Finite Group Databasebeta

← #28611152

Subgroups — #28611152

Order 15104.

982 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×#16837607C211yesmaximal, characteristic
2C2×#16837502C211yesmaximal, characteristic
3C2×C4×#11745670C211yesmaximal, characteristic
4#21427152C211yesmaximal, characteristic, complemented
5#21424915C211yesmaximal, characteristic, complemented
6#21424862C211yesmaximal, characteristic, complemented
7#21424785C211yesmaximal, characteristic, complemented
8#21424771C211yesmaximal, characteristic, complemented
9#21424430C211yesmaximal, characteristic
10#21424430C211yesmaximal, characteristic
11#21424363C211yesmaximal, characteristic, complemented
12#21424349C211yesmaximal, characteristic, complemented
13#21423971C211yesmaximal, characteristic, complemented
14#21423921C211yesmaximal, characteristic
15C59×#2349C211yesmaximal, characteristic
16C22×#11745670C2211yescharacteristic
17C22×#11745670C2211yescharacteristic
18C2×#14273727C2211yescharacteristic
19C2×C4×Dic118C2211yescharacteristic
20C2×C4×Dic118C2211yescharacteristic
21#16837607C418yescomplemented
22#16837502—24no—
23C4×#11745670—24no—
24C2×#14273660C2211yescharacteristic
25C2×#14273658C2211yescharacteristic
26C2×C4⋊(C59⋊C8)C2211yescharacteristic
27C2×#14273650C2211yescharacteristic
28C2×#14273650C2211yescharacteristic
29C2×#14273650C2211yescharacteristic
30C2×#14273650C2211yescharacteristic
31C2×#14273650C2211yescharacteristic
32C2×#14273650C2211yescharacteristic
33C2×#14273648C2211yescharacteristic
34C2×#14273648C2211yescharacteristic
35C2×#14273638C2211yescharacteristic
36C2×#14273638C2211yescharacteristic
37C2×C4×C59⋊C8C2211yescharacteristic
38#16837308C2211yescharacteristic
39#16837308C2211yescharacteristic
40#16837298—21no—
41#16837285—21no—
42#16837285—21no—
43C4⋊C4×Dic59—21no—
44C4×C4⋊Dic59C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837164C2211yescharacteristic
49#16837157C2211yescharacteristic
50#16837157C2211yescharacteristic
51C2×C4×D4×C59C2211yescharacteristic
52C2×C4×D4×C59C2211yescharacteristic
53C2×C4×D4×C59C2211yescharacteristic
54C59×#389C2211yescharacteristic
55C59×#389C2211yescharacteristic
56C59×#385—21no—
57C59×#385—21no—
58C59×#382C2211yescharacteristic
59C59×#380—21no—
60C4×C4⋊C4×C59C2211yescharacteristic
61C2×#95×C59—21no—
62C22×Dic118C2311yescharacteristic
63C22×Dic118C2311yescharacteristic
64C2×#11745670—22no—
65C2×#11745670—22no—
66C2×#11745670—22no—
67C2×#11745670—22no—
68C2×#11745670C2×C414yes—
69C2×#11745670C2×C414yes—
70C22×C59⋊C8C2311yescharacteristic
71C22×C59⋊C8C2311yescharacteristic
