Finite Group Databasebeta

← #28598680

Subgroups — #28598680

Order 15104.

1014 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16838012C211yesmaximal, characteristic
2C2×#16837428C211yesmaximal, characteristic
3#21426703C211yesmaximal, characteristic
4#21425052C211yesmaximal, characteristic, complemented
5#21425050C211yesmaximal, characteristic, complemented
6#21425017C211yesmaximal, characteristic, complemented
7#21425017C211yesmaximal, characteristic, complemented
8#21424530C211yesmaximal, characteristic
9#21424527C211yesmaximal, characteristic
10#21424519C211yesmaximal, characteristic
11#21424110C211yesmaximal, characteristic, complemented
12#21424109C211yesmaximal, characteristic, complemented
13#21424100C211yesmaximal, characteristic, complemented
14#21424099C211yesmaximal, characteristic, complemented
15C59×#1892C211yesmaximal, characteristic
16C22×#11745664C2211yescharacteristic
17#16838012—24no—
18#16838012C418yescomplemented
19C2×#14273725C2211yescharacteristic
20C2×#14273720—21no—
21C2×#14273720—21no—
22C2×#14273718—21no—
23C2×#14273718C2211yescharacteristic
24C2×#14273717—21no—
25C2×#14273706C2211yescharacteristic
26C2×Q8×Dic59C2211yescharacteristic
27C2×#14273692C2211yescharacteristic
28C2×#14273686C2211yescharacteristic
29#16837428—24no—
30C2×Dic59⋊C8C2211yescharacteristic
31C2×Dic59⋊C8C2211yescharacteristic
32C2×Dic59⋊C8C2211yescharacteristic
33C2×Dic59⋊C8C2211yescharacteristic
34C2×#14273652C2211yescharacteristic
35C2×#14273652C2211yescharacteristic
36C2×#14273651C2211yescharacteristic
37C2×#14273651C2211yescharacteristic
38C2×#14273650C2211yescharacteristic
39C2×#14273650C2211yescharacteristic
40#16837296—21no—
41#16837290—21no—
42#16837287—21no—
43#16837285—21no—
44#16837271C2211yescharacteristic
45#16837267C2211yescharacteristic
46#16837262C2211yescharacteristic
47C4⋊C4×Dic59C2211yescharacteristic
48C4⋊C4×Dic59C2211yescharacteristic
49#16837165C2211yescharacteristic
50#16837165C2211yescharacteristic
51#16837164C2211yescharacteristic
52#16837164C2211yescharacteristic
53#16837164C2211yescharacteristic
54#16837164C2211yescharacteristic
55C23×C8×C59C2211yescharacteristic
56C2×C22⋊Q8×C59C2211yescharacteristic
57C2×#103×C59C2211yescharacteristic
58C2×#103×C59C2211yescharacteristic
59C59×#380C2211yescharacteristic
60C59×#340C2211yescharacteristic
61C59×#340C2211yescharacteristic
62C22×Dic118C2311yescharacteristic
63C2×#11745664—22no—
64C2×#11745664—41no—
65C2×#11745664—41no—
66C22×C59⋊C8C2311yescharacteristic
67C22×C59⋊C8C2311yescharacteristic
68#14273725—22no—
