Finite Group Databasebeta

← #28559832

Subgroups — #28559832

Order 15104.

592 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21433116C211yesmaximal, characteristic
2#21433045C211yesmaximal, characteristic, complemented
3#21422999C211yesmaximal, characteristic, complemented
4#21422987C211yesmaximal, characteristic, complemented
5#21422975C211yesmaximal, characteristic, complemented
6#21422968C211yesmaximal, characteristic, complemented
7#21422947C211yesmaximal, characteristic, complemented
8#21422940C211yesmaximal, characteristic, complemented
9#21422915C211yesmaximal, characteristic, complemented
10Q16×C59⋊C8C211yesmaximal, characteristic, complemented
11#21422881C211yesmaximal, characteristic, complemented
12#21422569C211yesmaximal, characteristic, complemented
13#21422483C211yesmaximal, characteristic
14#21422391C211yesmaximal, characteristic, complemented
15C59×#3107C211yesmaximal, characteristic
16#16837533—21no—
17#16837533—21no—
18#16837533C2211yescharacteristic
19#16837533C2211yescharacteristic
20Q8×C59⋊C8C2211yescharacteristic
21Q8×C59⋊C8C2211yescharacteristic
22Q8×C59⋊C8C2211yescharacteristic
23Q8×C59⋊C8C2211yescharacteristic
24#16837480—21no—
25#16837480—21no—
26#16837480C2211yescharacteristic
27#16837480C2211yescharacteristic
28#16837459—21no—
29#16837459C2211yescharacteristic
30#16837455C2211yescharacteristic
31#16837429—21no—
32C2×C4⋊(C59⋊C8)C2211yescharacteristic
33#16837358—21no—
34C4×#11745664—21no—
35C4×#11745664C2211yescharacteristic
36#16837155C2211yescharacteristic
37#16837155C2211yescharacteristic
38#16837153C2211yescharacteristic
39#16837153C2211yescharacteristic
40Q8⋊(C59⋊C8)C2211yescharacteristic
41Q8⋊(C59⋊C8)C2211yescharacteristic
42Q8⋊(C59⋊C8)C2211yescharacteristic
43Q8⋊(C59⋊C8)C2211yescharacteristic
44Q8⋊(C59⋊C8)C2211yescharacteristic
45Q8⋊(C59⋊C8)C2211yescharacteristic
46D4⋊(C59⋊C8)C2211yescharacteristic
47D4⋊(C59⋊C8)C2211yescharacteristic
48#16837124C2211yescharacteristic
49#16837124C2211yescharacteristic
50#16837123C2211yescharacteristic
51C8×C59⋊C8C2211yescharacteristic
52C59×#533C2211yescharacteristic
53C4×#41×C59C2211yescharacteristic
54C59×#485C411yescharacteristic
55C59×#485C411yescharacteristic
56C59×#475C411yescharacteristic
57C59×#474C411yescharacteristic
58C59×#462C411yescharacteristic
59C59×#461C411yescharacteristic
60C59×#452C411yescharacteristic
61C59×#452C411yescharacteristic
62C4×Q16×C59C2211yescharacteristic
63C4×Q16×C59C2211yescharacteristic
64C4×SD16×C59C2211yescharacteristic
65C4×SD16×C59C2211yescharacteristic
66C59×#431C2211yescharacteristic
67C2×#11745664—21no—
68C2×#11745664—21no—
69C2×#11745664D411yescharacteristic
70C2×#11745664D411yescharacteristic
71C22×C59⋊C8—21no—
72C4⋊(C59⋊C8)—21no—
73C4⋊(C59⋊C8)—21no—
74C4⋊(C59⋊C8)—21no—
75C4⋊(C59⋊C8)—21no—
76C4⋊(C59⋊C8)—21no—
77C4⋊(C59⋊C8)—21no—
78C4⋊(C59⋊C8)—21no—
79C4⋊(C59⋊C8)—21no—
80C4⋊(C59⋊C8)C2311yescharacteristic
81C4⋊(C59⋊C8)C2311yescharacteristic
82C4⋊(C59⋊C8)C2311yescharacteristic
