Finite Group Databasebeta

← #28691905

Subgroups — #28691905

Order 15104.

674 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21426740C211yesmaximal, characteristic
2#21425598C211yesmaximal, characteristic, complemented
3#21425491C211yesmaximal, characteristic, complemented
4#21425481C211yesmaximal, characteristic
5#21425411C211yesmaximal, characteristic
6#21424953C211yesmaximal, characteristic
7#21424826C211yesmaximal, characteristic, complemented
8#21424792C211yesmaximal, characteristic
9#21424789C211yesmaximal, characteristic, complemented
10#21424734C211yesmaximal, characteristic, complemented
11#21424678C211yesmaximal, characteristic, complemented
12#21424433C211yesmaximal, characteristic
13#21424342C211yesmaximal, characteristic
14#21423953C211yesmaximal, characteristic, complemented
15C59×#2645C211yesmaximal, characteristic, complemented
16C22×#11745671C2211yescharacteristic
17C2×C4×D236C2211yescharacteristic
18C2×#14273660C2211yescharacteristic
19C2×#14273660C2211yescharacteristic
20C2×#14273658C2211yescharacteristic
21C2×C4⋊(C59⋊C8)C2211yescharacteristic
22C2×C4⋊(C59⋊C8)C2211yescharacteristic
23C2×#14273651C2211yescharacteristic
24C2×#14273651C2211yescharacteristic
25C2×#14273651C2211yescharacteristic
26C2×#14273647C2211yescharacteristic
27C2×#14273647C2211yescharacteristic
28C2×#14273647C2211yescharacteristic
29C2×#14273638C2211yescharacteristic
30C2×C4×C59⋊C8C2211yescharacteristic
31#16837304C2211yescharacteristic
32#16837304C2211yescharacteristic
33#16837300C2211yescharacteristic
34#16837300C2211yescharacteristic
35#16837291—21no—
36#16837291—21no—
37#16837264—21no—
38#16837261—21no—
39C4×C4⋊Dic59C2211yescharacteristic
40#16837164C2211yescharacteristic
41#16837164C2211yescharacteristic
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837157C2211yescharacteristic
47#16837157C2211yescharacteristic
48C59×#401—21no—
49C59×#400—21no—
50C59×#400—21no—
51C59×#400—21no—
52C59×#391C2211yescharacteristic
53C59×#391C2211yescharacteristic
54C59×#391C2211yescharacteristic
55C59×#389C2211yescharacteristic
56C59×#389C2211yescharacteristic
57C59×#389C2211yescharacteristic
58C59×#376C2211yescharacteristic
59C22×D236C2311yescharacteristic
60C22×C4×D59—21no—
61C2×#11745671—22no—
62C2×#11745671—22no—
63C2×#11745671—41no—
64C22×C59⋊C8C2311yescharacteristic
65C22×C59⋊C8C2311yescharacteristic
66C22×C59⋊C8C2311yescharacteristic
67C22×C59⋊C8C2311yescharacteristic
68C4×D236—22no—
69C4×D236—22no—
70C2×C4⋊Dic59C2311yescharacteristic
71C2×C4⋊Dic59C2311yescharacteristic
72C2×C4⋊Dic59C2311yescharacteristic
73C2×Dic59⋊C4—21no—
74C2×Dic59⋊C4—21no—
75C2×(C2×D59)⋊C4—21no—
76C2×(C2×D59)⋊C4—21no—
77C2×(C2×D59)⋊C4—21no—
78C2×(C2×D59)⋊C4—21no—
