Finite Group Databasebeta

← #28647493

Subgroups — #28647493

Order 15104.

826 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837806C211yesmaximal, characteristic, complemented
2C2×#16837445C211yesmaximal, characteristic
3#21427799C211yesmaximal, characteristic
4#21427781C211yesmaximal, characteristic
5#21425583C211yesmaximal, characteristic, complemented
6#21425570C211yesmaximal, characteristic, complemented
7#21425427C211yesmaximal, characteristic, complemented
8#21425423C211yesmaximal, characteristic, complemented
9#21424833C211yesmaximal, characteristic, complemented
10#21424823C211yesmaximal, characteristic, complemented
11#21424286C211yesmaximal, characteristic
12#21424261C211yesmaximal, characteristic, complemented
13#21424127C211yesmaximal, characteristic
14#21423882C211yesmaximal, characteristic
15C59×#1854C211yesmaximal, characteristic
16C22×#11745664C2211yescharacteristic
17C2×#14273726C2211yescharacteristic
18C2×#14273726C2211yescharacteristic
19C2×#14273704—21no—
20C22×C4⋊Dic59C2211yescharacteristic
21#16837806—24no—
22#16837445—24no—
23C2×#14273661C2211yescharacteristic
24C2×#14273661C2211yescharacteristic
25C2×#14273660C2211yescharacteristic
26C2×#14273660C2211yescharacteristic
27C2×#14273660C2211yescharacteristic
28C2×#14273658C2211yescharacteristic
29C2×#14273658C2211yescharacteristic
30C2×#14273658C2211yescharacteristic
31C2×#14273637C2211yescharacteristic
32C2×C8×Dic59C2211yescharacteristic
33#16837336—21no—
34#16837325—21no—
35#16837323—21no—
36#16837314—21no—
37#16837308—21no—
38#16837301C2211yescharacteristic
39#16837301C2211yescharacteristic
40#16837298C2211yescharacteristic
41#16837298C2211yescharacteristic
42#16837286—21no—
43#16837274—21no—
44#16837271C2211yescharacteristic
45#16837267C2211yescharacteristic
46#16837262C2211yescharacteristic
47C4×C22⋊Dic59C2211yescharacteristic
48#16837165C2211yescharacteristic
49#16837165C2211yescharacteristic
50#16837164C2211yescharacteristic
51#16837164C2211yescharacteristic
52#16837156C2211yescharacteristic
53#16837156C2211yescharacteristic
54C2×#117×C59C2211yescharacteristic
55C2×#107×C59C2211yescharacteristic
56C2×#106×C59C2211yescharacteristic
57C2×C22⋊C8×C59C2211yescharacteristic
58C59×#380C2211yescharacteristic
59C59×#340C2211yescharacteristic
60C59×#340C2211yescharacteristic
61C22×Dic118—21no—
62C22×Dic118—21no—
63C23×Dic59—21no—
64C2×#11745664—22no—
65C2×#11745664—22no—
66C2×#11745664Q814yescomplemented
67C22×C59⋊C8C2311yescharacteristic
68C22×C59⋊C8C2311yescharacteristic
69#14273726—22no—
70#14273726—22no—
71#14273726—41no—
72#14273726—41no—
73#14273704—44no—
74C2×C4⋊Dic59—22no—
75C2×C4⋊Dic59—21no—
76C2×C4⋊Dic59—21no—
77C2×C4⋊Dic59—21no—
78C2×C4⋊Dic59—41no—
79C2×C4⋊Dic59C2311yescharacteristic
80C2×C4⋊Dic59C2311yescharacteristic
81C2×C4⋊Dic59C2311yescharacteristic
82C2×C4⋊Dic59C2311yescharacteristic
83C2×Dic59⋊C4—21no—
