Finite Group Databasebeta

← #28647561

Subgroups — #28647561

Order 15104.

782 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837811C211yesmaximal, characteristic, complemented
2C2×#16837444C211yesmaximal, characteristic
3#21427820C211yesmaximal, characteristic
4#21427787C211yesmaximal, characteristic
5#21425579C211yesmaximal, characteristic, complemented
6#21425579C211yesmaximal, characteristic, complemented
7#21425435C211yesmaximal, characteristic, complemented
8#21425425C211yesmaximal, characteristic, complemented
9#21424834C211yesmaximal, characteristic, complemented
10#21424829C211yesmaximal, characteristic, complemented
11#21424297C211yesmaximal, characteristic
12#21424286C211yesmaximal, characteristic
13#21424260C211yesmaximal, characteristic, complemented
14#21424129C211yesmaximal, characteristic
15C59×#1753C211yesmaximal, characteristic
16C2×#14273727C2211yescharacteristic
17C2×#14273719C2211yescharacteristic
18C2×#14273686C2211yescharacteristic
19C22×C4⋊Dic59C2211yescharacteristic
20#16837811—24no—
21#16837444—24no—
22C2×#14273661C2211yescharacteristic
23C2×#14273661C2211yescharacteristic
24C2×#14273661C2211yescharacteristic
25C2×#14273661C2211yescharacteristic
26C2×#14273661C2211yescharacteristic
27C2×#14273661C2211yescharacteristic
28C2×#14273659C2211yescharacteristic
29C2×#14273659C2211yescharacteristic
30C2×#14273659C2211yescharacteristic
31C2×#14273659C2211yescharacteristic
32#16837344—21no—
33#16837336—21no—
34#16837325—21no—
35#16837314—21no—
36#16837306—21no—
37#16837300C2211yescharacteristic
38#16837300C2211yescharacteristic
39#16837300C2211yescharacteristic
40#16837300C2211yescharacteristic
41#16837289—21no—
42#16837274—21no—
43#16837268C2211yescharacteristic
44#16837268C2211yescharacteristic
45#16837262C2211yescharacteristic
46#16837262C2211yescharacteristic
47#16837260—21no—
48#16837165C2211yescharacteristic
49#16837165C2211yescharacteristic
50#16837165C2211yescharacteristic
51#16837165C2211yescharacteristic
52#16837165C2211yescharacteristic
53#16837165C2211yescharacteristic
54C22×M16×C59C2211yescharacteristic
55C2×C22⋊C8×C59C2211yescharacteristic
56C2×#97×C59C2211yescharacteristic
57C2×C4×C8×C59C2211yescharacteristic
58C4×C22⋊C4×C59C2211yescharacteristic
59C59×#336C2211yescharacteristic
60C59×#336C2211yescharacteristic
61C22×Dic118—21no—
62C23×Dic59—21no—
63#14273727—22no—
64#14273727—41no—
65#14273719—22no—
66#14273719—41no—
67#14273686—22no—
68#14273686—41no—
69C2×C4⋊Dic59—22no—
70C2×C4⋊Dic59—21no—
71C2×C4⋊Dic59—21no—
72C2×C4⋊Dic59—41no—
73C2×C4⋊Dic59C2311yescharacteristic
74C2×C4⋊Dic59C2311yescharacteristic
75C2×C4⋊Dic59C2311yescharacteristic
76C2×C4⋊Dic59C2311yescharacteristic
77C2×C4⋊Dic59C2311yescharacteristic
78C2×C4⋊Dic59C2311yescharacteristic
79C2×C4⋊Dic59C2311yescharacteristic
80C2×C4⋊Dic59C2311yescharacteristic
81C2×Dic59⋊C4—21no—
82C2×Dic59⋊C4—21no—
83C2×Dic59⋊C4—21no—
