Finite Group Databasebeta

← #28496525

Subgroups — #28496525

Order 15104.

866 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837404C211yesmaximal, characteristic
2C2×#16837401C211yesmaximal, characteristic
3C2×#16837370C211yesmaximal, characteristic
4C2×#16837170C211yesmaximal, characteristic, complemented
5C2×#16837166C211yesmaximal, characteristic, complemented
6C2×#16837164C211yesmaximal, characteristic, complemented
7C2×C59×#343C211yesmaximal, characteristic, complemented
8C22×#11745685C2211yescharacteristic
9C22×#11745664C2211yescharacteristic
10C23×C59⋊C8C2211yescharacteristic
11C22×C4⋊Dic59C2211yescharacteristic
12#16837404—22no—
13#16837404—22no—
14#16837401—22no—
15#16837401C414yescomplemented
16C2×#14273652—21no—
17C2×#14273650—21no—
18C2×#14273649—21no—
19C2×#14273647—21no—
20#16837370—22no—
21#16837370—22no—
22C2×(C2×D59)⋊C8—22no—
23#16837170—22no—
24#16837166—22no—
25#16837164—22no—
26C22×M16×C59C2211yescharacteristic
27C22×M16×C59C2211yescharacteristic
28C22×C4⋊C4×C59C2211yescharacteristic
29C59×#343—22no—
30C2×#11745685—21no—
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×#11745685C2×C412yes—
41C2×#11745685D412yes—
42C22×Dic118D411yescharacteristic
43C22×D236D411yescharacteristic
44C22×C22⋊D59—21no—
45C23×Dic59—21no—
46C22×C4×D59—21no—
47C2×#11745664—21no—
48C2×#11745664—21no—
49C2×#11745664—21no—
50C2×#11745664—21no—
51C2×#11745664—21no—
52C2×#11745664C2×C412yes—
53C22×C59⋊C8—21no—
54C22×C59⋊C8—21no—
55C22×C59⋊C8—21no—
56C22×C59⋊C8—21no—
57C22×C59⋊C8—21no—
58C22×C59⋊C8—21no—
59C22×C59⋊C8—21no—
60C22×C59⋊C8D412yes—
61C2×C4⋊Dic59—21no—
62C2×C4⋊Dic59—21no—
63C2×C4⋊Dic59—21no—
64C2×C4⋊Dic59—21no—
65C2×C4⋊Dic59—21no—
66C2×C4⋊Dic59D412yes—
67#14273652—24no—
68#14273650—24no—
69#14273649—24no—
70#14273647—24no—
71(C2×D59)⋊C8—28no—
72C23×C4×C59—21no—
73C23×C4×C59C2311yescharacteristic
74C2×M16×C59—21no—
75C2×M16×C59—21no—
76C2×M16×C59—21no—
77C2×M16×C59—21no—
78C2×M16×C59—21no—
79C2×M16×C59—21no—
80C2×M16×C59—21no—
81C2×M16×C59—21no—
82C2×M16×C59—21no—
83C2×M16×C59—21no—
84C2×M16×C59D412yes—
85C2×M16×C59D412yes—
86C22×C8×C59—21no—
87C22×C8×C59—21no—
88C2×C4⋊C4×C59—21no—
89C2×C4⋊C4×C59—21no—
90C2×C4⋊C4×C59—21no—
91C2×C4⋊C4×C59—21no—
92C2×C4⋊C4×C59—21no—
93C2×C4⋊C4×C59C2×C412yes—
94C24×C59#4111yescharacteristic
95C23×D59—21no—
96C22×C4×C59—22no—
97C22×C4×C59—41no—
98C22×C4×C59—41no—
99C22×C4×C59—81no—
100C22×C4×C59—81no—
101C22×C4×C59C22×C411yescharacteristic
102C22×C4×C59C2×D411yescharacteristic
103C22×C4×C59C2×D411yescharacteristic
104C22×C4×C59C2×D411yescharacteristic
105C22×C4×C59C2×D411yescharacteristic
106C22×C4×C59C2×D411yescharacteristic
107C22×C4×C59C2×D411yescharacteristic
108C22×C4×C59#4111yescharacteristic
109C22×C4×C59#4111yescharacteristic
110C22×C4×C59#4111yescharacteristic
111C22×C4×C59#4111yescharacteristic
112C22×C4×C59#4111yescharacteristic
113C22×C4×C59#4111yescharacteristic
114C22×C4×C59#4111yescharacteristic
115C2×C22⋊D59—22no—
116C2×C22⋊D59—22no—
117C2×C22⋊D59—41no—
118C2×C22⋊D59—41no—
119C2×C22⋊D59—41no—
120C2×C22⋊D59—41no—
121C22×Dic59—21no—
122C22×Dic59—22no—
