Finite Group Databasebeta

← #29460251

Subgroups — #29460251

Order 15104.

2098 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16838023C211yesmaximal, characteristic, complemented
2#21435205C218yesmaximal, complemented
3#21435190C218yesmaximal, complemented
4#21434564C212yesmaximal, complemented
5#21434489C212yesmaximal, complemented
6#21433564C212yesmaximal, complemented
7#21433556C212yesmaximal, complemented
8#21433356C212yesmaximal, complemented
9#21433348C212yesmaximal, complemented
10#21432734C211yesmaximal, characteristic, complemented
11C59×#3449C211yesmaximal, characteristic, complemented
12C2×#14273757C2214yescomplemented
13C2×#14273752C2214yescomplemented
14#16838023—24no—
15#16838023C2218yescomplemented
16C2×#14273727C2211yescharacteristic, complemented
17C2×#14273722—21no—
18C2×#14273715C2211yescharacteristic, complemented
19C2×#14273713—21no—
20C2×#14273709—21no—
21C2×#14273706C2211yescharacteristic
22C2×#14273705—21no—
23C2×#14273701C2211yescharacteristic
24C2×C4×Dic118C2211yescharacteristic, complemented
25C2×C4×D236C2211yescharacteristic, complemented
26#16837794C2214yescomplemented
27#16837789C2214yescomplemented
28#16837784C2218yescomplemented
29#16837761C2218yescomplemented
30#16837752C2214yescomplemented
31#16837748C2214yescomplemented
32#16837621C2218yescomplemented
33#16837614C2218yescomplemented
34#16837609C2214yescomplemented
35#16837604C2214yescomplemented
36#16837599C2214yescomplemented
37#16837594C2214yescomplemented
38#16837564C2218yescomplemented
39#16837558C2214yescomplemented
40#16837553C2214yescomplemented
41#16837547C2218yes—
42#16837541C2214yescomplemented
43#16837536C2214yescomplemented
44#16837463C2218yescomplemented
45#16837426C2212yescomplemented
46#16837403C2214yescomplemented
47#16837403C2214yescomplemented
48#16837366C2212yes—
49C4×#11745664C2212yescomplemented
50C22×#41×C59C2212yes—
51C59×#537—21no—
52C59×#537—21no—
53C59×#536C2214yescomplemented
54C59×#535C2214yescomplemented
55C59×#534—21no—
56C59×#534—21no—
57C59×#532C2212yes—
58C59×#532C2212yes—
59C59×#525—24no—
60C2×#117×C59C2211yescharacteristic
61C22×Dic118—21no—
62C22×Dic118C2311yescharacteristic
63C22×D236C2311yescharacteristic
64C22×C22⋊D59—21no—
65C22×C4×D59—21no—
66#14273757—48no—
67#14273752—48no—
68C2×#11745672C2318yes—
69C2×#11745670C2318yes—
70C2×#11745671C2318yes—
71C2×#11745669C2318yes—
72C2×#11745664C2314yes—
73#14273727—22no—
74#14273727C2314yes—
75#14273722—28no—
76#14273715—22no—
77#14273715C2314yes—
78#14273713—28no—
79#14273709—28no—
80#14273706—22no—
81#14273706C2314yes—
