Finite Group Databasebeta

← #28617617

Subgroups — #28617617

Order 15104.

892 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21432660C211yesmaximal, characteristic, complemented
2C2×#16837420C211yesmaximal, characteristic
3#21432613C211yesmaximal, characteristic
4C2×#16837384C211yesmaximal, characteristic
5#21425548C218yesmaximal, complemented
6C2×#16837172C211yesmaximal, characteristic, complemented
7#21423822C211yesmaximal, characteristic, complemented
8C2×C59×H(3,ℤ/4)C211yesmaximal, characteristic, complemented
9C22×#11745685C2211yescharacteristic
10C22×#11745664C2211yescharacteristic
11C22×#11745664C2211yescharacteristic
12C22×#11745664C2211yescharacteristic
13#16837429—22no—
14#16837429C2214yescomplemented
15#16837426—22no—
16#16837420—22no—
17#16837420C2214yes—
18C2×#14273655—22no—
19#16837410—21no—
20#16837406—21no—
21#16837401—22no—
22#16837396C2214yes—
23#16837384—22no—
24#16837384C2214yes—
25C2×#14273645—21no—
26C2×#14273643—21no—
27#16837172C2214yescomplemented
28#16837166C2214yescomplemented
29C22×M16×C59C2211yescharacteristic
30C2×#117×C59C2211yescharacteristic
31C2×#117×C59C2211yescharacteristic
32C59×H(3,ℤ/4)C2214yescomplemented
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×#11745685—21no—
42C2×#11745685—21no—
43C2×#11745685D412yes—
44C2×#11745685C2312yes—
45C22×Dic118D411yescharacteristic
46C22×D236D411yescharacteristic
47C22×C22⋊D59—21no—
48C22×C4×D59—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×#11745664—21no—
59C2×#11745664—21no—
60C2×#11745664—21no—
61C2×#11745664—21no—
62C2×#11745664—21no—
63C2×#11745664—21no—
64C2×#11745664—21no—
65C2×#11745664C2312yes—
66C2×#11745664C2312yes—
67C22×C59⋊C8—21no—
68C22×C59⋊C8—21no—
69C22×C59⋊C8C2312yes—
70C4⋊(C59⋊C8)—28no—
71#14273655—28no—
72#14273650—24no—
73#14273647—24no—
74#14273645—24no—
75#14273643—24no—
76C23×C4×C59C2311yescharacteristic
77C2×M16×C59—21no—
78C2×M16×C59—21no—
79C2×M16×C59—21no—
80C2×M16×C59—21no—
81C2×M16×C59—21no—
82C2×M16×C59C2312yes—
83C22×C8×C59—21no—
84#117×C59—21no—
85#117×C59—21no—
86#117×C59—21no—
87#117×C59—21no—
88#117×C59—21no—
89#117×C59D412yes—
90#117×C59C2312yes—
91#117×C59C2312yes—
92C2×C4⋊C4×C59—21no—
93C2×C4⋊C4×C59—21no—
94C2×C22⋊C4×C59—21no—
95C2×C22⋊C4×C59—21no—
96C2×C42×C59—21no—
97C2×C42×C59D412yes—
98C24×C59#4111yescharacteristic
99C23×D59—21no—
100C22×C4×C59—22no—
101C22×C4×C59—41no—
102C22×C4×C59C2×D411yescharacteristic
103C22×C4×C59C2×D411yescharacteristic
104C22×C4×C59C2×D411yescharacteristic
105C22×C4×C59C2×D411yescharacteristic
106C22×C4×C59#4111yescharacteristic
107C22×C4×C59#4111yescharacteristic
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
115C22×C4×C59C2411yescharacteristic
116C2×C22⋊D59—22no—
117C2×C22⋊D59—22no—
118C2×C22⋊D59—41no—
119C2×C22⋊D59—41no—
120C2×C22⋊D59—41no—
121C2×C22⋊D59—41no—
122C22×Dic59—21no—
123#11745685—22no—
124#11745685—22no—
