Finite Group Databasebeta

← #28495680

Subgroups — #28495680

Order 15104.

910 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837420C211yesmaximal, characteristic
2C2×#16837401C211yesmaximal, characteristic
3C2×#16837375C211yesmaximal, characteristic
4C2×#16837172C211yesmaximal, characteristic, complemented
5#21423815C211yesmaximal, characteristic, complemented
6C2×#16837158C211yesmaximal, characteristic, complemented
7C59×#2316C211yesmaximal, characteristic, complemented
8C22×#11745685C2211yescharacteristic
9C22×#11745664C2211yescharacteristic
10C22×#11745664C2211yescharacteristic
11C2×#14273686C2211yescharacteristic
12#16837420—22no—
13#16837420—22no—
14C2×#14273655—22no—
15#16837401—22no—
16#16837401—22no—
17C2×#14273650C412yescomplemented
18C2×#14273647C412yescomplemented
19#16837375—22no—
20#16837375—22no—
21C2×#14273641—22no—
22#16837172—22no—
23#16837164—22no—
24#16837158—22no—
25C2×#117×C59C2211yescharacteristic
26C2×#117×C59C2211yescharacteristic
27C22×C4⋊C4×C59C2211yescharacteristic
28C59×#389—22no—
29C59×#389—22no—
30C59×#380—22no—
31C2×#11745685—21no—
32C2×#11745685—21no—
33C2×#11745685—21no—
34C2×#11745685—21no—
35C2×#11745685—21no—
36C2×#11745685—21no—
37C2×#11745685—21no—
38C2×#11745685—21no—
39C2×#11745685—21no—
40C2×#11745685—21no—
41C2×#11745685D412yes—
42C2×#11745685Q812yes—
43C22×Dic118C2×C411yescharacteristic
44C22×D236C2×C411yescharacteristic
45C22×C22⋊D59—21no—
46C22×C4×D59—21no—
47C2×#11745664—21no—
48C2×#11745664—21no—
49C2×#11745664—21no—
50C2×#11745664—21no—
51C2×#11745664—21no—
52C2×#11745664—21no—
53C2×#11745664—21no—
54C2×#11745664—21no—
55C2×#11745664—21no—
56C2×#11745664—21no—
57C2×#11745664—21no—
58C2×#11745664D412yes—
59C22×C59⋊C8—21no—
60C22×C59⋊C8C2×C412yes—
61#14273686—21no—
62#14273686—21no—
63#14273686—21no—
64#14273686D412yes—
65C2×C4⋊Dic59—21no—
66C2×C22⋊Dic59—21no—
67C2×C4×Dic59—21no—
68#14273655—28no—
69#14273650—24no—
70#14273647—24no—
71#14273641—28no—
72C23×C4×C59—21no—
73C23×C4×C59C2311yescharacteristic
74#117×C59—21no—
75#117×C59—21no—
76#117×C59—21no—
77#117×C59—21no—
78#117×C59—21no—
79#117×C59—21no—
80#117×C59—21no—
81#117×C59—21no—
82C2×C4⋊C4×C59—21no—
83C2×C4⋊C4×C59—21no—
84C2×C4⋊C4×C59—21no—
85C2×C4⋊C4×C59—21no—
86C2×C4⋊C4×C59—21no—
87C2×C4⋊C4×C59—24no—
88C2×C4⋊C4×C59C2×C412yes—
89C2×C4⋊C4×C59D412yes—
90C2×C4⋊C4×C59D412yes—
91C2×C22⋊C4×C59—21no—
92C2×C22⋊C4×C59D412yes—
93C2×C42×C59—21no—
94C2×C42×C59D412yes—
95#95×C59—24no—
96C24×C59C2×D411yescharacteristic
97C23×D59—21no—
98C22×C4×C59—22no—
99C22×C4×C59—22no—
100C22×C4×C59—22no—
101C22×C4×C59—22no—
102C22×C4×C59—22no—
103C22×C4×C59—82no—
104C22×C4×C59C22×C411yescharacteristic
105C22×C4×C59C2×D411yescharacteristic
106C22×C4×C59C2×D411yescharacteristic
107C22×C4×C59C2×D411yescharacteristic
108C22×C4×C59C2×D411yescharacteristic
109C22×C4×C59C2×D411yescharacteristic
110C22×C4×C59C2×D411yescharacteristic
111C22×C4×C59C2×Q811yescharacteristic
112C22×C4×C59#4111yescharacteristic
113C22×C4×C59#4111yescharacteristic
114C22×C4×C59#4111yescharacteristic