82#14273705—28no—
83#14273701—22no—
84#14273701C2314yes—
85C4×Dic118—22no—
86C4×Dic118C2314yes—
87C4×D236—22no—
88C4×D236C2314yes—
89C2×C4⋊Dic59—21no—
90C2×C4⋊Dic59C2311yescharacteristic
91C2×C4⋊Dic59C2311yescharacteristic
92C2×Dic59⋊C4—21no—
93C2×Dic59⋊C4—21no—
94C2×C22⋊Dic59—21no—
95C2×(C2×D59)⋊C4—21no—
96C2×(C2×D59)⋊C4—21no—
97C2×(C2×D59)⋊C4—21no—
98C2×(C2×D59)⋊C4—21no—
99C2×C4×Dic59—21no—
100C4⋊(C59⋊C8)C2314yes—
101C4⋊(C59⋊C8)C2314yes—
102#14273651C2318yes—
103#14273651C2318yes—
104#14273648C2318yes—
105#14273648C2318yes—
106C22⋊(C59⋊C8)C2314yes—
107C22⋊(C59⋊C8)C2314yes—
108#14273638C2314yes—
109C4×C59⋊C8C2314yes—
110C2×#41×C59—22no—
111C2×#41×C59—22no—
112C2×#41×C59—48no—
113C2×#41×C59C2314yes—
114C2×#41×C59C2314yes—
115C22×Q8×C59C2312yes—
116C22×D4×C59—22no—
117C22×D4×C59C2312yes—
118C23×C4×C59—22no—
119C23×C4×C59C2311yescharacteristic
120C4⋊Q8×C59C2312yes—
121C4⋊Q8×C59C2312yes—
122#127×C59C2312yes—
123#127×C59C2312yes—
124#124×C59C2314yes—
125#124×C59C2314yes—
126#123×C59—24no—
127#123×C59—24no—
128C22⋊Q8×C59—24no—
129C22⋊Q8×C59—24no—
130#121×C59—22no—
131#121×C59—22no—
132#121×C59—22no—
133#121×C59—22no—
134#121×C59—24no—
135#121×C59—24no—
136#120×C59—24no—
137#120×C59—24no—
138C4×D4×C59—22no—
139C4×D4×C59—22no—
140C4×D4×C59—22no—
141C4×D4×C59—22no—
142#117×C59D414yes—
143#117×C59C2314yes—
144C2×C4⋊C4×C59D411yescharacteristic
145C2×C4⋊C4×C59D411yescharacteristic
146C2×C22⋊C4×C59D411yescharacteristic
147C2×C22⋊C4×C59D411yescharacteristic
148C2×C42×C59C2311yescharacteristic
149C2×C42×C59C2311yescharacteristic
150C24×C59—22no—
151C24×C59C2×D411yescharacteristic
152C23×D59—21no—
153#41×C59—28no—
154#41×C59—28no—
155#41×C59—44no—
156#41×C59—44no—
157#41×C59—48no—
158#41×C59—48no—
159C2×Q8×C59—22no—
160C2×Q8×C59—44no—
161C2×Q8×C59C2414yes—
162C2×D4×C59—24no—
163C2×D4×C59—24no—
164C2×D4×C59—22no—
165C2×D4×C59—22no—
166C2×D4×C59—22no—
167C2×D4×C59—22no—
168C2×D4×C59—22no—
169C2×D4×C59—44no—
170C2×D4×C59—48no—
171C2×D4×C59C2414yes—
172C22×C4×C59—24no—
173C22×C4×C59—22no—
174C22×C4×C59—24no—
175C22×C4×C59—24no—
176C22×C4×C59—48no—
177C22×C4×C59C2×D411yescharacteristic
178C22×C4×C59C2×D412yes—
179C22×C4×C59C2×D411yescharacteristic
180C22×C4×C59C2×D411yescharacteristic
181C22×C4×C59C2×D411yescharacteristic
182C22×C4×C59C2×D411yescharacteristic
183C22×C4×C59C2412yes—
184C22×C4×C59C2411yescharacteristic
185C2×C22⋊D59—24no—
186C2×C22⋊D59—82no—
187C22×Dic59—21no—
188C22×Dic59—21no—
189C22×Dic59—21no—
190C2×D236—24no—
191C2×D236—21no—
192C2×D236C2412yes—
193C2×C4×D59—24no—
194C2×C4×D59—82no—
195C2×Dic118—24no—