125#11745685—22no—
126#11745685—22no—
127#11745685—22no—
128#11745685—22no—
129#11745685—22no—
130#11745685—22no—
131#11745685—41no—
132#11745685—41no—
133#11745685—41no—
134#11745685—41no—
135#11745685—41no—
136#11745685—41no—
137#11745685—41no—
138#11745685—41no—
139C2×D236—21no—
140C2×D236—21no—
141C2×D236—21no—
142C2×D236—21no—
143C2×D236—21no—
144C2×D236C2×D412yes—
145C2×C4×D59—22no—
146C2×C4×D59—22no—
147C2×C4×D59—41no—
148C2×C4×D59—41no—
149C2×C4×D59—41no—
150C2×C4×D59—41no—
151C2×Dic118—21no—
152C2×Dic118—21no—
153C2×Dic118—21no—
154C2×Dic118—21no—
155C2×Dic118—21no—
156C2×Dic118C2×D412yes—
157M16×C59—22no—
158M16×C59—22no—
159M16×C59—41no—
160M16×C59—41no—
161C2×C8×C59—22no—
162C2×C8×C59—41no—
163C2×C8×C59—41no—
164C4⋊C4×C59—22no—
165C4⋊C4×C59—22no—
166C4⋊C4×C59—22no—
167C4⋊C4×C59—41no—
168C22⋊C4×C59—22no—
169C22⋊C4×C59—22no—
170C22⋊C4×C59—22no—
171C22⋊C4×C59—41no—
172C42×C59—22no—
173C42×C59—22no—
174C42×C59—41no—
175C42×C59C2×D414yes—
176#11745664—22no—
177#11745664—22no—
178#11745664—22no—
179#11745664—22no—
180#11745664—41no—
181#11745664—41no—
182#11745664—41no—
183#11745664—41no—
184#11745664—41no—
185#11745664—41no—
186#11745664—41no—
187#11745664—41no—
188C2×C59⋊C8—22no—
189C2×C59⋊C8—22no—
190C2×C59⋊C8—22no—
191C2×C59⋊C8—22no—
192C2×C59⋊C8—22no—
193C2×C59⋊C8—41no—
194C2×C59⋊C8—41no—
195C2×C59⋊C8—41no—
196C2×C59⋊C8—41no—
197C23×C59—81no—
198C23×C59#12311yescharacteristic
199C23×C59#12311yescharacteristic
200C23×C59#12311yescharacteristic
201C23×C59#12311yescharacteristic
202C23×C59#12611yescharacteristic
203C23×C59#12611yescharacteristic
204C23×C59C2×#4111yescharacteristic
205C22×D59—22no—
206C22×D59—41no—
207C22×D59—41no—
208C22×D59—82no—
209C2×C4×C59—24no—
210C2×C4×C59—42no—
211C2×C4×C59—82no—
212C2×C4×C59—82no—
213C2×C4×C59#12111yescharacteristic
214C2×C4×C59#12111yescharacteristic
215C2×C4×C59#12111yescharacteristic
216C2×C4×C59#12111yescharacteristic
217C2×C4×C59#12311yescharacteristic
218C2×C4×C59#12311yescharacteristic
219C2×C4×C59#12311yescharacteristic
220C2×C4×C59#12311yescharacteristic
221C2×C4×C59#12311yescharacteristic
222C2×C4×C59#12311yescharacteristic
223C2×C4×C59#12311yescharacteristic
224C2×C4×C59#12311yescharacteristic
225C2×C4×C59#12411yescharacteristic
226C2×C4×C59#12411yescharacteristic
227C2×C4×C59#12411yescharacteristic
228C2×C4×C59#12411yescharacteristic
229C2×C4×C59#12611yescharacteristic
230C2×C4×C59#12611yescharacteristic
231C2×C4×C59#12611yescharacteristic
232C2×C4×C59#12611yescharacteristic
233C2×C4×C59#12611yescharacteristic
234C2×C4×C59#12611yescharacteristic
235C2×C4×C59C22×D411yescharacteristic
236C2×C4×C59C22×D411yescharacteristic
237C2×C4×C59C2×#4111yescharacteristic
238C2×C4×C59C2×#4111yescharacteristic
239C2×C4×C59C2×#4111yescharacteristic
240C2×C4×C59C2×#4111yescharacteristic
241C22⋊D59—42no—
242C22⋊D59—42no—
243C22⋊D59—42no—
