—22no—
69#14273734—22no—
70#14273734C2×C414yescomplemented
71#14273734C2×C414yescomplemented
72C2×#11745664—21no—
73C2×#11745664—21no—
74C2×#11745664C2311yescharacteristic
75C2×#11745664C2311yescharacteristic
76C2×#11745664C2311yescharacteristic
77C22×C59⋊C8—21no—
78C22×C59⋊C8—21no—
79C22×C59⋊C8C2311yescharacteristic
80Q8×Dic59—21no—
81Q8×Dic59—21no—
82D4×Dic59—21no—
83D4×Dic59—21no—
84D4×Dic59—41no—
85D4×Dic59—41no—
86#14273686—21no—
87#14273686—21no—
88#14273686C2311yescharacteristic
89C2×C4⋊Dic59—21no—
90C2×C4⋊Dic59—21no—
91C2×C4⋊Dic59C2311yescharacteristic
92C2×C4⋊Dic59C2311yescharacteristic
93C2×Dic59⋊C4—41no—
94C2×C4×Dic59C2311yescharacteristic
95#14273663C2×C414yescomplemented
96#14273662—22no—
97#14273661—21no—
98#14273661—21no—
99#14273660—21no—
100#14273660—21no—
101#14273660—21no—
102#14273660—21no—
103#14273660—21no—
104#14273660—21no—
105#14273660D411yescharacteristic
106#14273660D411yescharacteristic
107#14273658—21no—
108#14273658—21no—
109#14273658—21no—
110#14273658—21no—
111#14273658—21no—
112#14273658—21no—
113#14273658D411yescharacteristic
114#14273658D411yescharacteristic
115#14273654C2×C414yescomplemented
116#14273654C2×C414yescomplemented
117#14273651—21no—
118#14273651—21no—
119#14273648—21no—
120#14273648—21no—
121#14273637—21no—
122#14273637C2×C414yescomplemented
123C8×Dic59—21no—
124C59⋊#95—41no—
125C2×#41×C59C2311yescharacteristic
126C2×#41×C59C2311yescharacteristic
127#135×C59C2×C414yes—
128#135×C59Q814yes—
129C2×Q16×C59D411yescharacteristic
130C2×SD16×C59D411yescharacteristic
131C2×SD16×C59D411yescharacteristic
132C2×D8×C59D411yescharacteristic
133C2×M16×C59C2311yescharacteristic
134C22×C8×C59C2311yescharacteristic
135C4×Q8×C59—21no—
136C4×Q8×C59—21no—
137C4×D4×C59—21no—
138C4×D4×C59—21no—
139C4×D4×C59—41no—
140C4×D4×C59—41no—
141#117×C59—21no—
142#117×C59—21no—
143#117×C59C2311yescharacteristic
144C2×C4⋊C4×C59—21no—
145C2×C4⋊C4×C59—21no—
146C2×C4⋊C4×C59C2311yescharacteristic
147C2×C4⋊C4×C59C2311yescharacteristic
148#106×C59—21no—
149#106×C59—21no—
150#103×C59—21no—
151#103×C59—21no—
152#102×C59—21no—
153#102×C59—21no—
154#97×C59—21no—
155C4×C8×C59—21no—
156#41×C59—21no—
157#41×C59—21no—
158#41×C59—21no—
159#41×C59—21no—
160#41×C59C22×C412yes—
161#41×C59C22×C412yes—
162#41×C59C2×Q812yes—
163#41×C59C2×Q812yes—
164C2×Q8×C59C2×D411yescharacteristic
165C2×Q8×C59C2×D411yescharacteristic
166C2×D4×C59—21no—
167C2×D4×C59—21no—
168C2×D4×C59C2×D411yescharacteristic
169C2×D4×C59C2×D411yescharacteristic
170C22×C4×C59—21no—
171C22×C4×C59—21no—
172C22×C4×C59—21no—
173C22×C4×C59—21no—
174C22×C4×C59—41no—
175C22×C4×C59—41no—
176C22×C4×C59C2411yescharacteristic
177C22×Dic59—21no—
178C22×Dic59—21no—
