Finite Group Databasebeta

← #28646822

Subgroups — #28646822

Order 15104.

734 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21429133C211yesmaximal, characteristic, complemented
2#21427044C211yesmaximal, characteristic
3#21425480C211yesmaximal, characteristic
4#21425242C211yesmaximal, characteristic
5#21425240C211yesmaximal, characteristic
6#21425052C211yesmaximal, characteristic, complemented
7#21425051C211yesmaximal, characteristic, complemented
8#21424851C211yesmaximal, characteristic, complemented
9#21424811C211yesmaximal, characteristic, complemented
10#21424690C211yesmaximal, characteristic, complemented
11#21424685C211yesmaximal, characteristic, complemented
12#21424222C211yesmaximal, characteristic, complemented
13#21424094C211yesmaximal, characteristic
14#102×Dic59C211yesmaximal, characteristic
15C59×#2067C211yesmaximal, characteristic
16C2×#14273719C2211yescharacteristic
17C2×#14273718C2211yescharacteristic
18C2×D4×Dic59C2211yescharacteristic
19C2×#14273683—21no—
20C2×#14273660C2211yescharacteristic
21C2×Dic59⋊C8C2211yescharacteristic
22C2×Dic59⋊C8C2211yescharacteristic
23C2×#14273648C2211yescharacteristic
24C2×#14273648C2211yescharacteristic
25C2×C8×Dic59C2211yescharacteristic
26#16837336—21no—
27#16837335—21no—
28#16837305C2211yescharacteristic
29#16837301C2211yescharacteristic
30#16837299—21no—
31#16837296—21no—
32#16837290—21no—
33#16837271—21no—
34#16837271C2211yescharacteristic
35#16837271C2211yescharacteristic
36#16837260—21no—
37C4⋊C4×Dic59C2211yescharacteristic
38C4⋊C4×Dic59C2211yescharacteristic
39C4×Dic59⋊C4C2211yescharacteristic
40C4×C4⋊Dic59C2211yescharacteristic
41#16837165C2211yescharacteristic
42#16837165C2211yescharacteristic
43#16837165C2211yescharacteristic
44#16837165C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837164C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837157C2211yescharacteristic
51#16837156C2211yescharacteristic
52C2×#107×C59C2211yescharacteristic
53C2×C4⋊C8×C59C2211yescharacteristic
54C2×#102×C59C2211yescharacteristic
55C2×#102×C59C2211yescharacteristic
56C59×#390C2211yescharacteristic
57C59×#389C2211yescharacteristic
58C59×#340C2211yescharacteristic
59C23×Dic59—21no—
60C22×C59⋊C8C2311yescharacteristic
61C22×C59⋊C8C2311yescharacteristic
62#14273719—22no—
63#14273719—41no—
64#14273718—22no—
65#14273718—41no—
66D4×Dic59—22no—
67D4×Dic59Q814yes—
68#14273683—44no—
69C2×C4⋊Dic59—21no—
70C2×C4⋊Dic59—21no—
71C2×C4⋊Dic59C2311yescharacteristic
72C2×C4⋊Dic59C2311yescharacteristic
73C2×C4⋊Dic59C2311yescharacteristic
74C2×C4⋊Dic59C2311yescharacteristic
75C2×Dic59⋊C4—21no—
76C2×Dic59⋊C4—21no—
77C2×Dic59⋊C4—21no—
78C2×Dic59⋊C4—21no—
79C2×Dic59⋊C4—21no—
80C2×Dic59⋊C4—21no—
81C2×Dic59⋊C4—21no—
82C2×Dic59⋊C4D411yescharacteristic