69#14273725C2×C414yes—
70#14273720—28no—
71#14273720—28no—
72#14273718—28no—
73#14273718—22no—
74#14273718C2×C414yes—
75#14273717—28no—
76#14273706—22no—
77#14273706C2×C414yes—
78Q8×Dic59—22no—
79Q8×Dic59C2×C414yes—
80#14273692—22no—
81#14273692C2×C414yes—
82#14273686C2×C414yes—
83#14273686D414yes—
84C2×C4⋊Dic59—21no—
85C2×C4⋊Dic59—21no—
86C2×C4⋊Dic59D411yescharacteristic
87C2×C4⋊Dic59D411yescharacteristic
88C2×C4⋊Dic59C2311yescharacteristic
89C2×C4⋊Dic59C2311yescharacteristic
90C2×C4⋊Dic59C2311yescharacteristic
91C2×Dic59⋊C4—21no—
92C2×Dic59⋊C4—21no—
93C2×Dic59⋊C4—21no—
94C2×Dic59⋊C4—21no—
95C2×Dic59⋊C4—21no—
96C2×Dic59⋊C4—21no—
97C2×Dic59⋊C4—21no—
98C2×C22⋊Dic59—21no—
99C2×C22⋊Dic59—21no—
100C2×C22⋊Dic59D411yescharacteristic
101C2×C22⋊Dic59D411yescharacteristic
102C2×C4×Dic59—21no—
103C2×C4×Dic59—21no—
104C2×C4×Dic59—21no—
105C2×C4×Dic59—21no—
106C2×C4×Dic59C2311yescharacteristic
107C2×C4×Dic59C2311yescharacteristic
108Dic59⋊C8—22no—
109Dic59⋊C8—22no—
110Dic59⋊C8—22no—
111Dic59⋊C8—22no—
112#14273652—41no—
113#14273652—41no—
114#14273651—41no—
115#14273651—41no—
116#14273650—41no—
117#14273650—41no—
118C59⋊#95—21no—
119C59⋊#95—21no—
120C59⋊#95—21no—
121C59⋊#95—21no—
122C59⋊#95—21no—
123C22×Q8×C59C2311yescharacteristic
124C23×C4×C59C2311yescharacteristic
125C22×C8×C59—22no—
126C22×C8×C59—41no—
127C22×C8×C59—41no—
128C22×C8×C59C2311yescharacteristic
129C22×C8×C59C2311yescharacteristic
130C22⋊Q8×C59—22no—
131C22⋊Q8×C59C2×C414yes—
132C2×C4⋊C4×C59—21no—
133C2×C4⋊C4×C59C2311yescharacteristic
134C2×C4⋊C4×C59C2311yescharacteristic
135C2×C4⋊C4×C59C2311yescharacteristic
136C2×C22⋊C4×C59—21no—
137C2×C22⋊C4×C59—21no—
138#103×C59—41no—
139#103×C59—41no—
140#95×C59—21no—
141C24×C59C2×D411yescharacteristic
142C2×Q8×C59—21no—
143C2×Q8×C59—21no—
144C2×Q8×C59—21no—
145C2×Q8×C59—21no—
146C2×Q8×C59—21no—
147C2×Q8×C59C22×C412yes—
148C22×C4×C59—21no—
149C22×C4×C59—21no—
150C22×C4×C59—21no—
151C22×C4×C59—21no—
152C22×C4×C59—21no—
153C22×C4×C59—21no—
154C22×C4×C59—21no—
155C22×C4×C59—21no—
156C22×C4×C59C22×C412yes—
157C22×C4×C59C2×D411yescharacteristic
158C22×C4×C59C2×D412yes—
159C22×C4×C59C2411yescharacteristic
160C22×Dic59—21no—
161C22×Dic59—21no—
162C22×Dic59—21no—
163C22×Dic59—21no—
164C22×Dic59C2×D411yescharacteristic
165C22×Dic59C2×D411yescharacteristic
166C22×Dic59C2×D411yescharacteristic
167C22×Dic59C2×D411yescharacteristic
168C2×Dic118—21no—
169C2×Dic118—21no—
170C2×Dic118—21no—
171C2×Dic118—21no—
172C2×Dic118—21no—
173C2×Dic118C22×C412yes—
174C2×C8×C59—21no—
175C2×C8×C59—21no—