196C2×Dic118—21no—
197C2×Dic118—24no—
198C2×Dic118—82no—
199C2×Dic118C2412yes—
200C4⋊C4×C59—22no—
201C4⋊C4×C59—22no—
202C4⋊C4×C59—22no—
203C4⋊C4×C59—22no—
204C4⋊C4×C59C2×D414yes—
205C4⋊C4×C59C2×D414yes—
206C22⋊C4×C59—24no—
207C22⋊C4×C59—24no—
208C22⋊C4×C59—24no—
209C22⋊C4×C59—24no—
210C22⋊C4×C59C2×D414yes—
211C22⋊C4×C59C2×D414yes—
212C42×C59C2×D412yes—
213C42×C59C2×D412yes—
214C42×C59C2412yes—
215C42×C59C2412yes—
216C22⋊Dic59—24no—
217#11745672—48no—
218#11745671—48no—
219#11745670—48no—
220#11745669—48no—
221(C2×D59)⋊C4—24no—
222(C2×D59)⋊C4—24no—
223(C2×D59)⋊C4—24no—
224(C2×D59)⋊C4—24no—
225C4⋊Dic59—24no—
226C4⋊Dic59—21no—
227C4⋊Dic59—21no—
228C4⋊Dic59C2412yes—
229C4⋊Dic59C2412yes—
230Dic59⋊C4—24no—
231Dic59⋊C4—24no—
232C4×Dic59—24no—
233#11745664—28no—
234C2×C59⋊C8C2418yes—
235C23×C59—22no—
236C23×C59—24no—
237C23×C59—41no—
238C23×C59—41no—
239C23×C59—44no—
240C23×C59—82no—
241C23×C59—82no—
242C23×C59C22×D412yes—
243C23×C59C22×D411yescharacteristic
244C22×D59—22no—
245C22×D59—42no—
246C22×D59—42no—
247C22×D59—42no—
248Q8×C59—24no—
249Q8×C59—24no—
250Q8×C59—44no—
251D4×C59—24no—
252D4×C59—24no—
253D4×C59—44no—
254D4×C59—48no—
255D4×C59—48no—
256D4×C59—82no—
257D4×C59—82no—
258D4×C59—82no—
259D4×C59—82no—
260C2×C4×C59—22no—
261C2×C4×C59—24no—
262C2×C4×C59—24no—
263C2×C4×C59—24no—
264C2×C4×C59—41no—
265C2×C4×C59—41no—
266C2×C4×C59—41no—
267C2×C4×C59—41no—
268C2×C4×C59—48no—
269C2×C4×C59—44no—
270C2×C4×C59—48no—
271C2×C4×C59C22×D412yes—
272C2×C4×C59C22×D412yes—
273C2×C4×C59C22×D412yes—
274C2×C4×C59C22×D412yes—
275C2×C4×C59C22×D412yes—
276C2×C4×C59C22×D411yescharacteristic
277C2×C4×C59H(5,𝔽2)14yes—
278C2×C4×C59C2511yescharacteristic
279C22⋊D59—84no—
280C2×Dic59—22no—
281C2×Dic59—22no—
282C2×Dic59—22no—
283C2×Dic59—42no—
284C2×Dic59—42no—
285C2×Dic59—42no—
286D236—24no—
287D236—42no—
288C4×D59—84no—
289Dic118—24no—
290Dic118—42no—
291Dic118—84no—
292C59⋊C8—28no—
293#60810—591nomaximal, Sylow
294C22×C59—21no—
295C22×C59—21no—
296C22×C59—22no—
297C22×C59—22no—
298C22×C59—22no—
299C22×C59—44no—
300C22×C59—44no—
301C22×C59—44no—
302C22×C59—82no—
303C22×C59—88no—
304C22×C59#53512yes—
305C22×C59C23×D411yescharacteristic
306C2×D59—42no—
307C2×D59—42no—
308C2×D59—42no—
309C2×D59—84no—
310C4×C59—22no—
311C4×C59—22no—
312C4×C59—22no—
313C4×C59—22no—
314C4×C59—22no—
315C4×C59—44no—
316C4×C59#53512yes—
317C4×C59C2×H(5,𝔽2)12yes—
318Dic59—42no—
319Dic59—42no—
320Dic59—42no—
321C2×#542—591no—
322#3449D5911yescharacteristic, complemented
323#3225—598no—
324#3224—598no—
325#3153—592no—
326#3141—592no—
327#3037—592no—
328#3036—592no—
