Finite Group Databasebeta

← #28629378

Subgroups — #28629378

Order 15104.

1016 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
1C2×#16837599C211yesmaximal, characteristic, complemented
2C2×#16837459C211yesmaximal, characteristic, complemented
3C4×#14273706C211yesmaximal, characteristic, complemented
4#21425275C212yesmaximal, complemented
5#21425099C212yesmaximal, complemented
6#21425024C211yesmaximal, characteristic, complemented
7#21424976C212yesmaximal, complemented
8#21424852C211yesmaximal, characteristic, complemented
9#21424796C212yesmaximal, complemented
10#21424098C211yesmaximal, characteristic, complemented
11C59×#2856C211yesmaximal, characteristic
12C22×#11745670C2212yes—
13C2×#14273706C2211yescharacteristic
14C2×C4×Dic118C2211yescharacteristic
15#16837599—24no—
16#16837459—24no—
17C2×#14273660C2212yes—
18C2×C4⋊(C59⋊C8)C2211yescharacteristic
19C2×C4⋊(C59⋊C8)C2211yescharacteristic
20C2×#14273650C2212yes—
21C2×#14273648C2212yes—
22C2×#14273648C2212yes—
23C2×#14273648C2212yes—
24C2×C22⋊(C59⋊C8)C2212yes—
25C2×#14273638C2212yes—
26#16837308C2211yescharacteristic
27#16837300C2211yescharacteristic
28#16837286—21no—
29#16837275—21no—
30#16837262C2211yescharacteristic
31C4×Dic59⋊C4—21no—
32C4×C4⋊Dic59C2211yescharacteristic
33C4×C22⋊Dic59—21no—
34#16837164C2212yes—
35#16837164C2212yes—
36C2×#127×C59C2211yescharacteristic
37C2×#121×C59—21no—
38C2×#121×C59—21no—
39C2×#121×C59C2212yes—
40C22×C42×C59C2211yescharacteristic
41C59×#390—21no—
42C59×#390—21no—
43C59×#390C2212yes—
44C59×#384C2211yescharacteristic
45C22×Dic118C2311yescharacteristic
46C2×#11745670—24no—
47C2×#11745670—24no—
48C2×#11745670—42no—
49C22×C59⋊C8C2312yes—
50C22×C59⋊C8C2312yes—
51#14273706—22no—
52#14273706—22no—
53C4×Dic118—22no—
54C4×Dic118—22no—
55C2×C4⋊Dic59C2311yescharacteristic
56C2×C4⋊Dic59C2311yescharacteristic
57C2×C4⋊Dic59C2311yescharacteristic
58C2×Dic59⋊C4—21no—
59C2×Dic59⋊C4—21no—
60C2×Dic59⋊C4—21no—
61C2×C22⋊Dic59—21no—
62C2×C22⋊Dic59—21no—
63C2×C4×Dic59—21no—
64C2×C4×Dic59—21no—
65#14273660—42no—
66C4⋊(C59⋊C8)—22no—
67C4⋊(C59⋊C8)D414yescomplemented
68#14273650—42no—
69#14273648—24no—
70#14273648—24no—
71#14273648—42no—
72C22⋊(C59⋊C8)—24no—
73#14273638—24no—
74C59⋊#95—21no—
75C59⋊#95—21no—
76C59⋊#95—21no—
77C59⋊#95—21no—
78C59⋊#95—21no—
79C22×D4×C59—22no—
80C22×D4×C59—21no—
81C22×D4×C59—21no—
82C22×D4×C59C2312yes—
83C23×C4×C59D411yescharacteristic
84C23×C4×C59D411yescharacteristic
85C23×C4×C59C2311yescharacteristic
86#127×C59—22no—
87#127×C59—22no—
88#121×C59—42no—
89#121×C59—42no—
90#121×C59—44no—
91#121×C59—44no—
92C2×C4⋊C4×C59—21no—
93C2×C4⋊C4×C59—21no—
94C2×C4⋊C4×C59C2312yes—
95C2×C22⋊C4×C59—22no—
96C2×C22⋊C4×C59—22no—
97C2×C22⋊C4×C59—22no—
98C2×C42×C59—22no—
99C2×C42×C59—22no—
100C2×C42×C59D411yescharacteristic
101C2×C42×C59D411yescharacteristic
102C2×C42×C59C2311yescharacteristic
103C2×C42×C59C2311yescharacteristic
104C24×C59—22no—
105C24×C59C2×D411yescharacteristic
106C2×D4×C59—22no—
107C2×D4×C59—22no—
108C2×D4×C59—22no—
109C2×D4×C59—22no—
110C2×D4×C59—22no—
111C2×D4×C59—22no—
112C2×D4×C59—42no—
113C2×D4×C59—42no—
114C2×D4×C59—44no—
115C2×D4×C59—44no—
116C2×D4×C59—44no—
117C2×D4×C59—42no—
118C2×D4×C59—42no—
119C2×D4×C59—42no—
120C2×D4×C59—42no—
121C22×C4×C59—22no—