83C4⋊(C59⋊C8)C2311yescharacteristic
84C22⋊(C59⋊C8)—21no—
85C22⋊(C59⋊C8)—21no—
86C22⋊(C59⋊C8)—21no—
87C22⋊(C59⋊C8)—21no—
88#14273638—21no—
89#14273638—21no—
90#14273638—21no—
91#14273638D411yescharacteristic
92#14273638D411yescharacteristic
93C4×C59⋊C8—21no—
94C4×C59⋊C8—21no—
95C4×C59⋊C8—21no—
96C4×C59⋊C8—21no—
97C4×C59⋊C8C2311yescharacteristic
98C4×C59⋊C8C2311yescharacteristic
99C4×C59⋊C8C2311yescharacteristic
100C4×C59⋊C8C2311yescharacteristic
101C2×#41×C59C2×C411yescharacteristic
102C2×Q16×C59C2×C411yescharacteristic
103C2×Q16×C59C2×C411yescharacteristic
104C2×SD16×C59C2×C411yescharacteristic
105C2×SD16×C59C2×C411yescharacteristic
106C4⋊Q8×C59C2×C411yescharacteristic
107C4⋊Q8×C59C2×C411yescharacteristic
108#125×C59—21no—
109C22⋊Q8×C59—21no—
110C22⋊Q8×C59C2×C411yescharacteristic
111C22⋊Q8×C59C2×C411yescharacteristic
112C4×Q8×C59—21no—
113C4×Q8×C59C2×C411yescharacteristic
114C4×Q8×C59C2311yescharacteristic
115C4×Q8×C59C2311yescharacteristic
116C4×Q8×C59C2311yescharacteristic
117C4×D4×C59—21no—
118C4×D4×C59—21no—
119C4×D4×C59C2×C411yescharacteristic
120C4×D4×C59C2311yescharacteristic
121#117×C59—21no—
122#117×C59—21no—
123#117×C59C2×C411yescharacteristic
124C2×C42×C59—21no—
125C2×C42×C59C2311yescharacteristic
126#107×C59C2×C411yescharacteristic
127#107×C59C2×C411yescharacteristic
128#106×C59C2×C411yescharacteristic
129#106×C59C2×C411yescharacteristic
130C4⋊C8×C59C2×C411yescharacteristic
131C4⋊C8×C59C2×C411yescharacteristic
132#103×C59C2×C411yescharacteristic
133#103×C59C2×C411yescharacteristic
134#103×C59C2×C411yescharacteristic
135#103×C59C2×C411yescharacteristic
136#103×C59C2×C411yescharacteristic
137#103×C59C2×C411yescharacteristic
138#102×C59C2×C411yescharacteristic
139#102×C59C2×C411yescharacteristic
140C22⋊C8×C59C2×C411yescharacteristic
141C22⋊C8×C59C2×C411yescharacteristic
142C4×C8×C59C2311yescharacteristic
143C4×C8×C59C2311yescharacteristic
144#41×C59—22no—
145#41×C59—41no—
146C2×Q8×C59—21no—
147C2×Q8×C59C22×C411yescharacteristic
148C2×Q8×C59C22×C411yescharacteristic
149C2×Q8×C59C22×C411yescharacteristic
150C2×D4×C59—21no—
151C2×D4×C59C22×C411yescharacteristic
152C22×C4×C59—21no—
153C22×C4×C59—21no—
154C22×C4×C59—21no—
155C22×C4×C59C22×C411yescharacteristic
156C22×C4×C59C2×D411yescharacteristic
157C22×C4×C59#4111yescharacteristic
158Q16×C59—22no—
159Q16×C59—22no—
160SD16×C59—41no—
161SD16×C59—41no—
162C2×C8×C59C22×C411yescharacteristic
163C2×C8×C59C22×C411yescharacteristic
164C2×C8×C59C22×C411yescharacteristic
165C2×C8×C59C22×C411yescharacteristic
166C4⋊C4×C59—21no—
167C4⋊C4×C59—21no—
168C4⋊C4×C59—21no—
169C4⋊C4×C59—21no—
170C4⋊C4×C59—21no—
171C4⋊C4×C59—21no—
172C4⋊C4×C59—21no—
173C4⋊C4×C59—21no—
174C4⋊C4×C59—21no—
175C4⋊C4×C59C22×C411yescharacteristic
176C4⋊C4×C59C22×C411yescharacteristic
177C4⋊C4×C59C22×C411yescharacteristic
178C4⋊C4×C59C22×C411yescharacteristic
179C22⋊C4×C59—21no—
180C22⋊C4×C59—21no—
181C22⋊C4×C59—21no—
182C22⋊C4×C59—21no—
183C22⋊C4×C59—21no—
