Finite Group Databasebeta

← #28636881

Subgroups — #28636881

Order 15104.

848 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837740C211yesmaximal, characteristic, complemented
2C2×#16837633C211yesmaximal, characteristic
3#21428558C211yesmaximal, characteristic
4#21425572C211yesmaximal, characteristic, complemented
5#21425514C211yesmaximal, characteristic, complemented
6#21425481C211yesmaximal, characteristic, complemented
7#21425411C211yesmaximal, characteristic, complemented
8#21424864C211yesmaximal, characteristic, complemented
9#21424723C211yesmaximal, characteristic, complemented
10#21424451C211yesmaximal, characteristic, complemented
11#21424418C211yesmaximal, characteristic
12#21424276C211yesmaximal, characteristic, complemented
13#21424101C211yesmaximal, characteristic, complemented
14#21423812C211yesmaximal, characteristic, complemented
15C59×#2362C211yesmaximal, characteristic, complemented
16C22×#11745671C2211yescharacteristic
17C23×C59⋊C8C2211yescharacteristic
18C2×#14273701C2211yescharacteristic
19C2×#14273686C2211yescharacteristic
20#16837740—24no—
21#16837633—24no—
22C2×#14273660C2211yescharacteristic
23C2×#14273658C2211yescharacteristic
24C2×#14273658C2211yescharacteristic
25C2×C4⋊(C59⋊C8)C2211yescharacteristic
26C2×C4⋊(C59⋊C8)C2211yescharacteristic
27C2×#14273651C2211yescharacteristic
28C2×#14273651C2211yescharacteristic
29C2×#14273651C2211yescharacteristic
30C2×#14273647C2211yescharacteristic
31C2×#14273647C2211yescharacteristic
32C2×#14273647C2211yescharacteristic
33C2×C22⋊(C59⋊C8)C2211yescharacteristic
34#16837318—21no—
35#16837318—21no—
36#16837304C2211yescharacteristic
37#16837304C2211yescharacteristic
38#16837294—21no—
39#16837294—21no—
40#16837272C2211yescharacteristic
41#16837268C2211yescharacteristic
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837157C2211yescharacteristic
49#16837157C2211yescharacteristic
50C2×C22⋊Q8×C59C2211yescharacteristic
51C2×C4×Q8×C59C2211yescharacteristic
52C2×#117×C59C2211yescharacteristic
53C59×#391C2211yescharacteristic
54C59×#389C2211yescharacteristic
55C59×#388—21no—
56C59×#388—21no—
57C59×#387—21no—
58C4×C4⋊C4×C59—21no—
59C4×C4⋊C4×C59C2211yescharacteristic
60C4×C22⋊C4×C59C2211yescharacteristic
61C22×D236C2311yescharacteristic
62C22×C22⋊D59—21no—
63C2×#11745671—22no—
64C2×#11745671—22no—
65C2×#11745671—41no—
66C22×C59⋊C8—22no—
67C22×C59⋊C8—22no—
68C22×C59⋊C8—41no—
69C22×C59⋊C8C2311yescharacteristic
70C22×C59⋊C8C2311yescharacteristic
71C22×C59⋊C8C2311yescharacteristic
72C22×C59⋊C8C2311yescharacteristic
73#14273701—22no—
74#14273701—22no—
75#14273686—22no—
76#14273686—22no—
77C2×C4⋊Dic59—21no—
78C2×C4⋊Dic59—21no—
79C2×C4⋊Dic59C2311yescharacteristic
80C2×C4⋊Dic59C2311yescharacteristic
81C2×C4⋊Dic59C2311yescharacteristic
82C2×C22⋊Dic59—21no—
83C2×(C2×D59)⋊C4—21no—
84C2×(C2×D59)⋊C4—21no—
85C2×(C2×D59)⋊C4—21no—
86C2×(C2×D59)⋊C4—21no—
87C2×(C2×D59)⋊C4—21no—
88C2×C4×Dic59—21no—
89#14273660—22no—
90#14273658—22no—
91#14273658—22no—