123C22×Dic59—41no—
124C22×Dic59—41no—
125C22×Dic59—81no—
126C22×Dic59—81no—
127#11745685—22no—
128#11745685—22no—
129#11745685—22no—
130#11745685—22no—
131#11745685—22no—
132#11745685—22no—
133#11745685—22no—
134#11745685—22no—
135#11745685—41no—
136#11745685—41no—
137#11745685—41no—
138#11745685—41no—
139#11745685—41no—
140#11745685—41no—
141#11745685—41no—
142#11745685—41no—
143C2×D236—21no—
144C2×D236—21no—
145C2×D236—21no—
146C2×D236—21no—
147C2×D236—21no—
148C2×D236C22⋊C412yes—
149C2×C4×D59—22no—
150C2×C4×D59—22no—
151C2×C4×D59—41no—
152C2×C4×D59—41no—
153C2×C4×D59—41no—
154C2×C4×D59—41no—
155C2×Dic118—21no—
156C2×Dic118—21no—
157C2×Dic118—21no—
158C2×Dic118—21no—
159C2×Dic118—21no—
160C2×Dic118C22⋊C412yes—
161M16×C59—22no—
162M16×C59—22no—
163M16×C59—22no—
164M16×C59—22no—
165M16×C59—41no—
166M16×C59—41no—
167M16×C59—41no—
168M16×C59—41no—
169C2×C8×C59—22no—
170C2×C8×C59—22no—
171C2×C8×C59—41no—
172C2×C8×C59—41no—
173C2×C8×C59—41no—
174C2×C8×C59—41no—
175C4⋊C4×C59—22no—
176C4⋊C4×C59—22no—
177C4⋊C4×C59—41no—
178C4⋊C4×C59—41no—
179C4⋊Dic59—22no—
180C4⋊Dic59—22no—
181C4⋊Dic59—41no—
182C4⋊Dic59—41no—
183#11745664—22no—
184#11745664—22no—
185#11745664—41no—
186#11745664—41no—
187C2×C59⋊C8—22no—
188C2×C59⋊C8—22no—
189C2×C59⋊C8—22no—
190C2×C59⋊C8—22no—
191C2×C59⋊C8—41no—
192C2×C59⋊C8—41no—
193C2×C59⋊C8—41no—
194C2×C59⋊C8—41no—
195C2×C59⋊C8—41no—
196C2×C59⋊C8—41no—
197C23×C59—21no—
198C23×C59—21no—
199C23×C59—21no—
200C23×C59—21no—
201C23×C59C4×D411yescharacteristic
202C23×C59#12111yescharacteristic
203C23×C59#12311yescharacteristic
204C23×C59#12311yescharacteristic
205C23×C59#12411yescharacteristic
206C23×C59#12611yescharacteristic
207C23×C59#12611yescharacteristic
208C22×D59—22no—
209C22×D59—41no—
210C22×D59—41no—
211C22×D59—41no—
212C22×D59—41no—
213C22×D59—41no—
214C22×D59—41no—
215C2×C4×C59—42no—
216C2×C4×C59—82no—
217C2×C4×C59—81no—
218C2×C4×C59—81no—
219C2×C4×C59—82no—
220C2×C4×C59C2×C22⋊C411yescharacteristic
221C2×C4×C59#11711yescharacteristic
222C2×C4×C59#11711yescharacteristic
223C2×C4×C59C4×D411yescharacteristic
224C2×C4×C59C4×D411yescharacteristic
225C2×C4×C59C4×D411yescharacteristic
226C2×C4×C59#12011yescharacteristic
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×C59#12111yescharacteristic
235C2×C4×C59#12311yescharacteristic
236C2×C4×C59#12311yescharacteristic
237C2×C4×C59#12311yescharacteristic
238C2×C4×C59#12311yescharacteristic
239C2×C4×C59#12311yescharacteristic
240C2×C4×C59#12311yescharacteristic
241C2×C4×C59#12311yescharacteristic
242C2×C4×C59#12311yescharacteristic
243C2×C4×C59#12411yescharacteristic
244C2×C4×C59#12411yescharacteristic
245C2×C4×C59#12411yescharacteristic
246C2×C4×C59#12611yescharacteristic
247C2×C4×C59#12611yescharacteristic
248C22⋊D59—42no—
249C22⋊D59—42no—
250C22⋊D59—42no—
251C22⋊D59—42no—
252C2×Dic59—22no—
253C2×Dic59—41no—
254C2×Dic59—41no—
255C2×Dic59—42no—
256C2×Dic59—82no—
257C2×Dic59—81no—
258C2×Dic59—81no—
259C2×Dic59—82no—
260D236—22no—
261D236—22no—
262D236—41no—
263D236—41no—
264C4×D59—42no—
265C4×D59—42no—
266C4×D59—42no—
267C4×D59—42no—
268Dic118—22no—
