Finite Group Databasebeta

← #28629418

Subgroups — #28629418

Order 15104.

814 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×#16837612C211yesmaximal, characteristic, complemented
2C2×#16837463C211yesmaximal, characteristic
3#21429862C211yesmaximal, characteristic
4#21425299C211yesmaximal, characteristic
5#21425292C211yesmaximal, characteristic
6#21425104C211yesmaximal, characteristic
7#21425104C211yesmaximal, characteristic
8#21425019C211yesmaximal, characteristic, complemented
9#21424966C211yesmaximal, characteristic, complemented
10#21424956C211yesmaximal, characteristic, complemented
11#21424864C211yesmaximal, characteristic, complemented
12#21424785C211yesmaximal, characteristic, complemented
13#21424771C211yesmaximal, characteristic, complemented
14#21424111C211yesmaximal, characteristic, complemented
15C59×#2323C211yesmaximal, characteristic
16C22×#11745672C2211yescharacteristic
17C22×#11745672C2211yescharacteristic
18C2×#14273727C2211yescharacteristic
19C2×#14273706C2211yescharacteristic
20C2×#14273706C2211yescharacteristic
21#16837612—24no—
22#16837463—24no—
23C2×#14273660C2211yescharacteristic
24C2×#14273658C2211yescharacteristic
25C2×C4⋊(C59⋊C8)C2211yescharacteristic
26C2×C4⋊(C59⋊C8)C2211yescharacteristic
27C2×#14273651C2211yescharacteristic
28C2×#14273651C2211yescharacteristic
29C2×#14273650C2211yescharacteristic
30C2×#14273650C2211yescharacteristic
31C2×#14273650C2211yescharacteristic
32C2×#14273650C2211yescharacteristic
33C2×#14273650C2211yescharacteristic
34C2×#14273650C2211yescharacteristic
35C2×C22⋊(C59⋊C8)C2211yescharacteristic
36C2×C22⋊(C59⋊C8)C2211yescharacteristic
37C2×#14273638C2211yescharacteristic
38C2×C4×C59⋊C8C2211yescharacteristic
39#16837332—21no—
40#16837332—21no—
41#16837327—21no—
42#16837327—21no—
43#16837308C2211yescharacteristic
44#16837308C2211yescharacteristic
45#16837268C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837164C2211yescharacteristic
49#16837164C2211yescharacteristic
50C2×C22⋊Q8×C59C2211yescharacteristic
51C2×C4×Q8×C59C2211yescharacteristic
52C2×#117×C59C2211yescharacteristic
53C59×#391C2211yescharacteristic
54C59×#389—21no—
55C59×#389C2211yescharacteristic
56C59×#388—21no—
57C59×#388—21no—
58C59×#387—21no—
59C59×#380C2211yescharacteristic
60C4×C4⋊C4×C59C2211yescharacteristic
61C22×Dic118C2311yescharacteristic
62C22×Dic118C2311yescharacteristic
63C2×#11745672—22no—
64C2×#11745672—22no—
65C2×#11745672—22no—
66C2×#11745672—22no—
67C2×#11745672—41no—
68C2×#11745672—41no—
69C22×C59⋊C8C2311yescharacteristic
70C22×C59⋊C8C2311yescharacteristic
71C22×C59⋊C8C2311yescharacteristic
72C22×C59⋊C8C2311yescharacteristic
73#14273727—22no—
74#14273727—22no—
75#14273706—41no—
76#14273706—41no—
77#14273706—41no—
78#14273706—41no—
79C2×C4⋊Dic59—21no—
80C2×C4⋊Dic59—21no—
