Finite Group Databasebeta

← #28560577

Subgroups — #28560577

Order 15104.

656 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21434691C211yesmaximal, characteristic
2#21433100C211yesmaximal, characteristic, complemented
3#21432842C211yesmaximal, characteristic, complemented
4#21423000C211yesmaximal, characteristic, complemented
5#21422982C211yesmaximal, characteristic, complemented
6#21422962C211yesmaximal, characteristic, complemented
7#21422955C211yesmaximal, characteristic, complemented
8#21422941C211yesmaximal, characteristic, complemented
9#21422927C211yesmaximal, characteristic, complemented
10#21422921C211yesmaximal, characteristic, complemented
11C8×#11745660C211yesmaximal, characteristic, complemented
12#21422848C211yesmaximal, characteristic, complemented
13#21422514C211yesmaximal, characteristic, complemented
14#21422510C211yesmaximal, characteristic, complemented
15C59×#1455C211yesmaximal, characteristic, complemented
16C2×#14273735—21no—
17#16837952C2211yescharacteristic
18#16837936C2211yescharacteristic
19#16837811C411yescharacteristic
20#16837791C411yescharacteristic
21#16837767C411yescharacteristic
22#16837762C411yescharacteristic
23#16837655C411yescharacteristic
24#16837655C411yescharacteristic
25#16837648C411yescharacteristic
26#16837648C411yescharacteristic
27#16837532C2211yescharacteristic
28C8×Dic118C2211yescharacteristic
29C4×#11745660C2211yescharacteristic
30C4×#11745660C2211yescharacteristic
31C4×D472C2211yescharacteristic
32C4×D472C2211yescharacteristic
33#16837481C2211yescharacteristic
34#16837481C2211yescharacteristic
35#16837481C2211yescharacteristic
36#16837478—21no—
37#16837475—21no—
38#16837475—21no—
39#16837471—21no—
40C8×D236C2211yescharacteristic
41C8×D236C2211yescharacteristic
42C8×D236C2211yescharacteristic
43#16837464—21no—
44#16837455C2211yescharacteristic
45C2×C4⋊(C59⋊C8)C2211yescharacteristic
46#16837370—21no—
47#16837359—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
60C4×M16×C59C2211yescharacteristic
61C4×M16×C59C2211yescharacteristic
62C4×M16×C59C2211yescharacteristic
63C59×#322C2211yescharacteristic
64C59×#322C2211yescharacteristic
65C59×#322C2211yescharacteristic
66C82×C59C2211yescharacteristic
67C2×#11745685—21no—
68C2×#11745660C2×C411yescharacteristic
69C2×#11745660C2×C411yescharacteristic
70C2×D472C2×C411yescharacteristic
71C2×D472C2×C411yescharacteristic
72#14273735—44no—
73C2×#11745659—21no—
74C2×#11745659—21no—
75C2×#11745659—21no—
76C22×C59⋊C8—21no—
77C2×C8×D59—21no—
78C2×C8×D59—21no—
79C2×C8×D59—21no—
80#14273727C2×C411yescharacteristic
81#14273724C2×C411yescharacteristic
82#14273721—21no—
83#14273719C2×C411yescharacteristic
84#14273715—21no—
85#14273715C2×C411yescharacteristic
86#14273710—21no—
87#14273706C2×C411yescharacteristic
88#14273701—21no—
89#14273701C2×C411yescharacteristic
90#14273701C2×C411yescharacteristic
91#14273701C2×C411yescharacteristic
92C4×Dic118C2311yescharacteristic
93C4×D236—21no—
94C4×D236C2311yescharacteristic
95C4×D236C2311yescharacteristic
96C4×D236C2311yescharacteristic
97C4×C22⋊D59—21no—
98#14273684—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#14273637—21no—
126C4×C59⋊C8—21no—
127C8×Dic59—21no—
128C2×M16×C59—21no—
129C2×M16×C59C2×C411yescharacteristic
130C2×M16×C59C2×C411yescharacteristic
131C2×M16×C59D411yescharacteristic
132C2×M16×C59D411yescharacteristic