115C22×C4×C59#4111yescharacteristic
116C22×C4×C59#4111yescharacteristic
117C22×C4×C59#4111yescharacteristic
118C2×C22⋊D59—22no—
119C2×C22⋊D59—22no—
120C2×C22⋊D59—41no—
121C2×C22⋊D59—41no—
122C2×C22⋊D59—41no—
123C2×C22⋊D59—41no—
124C22×Dic59—21no—
125C22×Dic59—41no—
126#11745685—22no—
127#11745685—22no—
128#11745685—22no—
129#11745685—22no—
130#11745685—22no—
131#11745685—22no—
132#11745685—22no—
133#11745685—22no—
134#11745685—41no—
135#11745685—41no—
136#11745685—41no—
137#11745685—41no—
138#11745685—41no—
139#11745685—41no—
140#11745685—41no—
141#11745685—41no—
142C2×D236—21no—
143C2×D236—21no—
144C2×D236—21no—
145C2×D236—21no—
146C2×D236—21no—
147C2×D236C4⋊C412yescomplemented
148C2×C4×D59—22no—
149C2×C4×D59—22no—
150C2×C4×D59—41no—
151C2×C4×D59—41no—
152C2×C4×D59—41no—
153C2×C4×D59—41no—
154C2×Dic118—21no—
155C2×Dic118—21no—
156C2×Dic118—21no—
157C2×Dic118—21no—
158C2×Dic118—21no—
159C2×Dic118C4⋊C412yescomplemented
160C4⋊C4×C59—22no—
161C4⋊C4×C59—22no—
162C4⋊C4×C59—22no—
163C4⋊C4×C59—41no—
164C4⋊C4×C59—41no—
165C4⋊C4×C59—41no—
166C4⋊C4×C59—41no—
167C4⋊C4×C59—84no—
168C4⋊C4×C59C22⋊C414yes—
169C22⋊C4×C59—22no—
170C22⋊C4×C59—22no—
171C22⋊C4×C59—41no—
172C22⋊C4×C59—41no—
173C42×C59—22no—
174C42×C59—22no—
175C42×C59—41no—
176C42×C59—41no—
177C22⋊Dic59—22no—
178C22⋊Dic59—41no—
179C4⋊Dic59—22no—
180C4⋊Dic59—41no—
181C4×Dic59—22no—
182C4×Dic59—41no—
183#11745664—22no—
184#11745664—22no—
185#11745664—41no—
186#11745664—41no—
187#11745664—41no—
188#11745664—41no—
189#11745664—41no—
190#11745664—41no—
191C2×C59⋊C8—22no—
192C2×C59⋊C8—22no—
193C2×C59⋊C8—22no—
194C2×C59⋊C8—22no—
195C2×C59⋊C8—41no—
196C2×C59⋊C8—41no—
197C23×C59—41no—
198C23×C59—41no—
199C23×C59C4×D411yescharacteristic
200C23×C59#12011yescharacteristic
201C23×C59#12011yescharacteristic
202C23×C59#12111yescharacteristic
203C23×C59#12111yescharacteristic
204C23×C59C22⋊Q811yescharacteristic
205C23×C59#12411yescharacteristic
206C22×D59—22no—
207C22×D59—41no—
208C22×D59—41no—
209C22×D59—81no—
210C22×D59—81no—
211C2×C4×C59—24no—
212C2×C4×C59—44no—
213C2×C4×C59—42no—
214C2×C4×C59—42no—
215C2×C4×C59—42no—
216C2×C4×C59—42no—
217C2×C4×C59—42no—
218C2×C4×C59—42no—
219C2×C4×C59—82no—
220C2×C4×C59—82no—
221C2×C4×C59C2×C22⋊C411yescharacteristic
222C2×C4×C59C2×C4⋊C411yescharacteristic
223C2×C4×C59#11711yescharacteristic
224C2×C4×C59C4×D411yescharacteristic
225C2×C4×C59C4×D411yescharacteristic
226C2×C4×C59C4×D411yescharacteristic
227C2×C4×C59#12011yescharacteristic
228C2×C4×C59#12111yescharacteristic
229C2×C4×C59#12111yescharacteristic
230C2×C4×C59#12111yescharacteristic
231C2×C4×C59#12111yescharacteristic
232C2×C4×C59#12111yescharacteristic
233C2×C4×C59#12111yescharacteristic
234C2×C4×C59C22⋊Q811yescharacteristic
235C2×C4×C59C22⋊Q811yescharacteristic
236C2×C4×C59C22⋊Q811yescharacteristic
237C2×C4×C59C22⋊Q811yescharacteristic
238C2×C4×C59C22⋊Q811yescharacteristic
239C2×C4×C59#12311yescharacteristic
240C2×C4×C59#12311yescharacteristic
241C2×C4×C59#12311yescharacteristic