81C2×C4⋊Dic59C2311yescharacteristic
82C2×C4⋊Dic59C2311yescharacteristic
83C2×Dic59⋊C4—21no—
84C2×Dic59⋊C4—21no—
85C2×C22⋊Dic59—21no—
86C2×C22⋊Dic59—21no—
87#14273660—41no—
88#14273658—41no—
89C4⋊(C59⋊C8)—22no—
90C4⋊(C59⋊C8)D414yes—
91#14273651—41no—
92#14273651—41no—
93#14273650—22no—
94#14273650—22no—
95#14273650—22no—
96#14273650—22no—
97#14273650—41no—
98#14273650—41no—
99C22⋊(C59⋊C8)—22no—
100C22⋊(C59⋊C8)—22no—
101#14273638—22no—
102C4×C59⋊C8—22no—
103C59⋊#95—21no—
104C59⋊#95—21no—
105C59⋊#95—21no—
106C59⋊#95—21no—
107C59⋊#95—21no—
108C59⋊#95—21no—
109C22×Q8×C59C2311yescharacteristic
110C23×C4×C59C2311yescharacteristic
111C22⋊Q8×C59—22no—
112C22⋊Q8×C59—22no—
113C4×Q8×C59—22no—
114C4×Q8×C59—22no—
115#117×C59—22no—
116#117×C59—22no—
117C2×C4⋊C4×C59—21no—
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×C59D411yescharacteristic
128C2×C22⋊C4×C59D411yescharacteristic
129C2×C42×C59—21no—
130C2×C42×C59—21no—
131C2×C42×C59C2311yescharacteristic
132C2×C42×C59C2311yescharacteristic
133#95×C59—21no—
134#95×C59—21no—
135#95×C59—21no—
136#95×C59—21no—
137#95×C59—21no—
138C24×C59C2×D411yescharacteristic
139C2×Q8×C59—21no—
140C2×Q8×C59—21no—
141C2×Q8×C59—21no—
142C2×Q8×C59—21no—
143C2×Q8×C59—21no—
144C2×Q8×C59#4112yes—
145C22×C4×C59—21no—
146C22×C4×C59—21no—
147C22×C4×C59—21no—
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×C59C2×D411yescharacteristic
156C22×C4×C59C2×D411yescharacteristic
157C22×C4×C59C2×D411yescharacteristic
158C22×C4×C59C2×D411yescharacteristic
159C22×C4×C59C2×D411yescharacteristic
160C22×C4×C59C2411yescharacteristic
161C22×Dic59—21no—
162C22×Dic59—21no—
163C22×Dic59—21no—
164C22×Dic59—21no—
165C2×Dic118—21no—
166C2×Dic118—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—
177C4⋊C4×C59—24no—
178C4⋊C4×C59—22no—
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—42no—
186C4⋊C4×C59—81no—
187C4⋊C4×C59#4112yes—
188C22⋊C4×C59—24no—
189C22⋊C4×C59—22no—
190C22⋊C4×C59—22no—
191C22⋊C4×C59—81no—
192C42×C59—21no—
193C42×C59—21no—
194C42×C59—42no—
195C42×C59—81no—
196C42×C59C2×D412yes—
197C42×C59#4112yes—
198C22⋊Dic59—42no—
199C22⋊Dic59—42no—
200#11745672—44no—
201#11745672—44no—
202C4⋊Dic59—21no—
203C4⋊Dic59—21no—
204C4⋊Dic59—21no—
205C4⋊Dic59—21no—
206C4⋊Dic59—42no—
207C4⋊Dic59—42no—
208Dic59⋊C4—42no—
209Dic59⋊C4—42no—
210C2×C59⋊C8—21no—
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⋊C8C2×D412yes—
221C2×C59⋊C8C2×D412yes—
222C23×C59—21no—
223C23×C59—21no—
224C23×C59—21no—
225C23×C59—21no—
226C23×C59—21no—
227C23×C59—21no—
228C23×C59—21no—
229C23×C59C22×D411yescharacteristic
230Q8×C59—22no—
231Q8×C59—22no—
232Q8×C59—41no—
233Q8×C59—41no—
234C2×C4×C59—21no—
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—22no—
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—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×C59—41no—