72C22×C59⋊C8C2311yescharacteristic
73C22×C59⋊C8C2311yescharacteristic
74#14273727—22no—
75#14273727C2×C414yescomplemented
76C4×Dic118—22no—
77C4×Dic118—22no—
78C4×Dic118—41no—
79C4×Dic118—41no—
80C2×C4⋊Dic59—21no—
81C2×C4⋊Dic59—21no—
82C2×C4⋊Dic59C2311yescharacteristic
83C2×C4⋊Dic59C2311yescharacteristic
84C2×Dic59⋊C4—21no—
85C2×Dic59⋊C4—21no—
86C2×C4×Dic59—21no—
87C2×C4×Dic59—21no—
88C2×C4×Dic59—21no—
89#14273660—22no—
90#14273658—22no—
91C4⋊(C59⋊C8)C2×C414yescomplemented
92#14273650—22no—
93#14273650—22no—
94#14273650—41no—
95#14273650—41no—
96#14273650C2×C414yescomplemented
97#14273650C2×C414yescomplemented
98#14273648—22no—
99#14273648D414yes—
100#14273638—22no—
101#14273638—22no—
102C4×C59⋊C8—22no—
103C59⋊#95—21no—
104C59⋊#95—21no—
105C59⋊#95—21no—
106C59⋊#95—21no—
107C59⋊#95—21no—
108C22×D4×C59C2311yescharacteristic
109C23×C4×C59—21no—
110C23×C4×C59—21no—
111C23×C4×C59—21no—
112C4×D4×C59—22no—
113C4×D4×C59—22no—
114C4×D4×C59—22no—
115C4×D4×C59—41no—
116C4×D4×C59—41no—
117C4×D4×C59C2×C414yes—
118C2×C4⋊C4×C59—21no—
119C2×C4⋊C4×C59—21no—
120C2×C4⋊C4×C59D411yescharacteristic
121C2×C4⋊C4×C59D411yescharacteristic
122C2×C4⋊C4×C59C2311yescharacteristic
123C2×C4⋊C4×C59C2311yescharacteristic
124C2×C4⋊C4×C59C2311yescharacteristic
125C2×C22⋊C4×C59—21no—
126C2×C22⋊C4×C59—21no—
127C2×C22⋊C4×C59—21no—
128C2×C42×C59C2311yescharacteristic
129C2×C42×C59C2311yescharacteristic
130C2×C42×C59C2311yescharacteristic
131#95×C59—21no—
132#95×C59—21no—
133#95×C59—21no—
134#95×C59—21no—
135#95×C59D411yescharacteristic
136#95×C59D411yescharacteristic
137C24×C59—21no—
138C2×D4×C59—21no—
139C2×D4×C59—21no—
140C2×D4×C59—21no—
141C2×D4×C59C22×C412yes—
142C2×D4×C59C2×D412yes—
143C2×D4×C59#4112yes—
144C22×C4×C59—21no—
145C22×C4×C59—24no—
146C22×C4×C59—21no—
147C22×C4×C59—21no—
148C22×C4×C59—42no—
149C22×C4×C59—42no—
150C22×C4×C59—42no—
151C22×C4×C59—42no—
152C22×C4×C59—81no—
153C22×C4×C59—81no—
154C22×C4×C59—81no—
155C22×C4×C59—81no—
156C22×C4×C59C2×D411yescharacteristic
157C22×C4×C59C2×D411yescharacteristic
158C22×C4×C59C2×D411yescharacteristic
159C22×C4×C59C2×D411yescharacteristic
160C22×C4×C59C2×D411yescharacteristic
161C22×C4×C59C2×D411yescharacteristic
162C22×C4×C59C2411yescharacteristic
163C22×Dic59—21no—
164C22×Dic59—21no—
165C22×Dic59—21no—
166C22×Dic59—21no—
167C2×Dic118—21no—
168C2×Dic118—21no—
169C2×Dic118—21no—
170C2×Dic118—21no—
171C2×Dic118—21no—
172C2×Dic118—21no—
173C2×Dic118—21no—
174C2×Dic118—21no—
175C2×Dic118—21no—
176C2×Dic118—21no—
177C2×Dic118C22×C412yes—
178C2×Dic118C22×C412yes—
179C4⋊C4×C59—22no—