329#3007—592no—
330#3006—592no—
331#2921—591no—
332C2×C59—22no—
333C2×C59—42no—
334C2×C59—84no—
335C2×C59C2×#53511yescharacteristic
336C2×C59#358511yescharacteristic
337C2×C59#358611yescharacteristic
338D59—82no—
339C22×#41C2×D5912yes—
340C2×#137—594no—
341C2×AGL(1,ℤ/8)—594no—
342#542—598no—
343#542—1184no—
344#537—21no—
345#537—21no—
346#536C2×D5914yescomplemented
347#535C2×D5914yescomplemented
348#534—21no—
349#534—21no—
350#532C2×D5912yes—
351#532C2×D5912yes—
352C2×C4⋊Q8—591no—
353C2×#126—1181no—
354C2×#124—591no—
355C2×#124—1181no—
356#525—24no—
357C2×#123—1181no—
358C2×C22⋊Q8—591no—
359C2×C22⋊Q8—1181no—
360C2×#121—591no—
361C2×C4×Q8—591no—
362C2×C4×D4—591no—
363C2×#117C2×D5911yescharacteristic
364#497—594no—
365#496—594no—
366#495—598no—
367#489—598no—
368#487—594no—
369#486—594no—
370#464—598no—
371#463—598no—
372#462—594no—
373#461—594no—
374#460—594no—
375#459—594no—
376#452—598no—
377#451—594no—
378#450—594no—
379#449—598no—
380#448—594no—
381#447—594no—
382#433—598no—
383#423—592no—
384#417—594no—
385#417—594no—
386#407—592no—
387C4×M16—592no—
388C59#6081011yescharacteristic, complemented, Sylow
389C2×#41—22no—
390C2×#41—22no—
391C2×#41—48no—
392C2×#41C22×D5914yes—
393C2×#41C22×D5914yes—
394C22×Q8—591no—
395C22×Q8—1181no—
396C22×Q8C22×D5912yes—
397C22×D4—22no—
398C22×D4—591no—
399C22×D4—1181no—
400C22×D4C22×D5912yes—
401C23×C4—22no—
402C23×C4—1181no—
403C23×C4C22×D5911yescharacteristic
404#137—2368no—
405AGL(1,ℤ/8)—2368no—
406C2×Q16—598no—
407C2×SD16—598no—
408C2×SD16—598no—
409C2×D8—598no—
410C2×M16—594no—
411C4⋊Q8—594no—
412C4⋊Q8—1182no—
413C4⋊Q8C22×D5912yes—
414C4⋊Q8C22×D5912yes—
415#127C22×D5912yes—
416#127C22×D5912yes—
417#126—1188no—
418#124—594no—
419#124—1188no—
420#124—1182no—
421#124C22×D5914yes—
422#124C22×D5914yes—
423#123—24no—
424#123—24no—
425#123—1188no—
426C22⋊Q8—24no—
427C22⋊Q8—24no—
428C22⋊Q8—594no—
429C22⋊Q8—1182no—
430C22⋊Q8—1188no—
431#121—22no—
432#121—22no—
433#121—22no—
434#121—22no—
435#121—24no—
436#121—24no—
437#121—594no—
438#121—1182no—
439#120—24no—
440#120—24no—
441C4×Q8—594no—
442C4×Q8—1182no—
443C4×D4—22no—
444C4×D4—22no—
445C4×D4—22no—
446C4×D4—22no—
447C4×D4—594no—
448C4×D4—1182no—
449#117C22⋊D5914yes—
450#117C22×D5914yes—
451C2×C4⋊C4—591no—
452C2×C4⋊C4—591no—
453C2×C4⋊C4—1181no—
454C2×C4⋊C4—1181no—
455C2×C4⋊C4—1181no—
456C2×C4⋊C4C22⋊D5911yescharacteristic
457C2×C4⋊C4C22⋊D5911yescharacteristic
458C2×C22⋊C4—1181no—
459C2×C22⋊C4—1181no—
460C2×C22⋊C4—1181no—
461C2×C22⋊C4—1181no—
462C2×C22⋊C4—1181no—
463C2×C22⋊C4C22⋊D5911yescharacteristic
464C2×C22⋊C4C22⋊D5911yescharacteristic