269Dic118—22no—
270Dic118—41no—
271Dic118—41no—
272C8×C59—42no—
273C8×C59—42no—
274C59⋊C8—42no—
275C59⋊C8—42no—
276#11192—591nomaximal, Sylow
277C22×C59—21no—
278C22×C59—21no—
279C22×C59—21no—
280C22×C59—21no—
281C22×C59—21no—
282C22×C59—21no—
283C22×C59—21no—
284C22×C59—21no—
285C22×C59—21no—
286C22×C59—21no—
287C22×C59—21no—
288C22×C59—21no—
289C22×C59#38611yescharacteristic
290C22×C59#38811yescharacteristic
291C22×C59#38811yescharacteristic
292C22×C59#39411yescharacteristic
293C22×C59#39711yescharacteristic
294C22×C59#39711yescharacteristic
295C22×C59#39711yescharacteristic
296C22×C59#47111yescharacteristic
297C22×C59#47111yescharacteristic
298C22×C59#47211yescharacteristic
299C22×C59#47311yescharacteristic
300C2×D59—42no—
301C2×D59—42no—
302C2×D59—42no—
303C2×D59—81no—
304C2×D59—81no—
305C2×D59—81no—
306C2×D59—81no—
307C4×C59—82no—
308C4×C59#38111yescharacteristic
309C4×C59#38611yescharacteristic
310C4×C59#38811yescharacteristic
311C4×C59#38811yescharacteristic
312C4×C59#39411yescharacteristic
313C4×C59#39411yescharacteristic
314C4×C59#39411yescharacteristic
315C4×C59#39711yescharacteristic
316Dic59—42no—
317Dic59—82no—
318C2×#417—591no—
319C2×#417—591no—
320C2×#408—591no—
321C2×#343—591no—
322C2×#343D5911yescharacteristic, complemented
323C2×#341—591no—
324C2×#340—591no—
325C2×C59—21no—
326C2×C59—21no—
327C2×C59—21no—
328C2×C59—21no—
329C2×C59#144011yescharacteristic
330C2×C59#190711yescharacteristic
331C2×C59#190811yescharacteristic
332C2×C59#205111yescharacteristic
333C2×C59#205211yescharacteristic
334C2×C59#207811yescharacteristic
335C2×C59#207911yescharacteristic
336D59—82no—
337C22×#41—591no—
338C22×M16—591no—
339C22×M16C2×D5911yescharacteristic
340C22×M16C2×D5911yescharacteristic
341C23×C8—591no—
342C22×C4⋊C4—591no—
343C22×C4⋊C4C2×D5911yescharacteristic
344#417—594no—
345#417—1182no—
346#417—1182no—
347#417—1182no—
348C2×#103—1181no—
349C2×#103—1181no—
350C2×#102—1181no—
351C2×#102—1181no—
352#408—1182no—
353#408—1182no—
354C2×C22⋊C8—1182no—
355#343—22no—
356#343—1182no—
357#341—1182no—
358#340—1182no—
359C59#1119211yescharacteristic, complemented, Sylow
360C2×#41—592no—
361C2×#41—592no—
362C2×#41—1181no—
363C2×#41—1181no—
364C2×#41—1181no—
365C2×#41—1181no—
366C2×#41—1181no—
367C2×#41—1181no—
368C2×#41—1181no—
369C2×#41—1181no—
370C2×#41—1181no—
371C2×#41—1181no—
372C22×Q8—591no—
373C22×D4—591no—
374C22×D4—1181no—
375C23×C4—21no—
376C23×C4—1181no—
377C23×C4—1181no—
378C23×C4C22×D5911yescharacteristic
379C2×M16—21no—
380C2×M16—21no—
381C2×M16—21no—
382C2×M16—21no—
383C2×M16—21no—
384C2×M16—21no—
385C2×M16—21no—
386C2×M16—21no—
387C2×M16—21no—
388C2×M16—21no—
389C2×M16—592no—
390C2×M16—1181no—
391C2×M16—1181no—
392C2×M16—1181no—
393C2×M16—1181no—
394C2×M16—1181no—
395C2×M16D23612yes—
396C2×M16C22⋊D5912yes—
397C22×C8—21no—
398C22×C8—21no—
399C22×C8—592no—
400C22×C8—1181no—
401C22×C8—1181no—
402C22×C8—1181no—
403C22×C8—1181no—
404C22×C8—1181no—
405C22×C8—1181no—
406C22×C8—1181no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—21no—
410C2×C4⋊C4—21no—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—592no—