83C2×Dic59⋊C4D411yescharacteristic
84C2×C22⋊Dic59—21no—
85C2×C22⋊Dic59—21no—
86C2×C22⋊Dic59—21no—
87C2×C4×Dic59—21no—
88C2×C4×Dic59—21no—
89C2×C4×Dic59—21no—
90C2×C4×Dic59C2311yescharacteristic
91C2×C4×Dic59C2311yescharacteristic
92#14273660—22no—
93Dic59⋊C8—22no—
94Dic59⋊C8—41no—
95#14273648—22no—
96#14273648—41no—
97C8×Dic59—22no—
98C59⋊#95—21no—
99C59⋊#95—21no—
100C59⋊#95—21no—
101C59⋊#95—21no—
102C59⋊#95D411yescharacteristic
103C59⋊#95D411yescharacteristic
104C22×D4×C59C2311yescharacteristic
105C22×C8×C59C2311yescharacteristic
106C22×C8×C59C2311yescharacteristic
107C2×C4⋊C4×C59—21no—
108C2×C4⋊C4×C59C2311yescharacteristic
109C2×C4⋊C4×C59C2311yescharacteristic
110C2×C4⋊C4×C59C2311yescharacteristic
111C2×C22⋊C4×C59—21no—
112C2×C22⋊C4×C59—21no—
113C2×C42×C59C2311yescharacteristic
114#107×C59—22no—
115C4⋊C8×C59—41no—
116#102×C59—22no—
117#102×C59—41no—
118#95×C59—21no—
119C24×C59—21no—
120C2×D4×C59—21no—
121C2×D4×C59—21no—
122C2×D4×C59—21no—
123C2×D4×C59—21no—
124C2×D4×C59—21no—
125C2×D4×C59C2×Q812yes—
126C22×C4×C59—21no—
127C22×C4×C59—21no—
128C22×C4×C59—21no—
129C22×C4×C59C2×D411yescharacteristic
130C22×C4×C59C2×D411yescharacteristic
131C22×C4×C59C2411yescharacteristic
132C22×Dic59—21no—
133C22×Dic59—21no—
134C22×Dic59—24no—
135C22×Dic59—21no—
136C22×Dic59—21no—
137C22×Dic59—42no—
138C22×Dic59—42no—
139C22×Dic59C2×D411yescharacteristic
140C22×Dic59C2×D411yescharacteristic
141C22×Dic59C2×D411yescharacteristic
142C22×Dic59C2×D411yescharacteristic
143C2×C8×C59—21no—
144C2×C8×C59—21no—
145C2×C8×C59—21no—
146C2×C8×C59—21no—
147C2×C8×C59—21no—
148C2×C8×C59—21no—
149C4⋊C4×C59—21no—
150C4⋊C4×C59—21no—
151C4⋊C4×C59—21no—
152C4⋊C4×C59—21no—
153C4⋊C4×C59—21no—
154C4⋊C4×C59—81no—
155C4⋊C4×C59#4112yes—
156C22⋊C4×C59—42no—
157C22⋊C4×C59—81no—
158C42×C59—21no—
159C42×C59—21no—
160C22⋊Dic59—24no—
161C22⋊Dic59—42no—
162C22⋊Dic59—81no—
163C4⋊Dic59—21no—
164C4⋊Dic59—21no—
165C4⋊Dic59—21no—
166C4⋊Dic59—21no—
167C4⋊Dic59—21no—
168C4⋊Dic59—21no—
169C4⋊Dic59—42no—
170C4⋊Dic59—81no—
171C4⋊Dic59C2×Q812yes—
172C4⋊Dic59#4112yes—
173Dic59⋊C4—22no—
174Dic59⋊C4—22no—
175Dic59⋊C4—42no—
176Dic59⋊C4—42no—
177Dic59⋊C4—42no—
178Dic59⋊C4—42no—
179Dic59⋊C4—42no—
180Dic59⋊C4—81no—
181Dic59⋊C4—81no—
182C4×Dic59—21no—
183C4×Dic59—21no—
184C4×Dic59—42no—
185C4×Dic59—42no—
186C4×Dic59—42no—
187C4×Dic59C2×Q812yes—
188C4×Dic59#4112yes—
189C2×C59⋊C8—21no—
190C2×C59⋊C8—21no—
191C2×C59⋊C8—21no—
192C2×C59⋊C8—21no—
193C2×C59⋊C8—21no—
194C2×C59⋊C8—21no—
195C23×C59—22no—
196C23×C59—41no—
197C23×C59—41no—
198C23×C59—41no—
199C23×C59—41no—
200C23×C59—41no—
201C23×C59—41no—
202C23×C59C22×D411yescharacteristic
203D4×C59—22no—
204D4×C59—22no—
205D4×C59—41no—
206D4×C59—41no—
207C2×C4×C59—21no—
208C2×C4×C59—21no—