465C2×C42—1181no—
466C2×C42C22×D5911yescharacteristic
467C2×C42C22×D5911yescharacteristic
468C4⋊C8—594no—
469C4⋊C8—594no—
470#103—598no—
471#103—598no—
472#102—598no—
473#102—598no—
474C22⋊C8—594no—
475C22⋊C8—594no—
476#97—594no—
477C4×C8—594no—
478C24—22no—
479C24—1181no—
480C24C2×C22⋊D5911yescharacteristic
481#41—28no—
482#41—28no—
483#41—44no—
484#41—44no—
485#41—48no—
486#41—48no—
487C2×Q8—22no—
488C2×Q8—44no—
489C2×Q8—592no—
490C2×Q8—1184no—
491C2×Q8—1181no—
492C2×Q8—1184no—
493C2×Q8—4722no—
494C2×Q8C23×D5914yes—
495C2×D4—24no—
496C2×D4—24no—
497C2×D4—22no—
498C2×D4—22no—
499C2×D4—22no—
500C2×D4—22no—
501C2×D4—22no—
502C2×D4—44no—
503C2×D4—48no—
504C2×D4—592no—
505C2×D4—1184no—
506C2×D4—1181no—
507C2×D4—1184no—
508C2×D4—4722no—
509C2×D4C23×D5914yes—
510C22×C4—24no—
511C22×C4—22no—
512C22×C4—24no—
513C22×C4—24no—
514C22×C4—48no—
515C22×C4—1181no—
516C22×C4—1181no—
517C22×C4—1181no—
518C22×C4—1184no—
519C22×C4—4722no—
520C22×C4C2×C22⋊D5911yescharacteristic
521C22×C4C2×C22⋊D5912yes—
522C22×C4C2×C22⋊D5911yescharacteristic
523C22×C4C2×C22⋊D5911yescharacteristic
524C22×C4C2×C22⋊D5911yescharacteristic
525C22×C4C2×C22⋊D5911yescharacteristic
526C22×C4C23×D5912yes—
527C22×C4C23×D5911yescharacteristic
528Q16—2368no—
529SD16—2368no—
530SD16—2368no—
531D8—2368no—
532M16—1188no—
533C2×C8—598no—
534C4⋊C4—22no—
535C4⋊C4—22no—
536C4⋊C4—22no—
537C4⋊C4—22no—
538C4⋊C4—592no—
539C4⋊C4—592no—
540C4⋊C4—1181no—
541C4⋊C4—1181no—
542C4⋊C4—1184no—
543C4⋊C4—1184no—
544C4⋊C4—1184no—
545C4⋊C4C2×C22⋊D5914yes—
546C4⋊C4C2×C22⋊D5914yes—
547C22⋊C4—24no—
548C22⋊C4—24no—
549C22⋊C4—24no—
550C22⋊C4—24no—
551C22⋊C4—1184no—
552C22⋊C4—1184no—
553C22⋊C4—1184no—
554C22⋊C4—1184no—
555C22⋊C4—1184no—
556C22⋊C4C2×C22⋊D5914yes—
557C22⋊C4C2×C22⋊D5914yes—
558C42—1184no—
559C42C2×C22⋊D5912yes—
560C42C2×C22⋊D5912yes—
561C42C23×D5912yes—
562C42C23×D5912yes—
563C23—22no—
564C23—24no—
565C23—41no—
566C23—41no—
567C23—44no—
568C23—82no—
569C23—82no—
570C23—1182no—
571C23—2362no—
572C23—2362no—
573C23—2362no—
574C23C22×C22⋊D5912yes—
575C23C22×C22⋊D5911yescharacteristic
576Q8—24no—
577Q8—24no—
578Q8—44no—
579Q8—1184no—
580Q8—4724no—
581Q8—2362no—
582D4—24no—
583D4—24no—
584D4—44no—
585D4—48no—
586D4—48no—
587D4—82no—
588D4—82no—
589D4—82no—
590D4—82no—
591D4—1184no—
592D4—4724no—
593D4—2362no—
594C2×C4—22no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—48no—
603C2×C4—44no—
604C2×C4—48no—
605C2×C4—1182no—
606C2×C4—1182no—
607C2×C4—1182no—
608C2×C4—4724no—
609C2×C4—2362no—
610C2×C4—2362no—
611C2×C4—2362no—
612C2×C4C22×C22⋊D5912yes—