79C2×(C2×D59)⋊C4—21no—
80C2×C4×Dic59—21no—
81#14273660—22no—
82#14273660—41no—
83#14273658—22no—
84C4⋊(C59⋊C8)—22no—
85C4⋊(C59⋊C8)—22no—
86#14273651—22no—
87#14273651—22no—
88#14273651—41no—
89#14273647—22no—
90#14273647—22no—
91#14273647—41no—
92#14273638—41no—
93C4×C59⋊C8—41no—
94C59⋊#95—21no—
95C59⋊#95—21no—
96C59⋊#95—21no—
97C22×Q8×C59C2311yescharacteristic
98C2×C4⋊C4×C59—21no—
99C2×C4⋊C4×C59—21no—
100C2×C4⋊C4×C59—21no—
101C2×C4⋊C4×C59C2311yescharacteristic
102C2×C4⋊C4×C59C2311yescharacteristic
103C2×C4⋊C4×C59C2311yescharacteristic
104C2×C42×C59C2311yescharacteristic
105C2×C42×C59C2311yescharacteristic
106C2×C42×C59C2311yescharacteristic
107#95×C59—21no—
108#95×C59—21no—
109#95×C59—21no—
110#95×C59—21no—
111#95×C59—21no—
112#95×C59—21no—
113#95×C59—21no—
114#95×C59—21no—
115#95×C59—21no—
116#95×C59D411yescharacteristic
117#95×C59D411yescharacteristic
118#95×C59D411yescharacteristic
119#95×C59D411yescharacteristic
120C23×D59—21no—
121C2×Q8×C59—21no—
122C2×Q8×C59—21no—
123C2×Q8×C59—21no—
124C2×Q8×C59—21no—
125C2×Q8×C59—21no—
126C2×Q8×C59—21no—
127C22×C4×C59—21no—
128C22×C4×C59—21no—
129C22×C4×C59—21no—
130C22×C4×C59—21no—
131C22×C4×C59C2×D411yescharacteristic
132C22×C4×C59C2×D411yescharacteristic
133C22×C4×C59C2×D411yescharacteristic
134C22×C4×C59C2×D411yescharacteristic
135C22×C4×C59C2×D411yescharacteristic
136C22×C4×C59C2×D411yescharacteristic
137C22×C4×C59C2411yescharacteristic
138C22×Dic59—21no—
139C22×Dic59—21no—
140C22×Dic59—21no—
141C2×D236—21no—
142C2×D236—21no—
143C2×D236—21no—
144C2×D236—21no—
145C2×D236—21no—
146C2×D236#4112yes—
147C2×C4×D59—24no—
148C2×C4×D59—81no—
149C2×C4×D59—81no—
150C4⋊C4×C59—21no—
151C4⋊C4×C59—21no—
152C4⋊C4×C59—21no—
153C4⋊C4×C59—21no—
154C4⋊C4×C59—21no—
155C4⋊C4×C59—21no—
156C4⋊C4×C59—81no—
157C4⋊C4×C59—81no—
158C4⋊C4×C59—81no—
159C42×C59—21no—
160C42×C59—21no—
161C42×C59—21no—
162C42×C59—21no—
163C42×C59—21no—
164C42×C59—21no—
165#11745671—44no—
166(C2×D59)⋊C4—24no—
167(C2×D59)⋊C4—81no—
168(C2×D59)⋊C4—81no—
169(C2×D59)⋊C4—81no—
170(C2×D59)⋊C4—81no—
171C4⋊Dic59—21no—
172C4⋊Dic59—21no—
173C4⋊Dic59—21no—
174C4⋊Dic59—21no—
175C4⋊Dic59—21no—
176C4⋊Dic59#4112yes—
177Dic59⋊C4—81no—
178Dic59⋊C4—81no—
179C4×Dic59—81no—
180C2×C59⋊C8—21no—
181C2×C59⋊C8—21no—
182C2×C59⋊C8—21no—
183C2×C59⋊C8—21no—
184C2×C59⋊C8—21no—
185C2×C59⋊C8—21no—
186C2×C59⋊C8—21no—
187C2×C59⋊C8—21no—
188C2×C59⋊C8—21no—
189C2×C59⋊C8—21no—
190C2×C59⋊C8#4112yes—
191C2×C59⋊C8#4112yes—
192C23×C59C22×D411yescharacteristic
193C22×D59—22no—
194C22×D59—41no—
195C22×D59—41no—
196C22×D59—41no—
197C22×D59—41no—
198C22×D59—41no—
199C22×D59—41no—
200Q8×C59—41no—
201Q8×C59—41no—
202Q8×C59—41no—
203Q8×C59—41no—
204C2×C4×C59—21no—
205C2×C4×C59—21no—