184C42×C59—21no—
185C42×C59—21no—
186C42×C59—21no—
187C42×C59—21no—
188C42×C59—21no—
189C42×C59C22×C411yescharacteristic
190C42×C59C2×D411yescharacteristic
191C42×C59#4111yescharacteristic
192C42×C59C2411yescharacteristic
193#11745664—22no—
194#11745664—41no—
195#11745664—42no—
196#11745664—42no—
197C2×C59⋊C8—21no—
198C2×C59⋊C8—21no—
199C2×C59⋊C8—21no—
200C2×C59⋊C8—21no—
201C2×C59⋊C8—21no—
202C2×C59⋊C8—21no—
203C2×C59⋊C8—42no—
204C2×C59⋊C8C2×D411yescharacteristic
205C2×C59⋊C8C2×D411yescharacteristic
206C2×C59⋊C8C2×D411yescharacteristic
207C2×C59⋊C8C2×D411yescharacteristic
208C23×C59—21no—
209C23×C59C4×D411yescharacteristic
210Q8×C59—21no—
211Q8×C59—21no—
212Q8×C59—21no—
213Q8×C59—21no—
214Q8×C59—21no—
215Q8×C59—81no—
216Q8×C59#13112yes—
217D4×C59—21no—
218D4×C59—21no—
219D4×C59—42no—
220C2×C4×C59—21no—
221C2×C4×C59—21no—
222C2×C4×C59—21no—
223C2×C4×C59—21no—
224C2×C4×C59—21no—
225C2×C4×C59—21no—
226C2×C4×C59—21no—
227C2×C4×C59—21no—
228C2×C4×C59—21no—
229C2×C4×C59—21no—
230C2×C4×C59—21no—
231C2×C4×C59—41no—
232C2×C4×C59—42no—
233C2×C4×C59—41no—
234C2×C4×C59—41no—
235C2×C4×C59—81no—
236C2×C4×C59C4×D411yescharacteristic
237C2×C4×C59C4×D411yescharacteristic
238C2×C4×C59C4×D411yescharacteristic
239C2×C4×C59C23×C411yescharacteristic
240C2×C4×C59C22×D411yescharacteristic
241C2×C4×C59C2×#4111yescharacteristic
242C8×C59—21no—
243C8×C59—21no—
244C8×C59—21no—
245C8×C59—21no—
246C59⋊C8—21no—
247C59⋊C8—21no—
248C59⋊C8—21no—
249C59⋊C8—41no—
250C59⋊C8—41no—
251C59⋊C8—41no—
252C59⋊C8—41no—
253C59⋊C8—41no—
254C59⋊C8—41no—
255C59⋊C8#13712yes—
256#18013—591nomaximal, Sylow
257C22×C59—21no—
258C22×C59—21no—
259C22×C59—21no—
260C22×C59—41no—
261C22×C59—41no—
262C22×C59—41no—
263C22×C59C2×C4×D411yescharacteristic
264C4×C59—21no—
265C4×C59—22no—
266C4×C59—21no—
267C4×C59—21no—
268C4×C59—21no—
269C4×C59—41no—
270C4×C59—41no—
271C4×C59—41no—
272C4×C59—41no—
273C4×C59—41no—
274C4×C59—41no—
275C4×C59C2×#13111yescharacteristic
276C4×C59#56811yescharacteristic
277C4×C59C2×#13711yescharacteristic
278C4×C59#57511yescharacteristic
279#3107D5911yescharacteristic
280#2971—591no—
281#2962—591no—
282#1592—591no—
283#1590—591no—
284#1588—591no—
285#1587—591no—
286#1584—591no—
287#1583—591no—
288#1579—591no—
289C8×Q16—591no—
290#1573—591no—
291#1497—591no—
292#1477—591no—
293#1453—591no—
294C2×C59—41no—
295C2×C59—81no—
296C2×C59#293311yescharacteristic
297C2×C59C4×#13711yescharacteristic
298C2×C59#295711yescharacteristic
299#533C2×D5911yescharacteristic
300C4×#41C2×D5911yescharacteristic
301#485Dic5911yescharacteristic
302#485Dic5911yescharacteristic
303#475Dic5911yescharacteristic
304#474Dic5911yescharacteristic
305#462Dic5911yescharacteristic
306#461Dic5911yescharacteristic
307#452Dic5911yescharacteristic
308#452Dic5911yescharacteristic
309#446—591no—
310#446—591no—
311#446—1181no—
312#446—1181no—
313C8×Q8—591no—
314C8×Q8—591no—
315C8×Q8—591no—
316C8×Q8—591no—
317C4×Q16C2×D5911yescharacteristic
318C4×Q16C2×D5911yescharacteristic