84C2×Dic59⋊C4—21no—
85C2×Dic59⋊C4—21no—
86C2×Dic59⋊C4—21no—
87C2×Dic59⋊C4—21no—
88C2×Dic59⋊C4—21no—
89C2×C22⋊Dic59—21no—
90C2×C22⋊Dic59D411yescharacteristic
91C2×C22⋊Dic59D411yescharacteristic
92C2×C4×Dic59C2311yescharacteristic
93C2×C4×Dic59C2311yescharacteristic
94#14273661—22no—
95#14273661—41no—
96#14273660—22no—
97#14273660—22no—
98#14273660D414yes—
99#14273658—22no—
100#14273658—22no—
101#14273658—22no—
102#14273637—22no—
103C8×Dic59—22no—
104C59⋊#95—21no—
105C59⋊#95—21no—
106C59⋊#95—21no—
107C59⋊#95—21no—
108C59⋊#95—21no—
109C59⋊#95—21no—
110C59⋊#95—21no—
111C59⋊#95D411yescharacteristic
112C59⋊#95D411yescharacteristic
113C23×C4×C59C2311yescharacteristic
114C22×C8×C59C2311yescharacteristic
115C22×C8×C59C2311yescharacteristic
116#117×C59—22no—
117#117×C59—41no—
118C2×C4⋊C4×C59C2311yescharacteristic
119C2×C4⋊C4×C59C2311yescharacteristic
120C2×C4⋊C4×C59C2311yescharacteristic
121C2×C4⋊C4×C59C2311yescharacteristic
122C2×C22⋊C4×C59—21no—
123C2×C22⋊C4×C59—21no—
124C2×C42×C59—21no—
125#107×C59—22no—
126#106×C59—41no—
127C22⋊C8×C59Q814yescomplemented
128#95×C59—21no—
129C24×C59C2×D411yescharacteristic
130C22×C4×C59—21no—
131C22×C4×C59—21no—
132C22×C4×C59—21no—
133C22×C4×C59—21no—
134C22×C4×C59—21no—
135C22×C4×C59—21no—
136C22×C4×C59—21no—
137C22×C4×C59—21no—
138C22×C4×C59C2×D411yescharacteristic
139C22×C4×C59C2×Q812yes—
140C22×C4×C59C2×Q812yes—
141C22×C4×C59C2411yescharacteristic
142C22×Dic59—21no—
143C22×Dic59—21no—
144C22×Dic59—21no—
145C22×Dic59—21no—
146C22×Dic59—42no—
147C22×Dic59—42no—
148C22×Dic59—81no—
149C22×Dic59C2×D411yescharacteristic
150C22×Dic59C2×D411yescharacteristic
151C22×Dic59C2×D411yescharacteristic
152C22×Dic59C2×D411yescharacteristic
153C2×Dic118—42no—
154C2×Dic118—42no—
155C2×Dic118—42no—
156C2×Dic118—42no—
157C2×Dic118—81no—
158C2×Dic118—81no—
159C2×C8×C59—21no—
160C2×C8×C59—21no—
161C2×C8×C59—21no—
162C2×C8×C59—21no—
163C2×C8×C59C2×Q812yes—
164C2×C8×C59C2×Q812yes—
165C4⋊C4×C59—21no—
166C4⋊C4×C59—21no—
167C4⋊C4×C59—21no—
168C4⋊C4×C59—21no—
169C4⋊C4×C59—21no—
170C4⋊C4×C59—21no—
171C4⋊C4×C59—21no—
172C4⋊C4×C59C2×D412yes—
173C22⋊C4×C59—42no—
174C22⋊C4×C59—42no—
175C42×C59—42no—
176C22⋊Dic59—22no—
177C22⋊Dic59—22no—
178C22⋊Dic59—81no—
179C4⋊Dic59—21no—
180C4⋊Dic59—21no—
181C4⋊Dic59—21no—
182C4⋊Dic59—21no—
183C4⋊Dic59—21no—
184C4⋊Dic59—21no—
185C4⋊Dic59—21no—
186C4⋊Dic59—42no—
187C4⋊Dic59—42no—
188C4⋊Dic59—81no—
189C4⋊Dic59C2×D412yes—
190Dic59⋊C4—42no—
191Dic59⋊C4—42no—
192Dic59⋊C4—42no—
193Dic59⋊C4—42no—
194Dic59⋊C4—42no—
195Dic59⋊C4—81no—
196C4×Dic59—21no—
197C4×Dic59—21no—
198C4×Dic59—21no—
199C4×Dic59—21no—
200#11745664—28no—
201C2×C59⋊C8—21no—
202C2×C59⋊C8—21no—
203C2×C59⋊C8C2×D412yes—
204C2×C59⋊C8C2×Q812yes—
205C2×C59⋊C8C2×Q812yes—
206C2×C59⋊C8#4112yes—
207C23×C59—21no—
208C23×C59—21no—
209C23×C59—21no—
210C23×C59—21no—
211C23×C59—21no—
212C23×C59—21no—