84C2×Dic59⋊C4—21no—
85C2×Dic59⋊C4—21no—
86C2×Dic59⋊C4—21no—
87C2×C22⋊Dic59—21no—
88C2×C22⋊Dic59—21no—
89C2×C22⋊Dic59—21no—
90C2×C4×Dic59—21no—
91#14273661—22no—
92#14273661—22no—
93#14273661—22no—
94#14273661—41no—
95#14273661—41no—
96#14273661D414yes—
97#14273659—22no—
98#14273659—22no—
99#14273659—22no—
100#14273659—22no—
101C59⋊#95—21no—
102C59⋊#95—21no—
103C59⋊#95—21no—
104C59⋊#95—21no—
105C59⋊#95—21no—
106C59⋊#95—21no—
107C59⋊#95—21no—
108C59⋊#95—21no—
109C59⋊#95—21no—
110C59⋊#95—21no—
111C23×C4×C59C2311yescharacteristic
112C2×M16×C59—22no—
113C2×M16×C59—22no—
114C2×M16×C59Q814yescomplemented
115C22×C8×C59C2311yescharacteristic
116C22×C8×C59C2311yescharacteristic
117C22×C8×C59C2311yescharacteristic
118C22×C8×C59C2311yescharacteristic
119C2×C22⋊C4×C59D411yescharacteristic
120C2×C22⋊C4×C59D411yescharacteristic
121C2×C42×C59C2311yescharacteristic
122C2×C42×C59C2311yescharacteristic
123C22⋊C8×C59Q814yescomplemented
124#97×C59—22no—
125C4×C8×C59—22no—
126#95×C59D411yescharacteristic
127#95×C59D411yescharacteristic
128C24×C59C2×D411yescharacteristic
129C22×C4×C59—21no—
130C22×C4×C59—21no—
131C22×C4×C59—21no—
132C22×C4×C59—21no—
133C22×C4×C59C2×D411yescharacteristic
134C22×C4×C59C2×D411yescharacteristic
135C22×C4×C59C2×D411yescharacteristic
136C22×C4×C59C2×D411yescharacteristic
137C22×C4×C59C2×D411yescharacteristic
138C22×C4×C59C2×Q812yes—
139C22×C4×C59C2×Q812yes—
140C22×C4×C59C2411yescharacteristic
141C22×Dic59—21no—
142C22×Dic59—21no—
143C22×Dic59—21no—
144C22×Dic59—21no—
145C22×Dic59—21no—
146C22×Dic59—21no—
147C22×Dic59—21no—
148C22×Dic59—21no—
149C22×Dic59—42no—
150C22×Dic59—42no—
151C22×Dic59—81no—
152C2×Dic118—42no—
153C2×Dic118—42no—
154C2×Dic118—81no—
155M16×C59—28no—
156C2×C8×C59—21no—
157C2×C8×C59—21no—
158C2×C8×C59—21no—
159C2×C8×C59—21no—
160C2×C8×C59—21no—
161C2×C8×C59—21no—
162C2×C8×C59C2×D412yes—
163C2×C8×C59C2×Q812yes—
164C2×C8×C59C2×Q812yes—
165C2×C8×C59C2×Q812yes—
166C2×C8×C59C2×Q812yes—
167C2×C8×C59#4112yes—
168C22⋊C4×C59—41no—
169C22⋊C4×C59—41no—
170C42×C59—21no—
171C42×C59—21no—
172C42×C59—21no—
173C42×C59—21no—
174C22⋊Dic59—42no—
175C22⋊Dic59—42no—
176C22⋊Dic59—81no—
177C4⋊Dic59—21no—
178C4⋊Dic59—21no—
179C4⋊Dic59—21no—
180C4⋊Dic59—21no—
181C4⋊Dic59—21no—
182C4⋊Dic59—21no—
183C4⋊Dic59—21no—
184C4⋊Dic59—21no—
185C4⋊Dic59—21no—
186C4⋊Dic59—21no—
187C4⋊Dic59—21no—
188C4⋊Dic59—21no—
189C4⋊Dic59—21no—
190C4⋊Dic59—21no—
191C4⋊Dic59—42no—
192C4⋊Dic59—42no—
193C4⋊Dic59C2×D412yes—
194C4⋊Dic59C2×D412yes—
195Dic59⋊C4—42no—
196Dic59⋊C4—42no—
197Dic59⋊C4—42no—
198Dic59⋊C4—81no—
199Dic59⋊C4—81no—
200Dic59⋊C4—81no—
201C4×Dic59—42no—
202C23×C59—21no—
203C23×C59—21no—
204C23×C59—21no—
205C23×C59—21no—
206C23×C59—21no—
207C23×C59—21no—
208C23×C59C4⋊Q812yes—
209C23×C59C22×D411yescharacteristic
210C2×C4×C59—21no—
211C2×C4×C59—21no—
212C2×C4×C59—22no—
213C2×C4×C59—22no—
214C2×C4×C59—22no—
215C2×C4×C59—22no—
216C2×C4×C59—21no—
