Finite Group Databasebeta

← #28593336

Subgroups — #28593336

Order 15104.

1338 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16838027C211yesmaximal, characteristic
2#21426709C211yesmaximal, characteristic
3#21426448C211yesmaximal, characteristic
4#21425082C211yesmaximal, characteristic, complemented
5#21425081C211yesmaximal, characteristic, complemented
6#21425080C211yesmaximal, characteristic, complemented
7#21425079C214yesmaximal, complemented
8#342×D59C214yesmaximal
9C59×#1996C211yesmaximal, characteristic, complemented
10#16838027—22no—
11#16838027—22no—
12#16838027—22no—
13#16838027—22no—
14C2×#14273707C412yescomplemented
15C2×#14273706C2211yescharacteristic
16C2×#14273705C412yescomplemented
17C2×#14273705C412yescomplemented
18C2×#14273704C2212yes—
19C2×#14273703C412yescomplemented
20C22×(C2×D59)⋊C4C2211yescharacteristic
21C2×C22⋊C4×D59C2212yes—
22C2×C22⋊C4×D59C2212yes—
23C2×C22⋊C4×D59C2212yes—
24#16837276—22no—
25#16837273—22no—
26#16837262C2211yescharacteristic
27#16837260C2212yes—
28C22⋊C4×Dic59C2212yes—
29C4×(C2×D59)⋊C4—22no—
30C4×C22⋊Dic59C2211yescharacteristic
31#16837249—22no—
32#16837169C2214yes—
33#16837168C2214yes—
34#16837168C2214yes—
35C2×C22⋊Q8×C59C2211yescharacteristic
36C59×#380C2211yescharacteristic
37C4×C22⋊C4×C59C2211yescharacteristic
38C59×#342C2214yes—
39C24×D59C2311yescharacteristic
40C22×Dic118C2×C412yescomplemented
41C22×Dic118C2×C411yescharacteristic, complemented
42C23×Dic59C2311yescharacteristic
43C22×C4×D59—22no—
44C22×C4×D59—22no—
45C22×C4×D59C2×C412yes—
46C22×C4×D59C2×C412yes—
47#14273707—24no—
48#14273707—24no—
49#14273706—21no—
50#14273706—21no—
51#14273706—21no—
52#14273706—21no—
53#14273705—24no—
54#14273705—24no—
55#14273705—24no—
56#14273705—24no—
57#14273704—24no—
58#14273704—24no—
59#14273703—24no—
60#14273703—24no—
61C2×C4⋊Dic59—21no—
62C2×C4⋊Dic59C2×C412yescomplemented
63C2×C4⋊Dic59C2×C411yescharacteristic, complemented
64C2×Dic59⋊C4—22no—
65C2×Dic59⋊C4C2×C412yescomplemented
66C2×Dic59⋊C4C2×C412yescomplemented
67C2×Dic59⋊C4C2×C412yescomplemented
68C2×C22⋊Dic59C2312yes—
69C2×C22⋊Dic59C2312yes—
70C2×C22⋊Dic59C2312yes—
71C2×(C2×D59)⋊C4—22no—
72C2×(C2×D59)⋊C4—22no—
73C2×(C2×D59)⋊C4C2×C412yes—
74C2×(C2×D59)⋊C4C2×C412yes—
75C2×(C2×D59)⋊C4C2×C412yes—
76C2×(C2×D59)⋊C4C2×C412yes—
77C2×(C2×D59)⋊C4D412yes—
78C2×(C2×D59)⋊C4D412yes—
79C2×(C2×D59)⋊C4D412yes—
80C2×(C2×D59)⋊C4D412yes—
81C22⋊C4×D59—24no—
82C22⋊C4×D59—24no—
83C22⋊C4×D59—24no—
84C22⋊C4×D59—28no—
85C22⋊C4×D59Q818yescomplemented
86C2×C4×Dic59—21no—
87C2×C4×Dic59—22no—
88C59⋊#95—21no—
89C59⋊#95—21no—
90C59⋊#95—22no—
91C59⋊#95—22no—
92C22×Q8×C59C2×C411yescharacteristic, complemented
93C23×C4×C59C2311yescharacteristic
94C22⋊Q8×C59—21no—
95C22⋊Q8×C59—21no—
96C22⋊Q8×C59—21no—
97C22⋊Q8×C59—21no—
98C2×C4⋊C4×C59—21no—
99C2×C4⋊C4×C59C2×C411yescharacteristic, complemented
100C2×C4⋊C4×C59C2×C412yescomplemented
101C2×C22⋊C4×C59C2312yes—
102C2×C22⋊C4×C59C2312yes—
103C2×C22⋊C4×C59C2312yes—
104C2×C42×C59—21no—
105#95×C59—21no—
106#95×C59—21no—
107C24×C59C2411yescharacteristic
108C23×D59—81no—
109C23×D59—81no—
110C23×D59C22×C412yes—
111C23×D59C2×D412yes—
112C23×D59C2×D412yes—
113C23×D59C2×Q814yes—