179C22×Dic59—21no—
180C22×Dic59—41no—
181C22×Dic59—41no—
182Q16×C59C4⋊C414yescomplemented
183SD16×C59C4⋊C414yescomplemented
184SD16×C59C4⋊C414yescomplemented
185D8×C59C4⋊C414yescomplemented
186M16×C59—21no—
187M16×C59—21no—
188C2×C8×C59—21no—
189C2×C8×C59C22×C412yes—
190C2×C8×C59C2×D411yescharacteristic
191C2×C8×C59C2×D411yescharacteristic
192C2×C8×C59C2×Q812yes—
193C4⋊C4×C59—21no—
194C4⋊C4×C59—21no—
195C4⋊C4×C59—21no—
196C4⋊C4×C59—21no—
197C4⋊C4×C59—41no—
198C4⋊C4×C59—41no—
199C4⋊C4×C59C2×D411yescharacteristic
200C4⋊C4×C59C2×D411yescharacteristic
201C22⋊C4×C59—21no—
202C22⋊C4×C59—41no—
203C22⋊C4×C59—41no—
204C42×C59—21no—
205C42×C59—41no—
206C42×C59—41no—
207C22⋊Dic59—21no—
208C22⋊Dic59—41no—
209C22⋊Dic59—41no—
210C4⋊Dic59—21no—
211C4⋊Dic59—21no—
212C4⋊Dic59—21no—
213C4⋊Dic59—21no—
214C4⋊Dic59—41no—
215C4⋊Dic59—41no—
216C4⋊Dic59C2×D411yescharacteristic
217C4⋊Dic59C2×D411yescharacteristic
218Dic59⋊C4—82no—
219C4×Dic59—21no—
220C4×Dic59—21no—
221C4×Dic59—41no—
222C4×Dic59—41no—
223C4×Dic59C22×C412yes—
224#11745664—21no—
225#11745664—21no—
226#11745664—24no—
227#11745664—24no—
228#11745664—21no—
229#11745664—21no—
230#11745664C22×C412yes—
231#11745664C22×C412yes—
232C2×C59⋊C8—21no—
233C2×C59⋊C8—21no—
234C2×C59⋊C8—24no—
235C2×C59⋊C8—24no—
236C2×C59⋊C8—21no—
237C2×C59⋊C8C22×C412yes—
238C2×C59⋊C8C2×D411yescharacteristic
239C2×C59⋊C8C2×D411yescharacteristic
240C2×C59⋊C8#4112yes—
241C23×C59—21no—
242C23×C59—21no—
243C23×C59H(5,𝔽2)11yescharacteristic
244Q8×C59—21no—
245Q8×C59—21no—
246Q8×C59C2×C4⋊C412yes—
247Q8×C59C2×C4⋊C412yes—
248D4×C59—22no—
249D4×C59—22no—
250D4×C59—22no—
251D4×C59—22no—
252D4×C59—21no—
253D4×C59—21no—
254D4×C59C2×C4⋊C412yes—
255D4×C59C2×C4⋊C412yes—
256C2×C4×C59—21no—
257C2×C4×C59—21no—
258C2×C4×C59—22no—
259C2×C4×C59—22no—
260C2×C4×C59—22no—
261C2×C4×C59—22no—
262C2×C4×C59—21no—
263C2×C4×C59—21no—
264C2×C4×C59—41no—
265C2×C4×C59—41no—
266C2×C4×C59—81no—
267C2×C4×C59—81no—
268C2×C4×C59—161no—
269C2×C4×C59—161no—
270C2×C4×C59C23×C411yescharacteristic
271C2×C4×C59C22×D411yescharacteristic
272C2×C4×C59C22×D411yescharacteristic
273C2×C4×C59C22×Q811yescharacteristic
274C2×C4×C59C2×#4111yescharacteristic
275C2×C4×C592−1+411yescharacteristic
276C2×Dic59—22no—
277C2×Dic59—21no—
278C2×Dic59—21no—
279C2×Dic59—41no—
280C2×Dic59—41no—
281C2×Dic59—41no—
282C2×Dic59—41no—
283C2×Dic59—81no—
284C2×Dic59—81no—
285C2×Dic59—161no—
286C2×Dic59—161no—
287C8×C59—41no—
288C8×C59C2×C4⋊C412yes—
289C8×C59C2×C4⋊C412yes—
290C59⋊C8—22no—
291C59⋊C8—22no—
292C59⋊C8—41no—
293C59⋊C8C4×D412yes—
294C59⋊C8C4×D412yes—
295#22693—591nomaximal, Sylow
296C22×C59—21no—
297C22×C59—21no—
298C22×C59—22no—
299C22×C59—22no—
300C22×C59—41no—
301C22×C59—41no—
