Finite Group Databasebeta

← #28558763

Subgroups — #28558763

Order 15104.

596 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21434180C211yesmaximal, characteristic
2#21433248C211yesmaximal, characteristic
3#21433175C211yesmaximal, characteristic, complemented
4#21422976C211yesmaximal, characteristic, complemented
5#21422944C211yesmaximal, characteristic, complemented
6#21422939C211yesmaximal, characteristic, complemented
7#21422912C211yesmaximal, characteristic, complemented
8#21422911C211yesmaximal, characteristic, complemented
9Q16×C59⋊C8C211yesmaximal, characteristic, complemented
10#21422895C211yesmaximal, characteristic, complemented
11#21422883C211yesmaximal, characteristic, complemented
12#21422523C211yesmaximal, characteristic, complemented
13#21422503C211yesmaximal, characteristic
14#21422397C211yesmaximal, characteristic, complemented
15C59×#1587C211yesmaximal, characteristic, complemented
16#16837939C2211yescharacteristic
17C4×Q8×D59C2211yescharacteristic
18#16837731C411yescharacteristic
19#16837717C411yescharacteristic
20#16837708C411yescharacteristic
21#16837707C411yescharacteristic
22#16837627C411yescharacteristic
23#16837626C411yescharacteristic
24#16837556C411yescharacteristic
25#16837542C411yescharacteristic
26Q8×C59⋊C8C2211yescharacteristic
27Q8×C59⋊C8C2211yescharacteristic
28C8×Dic118—21no—
29C8×Dic118C2211yescharacteristic
30#16837526—21no—
31#16837526C2211yescharacteristic
32C4×#11745672C2211yescharacteristic
33C4×#11745660C2211yescharacteristic
34C4×#11745671C2211yescharacteristic
35#16837481C2211yescharacteristic
36#16837479—21no—
37#16837475—21no—
38#16837467C2211yescharacteristic
39#16837464—21no—
40#16837454—21no—
41#16837454C2211yescharacteristic
42#16837453—21no—
43#16837453C2211yescharacteristic
44#16837453C2211yescharacteristic
45#16837453C2211yescharacteristic
46#16837427—21no—
47#16837424C2211yescharacteristic
48#16837155C2211yescharacteristic
49#16837154C2211yescharacteristic
50#16837152C2211yescharacteristic
51#16837134C2211yescharacteristic
52Q8⋊(C59⋊C8)C2211yescharacteristic
53Q8⋊(C59⋊C8)C2211yescharacteristic
54#16837132C2211yescharacteristic
55#16837131C2211yescharacteristic
56#16837129C2211yescharacteristic
57#16837124C2211yescharacteristic
58#16837123C2211yescharacteristic
59C8×C59⋊C8C2211yescharacteristic
60C59×#446C2211yescharacteristic
61C8×Q8×C59C2211yescharacteristic
62C4×Q16×C59C2211yescharacteristic
63C59×#334C2211yescharacteristic
64C59×Q8⋊C8C2211yescharacteristic
65C59×Q8⋊C8C2211yescharacteristic
66C59×#322C2211yescharacteristic
67C2×Q8×D59C2×C411yescharacteristic
68C2×#11745672C2×C411yescharacteristic
69C2×#11745660C2×C411yescharacteristic
70C2×#11745671C2×C411yescharacteristic
71C2×#11745659—21no—
72C2×#11745659—21no—
73C2×C8×D59—21no—
74#14273722—21no—
75#14273718C2×C411yescharacteristic