114C23×D59#4114yes—
115C2×Q8×C59—21no—
116C2×Q8×C59—21no—
117C2×Q8×C59—81no—
118C22×C4×C59—22no—
119C22×C4×C59—22no—
120C22×C4×C59—81no—
121C22×C4×C59C22×C412yes—
122C22×C4×C59C22×C412yes—
123C22×C4×C59C22×C411yescharacteristic
124C22×C4×C59C22×C411yescharacteristic
125C22×C4×C59C2×D411yescharacteristic
126C22×C4×C59C2×D411yescharacteristic
127C22×C4×C59C2×D411yescharacteristic
128C22×C4×C59C2×D411yescharacteristic
129C22×Dic59—22no—
130C22×Dic59—22no—
131C22×Dic59—81no—
132C22×Dic59C22×C412yes—
133C22×Dic59C22×C412yes—
134C22×Dic59C22×C412yes—
135C22×Dic59C2×D412yes—
136C22×Dic59C2×D412yes—
137C2×C4×D59—24no—
138C2×C4×D59—24no—
139C2×C4×D59—28no—
140C2×C4×D59—44no—
141C2×C4×D59—82no—
142C2×C4×D59—82no—
143C2×C4×D59—82no—
144C2×C4×D59—82no—
145C2×C4×D59—82no—
146C2×C4×D59—82no—
147C2×Dic118—21no—
148C2×Dic118—21no—
149C2×Dic118—24no—
150C2×Dic118—82no—
151C2×Dic118—81no—
152C4⋊C4×C59—21no—
153C4⋊C4×C59—21no—
154C4⋊C4×C59—22no—
155C4⋊C4×C59—22no—
156C4⋊C4×C59—81no—
157C22⋊C4×C59—22no—
158C22⋊C4×C59—22no—
159C22⋊C4×C59—22no—
160C22⋊C4×C59—24no—
161C22⋊C4×C59C2×Q814yes—
162C42×C59—81no—
163C22⋊Dic59—22no—
164C22⋊Dic59—22no—
165C22⋊Dic59—22no—
166C22⋊Dic59—24no—
167C22⋊Dic59C2×Q814yes—
168(C2×D59)⋊C4—24no—
169(C2×D59)⋊C4—24no—
170(C2×D59)⋊C4—28no—
171(C2×D59)⋊C4—44no—
172(C2×D59)⋊C4C22⋊C414yes—
173(C2×D59)⋊C4C22⋊C414yes—
174(C2×D59)⋊C4C22⋊C414yes—
175(C2×D59)⋊C4C22⋊C414yes—
176C4⋊Dic59—21no—
177C4⋊Dic59—21no—
178C4⋊Dic59—24no—
179C4⋊Dic59—81no—
180Dic59⋊C4—22no—
181Dic59⋊C4—22no—
182Dic59⋊C4—24no—
183Dic59⋊C4—24no—
184Dic59⋊C4—82no—
185C4×Dic59—82no—
186C4×Dic59—81no—
187C23×C59—41no—
188C23×C59—41no—
189C23×C59C23×C411yescharacteristic
190C23×C59C22×D411yescharacteristic
191C23×C59C22×D411yescharacteristic
192C23×C59C22×Q812yes—
193C23×C59C2×#4112yes—
194C22×D59—42no—
195C22×D59—42no—
196C22×D59—42no—
197C22×D59—42no—
198C22×D59—42no—
199C22×D59—42no—
200C22×D59—84no—
201C22×D59—84no—
202C22×D59C2×C22⋊C414yes—
203C22×D59C4×Q818yes—
204C22×D59C22⋊Q814yes—
205C22×D59C22⋊Q814yes—
206C22×D59#12414yes—
207C22×D59C4⋊Q814yes—
208Q8×C59—81no—
209Q8×C59—81no—
210C2×C4×C59—22no—
211C2×C4×C59—22no—
212C2×C4×C59—24no—
213C2×C4×C59—41no—
214C2×C4×C59—41no—
215C2×C4×C59—41no—
216C2×C4×C59—41no—
217C2×C4×C59—41no—
218C2×C4×C59—41no—
219C2×C4×C59—42no—
220C2×C4×C59—42no—
221C2×C4×C59—42no—
222C2×C4×C59—42no—
223C2×C4×C59—42no—
224C2×C4×C59—42no—
225C2×C4×C59—42no—
226C2×C4×C59C2×C22⋊C411yescharacteristic
227C2×C4×C59C2×C22⋊C411yescharacteristic
228C2×C4×C59C2×C22⋊C411yescharacteristic
229C2×C4×C59C2×C22⋊C411yescharacteristic
230C2×Dic59—22no—
231C2×Dic59—22no—
232C2×Dic59—24no—
233C2×Dic59—42no—
234C2×Dic59—42no—
235C2×Dic59—42no—
236C2×Dic59—42no—
237C2×Dic59—42no—
238C2×Dic59—42no—
239C2×Dic59—42no—
240C2×Dic59—42no—
241C2×Dic59—42no—
242C2×Dic59—42no—
243C2×Dic59C2×C22⋊C414yes—
244C4×D59—84no—
245C4×D59—84no—
246C4×D59—88no—
247C4×D59—88no—
248Dic118—84no—
249Dic118—81no—
250Dic118—81no—
251#21333—591nomaximal, Sylow
252C22×C59—21no—
253C22×C59—21no—
254C22×C59—21no—
255C22×C59—21no—
256C22×C59—21no—
257C22×C59—21no—
258C22×C59—42no—
259C22×C59—42no—
260C22×C59C22×C22⋊C411yescharacteristic
