Finite Group Databasebeta

← #28640747

Subgroups — #28640747

Order 15104.

790 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837689C211yesmaximal, characteristic
2C2×#16837533C211yesmaximal, characteristic, complemented
3#21429454C211yesmaximal, characteristic
4#21425598C211yesmaximal, characteristic, complemented
5#21425597C211yesmaximal, characteristic, complemented
6#21425481C211yesmaximal, characteristic
7#21425467C211yesmaximal, characteristic
8#21425070C211yesmaximal, characteristic, complemented
9#21424864C211yesmaximal, characteristic, complemented
10#21424831C211yesmaximal, characteristic, complemented
11#21424821C211yesmaximal, characteristic, complemented
12#21424792C211yesmaximal, characteristic
13#21424778C211yesmaximal, characteristic
14#21423924C211yesmaximal, characteristic
15C59×#2546C211yesmaximal, characteristic, complemented
16C2×C4×D236C2211yescharacteristic
17#16837689—24no—
18#16837533—24no—
19C2×#14273660C2211yescharacteristic
20C2×#14273658C2211yescharacteristic
21C2×#14273658C2211yescharacteristic
22C2×#14273658C2211yescharacteristic
23C2×C4⋊(C59⋊C8)C2211yescharacteristic
24C2×C4⋊(C59⋊C8)C2211yescharacteristic
25C2×C4⋊(C59⋊C8)C2211yescharacteristic
26C2×#14273647C2211yescharacteristic
27C2×#14273647C2211yescharacteristic
28C2×#14273647C2211yescharacteristic
29C2×#14273647C2211yescharacteristic
30C2×#14273638C2211yescharacteristic
31C2×#14273638C2211yescharacteristic
32C2×C4×C59⋊C8C2211yescharacteristic
33#16837337—21no—
34#16837313—21no—
35#16837304C2211yescharacteristic
36#16837304C2211yescharacteristic
37#16837300C2211yescharacteristic
38#16837300C2211yescharacteristic
39#16837273—21no—
40#16837273—21no—
41#16837268C2211yescharacteristic
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837164C2211yescharacteristic
49#16837164C2211yescharacteristic
50C2×C4⋊Q8×C59—21no—
51C2×C4⋊Q8×C59C2211yescharacteristic
52C2×C4×Q8×C59C2211yescharacteristic
53C59×#398—21no—
54C59×#398—21no—
55C59×#391—21no—
56C59×#389C2211yescharacteristic
57C59×#389C2211yescharacteristic
58C59×#389C2211yescharacteristic
59C59×#389C2211yescharacteristic
60C4×C4⋊C4×C59C2211yescharacteristic
61C22×D236C2311yescharacteristic
62C22×C4×D59—21no—
63C22×C59⋊C8C2311yescharacteristic
64C22×C59⋊C8C2311yescharacteristic
65C22×C59⋊C8C2311yescharacteristic
66C22×C59⋊C8C2311yescharacteristic
67C4×D236—22no—
68C4×D236—22no—
69C2×C4⋊Dic59C2311yescharacteristic
70C2×C4⋊Dic59C2311yescharacteristic
71C2×C4⋊Dic59C2311yescharacteristic
72C2×Dic59⋊C4—21no—
73C2×Dic59⋊C4—21no—
74C2×(C2×D59)⋊C4—21no—
75C2×(C2×D59)⋊C4—21no—
76C2×(C2×D59)⋊C4—21no—
77C2×(C2×D59)⋊C4—21no—
78C2×(C2×D59)⋊C4—21no—
79#14273660—22no—
80#14273658—22no—
81#14273658—22no—
82#14273658—22no—
83C4⋊(C59⋊C8)—22no—
84C4⋊(C59⋊C8)—22no—
85C4⋊(C59⋊C8)D414yes—
86#14273647—22no—
87#14273647—22no—
88#14273647—22no—
89#14273647Q814yescomplemented
90#14273638—22no—
91#14273638—22no—
92C4×C59⋊C8—22no—
93C59⋊#95—21no—
94C59⋊#95—21no—
95C59⋊#95—21no—
96C59⋊#95—21no—
97C22×Q8×C59—21no—
98C22×Q8×C59D411yescharacteristic
99C4⋊Q8×C59—22no—
100C4⋊Q8×C59—22no—
101C4⋊Q8×C59—44no—