92C4⋊(C59⋊C8)—22no—
93C4⋊(C59⋊C8)—41no—
94#14273651—22no—
95#14273651—22no—
96#14273651—22no—
97#14273647—22no—
98#14273647—22no—
99#14273647—41no—
100C22⋊(C59⋊C8)—22no—
101C59⋊#95—21no—
102C59⋊#95—21no—
103C22×Q8×C59C2311yescharacteristic
104C23×C4×C59C2311yescharacteristic
105C22⋊Q8×C59—22no—
106C22⋊Q8×C59D414yes—
107C4×Q8×C59—22no—
108C4×Q8×C59—41no—
109#117×C59—22no—
110#117×C59—41no—
111C2×C4⋊C4×C59—21no—
112C2×C4⋊C4×C59—21no—
113C2×C4⋊C4×C59—21no—
114C2×C4⋊C4×C59C2311yescharacteristic
115C2×C4⋊C4×C59C2311yescharacteristic
116C2×C4⋊C4×C59C2311yescharacteristic
117C2×C22⋊C4×C59—21no—
118C2×C22⋊C4×C59—21no—
119C2×C22⋊C4×C59D411yescharacteristic
120C2×C22⋊C4×C59D411yescharacteristic
121C2×C42×C59—21no—
122C2×C42×C59—21no—
123C2×C42×C59—21no—
124C2×C42×C59C2311yescharacteristic
125C2×C42×C59C2311yescharacteristic
126#95×C59—21no—
127#95×C59—21no—
128#95×C59—21no—
129#95×C59—21no—
130#95×C59D411yescharacteristic
131#95×C59D411yescharacteristic
132C24×C59C2×D411yescharacteristic
133C23×D59—21no—
134C2×Q8×C59—21no—
135C2×Q8×C59—21no—
136C2×Q8×C59—21no—
137C2×Q8×C59—21no—
138C2×Q8×C59—21no—
139C2×Q8×C59C2×D412yes—
140C22×C4×C59—21no—
141C22×C4×C59—21no—
142C22×C4×C59—21no—
143C22×C4×C59—21no—
144C22×C4×C59—21no—
145C22×C4×C59—21no—
146C22×C4×C59—21no—
147C22×C4×C59—21no—
148C22×C4×C59C2×D411yescharacteristic
149C22×C4×C59C2×D411yescharacteristic
150C22×C4×C59C2×D411yescharacteristic
151C22×C4×C59C2×D411yescharacteristic
152C22×C4×C59C2×D412yes—
153C22×C4×C59C2×D411yescharacteristic
154C22×C4×C59#4112yes—
155C22×C4×C59C2411yescharacteristic
156C2×C22⋊D59—42no—
157C2×C22⋊D59—81no—
158C2×C22⋊D59—81no—
159C22×Dic59—21no—
160C22×Dic59—21no—
161C22×Dic59—21no—
162C2×D236—21no—
163C2×D236—21no—
164C2×D236—21no—
165C2×D236—21no—
166C2×D236—21no—
167C2×D236—21no—
168C4⋊C4×C59—24no—
169C4⋊C4×C59—21no—
170C4⋊C4×C59—21no—
171C4⋊C4×C59—21no—
172C4⋊C4×C59—21no—
173C4⋊C4×C59—42no—
174C4⋊C4×C59—42no—
175C4⋊C4×C59C2×D412yes—
176C4⋊C4×C59#4112yes—
177C22⋊C4×C59—24no—
178C22⋊C4×C59—22no—
179C22⋊C4×C59—41no—
180C22⋊C4×C59—42no—
181C42×C59—21no—
182C42×C59—21no—
183C42×C59—21no—
184C42×C59—21no—
185C42×C59—42no—
186C42×C59—42no—
187C42×C59—42no—
188C22⋊Dic59—42no—
189#11745671—44no—
190(C2×D59)⋊C4—42no—
191(C2×D59)⋊C4—42no—
192(C2×D59)⋊C4—42no—
193(C2×D59)⋊C4—81no—
194(C2×D59)⋊C4—81no—
195C4⋊Dic59—21no—
196C4⋊Dic59—21no—
197C4⋊Dic59—21no—
198C4⋊Dic59—21no—
199C4⋊Dic59—21no—
200C4⋊Dic59—21no—
201C4⋊Dic59—81no—
202C4⋊Dic59—81no—
203C4×Dic59—42no—
204C2×C59⋊C8—21no—
205C2×C59⋊C8—21no—
206C2×C59⋊C8—21no—
207C2×C59⋊C8—21no—
208C2×C59⋊C8—21no—
209C2×C59⋊C8—21no—
210C2×C59⋊C8—21no—
211C2×C59⋊C8—21no—
212C2×C59⋊C8—21no—
213C2×C59⋊C8—21no—
214C2×C59⋊C8—21no—
215C2×C59⋊C8—44no—
216C2×C59⋊C8#4112yes—
217C23×C59—21no—
218C23×C59—21no—
219C23×C59—21no—
220C23×C59—21no—
221C23×C59—21no—
222C23×C59—21no—
223C23×C59#12112yes—
224C23×C59C22×D411yescharacteristic
225C22×D59—41no—
226C22×D59—41no—
227C22×D59—41no—
228C22×D59—41no—