122C22×C4×C59—22no—
123C22×C4×C59—22no—
124C22×C4×C59—22no—
125C22×C4×C59—22no—
126C22×C4×C59—21no—
127C22×C4×C59—21no—
128C22×C4×C59—21no—
129C22×C4×C59—21no—
130C22×C4×C59—21no—
131C22×C4×C59—21no—
132C22×C4×C59—22no—
133C22×C4×C59—41no—
134C22×C4×C59C2×D411yescharacteristic
135C22×C4×C59C2×D411yescharacteristic
136C22×C4×C59C2×D411yescharacteristic
137C22×C4×C59C2×D411yescharacteristic
138C22×C4×C59C2×D411yescharacteristic
139C22×C4×C59C2411yescharacteristic
140C22×Dic59—21no—
141C22×Dic59—21no—
142C22×Dic59—21no—
143C22×Dic59—21no—
144C2×Dic118—21no—
145C2×Dic118—21no—
146C2×Dic118—21no—
147C2×Dic118—21no—
148C2×Dic118—21no—
149C2×Dic118#4112yes—
150C4⋊C4×C59—22no—
151C4⋊C4×C59—22no—
152C4⋊C4×C59—42no—
153C4⋊C4×C59—42no—
154C22⋊C4×C59—44no—
155C22⋊C4×C59—44no—
156C22⋊C4×C59—44no—
157C42×C59—22no—
158C42×C59—22no—
159C42×C59—21no—
160C42×C59—21no—
161C42×C59C2×D412yes—
162C42×C59#4112yes—
163C22⋊Dic59—24no—
164C22⋊Dic59—42no—
165#11745670—48no—
166C4⋊Dic59—21no—
167C4⋊Dic59—21no—
168C4⋊Dic59—21no—
169C4⋊Dic59—21no—
170C4⋊Dic59—21no—
171C4⋊Dic59#4112yes—
172Dic59⋊C4—24no—
173Dic59⋊C4—42no—
174Dic59⋊C4—42no—
175C4×Dic59—42no—
176C4×Dic59—42no—
177C2×C59⋊C8—22no—
178C2×C59⋊C8—22no—
179C2×C59⋊C8—22no—
180C2×C59⋊C8—22no—
181C2×C59⋊C8—22no—
182C2×C59⋊C8C2×D414yes—
183C23×C59—21no—
184C23×C59—21no—
185C23×C59—21no—
186C23×C59—21no—
187C23×C59—21no—
188C23×C59—21no—
189C23×C59—42no—
190C23×C59—42no—
191C23×C59—42no—
192C23×C59—42no—
193C23×C59—42no—
194C23×C59—42no—
195C23×C59—42no—
196C23×C59AGL(1,ℤ/8)12yes—
197C23×C59C22×D411yescharacteristic
198D4×C59—42no—
199D4×C59—42no—
200D4×C59—42no—
201D4×C59—42no—
202D4×C59—84no—
203D4×C59—88no—
204D4×C59—84no—
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—44no—
217C2×C4×C59—41no—
218C2×C4×C59—41no—
219C2×C4×C59—41no—
220C2×C4×C59—41no—
221C2×C4×C59—44no—
222C2×C4×C59—42no—
223C2×C4×C59—42no—
224C2×C4×C59—42no—
225C2×C4×C59#12112yes—
226C2×C4×C59#12112yes—
227C2×C4×C59AGL(1,ℤ/8)12yes—
228C2×C4×C59#13712yes—
229C2×C4×C59C22×D411yescharacteristic
230C2×C4×C59C2×#4111yescharacteristic
231C2×C4×C59C2×#4111yescharacteristic
232C2×C4×C59C2×#4111yescharacteristic
233C2×C4×C59H(5,𝔽2)11yescharacteristic
234C2×C4×C59H(5,𝔽2)11yescharacteristic
235C2×Dic59—22no—
236C2×Dic59—22no—
237C2×Dic59—41no—
238C2×Dic59—41no—
239C2×Dic59—41no—
240C2×Dic59—41no—
241C2×Dic59—41no—
242C2×Dic59—41no—
243C2×Dic59—41no—
244C2×Dic59—41no—
245C2×Dic59—41no—
246C2×Dic59—41no—
247Dic118—22no—
248Dic118—22no—
249Dic118—41no—
250Dic118—41no—
251C59⋊C8—24no—
252C59⋊C8—42no—
253#24661—591nomaximal, Sylow
254C22×C59—22no—
255C22×C59—22no—
256C22×C59—22no—
257C22×C59—41no—
258C22×C59—41no—
259C22×C59—41no—
260C22×C59—41no—
261C22×C59—82no—
262C22×C59—82no—
263C22×C59—82no—
264C22×C59—82no—
265C22×C59—82no—
266C22×C59—82no—
267C22×C59—82no—
268C22×C59C2×#12111yescharacteristic
269C22×C59#52511yescharacteristic
270C22×C59#53511yescharacteristic
271C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
272C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
273C22×C59C2×#13711yescharacteristic
274C22×C59#57511yescharacteristic
275C4×C59—22no—
276C4×C59—41no—
277C4×C59—41no—
278C4×C59—41no—
279C4×C59—82no—
280C4×C59#46814yes—
281C4×C59#54611yescharacteristic