209C2×C4×C59—22no—
210C2×C4×C59—21no—
211C2×C4×C59—21no—
212C2×C4×C59—21no—
213C2×C4×C59—21no—
214C2×C4×C59—41no—
215C2×C4×C59—41no—
216C2×C4×C59—41no—
217C2×C4×C59—41no—
218C2×C4×C59—41no—
219C2×C4×C59—41no—
220C2×C4×C59—41no—
221C2×C4×C59—41no—
222C2×C4×C59C22×Q811yescharacteristic
223C2×C4×C59C2×#4111yescharacteristic
224C2×C4×C59C2×#4111yescharacteristic
225C2×C4×C59C2×#4111yescharacteristic
226C2×C4×C59H(5,𝔽2)11yescharacteristic
227C2×C4×C592−1+411yescharacteristic
228C2×Dic59—21no—
229C2×Dic59—21no—
230C2×Dic59—22no—
231C2×Dic59—21no—
232C2×Dic59—21no—
233C2×Dic59—22no—
234C2×Dic59—21no—
235C2×Dic59—21no—
236C2×Dic59—21no—
237C2×Dic59—21no—
238C2×Dic59—41no—
239C2×Dic59—41no—
240C2×Dic59—41no—
241C2×Dic59—41no—
242C2×Dic59—48no—
243C2×Dic59—41no—
244C2×Dic59—41no—
245C2×Dic59—41no—
246C2×Dic59—41no—
247C2×Dic59—41no—
248C2×Dic59—41no—
249C2×Dic59C22⋊Q812yes—
250C2×Dic59C22⋊Q812yes—
251C2×Dic59#13512yes—
252C2×Dic59AGL(1,ℤ/8)12yes—
253C8×C59—41no—
254C8×C59—41no—
255C59⋊C8—41no—
256C59⋊C8—41no—
257#26025—591nomaximal, Sylow
258C22×C59—42no—
259C22×C59—42no—
260C22×C59—42no—
261C22×C59—81no—
262C22×C59—81no—
263C22×C59—81no—
264C22×C59—81no—
265C22×C59C2×C22⋊Q811yescharacteristic
266C22×C59#54611yescharacteristic
267C22×C59#54811yescharacteristic
268C22×C59C2×#13511yescharacteristic
269C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
270C22×C59#57611yescharacteristic
271C22×C59#57811yescharacteristic
272C4×C59—41no—
273C4×C59—42no—
274C4×C59—41no—
275C4×C59—81no—
276C4×C59—81no—
277C4×C59#53311yescharacteristic
278C4×C59#54411yescharacteristic
279C4×C59#55511yescharacteristic
280C4×C59#55611yescharacteristic
281Dic59—22no—
282Dic59—42no—
283Dic59—41no—
284Dic59—42no—
285Dic59—41no—
286Dic59—81no—
287Dic59—81no—
288Dic59#47814yes—
289#2552—591no—
290#2354—591no—
291#2067—591no—
292#2067D5911yescharacteristic
293#2025—591no—
294#2025—591no—
295#1989—591no—
296#1989—591no—
297#1947—591no—
298#1939—591no—
299#1921—591no—
300#1921—591no—
301#1834—591no—
302#1802—591no—
303C4×#102—591no—
304C2×C59—82no—
305C2×C59#243911yescharacteristic
306C2×C59C2×#47811yescharacteristic
307C2×C59#308311yescharacteristic
308C2×C59#330511yescharacteristic
309C2×C59#333011yescharacteristic
310C2×C59#336411yescharacteristic
311C2×C59#336611yescharacteristic
312C2×#125—591no—
313C2×#125—591no—
314C2×C4×D4—591no—
315C2×#117—1181no—
316C2×#107—591no—
317C2×#107C2×D5911yescharacteristic
318C2×C4⋊C8—591no—
319C2×C4⋊C8—591no—
320C2×C4⋊C8C2×D5911yescharacteristic
321C2×#102—591no—
322C2×#102—591no—
323C2×#102C2×D5911yescharacteristic
324C2×#102C2×D5911yescharacteristic
325C2×C4×C8—591no—
326#398—1181no—
327#398—1181no—
328#390—591no—
329#390C2×D5911yescharacteristic
330#389—591no—
331#389—1181no—