229C22×D59—41no—
230C22×D59—41no—
231C22×D59—41no—
232Q8×C59—22no—
233Q8×C59—22no—
234Q8×C59—41no—
235Q8×C59—41no—
236C2×C4×C59—21no—
237C2×C4×C59—21no—
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—21no—
247C2×C4×C59—22no—
248C2×C4×C59—21no—
249C2×C4×C59—21no—
250C2×C4×C59—21no—
251C2×C4×C59—21no—
252C2×C4×C59—21no—
253C2×C4×C59—21no—
254C2×C4×C59—21no—
255C2×C4×C59—21no—
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—41no—
264C2×C4×C59—41no—
265C2×C4×C59#12112yes—
266C2×C4×C59#13512yes—
267C2×C4×C59C22×D411yescharacteristic
268C2×C4×C59C2×#4111yescharacteristic
269C2×C4×C59C2×#4111yescharacteristic
270C2×C4×C59C2×#4111yescharacteristic
271C2×C4×C59H(5,𝔽2)11yescharacteristic
272C2×C4×C592−1+411yescharacteristic
273C22⋊D59—84no—
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—
283D236—41no—
284D236—41no—
285D236—41no—
286D236—41no—
287C59⋊C8—22no—
288C59⋊C8—41no—
289C59⋊C8—41no—
290C59⋊C8—41no—
291#25203—591nomaximal, Sylow
292C22×C59—22no—
293C22×C59—22no—
294C22×C59—22no—
295C22×C59—41no—
296C22×C59—41no—
297C22×C59—41no—
298C22×C59—41no—
299C22×C59C2×#12111yescharacteristic
300C22×C59#54611yescharacteristic
301C22×C59#54711yescharacteristic
302C22×C59C2×#13511yescharacteristic
303C22×C59#57511yescharacteristic
304C22×C59#57711yescharacteristic
305C22×C59#57711yescharacteristic
306C2×D59—81no—
307C2×D59—81no—
308C2×D59—81no—
309C2×D59—81no—
310C2×D59—81no—
311C2×D59—81no—
312C2×D59—81no—
313C4×C59—22no—
314C4×C59—22no—
315C4×C59—41no—
316C4×C59—41no—
317C4×C59—42no—
318C4×C59—42no—
319C4×C59—41no—
320C4×C59—42no—
321C4×C59—81no—
322C4×C59#52911yescharacteristic
323C4×C59#53911yescharacteristic
324C4×C59#54611yescharacteristic
325C4×C59#54711yescharacteristic
326Dic59—81no—
327Dic59—81no—
328Dic59—81no—
329C2×#485—591no—
330C2×#465—591no—
331#2500—591no—
332#2362D5911yescharacteristic, complemented
333#2084—591no—
334#2074—591no—
335#2067—591no—
336#2055—591no—
337#1949—591no—
338#1926—591no—
339#1880—591no—
340#1875—591no—
341#1850—591no—
342#1803—591no—
343#1735—591no—
344C2×C59—42no—
345C2×C59#242511yescharacteristic
346C2×C59#303511yescharacteristic
347C2×C59#304311yescharacteristic
348C2×C59#330111yescharacteristic
349C2×C59#330711yescharacteristic
350C2×C59#334011yescharacteristic
351C2×C59#334811yescharacteristic
352D59—161no—
353C22×SD16—591no—
354C23×C8—591no—
355C2×C22⋊Q8C2×D5911yescharacteristic
356C2×#121—591no—
357C2×C4×Q8C2×D5911yescharacteristic
358C2×#117—591no—
359C2×#117C2×D5911yescharacteristic
360#485—1184no—
361#465—1184no—
362C2×#107—591no—
363C2×#106—591no—
364C2×#106—591no—
365C2×C4⋊C8—591no—
366C2×C4⋊C8—591no—
367C2×#103—591no—
368C2×#103—591no—
369C2×#103—591no—
370C2×#102—591no—
371C2×#102—591no—
372C2×#102—591no—
373C2×C22⋊C8—591no—
374#394—1181no—
375#394—1181no—
376#391C2×D5911yescharacteristic
377#390—591no—
378#390—591no—
379#389C2×D5911yescharacteristic
380#388—21no—
381#388—21no—
382#388—1181no—
383#388—1181no—
384#387—21no—
385#384—591no—