180C4⋊C4×C59—21no—
181C4⋊C4×C59—21no—
182C4⋊C4×C59—21no—
183C4⋊C4×C59—21no—
184C4⋊C4×C59—21no—
185C4⋊C4×C59—41no—
186C4⋊C4×C59—81no—
187C4⋊C4×C59—81no—
188C4⋊C4×C59C22×C412yes—
189C22⋊C4×C59—24no—
190C22⋊C4×C59—42no—
191C22⋊C4×C59—42no—
192C42×C59—21no—
193C42×C59—21no—
194C42×C59—21no—
195C42×C59—21no—
196C42×C59C22×C412yes—
197C42×C59#4112yes—
198#11745670—28no—
199#11745670—28no—
200C4⋊Dic59—24no—
201C4⋊Dic59—24no—
202C4⋊Dic59—21no—
203C4⋊Dic59—21no—
204C4⋊Dic59—21no—
205C4⋊Dic59C2×D412yes—
206Dic59⋊C4—42no—
207Dic59⋊C4—42no—
208C4×Dic59—42no—
209C4×Dic59—42no—
210C4×Dic59—42no—
211C2×C59⋊C8—21no—
212C2×C59⋊C8—21no—
213C2×C59⋊C8—21no—
214C2×C59⋊C8—21no—
215C2×C59⋊C8—21no—
216C2×C59⋊C8—21no—
217C2×C59⋊C8—21no—
218C2×C59⋊C8—21no—
219C2×C59⋊C8—21no—
220C2×C59⋊C8C22×C412yes—
221C2×C59⋊C8C22×C412yes—
222C2×C59⋊C8C2×D412yes—
223C23×C59—22no—
224C23×C59—22no—
225C23×C59—22no—
226C23×C59—41no—
227C23×C59—41no—
228C23×C59—41no—
229C23×C59—41no—
230C23×C59C22×D411yescharacteristic
231D4×C59—22no—
232D4×C59—22no—
233D4×C59—22no—
234D4×C59C4×D414yes—
235C2×C4×C59—21no—
236C2×C4×C59—21no—
237C2×C4×C59—21no—
238C2×C4×C59—21no—
239C2×C4×C59—21no—
240C2×C4×C59—21no—
241C2×C4×C59—22no—
242C2×C4×C59—21no—
243C2×C4×C59—21no—
244C2×C4×C59—21no—
245C2×C4×C59—21no—
246C2×C4×C59—21no—
247C2×C4×C59—21no—
248C2×C4×C59—21no—
249C2×C4×C59—21no—
250C2×C4×C59—21no—
251C2×C4×C59—41no—
252C2×C4×C59—41no—
253C2×C4×C59—41no—
254C2×C4×C59—41no—
255C2×C4×C59—41no—
256C2×C4×C59—41no—
257C2×C4×C59—41no—
258C2×C4×C59—41no—
259C2×C4×C59—84no—
260C2×C4×C59—84no—
261C2×C4×C59—84no—
262C2×C4×C59C4×D412yes—
263C2×C4×C59C4×D412yes—
264C2×C4×C59#13712yes—
265C2×C4×C59C23×C411yescharacteristic
266C2×C4×C59C22×D411yescharacteristic
267C2×C4×C59C2×#4111yescharacteristic
268C2×C4×C59C2×#4111yescharacteristic
269C2×C4×C59H(5,𝔽2)11yescharacteristic
270C2×C4×C592−1+411yescharacteristic
271C2×Dic59—22no—
272C2×Dic59—22no—
273C2×Dic59—22no—
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—
283Dic118—22no—
284Dic118—22no—
285Dic118—22no—
286Dic118—22no—
287Dic118—41no—
288Dic118—41no—
289Dic118—41no—
290Dic118—41no—
291C59⋊C8—22no—
292C59⋊C8—22no—
293C59⋊C8—22no—
294C59⋊C8—41no—
295#23012—591nomaximal, Sylow
296C22×C59—24no—
297C22×C59—42no—
298C22×C59—42no—
299C22×C59—42no—
300C22×C59—42no—
301C22×C59—42no—
302C22×C59—42no—
303C22×C59C2×C4×D411yescharacteristic
304C22×C59D4211yescharacteristic
305C22×C59#54811yescharacteristic
306C22×C59C2×#13711yescharacteristic
307C22×C59#57511yescharacteristic
308C22×C59#57711yescharacteristic