261C22×C59C2×C4×Q812yes—
262C22×C59C2×C22⋊Q812yes—
263C22×C59C2×#12411yescharacteristic
264C22×C59C2×C4⋊Q811yescharacteristic
265C2×D59—24no—
266C2×D59—24no—
267C2×D59—44no—
268C2×D59—44no—
269C2×D59—48no—
270C2×D59—44no—
271C2×D59—44no—
272C2×D59#39118yes—
273C4×C59—21no—
274C4×C59—21no—
275C4×C59—21no—
276C4×C59—21no—
277C4×C59—44no—
278C4×C59—42no—
279C4×C59—42no—
280C4×C59—82no—
281Dic59—24no—
282Dic59—44no—
283Dic59—42no—
284Dic59—42no—
285Dic59—82no—
286C2×#543—591no—
287#2315—591no—
288#2273—591no—
289#1996—591no—
290#1996—591no—
291#1996—594no—
292#1996—591no—
293#1996D5911yescharacteristic, complemented
294C2×#342—594no—
295C2×C59—22no—
296C2×C59—21no—
297C2×C59—22no—
298C2×C59—21no—
299C2×C59C2×#39111yescharacteristic
300C2×C59#288311yescharacteristic
301C2×C59#288311yescharacteristic
302D59—28no—
303#543—1182no—
304#543—1182no—
305#543—1182no—
306#543—1182no—
307C2×C22⋊Q8—592no—
308C2×C22⋊Q8—591no—
309C2×C22⋊Q8—592no—
310C2×C22⋊Q8—592no—
311C2×C22⋊Q8—592no—
312C2×C22⋊Q8—592no—
313C2×C22⋊Q8C2×D5911yescharacteristic
314C22×C22⋊C4—591no—
315C22×C22⋊C4—592no—
316C22×C22⋊C4—592no—
317C22×C22⋊C4—592no—
318#385—1182no—
319#385—1182no—
320#380—591no—
321#380—592no—
322#380C2×D5911yescharacteristic
323C4×C22⋊C4—591no—
324C4×C22⋊C4—592no—
325C4×C22⋊C4—1182no—
326C4×C22⋊C4—1182no—
327C4×C22⋊C4C2×D5911yescharacteristic
328#342—594no—
329#342—594no—
330#342—594no—
331#342C2×D5914yes—
332C59#2133311yescharacteristic, complemented, Sylow
333C25—591no—
334C22×Q8—592no—
335C22×Q8—591no—
336C22×Q8C4×D5911yescharacteristic, complemented
337C23×C4—591no—
338C23×C4—592no—
339C23×C4—592no—
340C23×C4—1182no—
341C23×C4—1182no—
342C23×C4C22×D5911yescharacteristic
343C22⋊Q8—21no—
344C22⋊Q8—21no—
345C22⋊Q8—21no—
346C22⋊Q8—21no—
347C22⋊Q8—1184no—
348C22⋊Q8—1184no—
349C22⋊Q8—1184no—
350C22⋊Q8—1184no—
351C22⋊Q8—1184no—
352C22⋊Q8—1184no—
353C22⋊Q8—1184no—
354C22⋊Q8—1184no—
355C22⋊Q8—1184no—
356C22⋊Q8—1184no—
357C22⋊Q8—1181no—
358C22⋊Q8—1181no—
359C22⋊Q8—1181no—
360C22⋊Q8—1181no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—592no—
363C2×C4⋊C4—592no—
364C2×C4⋊C4—592no—
365C2×C4⋊C4—592no—
366C2×C4⋊C4—591no—
367C2×C4⋊C4—1182no—
368C2×C4⋊C4—1181no—
369C2×C4⋊C4C4×D5912yescomplemented
370C2×C4⋊C4C4×D5911yescharacteristic, complemented
371C2×C22⋊C4—592no—
372C2×C22⋊C4—592no—
373C2×C22⋊C4—592no—
374C2×C22⋊C4—592no—
375C2×C22⋊C4—592no—
376C2×C22⋊C4—592no—
377C2×C22⋊C4—592no—
378C2×C22⋊C4—598no—
379C2×C22⋊C4—592no—
380C2×C22⋊C4—592no—
381C2×C22⋊C4—592no—
382C2×C22⋊C4—592no—
383C2×C22⋊C4—1184no—
384C2×C22⋊C4—1184no—
385C2×C22⋊C4—1184no—
386C2×C22⋊C4—1188no—
387C2×C22⋊C4—1182no—
388C2×C22⋊C4—1182no—
389C2×C22⋊C4C22×D5912yes—
390C2×C22⋊C4C22×D5912yes—
391C2×C22⋊C4C22×D5912yes—
392C2×C42—21no—
393C2×C42—1182no—
394C2×C42—1181no—
395#95—21no—
396#95—21no—
397#95—1182no—
398#95—1182no—
399#95—1181no—
400#95—1181no—
401C24—594no—
402C24—594no—
403C24—592no—
404C24—592no—
405C24—592no—
406C24—4721no—
407C24—4721no—
408C24C23×D5911yescharacteristic
409C2×Q8—21no—
410C2×Q8—21no—
411C2×Q8—81no—
412C2×Q8—1184no—
413C2×Q8—1181no—
414C2×Q8—1181no—
415C2×Q8—4721no—