332#389C2×D5911yescharacteristic
333#388—1181no—
334#388—1181no—
335#384—591no—
336#384—591no—
337#384—1181no—
338#380—1181no—
339C4×C4⋊C4—591no—
340C4×C4⋊C4—591no—
341C4×C4⋊C4—591no—
342C4×C4⋊C4—591no—
343#340—591no—
344#340—591no—
345#340—591no—
346#340—591no—
347#340—591no—
348#340—591no—
349#340—591no—
350#340—591no—
351#340—591no—
352#340C2×D5911yescharacteristic
353#336—591no—
354#336—591no—
355C59#2602511yescharacteristic, complemented, Sylow
356C22×D4C22×D5911yescharacteristic
357C23×C4—1181no—
358C22×C8—591no—
359C22×C8—591no—
360C22×C8C22×D5911yescharacteristic
361C22×C8C22×D5911yescharacteristic
362#125—1182no—
363#125—1182no—
364#125—2361no—
365#125—2361no—
366C4×D4—594no—
367C4×D4—1182no—
368#117—2364no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—591no—
371C2×C4⋊C4—591no—
372C2×C4⋊C4—591no—
373C2×C4⋊C4—591no—
374C2×C4⋊C4—591no—
375C2×C4⋊C4—591no—
376C2×C4⋊C4—1181no—
377C2×C4⋊C4—1181no—
378C2×C4⋊C4—1181no—
379C2×C4⋊C4—1181no—
380C2×C4⋊C4—1181no—
381C2×C4⋊C4—1181no—
382C2×C4⋊C4—1181no—
383C2×C4⋊C4—1181no—
384C2×C4⋊C4—1181no—
385C2×C4⋊C4C22×D5911yescharacteristic
386C2×C4⋊C4C22×D5911yescharacteristic
387C2×C4⋊C4C22×D5911yescharacteristic
388C2×C22⋊C4—21no—
389C2×C22⋊C4—21no—
390C2×C22⋊C4—1181no—
391C2×C22⋊C4—1181no—
392C2×C22⋊C4—1181no—
393C2×C42—591no—
394C2×C42—591no—
395C2×C42—1181no—
396C2×C42—1181no—
397C2×C42—1181no—
398C2×C42C22×D5911yescharacteristic
399#107—22no—
400#107—1182no—
401C4⋊C8—41no—
402C4⋊C8—1182no—
403C4⋊C8—2361no—
404#102—22no—
405#102—41no—
406#102—1182no—
407#102—2361no—
408C4×C8—1182no—
409#95—21no—
410#95—591no—
411#95—591no—
412#95—1181no—
413#95—1181no—
414#95—1181no—
415#95—1181no—
416C24—21no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4Q8×D5912yes—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—591no—
427C22×C4—591no—
428C22×C4—591no—
429C22×C4—591no—
430C22×C4—1184no—
431C22×C4—1181no—
432C22×C4—1181no—
433C22×C4—1181no—
434C22×C4—1181no—
435C22×C4—2362no—
436C22×C4—2362no—
437C22×C4D4×D5911yescharacteristic
438C22×C4D4×D5911yescharacteristic
439C22×C4C23×D5911yescharacteristic
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—1181no—
447C2×C8—1181no—
448C2×C8—1181no—
449C2×C8—1181no—
450C2×C8—1181no—
451C2×C8—1181no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—81no—
458C4⋊C4—592no—
459C4⋊C4—592no—
460C4⋊C4—1182no—
461C4⋊C4—1182no—
462C4⋊C4—1181no—
463C4⋊C4—1181no—
464C4⋊C4—1181no—
465C4⋊C4—1181no—
466C4⋊C4—1181no—
467C4⋊C4—1181no—
468C4⋊C4—4721no—
469C4⋊C4—4721no—
470C4⋊C4—4721no—
471C4⋊C4—2362no—
472C4⋊C4—2362no—
473C4⋊C4—2362no—
474C4⋊C4—2362no—
475C4⋊C4—2362no—
476C4⋊C4—2362no—
477C4⋊C4#1174568712yes—
478C22⋊C4—42no—
479C22⋊C4—81no—
480C22⋊C4—1184no—
481C22⋊C4—4721no—
482C22⋊C4—2362no—