309C22×C59#57711yescharacteristic
310C4×C59—22no—
311C4×C59—22no—
312C4×C59—22no—
313C4×C59—42no—
314C4×C59—41no—
315C4×C59—41no—
316C4×C59—41no—
317C4×C59—81no—
318C4×C59—81no—
319C4×C59C2×C4×D411yescharacteristic
320C4×C59#52011yescharacteristic
321C4×C59#54711yescharacteristic
322C4×C59#55311yescharacteristic
323Dic59—42no—
324Dic59—42no—
325Dic59—42no—
326Dic59—81no—
327C2×#461—591no—
328C2×#440—591no—
329C2×C4×SD16—591no—
330#2365—591no—
331#2349D5911yescharacteristic
332#1960—591no—
333#1949—591no—
334#1935—591no—
335#1933—591no—
336#1877—591no—
337#1877—591no—
338#1865—591no—
339#1863—591no—
340#1771—591no—
341#1762—591no—
342C2×C59—44no—
343C2×C59#236111yescharacteristic
344C2×C59C4×#13711yescharacteristic
345C2×C59#294911yescharacteristic
346C2×C59#326611yescharacteristic
347C2×C59#326711yescharacteristic
348C2×C59#337211yescharacteristic
349C2×C59#337611yescharacteristic
350C22×SD16—591no—
351C22×SD16—591no—
352C2×C4⋊Q8—591no—
353C2×C4×Q8—591no—
354C2×C4×Q8—591no—
355C2×C4×D4C2×D5911yescharacteristic
356C2×C4×D4C2×D5911yescharacteristic
357C2×C4×D4C2×D5911yescharacteristic
358#461—598no—
359#440—1184no—
360C4×SD16—1184no—
361C2×#107—591no—
362C2×#106—591no—
363C2×C4⋊C8—591no—
364C2×#103—591no—
365C2×#103—591no—
366C2×#103—591no—
367C2×#103—591no—
368C2×#103—591no—
369C2×#103—591no—
370C2×#102—591no—
371C2×#102—591no—
372C2×#97—591no—
373C2×#97—591no—
374C2×C4×C8—591no—
375#391—591no—
376#391—591no—
377#389—1181no—
378#389C2×D5911yescharacteristic
379#389C2×D5911yescharacteristic
380#387—1181no—
381#387—1181no—
382#385—21no—
383#385—21no—
384#382C2×D5911yescharacteristic
385#380—21no—
386C4×C4⋊C4—591no—
387C4×C4⋊C4—1181no—
388C4×C4⋊C4C2×D5911yescharacteristic
389C2×#95—21no—
390#340—591no—
391#340—591no—
392#340—591no—
393#340—591no—
394#336—591no—
395#336—591no—
396C59#2301211yescharacteristic, complemented, Sylow
397C22×Q8—591no—
398C22×Q8—591no—
399C22×D4C22×D5911yescharacteristic
400C23×C4—21no—
401C23×C4—21no—
402C23×C4—21no—
403C2×SD16—594no—
404C2×SD16—594no—
405C2×SD16—1182no—
406C2×SD16—1182no—
407C2×SD16—1182no—
408C2×SD16—1182no—
409C22×C8—591no—
410C22×C8—591no—
411C22×C8—591no—
412C22×C8—591no—
413C4⋊Q8—594no—
414C4⋊Q8—1182no—
415C4×Q8—1182no—
416C4×Q8—1182no—
417C4×Q8—2361no—
418C4×Q8—2361no—
419C4×D4—22no—
420C4×D4—22no—
421C4×D4—22no—
422C4×D4—41no—
423C4×D4—41no—
424C4×D4C4×D5914yes—
425C2×C4⋊C4—21no—
426C2×C4⋊C4—21no—
427C2×C4⋊C4—591no—
428C2×C4⋊C4—591no—
429C2×C4⋊C4—1181no—
430C2×C4⋊C4—1181no—
431C2×C4⋊C4—1181no—
432C2×C4⋊C4—1181no—
433C2×C4⋊C4C22⋊D5911yescharacteristic