242C2×C4×C59#12311yescharacteristic
243C2×C4×C59#12311yescharacteristic
244C2×C4×C59#12411yescharacteristic
245C2×C4×C59#12411yescharacteristic
246C2×C4×C59#12611yescharacteristic
247C2×C4×C59#12611yescharacteristic
248C2×C4×C59#12711yescharacteristic
249C22⋊D59—42no—
250C22⋊D59—42no—
251C22⋊D59—42no—
252C22⋊D59—42no—
253C2×Dic59—22no—
254C2×Dic59—41no—
255C2×Dic59—41no—
256C2×Dic59—42no—
257C2×Dic59—81no—
258C2×Dic59—81no—
259D236—22no—
260D236—22no—
261D236—41no—
262D236—41no—
263C4×D59—42no—
264C4×D59—42no—
265C4×D59—42no—
266C4×D59—42no—
267Dic118—22no—
268Dic118—22no—
269Dic118—41no—
270Dic118—41no—
271C59⋊C8—42no—
272C59⋊C8—42no—
273#11057—591nomaximal, Sylow
274C22×C59—21no—
275C22×C59—21no—
276C22×C59—21no—
277C22×C59—21no—
278C22×C59—21no—
279C22×C59—21no—
280C22×C59—41no—
281C22×C59—41no—
282C22×C59—41no—
283C22×C59—41no—
284C22×C59#38511yescharacteristic
285C22×C59#38611yescharacteristic
286C22×C59#38811yescharacteristic
287C22×C59#39211yescharacteristic
288C22×C59#39311yescharacteristic
289C22×C59#39411yescharacteristic
290C22×C59#39611yescharacteristic
291C2×D59—42no—
292C2×D59—82no—
293C2×D59—81no—
294C2×D59—81no—
295C2×D59—82no—
296C4×C59—44no—
297C4×C59—82no—
298C4×C59—82no—
299C4×C59#38011yescharacteristic
300C4×C59#38511yescharacteristic
301C4×C59#38811yescharacteristic
302C4×C59#39011yescharacteristic
303C4×C59#39411yescharacteristic
304C4×C59#39611yescharacteristic
305C4×C59#39711yescharacteristic
306C4×C59#39911yescharacteristic
307Dic59—42no—
308Dic59—82no—
309C2×#421—591no—
310C2×#417—591no—
311C2×#410—591no—
312#2316D5911yescharacteristic, complemented
313C2×#344—591no—
314#1736—591no—
315C2×H(3,ℤ/4)—591no—
316C2×C59—21no—
317C2×C59—21no—
318C2×C59—21no—
319C2×C59—21no—
320C2×C59—21no—
321C2×C59—21no—
322C2×C59#143911yescharacteristic
323C2×C59#190411yescharacteristic
324C2×C59#190511yescharacteristic
325D59—82no—
326C22×#41—591no—
327C22×M16—591no—
328C22×M16—591no—
329C2×#117—591no—
330C2×#117C2×D5911yescharacteristic
331C2×#117C2×D5911yescharacteristic
332C22×C4⋊C4C2×D5911yescharacteristic
333#421—1182no—
334#421—1182no—
335C2×#104—1182no—
336#417—1182no—
337#417—1182no—
338C2×#103—592no—
339C2×#102—592no—
340#410—1182no—
341#410—1182no—
342C2×#99—1182no—
343#389—22no—
344#389—22no—
345#380—22no—
346#344—1182no—
347#340—1182no—
348H(3,ℤ/4)—1182no—
349C59#1105711yescharacteristic, complemented, Sylow
350C2×#41—592no—
351C2×#41—592no—
352C2×#41—1181no—
353C2×#41—1181no—
354C2×#41—1181no—
355C2×#41—1181no—
356C2×#41—1181no—
357C2×#41—1181no—
358C2×#41—1181no—
359C2×#41—1181no—
360C2×#41—1181no—
361C2×#41—1181no—
362C22×Q8—591no—
363C22×D4—591no—
364C22×D4—1181no—
365C23×C4—21no—
366C23×C4—1181no—
367C23×C4C22×D5911yescharacteristic
368C2×M16—592no—
369C2×M16—1181no—
370C2×M16—1181no—
371C2×M16—1181no—
372C2×M16—1181no—
373C2×M16—1181no—
374C2×M16—1181no—
375C2×M16—1181no—
376C2×M16—1181no—
377C2×M16—1181no—
378C2×M16—1181no—
379C2×M16—1181no—
380C22×C8—592no—
381C22×C8—1181no—
382#117—21no—
383#117—21no—
384#117—21no—
385#117—21no—