416C2×Q8—4722no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—81no—
420C22×C4—592no—
421C22×C4—592no—
422C22×C4—592no—
423C22×C4—592no—
424C22×C4—592no—
425C22×C4—1188no—
426C22×C4—1184no—
427C22×C4—1184no—
428C22×C4—1182no—
429C22×C4—1182no—
430C22×C4—4722no—
431C22×C4—4722no—
432C22×C4—4722no—
433C22×C4—4722no—
434C22×C4—4722no—
435C22×C4—4722no—
436C22×C4—4721no—
437C22×C4—2364no—
438C22×C4C2×C4×D5912yes—
439C22×C4C2×C4×D5912yes—
440C22×C4C2×C4×D5911yescharacteristic
441C22×C4C2×C4×D5911yescharacteristic
442C22×C4D4×D5911yescharacteristic
443C22×C4D4×D5911yescharacteristic
444C22×C4D4×D5911yescharacteristic
445C22×C4D4×D5911yescharacteristic
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—81no—
451C4⋊C4—1184no—
452C4⋊C4—1184no—
453C4⋊C4—1184no—
454C4⋊C4—1182no—
455C4⋊C4—1182no—
456C4⋊C4—1181no—
457C4⋊C4—1181no—
458C4⋊C4—4722no—
459C4⋊C4—4721no—
460C22⋊C4—22no—
461C22⋊C4—22no—
462C22⋊C4—22no—
463C22⋊C4—24no—
464C22⋊C4—594no—
465C22⋊C4—594no—
466C22⋊C4—594no—
467C22⋊C4—594no—
468C22⋊C4—594no—
469C22⋊C4—1184no—
470C22⋊C4—1184no—
471C22⋊C4—1188no—
472C22⋊C4—1182no—
473C22⋊C4—1182no—
474C22⋊C4—1182no—
475C22⋊C4—1184no—
476C22⋊C4—2364no—
477C22⋊C4Q8×D5914yes—
478C42—81no—
479C42—4721no—
480C42—4722no—
481C23—41no—
482C23—41no—
483C23—594no—
484C23—594no—
485C23—594no—
486C23—594no—
487C23—594no—
488C23—598no—
489C23—4724no—
490C23—4724no—
491C23—2362no—
492C23—2362no—
493C23—2362no—
494C23—2362no—
495C23—2362no—
496C23—2362no—
497C23C22×C4×D5911yescharacteristic
498C23C2×D4×D5911yescharacteristic
499C23C2×D4×D5911yescharacteristic
500C23C2×Q8×D5912yes—
501C23#41×D5912yes—
502Q8—81no—
503Q8—81no—
504Q8—4721no—
505Q8—4721no—
506Q8—4724no—
507C2×C4—24no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—594no—
524C2×C4—1184no—
525C2×C4—1182no—
526C2×C4—1182no—
527C2×C4—4724no—
528C2×C4—4724no—
529C2×C4—4728no—
530C2×C4—4728no—
531C2×C4—2362no—
532C2×C4—2362no—
533C2×C4—2362no—
534C2×C4—2362no—
535C2×C4—2362no—
536C2×C4—2362no—
537C2×C4—2362no—
538C2×C4—2362no—
539C2×C4—2362no—
540C2×C4—2362no—
541C2×C4C22⋊C4×D5911yescharacteristic
542C2×C4C22⋊C4×D5911yescharacteristic
543C2×C4C22⋊C4×D5911yescharacteristic
544C2×C4C22⋊C4×D5911yescharacteristic
545C22—21no—
546C22—21no—
547C22—21no—
548C22—21no—
549C22—21no—
550C22—21no—
551C22—42no—
552C22—42no—
553C22—598no—
554C22—1184no—
555C22—1184no—
556C22—2364no—
557C22—2364no—
558C22—2364no—
559C22—2364no—
560C22—2368no—
561C22C2×C22⋊C4×D5911yescharacteristic
562C22C4×Q8×D5912yes—
563C22C22⋊Q8×D5912yes—
564C22#124×D5911yescharacteristic
565C22C4⋊Q8×D5911yescharacteristic
566C4—21no—
567C4—21no—
568C4—21no—
569C4—21no—
570C4—44no—
571C4—42no—
572C4—42no—
573C4—82no—
574C4—1184no—
575C4—4722no—
576C4—2364no—
577C4—2362no—
578C4—2362no—
579C2—22no—
580C2—21no—
581C2—21no—
582C2—22no—
583C2—1188no—
584C2#391×D5911yescharacteristic
585C2#2143255411yescharacteristic
586C2#2143255511yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D59
commutatorC23×C59C23×C4
FrattiniC24C23×D59
FittingC59×#1996C2
socleC22×C59C22×C22⋊C4
prime radicalC24×D59C23
cosocleC24×C59C24
hypercenter#1996D59
hypocenterC59#21333
LCSG ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1996