133C2×C42×C59C2311yescharacteristic
134#97×C59—21no—
135#97×C59C2×C411yescharacteristic
136#97×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—42no—
152#11745685—42no—
153C2×D236—21no—
154C2×D236—21no—
155C2×D236C22×C411yescharacteristic
156C2×D236C22×C411yescharacteristic
157C2×D236C22×C411yescharacteristic
158C2×C4×D59—21no—
159C2×C4×D59—21no—
160C2×C4×D59—21no—
161C2×Dic118—21no—
162C2×Dic118C22×C411yescharacteristic
163M16×C59—22no—
164M16×C59—22no—
165M16×C59—41no—
166M16×C59—41no—
167M16×C59—42no—
168C2×C8×C59—21no—
169C2×C8×C59—21no—
170C2×C8×C59C22×C411yescharacteristic
171C2×C8×C59C22×C411yescharacteristic
172C2×C8×C59C22×C411yescharacteristic
173C2×C8×C59C22×C411yescharacteristic
174C2×C8×C59C2×D411yescharacteristic
175C2×C8×C59C2×D411yescharacteristic
176C2×C8×C59C2×D411yescharacteristic
177C2×C8×C59C2×D411yescharacteristic
178C42×C59C22×C411yescharacteristic
179C42×C59C2×D411yescharacteristic
180C42×C59#4111yescharacteristic
181C42×C59C2411yescharacteristic
182C22⋊Dic59—21no—
183(C2×D59)⋊C4—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—
190(C2×D59)⋊C4—21no—
191C4⋊Dic59—21no—
192C4⋊Dic59C22×C411yescharacteristic
193C4⋊Dic59C22×C411yescharacteristic
194C4⋊Dic59C22×C411yescharacteristic
195C4⋊Dic59C22×C411yescharacteristic
196Dic59⋊C4—21no—
197Dic59⋊C4—21no—
198Dic59⋊C4—21no—
199Dic59⋊C4—21no—
200C4×Dic59—21no—
201C2×C59⋊C8—21no—
202C2×C59⋊C8—21no—
203C2×C59⋊C8—21no—
204C2×C59⋊C8—21no—
205C2×C59⋊C8—42no—
206D472—22no—
207D472—41no—
208#11745660—22no—
209#11745660—41no—
210#11745659—42no—
211#11745659—42no—
212#11745659—42no—
213C8×D59—42no—
214C8×D59—42no—
215C8×D59—42no—
216C23×C59C4×D411yescharacteristic
217C22×D59—21no—
218C22×D59—21no—
219C22×D59—21no—
220C2×C4×C59—21no—
221C2×C4×C59—21no—
222C2×C4×C59—21no—
223C2×C4×C59—21no—
224C2×C4×C59—41no—
225C2×C4×C59C4×D411yescharacteristic
226C2×C4×C59C4×D411yescharacteristic
227C2×C4×C59C4×D411yescharacteristic
228C2×C4×C59C23×C411yescharacteristic
229C2×C4×C59C22×D411yescharacteristic
230C2×C4×C59C2×#4111yescharacteristic
231C22⋊D59—42no—
232C22⋊D59—42no—
233C22⋊D59—81no—
234C2×Dic59—21no—
235C2×Dic59—21no—
236C2×Dic59—21no—
237C2×Dic59—21no—
238C2×Dic59—21no—
239D236—21no—
240D236—21no—
241D236—21no—
242D236—21no—
243D236—21no—
244D236—21no—
245D236—42no—
246D236—81no—
247C4×D59—41no—
248C4×D59—41no—
249C4×D59—41no—
250C4×D59—41no—
251C4×D59—41no—
252C4×D59—41no—
253Dic118—21no—
254Dic118—21no—
255Dic118—42no—
256C8×C59—22no—
257C8×C59—21no—
258C8×C59—21no—
259C8×C59—21no—
260C8×C59—21no—
261C8×C59—21no—
262C8×C59—21no—
263C8×C59—41no—
264C8×C59#13112yescomplemented
265C8×C59AGL(1,ℤ/8)12yes—
266C59⋊C8—41no—
267C59⋊C8—41no—
268C59⋊C8—41no—
269C59⋊C8—41no—
270#18078—591nomaximal, Sylow
271C22×C59—21no—
272C22×C59—21no—
273C22×C59—21no—
274C22×C59C2×C4×D411yescharacteristic
275C2×D59—41no—
276C2×D59—41no—
277C2×D59—41no—
278C2×D59—41no—
279C2×D59—41no—
280C2×D59—41no—
281C2×D59—41no—
282C2×D59—41no—
283C2×D59—41no—
284C4×C59—21no—
285C4×C59—21no—
286C4×C59—21no—
287C4×C59—21no—
288C4×C59C2×#13111yescharacteristic
289C4×C59#56811yescharacteristic
290C4×C59C2×AGL(1,ℤ/8)11yescharacteristic
291C4×C59#57511yescharacteristic
292Dic59—41no—
293Dic59—41no—
294Dic59—41no—
295Dic59—41no—
296Dic59—41no—
297#3171—591no—
298#2969—591no—
299#2935—591no—
300#1592—591no—
301#1589—591no—
302#1586—591no—
303#1585—591no—
304#1583—591no—