282C4×C59#55011yescharacteristic
283C4×C59#55311yescharacteristic
284C4×C59#55511yescharacteristic
285Dic59—42no—
286Dic59—42no—
287Dic59—81no—
288Dic59—81no—
289C2×#460—591no—
290C2×#432—591no—
291#2856D5911yescharacteristic
292C4×C22⋊Q8—591no—
293#2033—592no—
294#2000—592no—
295#1982—591no—
296#1973—592no—
297#1947—591no—
298#1936—592no—
299#1803—591no—
300C2×C59—42no—
301C2×C59—162no—
302C2×C59#241411yescharacteristic
303C2×C59C2×#46811yescharacteristic
304C2×C59#305511yescharacteristic
305C2×C59#311211yescharacteristic
306C2×C59#311511yescharacteristic
307C2×C59#321611yescharacteristic
308C2×C59#321711yescharacteristic
309C22×SD16—592no—
310C2×#127C2×D5911yescharacteristic
311C2×C22⋊Q8—591no—
312C2×#121—21no—
313C2×#121—21no—
314C2×#121C2×D5912yes—
315C2×C4×Q8—591no—
316C22×C42C2×D5911yescharacteristic
317#460—1184no—
318#432—1184no—
319C2×#107—592no—
320C2×C4⋊C8—591no—
321C2×C4⋊C8—591no—
322C2×#103—592no—
323C2×#102—592no—
324C2×#102—592no—
325C2×#102—592no—
326C2×C22⋊C8—592no—
327C2×#97—592no—
328#391—591no—
329#390—21no—
330#390—21no—
331#390C2×D5912yes—
332#389—591no—
333#387—1181no—
334#385—1181no—
335#384C2×D5911yescharacteristic
336#380—591no—
337C4×C4⋊C4—591no—
338C4×C4⋊C4—1181no—
339C4×C22⋊C4—1181no—
340#340—592no—
341#340—592no—
342C59#2466111yescharacteristic, complemented, Sylow
343C22×Q8—591no—
344C22×D4—22no—
345C22×D4—21no—
346C22×D4—21no—
347C22×D4C22×D5912yes—
348C23×C4C22⋊D5911yescharacteristic
349C23×C4C22⋊D5911yescharacteristic
350C23×C4C22×D5911yescharacteristic
351C2×SD16—1184no—
352C2×SD16—1184no—
353C2×SD16—2362no—
354C22×C8—592no—
355C22×C8—592no—
356#127—22no—
357#127—22no—
358C22⋊Q8—1182no—
359C22⋊Q8—1182no—
360#121—42no—
361#121—42no—
362#121—44no—
363#121—44no—
364C4×Q8—1182no—
365C4×Q8—1182no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—591no—
369C2×C4⋊C4—591no—
370C2×C4⋊C4—591no—
371C2×C4⋊C4—1181no—
372C2×C4⋊C4—1181no—
373C2×C4⋊C4—1181no—
374C2×C4⋊C4C22×D5912yes—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—22no—
377C2×C22⋊C4—22no—
378C2×C22⋊C4—1181no—
379C2×C22⋊C4—1181no—
380C2×C42—22no—
381C2×C42—22no—
382C2×C42—1181no—
383C2×C42—1181no—
384C2×C42C22⋊D5911yescharacteristic
385C2×C42C22⋊D5911yescharacteristic
386C2×C42C22×D5911yescharacteristic
387C2×C42C22×D5911yescharacteristic
388#107—2362no—
389C4⋊C8—594no—
390C4⋊C8—1182no—
391#103—2362no—
392#102—1184no—
393#102—1184no—
394#102—2362no—
395C22⋊C8—1184no—
396#97—1184no—
397#95—1181no—
398#95—1181no—
399#95—1181no—
400#95—1181no—
401#95—1181no—
402C24—22no—
403C24C2×C22⋊D5911yescharacteristic
404C2×Q8—592no—
405C2×Q8—1181no—
406C2×Q8—1181no—
407C2×Q8—1181no—
408C2×Q8—1181no—
409C2×Q8—1181no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—42no—
417C2×D4—42no—
418C2×D4—44no—
419C2×D4—44no—
420C2×D4—44no—
421C2×D4—42no—
422C2×D4—42no—
423C2×D4—42no—
424C2×D4—42no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—41no—
438C22×C4—1181no—
439C22×C4—1181no—
440C22×C4—1181no—
441C22×C4—1181no—
442C22×C4C2×C22⋊D5911yescharacteristic
443C22×C4C2×C22⋊D5911yescharacteristic
444C22×C4C2×C22⋊D5911yescharacteristic
445C22×C4C2×C22⋊D5911yescharacteristic
446C22×C4C2×C22⋊D5911yescharacteristic
447C22×C4C23×D5911yescharacteristic
448SD16—2368no—