76#14273716C2×C411yescharacteristic
77#14273712—21no—
78#14273703C2×C411yescharacteristic
79#14273702C2×C411yescharacteristic
80Q8×Dic59C2×C411yescharacteristic
81C4×Dic118—21no—
82C4×Dic118C2311yescharacteristic
83#14273692—21no—
84#14273692C2×C411yescharacteristic
85C4×D236—21no—
86C4×D236C2311yescharacteristic
87#14273684—21no—
88C4⋊C4×D59—21no—
89C4⋊C4×D59C2×C411yescharacteristic
90C42×D59—21no—
91C42×D59C2311yescharacteristic
92#14273660C2×C411yescharacteristic
93#14273659C2×C411yescharacteristic
94#14273658C2×C411yescharacteristic
95C4⋊(C59⋊C8)—21no—
96C4⋊(C59⋊C8)—21no—
97C4⋊(C59⋊C8)C2311yescharacteristic
98C4⋊(C59⋊C8)C2311yescharacteristic
99Dic59⋊C8—21no—
100Dic59⋊C8—21no—
101Dic59⋊C8—21no—
102Dic59⋊C8—21no—
103Dic59⋊C8C2×C411yescharacteristic
104Dic59⋊C8C2×C411yescharacteristic
105Dic59⋊C8D411yescharacteristic
106Dic59⋊C8D411yescharacteristic
107#14273652C2×C411yescharacteristic
108#14273651C2×C411yescharacteristic
109#14273651C2×C411yescharacteristic
110#14273650C2×C411yescharacteristic
111#14273649C2×C411yescharacteristic
112#14273647C2×C411yescharacteristic
113(C2×D59)⋊C8—21no—
114(C2×D59)⋊C8—21no—
115(C2×D59)⋊C8—21no—
116(C2×D59)⋊C8—21no—
117(C2×D59)⋊C8C2×C411yescharacteristic
118(C2×D59)⋊C8C2×C411yescharacteristic
119(C2×D59)⋊C8D411yescharacteristic
120(C2×D59)⋊C8D411yescharacteristic
121C4×C59⋊C8—21no—
122C4×C59⋊C8—21no—
123C4×C59⋊C8C2311yescharacteristic
124C4×C59⋊C8C2311yescharacteristic
125C4×C59⋊C8C2311yescharacteristic
126C8×Dic59—21no—
127C8×Dic59—21no—
128C8×Dic59—21no—
129C2×Q16×C59C2×C411yescharacteristic
130C4×Q8×C59C2311yescharacteristic
131C4×Q8×C59C2311yescharacteristic
132#107×C59C2×C411yescharacteristic
133C4⋊C8×C59—21no—
134C4⋊C8×C59—21no—
135C4⋊C8×C59C2311yescharacteristic
136C4⋊C8×C59C2311yescharacteristic
137#103×C59C2×C411yescharacteristic
138#103×C59C2×C411yescharacteristic
139#97×C59—21no—
140C4×C8×C59—21no—
141C4×C8×C59C2311yescharacteristic
142C4×C8×C59C2311yescharacteristic
143C4×C8×C59C2311yescharacteristic
144C2×Q8×C59C22×C411yescharacteristic
145C2×Q8×C59C22×C411yescharacteristic
146Q8×D59—22no—
147Q8×D59—22no—
148C2×D236—21no—
149C2×D236C22×C411yescharacteristic
150C2×C4×D59—21no—
151C2×C4×D59—21no—
152C2×C4×D59—21no—
153C2×C4×D59C22×C411yescharacteristic
154C2×C4×D59C2×D411yescharacteristic
155C2×C4×D59#4111yescharacteristic
156C2×Dic118—21no—
157C2×Dic118C22×C411yescharacteristic
158Q16×C59—22no—
159C2×C8×C59—21no—
160C2×C8×C59—21no—
161C2×C8×C59—21no—
162C2×C8×C59C22×C411yescharacteristic
163C2×C8×C59C22×C411yescharacteristic
164C2×C8×C59C2×D411yescharacteristic
165C2×C8×C59C2×D411yescharacteristic
166C4⋊C4×C59—21no—
167C4⋊C4×C59—21no—
168C4⋊C4×C59C22×C411yescharacteristic
169C4⋊C4×C59C22×C411yescharacteristic
170C42×C59—21no—
171C42×C59—21no—
172C42×C59C2411yescharacteristic
173#11745672—22no—
174#11745671—22no—