305#1581—591no—
306#1580—591no—
307C8×SD16—591no—
308#1565—591no—
309#1484—591no—
310#1483—591no—
311#1455D5911yescharacteristic, complemented
312C2×C59—41no—
313C2×C59#293511yescharacteristic
314C2×C59C4×AGL(1,ℤ/8)11yescharacteristic
315C2×C59#295611yescharacteristic
316D59—81no—
317D59—81no—
318D59—81no—
319C2×#131—1181no—
320#532—591no—
321#529—591no—
322#501—591no—
323#496—591no—
324#490—591no—
325#489—591no—
326#469—591no—
327#469—591no—
328#468—591no—
329#468—591no—
330#446—591no—
331C8×Q8—591no—
332C4×SD16—591no—
333C4×SD16—591no—
334C4×D8—591no—
335C4×D8—591no—
336#436—591no—
337#436—591no—
338#436—591no—
339#436—1181no—
340#435—1181no—
341#435—1181no—
342#435—1181no—
343C8×D4—591no—
344C8×D4—591no—
345C8×D4—591no—
346#433—1181no—
347#431—591no—
348C2×C4⋊C8—591no—
349#408—1181no—
350#405—1181no—
351C4×M16C2×D5911yescharacteristic
352C4×M16C2×D5911yescharacteristic
353C4×M16C2×D5911yescharacteristic
354#335—591no—
355#335—591no—
356#334—591no—
357#334—591no—
358Q8⋊C8—591no—
359Q8⋊C8—591no—
360D4⋊C8—591no—
361D4⋊C8—591no—
362D4⋊C8—591no—
363D4⋊C8—591no—
364D4⋊C8—591no—
365D4⋊C8—591no—
366#322C2×D5911yescharacteristic
367#322C2×D5911yescharacteristic
368#322C2×D5911yescharacteristic
369C82C2×D5911yescharacteristic
370C59#1807811yescharacteristic, complemented, Sylow
371C2×#41—1181no—
372C2×SD16—591no—
373C2×SD16—591no—
374C2×D8—591no—
375C2×D8—591no—
376#131—2364no—
377C2×M16—21no—
378C2×M16—1181no—
379C2×M16—1181no—
380C2×M16—1181no—
381C2×M16C4×D5911yescharacteristic
382C2×M16C4×D5911yescharacteristic
383C2×M16D23611yescharacteristic
384C2×M16D23611yescharacteristic
385C22×C8—1181no—
386C22×C8—1181no—
387C22×C8—1181no—
388C22×C8—1181no—
389C4⋊Q8—591no—
390#127—591no—
391#126—1181no—
392#125—591no—
393#124—591no—
394#124—1181no—
395#123—1181no—
396C22⋊Q8—591no—
397#121—591no—
398#121—591no—
399#121—591no—
400#121—1181no—
401C4×Q8—591no—
402C4×D4—591no—
403C4×D4—591no—
404C4×D4—591no—
405C4×D4—1181no—
406C4×D4—1181no—
407#117—1181no—
408C2×C42C22×D5911yescharacteristic
409#107—591no—
410#107—591no—
411#106—591no—
412#106—591no—
413C4⋊C8—591no—
414C4⋊C8—591no—
415C4⋊C8—591no—
416C4⋊C8—591no—
417C4⋊C8—1181no—
418C4⋊C8—1181no—
419C4⋊C8—1181no—
420#103—591no—
421#103—591no—
422#102—591no—
423#102—591no—
424#102—591no—
425#102—591no—
426#102—591no—
427#102—591no—
428C22⋊C8—1181no—
429C22⋊C8—1181no—
430C22⋊C8—1181no—
431C22⋊C8—1181no—
432C22⋊C8—1181no—
433C22⋊C8—1181no—
434C22⋊C8—1181no—
435#97—21no—
436#97—1181no—
437#97C4×D5911yescharacteristic
438#97C4×D5911yescharacteristic
439#97D23611yescharacteristic
440#97D23611yescharacteristic
441C4×C8—1181no—
442C4×C8—1181no—
443C4×C8C22×D5911yescharacteristic
444C4×C8C22×D5911yescharacteristic
445C4×C8C22×D5911yescharacteristic
446C4×C8C22×D5911yescharacteristic
447C4×C8C22×D5911yescharacteristic
448C4×C8C22×D5911yescharacteristic
449#41—2362no—
450#41—2362no—
451C2×Q8—591no—
452C2×Q8—1181no—
453C2×D4—591no—
454C2×D4—591no—
455C2×D4—591no—
456C2×D4—1181no—
457C2×D4—1181no—
458C2×D4—1181no—
459C2×D4—1181no—
460C2×D4—1181no—
461C22×C4—1181no—
462C22×C4—1181no—
463C22×C4—1181no—
464C22×C4C2×C4×D5911yescharacteristic
465C22×C4C2×D23611yescharacteristic
466C22×C4#1174568511yescharacteristic
467SD16—1182no—
468SD16—2361no—
469D8—1182no—
470D8—2361no—
471M16—22no—
472M16—22no—
473M16—41no—
474M16—41no—
475M16—42no—
476M16—2362no—
477M16—2362no—