319C4×SD16C2×D5911yescharacteristic
320C4×SD16C2×D5911yescharacteristic
321#436—591no—
322#436—591no—
323#436—1181no—
324#436—1181no—
325#432—591no—
326#432—1181no—
327#431—591no—
328#431C2×D5911yescharacteristic
329#424—1181no—
330C2×C4⋊C8—591no—
331#405—1181no—
332C4×M16—591no—
333C4×M16—1181no—
334#335—591no—
335#335—591no—
336#334—591no—
337#334—591no—
338Q8⋊C8—591no—
339Q8⋊C8—591no—
340Q8⋊C8—591no—
341Q8⋊C8—591no—
342Q8⋊C8—591no—
343Q8⋊C8—591no—
344D4⋊C8—591no—
345D4⋊C8—591no—
346#322—591no—
347#322—591no—
348#322—591no—
349C82—591no—
350C59#1801311yescharacteristic, complemented, Sylow
351C2×#41C2×Dic5911yescharacteristic
352C2×Q16C2×Dic5911yescharacteristic
353C2×Q16C2×Dic5911yescharacteristic
354C2×SD16C2×Dic5911yescharacteristic
355C2×SD16C2×Dic5911yescharacteristic
356C2×M16—591no—
357C2×M16—591no—
358C2×M16—1181no—
359C2×M16—1181no—
360C22×C8—1181no—
361C4⋊Q8C2×Dic5911yescharacteristic
362C4⋊Q8C2×Dic5911yescharacteristic
363#125—21no—
364C22⋊Q8—21no—
365C22⋊Q8C2×Dic5911yescharacteristic
366C22⋊Q8C2×Dic5911yescharacteristic
367C4×Q8—21no—
368C4×Q8C2×Dic5911yescharacteristic
369C4×Q8C22×D5911yescharacteristic
370C4×Q8C22×D5911yescharacteristic
371C4×Q8C22×D5911yescharacteristic
372C4×D4—21no—
373C4×D4—21no—
374C4×D4C2×Dic5911yescharacteristic
375C4×D4C22×D5911yescharacteristic
376#117—21no—
377#117—21no—
378#117C2×Dic5911yescharacteristic
379C2×C42—21no—
380C2×C42C22×D5911yescharacteristic
381#107C2×Dic5911yescharacteristic
382#107C2×Dic5911yescharacteristic
383#106C2×Dic5911yescharacteristic
384#106C2×Dic5911yescharacteristic
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⋊C8—1181no—
396C4⋊C8—1181no—
397C4⋊C8C2×Dic5911yescharacteristic
398C4⋊C8C2×Dic5911yescharacteristic
399#103C2×Dic5911yescharacteristic
400#103C2×Dic5911yescharacteristic
401#103C2×Dic5911yescharacteristic
402#103C2×Dic5911yescharacteristic
403#103C2×Dic5911yescharacteristic
404#103C2×Dic5911yescharacteristic
405#102C2×Dic5911yescharacteristic
406#102C2×Dic5911yescharacteristic
407C22⋊C8—1181no—
408C22⋊C8—1181no—
409C22⋊C8—1181no—
410C22⋊C8—1181no—
411C22⋊C8C2×Dic5911yescharacteristic
412C22⋊C8C2×Dic5911yescharacteristic
413#97—591no—
414#97—591no—
415#97—1181no—
416#97—1181no—
417#97—1181no—
418C4×C8—591no—
419C4×C8—591no—
420C4×C8—591no—
421C4×C8—591no—
422C4×C8—1181no—
423C4×C8—1181no—
424C4×C8—1181no—
425C4×C8—1181no—
426C4×C8C22×D5911yescharacteristic
427C4×C8C22×D5911yescharacteristic
428#41—22no—
429#41—41no—
430C2×Q8—21no—
431C2×Q8C22×Dic5911yescharacteristic
432C2×Q8C22×Dic5911yescharacteristic
433C2×Q8C22×Dic5911yescharacteristic
434C2×D4—21no—
435C2×D4C22×Dic5911yescharacteristic
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4D4×D5911yescharacteristic
440C22×C4#1174568711yescharacteristic
441C22×C4C22×Dic5911yescharacteristic
442Q16—22no—
443Q16—22no—
444SD16—41no—
445SD16—41no—
446M16—1182no—
447M16—2362no—
448M16—2361no—
449M16—2362no—
450C2×C8—591no—
451C2×C8—591no—
452C2×C8—591no—