102C4×Q8×C59—22no—
103C4×Q8×C59—22no—
104C2×C4⋊C4×C59—21no—
105C2×C4⋊C4×C59—21no—
106C2×C4⋊C4×C59—21no—
107C2×C4⋊C4×C59—21no—
108C2×C4⋊C4×C59—21no—
109C2×C4⋊C4×C59D411yescharacteristic
110C2×C4⋊C4×C59D411yescharacteristic
111C2×C4⋊C4×C59D411yescharacteristic
112C2×C4⋊C4×C59C2311yescharacteristic
113C2×C4⋊C4×C59C2311yescharacteristic
114C2×C4⋊C4×C59C2311yescharacteristic
115C2×C4⋊C4×C59C2311yescharacteristic
116C2×C42×C59—21no—
117C2×C42×C59C2311yescharacteristic
118C2×C42×C59C2311yescharacteristic
119C2×C42×C59C2311yescharacteristic
120#95×C59—21no—
121#95×C59—21no—
122#95×C59—21no—
123#95×C59—21no—
124#95×C59—21no—
125C23×D59—21no—
126C2×Q8×C59—22no—
127C2×Q8×C59—22no—
128C2×Q8×C59—41no—
129C2×Q8×C59—42no—
130C2×Q8×C59—42no—
131C2×Q8×C59—42no—
132C22×C4×C59—21no—
133C22×C4×C59—21no—
134C22×C4×C59—21no—
135C22×C4×C59—21no—
136C22×C4×C59C2×D411yescharacteristic
137C22×C4×C59C2×D411yescharacteristic
138C22×C4×C59C2×D411yescharacteristic
139C22×C4×C59C2×D411yescharacteristic
140C22×C4×C59C2×D411yescharacteristic
141C22×C4×C59C2×D411yescharacteristic
142C22×C4×C59C2411yescharacteristic
143C22×Dic59—21no—
144C22×Dic59—21no—
145C22×Dic59—21no—
146C2×D236—21no—
147C2×D236—21no—
148C2×D236—21no—
149C2×D236—21no—
150C2×D236—21no—
151C2×D236C2×Q812yes—
152C2×C4×D59—42no—
153C2×C4×D59—81no—
154C2×C4×D59—81no—
155C4⋊C4×C59—22no—
156C4⋊C4×C59—22no—
157C4⋊C4×C59—22no—
158C4⋊C4×C59—21no—
159C4⋊C4×C59—21no—
160C4⋊C4×C59—21no—
161C4⋊C4×C59—21no—
162C4⋊C4×C59—21no—
163C4⋊C4×C59—21no—
164C4⋊C4×C59—42no—
165C4⋊C4×C59—42no—
166C4⋊C4×C59—42no—
167C4⋊C4×C59—42no—
168C4⋊C4×C59—42no—
169C4⋊C4×C59C2×Q812yes—
170C4⋊C4×C59#4112yes—
171C42×C59—21no—
172C42×C59—21no—
173C42×C59—21no—
174C42×C59—21no—
175C42×C59—42no—
176C42×C59C2×D412yes—
177C42×C59#4112yes—
178(C2×D59)⋊C4—42no—
179(C2×D59)⋊C4—42no—
180(C2×D59)⋊C4—42no—
181(C2×D59)⋊C4—81no—
182(C2×D59)⋊C4—81no—
183C4⋊Dic59—21no—
184C4⋊Dic59—21no—
185C4⋊Dic59—21no—
186C4⋊Dic59—21no—
187C4⋊Dic59—21no—
188C4⋊Dic59—21no—
189Dic59⋊C4—81no—
190Dic59⋊C4—81no—
191C2×C59⋊C8—21no—
192C2×C59⋊C8—21no—
193C2×C59⋊C8—21no—
194C2×C59⋊C8—21no—
195C2×C59⋊C8—21no—
196C2×C59⋊C8—21no—
197C2×C59⋊C8—21no—
198C2×C59⋊C8—21no—
199C2×C59⋊C8—21no—
200C2×C59⋊C8C2×D412yes—
201C2×C59⋊C8C2×D412yes—
202C2×C59⋊C8C2×Q812yes—
203C23×C59C22×D411yescharacteristic
204C22×D59—22no—
205C22×D59—41no—
206C22×D59—41no—
207C22×D59—41no—
208C22×D59—41no—
209C22×D59—41no—
210C22×D59—41no—
211Q8×C59—44no—
212Q8×C59—84no—
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—22no—
223C2×C4×C59—22no—
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—41no—
231C2×C4×C59—41no—
232C2×C4×C59—41no—
233C2×C4×C59—41no—
234C2×C4×C59—41no—
235C2×C4×C59—41no—
236C2×C4×C59—41no—
237C2×C4×C59—41no—
238C2×C4×C59—41no—
239C2×C4×C59—41no—
240C2×C4×C59#12112yes—
241C2×C4×C59#12112yes—
242C2×C4×C59AGL(1,ℤ/8)12yes—
243C2×C4×C59C22×D411yescharacteristic
244C2×C4×C59C22×Q811yescharacteristic
245C2×C4×C59C2×#4111yescharacteristic
246C2×C4×C59C2×#4111yescharacteristic
247C2×C4×C592−1+411yescharacteristic
248C2×C4×C592−1+411yescharacteristic
249C2×Dic59—41no—