386#382—591no—
387C4×C4⋊C4—21no—
388C4×C4⋊C4C2×D5911yescharacteristic
389C4×C22⋊C4C2×D5911yescharacteristic
390#340—591no—
391#340—591no—
392#340—591no—
393#340—591no—
394#340—591no—
395#340—591no—
396#336—591no—
397#336—591no—
398C59#2520311yescharacteristic, complemented, Sylow
399C22×Q8C22×D5911yescharacteristic
400C22×D4—591no—
401C22×D4—1181no—
402C23×C4C22×D5911yescharacteristic
403C2×SD16—1182no—
404C2×SD16—1182no—
405C2×SD16—2361no—
406C22×C8—591no—
407C22×C8—591no—
408C22×C8—591no—
409C22×C8—591no—
410C22×C8—1182no—
411C22×C8—1182no—
412C22×C8—2361no—
413C22⋊Q8—22no—
414C22⋊Q8D23614yes—
415#121—1182no—
416#121—1182no—
417C4×Q8—22no—
418C4×Q8—41no—
419#117—22no—
420#117—41no—
421#117—1182no—
422#117—1182no—
423C2×C4⋊C4—21no—
424C2×C4⋊C4—21no—
425C2×C4⋊C4—21no—
426C2×C4⋊C4—591no—
427C2×C4⋊C4—591no—
428C2×C4⋊C4—591no—
429C2×C4⋊C4—1181no—
430C2×C4⋊C4—1181no—
431C2×C4⋊C4C22×D5911yescharacteristic
432C2×C4⋊C4C22×D5911yescharacteristic
433C2×C4⋊C4C22×D5911yescharacteristic
434C2×C22⋊C4—21no—
435C2×C22⋊C4—21no—
436C2×C22⋊C4—1181no—
437C2×C22⋊C4—1181no—
438C2×C22⋊C4—1181no—
439C2×C22⋊C4—1181no—
440C2×C22⋊C4—1181no—
441C2×C22⋊C4—1181no—
442C2×C22⋊C4C22⋊D5911yescharacteristic
443C2×C22⋊C4C22⋊D5911yescharacteristic
444C2×C42—21no—
445C2×C42—21no—
446C2×C42—21no—
447C2×C42—1181no—
448C2×C42C22×D5911yescharacteristic
449C2×C42C22×D5911yescharacteristic
450#107—1182no—
451#106—1182no—
452#106—1182no—
453C4⋊C8—1182no—
454C4⋊C8—2361no—
455#103—1182no—
456#103—1182no—
457#103—1182no—
458#102—1182no—
459#102—1182no—
460#102—2361no—
461C22⋊C8—1182no—
462#95—21no—
463#95—21no—
464#95—21no—
465#95—21no—
466#95—1181no—
467#95—1181no—
468#95C22⋊D5911yescharacteristic
469#95C22⋊D5911yescharacteristic
470C24—1181no—
471C24C2×C22⋊D5911yescharacteristic
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—21no—
475C2×Q8—21no—
476C2×Q8—21no—
477C2×Q8C2×D23612yes—
478C2×D4—1181no—
479C2×D4—1181no—
480C2×D4—1181no—
481C2×D4—1181no—
482C2×D4—1181no—
483C2×D4—1181no—
484C2×D4—4721no—
485C2×D4—4721no—
486C2×D4—2362no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—1181no—
496C22×C4—1181no—
497C22×C4—1181no—
498C22×C4C2×D23612yes—
499C22×C4#1174568512yes—
500C22×C4C2×C22⋊D5911yescharacteristic
501C22×C4C2×C22⋊D5911yescharacteristic
502C22×C4C2×C22⋊D5911yescharacteristic
503C22×C4C2×C22⋊D5911yescharacteristic
504C22×C4C2×C22⋊D5911yescharacteristic
505C22×C4C23×D5911yescharacteristic
506SD16—2364no—
507C2×C8—592no—
508C2×C8—1181no—
509C2×C8—1181no—
510C2×C8—1181no—
511C2×C8—1181no—
512C2×C8—1181no—
513C2×C8—1181no—
514C2×C8—1181no—
515C2×C8—1181no—
516C2×C8—1181no—
517C2×C8—1181no—
518C2×C8—1181no—
519C2×C8—2364no—
520C4⋊C4—24no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—42no—
526C4⋊C4—42no—
527C4⋊C4—1181no—
528C4⋊C4—1181no—
529C4⋊C4—1181no—
530C4⋊C4—1181no—
531C4⋊C4—1181no—
532C4⋊C4—1181no—
533C4⋊C4—4721no—
534C4⋊C4—4721no—
535C4⋊C4C2×D23612yes—
536C4⋊C4#1174568712yes—
537C22⋊C4—24no—
538C22⋊C4—22no—