244C22⋊D59—42no—
245C2×Dic59—22no—
246C2×Dic59—41no—
247C2×Dic59—41no—
248D236—22no—
249D236—22no—
250D236—41no—
251D236—41no—
252C4×D59—42no—
253C4×D59—42no—
254C4×D59—42no—
255C4×D59—42no—
256Dic118—22no—
257Dic118—22no—
258Dic118—41no—
259Dic118—41no—
260C8×C59—42no—
261C59⋊C8—42no—
262C59⋊C8—42no—
263C59⋊C8—42no—
264#23649—591nomaximal, Sylow
265C22×C59—41no—
266C22×C59—41no—
267C22×C59—41no—
268C22×C59—41no—
269C22×C59—41no—
270C22×C59—41no—
271C22×C59—41no—
272C22×C59#39711yescharacteristic
273C22×C59#39711yescharacteristic
274C22×C59#39711yescharacteristic
275C22×C59#39711yescharacteristic
276C22×C59C2×#12311yescharacteristic
277C22×C59C2×#12311yescharacteristic
278C22×C59C2×#12611yescharacteristic
279C2×D59—42no—
280C2×D59—82no—
281C2×D59—82no—
282C2×D59—82no—
283C4×C59—44no—
284C4×C59—82no—
285C4×C59#39711yescharacteristic
286C4×C59#39711yescharacteristic
287C4×C59#39711yescharacteristic
288C4×C59#39711yescharacteristic
289C4×C59C2×#12111yescharacteristic
290C4×C59C2×#12311yescharacteristic
291C4×C59C2×#12411yescharacteristic
292C4×C59C2×#12611yescharacteristic
293Dic59—42no—
294#2905—591no—
295C2×#421—591no—
296#2896—591no—
297C2×#413—591no—
298#2080—598no—
299C2×#344—591no—
300#1738—591no—
301C2×H(3,ℤ/4)D5911yescharacteristic, complemented
302C2×C59—21no—
303C2×C59—21no—
304C2×C59—21no—
305C2×C59—21no—
306C2×C59—21no—
307C2×C59—21no—
308C2×C59—21no—
309C2×C59C2×#39711yescharacteristic
310D59—82no—
311C22×#41—591no—
312C22×M16—591no—
313C22×M16—591no—
314C22×M16—591no—
315C22×M16C2×D5911yescharacteristic
316C2×#117C2×D5911yescharacteristic
317C2×#117C2×D5911yescharacteristic
318#424—594no—
319#424—1182no—
320#423—1182no—
321#421—594no—
322#421—1182no—
323C2×#104—1182no—
324#419—1181no—
325#418—1181no—
326#417—1182no—
327#416—594no—
328#413—594no—
329#413—1182no—
330C2×#101—1181no—
331C2×#100—1181no—
332#344—594no—
333#341—594no—
334H(3,ℤ/4)C2×D5914yescomplemented
335C59#2364911yescharacteristic, complemented, Sylow
336C2×#41—592no—
337C2×#41—592no—
338C2×#41—1181no—
339C2×#41—1181no—
340C2×#41—1181no—
341C2×#41—1181no—
342C2×#41—1181no—
343C2×#41—1181no—
344C2×#41—1181no—
345C2×#41—1181no—
346C2×#41—1181no—
347C2×#41—1181no—
348C22×Q8—591no—
349C22×D4—591no—
350C22×D4—1181no—
351C23×C4—1181no—
352C23×C4C22×D5911yescharacteristic
353C2×M16—21no—
354C2×M16—21no—
355C2×M16—21no—
356C2×M16—21no—
357C2×M16—21no—
358C2×M16—592no—
359C2×M16—592no—
360C2×M16—1181no—
361C2×M16—1181no—
362C2×M16—1181no—
363C2×M16—1181no—
364C2×M16—1181no—
365C2×M16—1181no—
366C2×M16—1181no—
367C2×M16—1181no—
368C2×M16—1181no—
369C2×M16—1181no—
370C2×M16—1181no—
371C2×M16—1181no—
372C2×M16—1181no—
373C2×M16—1181no—
374C2×M16—1181no—
375C2×M16—1181no—
376C2×M16C22×D5912yes—
377C22×C8—21no—
378C22×C8—592no—
379C22×C8—1181no—
380C22×C8—1181no—
381#117—21no—
382#117—21no—
383#117—21no—
384#117—21no—