302C22×C59—41no—
303C22×C59—41no—
304C22×C59#52011yescharacteristic
305C22×C59D4211yescharacteristic
306C22×C59#55411yescharacteristic
307C4×C59—22no—
308C4×C59—22no—
309C4×C59—41no—
310C4×C59—41no—
311C4×C59—81no—
312C4×C59—81no—
313C4×C59C22×C4⋊C411yescharacteristic
314C4×C59C2×C4×D411yescharacteristic
315C4×C59#54711yescharacteristic
316C4×C59D4×Q811yescharacteristic
317Dic59—42no—
318Dic59—41no—
319Dic59—41no—
320Dic59—81no—
321Dic59—81no—
322C2×#444—591no—
323#2952D5911yescharacteristic
324#2951—591no—
325#2913—591no—
326#2912—591no—
327#1952—591no—
328#1951—591no—
329#1860—591no—
330#1860—591no—
331#1859—591no—
332#1859—591no—
333#1847—591no—
334#1816—591no—
335#1815—591no—
336#1773—591no—
337C2×C59—21no—
338C2×C59—21no—
339C2×C59—42no—
340C2×C59—42no—
341C2×C59D4×C4⋊C411yescharacteristic
342C2×C59#328811yescharacteristic
343C2×C59#328911yescharacteristic
344C2×#135C2×D5911yescharacteristic
345C2×#131—591no—
346C2×#131—591no—
347#520—591no—
348#520—591no—
349#520C2×D5911yescharacteristic
350#520C2×D5911yescharacteristic
351#444—598no—
352#442—21no—
353#441—21no—
354#440—1181no—
355#440—1181no—
356C4×Q16—1181no—
357C4×SD16—21no—
358C4×SD16—21no—
359C4×D8—1181no—
360C2×#108—591no—
361C2×#108—591no—
362#428—591no—
363#428—591no—
364#428—591no—
365#428—591no—
366#428—1181no—
367#428—1181no—
368#428—1181no—
369#428—1181no—
370#427—591no—
371#427—1181no—
372#427—1181no—
373C2×#107—591no—
374C2×#107—591no—
375C2×#107—1181no—
376C2×#106—591no—
377C2×#106—1181no—
378C2×#106C2×D5911yescharacteristic
379C2×#104—591no—
380C2×#104—591no—
381#417—591no—
382#417—591no—
383#417C2×D5911yescharacteristic
384#417C2×D5911yescharacteristic
385#405—591no—
386#405C2×D5911yescharacteristic
387C2×#97—591no—
388#384—591no—
389#382—591no—
390#343—591no—
391#343—591no—
392#343—591no—
393#343—591no—
394#339—591no—
395#339—591no—
396C59#2269311yescharacteristic, complemented, Sylow
397C2×#41C22×D5911yescharacteristic
398C2×#41C22×D5911yescharacteristic
399#135Dic11814yes—
400#135C4×D5914yes—
401C2×Q16C22⋊D5911yescharacteristic
402C2×SD16C22⋊D5911yescharacteristic
403C2×SD16C22⋊D5911yescharacteristic
404C2×D8C22⋊D5911yescharacteristic
405#131—594no—
406#131—594no—
407#131—1182no—
408#131—1182no—
409C2×M16—591no—
410C2×M16—591no—
411C2×M16—591no—
412C2×M16—1181no—
413C2×M16—1181no—
414C2×M16C22×D5911yescharacteristic
415C22×C8—591no—
416C22×C8—1181no—
417C22×C8—1181no—
418C22×C8C22×D5911yescharacteristic
419C4×Q8—21no—
420C4×Q8—21no—
421C4×Q8—1181no—
422C4×Q8—1181no—
423C4×D4—21no—
424C4×D4—21no—
425C4×D4—41no—
426C4×D4—41no—
427C4×D4—1181no—
428C4×D4—1181no—
429C4×D4—2361no—
430C4×D4—2361no—
431#117—21no—
432#117—21no—
433#117—591no—
434#117—1181no—