257C2×C4×C59—41no—
258C2×C4×C59—41no—
259C2×C4×C59—41no—
260C2×C4×C59—41no—
261C2×C4×C59—41no—
262C2×C4×C59—41no—
263C2×C4×C59#12112yes—
264C2×C4×C59#12112yes—
265C2×C4×C59C22×D411yescharacteristic
266C2×C4×C59C2×#4111yescharacteristic
267C2×C4×C59C2×#4111yescharacteristic
268C2×C4×C59C2×#4111yescharacteristic
269C2×C4×C59H(5,𝔽2)11yescharacteristic
270C2×C4×C59H(5,𝔽2)11yescharacteristic
271C2×Dic59—41no—
272C2×Dic59—41no—
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—
283Dic118—41no—
284Dic118—41no—
285Dic118—41no—
286Dic118—41no—
287Dic118—41no—
288Dic118—41no—
289Dic118—41no—
290Dic118—41no—
291C59⋊C8—22no—
292C59⋊C8—22no—
293C59⋊C8—41no—
294C59⋊C8—41no—
295#24664—591nomaximal, Sylow
296C22×C59—41no—
297C22×C59—41no—
298C22×C59—41no—
299C22×C59—41no—
300C22×C59—41no—
301C22×C59—41no—
302C22×C59—41no—
303C22×C59C2×#12111yescharacteristic
304C22×C59#52511yescharacteristic
305C22×C59#53511yescharacteristic
306C22×C59#57711yescharacteristic
307C22×C59#57811yescharacteristic
308C22×C59#57811yescharacteristic
309C22×C59#57811yescharacteristic
310C4×C59—22no—
311C4×C59—22no—
312C4×C59—42no—
313C4×C59—41no—
314C4×C59—42no—
315C4×C59—41no—
316C4×C59—41no—
317C4×C59—81no—
318C4×C59—81no—
319C4×C59#54611yescharacteristic
320C4×C59#55011yescharacteristic
321C4×C59#55311yescharacteristic
322C4×C59#55511yescharacteristic
323Dic59—81no—
324Dic59—81no—
325Dic59—81no—
326Dic59—81no—
327C2×#462—591no—
328C2×#433—591no—
329#2621—591no—
330#2323D5911yescharacteristic
331#2036—591no—
332#2035—591no—
333#2001—591no—
334#2001—591no—
335#1981—591no—
336#1971—591no—
337#1969—591no—
338#1949—591no—
339#1935—591no—
340#1933—591no—
341#1805—591no—
342C2×C59—81no—
343C2×C59#241411yescharacteristic
344C2×C59#305911yescharacteristic
345C2×C59#306311yescharacteristic
346C2×C59#312011yescharacteristic
347C2×C59#312711yescharacteristic
348C2×C59#322311yescharacteristic
349C2×C59#322511yescharacteristic
350C22×Q16—591no—
351C22×Q16—591no—
352C2×C4⋊Q8—591no—
353C2×C22⋊Q8—591no—
354C2×C22⋊Q8—591no—
355C2×C22⋊Q8C2×D5911yescharacteristic
356C2×C4×Q8C2×D5911yescharacteristic
357C2×#117C2×D5911yescharacteristic
358#462—1184no—
359#433—1184no—
360C2×#107—591no—
361C2×#106—591no—
362C2×C4⋊C8—591no—
363C2×C4⋊C8—591no—
364C2×#103—591no—
365C2×#103—591no—
366C2×#103—591no—
367C2×#103—591no—
368C2×#103—591no—
369C2×#103—591no—
370C2×#103—591no—
371C2×#103—591no—
372C2×C22⋊C8—591no—
373C2×C22⋊C8—591no—
374C2×#97—591no—
375C2×C4×C8—591no—
376#398—1181no—
377#398—1181no—
378#397—1181no—
379#397—1181no—
380#391—591no—
381#391—591no—
382#391C2×D5911yescharacteristic
383#389—21no—
384#389C2×D5911yescharacteristic
385#388—21no—
386#388—21no—
387#387—21no—
388#382—591no—
389#380C2×D5911yescharacteristic
390C4×C4⋊C4C2×D5911yescharacteristic
