Finite Group Databasebeta

← #28559804

Subgroups — #28559804

Order 15104.

660 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21434157C211yesmaximal, characteristic
2#21433114C211yesmaximal, characteristic, complemented
3#21432842C211yesmaximal, characteristic, complemented
4#21423011C211yesmaximal, characteristic, complemented
5#21422982C211yesmaximal, characteristic, complemented
6#21422976C211yesmaximal, characteristic, complemented
7#21422962C211yesmaximal, characteristic, complemented
8#21422941C211yesmaximal, characteristic, complemented
9#21422927C211yesmaximal, characteristic, complemented
10#21422910C211yesmaximal, characteristic, complemented
11C8×#11745660C211yesmaximal, characteristic, complemented
12#21422881C211yesmaximal, characteristic, complemented
13#21422566C211yesmaximal, characteristic, complemented
14#21422484C211yesmaximal, characteristic, complemented
15C59×#1453C211yesmaximal, characteristic, complemented
16C2×#14273735—21no—
17#16837952C2211yescharacteristic
18C4×#11745685C2211yescharacteristic
19#16837727C411yescharacteristic
20#16837727C411yescharacteristic
21#16837698C411yescharacteristic
22#16837693C411yescharacteristic
23#16837603C411yescharacteristic
24#16837598C411yescharacteristic
25#16837549C411yescharacteristic
26#16837549C411yescharacteristic
27C8×Dic118C2211yescharacteristic
28C8×Dic118C2211yescharacteristic
29C4×#11745660C2211yescharacteristic
30C4×#11745660C2211yescharacteristic
31C4×D472C2211yescharacteristic
32C4×D472C2211yescharacteristic
33#16837481C2211yescharacteristic
34#16837481C2211yescharacteristic
35#16837481C2211yescharacteristic
36#16837481C2211yescharacteristic
37#16837478—21no—
38#16837478—21no—
39#16837475—21no—
40#16837475—21no—
41C8×D236C2211yescharacteristic
42C8×D236C2211yescharacteristic
43#16837459C2211yescharacteristic
44#16837457—21no—
45C2×C4⋊(C59⋊C8)C2211yescharacteristic
46#16837370—21no—
47#16837354—21no—
48#16837155C2211yescharacteristic
49#16837155C2211yescharacteristic
50#16837153C2211yescharacteristic
51#16837153C2211yescharacteristic
52#16837134C2211yescharacteristic
53#16837134C2211yescharacteristic
54#16837131C2211yescharacteristic
55#16837131C2211yescharacteristic
56#16837131C2211yescharacteristic
57#16837131C2211yescharacteristic
58#16837131C2211yescharacteristic
59#16837131C2211yescharacteristic
60C59×#431C2211yescharacteristic
61C59×#431C2211yescharacteristic
62C4×M16×C59C2211yescharacteristic
63C59×#322C2211yescharacteristic
64C59×#322C2211yescharacteristic
65C59×#322C2211yescharacteristic
66C82×C59C2211yescharacteristic
67C2×#11745685—21no—
68C2×#11745685C2×C411yescharacteristic
69C2×#11745660C2×C411yescharacteristic
70C2×#11745660C2×C411yescharacteristic
71C2×D472C2×C411yescharacteristic
72C2×D472C2×C411yescharacteristic
73#14273735—44no—
74C2×#11745659—21no—
75C2×#11745659—21no—
76C2×#11745659—21no—
77C2×#11745659—21no—
78C22×C59⋊C8—21no—
79C2×C8×D59—21no—
80C2×C8×D59—21no—
81#14273727C2×C411yescharacteristic
82#14273724C2×C411yescharacteristic
83#14273715—21no—
84#14273701—21no—
85#14273701C2×C411yescharacteristic