478M16—2362no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—1181no—
482C2×C8—1181no—
483C2×C8—1181no—
484C2×C8—1181no—
485C2×C8—2362no—
486C2×C8—2362no—
487C2×C8—2362no—
488C2×C8—2362no—
489C2×C8C2×C4×D5911yescharacteristic
490C2×C8C2×C4×D5911yescharacteristic
491C2×C8C2×C4×D5911yescharacteristic
492C2×C8C2×C4×D5911yescharacteristic
493C2×C8C2×D23611yescharacteristic
494C2×C8C2×D23611yescharacteristic
495C2×C8C2×D23611yescharacteristic
496C2×C8C2×D23611yescharacteristic
497C4⋊C4—591no—
498C4⋊C4—591no—
499C4⋊C4—591no—
500C4⋊C4—591no—
501C4⋊C4—1181no—
502C4⋊C4—1181no—
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—
513C22⋊C4—1181no—
514C22⋊C4—1181no—
515C42—1181no—
516C42C2×C4×D5911yescharacteristic
517C42C2×D23611yescharacteristic
518C42#1174568511yescharacteristic
519C42C23×D5911yescharacteristic
520C23—1181no—
521C23—1181no—
522C23—1181no—
523C23C4×D23611yescharacteristic
524Q8—1181no—
525Q8—1181no—
526Q8—2362no—
527D4—1181no—
528D4—1181no—
529D4—1181no—
530D4—1181no—
531D4—1181no—
532D4—1181no—
533D4—4721no—
534D4—4721no—
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—2361no—
549C2×C4—2361no—
550C2×C4—2361no—
551C2×C4—2361no—
552C2×C4—2361no—
553C2×C4—2361no—
554C2×C4C4×D23611yescharacteristic
555C2×C4C4×D23611yescharacteristic
556C2×C4C4×D23611yescharacteristic
557C2×C4C22×C4×D5911yescharacteristic
558C2×C4C22×D23611yescharacteristic
559C2×C4C2×#1174568511yescharacteristic
560C8—22no—
561C8—21no—
562C8—21no—
563C8—21no—
564C8—21no—
565C8—21no—
566C8—21no—
567C8—41no—
568C8—2361no—
569C8—2361no—
570C8—2361no—
571C8—2361no—
572C8#1427373512yescomplemented
573C8#1427375512yes—
574C22—21no—
575C22—21no—
576C22—21no—
577C22—2361no—
578C22—2361no—
579C22—2361no—
580C22—2361no—
581C22—2361no—
582C22—2361no—
583C22—2361no—
584C22—2361no—
585C22—2361no—
586C22C2×C4×D23611yescharacteristic
587C4—21no—
588C4—21no—
589C4—21no—
590C4—21no—
591C4—2361no—
592C4—2361no—
593C4—2361no—
594C4—2361no—
595C4—2361no—
596C4C2×#1427373511yescharacteristic
597C4#1683816811yescharacteristic
598C4C2×#1427375511yescharacteristic
599C4#1683820211yescharacteristic
600C2—41no—
601C2—4721no—
602C2—4721no—
603C2—4721no—
604C2#2143284211yescharacteristic
605C2C4×#1427375511yescharacteristic
606C2#2143300711yescharacteristic
subgroupG/H
centerC2×C4C22×D236
commutatorC2×C4×C59C23×C4
FrattiniC42C23×D59
FittingC59×#1455C2
socleC22×C59C2×C4×D4
prime radical#21434691C2
cosocleC42×C59C24
hypercenter#1455D59
hypocenterC59#18078
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C4×M16 ◁ #1455

Complements

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

NH ≅ G/Ntype
2. #21433100C2semidirect
3. #21432842C2semidirect
4. #21423000C2semidirect
5. #21422982C2semidirect
6. #21422962C2semidirect
7. #21422955C2semidirect
8. #21422941C2semidirect
9. #21422927C2semidirect
10. #21422921C2semidirect
11. C8×#11745660C2semidirect
12. #21422848C2semidirect
13. #21422514C2semidirect
14. #21422510C2semidirect
15. C59×#1455C2semidirect
264. C8×C59#131semidirect
311. #1455D59semidirect
370. C59#18078semidirect
572. C8#14273735semidirect

Maximal subgroups

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

subgroupindex
#214346912
#214331002
#214328422
#214230002
#214229822
#214229622
#214229552
#214229412
#214229272
#214229212
C8×#117456602
#214228482
#214225142
#214225102
C59×#14552
#1807859