386#117—21no—
387#117—21no—
388#117—21no—
389#117—21no—
390#117—592no—
391#117—1181no—
392#117—1181no—
393#117—1181no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—24no—
400C2×C4⋊C4—1181no—
401C2×C4⋊C4C4×D5912yes—
402C2×C4⋊C4D23612yes—
403C2×C4⋊C4C22⋊D5912yes—
404C2×C22⋊C4—21no—
405C2×C22⋊C4—1181no—
406C2×C22⋊C4C22⋊D5912yes—
407C2×C42—21no—
408C2×C42—1181no—
409C2×C42C22⋊D5912yes—
410#104—1188no—
411#103—1184no—
412#102—1184no—
413#99—1188no—
414#95—24no—
415C24—1181no—
416C24C2×C22⋊D5911yescharacteristic
417#41—1182no—
418#41—1182no—
419#41—1182no—
420#41—1182no—
421#41—1182no—
422#41—1182no—
423#41—1182no—
424#41—1182no—
425#41—2361no—
426#41—2361no—
427#41—2361no—
428#41—2361no—
429#41—2361no—
430#41—2361no—
431#41—2361no—
432#41—2361no—
433C2×Q8—592no—
434C2×Q8—1181no—
435C2×Q8—1181no—
436C2×Q8—1181no—
437C2×Q8—1181no—
438C2×Q8—1181no—
439C2×D4—592no—
440C2×D4—1182no—
441C2×D4—1182no—
442C2×D4—1181no—
443C2×D4—1181no—
444C2×D4—1181no—
445C2×D4—1181no—
446C2×D4—1181no—
447C2×D4—2361no—
448C2×D4—2361no—
449C2×D4—2361no—
450C2×D4—2361no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—82no—
457C22×C4—1181no—
458C22×C4—1182no—
459C22×C4—1182no—
460C22×C4—2361no—
461C22×C4—2361no—
462C22×C4—2361no—
463C22×C4—2361no—
464C22×C4—2361no—
465C22×C4C2×C4×D5911yescharacteristic
466C22×C4C2×D23611yescharacteristic
467C22×C4#1174568511yescharacteristic
468C22×C4#1174568511yescharacteristic
469C22×C4D4×D5911yescharacteristic
470C22×C4D4×D5911yescharacteristic
471C22×C4D4×D5911yescharacteristic
472C22×C4#1174568711yescharacteristic
473C22×C4Q8×D5911yescharacteristic
474C22×C4#1174568911yescharacteristic
475C22×C4#1174568911yescharacteristic
476C22×C4#1174568911yescharacteristic
477C22×C4C2×C22⋊D5911yescharacteristic
478C22×C4C2×C22⋊D5911yescharacteristic
479M16—1182no—
480M16—1182no—
481M16—2361no—
482M16—2361no—
483M16—2361no—
484M16—2361no—
485M16—2361no—
486M16—2361no—
487C2×C8—1182no—
488C2×C8—1182no—
489C2×C8—1182no—
490C2×C8—1182no—
491C2×C8—2361no—
492C2×C8—2361no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—41no—
497C4⋊C4—41no—
498C4⋊C4—41no—
499C4⋊C4—41no—
500C4⋊C4—84no—
501C4⋊C4—1182no—
502C4⋊C4—2361no—
503C4⋊C4(C2×D59)⋊C414yes—
504C22⋊C4—22no—
505C22⋊C4—22no—
506C22⋊C4—41no—
507C22⋊C4—41no—
508C22⋊C4—1182no—
509C22⋊C4—2361no—
510C42—22no—
511C42—22no—
512C42—41no—
513C42—41no—
514C42—1182no—
515C42—2361no—
516C23—41no—
517C23—41no—
518C23—1182no—
519C23—4721no—
520C23—4721no—
521C23—2361no—
522C23—2361no—
523C23C4×C22⋊D5911yescharacteristic
524C23#1427369511yescharacteristic
525C23#1427369611yescharacteristic
526C23#1427369911yescharacteristic
527C23#1427370111yescharacteristic
528C23#1427370311yescharacteristic
529C23#1427371611yescharacteristic
530Q8—1182no—
531Q8—1182no—
532Q8—2361no—
533Q8—2361no—
534D4—1182no—
535D4—1182no—
536D4—2362no—
537D4—2362no—
538D4—2362no—
539D4—2362no—
540D4—2361no—
541D4—2361no—
542C2×C4—24no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—82no—
551C2×C4—82no—
552C2×C4—1182no—