176C2×C8×C59—21no—
177C2×C8×C59—21no—
178C2×C8×C59—21no—
179C2×C8×C59—21no—
180C2×C8×C59—44no—
181C4⋊C4×C59—24no—
182C4⋊C4×C59—21no—
183C4⋊C4×C59—21no—
184C4⋊C4×C59—21no—
185C4⋊C4×C59—21no—
186C4⋊C4×C59—21no—
187C4⋊C4×C59C22×C412yes—
188C22⋊C4×C59—24no—
189C22⋊C4×C59—81no—
190C22⋊Dic59—24no—
191C22⋊Dic59—81no—
192C22⋊Dic59C22⋊C414yes—
193C22⋊Dic59C22⋊C414yes—
194C4⋊Dic59—21no—
195C4⋊Dic59—24no—
196C4⋊Dic59—24no—
197C4⋊Dic59—21no—
198C4⋊Dic59—21no—
199C4⋊Dic59—21no—
200C4⋊Dic59—21no—
201C4⋊Dic59C22⋊C414yes—
202C4⋊Dic59C22⋊C414yes—
203C4⋊Dic59C22×C412yes—
204Dic59⋊C4—24no—
205Dic59⋊C4—24no—
206Dic59⋊C4—24no—
207Dic59⋊C4—24no—
208Dic59⋊C4—24no—
209Dic59⋊C4—81no—
210Dic59⋊C4—81no—
211C4×Dic59—24no—
212C4×Dic59—24no—
213C4×Dic59—81no—
214C4×Dic59—81no—
215C4×Dic59C22×C412yes—
216C4×Dic59C22×C412yes—
217C4×Dic59C2×D412yes—
218C4×Dic59C2×D412yes—
219#11745664—44no—
220C2×C59⋊C8—21no—
221C2×C59⋊C8—21no—
222C2×C59⋊C8—21no—
223C2×C59⋊C8—21no—
224C2×C59⋊C8—21no—
225C2×C59⋊C8—21no—
226C23×C59—21no—
227C23×C59—21no—
228C23×C59—21no—
229C23×C59—21no—
230C23×C59—21no—
231C23×C59—21no—
232C23×C59C2×C22⋊C412yes—
233C23×C59C22×D411yescharacteristic
234Q8×C59—22no—
235Q8×C59—22no—
236Q8×C59—41no—
237Q8×C59—41no—
238C2×C4×C59—21no—
239C2×C4×C59—21no—
240C2×C4×C59—22no—
241C2×C4×C59—22no—
242C2×C4×C59—22no—
243C2×C4×C59—22no—
244C2×C4×C59—22no—
245C2×C4×C59—22no—
246C2×C4×C59—41no—
247C2×C4×C59—41no—
248C2×C4×C59—41no—
249C2×C4×C59—41no—
250C2×C4×C59—41no—
251C2×C4×C59—41no—
252C2×C4×C59—41no—
253C2×C4×C59—41no—
254C2×C4×C59—41no—
255C2×C4×C59—41no—
256C2×C4×C59C2×C22⋊C412yes—
257C2×C4×C59C23×C411yescharacteristic
258C2×C4×C59C22×D411yescharacteristic
259C2×C4×C59H(5,𝔽2)11yescharacteristic
260C2×C4×C59H(5,𝔽2)11yescharacteristic
261C2×C4×C592−1+411yescharacteristic
262C2×C4×C592−1+411yescharacteristic
263C2×Dic59—21no—
264C2×Dic59—21no—
265C2×Dic59—21no—
266C2×Dic59—21no—
267C2×Dic59—21no—
268C2×Dic59—21no—
269C2×Dic59—21no—
270C2×Dic59—21no—
271C2×Dic59—22no—
272C2×Dic59—22no—
273C2×Dic59—41no—
274C2×Dic59—41no—
275C2×Dic59—41no—
276C2×Dic59—41no—
277C2×Dic59—41no—
278C2×Dic59—41no—
279C2×Dic59—41no—
280C2×Dic59—41no—
281C2×Dic59—41no—
282C2×Dic59—41no—
283C2×Dic59C2×C22⋊C412yes—
284C2×Dic59C2×C22⋊C412yes—
285C2×Dic59C2×C22⋊C412yes—
286C2×Dic59C2×C22⋊C412yes—
287Dic118—22no—
288Dic118—22no—
289Dic118—41no—
290Dic118—41no—
291C8×C59—41no—
292C8×C59—41no—
293C59⋊C8—41no—
294C59⋊C8—41no—
295#21898—591nomaximal, Sylow
296C22×C59—22no—
297C22×C59—22no—
298C22×C59—22no—
299C22×C59—41no—
300C22×C59—41no—