217C2×C4×C59—21no—
218C2×C4×C59—21no—
219C2×C4×C59—21no—
220C2×C4×C59—21no—
221C2×C4×C59—21no—
222C2×C4×C59—21no—
223C2×C4×C59—21no—
224C2×C4×C59—21no—
225C2×C4×C59—21no—
226C2×C4×C59—21no—
227C2×C4×C59—21no—
228C2×C4×C59C4⋊Q812yes—
229C2×C4×C59C22×D411yescharacteristic
230C2×C4×C59C22×Q811yescharacteristic
231C2×C4×C59C22×Q811yescharacteristic
232C2×C4×C59C2×#4111yescharacteristic
233C2×C4×C59H(5,𝔽2)11yescharacteristic
234C2×C4×C592−1+411yescharacteristic
235C2×Dic59—22no—
236C2×Dic59—22no—
237C2×Dic59—41no—
238C2×Dic59—41no—
239C2×Dic59—41no—
240C2×Dic59—41no—
241C2×Dic59—41no—
242C2×Dic59—41no—
243C2×Dic59—41no—
244C2×Dic59—41no—
245C2×Dic59—41no—
246C2×Dic59—41no—
247C2×Dic59—41no—
248C2×Dic59—41no—
249C2×Dic59—41no—
250C2×Dic59—41no—
251C2×Dic59—41no—
252C2×Dic59—41no—
253C2×Dic59—41no—
254C2×Dic59—41no—
255C2×Dic59—41no—
256C2×Dic59—41no—
257C2×Dic59—41no—
258C2×Dic59—41no—
259C2×Dic59—84no—
260Dic118—84no—
261C8×C59—22no—
262C8×C59—22no—
263C8×C59—22no—
264C8×C59C22⋊Q814yes—
265#26079—591nomaximal, Sylow
266C22×C59—22no—
267C22×C59—22no—
268C22×C59—22no—
269C22×C59—41no—
270C22×C59—41no—
271C22×C59—41no—
272C22×C59—41no—
273C22×C59C2×C4⋊Q811yescharacteristic
274C22×C59D4211yescharacteristic
275C22×C59#54711yescharacteristic
276C22×C59#57511yescharacteristic
277C22×C59#57511yescharacteristic
278C22×C59#57611yescharacteristic
279C22×C59#57811yescharacteristic
280C4×C59—22no—
281C4×C59—41no—
282C4×C59—41no—
283C4×C59—41no—
284C4×C59—41no—
285C4×C59C2×C22⋊Q811yescharacteristic
286C4×C59#54411yescharacteristic
287C4×C59D4×Q811yescharacteristic
288C4×C59#55411yescharacteristic
289Dic59—42no—
290Dic59—42no—
291Dic59—81no—
292Dic59—81no—
293Dic59—81no—
294Dic59—81no—
295Dic59—81no—
296Dic59—81no—
297C2×#501—591no—
298C2×#428—591no—
299#2442—591no—
300#2439—591no—
301#2085—591no—
302#2085—591no—
303#2059—591no—
304#2057—591no—
305#1943—591no—
306#1942—591no—
307#1854—591no—
308#1852—591no—
309#1846—591no—
310#1809—591no—
311#1753D5911yescharacteristic
312C2×C59—42no—
313C2×C59#244011yescharacteristic
314C2×C59#316411yescharacteristic
315C2×C59#316611yescharacteristic
316C2×C59#327311yescharacteristic
317C2×C59#327711yescharacteristic
318C2×C59#334711yescharacteristic
319C2×C59#334911yescharacteristic
320C22×M16C2×D5911yescharacteristic
321C2×C4⋊Q8—591no—
322C2×#125—591no—
323C2×#117—591no—
324C22×C4⋊C4—591no—
325#501—1184no—
326#428—1184no—
327C2×#107—591no—
328C2×#107—591no—
329C2×#107—591no—
330C2×#107—591no—
331C2×#107—591no—
332C2×#107—591no—
333C2×#106—591no—
334C2×#106—591no—
335C2×#106—591no—
336C2×#106—591no—
337C2×C22⋊C8C2×D5911yescharacteristic
338C2×#97C2×D5911yescharacteristic
339C2×C4×C8C2×D5911yescharacteristic
340#400—1181no—
341#398—1181no—
342#396—1181no—
343#393—1181no—
344#391—1181no—
345#389—591no—
346#389—591no—
347#389—591no—
348#389—591no—
349#387—1181no—
350#385—1181no—
351#382—591no—