553C2×C4—4721no—
554C2×C4—4721no—
555C2×C4—2362no—
556C2×C4—2361no—
557C2×C4—2361no—
558C2×C4—2362no—
559C2×C4—2362no—
560C2×C4—2362no—
561C2×C4—2362no—
562C2×C4C2×(C2×D59)⋊C411yescharacteristic
563C2×C4C4⋊C4×D5911yescharacteristic
564C2×C4#1427368511yescharacteristic
565C2×C4C4×C22⋊D5911yescharacteristic
566C2×C4#1427368911yescharacteristic
567C2×C4#1427368911yescharacteristic
568C2×C4#1427369511yescharacteristic
569C2×C4#1427369811yescharacteristic
570C2×C4#1427369911yescharacteristic
571C2×C4#1427370111yescharacteristic
572C2×C4#1427370211yescharacteristic
573C2×C4#1427370211yescharacteristic
574C2×C4#1427370211yescharacteristic
575C2×C4#1427370311yescharacteristic
576C2×C4#1427370311yescharacteristic
577C2×C4#1427370511yescharacteristic
578C2×C4#1427370511yescharacteristic
579C2×C4#1427370711yescharacteristic
580C2×C4#1427371011yescharacteristic
581C2×C4#1427371211yescharacteristic
582C2×C4#1427371211yescharacteristic
583C2×C4#1427371211yescharacteristic
584C2×C4#1427371211yescharacteristic
585C2×C4#1427371611yescharacteristic
586C2×C4#1427371611yescharacteristic
587C2×C4#1427372211yescharacteristic
588C2×C4#1427372211yescharacteristic
589C2×C4#1427372311yescharacteristic
590C8—2362no—
591C8—2362no—
592C22—21no—
593C22—21no—
594C22—21no—
595C22—21no—
596C22—21no—
597C22—21no—
598C22—41no—
599C22—41no—
600C22—41no—
601C22—41no—
602C22—4722no—
603C22—4722no—
604C22—4721no—
605C22—4721no—
606C22—2362no—
607C22#1683727311yescharacteristic
608C22#1683728111yescharacteristic
609C22#1683729211yescharacteristic
610C22#1683731011yescharacteristic
611C22#1683731311yescharacteristic
612C22#1683731911yescharacteristic
613C22#1683732411yescharacteristic
614C4—44no—
615C4—82no—
616C4—82no—
617C4—4722no—
618C4—2362no—
619C4#1683725911yescharacteristic
620C4#1683727311yescharacteristic
621C4#1683729211yescharacteristic
622C4#1683730311yescharacteristic
623C4#1683731911yescharacteristic
624C4#1683732411yescharacteristic
625C4#1683732811yescharacteristic
626C4#1683733711yescharacteristic
627C2—21no—
628C2—21no—
629C2—21no—
630C2—21no—
631C2—21no—
632C2—21no—
633C2—4722no—
634C2#2142233811yescharacteristic
635C2#2142458311yescharacteristic
636C2#2142459011yescharacteristic
subgroupG/H
centerC22#16837281
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC59×#2316C2
socleC22×C59#386
prime radicalC22×#11745685C22
cosocleC23×C4×C59C23
hypercenter#2316D59
hypocenterC59#11057
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2316

Complements

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

NH ≅ G/Ntype
4. C2×#16837172C2semidirect
5. #21423815C2semidirect
6. C2×#16837158C2semidirect
7. C59×#2316C2semidirect
17. C2×#14273650C4semidirect
18. C2×#14273647C4semidirect
147. C2×D236C4⋊C4semidirect
159. C2×Dic118C4⋊C4semidirect
312. #2316D59semidirect
349. C59#11057semidirect

Maximal subgroups

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

subgroupindex
C2×#168374202
C2×#168374012
C2×#168373752
C2×#168371722
#214238152
C2×#168371582
C59×#23162
#1105759