449C2×C8—594no—
450C2×C8—1182no—
451C2×C8—1182no—
452C2×C8—1182no—
453C2×C8—1182no—
454C2×C8—1182no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—592no—
460C4⋊C4—1184no—
461C4⋊C4—1181no—
462C4⋊C4—1181no—
463C4⋊C4—1181no—
464C4⋊C4—1181no—
465C4⋊C4—1181no—
466C4⋊C4—2362no—
467C4⋊C4—2362no—
468C22⋊C4—44no—
469C22⋊C4—44no—
470C22⋊C4—44no—
471C22⋊C4—1184no—
472C22⋊C4—2362no—
473C42—22no—
474C42—22no—
475C42—21no—
476C42—21no—
477C42—2362no—
478C42—2362no—
479C42D4×D5912yes—
480C42#1174568712yes—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23#1427375312yes—
495C23C22×C22⋊D5911yescharacteristic
496Q8—1182no—
497Q8—1182no—
498Q8—2361no—
499Q8—2361no—
500D4—42no—
501D4—42no—
502D4—42no—
503D4—42no—
504D4—84no—
505D4—88no—
506D4—84no—
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—44no—
519C2×C4—44no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—1182no—
528C2×C4—1182no—
529C2×C4—2361no—
530C2×C4—2361no—
531C2×C4—2361no—
532C2×C4—2361no—
533C2×C4—2361no—
534C2×C4—2361no—
535C2×C4—2361no—
536C2×C4—2361no—
537C2×C4—2361no—
538C2×C4—2361no—
539C2×C4#1427369912yes—
540C2×C4#1427369912yes—
541C2×C4#1427375312yes—
542C2×C4#1427375712yes—
543C2×C4C2×D4×D5911yescharacteristic
544C2×C4#41×D5911yescharacteristic
545C2×C4#41×D5911yescharacteristic
546C2×C4C2×#1174568711yescharacteristic
547C2×C4#1427377211yescharacteristic
548C2×C4#1427377211yescharacteristic
549C8—1184no—
550C8—2362no—
551C22—22no—
552C22—22no—
553C22—22no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22—41no—
558C22—82no—
559C22—82no—
560C22—82no—
561C22—82no—
562C22—82no—
563C22—82no—
564C22—82no—
565C22C2×#1427369911yescharacteristic
566C22#1683790911yescharacteristic
567C22#1683796911yescharacteristic
568C22C2×#1427375311yescharacteristic
569C22C2×#1427375311yescharacteristic
570C22C2×#1427375711yescharacteristic
571C22#1683820011yescharacteristic
572C4—22no—
573C4—41no—
574C4—41no—
575C4—41no—
576C4—82no—
577C4—4721no—
578C4—4721no—
579C4—2362no—
580C4—2362no—
581C4#1683764614yes—
582C4#1683804911yescharacteristic
583C4#1683807011yescharacteristic
584C4#1683809911yescharacteristic
585C4#1683811611yescharacteristic
586C2—42no—
587C2—162no—
588C2#2142751511yescharacteristic
589C2C2×#1683764611yescharacteristic
590C2#2143371111yescharacteristic
591C2#2143421411yescharacteristic
592C2#2143424111yescharacteristic
593C2#2143510711yescharacteristic
594C2#2143511411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
Fitting, prime radicalC59×#2856C2
socleC23×C59C22×D4
hypercenter#2856D59
hypocenterC59#24661
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2856

Complements

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

NH ≅ G/Ntype
1. C2×#16837599C2semidirect
2. C2×#16837459C2semidirect
3. C4×#14273706C2semidirect
4. #21425275C2semidirect
5. #21425099C2semidirect
6. #21425024C2semidirect
7. #21424976C2semidirect
8. #21424852C2semidirect
9. #21424796C2semidirect
10. #21424098C2semidirect
67. C4⋊(C59⋊C8)D4semidirect
342. C59#24661semidirect

Maximal subgroups

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

subgroupindex
C2×#168375992
C2×#168374592
C4×#142737062
#214252752
#214250992
#214250242
#214249762
#214248522
#214247962
#214240982
C59×#28562
#2466159