352#382—591no—
353#380—591no—
354#380—591no—
355#380—1181no—
356C4×C22⋊C4C2×D5911yescharacteristic
357#340—591no—
358#340—591no—
359#340—591no—
360#340—591no—
361#340—591no—
362#340—591no—
363#336C2×D5911yescharacteristic
364#336C2×D5911yescharacteristic
365C59#2607911yescharacteristic, complemented, Sylow
366C22×Q8—1181no—
367C23×C4—1181no—
368C23×C4C22×D5911yescharacteristic
369C2×M16—22no—
370C2×M16—22no—
371C2×M16Dic11814yescomplemented
372C22×C8C22×D5911yescharacteristic
373C22×C8C22×D5911yescharacteristic
374C22×C8C22×D5911yescharacteristic
375C22×C8C22×D5911yescharacteristic
376C4⋊Q8—1182no—
377C4⋊Q8—2361no—
378#125—1182no—
379#125—2361no—
380#117—1182no—
381#117—2361no—
382C2×C4⋊C4—591no—
383C2×C4⋊C4—591no—
384C2×C4⋊C4—591no—
385C2×C4⋊C4—591no—
386C2×C4⋊C4—591no—
387C2×C4⋊C4—591no—
388C2×C4⋊C4—591no—
389C2×C4⋊C4—591no—
390C2×C4⋊C4—1182no—
391C2×C4⋊C4—1181no—
392C2×C4⋊C4—1181no—
393C2×C4⋊C4—1181no—
394C2×C4⋊C4—1181no—
395C2×C4⋊C4—1181no—
396C2×C4⋊C4—1181no—
397C2×C4⋊C4—1181no—
398C2×C4⋊C4—1181no—
399C2×C4⋊C4—2361no—
400C2×C22⋊C4—1181no—
401C2×C22⋊C4—1181no—
402C2×C22⋊C4—1181no—
403C2×C22⋊C4D23611yescharacteristic
404C2×C22⋊C4D23611yescharacteristic
405C2×C42—1181no—
406C2×C42C22×D5911yescharacteristic
407C2×C42C22×D5911yescharacteristic
408#107—594no—
409#107—1182no—
410#107—1182no—
411#107—1182no—
412#107—2361no—
413#107—2361no—
414#106—1182no—
415#106—1182no—
416#106—1182no—
417#106—1182no—
418C22⋊C8Dic11814yescomplemented
419#97—22no—
420C4×C8—22no—
421#95—1181no—
422#95—1181no—
423#95—1181no—
424#95—1181no—
425#95—1181no—
426#95—1181no—
427#95—1181no—
428#95—1181no—
429#95—1181no—
430#95—1181no—
431#95D23611yescharacteristic
432#95D23611yescharacteristic
433C24C2×D23611yescharacteristic
434C2×Q8—4721no—
435C2×Q8—2362no—
436C2×Q8—2362no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—1181no—
442C22×C4—1181no—
443C22×C4—1181no—
444C22×C4—1181no—
445C22×C4—1181no—
446C22×C4—1181no—
447C22×C4—1181no—
448C22×C4—1181no—
449C22×C4—4721no—
450C22×C4—2362no—
451C22×C4—2362no—
452C22×C4C2×Dic11812yes—
453C22×C4C2×Dic11812yes—
454C22×C4C2×D23611yescharacteristic
455C22×C4C2×D23611yescharacteristic
456C22×C4C2×D23611yescharacteristic
457C22×C4C2×D23611yescharacteristic
458C22×C4C2×D23611yescharacteristic
459C22×C4C23×D5911yescharacteristic
460M16—28no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8C2×Dic11812yes—
468C2×C8C2×Dic11812yes—
469C2×C8C2×Dic11812yes—
470C2×C8C2×Dic11812yes—
471C2×C8D4×D5912yes—
472C2×C8#1174568712yes—
473C4⋊C4—592no—
474C4⋊C4—592no—
475C4⋊C4—1181no—
476C4⋊C4—1181no—
477C4⋊C4—1181no—
478C4⋊C4—1181no—
479C4⋊C4—1181no—
480C4⋊C4—1181no—
481C4⋊C4—1181no—
482C4⋊C4—1181no—
483C4⋊C4—1181no—
484C4⋊C4—1181no—
485C4⋊C4—1181no—
486C4⋊C4—1181no—
487C4⋊C4—1181no—
488C4⋊C4—1181no—
489C4⋊C4—4721no—
490C4⋊C4—4721no—
491C4⋊C4—4721no—
492C4⋊C4—2362no—
493C4⋊C4—2362no—
494C4⋊C4—2362no—
495C4⋊C4—2362no—