206C2×C4×C59—21no—
207C2×C4×C59—21no—
208C2×C4×C59—21no—
209C2×C4×C59—21no—
210C2×C4×C59—21no—
211C2×C4×C59—21no—
212C2×C4×C59—21no—
213C2×C4×C59—21no—
214C2×C4×C59—21no—
215C2×C4×C59—21no—
216C2×C4×C59—21no—
217C2×C4×C59—21no—
218C2×C4×C59—21no—
219C2×C4×C59—21no—
220C2×C4×C59—21no—
221C2×C4×C59—21no—
222C2×C4×C59—41no—
223C2×C4×C59—41no—
224C2×C4×C59—41no—
225C2×C4×C59—41no—
226C2×C4×C59—41no—
227C2×C4×C59—41no—
228C2×C4×C59—41no—
229C2×C4×C59—41no—
230C2×C4×C59—41no—
231C2×C4×C59—41no—
232C2×C4×C59—41no—
233C2×C4×C59—41no—
234C2×C4×C59C2×#4111yescharacteristic
235C2×C4×C59C2×#4111yescharacteristic
236C2×C4×C59C2×#4111yescharacteristic
237C2×C4×C59C2×#4111yescharacteristic
238C2×C4×C59H(5,𝔽2)11yescharacteristic
239C2×C4×C592−1+411yescharacteristic
240C2×Dic59—22no—
241C2×Dic59—41no—
242C2×Dic59—41no—
243C2×Dic59—41no—
244C2×Dic59—41no—
245C2×Dic59—41no—
246C2×Dic59—41no—
247C2×Dic59—41no—
248C2×Dic59—41no—
249D236—22no—
250D236—22no—
251D236—41no—
252D236—41no—
253C4×D59—84no—
254C59⋊C8—22no—
255C59⋊C8—22no—
256C59⋊C8—41no—
257C59⋊C8—41no—
258#29441—591nomaximal, Sylow
259C22×C59#52511yescharacteristic
260C22×C59#53211yescharacteristic
261C22×C59#53711yescharacteristic
262C22×C59#57611yescharacteristic
263C22×C59#57711yescharacteristic
264C22×C59#57711yescharacteristic
265C22×C59#57811yescharacteristic
266C2×D59—42no—
267C2×D59—42no—
268C2×D59—42no—
269C2×D59—81no—
270C2×D59—81no—
271C2×D59—81no—
272C2×D59—81no—
273C4×C59—41no—
274C4×C59—41no—
275C4×C59—41no—
276C4×C59—41no—
277C4×C59—41no—
278C4×C59—41no—
279C4×C59—81no—
280C4×C59—81no—
281C4×C59—81no—
282C4×C59—81no—
283C4×C59#55111yescharacteristic
284C4×C59#55211yescharacteristic
285C4×C59#55311yescharacteristic
286C4×C59#55611yescharacteristic
287Dic59—42no—
288Dic59—81no—
289Dic59—81no—
290#2645D5911yescharacteristic, complemented
291#2319—591no—
292#2088—591no—
293#2068—591no—
294#2067—591no—
295#2055—591no—
296#1968—591no—
297#1942—591no—
298#1936—591no—
299#1935—591no—
300#1928—591no—
301#1920—591no—
302#1877—591no—
303#1862—591no—
304#1768—591no—
305C2×C59#260111yescharacteristic
306C2×C59#311811yescharacteristic
307C2×C59#312011yescharacteristic
308C2×C59#317411yescharacteristic
309C2×C59#317511yescharacteristic
310C2×C59#325711yescharacteristic
311C2×C59#325811yescharacteristic
312D59—82no—
313C22×SD16—591no—
314C2×C4×D4—591no—
315C2×#107—591no—
316C2×#107—591no—
317C2×#106—591no—
318C2×C4⋊C8—591no—
319C2×C4⋊C8—591no—
320C2×#103—591no—
321C2×#103—591no—
322C2×#103—591no—
323C2×#102—591no—
324C2×#102—591no—
325C2×#102—591no—
326C2×#97—591no—
327C2×C4×C8—591no—
328#401—21no—
329#400—21no—
330#400—21no—
331#400—21no—
332#391C2×D5911yescharacteristic
333#391C2×D5911yescharacteristic