434C2×C4⋊C4C22⋊D5911yescharacteristic
435C2×C4⋊C4C22×D5911yescharacteristic
436C2×C4⋊C4C22×D5911yescharacteristic
437C2×C4⋊C4C22×D5911yescharacteristic
438C2×C22⋊C4—21no—
439C2×C22⋊C4—21no—
440C2×C22⋊C4—21no—
441C2×C42—1181no—
442C2×C42—1181no—
443C2×C42—1181no—
444C2×C42C22×D5911yescharacteristic
445C2×C42C22×D5911yescharacteristic
446C2×C42C22×D5911yescharacteristic
447#107—1182no—
448#106—1182no—
449C4⋊C8—594no—
450#103—594no—
451#103—594no—
452#103—1182no—
453#103—1182no—
454#103—2361no—
455#103—2361no—
456#102—594no—
457#102—1182no—
458#97—1182no—
459#97—1182no—
460C4×C8—1182no—
461#95—21no—
462#95—21no—
463#95—21no—
464#95—21no—
465#95—1181no—
466#95—1181no—
467#95—1181no—
468#95—1181no—
469#95—1181no—
470#95C22⋊D5911yescharacteristic
471#95C22⋊D5911yescharacteristic
472C24—21no—
473C2×Q8—592no—
474C2×Q8—592no—
475C2×Q8—1181no—
476C2×Q8—1181no—
477C2×Q8—1181no—
478C2×Q8—1181no—
479C2×Q8—1181no—
480C2×Q8—1181no—
481C2×Q8—1181no—
482C2×Q8—1181no—
483C2×Q8—1181no—
484C2×Q8—1181no—
485C2×D4—21no—
486C2×D4—21no—
487C2×D4—21no—
488C2×D4C2×C4×D5912yes—
489C2×D4D4×D5912yes—
490C2×D4#1174568912yes—
491C22×C4—21no—
492C22×C4—24no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—42no—
496C22×C4—42no—
497C22×C4—42no—
498C22×C4—42no—
499C22×C4—81no—
500C22×C4—81no—
501C22×C4—81no—
502C22×C4—81no—
503C22×C4—1181no—
504C22×C4—1181no—
505C22×C4—1181no—
506C22×C4—1181no—
507C22×C4C2×C22⋊D5911yescharacteristic
508C22×C4C2×C22⋊D5911yescharacteristic
509C22×C4C2×C22⋊D5911yescharacteristic
510C22×C4C2×C22⋊D5911yescharacteristic
511C22×C4C2×C22⋊D5911yescharacteristic
512C22×C4C2×C22⋊D5911yescharacteristic
513C22×C4C23×D5911yescharacteristic
514SD16—1188no—
515SD16—1188no—
516C2×C8—592no—
517C2×C8—592no—
518C2×C8—592no—
519C2×C8—1181no—
520C2×C8—1181no—
521C2×C8—1181no—
522C2×C8—1181no—
523C2×C8—1181no—
524C2×C8—1181no—
525C2×C8—1181no—
526C2×C8—1181no—
527C2×C8—1181no—
528C4⋊C4—22no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—21no—
533C4⋊C4—21no—
534C4⋊C4—41no—
535C4⋊C4—81no—
536C4⋊C4—81no—
537C4⋊C4—592no—
538C4⋊C4—1184no—
539C4⋊C4—1184no—
540C4⋊C4—1181no—
541C4⋊C4—1181no—
542C4⋊C4—1181no—
543C4⋊C4—2362no—
544C4⋊C4—2362no—
545C4⋊C4C2×C4×D5912yes—
546C22⋊C4—24no—
547C22⋊C4—42no—
548C22⋊C4—42no—
549C42—21no—
550C42—21no—
551C42—21no—
552C42—21no—
553C42—2362no—
554C42—2362no—
555C42—2362no—
556C42C2×C4×D5912yes—
557C42#1174568512yes—
558C23—22no—
559C23—22no—
560C23—22no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23C22×C22⋊D5911yescharacteristic
566Q8—1182no—
567Q8—1182no—
568Q8—1182no—
569Q8—1182no—
570Q8—2361no—