391#340—591no—
392#340—591no—
393#340—591no—
394#340—591no—
395C59#2466411yescharacteristic, complemented, Sylow
396C22×Q8—591no—
397C22×Q8—591no—
398C22×Q8C22×D5911yescharacteristic
399C23×C4C22×D5911yescharacteristic
400C2×Q16—1182no—
401C2×Q16—1182no—
402C2×Q16—1182no—
403C2×Q16—1182no—
404C2×Q16—2361no—
405C2×Q16—2361no—
406C22×C8—591no—
407C22×C8—591no—
408C22×C8—591no—
409C22×C8—591no—
410C4⋊Q8—1182no—
411C4⋊Q8—1182no—
412C22⋊Q8—22no—
413C22⋊Q8—22no—
414C22⋊Q8—2361no—
415C22⋊Q8—2361no—
416C22⋊Q8—2361no—
417C22⋊Q8—2361no—
418C4×Q8—22no—
419C4×Q8—22no—
420#117—22no—
421#117—22no—
422C2×C4⋊C4—21no—
423C2×C4⋊C4—21no—
424C2×C4⋊C4—21no—
425C2×C4⋊C4—591no—
426C2×C4⋊C4—591no—
427C2×C4⋊C4—1181no—
428C2×C4⋊C4—1181no—
429C2×C4⋊C4—1181no—
430C2×C4⋊C4—1181no—
431C2×C4⋊C4C22⋊D5911yescharacteristic
432C2×C4⋊C4C22⋊D5911yescharacteristic
433C2×C4⋊C4C22×D5911yescharacteristic
434C2×C4⋊C4C22×D5911yescharacteristic
435C2×C4⋊C4C22×D5911yescharacteristic
436C2×C22⋊C4—21no—
437C2×C22⋊C4—21no—
438C2×C22⋊C4—1181no—
439C2×C22⋊C4—1181no—
440C2×C22⋊C4C22⋊D5911yescharacteristic
441C2×C22⋊C4C22⋊D5911yescharacteristic
442C2×C42—21no—
443C2×C42—21no—
444C2×C42C22×D5911yescharacteristic
445C2×C42C22×D5911yescharacteristic
446#107—2361no—
447#106—2361no—
448C4⋊C8—594no—
449C4⋊C8—1182no—
450#103—1182no—
451#103—1182no—
452#103—1182no—
453#103—1182no—
454#103—2361no—
455#103—2361no—
456#103—2361no—
457#103—2361no—
458C22⋊C8—1182no—
459C22⋊C8—1182no—
460#97—1182no—
461C4×C8—1182no—
462#95—21no—
463#95—21no—
464#95—21no—
465#95—21no—
466#95—21no—
467#95—1181no—
468#95—1181no—
469#95—1181no—
470#95—1181no—
471#95—1181no—
472#95—1181no—
473C24C2×C22⋊D5911yescharacteristic
474C2×Q8—21no—
475C2×Q8—21no—
476C2×Q8—21no—
477C2×Q8—21no—
478C2×Q8—21no—
479C2×Q8—1181no—
480C2×Q8—1181no—
481C2×Q8—1181no—
482C2×Q8—1181no—
483C2×Q8—1181no—
484C2×Q8—1181no—
485C2×Q8—1181no—
486C2×Q8—1181no—
487C2×Q8—1181no—
488C2×Q8—1181no—
489C2×Q8—1181no—
490C2×Q8—1181no—
491C2×Q8#1174568512yes—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—1181no—
503C22×C4—1181no—
504C22×C4—1181no—
505C22×C4—1181no—
506C22×C4C2×C22⋊D5911yescharacteristic
507C22×C4C2×C22⋊D5911yescharacteristic
508C22×C4C2×C22⋊D5911yescharacteristic
509C22×C4C2×C22⋊D5911yescharacteristic
510C22×C4C2×C22⋊D5911yescharacteristic
511C22×C4C23×D5911yescharacteristic
512Q16—2364no—
513Q16—2364no—
514C2×C8—592no—
515C2×C8—592no—
516C2×C8—1181no—
517C2×C8—1181no—
518C2×C8—1181no—
519C2×C8—1181no—
520C2×C8—1181no—
521C2×C8—1181no—
522C2×C8—1181no—
523C2×C8—1181no—
524C2×C8—1181no—
525C2×C8—1181no—
526C4⋊C4—24no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—21no—
533C4⋊C4—21no—
534C4⋊C4—42no—
535C4⋊C4—81no—
536C4⋊C4—1181no—
537C4⋊C4—1181no—
538C4⋊C4—1181no—
539C4⋊C4—1181no—
540C4⋊C4—2362no—