86#14273701C2×C411yescharacteristic
87C4×Dic118C2×C411yescharacteristic
88C4×Dic118C2311yescharacteristic
89C4×D236—21no—
90C4×D236C2×C411yescharacteristic
91C4×D236C2311yescharacteristic
92C4×D236C2311yescharacteristic
93C4×D236C2311yescharacteristic
94C4×C22⋊D59—21no—
95C4×C22⋊D59—21no—
96#14273686C2×C411yescharacteristic
97#14273684—21no—
98C42×D59—21no—
99#14273661C2×C411yescharacteristic
100#14273661C2×C411yescharacteristic
101#14273659C2×C411yescharacteristic
102#14273659C2×C411yescharacteristic
103C4⋊(C59⋊C8)—21no—
104C4⋊(C59⋊C8)C2311yescharacteristic
105C4⋊(C59⋊C8)C2311yescharacteristic
106C4⋊(C59⋊C8)C2311yescharacteristic
107C4⋊(C59⋊C8)C2311yescharacteristic
108Dic59⋊C8—21no—
109Dic59⋊C8—21no—
110#14273652C2×C411yescharacteristic
111#14273652C2×C411yescharacteristic
112#14273649C2×C411yescharacteristic
113#14273649C2×C411yescharacteristic
114#14273649C2×C411yescharacteristic
115#14273649C2×C411yescharacteristic
116#14273649C2×C411yescharacteristic
117#14273649C2×C411yescharacteristic
118C22⋊(C59⋊C8)—21no—
119(C2×D59)⋊C8—21no—
120(C2×D59)⋊C8—21no—
121(C2×D59)⋊C8—21no—
122(C2×D59)⋊C8—21no—
123(C2×D59)⋊C8—21no—
124(C2×D59)⋊C8—21no—
125#14273638—21no—
126C8×Dic59—21no—
127C8×Dic59—21no—
128C2×M16×C59D411yescharacteristic
129C2×M16×C59D411yescharacteristic
130C2×C42×C59C2311yescharacteristic
131C4⋊C8×C59—21no—
132C4⋊C8×C59C2×C411yescharacteristic
133C4⋊C8×C59C2×C411yescharacteristic
134C22⋊C8×C59—21no—
135C22⋊C8×C59C2×C411yescharacteristic
136C22⋊C8×C59C2×C411yescharacteristic
137#97×C59D411yescharacteristic
138#97×C59D411yescharacteristic
139C4×C8×C59C2311yescharacteristic
140C4×C8×C59C2311yescharacteristic
141C4×C8×C59C2311yescharacteristic
142C4×C8×C59C2311yescharacteristic
143C4×C8×C59C2311yescharacteristic
144C4×C8×C59C2311yescharacteristic
145C22×C4×C59C22×C411yescharacteristic
146C22×C4×C59C2×D411yescharacteristic
147C22×C4×C59#4111yescharacteristic
148C2×C22⋊D59—21no—
149C2×C22⋊D59—21no—
150C2×C22⋊D59—21no—
151#11745685—22no—
152#11745685—41no—
153#11745685—42no—
154#11745685—42no—
155C2×D236—21no—
156C2×D236—21no—
157C2×D236C22×C411yescharacteristic
158C2×D236C22×C411yescharacteristic
159C2×D236C22×C411yescharacteristic
160C2×C4×D59—21no—
161C2×C4×D59—21no—
162C2×C4×D59—21no—
163C2×C4×D59—21no—
164C2×C4×D59—21no—
165C2×Dic118—21no—
166C2×Dic118C22×C411yescharacteristic
167M16×C59—22no—
168M16×C59—41no—
169C2×C8×C59—21no—
170C2×C8×C59—21no—
171C2×C8×C59C22×C411yescharacteristic
172C2×C8×C59C22×C411yescharacteristic
173C2×C8×C59C22×C411yescharacteristic
174C2×C8×C59C22×C411yescharacteristic
175C2×C8×C59C2×D411yescharacteristic
176C2×C8×C59C2×D411yescharacteristic
177C2×C8×C59C2×D411yescharacteristic
178C2×C8×C59C2×D411yescharacteristic
179C42×C59C22×C411yescharacteristic
180C42×C59C2×D411yescharacteristic
181C42×C59#4111yescharacteristic
182C42×C59C2411yescharacteristic
183C22⋊Dic59—21no—
184(C2×D59)⋊C4—21no—
185(C2×D59)⋊C4—21no—
186(C2×D59)⋊C4—21no—
187(C2×D59)⋊C4—21no—
188(C2×D59)⋊C4—21no—
189(C2×D59)⋊C4—21no—
190C4⋊Dic59—21no—
191C4⋊Dic59C22×C411yescharacteristic