435#117—1181no—
436#117C22×D5911yescharacteristic
437C2×C4⋊C4—21no—
438C2×C4⋊C4—21no—
439C2×C4⋊C4—591no—
440C2×C4⋊C4—591no—
441C2×C4⋊C4—1181no—
442C2×C4⋊C4—1181no—
443C2×C4⋊C4—2361no—
444C2×C4⋊C4C22×D5911yescharacteristic
445C2×C4⋊C4C22×D5911yescharacteristic
446C2×C42—591no—
447#108—594no—
448#108—1182no—
449#107—591no—
450#107—591no—
451#107—1181no—
452#107—1181no—
453#107—1181no—
454#107—1181no—
455#107—1181no—
456#107—1181no—
457#107—1181no—
458#107—1181no—
459#106—21no—
460#106—21no—
461#106—591no—
462#106—591no—
463#106—1181no—
464#106—1181no—
465#106—1181no—
466#106—1181no—
467#106—1181no—
468#106—1181no—
469#104—594no—
470#104—594no—
471#103—21no—
472#103—21no—
473#103—1181no—
474#103—1181no—
475#102—21no—
476#102—21no—
477#102—1181no—
478#102—1181no—
479#97—21no—
480#97—594no—
481#97—1181no—
482C4×C8—21no—
483C4×C8—1181no—
484#95—2361no—
485#41—21no—
486#41—21no—
487#41—21no—
488#41—21no—
489#41C2×Dic11812yes—
490#41C2×Dic11812yes—
491#41C2×C4×D5912yes—
492#41C2×C4×D5912yes—
493C2×Q8C2×C22⋊D5911yescharacteristic
494C2×Q8C2×C22⋊D5911yescharacteristic
495C2×D4—21no—
496C2×D4—21no—
497C2×D4C2×C22⋊D5911yescharacteristic
498C2×D4C2×C22⋊D5911yescharacteristic
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—21no—
503C22×C4—41no—
504C22×C4—41no—
505C22×C4—1181no—
506C22×C4—1181no—
507C22×C4—1181no—
508C22×C4—2361no—
509C22×C4—2361no—
510C22×C4C23×D5911yescharacteristic
511Q16Dic59⋊C414yescomplemented
512SD16Dic59⋊C414yescomplemented
513SD16Dic59⋊C414yescomplemented
514D8Dic59⋊C414yescomplemented
515M16—21no—
516M16—21no—
517M16—592no—
518M16—592no—
519M16—1184no—
520M16—1184no—
521M16—1181no—
522M16—1181no—
523M16—1181no—
524M16—1181no—
525C2×C8—21no—
526C2×C8—592no—
527C2×C8—591no—
528C2×C8—591no—
529C2×C8—592no—
530C2×C8—1184no—
531C2×C8—1184no—
532C2×C8—1181no—
533C2×C8—1181no—
534C2×C8—1181no—
535C2×C8C2×Dic11812yes—
536C2×C8C2×C4×D5912yes—
537C2×C8C2×C22⋊D5911yescharacteristic
538C2×C8C2×C22⋊D5911yescharacteristic
539C4⋊C4—21no—
540C4⋊C4—21no—
541C4⋊C4—21no—
542C4⋊C4—21no—
543C4⋊C4—41no—
544C4⋊C4—41no—
545C4⋊C4—591no—
546C4⋊C4—591no—
547C4⋊C4—1181no—
548C4⋊C4—1181no—
549C4⋊C4—1181no—
550C4⋊C4—1181no—
551C4⋊C4—4722no—
552C4⋊C4—2361no—
553C4⋊C4—2361no—
554C4⋊C4D4×D5911yescharacteristic
555C4⋊C4D4×D5911yescharacteristic
556C22⋊C4—21no—
557C22⋊C4—41no—
558C22⋊C4—41no—
559C22⋊C4—1181no—
560C22⋊C4—2361no—
561C22⋊C4—2361no—
562C42—21no—
563C42—41no—
564C42—41no—
565C42—592no—
566C42—1181no—
567C42—1181no—
568C42—2361no—
569C42—2361no—
570C23—21no—
571C23—21no—
572C23#1427377211yescharacteristic
573Q8—21no—
574Q8—21no—
575Q8C2×Dic59⋊C412yes—
576Q8C2×Dic59⋊C412yes—
577D4—22no—
578D4—22no—
579D4—22no—
580D4—22no—