413C2×C4⋊C4—1181no—
414C2×C4⋊C4—1181no—
415C2×C4⋊C4—1181no—
416C2×C4⋊C4—1181no—
417C2×C4⋊C4—1181no—
418C2×C4⋊C4C4×D5912yes—
419#103—1184no—
420#103—1184no—
421#102—1184no—
422#102—1184no—
423C22⋊C8—1188no—
424C24—1181no—
425C24#1174568511yescharacteristic
426#41—1182no—
427#41—1182no—
428#41—1182no—
429#41—1182no—
430#41—1182no—
431#41—1182no—
432#41—1182no—
433#41—1182no—
434#41—2361no—
435#41—2361no—
436#41—2361no—
437#41—2361no—
438#41—2361no—
439#41—2361no—
440#41—2361no—
441#41—2361no—
442C2×Q8—592no—
443C2×Q8—1181no—
444C2×Q8—1181no—
445C2×Q8—1181no—
446C2×Q8—1181no—
447C2×Q8—1181no—
448C2×D4—592no—
449C2×D4—1182no—
450C2×D4—1182no—
451C2×D4—1181no—
452C2×D4—1181no—
453C2×D4—1181no—
454C2×D4—1181no—
455C2×D4—1181no—
456C2×D4—2361no—
457C2×D4—2361no—
458C2×D4—2361no—
459C2×D4—2361no—
460C22×C4—22no—
461C22×C4—41no—
462C22×C4—41no—
463C22×C4—81no—
464C22×C4—81no—
465C22×C4—1181no—
466C22×C4—1182no—
467C22×C4—1182no—
468C22×C4—1182no—
469C22×C4—4721no—
470C22×C4—4721no—
471C22×C4—2361no—
472C22×C4—2361no—
473C22×C4—2361no—
474C22×C4—2361no—
475C22×C4—2361no—
476C22×C4—2361no—
477C22×C4C2×C4×D5911yescharacteristic
478C22×C4C2×D23611yescharacteristic
479C22×C4#1174568511yescharacteristic
480C22×C4#1174568511yescharacteristic
481C22×C4#1174568511yescharacteristic
482C22×C4D4×D5911yescharacteristic
483C22×C4D4×D5911yescharacteristic
484C22×C4D4×D5911yescharacteristic
485C22×C4D4×D5911yescharacteristic
486C22×C4#1174568711yescharacteristic
487C22×C4#1174568711yescharacteristic
488C22×C4#1174568911yescharacteristic
489C22×C4#1174568911yescharacteristic
490C22×C4C2×C22⋊D5911yescharacteristic
491M16—22no—
492M16—22no—
493M16—22no—
494M16—22no—
495M16—41no—
496M16—41no—
497M16—41no—
498M16—41no—
499M16—1182no—
500M16—1182no—
501M16—2361no—
502M16—2361no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—41no—
506C2×C8—41no—
507C2×C8—41no—
508C2×C8—41no—
509C2×C8—1182no—
510C2×C8—1182no—
511C2×C8—1182no—
512C2×C8—1182no—
513C2×C8—2361no—
514C2×C8—2361no—
515C2×C8—2361no—
516C2×C8—2361no—
517C2×C8—2361no—
518C2×C8—2361no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—41no—
522C4⋊C4—41no—
523C4⋊C4—1182no—
524C4⋊C4—1182no—
525C4⋊C4—2361no—
526C4⋊C4—2361no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—1182no—
532C23—2361no—
533C23—2361no—
534C23—2361no—
535C23—2361no—
536C23—2361no—
537C23—2361no—
538C23C4×C22⋊D5911yescharacteristic
539C23#1427370011yescharacteristic
540C23#1427370911yescharacteristic
541C23#1427371211yescharacteristic
542C23#1427371511yescharacteristic
543C23#1427372111yescharacteristic
544C23#1427372211yescharacteristic
545Q8—1182no—
546Q8—1182no—
547Q8—2361no—
548Q8—2361no—
549D4—1182no—
550D4—1182no—
551D4—2362no—
552D4—2362no—
553D4—2362no—
554D4—2362no—
555D4—2361no—
556D4—2361no—
557C2×C4—42no—
558C2×C4—82no—
559C2×C4—81no—
560C2×C4—81no—
561C2×C4—82no—
562C2×C4—1182no—
563C2×C4—4722no—
564C2×C4—4721no—
565C2×C4—4721no—
566C2×C4—4722no—
567C2×C4—2362no—
568C2×C4—2361no—
569C2×C4—2361no—
570C2×C4—2362no—
571C2×C4—2362no—
572C2×C4—2362no—
573C2×C4—2362no—