483C42—21no—
484C42—21no—
485C42—592no—
486C42—592no—
487C42—1181no—
488C42—1181no—
489C42—2362no—
490C42—2362no—
491C42—2362no—
492C23—22no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23C2×D4×D5911yescharacteristic
500D4—22no—
501D4—22no—
502D4—41no—
503D4—41no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—22no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—592no—
520C2×C4—592no—
521C2×C4—592no—
522C2×C4—592no—
523C2×C4—1181no—
524C2×C4—1181no—
525C2×C4—1182no—
526C2×C4—1181no—
527C2×C4—1181no—
528C2×C4—1181no—
529C2×C4—1181no—
530C2×C4—1181no—
531C2×C4—1181no—
532C2×C4—1182no—
533C2×C4—2368no—
534C2×C4—2361no—
535C2×C4—2361no—
536C2×C4—2361no—
537C2×C4—2361no—
538C2×C4—2361no—
539C2×C4—2361no—
540C2×C4—2361no—
541C2×C4—2361no—
542C2×C4—2361no—
543C2×C4—2361no—
544C2×C4C2×Q8×D5911yescharacteristic
545C2×C4#41×D5911yescharacteristic
546C2×C4#41×D5911yescharacteristic
547C2×C4C2×#1174568711yescharacteristic
548C2×C4#1427377211yescharacteristic
549C2×C4#1427377411yescharacteristic
550C8—41no—
551C8—41no—
552C8—2361no—
553C8—2361no—
554C22—42no—
555C22—42no—
556C22—42no—
557C22—81no—
558C22—81no—
559C22—81no—
560C22—81no—
561C22C22⋊Q8×D5911yescharacteristic
562C22#1683804111yescharacteristic
563C22#1683805811yescharacteristic
564C22#135×D5911yescharacteristic
565C22AGL(1,ℤ/8)×D5911yescharacteristic
566C22#1683820311yescharacteristic
567C22#1683821811yescharacteristic
568C4—42no—
569C4—41no—
570C4—41no—
571C4—81no—
572C4—81no—
573C4—594no—
574C4—1182no—
575C4—4721no—
576C4—4721no—
577C4—2361no—
578C4—2361no—
579C4—2362no—
580C4—2362no—
581C4#1683795411yescharacteristic
582C4#1683803111yescharacteristic
583C4#1683811611yescharacteristic
584C4#1683811911yescharacteristic
585C2—82no—
586C2#2142778211yescharacteristic
587C2#478×D5911yescharacteristic
588C2#2143394211yescharacteristic
589C2#2143604111yescharacteristic
590C2#2143639611yescharacteristic
591C2#2143677611yescharacteristic
592C2#2143679811yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2067C2
socleC23×C59C22×D4
prime radicalC22×D4×C59C23
hypercenter#2067D59
hypocenterC59#26025
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2067

Complements

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

NH ≅ G/Ntype
1. #21429133C2semidirect
6. #21425052C2semidirect
7. #21425051C2semidirect
8. #21424851C2semidirect
9. #21424811C2semidirect
10. #21424690C2semidirect
11. #21424685C2semidirect
12. #21424222C2semidirect
355. C59#26025semidirect

Maximal subgroups

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

subgroupindex
#214291332
#214270442
#214254802
#214252422
#214252402
#214250522
#214250512
#214248512
#214248112
#214246902
#214246852
#214242222
#214240942
#102×Dic592
C59×#20672
#2602559