581D4—21no—
582D4—21no—
583D4C2×Dic59⋊C412yes—
584D4C2×Dic59⋊C412yes—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—81no—
596C2×C4—81no—
597C2×C4—161no—
598C2×C4—161no—
599C2×C4—1181no—
600C2×C4—1181no—
601C2×C4—1182no—
602C2×C4—9441no—
603C2×C4—9441no—
604C2×C4—4721no—
605C2×C4—4721no—
606C2×C4—2361no—
607C2×C4—2361no—
608C2×C4—2361no—
609C2×C4—2361no—
610C2×C4C22×C4×D5911yescharacteristic
611C2×C4C2×D4×D5911yescharacteristic
612C2×C4C22×C22⋊D5911yescharacteristic
613C2×C4C22×Dic11811yescharacteristic
614C2×C4#41×D5911yescharacteristic
615C2×C4#1427377411yescharacteristic
616C8—41no—
617C8—592no—
618C8—592no—
619C8—1182no—
620C8—1182no—
621C8—2361no—
622C8C2×Dic59⋊C412yes—
623C8C2×Dic59⋊C412yes—
624C22—21no—
625C22—21no—
626C22—22no—
627C22—22no—
628C22—41no—
629C22—41no—
630C22—41no—
631C22—41no—
632C22#1683788111yescharacteristic
633C22D4×C22⋊D5911yescharacteristic
634C22#1683811111yescharacteristic
635C4—22no—
636C4—22no—
637C4—41no—
638C4—41no—
639C4—81no—
640C4—81no—
641C4—4721no—
642C4—4721no—
643C4—2362no—
644C4—2361no—
645C4—2361no—
646C4C22×Dic59⋊C411yescharacteristic
647C4C4×D4×D5911yescharacteristic
648C4#1683805111yescharacteristic
649C4D4×Dic11811yescharacteristic
650C2—21no—
651C2—21no—
652C2—42no—
653C2—42no—
654C2D4×Dic59⋊C411yescharacteristic
655C2#2143585811yescharacteristic
656C2#2143586911yescharacteristic
subgroupG/H
centerC22D4×C22⋊D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#2952C2
socleC22×C59D42
prime radicalC2×#135×C59C22
cosocleC22×C4×C59C24
hypercenter#2952D59
hypocenterC59#22693
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ #2952

Complements

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

NH ≅ G/Ntype
3. #21432694C2semidirect
4. #21432689C2semidirect
5. #21424881C2semidirect
6. #21424874C2semidirect
7. #21424329C2semidirect
8. #21424327C2semidirect
9. #21424322C2semidirect
10. #21424320C2semidirect
11. #21424261C2semidirect
12. #21424158C2semidirect
13. #21424153C2semidirect
14. #21423981C2semidirect
20. #16837524C4semidirect
70. #14273734C2×C4semidirect
71. #14273734C2×C4semidirect
95. #14273663C2×C4semidirect
115. #14273654C2×C4semidirect
116. #14273654C2×C4semidirect
122. #14273637C2×C4semidirect
182. Q16×C59C4⋊C4semidirect
183. SD16×C59C4⋊C4semidirect
184. SD16×C59C4⋊C4semidirect
185. D8×C59C4⋊C4semidirect
396. C59#22693semidirect
511. Q16Dic59⋊C4semidirect
512. SD16Dic59⋊C4semidirect
513. SD16Dic59⋊C4semidirect
514. D8Dic59⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#168375242
#214329722
#214326942
#214326892
#214248812
#214248742
#214243292
#214243272
#214243222
#214243202
#214242612
#214241582
#214241532
#214239812
C59×#29522
#2269359