539C22⋊C4—41no—
540C22⋊C4—42no—
541C22⋊C4—4721no—
542C22⋊C4—4721no—
543C22⋊C4—2362no—
544C22⋊C4—2362no—
545C22⋊C4—2362no—
546C22⋊C4—2362no—
547C42—21no—
548C42—21no—
549C42—21no—
550C42—21no—
551C42—42no—
552C42—42no—
553C42—42no—
554C42—2362no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—2361no—
562C23—2361no—
563C23—2361no—
564C23—2361no—
565C23—2361no—
566C23—2361no—
567C23—2361no—
568C23#1427370112yes—
569C23C22×C22⋊D5911yescharacteristic
570Q8—22no—
571Q8—22no—
572Q8—41no—
573Q8—41no—
574D4—4724no—
575D4—2361no—
576D4—2361no—
577D4—2361no—
578D4—2361no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—22no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
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#1427370112yes—
618C2×C4#1427374712yes—
619C2×C4C22×D23611yescharacteristic
620C2×C4#41×D5911yescharacteristic
621C2×C4C2×#1174568711yescharacteristic
622C2×C4C2×#1174568511yescharacteristic
623C2×C4#1427377211yescharacteristic
624C2×C4#1427377411yescharacteristic
625C8—1182no—
626C8—2361no—
627C8—2361no—
628C8—2361no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—41no—
633C22—41no—
634C22—41no—
635C22—41no—
636C22—4721no—
637C22—4721no—
638C22—4721no—
639C22—4721no—
640C22—4721no—
641C22—4721no—
642C22—4721no—
643C22C2×#1427370111yescharacteristic
644C22#1683804611yescharacteristic
645C22#1683805111yescharacteristic
646C22C2×#1427374711yescharacteristic
647C22#1683820011yescharacteristic
648C22#1683821011yescharacteristic
649C22#1683821011yescharacteristic
650C4—22no—
651C4—22no—
652C4—41no—
653C4—41no—
654C4—42no—
655C4—42no—
656C4—41no—
657C4—42no—
658C4—81no—
659C4—4721no—
660C4—4721no—
661C4—4721no—
662C4#1683793211yescharacteristic
663C4#1683799411yescharacteristic
664C4#1683804711yescharacteristic
665C4#1683805611yescharacteristic
666C2—42no—
667C2—9441no—
668C2#2142764111yescharacteristic
669C2#2143355111yescharacteristic
670C2#2143361711yescharacteristic
671C2#2143599811yescharacteristic
672C2#2143607211yescharacteristic
673C2#2143650311yescharacteristic
674C2#2143660511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2362C2
socleC23×C59C22×D4
prime radicalC2×#14273701C22
hypercenter#2362D59
hypocenterC59#25203
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2362

Complements

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

NH ≅ G/Ntype
1. C2×#16837740C2semidirect
4. #21425572C2semidirect
5. #21425514C2semidirect
6. #21425481C2semidirect
7. #21425411C2semidirect
8. #21424864C2semidirect
9. #21424723C2semidirect
10. #21424451C2semidirect
12. #21424276C2semidirect
13. #21424101C2semidirect
14. #21423812C2semidirect
15. C59×#2362C2semidirect
332. #2362D59semidirect
398. C59#25203semidirect

Maximal subgroups

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

subgroupindex
C2×#168377402
C2×#168376332
#214285582
#214255722
#214255142
#214254812
#214254112
#214248642
#214247232
#214244512
#214244182
#214242762
#214241012
#214238122
C59×#23622
#2520359