574C2×C4C22⋊C4×D5911yescharacteristic
575C2×C4#1427368411yescharacteristic
576C2×C4#1427368511yescharacteristic
577C2×C4(C22⋊D59)⋊C411yescharacteristic
578C2×C4#1427368911yescharacteristic
579C2×C4C4×D23611yescharacteristic
580C2×C4#1427369511yescharacteristic
581C2×C4#1427369711yescharacteristic
582C2×C4#1427369811yescharacteristic
583C2×C4#1427369911yescharacteristic
584C2×C4#1427370011yescharacteristic
585C2×C4#1427370011yescharacteristic
586C2×C4#1427370111yescharacteristic
587C2×C4#1427370211yescharacteristic
588C2×C4#1427370211yescharacteristic
589C2×C4#1427370911yescharacteristic
590C2×C4#1427370911yescharacteristic
591C2×C4#1427371011yescharacteristic
592C2×C4#1427371111yescharacteristic
593C2×C4#1427371211yescharacteristic
594C2×C4#1427371211yescharacteristic
595C2×C4#1427371211yescharacteristic
596C2×C4#1427371211yescharacteristic
597C2×C4#1427371311yescharacteristic
598C2×C4#1427371311yescharacteristic
599C2×C4#1427371611yescharacteristic
600C2×C4#1427372211yescharacteristic
601C2×C4#1427372211yescharacteristic
602C8—42no—
603C8—42no—
604C8—2362no—
605C8—2362no—
606C22—21no—
607C22—21no—
608C22—21no—
609C22—21no—
610C22—21no—
611C22—21no—
612C22—21no—
613C22—21no—
614C22—21no—
615C22—21no—
616C22—21no—
617C22—21no—
618C22—4721no—
619C22—4721no—
620C22—4721no—
621C22—4721no—
622C22—2362no—
623C22—2362no—
624C22—2362no—
625C22#1683727811yescharacteristic
626C22#1683729211yescharacteristic
627C22#1683729311yescharacteristic
628C22#1683732011yescharacteristic
629C22#1683732911yescharacteristic
630C22#1683733111yescharacteristic
631C22#1683733111yescharacteristic
632C22#1683766411yescharacteristic
633C22#1683766811yescharacteristic
634C22#1683767111yescharacteristic
635C22#1683767611yescharacteristic
636C4—82no—
637C4—4722no—
638C4—2362no—
639C4#1683726411yescharacteristic
640C4#1683728011yescharacteristic
641C4#1683729111yescharacteristic
642C4#1683729411yescharacteristic
643C4#1683731711yescharacteristic
644C4#1683731811yescharacteristic
645C4#1683731911yescharacteristic
646C4#1683732811yescharacteristic
647C2—21no—
648C2—21no—
649C2—21no—
650C2—21no—
651C2—4722no—
652C2#2142234411yescharacteristic
653C2#2142460111yescharacteristic
654C2#2142460811yescharacteristic
655C2#2142539111yescharacteristic
656C2#2142539811yescharacteristic
657C2#2142554011yescharacteristic
658C2#2142554511yescharacteristic
subgroupG/H
centerC23#14273700
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC2×C59×#343C2
socleC23×C59#121
prime radicalC22×#11745685C22
cosocleC23×C4×C59C23
hypercenterC2×#343D59
hypocenterC59#11192
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#343

Complements

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

NH ≅ G/Ntype
4. C2×#16837170C2semidirect
5. C2×#16837166C2semidirect
6. C2×#16837164C2semidirect
7. C2×C59×#343C2semidirect
15. #16837401C4semidirect
322. C2×#343D59semidirect
359. C59#11192semidirect

Maximal subgroups

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

subgroupindex
C2×#168374042
C2×#168374012
C2×#168373702
C2×#168371702
C2×#168371662
C2×#168371642
C2×C59×#3432
#1119259