334#391C2×D5911yescharacteristic
335#390—591no—
336#390—591no—
337#389—591no—
338#389—591no—
339#389C2×D5911yescharacteristic
340#389C2×D5911yescharacteristic
341#389C2×D5911yescharacteristic
342#388—1181no—
343#388—1181no—
344#381—1181no—
345#380—1181no—
346C4×C4⋊C4—591no—
347#376C2×D5911yescharacteristic
348#340—591no—
349#340—591no—
350#340—591no—
351#340—591no—
352#340—591no—
353#340—591no—
354#336—591no—
355#336—591no—
356C59#2944111yescharacteristic, complemented, Sylow
357C22×Q8C22×D5911yescharacteristic
358C22×D4—591no—
359C23×C4—1181no—
360C2×SD16—1182no—
361C2×SD16—1182no—
362C2×SD16—2361no—
363C22×C8—591no—
364C22×C8—591no—
365C22×C8—591no—
366C22×C8—591no—
367C4×D4—1182no—
368C4×D4—1182no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—591no—
373C2×C4⋊C4—591no—
374C2×C4⋊C4—591no—
375C2×C4⋊C4—1181no—
376C2×C4⋊C4—1181no—
377C2×C4⋊C4C22×D5911yescharacteristic
378C2×C4⋊C4C22×D5911yescharacteristic
379C2×C4⋊C4C22×D5911yescharacteristic
380C2×C22⋊C4—1181no—
381C2×C22⋊C4—1181no—
382C2×C22⋊C4—1181no—
383C2×C22⋊C4—1181no—
384C2×C22⋊C4—1181no—
385C2×C42—1181no—
386C2×C42C22×D5911yescharacteristic
387C2×C42C22×D5911yescharacteristic
388C2×C42C22×D5911yescharacteristic
389#107—1182no—
390#107—2361no—
391#106—1182no—
392C4⋊C8—1182no—
393C4⋊C8—1182no—
394#103—1182no—
395#103—1182no—
396#103—2361no—
397#102—1182no—
398#102—1182no—
399#102—2361no—
400#97—2361no—
401C4×C8—2361no—
402#95—21no—
403#95—21no—
404#95—21no—
405#95—21no—
406#95—21no—
407#95—21no—
408#95—21no—
409#95—21no—
410#95—21no—
411#95—1181no—
412#95—1181no—
413#95—1181no—
414#95C22⋊D5911yescharacteristic
415#95C22⋊D5911yescharacteristic
416#95C22⋊D5911yescharacteristic
417#95C22⋊D5911yescharacteristic
418C24—1181no—
419C2×Q8—21no—
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×D4—592no—
426C2×D4—1181no—
427C2×D4—1181no—
428C2×D4—1181no—
429C2×D4—1181no—
430C2×D4—1181no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—1184no—
436C22×C4—1181no—
437C22×C4—1181no—
438C22×C4—1181no—
439C22×C4—4721no—
440C22×C4—4721no—
441C22×C4C2×C22⋊D5911yescharacteristic
442C22×C4C2×C22⋊D5911yescharacteristic
443C22×C4C2×C22⋊D5911yescharacteristic
444C22×C4C2×C22⋊D5911yescharacteristic
445C22×C4C2×C22⋊D5911yescharacteristic
446C22×C4C2×C22⋊D5911yescharacteristic
447C22×C4C23×D5911yescharacteristic
448SD16—2364no—
449C2×C8—592no—
450C2×C8—592no—
451C2×C8—1181no—
452C2×C8—1181no—
453C2×C8—1181no—
454C2×C8—1181no—
455C2×C8—1181no—
456C2×C8—1181no—
457C2×C8—1181no—
458C2×C8—1181no—
459C2×C8—1181no—
460C2×C8—1181no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—81no—
468C4⋊C4—81no—
469C4⋊C4—81no—
470C4⋊C4—592no—
471C4⋊C4—1181no—
472C4⋊C4—1181no—
473C4⋊C4—1181no—