175(C2×D59)⋊C4—21no—
176(C2×D59)⋊C4—21no—
177(C2×D59)⋊C4—21no—
178(C2×D59)⋊C4—21no—
179(C2×D59)⋊C4—21no—
180C4⋊Dic59—21no—
181C4⋊Dic59—21no—
182C4⋊Dic59C22×C411yescharacteristic
183C4⋊Dic59C22×C411yescharacteristic
184Dic59⋊C4—21no—
185Dic59⋊C4—21no—
186Dic59⋊C4—21no—
187Dic59⋊C4—21no—
188Dic59⋊C4—21no—
189C4×Dic59—21no—
190C4×Dic59—21no—
191C4×Dic59—21no—
192C4×Dic59C22×C411yescharacteristic
193C4×Dic59C2×D411yescharacteristic
194C4×Dic59#4111yescharacteristic
195C2×C59⋊C8—21no—
196C2×C59⋊C8—21no—
197C2×C59⋊C8—21no—
198C2×C59⋊C8C22×C411yescharacteristic
199C2×C59⋊C8C22×C411yescharacteristic
200C2×C59⋊C8C2×D411yescharacteristic
201C2×C59⋊C8C2×D411yescharacteristic
202#11745660—22no—
203#11745659—81no—
204#11745659—81no—
205C8×D59—81no—
206C22×D59—21no—
207C22×D59C4×D411yescharacteristic
208Q8×C59—21no—
209Q8×C59—21no—
210Q8×C59—21no—
211Q8×C59#13112yes—
212C2×C4×C59—21no—
213C2×C4×C59—21no—
214C2×C4×C59—21no—
215C2×C4×C59—21no—
216C2×C4×C59C23×C411yescharacteristic
217C2×C4×C59C22×D411yescharacteristic
218C2×C4×C59C2×#4111yescharacteristic
219C2×Dic59—21no—
220C2×Dic59—21no—
221C2×Dic59—21no—
222C2×Dic59C4×D411yescharacteristic
223C2×Dic59C4×D411yescharacteristic
224C2×Dic59C4×D411yescharacteristic
225D236—21no—
226D236—21no—
227D236—81no—
228C4×D59—24no—
229C4×D59—21no—
230C4×D59—21no—
231C4×D59—21no—
232C4×D59—21no—
233C4×D59—41no—
234C4×D59—41no—
235C4×D59—41no—
236C4×D59—81no—
237Dic118—24no—
238Dic118—21no—
239Dic118#13112yes—
240C8×C59—21no—
241C8×C59—21no—
242C8×C59—21no—
243C8×C59—21no—
244C8×C59—41no—
245C8×C59—41no—
246C8×C59—41no—
247C59⋊C8—21no—
248C59⋊C8—21no—
249C59⋊C8—21no—
250C59⋊C8—21no—
251C59⋊C8—41no—
252C59⋊C8—41no—
253C59⋊C8—41no—
254#17919—591nomaximal, Sylow
255C22×C59C2×C4×D411yescharacteristic
256C2×D59—21no—
257C2×D59—21no—
258C2×D59—21no—
259C2×D59—41no—
260C2×D59—41no—
261C2×D59—41no—
262C4×C59—22no—
263C4×C59—21no—
264C4×C59—41no—
265C4×C59—41no—
266C4×C59—41no—
267C4×C59C2×#13111yescharacteristic
268C4×C59C2×#13111yescharacteristic
269C4×C59#57711yescharacteristic
270C4×C59#57811yescharacteristic
271Dic59—21no—
272Dic59—22no—
273Dic59—21no—
274Dic59—21no—
275Dic59—41no—
276Dic59—41no—
277#3108—591no—
278#2993—591no—
279#2982—591no—
280#1588—591no—
281#1587D5911yescharacteristic, complemented
282#1584—591no—
283#1583—591no—
284#1579—591no—
285SD16⋊C8—591no—
286C8×Q16—591no—
287C8×SD16—591no—
288#1574—591no—
289#1487—591no—
290#1482—591no—
291#1454—591no—
292C2×C59#292911yescharacteristic
293C2×C59#294711yescharacteristic
294C2×C59#295711yescharacteristic
295D59—41no—
296D59—81no—
297#529—591no—
298C2×C4×Q8—591no—
299#483—591no—
300#480—591no—
301#479—591no—
302#479—591no—
303#464—591no—
304#464—591no—
305#451—591no—
306#448—591no—
307#446C2×D5911yescharacteristic
308C8×Q8—591no—
309C8×Q8—591no—
310C8×Q8—591no—
311C8×Q8—591no—