301C22×C59—41no—
302C22×C59—41no—
303C22×C59C22×C22⋊C411yescharacteristic
304C22×C59#53711yescharacteristic
305C22×C59#53711yescharacteristic
306C22×C59#57711yescharacteristic
307C22×C59#57711yescharacteristic
308C22×C59#57811yescharacteristic
309C22×C59#57811yescharacteristic
310C4×C59—22no—
311C4×C59—42no—
312C4×C59—42no—
313C4×C59—81no—
314C4×C59—81no—
315C4×C59#52011yescharacteristic
316C4×C59#52011yescharacteristic
317C4×C59#53511yescharacteristic
318C4×C59#53611yescharacteristic
319Dic59—22no—
320Dic59—22no—
321Dic59—22no—
322Dic59—22no—
323Dic59—42no—
324Dic59—42no—
325Dic59—81no—
326Dic59—81no—
327C2×#541—591no—
328C2×#424—591no—
329#2314—591no—
330#1989—591no—
331#1989—591no—
332#1981—591no—
333#1981—591no—
334#1894—591no—
335#1894—591no—
336#1892—591no—
337#1892D5911yescharacteristic
338#1805—591no—
339#1805—591no—
340#1803—591no—
341#1803—591no—
342C2×C59—42no—
343C2×C59#231111yescharacteristic
344C2×C59#289811yescharacteristic
345C2×C59#289811yescharacteristic
346C2×C59#325811yescharacteristic
347C2×C59#325811yescharacteristic
348C2×C59#326011yescharacteristic
349C2×C59#326111yescharacteristic
350C22×M16—591no—
351C23×C8C2×D5911yescharacteristic
352#541—598no—
353#541—1184no—
354C2×C4⋊Q8—591no—
355C2×#126—1181no—
356C2×#126—1181no—
357C2×#125—591no—
358C2×#125—1181no—
359C2×#125—1181no—
360C2×C22⋊Q8—591no—
361C2×C22⋊Q8C2×D5911yescharacteristic
362C2×C4×Q8—591no—
363C2×C4×Q8—591no—
364C2×#117—591no—
365#424—1184no—
366C2×C4⋊C8—591no—
367C2×C4⋊C8—591no—
368C2×C4⋊C8—591no—
369C2×C4⋊C8—591no—
370C2×#103—591no—
371C2×#103—591no—
372C2×#103—591no—
373C2×#103—591no—
374C2×#103—591no—
375C2×#103—591no—
376C2×#103C2×D5911yescharacteristic
377C2×#103C2×D5911yescharacteristic
378#388—1181no—
379#388—1181no—
380#387—1181no—
381#387—1181no—
382#384—591no—
383#382—591no—
384#380—591no—
385#380C2×D5911yescharacteristic
386C4×C4⋊C4—591no—
387C4×C4⋊C4—591no—
388#340—591no—
389#340—591no—
390#340—591no—
391#340—591no—
392#340—591no—
393#340—591no—
394#340C2×D5911yescharacteristic
395#340C2×D5911yescharacteristic
396C59#2189811yescharacteristic, complemented, Sylow
397C22×Q8—591no—
398C22×Q8C22×D5911yescharacteristic
399C23×C4C22×D5911yescharacteristic
400C2×M16—1182no—
401C2×M16—2361no—
402C2×M16—2361no—
403C22×C8—22no—
404C22×C8—41no—
405C22×C8—41no—
406C22×C8—591no—
407C22×C8—591no—
408C22×C8C22×D5911yescharacteristic
409C22×C8C22×D5911yescharacteristic
410C4⋊Q8—594no—
411C4⋊Q8—1182no—
412#126—1188no—
413#126—1188no—
414#125—594no—
415#125—1182no—
416#125—1188no—
417#125—1188no—
418C22⋊Q8—22no—