474C4⋊C4—1181no—
475C4⋊C4—1181no—
476C4⋊C4—4721no—
477C4⋊C4—4721no—
478C22⋊C4—1184no—
479C22⋊C4—4721no—
480C22⋊C4—4721no—
481C22⋊C4—4721no—
482C22⋊C4—4721no—
483C42—21no—
484C42—21no—
485C42—21no—
486C42—21no—
487C42—21no—
488C42—21no—
489C42—4721no—
490C23—1182no—
491C23—2361no—
492C23—2361no—
493C23—2361no—
494C23—2361no—
495C23—2361no—
496C23—2361no—
497C23C22×C22⋊D5911yescharacteristic
498Q8—41no—
499Q8—41no—
500Q8—41no—
501Q8—41no—
502D4—1182no—
503D4—1182no—
504D4—2361no—
505D4—2361no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—1182no—
537C2×C4—4724no—
538C2×C4—2361no—
539C2×C4—2361no—
540C2×C4—2361no—
541C2×C4—2361no—
542C2×C4—2361no—
543C2×C4—2361no—
544C2×C4—2361no—
545C2×C4—2361no—
546C2×C4#41×D5911yescharacteristic
547C2×C4#41×D5911yescharacteristic
548C2×C4#41×D5911yescharacteristic
549C2×C4#41×D5911yescharacteristic
550C2×C4#1427377311yescharacteristic
551C2×C4#1427377411yescharacteristic
552C8—1182no—
553C8—1182no—
554C8—2361no—
555C8—2361no—
556C22—4721no—
557C22—4721no—
558C22—4721no—
559C22—4721no—
560C22—2362no—
561C22—2362no—
562C22—2362no—
563C22#1683790911yescharacteristic
564C22#1683794811yescharacteristic
565C22#1683798011yescharacteristic
566C22#1683820511yescharacteristic
567C22#1683821011yescharacteristic
568C22#1683821011yescharacteristic
569C22#1683821711yescharacteristic
570C4—41no—
571C4—41no—
572C4—41no—
573C4—41no—
574C4—41no—
575C4—41no—
576C4—81no—
577C4—81no—
578C4—81no—
579C4—81no—
580C4—4721no—
581C4—4721no—
582C4—2362no—
583C4#1683807711yescharacteristic
584C4#1683808811yescharacteristic
585C4#1683810311yescharacteristic
586C4#1683811911yescharacteristic
587C2—4722no—
588C2#2142969511yescharacteristic
589C2#2143427411yescharacteristic
590C2#2143430411yescharacteristic
591C2#2143472511yescharacteristic
592C2#2143472911yescharacteristic
593C2#2143550711yescharacteristic
594C2#2143551611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2645C2
socleC23×C59C22×D4
prime radicalC22×D236C23
hypercenter#2645D59
hypocenterC59#29441
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ #376 ◁ #2645

Complements

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

NH ≅ G/Ntype
2. #21425598C2semidirect
3. #21425491C2semidirect
7. #21424826C2semidirect
9. #21424789C2semidirect
10. #21424734C2semidirect
11. #21424678C2semidirect
14. #21423953C2semidirect
15. C59×#2645C2semidirect
290. #2645D59semidirect
356. C59#29441semidirect

Maximal subgroups

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

subgroupindex
#214267402
#214255982
#214254912
#214254812
#214254112
#214249532
#214248262
#214247922
#214247892
#214247342
#214246782
#214244332
#214243422
#214239532
C59×#26452
#2944159