613C2×C4C22×C22⋊D5912yes—
614C2×C4C22×C22⋊D5912yes—
615C2×C4C22×C22⋊D5912yes—
616C2×C4C22×C22⋊D5912yes—
617C2×C4C22×C22⋊D5911yescharacteristic
618C2×C4#1427377214yes—
619C2×C4C24×D5911yescharacteristic
620C8—1188no—
621C22—21no—
622C22—21no—
623C22—22no—
624C22—22no—
625C22—22no—
626C22—44no—
627C22—44no—
628C22—44no—
629C22—82no—
630C22—88no—
631C22—4724no—
632C22—2362no—
633C22—2362no—
634C22—2362no—
635C22#1683796912yes—
636C22C23×C22⋊D5911yescharacteristic
637C4—22no—
638C4—22no—
639C4—22no—
640C4—22no—
641C4—22no—
642C4—44no—
643C4—2362no—
644C4—2362no—
645C4—2362no—
646C4#1683796912yes—
647C4C2×#1427377212yes—
648C2—22no—
649C2—42no—
650C2—84no—
651C2—4722no—
652C2C2×#1683796911yescharacteristic
653C2#2143887511yescharacteristic
654C2#2143888211yescharacteristic
subgroupG/H
centerC22C23×C22⋊D59
commutator, cosocleC2×C4×C59C25
FrattiniC2×C4C24×D59
FittingC59×#3449C2
socleC22×C59C23×D4
hypercenter#3449D59
hypocenterC59#60810
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×#117 ◁ #3449

Complements

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

NH ≅ G/Ntype
1. C2×#16838023C2semidirect
2. #21435205C2semidirect
3. #21435190C2semidirect
4. #21434564C2semidirect
5. #21434489C2semidirect
6. #21433564C2semidirect
7. #21433556C2semidirect
8. #21433356C2semidirect
9. #21433348C2semidirect
10. #21432734C2semidirect
11. C59×#3449C2semidirect
12. C2×#14273757C22semidirect
13. C2×#14273752C22semidirect
15. #16838023C22semidirect
16. C2×#14273727C22semidirect
18. C2×#14273715C22semidirect
24. C2×C4×Dic118C22semidirect
25. C2×C4×D236C22semidirect
26. #16837794C22semidirect
27. #16837789C22semidirect
28. #16837784C22semidirect
29. #16837761C22semidirect
30. #16837752C22semidirect
31. #16837748C22semidirect
32. #16837621C22semidirect
33. #16837614C22semidirect
34. #16837609C22semidirect
35. #16837604C22semidirect
36. #16837599C22semidirect
37. #16837594C22semidirect
38. #16837564C22semidirect
39. #16837558C22semidirect
40. #16837553C22semidirect
42. #16837541C22semidirect
43. #16837536C22semidirect
44. #16837463C22semidirect
45. #16837426C22semidirect
46. #16837403C22semidirect
47. #16837403C22semidirect
49. C4×#11745664C22semidirect
53. C59×#536C22semidirect
54. C59×#535C22semidirect
322. #3449D59semidirect
346. #536C2×D59semidirect
347. #535C2×D59semidirect
388. C59#60810semidirect

Maximal subgroups

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

subgroupindex
C2×#168380232
#214352052
#214351902
#214345642
#214344892
#214335642
#214335562
#214333562
#214333482
#214327342
C59×#34492
#6081059