419C22⋊Q8—594no—
420C22⋊Q8—1182no—
421C22⋊Q8C4×D5914yes—
422C4×Q8—594no—
423C4×Q8—594no—
424C4×Q8—1182no—
425C4×Q8—1182no—
426#117—594no—
427#117—594no—
428C2×C4⋊C4—21no—
429C2×C4⋊C4—591no—
430C2×C4⋊C4—591no—
431C2×C4⋊C4—591no—
432C2×C4⋊C4—591no—
433C2×C4⋊C4—591no—
434C2×C4⋊C4—1181no—
435C2×C4⋊C4—1181no—
436C2×C4⋊C4—1181no—
437C2×C4⋊C4—1181no—
438C2×C4⋊C4—1181no—
439C2×C4⋊C4—1181no—
440C2×C4⋊C4—1181no—
441C2×C4⋊C4—1181no—
442C2×C4⋊C4—1181no—
443C2×C4⋊C4C22×D5911yescharacteristic
444C2×C4⋊C4C22×D5911yescharacteristic
445C2×C4⋊C4C22×D5911yescharacteristic
446C2×C22⋊C4—21no—
447C2×C22⋊C4—21no—
448C2×C22⋊C4—591no—
449C2×C22⋊C4—591no—
450C2×C22⋊C4—1181no—
451C2×C22⋊C4—1181no—
452C2×C42—591no—
453C2×C42—591no—
454C2×C42—1181no—
455C2×C42—1181no—
456C2×C42—1181no—
457C2×C42—1181no—
458C4⋊C8—1182no—
459C4⋊C8—1182no—
460C4⋊C8—1182no—
461C4⋊C8—1182no—
462#103—41no—
463#103—41no—
464#103—2361no—
465#103—2361no—
466#103—2361no—
467#103—2361no—
468#103—2361no—
469#103—2361no—
470#95—21no—
471#95—1181no—
472#95—1181no—
473#95—1181no—
474#95—1181no—
475#95—1181no—
476C24D4×D5911yescharacteristic
477C2×Q8—21no—
478C2×Q8—21no—
479C2×Q8—21no—
480C2×Q8—21no—
481C2×Q8—21no—
482C2×Q8—592no—
483C2×Q8—1181no—
484C2×Q8—1181no—
485C2×Q8—1181no—
486C2×Q8—1181no—
487C2×Q8—1181no—
488C2×Q8C2×C4×D5912yes—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—591no—
498C22×C4—591no—
499C22×C4—591no—
500C22×C4—591no—
501C22×C4—1181no—
502C22×C4—1181no—
503C22×C4—1181no—
504C22×C4—1181no—
505C22×C4C2×C4×D5912yes—
506C22×C4D4×D5911yescharacteristic
507C22×C4D4×D5912yes—
508C22×C4C23×D5911yescharacteristic
509M16—2364no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—21no—
516C2×C8—44no—
517C2×C8—1181no—
518C2×C8—1181no—
519C2×C8—1181no—
520C2×C8—1181no—
521C2×C8—1181no—
522C2×C8—1181no—
523C4⋊C4—24no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—594no—
530C4⋊C4—594no—
531C4⋊C4—592no—
532C4⋊C4—1184no—
533C4⋊C4—1184no—
534C4⋊C4—1184no—
535C4⋊C4—1184no—
536C4⋊C4—1184no—
537C4⋊C4—1184no—
538C4⋊C4—1184no—
539C4⋊C4—1181no—
540C4⋊C4—1181no—
541C4⋊C4—1181no—
542C4⋊C4—1181no—
543C4⋊C4—1181no—
544C4⋊C4—4721no—
545C4⋊C4—4721no—
546C4⋊C4C2×C4×D5912yes—
547C22⋊C4—24no—
548C22⋊C4—81no—
549C22⋊C4—594no—
550C22⋊C4—594no—
551C22⋊C4—1184no—
552C22⋊C4—4721no—
553C42—592no—
554C42—592no—
555C42—592no—
556C42—592no—
557C42—1184no—
558C42—1184no—
559C42—4721no—
560C42—4721no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23C22⋊C4×D5912yes—
568C23C2×D4×D5911yescharacteristic
569Q8—22no—
570Q8—22no—
571Q8—41no—