192C4⋊Dic59C22×C411yescharacteristic
193C4⋊Dic59C22×C411yescharacteristic
194C4⋊Dic59C22×C411yescharacteristic
195Dic59⋊C4—21no—
196Dic59⋊C4—21no—
197C4×Dic59—21no—
198C4×Dic59—21no—
199C4×Dic59—21no—
200C2×C59⋊C8—21no—
201C2×C59⋊C8—21no—
202C2×C59⋊C8—21no—
203C2×C59⋊C8—21no—
204C2×C59⋊C8—42no—
205D472—41no—
206D472—41no—
207#11745660—22no—
208#11745660—22no—
209#11745659—42no—
210#11745659—42no—
211#11745659—42no—
212#11745659—42no—
213C8×D59—42no—
214C8×D59—42no—
215C23×C59C4×D411yescharacteristic
216C22×D59—21no—
217C22×D59—21no—
218C22×D59—21no—
219C2×C4×C59—21no—
220C2×C4×C59—21no—
221C2×C4×C59—21no—
222C2×C4×C59—21no—
223C2×C4×C59—41no—
224C2×C4×C59C4×D411yescharacteristic
225C2×C4×C59C4×D411yescharacteristic
226C2×C4×C59C4×D411yescharacteristic
227C2×C4×C59C23×C411yescharacteristic
228C2×C4×C59C22×D411yescharacteristic
229C2×C4×C59C2×#4111yescharacteristic
230C22⋊D59—42no—
231C22⋊D59—42no—
232C22⋊D59—42no—
233C2×Dic59—21no—
234C2×Dic59—21no—
235C2×Dic59—21no—
236C2×Dic59—21no—
237C2×Dic59—21no—
238D236—21no—
239D236—21no—
240D236—21no—
241D236—21no—
242D236—21no—
243D236—21no—
244D236—42no—
245D236—81no—
246C4×D59—42no—
247C4×D59—41no—
248C4×D59—41no—
249C4×D59—41no—
250C4×D59—41no—
251C4×D59—41no—
252C4×D59—41no—
253C4×D59—81no—
254Dic118—21no—
255Dic118—42no—
256Dic118#13112yescomplemented
257C8×C59—21no—
258C8×C59—21no—
259C8×C59—21no—
260C8×C59—21no—
261C8×C59—21no—
262C8×C59—21no—
263C8×C59—21no—
264C8×C59—41no—
265C8×C59—41no—
266C8×C59AGL(1,ℤ/8)12yes—
267C59⋊C8—41no—
268C59⋊C8—41no—
269C59⋊C8—41no—
270C59⋊C8—41no—
271#18011—591nomaximal, Sylow
272C22×C59—21no—
273C22×C59—21no—
274C22×C59—21no—
275C22×C59C2×C4×D411yescharacteristic
276C2×D59—41no—
277C2×D59—41no—
278C2×D59—41no—
279C2×D59—41no—
280C2×D59—41no—
281C2×D59—41no—
282C2×D59—41no—
283C2×D59—41no—
284C2×D59—41no—
285C4×C59—21no—
286C4×C59—21no—
287C4×C59—21no—
288C4×C59—21no—
289C4×C59C2×#13111yescharacteristic
290C4×C59#56811yescharacteristic
291C4×C59C2×AGL(1,ℤ/8)11yescharacteristic
292C4×C59#57511yescharacteristic
293Dic59—22no—
294Dic59—41no—
295Dic59—41no—
296Dic59—41no—
297Dic59—41no—
298#3106—591no—
299#2971—591no—
300#2935—591no—
301#1594—591no—
302#1589—591no—
303#1588—591no—
304#1586—591no—
305#1583—591no—
306#1581—591no—
307SD16⋊C8—591no—
308C8×SD16—591no—
309#1573—591no—
310#1496—591no—
311#1477—591no—
312#1453D5911yescharacteristic, complemented
313C2×C59—41no—
314C2×C59#293311yescharacteristic
315C2×C59C4×AGL(1,ℤ/8)11yescharacteristic
316C2×C59#295711yescharacteristic
317D59—81no—
318D59—81no—
319D59—81no—
320C2×#131—1181no—
321#532—591no—
322C4×#41—591no—
323#482—591no—
324#482—591no—
325#477—591no—
326#476—591no—
327#460—591no—
328#459—591no—
329#449—591no—
330#449—591no—
331C8×Q8—591no—
332C8×Q8—591no—
333C4×SD16—591no—
334C4×SD16—591no—
335C4×D8—591no—
336C4×D8—591no—
337#436—591no—
338#436—591no—
339#436—591no—
340#436—591no—
341#436—1181no—
342#436—1181no—
343#435—1181no—
344#435—1181no—
345C8×D4—591no—
346C8×D4—591no—
347#432—591no—
348#432—1181no—
349#431C2×D5911yescharacteristic
350#431C2×D5911yescharacteristic