385#117—21no—
386#117D23612yes—
387#117C22×D5912yes—
388#117C22×D5912yes—
389C2×C4⋊C4—21no—
390C2×C4⋊C4—21no—
391C2×C22⋊C4—21no—
392C2×C22⋊C4—21no—
393C2×C42—21no—
394C2×C42D23612yes—
395C4⋊C8—1188no—
396#104—1188no—
397#103—1184no—
398#102—1184no—
399#101—1184no—
400#100—1184no—
401C24—1181no—
402C24#1174568711yescharacteristic
403#41—1182no—
404#41—1182no—
405#41—1182no—
406#41—1182no—
407#41—1182no—
408#41—1182no—
409#41—1182no—
410#41—1182no—
411#41—2361no—
412#41—2361no—
413#41—2361no—
414#41—2361no—
415#41—2361no—
416#41—2361no—
417#41—2361no—
418#41—2361no—
419C2×Q8—592no—
420C2×Q8—1181no—
421C2×Q8—1181no—
422C2×Q8—1181no—
423C2×Q8—1181no—
424C2×Q8—1181no—
425C2×D4—592no—
426C2×D4—1182no—
427C2×D4—1182no—
428C2×D4—1181no—
429C2×D4—1181no—
430C2×D4—1181no—
431C2×D4—1181no—
432C2×D4—1181no—
433C2×D4—2361no—
434C2×D4—2361no—
435C2×D4—2361no—
436C2×D4—2361no—
437C22×C4—22no—
438C22×C4—41no—
439C22×C4—1181no—
440C22×C4—1182no—
441C22×C4—1182no—
442C22×C4—2361no—
443C22×C4—2361no—
444C22×C4—2361no—
445C22×C4—2361no—
446C22×C4C2×D23611yescharacteristic
447C22×C4C2×D23611yescharacteristic
448C22×C4#1174568511yescharacteristic
449C22×C4#1174568511yescharacteristic
450C22×C4#1174568511yescharacteristic
451C22×C4#1174568511yescharacteristic
452C22×C4D4×D5911yescharacteristic
453C22×C4D4×D5911yescharacteristic
454C22×C4#1174568711yescharacteristic
455C22×C4#1174568711yescharacteristic
456C22×C4#1174568711yescharacteristic
457C22×C4#1174568911yescharacteristic
458C22×C4#1174568911yescharacteristic
459C22×C4C23×D5911yescharacteristic
460M16—22no—
461M16—22no—
462M16—41no—
463M16—41no—
464M16—1182no—
465M16—1182no—
466M16—1182no—
467M16—1182no—
468M16—2361no—
469M16—2361no—
470M16—2361no—
471M16—2361no—
472M16—2361no—
473M16—2361no—
474M16—2361no—
475M16—2361no—
476C2×C8—22no—
477C2×C8—41no—
478C2×C8—41no—
479C2×C8—1182no—
480C2×C8—1182no—
481C2×C8—1182no—
482C2×C8—1182no—
483C2×C8—1182no—
484C2×C8—2361no—
485C2×C8—2361no—
486C2×C8—2361no—
487C2×C8—2361no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—41no—
492C22⋊C4—22no—
493C22⋊C4—22no—
494C22⋊C4—22no—
495C22⋊C4—41no—
496C42—22no—
497C42—22no—
498C42—41no—
499C42C2×D23614yes—
500C23—81no—
501C23—1182no—
502C23—4722no—
503C23—2361no—
504C23—2361no—
505C23#1427370911yescharacteristic
506C23#1427370911yescharacteristic
507C23#1427371111yescharacteristic
508C23#1427371111yescharacteristic
509C23#1427372211yescharacteristic
510C23#1427372211yescharacteristic
511C23C2×#1174568711yescharacteristic
512Q8—1182no—
513Q8—1182no—
514Q8—2361no—
515Q8—2361no—
516D4—1182no—
517D4—1182no—
518D4—2362no—
519D4—2362no—
520D4—2362no—
521D4—2362no—
522D4—2361no—
523D4—2361no—
524C2×C4—24no—
525C2×C4—42no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—1182no—
529C2×C4—2361no—
530C2×C4—2361no—
531C2×C4—2362no—
532C2×C4—2362no—
533C2×C4—2362no—
534C2×C4—2362no—