572Q8—41no—
573Q8—1182no—
574Q8—1182no—
575Q8—2361no—
576Q8—2361no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—592no—
596C2×C4—592no—
597C2×C4—592no—
598C2×C4—592no—
599C2×C4—1181no—
600C2×C4—1181no—
601C2×C4—1181no—
602C2×C4—1181no—
603C2×C4—1181no—
604C2×C4—1181no—
605C2×C4—1181no—
606C2×C4—1181no—
607C2×C4—1182no—
608C2×C4—1182no—
609C2×C4—2361no—
610C2×C4—2361no—
611C2×C4—2361no—
612C2×C4—2361no—
613C2×C4—2361no—
614C2×C4—2361no—
615C2×C4—2361no—
616C2×C4—2361no—
617C2×C4—2361no—
618C2×C4—2361no—
619C2×C4C22⋊C4×D5912yes—
620C2×C4C22×C4×D5911yescharacteristic
621C2×C4C2×D4×D5911yescharacteristic
622C2×C4#1427377211yescharacteristic
623C2×C4#1427377211yescharacteristic
624C2×C4#1427377411yescharacteristic
625C2×C4#1427377411yescharacteristic
626C8—41no—
627C8—41no—
628C8—2361no—
629C8—2361no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—41no—
634C22—41no—
635C22—41no—
636C22—41no—
637C22C2×C22⋊C4×D5911yescharacteristic
638C22#1683797911yescharacteristic
639C22#1683797911yescharacteristic
640C22#1683820811yescharacteristic
641C22#1683820911yescharacteristic
642C22#1683821511yescharacteristic
643C22#1683821611yescharacteristic
644C4—22no—
645C4—42no—
646C4—42no—
647C4—81no—
648C4—81no—
649C4—1182no—
650C4—1182no—
651C4—1182no—
652C4—1182no—
653C4—4721no—
654C4—4721no—
655C4—2362no—
656C4—2362no—
657C4#1683788111yescharacteristic
658C4#1683788111yescharacteristic
659C4#1683796711yescharacteristic
660C4#1683797411yescharacteristic
661C2—42no—
662C2#2142668311yescharacteristic
663C2#2143262411yescharacteristic
664C2#2143262511yescharacteristic
665C2#2143551511yescharacteristic
666C2#2143552511yescharacteristic
667C2#2143553611yescharacteristic
668C2#2143554511yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1892C2
socleC23×C59C22×D4
prime radicalC24×C59C2×D4
cosocleC22×C4×C59C24
hypercenter#1892D59
hypocenterC59#21898
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1892

Complements

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

NH ≅ G/Ntype
4. #21425052C2semidirect
5. #21425050C2semidirect
6. #21425017C2semidirect
7. #21425017C2semidirect
11. #21424110C2semidirect
12. #21424109C2semidirect
13. #21424100C2semidirect
14. #21424099C2semidirect
18. #16838012C4semidirect
396. C59#21898semidirect

Maximal subgroups

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

subgroupindex
C2×#168380122
C2×#168374282
#214267032
#214250522
#214250502
#214250172
#214245302
#214245272
#214245192
#214241102
#214241092
#214241002
#214240992
C59×#18922
#2189859