312C8×Q8—1181no—
313C8×Q8—1181no—
314C8×Q8C2×D5911yescharacteristic
315C4×Q16—591no—
316C4×Q16C2×D5911yescharacteristic
317C4×SD16—591no—
318C4×SD16—591no—
319#436—591no—
320#436—1181no—
321#435—1181no—
322C8×D4—591no—
323#433—1181no—
324#431—591no—
325#431—591no—
326#431—591no—
327#431—591no—
328#431—1181no—
329#431—1181no—
330#424—1181no—
331#423—591no—
332#335—591no—
333#335—591no—
334#334—591no—
335#334C2×D5911yescharacteristic
336Q8⋊C8—591no—
337Q8⋊C8—591no—
338Q8⋊C8—591no—
339Q8⋊C8—591no—
340Q8⋊C8C2×D5911yescharacteristic
341Q8⋊C8C2×D5911yescharacteristic
342D4⋊C8—591no—
343D4⋊C8—591no—
344#322—591no—
345#322—591no—
346#322C2×D5911yescharacteristic
347C82—591no—
348C59#1791911yescharacteristic, complemented, Sylow
349C22×Q8—591no—
350C2×Q16—591no—
351C2×Q16C4×D5911yescharacteristic
352C2×SD16—591no—
353C2×SD16—591no—
354C2×M16—1181no—
355C2×M16—1181no—
356C22×C8—1181no—
357#126—1181no—
358#125—591no—
359#124—591no—
360#123—1181no—
361C22⋊Q8—591no—
362#121—591no—
363C4×Q8—591no—
364C4×Q8—591no—
365C4×Q8—591no—
366C4×Q8—1181no—
367C4×Q8—1181no—
368C4×Q8C22×D5911yescharacteristic
369C4×Q8C22×D5911yescharacteristic
370C4×D4—591no—
371C4×D4—1181no—
372#117—1181no—
373C2×C4⋊C4—591no—
374C2×C4⋊C4—1181no—
375C2×C42—591no—
376C2×C42—1181no—
377#107—591no—
378#107C4×D5911yescharacteristic
379#106—591no—
380#106—591no—
381C4⋊C8—21no—
382C4⋊C8—21no—
383C4⋊C8—591no—
384C4⋊C8—591no—
385C4⋊C8—591no—
386C4⋊C8—591no—
387C4⋊C8—591no—
388C4⋊C8—591no—
389C4⋊C8—1181no—
390C4⋊C8—1181no—
391C4⋊C8—1181no—
392C4⋊C8—1181no—
393C4⋊C8—1181no—
394C4⋊C8—1181no—
395C4⋊C8C22×D5911yescharacteristic
396C4⋊C8C22×D5911yescharacteristic
397#103—591no—
398#103—591no—
399#103—591no—
400#103—591no—
401#103C4×D5911yescharacteristic
402#103C4×D5911yescharacteristic
403#102—591no—
404#102—591no—
405C22⋊C8—591no—
406C22⋊C8—591no—
407C22⋊C8—591no—
408C22⋊C8—591no—
409C22⋊C8—1181no—
410C22⋊C8—1181no—
411C22⋊C8—1181no—
412C22⋊C8—1181no—
413#97—21no—
414C4×C8—21no—
415C4×C8—591no—
416C4×C8—591no—
417C4×C8—591no—
418C4×C8—1181no—
419C4×C8—1181no—
420C4×C8—1181no—
421C4×C8—1181no—
422C4×C8—1181no—
423C4×C8C22×D5911yescharacteristic
424C4×C8C22×D5911yescharacteristic
425C4×C8C22×D5911yescharacteristic
426C2×Q8—591no—
427C2×Q8—1181no—
428C2×Q8—1182no—
429C2×Q8—1182no—
430C2×Q8C2×C4×D5911yescharacteristic
431C2×Q8C2×C4×D5911yescharacteristic
432C2×D4—591no—
433C2×D4—1181no—
434C22×C4—591no—
435C22×C4—591no—
436C22×C4—591no—
437C22×C4—1181no—
438C22×C4—1181no—
439C22×C4—1181no—
440Q16—22no—
441Q16—1182no—
442SD16—1182no—
443SD16—1182no—
444M16—4721no—
445M16—4721no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—591no—
450C2×C8—591no—
451C2×C8—591no—
452C2×C8—591no—
453C2×C8—1181no—
454C2×C8—1181no—
455C2×C8—1181no—
456C2×C8—4721no—
457C2×C8C2×C4×D5911yescharacteristic
458C2×C8C2×C4×D5911yescharacteristic