351C2×C4⋊C8—591no—
352#408—1181no—
353C4×M16—1181no—
354C4×M16C2×D5911yescharacteristic
355#335—591no—
356#335—591no—
357#334—591no—
358#334—591no—
359Q8⋊C8—591no—
360Q8⋊C8—591no—
361D4⋊C8—591no—
362D4⋊C8—591no—
363D4⋊C8—591no—
364D4⋊C8—591no—
365D4⋊C8—591no—
366D4⋊C8—591no—
367#322C2×D5911yescharacteristic
368#322C2×D5911yescharacteristic
369#322C2×D5911yescharacteristic
370C82C2×D5911yescharacteristic
371C59#1801111yescharacteristic, complemented, Sylow
372C2×#41—591no—
373C2×#41—1181no—
374C2×SD16—591no—
375C2×SD16—591no—
376C2×D8—591no—
377C2×D8—591no—
378#131—2364no—
379C2×M16—1181no—
380C2×M16—1181no—
381C2×M16—1181no—
382C2×M16—1181no—
383C2×M16D23611yescharacteristic
384C2×M16D23611yescharacteristic
385C22×C8—1181no—
386C22×C8—1181no—
387C22×C8—1181no—
388C4⋊Q8—591no—
389#127—591no—
390#124—1181no—
391#121—591no—
392#121—591no—
393#121—1181no—
394C4×Q8—591no—
395C4×Q8—591no—
396C4×D4—591no—
397C4×D4—591no—
398C4×D4—591no—
399C4×D4—591no—
400C4×D4—1181no—
401C4×D4—1181no—
402C4×D4—1181no—
403#117—591no—
404#117—1181no—
405C2×C42—1181no—
406C2×C42C22×D5911yescharacteristic
407#107—591no—
408#107—591no—
409#106—591no—
410#106—591no—
411C4⋊C8—21no—
412C4⋊C8—591no—
413C4⋊C8—591no—
414C4⋊C8—591no—
415C4⋊C8—591no—
416C4⋊C8—1181no—
417C4⋊C8—1181no—
418C4⋊C8—1181no—
419C4⋊C8C4×D5911yescharacteristic
420C4⋊C8C4×D5911yescharacteristic
421#103—591no—
422#103—591no—
423#102—591no—
424#102—591no—
425#102—591no—
426#102—591no—
427#102—591no—
428#102—591no—
429C22⋊C8—21no—
430C22⋊C8—1181no—
431C22⋊C8—1181no—
432C22⋊C8—1181no—
433C22⋊C8—1181no—
434C22⋊C8—1181no—
435C22⋊C8—1181no—
436C22⋊C8—1181no—
437C22⋊C8C4×D5911yescharacteristic
438C22⋊C8C4×D5911yescharacteristic
439#97—1181no—
440#97D23611yescharacteristic
441#97D23611yescharacteristic
442C4×C8—1181no—
443C4×C8—1181no—
444C4×C8C22×D5911yescharacteristic
445C4×C8C22×D5911yescharacteristic
446C4×C8C22×D5911yescharacteristic
447C4×C8C22×D5911yescharacteristic
448C4×C8C22×D5911yescharacteristic
449C4×C8C22×D5911yescharacteristic
450#41—1182no—
451#41—2361no—
452#41—2362no—
453#41—2362no—
454C2×Q8—591no—
455C2×Q8—1181no—
456C2×D4—591no—
457C2×D4—591no—
458C2×D4—591no—
459C2×D4—1181no—
460C2×D4—1181no—
461C2×D4—1181no—
462C2×D4—1181no—
463C2×D4—1181no—
464C22×C4—1181no—
465C22×C4—1181no—
466C22×C4—1181no—
467C22×C4—1181no—
468C22×C4—1181no—
469C22×C4C2×C4×D5911yescharacteristic
470C22×C4C2×D23611yescharacteristic
471C22×C4#1174568511yescharacteristic
472SD16—1182no—
473SD16—1182no—
474D8—2361no—
475D8—2361no—
476M16—22no—
477M16—41no—
478M16—2362no—
479M16—2362no—
480M16—2362no—
481M16—2362no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—1181no—
485C2×C8—1181no—
486C2×C8—1181no—
487C2×C8—1181no—
488C2×C8—2362no—
489C2×C8—2362no—
490C2×C8—2362no—
491C2×C8C2×C4×D5911yescharacteristic
492C2×C8C2×C4×D5911yescharacteristic
493C2×C8C2×C4×D5911yescharacteristic
494C2×C8C2×C4×D5911yescharacteristic
495C2×C8C2×D23611yescharacteristic
496C2×C8C2×D23611yescharacteristic
497C2×C8C2×D23611yescharacteristic
498C2×C8C2×D23611yescharacteristic