541C4⋊C4—2362no—
542C4⋊C4—2362no—
543C4⋊C4—2362no—
544C4⋊C4#1174568512yes—
545C22⋊C4—24no—
546C22⋊C4—22no—
547C22⋊C4—22no—
548C22⋊C4—81no—
549C22⋊C4—2362no—
550C22⋊C4—2362no—
551C42—21no—
552C42—21no—
553C42—42no—
554C42—81no—
555C42D4×D5912yes—
556C42#1174568712yes—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23C22×C22⋊D5911yescharacteristic
565Q8—22no—
566Q8—22no—
567Q8—41no—
568Q8—41no—
569Q8—2361no—
570Q8—2361no—
571Q8—2361no—
572Q8—2361no—
573Q8—2361no—
574Q8—2361no—
575Q8—2361no—
576Q8—2361no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—41no—
593C2×C4—41no—
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—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—2361no—
607C2×C4—2361no—
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×C4—2361no—
618C2×C4#1427369912yes—
619C2×C4#1427369912yes—
620C2×C4C2×D4×D5911yescharacteristic
621C2×C4C2×#1174568711yescharacteristic
622C2×C4C2×#1174568511yescharacteristic
623C2×C4C2×#1174568511yescharacteristic
624C2×C4#1427377311yescharacteristic
625C2×C4#1427377311yescharacteristic
626C8—1182no—
627C8—1182no—
628C8—2361no—
629C8—2361no—
630C22—41no—
631C22—41no—
632C22—41no—
633C22—41no—
634C22—41no—
635C22—41no—
636C22—41no—
637C22C2×#1427369911yescharacteristic
638C22#1683791311yescharacteristic
639C22#1683796611yescharacteristic
640C22#1683821311yescharacteristic
641C22#1683821711yescharacteristic
642C22#1683821711yescharacteristic
643C22#1683821711yescharacteristic
644C4—22no—
645C4—22no—
646C4—41no—
647C4—42no—
648C4—41no—
649C4—41no—
650C4—42no—
651C4—81no—
652C4—81no—
653C4—4721no—
654C4—4721no—
655C4—4721no—
656C4—4721no—
657C4#1683804811yescharacteristic
658C4#1683807311yescharacteristic
659C4#1683809811yescharacteristic
660C4#1683811711yescharacteristic
661C2—81no—
662C2#2142751411yescharacteristic
663C2#2143374811yescharacteristic
664C2#2143378911yescharacteristic
665C2#2143431011yescharacteristic
666C2#2143439611yescharacteristic
667C2#2143517311yescharacteristic
668C2#2143519911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2323C2
socleC23×C59C22×D4
prime radicalC24×C59C2×D4
hypercenter#2323D59
hypocenterC59#24664
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2323

Complements

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

NH ≅ G/Ntype
1. C2×#16837612C2semidirect
8. #21425019C2semidirect
9. #21424966C2semidirect
10. #21424956C2semidirect
11. #21424864C2semidirect
12. #21424785C2semidirect
13. #21424771C2semidirect
14. #21424111C2semidirect
395. C59#24664semidirect

Maximal subgroups

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

subgroupindex
C2×#168376122
C2×#168374632
#214298622
#214252992
#214252922
#214251042
#214250192
#214249662
#214249562
#214248642
#214247852
#214247712
#214241112
C59×#23232
#2466459