Complements

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

NH ≅ G/Ntype
4. #21425082C2semidirect
5. #21425081C2semidirect
6. #21425080C2semidirect
7. #21425079C2semidirect
9. C59×#1996C2semidirect
14. C2×#14273707C4semidirect
16. C2×#14273705C4semidirect
17. C2×#14273705C4semidirect
19. C2×#14273703C4semidirect
40. C22×Dic118C2×C4semidirect
41. C22×Dic118C2×C4semidirect
62. C2×C4⋊Dic59C2×C4semidirect
63. C2×C4⋊Dic59C2×C4semidirect
65. C2×Dic59⋊C4C2×C4semidirect
66. C2×Dic59⋊C4C2×C4semidirect
67. C2×Dic59⋊C4C2×C4semidirect
85. C22⋊C4×D59Q8semidirect
92. C22×Q8×C59C2×C4semidirect
99. C2×C4⋊C4×C59C2×C4semidirect
100. C2×C4⋊C4×C59C2×C4semidirect
293. #1996D59semidirect
332. C59#21333semidirect
336. C22×Q8C4×D59semidirect
369. C2×C4⋊C4C4×D59semidirect
370. C2×C4⋊C4C4×D59semidirect

Maximal subgroups

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

subgroupindex
C2×#168380272
#214267092
#214264482
#214250822
#214250812
#214250802
#214250792
#342×D592
C59×#19962
#2133359