453C2×C8—591no—
454C2×C8—1181no—
455C2×C8—1181no—
456C2×C8—1181no—
457C2×C8—1181no—
458C2×C8—1181no—
459C2×C8—1181no—
460C2×C8—2362no—
461C2×C8C22×Dic5911yescharacteristic
462C2×C8C22×Dic5911yescharacteristic
463C2×C8C22×Dic5911yescharacteristic
464C2×C8C22×Dic5911yescharacteristic
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4C22×Dic5911yescharacteristic
475C4⋊C4C22×Dic5911yescharacteristic
476C4⋊C4C22×Dic5911yescharacteristic
477C4⋊C4C22×Dic5911yescharacteristic
478C22⋊C4—21no—
479C22⋊C4—21no—
480C22⋊C4—21no—
481C22⋊C4—21no—
482C22⋊C4—21no—
483C42—21no—
484C42—21no—
485C42—21no—
486C42—21no—
487C42—21no—
488C42D4×D5911yescharacteristic
489C42#1174568711yescharacteristic
490C42C22×Dic5911yescharacteristic
491C42C23×D5911yescharacteristic
492C23—21no—
493C23D4×Dic5911yescharacteristic
494Q8—21no—
495Q8—21no—
496Q8—21no—
497Q8—21no—
498Q8—21no—
499Q8—81no—
500Q8#1427373412yes—
501D4—21no—
502D4—21no—
503D4—42no—
504C2×C4—21no—
505C2×C4—21no—
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—41no—
516C2×C4—42no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—81no—
520C2×C4D4×Dic5911yescharacteristic
521C2×C4D4×Dic5911yescharacteristic
522C2×C4D4×Dic5911yescharacteristic
523C2×C4C23×Dic5911yescharacteristic
524C2×C4C2×D4×D5911yescharacteristic
525C2×C4C2×#1174568711yescharacteristic
526C8—21no—
527C8—21no—
528C8—21no—
529C8—21no—
530C8—592no—
531C8—1181no—
532C8—1181no—
533C8—1181no—
534C8—2361no—
535C8—2361no—
536C8—2361no—
537C8—2361no—
538C8—2361no—
539C8—2361no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—41no—
544C22—41no—
545C22—41no—
546C22C2×D4×Dic5911yescharacteristic
547C4—21no—
548C4—21no—
549C4—21no—
550C4—21no—
551C4—22no—
552C4—41no—
553C4—41no—
554C4—41no—
555C4—41no—
556C4—41no—
557C4—41no—
558C4C2×#1427373411yescharacteristic
559C4#1683816911yescharacteristic
560C4#137×D5911yescharacteristic
561C4#1683820111yescharacteristic
562C2—41no—
563C2—81no—
564C2#2143282311yescharacteristic
565C2#137×Dic5911yescharacteristic
566C2#2143301311yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC42C23×D59
FittingC59×#3107C2
socleC22×C59C2×C4×D4
prime radicalC2×#41×C59C2×C4
cosocleC42×C59C24
hypercenter#3107D59
hypocenterC59#18013
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3107

Complements

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

NH ≅ G/Ntype
2. #21433045C2semidirect
3. #21422999C2semidirect
4. #21422987C2semidirect
5. #21422975C2semidirect
6. #21422968C2semidirect
7. #21422947C2semidirect
8. #21422940C2semidirect
9. #21422915C2semidirect
10. Q16×C59⋊C8C2semidirect
11. #21422881C2semidirect
12. #21422569C2semidirect
14. #21422391C2semidirect
350. C59#18013semidirect

Maximal subgroups

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

subgroupindex
#214331162
#214330452
#214229992
#214229872
#214229752
#214229682
#214229472
#214229402
#214229152
Q16×C59⋊C82
#214228812
#214225692
#214224832
#214223912
C59×#31072
#1801359