213C23×C59C4⋊Q812yes—
214C23×C59C22×D411yescharacteristic
215C2×C4×C59—21no—
216C2×C4×C59—21no—
217C2×C4×C59—22no—
218C2×C4×C59—22no—
219C2×C4×C59—22no—
220C2×C4×C59—22no—
221C2×C4×C59—22no—
222C2×C4×C59—41no—
223C2×C4×C59—41no—
224C2×C4×C59—41no—
225C2×C4×C59—41no—
226C2×C4×C59—41no—
227C2×C4×C59—41no—
228C2×C4×C59—41no—
229C2×C4×C59—41no—
230C2×C4×C59—41no—
231C2×C4×C59—41no—
232C2×C4×C59—41no—
233C2×C4×C59C4⋊Q812yes—
234C2×C4×C59C22×D411yescharacteristic
235C2×C4×C59C22×Q811yescharacteristic
236C2×C4×C59C22×Q811yescharacteristic
237C2×C4×C59C2×#4111yescharacteristic
238C2×C4×C59H(5,𝔽2)11yescharacteristic
239C2×C4×C592−1+411yescharacteristic
240C2×Dic59—21no—
241C2×Dic59—21no—
242C2×Dic59—21no—
243C2×Dic59—21no—
244C2×Dic59—22no—
245C2×Dic59—21no—
246C2×Dic59—21no—
247C2×Dic59—21no—
248C2×Dic59—21no—
249C2×Dic59—21no—
250C2×Dic59—21no—
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—41no—
260C2×Dic59—41no—
261C2×Dic59—41no—
262C2×Dic59—84no—
263C2×Dic59AGL(1,ℤ/8)12yes—
264C2×Dic59#13712yes—
265Dic118—84no—
266Dic118—84no—
267C8×C59—22no—
268C8×C59—22no—
269C59⋊C8—22no—
270C59⋊C8C22⋊Q814yescomplemented
271#26074—591nomaximal, Sylow
272C22×C59—22no—
273C22×C59—22no—
274C22×C59—22no—
275C22×C59—41no—
276C22×C59—41no—
277C22×C59—41no—
278C22×C59—41no—
279C22×C59C2×C4⋊Q811yescharacteristic
280C22×C59D4211yescharacteristic
281C22×C59#54711yescharacteristic
282C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
283C22×C59C2×#13711yescharacteristic
284C22×C59#57611yescharacteristic
285C22×C59#57811yescharacteristic
286C4×C59—22no—
287C4×C59—42no—
288C4×C59—81no—
289C4×C59—81no—
290C4×C59—81no—
291C4×C59C2×C22⋊Q811yescharacteristic
292C4×C59#54411yescharacteristic
293C4×C59D4×Q811yescharacteristic
294C4×C59#55411yescharacteristic
295Dic59—22no—
296Dic59—22no—
297Dic59—41no—
298Dic59—41no—
299Dic59—42no—
300Dic59—81no—
301Dic59—81no—
302Dic59—81no—
303C2×#500—591no—
304C2×#428—591no—
305#2440—591no—
306#2439—591no—
307#2085—591no—
308#2083—591no—
309#2058—591no—
310#2057—591no—
311#1943—591no—
312#1941—591no—
313#1854D5911yescharacteristic
314#1852—591no—
315#1847—591no—
316#1809—591no—
317#1753—591no—
318C2×C59—42no—
319C2×C59#244011yescharacteristic
320C2×C59#316411yescharacteristic
321C2×C59#316511yescharacteristic
322C2×C59#326411yescharacteristic
323C2×C59#327011yescharacteristic
324C2×C59#335311yescharacteristic
325C2×C59#335711yescharacteristic
326C22×M16—591no—
327C2×C4⋊Q8—591no—
328C2×C4⋊Q8—591no—
329C2×C22⋊Q8—1181no—
330C2×#117C2×D5911yescharacteristic
331C22×C4⋊C4—591no—
332#500—1184no—
333#428—1184no—
334C2×#107—591no—
335C2×#107—591no—
336C2×#107—591no—
337C2×#107—591no—
338C2×#107—591no—
339C2×#107C2×D5911yescharacteristic
340C2×#106—591no—
341C2×#106—591no—
342C2×#106—591no—
343C2×#106C2×D5911yescharacteristic
344C2×C22⋊C8C2×D5911yescharacteristic
345C2×#97—591no—
346C2×C4×C8—591no—