496C4⋊C4—2362no—
497C22⋊C4—41no—
498C22⋊C4—41no—
499C22⋊C4—4721no—
500C22⋊C4—2362no—
501C22⋊C4—2362no—
502C42—21no—
503C42—21no—
504C42—21no—
505C42—21no—
506C42—2362no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23#1427372712yes—
514C23C22×D23611yescharacteristic
515Q8—4724no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—1182no—
535C2×C4—1182no—
536C2×C4—4724no—
537C2×C4—2361no—
538C2×C4—2361no—
539C2×C4—2361no—
540C2×C4—2361no—
541C2×C4—2361no—
542C2×C4—2361no—
543C2×C4—2361no—
544C2×C4—2361no—
545C2×C4—2361no—
546C2×C4—2361no—
547C2×C4—2361no—
548C2×C4—2361no—
549C2×C4—2361no—
550C2×C4—2361no—
551C2×C4—2361no—
552C2×C4—2361no—
553C2×C4—2361no—
554C2×C4—2361no—
555C2×C4—2361no—
556C2×C4—2361no—
557C2×C4—2361no—
558C2×C4—2361no—
559C2×C4#1427372712yes—
560C2×C4C2×D4×D5911yescharacteristic
561C2×C4C22×Dic11811yescharacteristic
562C2×C4C22×Dic11811yescharacteristic
563C2×C4C2×#1174568711yescharacteristic
564C2×C4#1427377311yescharacteristic
565C2×C4#1427377411yescharacteristic
566C8—22no—
567C8—22no—
568C8—22no—
569C8#1427370414yes—
570C22—22no—
571C22—22no—
572C22—22no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22C2×#1427372711yescharacteristic
578C22D4×D23611yescharacteristic
579C22#1683805611yescharacteristic
580C22#1683820211yescharacteristic
581C22#1683820211yescharacteristic
582C22#1683820711yescharacteristic
583C22#1683821911yescharacteristic
584C4—22no—
585C4—41no—
586C4—41no—
587C4—41no—
588C4—41no—
589C4—4721no—
590C4—4721no—
591C4—4721no—
592C4—4721no—
593C4—4721no—
594C4—4721no—
595C4—2362no—
596C4—2362no—
597C4C2×#1427370411yescharacteristic
598C4#1683803011yescharacteristic
599C4D4×Dic11811yescharacteristic
600C4#1683810911yescharacteristic
601C2—42no—
602C2#2142779811yescharacteristic
603C2#2143463511yescharacteristic
604C2#2143464811yescharacteristic
605C2#2143570811yescharacteristic
606C2#2143575411yescharacteristic
607C2#2143659811yescharacteristic
608C2#2143662811yescharacteristic
subgroupG/H
centerC23C22×D236
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#1753C2
socleC23×C59C22×D4
prime radicalC24×C59C2×D4
hypercenter#1753D59
hypocenterC59#26079
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1753

Complements

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

NH ≅ G/Ntype
1. C2×#16837811C2semidirect
5. #21425579C2semidirect
6. #21425579C2semidirect
7. #21425435C2semidirect
8. #21425425C2semidirect
9. #21424834C2semidirect
10. #21424829C2semidirect
13. #21424260C2semidirect
114. C2×M16×C59Q8semidirect
123. C22⋊C8×C59Q8semidirect
365. C59#26079semidirect
371. C2×M16Dic118semidirect
418. C22⋊C8Dic118semidirect

Maximal subgroups

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

subgroupindex
C2×#168378112
C2×#168374442
#214278202
#214277872
#214255792
#214254352
#214254252
#214248342
#214248292
#214242972
#214242862
#214242602
#214241292
C59×#17532
#2607959