571Q8—2361no—
572Q8—2361no—
573Q8—2361no—
574D4—22no—
575D4—22no—
576D4—22no—
577D4#1427368914yes—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—22no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—84no—
603C2×C4—84no—
604C2×C4—84no—
605C2×C4—1182no—
606C2×C4—1182no—
607C2×C4—1182no—
608C2×C4—2361no—
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×C4C4×C22⋊D5912yes—
618C2×C4C4×C22⋊D5912yes—
619C2×C4#1427375712yes—
620C2×C4C22×C4×D5911yescharacteristic
621C2×C4C2×D4×D5911yescharacteristic
622C2×C4C2×#1174568511yescharacteristic
623C2×C4C2×#1174568911yescharacteristic
624C2×C4#1427377211yescharacteristic
625C2×C4#1427377511yescharacteristic
626C8—1182no—
627C8—1182no—
628C8—1182no—
629C8—2361no—
630C22—24no—
631C22—42no—
632C22—42no—
633C22—42no—
634C22—42no—
635C22—42no—
636C22—42no—
637C22C2×C4×C22⋊D5911yescharacteristic
638C22D4×C22⋊D5911yescharacteristic
639C22#1683806311yescharacteristic
640C22C2×#1427375711yescharacteristic
641C22#1683820011yescharacteristic
642C22#1683821311yescharacteristic
643C22#1683821311yescharacteristic
644C4—22no—
645C4—22no—
646C4—22no—
647C4—42no—
648C4—41no—
649C4—41no—
650C4—41no—
651C4—81no—
652C4—81no—
653C4—4721no—
654C4—2362no—
655C4—2362no—
656C4—2362no—
657C4C2×#1427368911yescharacteristic
658C4#1683788311yescharacteristic
659C4#1683805311yescharacteristic
660C4#1683810011yescharacteristic
661C2—44no—
662C2#2142711211yescharacteristic
663C2C4×#1427375711yescharacteristic
664C2#2143295311yescharacteristic
665C2#2143560211yescharacteristic
666C2#2143561711yescharacteristic
667C2#2143687711yescharacteristic
668C2#2143692311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#2349C2
socleC23×C59C22×D4
prime radicalC22×D4×C59C23
cosocleC22×C4×C59C24
hypercenter#2349D59
hypocenterC59#23012
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2349

Complements

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

NH ≅ G/Ntype
4. #21427152C2semidirect
5. #21424915C2semidirect
6. #21424862C2semidirect
7. #21424785C2semidirect
8. #21424771C2semidirect
11. #21424363C2semidirect
12. #21424349C2semidirect
13. #21423971C2semidirect
21. #16837607C4semidirect
75. #14273727C2×C4semidirect
91. C4⋊(C59⋊C8)C2×C4semidirect
96. #14273650C2×C4semidirect
97. #14273650C2×C4semidirect
396. C59#23012semidirect

Maximal subgroups

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

subgroupindex
C2×#168376072
C2×#168375022
C2×C4×#117456702
#214271522
#214249152
#214248622
#214247852
#214247712
#214244302
#214243632
#214243492
#214239712
#214239212
C59×#23492
#2301259