347#398—1181no—
348#396—1181no—
349#395—1181no—
350#393—1181no—
351#391—1181no—
352#389—591no—
353#389—591no—
354#389—591no—
355#389—591no—
356#387—1181no—
357#385—1181no—
358#384—591no—
359#382—591no—
360#380—591no—
361#380C2×D5911yescharacteristic
362C4×C22⋊C4—591no—
363#340—591no—
364#340—591no—
365#340—591no—
366#340—591no—
367#340C2×D5911yescharacteristic
368#340C2×D5911yescharacteristic
369#336—591no—
370#336—591no—
371C59#2607411yescharacteristic, complemented, Sylow
372C22×Q8—1181no—
373C22×Q8—1181no—
374C23×C4—1181no—
375C23×C4C22×D5911yescharacteristic
376C2×M16—594no—
377C2×M16—1182no—
378C2×M16—1182no—
379C22×C8—591no—
380C22×C8—591no—
381C22×C8C22×D5911yescharacteristic
382C22×C8C22×D5911yescharacteristic
383C4⋊Q8—1182no—
384C4⋊Q8—1182no—
385C4⋊Q8—2361no—
386C4⋊Q8—2361no—
387C22⋊Q8—2364no—
388#117—22no—
389#117—41no—
390C2×C4⋊C4—591no—
391C2×C4⋊C4—591no—
392C2×C4⋊C4—591no—
393C2×C4⋊C4—591no—
394C2×C4⋊C4—1182no—
395C2×C4⋊C4—1181no—
396C2×C4⋊C4—1181no—
397C2×C4⋊C4—1181no—
398C2×C4⋊C4—1181no—
399C2×C4⋊C4—1181no—
400C2×C4⋊C4—1181no—
401C2×C4⋊C4—1181no—
402C2×C4⋊C4—1181no—
403C2×C4⋊C4—1181no—
404C2×C4⋊C4—2361no—
405C2×C4⋊C4C22×D5911yescharacteristic
406C2×C4⋊C4C22×D5911yescharacteristic
407C2×C4⋊C4C22×D5911yescharacteristic
408C2×C4⋊C4C22×D5911yescharacteristic
409C2×C22⋊C4—21no—
410C2×C22⋊C4—21no—
411C2×C22⋊C4—591no—
412C2×C22⋊C4—591no—
413C2×C22⋊C4—1181no—
414C2×C42—21no—
415C2×C42—591no—
416C2×C42—591no—
417#107—22no—
418#107—594no—
419#107—1182no—
420#107—1182no—
421#107—1182no—
422#107—2361no—
423#106—41no—
424#106—1182no—
425#106—1182no—
426#106—1182no—
427C22⋊C8Dic11814yescomplemented
428#97—1182no—
429C4×C8—1182no—
430#95—21no—
431#95—591no—
432#95—591no—
433#95—1181no—
434#95—1181no—
435#95—1181no—
436#95—1181no—
437#95—1181no—
438#95—1181no—
439#95—1181no—
440C24D4×D5911yescharacteristic
441C2×Q8—4721no—
442C2×Q8—4721no—
443C2×Q8—2362no—
444C2×Q8—2362no—
445C2×Q8—2362no—
446C2×Q8—2362no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—591no—
456C22×C4—591no—
457C22×C4—591no—
458C22×C4—591no—
459C22×C4—1181no—
460C22×C4—1181no—
461C22×C4—1181no—
462C22×C4—1181no—
463C22×C4—4721no—
464C22×C4—2362no—
465C22×C4—2362no—
466C22×C4C2×Dic11812yes—
467C22×C4D4×D5911yescharacteristic
468C22×C4Q8×D5912yes—
469C22×C4C23×D5911yescharacteristic
470M16—1188no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—592no—
476C2×C8—592no—
477C2×C8—592no—
478C2×C8—592no—
479C2×C8—1181no—
480C2×C8—1181no—
481C2×C8C2×Dic11812yes—
482C2×C8C2×Dic11812yes—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—592no—
491C4⋊C4—1181no—
492C4⋊C4—1181no—
493C4⋊C4—1181no—
494C4⋊C4—1181no—
495C4⋊C4—1181no—
496C4⋊C4—1181no—
497C4⋊C4—1181no—
498C4⋊C4—4721no—
499C4⋊C4—4721no—
500C4⋊C4—2362no—
501C4⋊C4—2362no—
502C4⋊C4—2362no—
503C4⋊C4—2362no—
504C4⋊C4—2362no—
505C4⋊C4—2362no—
506C4⋊C4—2362no—
507C4⋊C4D4×D5912yes—