459C2×C8D4×D5911yescharacteristic
460C2×C8D4×D5911yescharacteristic
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—591no—
464C4⋊C4—591no—
465C4⋊C4—1181no—
466C4⋊C4—1181no—
467C4⋊C4—1181no—
468C4⋊C4—1181no—
469C4⋊C4—1181no—
470C4⋊C4—1181no—
471C4⋊C4—1181no—
472C4⋊C4C2×C4×D5911yescharacteristic
473C4⋊C4C2×C4×D5911yescharacteristic
474C22⋊C4—1181no—
475C22⋊C4—1181no—
476C22⋊C4—1181no—
477C22⋊C4—1181no—
478C22⋊C4—1181no—
479C42—21no—
480C42—21no—
481C42—591no—
482C42—591no—
483C42—591no—
484C42—1181no—
485C42—1181no—
486C42—1181no—
487C42C23×D5911yescharacteristic
488C23—591no—
489C23—1181no—
490Q8—21no—
491Q8—21no—
492Q8—21no—
493Q8—592no—
494Q8—1184no—
495Q8—1181no—
496Q8#1427373512yes—
497D4—1181no—
498D4—1181no—
499D4—4721no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—591no—
505C2×C4—591no—
506C2×C4—591no—
507C2×C4—1184no—
508C2×C4—1181no—
509C2×C4—1181no—
510C2×C4—1181no—
511C2×C4—1181no—
512C2×C4—1181no—
513C2×C4—1181no—
514C2×C4—1181no—
515C2×C4—4721no—
516C2×C4—2361no—
517C2×C4—2361no—
518C2×C4—2361no—
519C2×C4C22×C4×D5911yescharacteristic
520C2×C4C2×D4×D5911yescharacteristic
521C2×C4#41×D5911yescharacteristic
522C8—21no—
523C8—21no—
524C8—21no—
525C8—21no—
526C8—41no—
527C8—41no—
528C8—41no—
529C8—1181no—
530C8—1181no—
531C8—1181no—
532C8—1181no—
533C8—2361no—
534C8—2361no—
535C8—2361no—
536C22—1181no—
537C22—1181no—
538C22—1181no—
539C22—2361no—
540C22—2361no—
541C22—2361no—
542C22C4×D4×D5911yescharacteristic
543C4—21no—
544C4—22no—
545C4—41no—
546C4—41no—
547C4—41no—
548C4—1181no—
549C4—1181no—
550C4—1181no—
551C4—1182no—
552C4—2361no—
553C4—2361no—
554C4#131×D5911yescharacteristic
555C4C2×#1427373511yescharacteristic
556C4#1683821111yescharacteristic
557C4#1683821811yescharacteristic
558C2—4721no—
559C2—2361no—
560C2#2143278711yescharacteristic
561C2#2143293811yescharacteristic
562C2#2143300811yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC42C23×D59
FittingC59×#1587C2
socleC22×C59C2×C4×D4
prime radical#14273702C2×C4
cosocleC42×C59C24
hypercenter#1587D59
hypocenterC59#17919
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1587

Complements

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

NH ≅ G/Ntype
3. #21433175C2semidirect
4. #21422976C2semidirect
5. #21422944C2semidirect
6. #21422939C2semidirect
7. #21422912C2semidirect
8. #21422911C2semidirect
9. Q16×C59⋊C8C2semidirect
10. #21422895C2semidirect
11. #21422883C2semidirect
12. #21422523C2semidirect
14. #21422397C2semidirect
15. C59×#1587C2semidirect
281. #1587D59semidirect
348. C59#17919semidirect

Maximal subgroups

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

subgroupindex
#214341802
#214332482
#214331752
#214229762
#214229442
#214229392
#214229122
#214229112
Q16×C59⋊C82
#214228952
#214228832
#214225232
#214225032
#214223972
C59×#15872
#1791959