499C4⋊C4—591no—
500C4⋊C4—591no—
501C4⋊C4—591no—
502C4⋊C4—591no—
503C4⋊C4—1181no—
504C4⋊C4—1181no—
505C4⋊C4—1181no—
506C22⋊C4—1181no—
507C22⋊C4—1181no—
508C22⋊C4—1181no—
509C22⋊C4—1181no—
510C22⋊C4—1181no—
511C22⋊C4—1181no—
512C22⋊C4—1181no—
513C42—1181no—
514C42—1181no—
515C42—1181no—
516C42C2×C4×D5911yescharacteristic
517C42C2×D23611yescharacteristic
518C42#1174568511yescharacteristic
519C42C23×D5911yescharacteristic
520C23—1181no—
521C23—1181no—
522C23—1181no—
523C23C4×D23611yescharacteristic
524Q8—592no—
525Q8—1181no—
526Q8—2362no—
527D4—1181no—
528D4—1181no—
529D4—1181no—
530D4—1181no—
531D4—1181no—
532D4—1181no—
533D4—4721no—
534D4—2362no—
535D4—2362no—
536D4—2362no—
537D4—2362no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—41no—
543C2×C4—1181no—
544C2×C4—1181no—
545C2×C4—1181no—
546C2×C4—1181no—
547C2×C4—1181no—
548C2×C4—4721no—
549C2×C4—2362no—
550C2×C4—2361no—
551C2×C4—2361no—
552C2×C4—2361no—
553C2×C4—2361no—
554C2×C4—2361no—
555C2×C4—2361no—
556C2×C4C4×D23611yescharacteristic
557C2×C4C4×D23611yescharacteristic
558C2×C4C4×D23611yescharacteristic
559C2×C4C22×C4×D5911yescharacteristic
560C2×C4C22×D23611yescharacteristic
561C2×C4C2×#1174568511yescharacteristic
562C8—21no—
563C8—21no—
564C8—21no—
565C8—21no—
566C8—21no—
567C8—21no—
568C8—21no—
569C8—41no—
570C8—41no—
571C8—2361no—
572C8—2361no—
573C8—2361no—
574C8—2361no—
575C8#1427375512yes—
576C22—21no—
577C22—21no—
578C22—21no—
579C22—2361no—
580C22—2361no—
581C22—2361no—
582C22—2361no—
583C22—2361no—
584C22—2361no—
585C22—2361no—
586C22—2361no—
587C22—2361no—
588C22C2×C4×D23611yescharacteristic
589C4—21no—
590C4—21no—
591C4—21no—
592C4—21no—
593C4—1182no—
594C4—2361no—
595C4—2361no—
596C4—2361no—
597C4—2361no—
598C4#131×D5911yescharacteristic
599C4#1683816711yescharacteristic
600C4C2×#1427375511yescharacteristic
601C4#1683820211yescharacteristic
602C2—41no—
603C2—4721no—
604C2—4721no—
605C2—4721no—
606C2#2143282411yescharacteristic
607C2C4×#1427375511yescharacteristic
608C2#2143301411yescharacteristic
subgroupG/H
centerC2×C4C22×D236
commutatorC2×C4×C59C23×C4
FrattiniC42C23×D59
FittingC59×#1453C2
socleC22×C59C2×C4×D4
prime radical#21434157C2
cosocleC42×C59C24
hypercenter#1453D59
hypocenterC59#18011
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C4×M16 ◁ #1453

Complements

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

NH ≅ G/Ntype
2. #21433114C2semidirect
3. #21432842C2semidirect
4. #21423011C2semidirect
5. #21422982C2semidirect
6. #21422976C2semidirect
7. #21422962C2semidirect
8. #21422941C2semidirect
9. #21422927C2semidirect
10. #21422910C2semidirect
11. C8×#11745660C2semidirect
12. #21422881C2semidirect
13. #21422566C2semidirect
14. #21422484C2semidirect
15. C59×#1453C2semidirect
256. Dic118#131semidirect
312. #1453D59semidirect
371. C59#18011semidirect

Maximal subgroups

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

subgroupindex
#214341572
#214331142
#214328422
#214230112
#214229822
#214229762
#214229622
#214229412
#214229272
#214229102
C8×#117456602
#214228812
#214225662
#214224842
C59×#14532
#1801159