508C22⋊C4—42no—
509C22⋊C4—42no—
510C22⋊C4—1182no—
511C22⋊C4—1182no—
512C22⋊C4—4721no—
513C42—42no—
514C42—1181no—
515C42—1181no—
516C42—1181no—
517C42—1181no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23#1427372612yes—
525C23C2×D4×D5911yescharacteristic
526Q8—4724no—
527Q8—4724no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—592no—
547C2×C4—592no—
548C2×C4—1181no—
549C2×C4—1181no—
550C2×C4—1181no—
551C2×C4—1181no—
552C2×C4—1181no—
553C2×C4—1181no—
554C2×C4—1181no—
555C2×C4—1181no—
556C2×C4—1182no—
557C2×C4—1181no—
558C2×C4—1181no—
559C2×C4—4724no—
560C2×C4—2361no—
561C2×C4—2361no—
562C2×C4—2361no—
563C2×C4—2361no—
564C2×C4—2361no—
565C2×C4—2361no—
566C2×C4—2361no—
567C2×C4—2361no—
568C2×C4—2361no—
569C2×C4—2361no—
570C2×C4—2361no—
571C2×C4#1427372612yes—
572C2×C4C2×D4×D5911yescharacteristic
573C2×C4C2×Q8×D5911yescharacteristic
574C2×C4C22×Dic11811yescharacteristic
575C2×C4#41×D5911yescharacteristic
576C2×C4#1427377211yescharacteristic
577C2×C4#1427377411yescharacteristic
578C8—22no—
579C8—22no—
580C8—594no—
581C8—1182no—
582C22—22no—
583C22—22no—
584C22—22no—
585C22—41no—
586C22—41no—
587C22—41no—
588C22—41no—
589C22C2×#1427372611yescharacteristic
590C22#1683803411yescharacteristic
591C22#1683805011yescharacteristic
592C22AGL(1,ℤ/8)×D5911yescharacteristic
593C22#137×D5911yescharacteristic
594C22#1683820411yescharacteristic
595C22#1683821611yescharacteristic
596C4—22no—
597C4—42no—
598C4—81no—
599C4—81no—
600C4—81no—
601C4—1182no—
602C4—1182no—
603C4—4721no—
604C4—4721no—
605C4—4721no—
606C4—2361no—
607C4—2361no—
608C4—2362no—
609C4C22⋊Q8×D5911yescharacteristic
610C4#1683803111yescharacteristic
611C4D4×Dic11811yescharacteristic
612C4#1683811111yescharacteristic
613C2—42no—
614C2#2142779711yescharacteristic
615C2#2143463611yescharacteristic
616C2#2143464011yescharacteristic
617C2#2143557911yescharacteristic
618C2#2143566411yescharacteristic
619C2#2143667811yescharacteristic
620C2#2143672011yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#1854C2
socleC23×C59C22×D4
prime radicalC24×C59C2×D4
hypercenter#1854D59
hypocenterC59#26074
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1854

Complements

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

NH ≅ G/Ntype
1. C2×#16837806C2semidirect
5. #21425583C2semidirect
6. #21425570C2semidirect
7. #21425427C2semidirect
8. #21425423C2semidirect
9. #21424833C2semidirect
10. #21424823C2semidirect
12. #21424261C2semidirect
66. C2×#11745664Q8semidirect
127. C22⋊C8×C59Q8semidirect
270. C59⋊C8C22⋊Q8semidirect
371. C59#26074semidirect
427. C22⋊C8Dic118semidirect

Maximal subgroups

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

subgroupindex
C2×#168378062
C2×#168374452
#214277992
#214277812
#214255832
#214255702
#214254272
#214254232
#214248332
#214248232
#214242862
#214242612
#214241272
#214238822
C59×#18542
#2607459