535C2×C4#1427370211yescharacteristic
536C2×C4#1427370211yescharacteristic
537C2×C4#1427370211yescharacteristic
538C2×C4#1427370211yescharacteristic
539C2×C4#1427370911yescharacteristic
540C2×C4#1427370911yescharacteristic
541C2×C4#1427370911yescharacteristic
542C2×C4#1427370911yescharacteristic
543C2×C4#1427370911yescharacteristic
544C2×C4#1427370911yescharacteristic
545C2×C4#1427371111yescharacteristic
546C2×C4#1427371111yescharacteristic
547C2×C4#1427371511yescharacteristic
548C2×C4#1427371511yescharacteristic
549C2×C4#1427371511yescharacteristic
550C2×C4#1427371511yescharacteristic
551C2×C4#1427372211yescharacteristic
552C2×C4#1427372211yescharacteristic
553C2×C4#1427372211yescharacteristic
554C2×C4#1427372211yescharacteristic
555C2×C4#1427372211yescharacteristic
556C2×C4#1427372211yescharacteristic
557C2×C4C2×D4×D5911yescharacteristic
558C2×C4C22×D23611yescharacteristic
559C2×C4C2×#1174568711yescharacteristic
560C2×C4C2×#1174568511yescharacteristic
561C2×C4C2×#1174568511yescharacteristic
562C2×C4C2×#1174568911yescharacteristic
563C8—42no—
564C8—2362no—
565C8—2362no—
566C8—2362no—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—41no—
573C22—41no—
574C22—4722no—
575C22—4722no—
576C22—4722no—
577C22—2362no—
578C22#1683732911yescharacteristic
579C22#1683732911yescharacteristic
580C22#1683732911yescharacteristic
581C22#1683732911yescharacteristic
582C22C2×#1427370911yescharacteristic
583C22C2×#1427371111yescharacteristic
584C22C2×#1427372211yescharacteristic
585C4—44no—
586C4—82no—
587C4—2362no—
588C4#1683732911yescharacteristic
589C4#1683732911yescharacteristic
590C4#1683732911yescharacteristic
591C4#1683732911yescharacteristic
592C4C2×#1427370211yescharacteristic
593C4C2×#1427370911yescharacteristic
594C4C2×#1427371511yescharacteristic
595C4C2×#1427372211yescharacteristic
596C2—21no—
597C2—21no—
598C2—21no—
599C2—21no—
600C2—21no—
601C2—21no—
602C2—21no—
603C2—4722no—
604C2C2×#1683732911yescharacteristic
subgroupG/H
centerC4C2×#14273715
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC2×C59×H(3,ℤ/4)C2
socleC2×C59C2×#397
prime radicalC22×#11745685C22
hypercenterC2×H(3,ℤ/4)D59
hypocenterC59#23649
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C4 ◁ C23×C4 ◁ C2×H(3,ℤ/4)

Complements

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

NH ≅ G/Ntype
1. #21432660C2semidirect
5. #21425548C2semidirect
6. C2×#16837172C2semidirect
7. #21423822C2semidirect
8. C2×C59×H(3,ℤ/4)C2semidirect
14. #16837429C22semidirect
27. #16837172C22semidirect
28. #16837166C22semidirect
32. C59×H(3,ℤ/4)C22semidirect
301. C2×H(3,ℤ/4)D59semidirect
334. H(3,ℤ/4)C2×D59semidirect
335. C59#23649semidirect

Maximal subgroups

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

subgroupindex
#214326602
C2×#168374202
#214326132
C2×#168373842
#214255482
C2×#168371722
#214238222
C2×C59×H(3,ℤ/4)2
#2364959