250C2×Dic59—41no—
251C2×Dic59—41no—
252C2×Dic59—41no—
253C2×Dic59—41no—
254C2×Dic59—41no—
255C2×Dic59—41no—
256C2×Dic59—41no—
257C2×Dic59—41no—
258D236—22no—
259D236—22no—
260D236—41no—
261D236—41no—
262C4×D59—84no—
263C59⋊C8—22no—
264C59⋊C8—22no—
265C59⋊C8—22no—
266C59⋊C8—41no—
267#25485—591nomaximal, Sylow
268C22×C59C2×#12111yescharacteristic
269C22×C59#54811yescharacteristic
270C22×C59D4×Q811yescharacteristic
271C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
272C22×C59#57511yescharacteristic
273C22×C59#57711yescharacteristic
274C22×C59#57711yescharacteristic
275C2×D59—42no—
276C2×D59—42no—
277C2×D59—42no—
278C2×D59—81no—
279C2×D59—81no—
280C2×D59—81no—
281C2×D59—81no—
282C4×C59—22no—
283C4×C59—22no—
284C4×C59—22no—
285C4×C59—42no—
286C4×C59—42no—
287C4×C59—41no—
288C4×C59—41no—
289C4×C59—41no—
290C4×C59—81no—
291C4×C59—81no—
292C4×C59#53311yescharacteristic
293C4×C59#54111yescharacteristic
294C4×C59#54711yescharacteristic
295C4×C59D4×Q811yescharacteristic
296Dic59—81no—
297Dic59—81no—
298Dic59—81no—
299C2×#476—591no—
300C2×#446—591no—
301#2580—591no—
302#2546D5911yescharacteristic, complemented
303#2088—591no—
304#2088—591no—
305#2067—591no—
306#2065—591no—
307#1993—591no—
308#1949—591no—
309#1943—591no—
310#1941—591no—
311#1936—591no—
312#1934—591no—
313#1762—591no—
314C2×C59#242911yescharacteristic
315C2×C59#305511yescharacteristic
316C2×C59#305711yescharacteristic
317C2×C59#337111yescharacteristic
318C2×C59#337311yescharacteristic
319C2×C59#338411yescharacteristic
320C2×C59#338811yescharacteristic
321D59—82no—
322C2×C4⋊Q8—21no—
323C2×C4⋊Q8C2×D5911yescharacteristic
324C2×C4×Q8C2×D5911yescharacteristic
325C2×C4×D4—591no—
326#476—1184no—
327#446—1184no—
328C2×#107—591no—
329C2×#106—591no—
330C2×#106—591no—
331C2×#106—591no—
332C2×C4⋊C8—591no—
333C2×C4⋊C8—591no—
334C2×C4⋊C8—591no—
335C2×#102—591no—
336C2×#102—591no—
337C2×#102—591no—
338C2×#102—591no—
339C2×#97—591no—
340C2×#97—591no—
341C2×C4×C8—591no—
342#399—1181no—
343#398—21no—
344#398—21no—
345#393—1181no—
346#391—21no—
347#390—591no—
348#390—591no—
349#389—591no—
350#389—591no—
351#389C2×D5911yescharacteristic
352#389C2×D5911yescharacteristic
353#389C2×D5911yescharacteristic
354#389C2×D5911yescharacteristic
355#385—1181no—
356#385—1181no—
357#382—591no—
358C4×C4⋊C4C2×D5911yescharacteristic
359#340—591no—
360#340—591no—
361#340—591no—
362#340—591no—
363#340—591no—
364#340—591no—
365#340—591no—
366#340—591no—
367C59#2548511yescharacteristic, complemented, Sylow
368C22×Q8—21no—
369C22×Q8C22⋊D5911yescharacteristic
370C22×D4—591no—
371C23×C4—1181no—
372C22×C8—591no—
373C22×C8—591no—
374C22×C8—591no—
375C22×C8—591no—
376C4⋊Q8—22no—
377C4⋊Q8—22no—
378C4⋊Q8—44no—
379C4×Q8—22no—
380C4×Q8—22no—
381C4×D4—1182no—
382C4×D4—1182no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—591no—
389C2×C4⋊C4—591no—
390C2×C4⋊C4—591no—
391C2×C4⋊C4—1181no—
392C2×C4⋊C4—1181no—
393C2×C4⋊C4C22⋊D5911yescharacteristic
394C2×C4⋊C4C22⋊D5911yescharacteristic
395C2×C4⋊C4C22⋊D5911yescharacteristic
396C2×C4⋊C4C22×D5911yescharacteristic
397C2×C4⋊C4C22×D5911yescharacteristic
398C2×C4⋊C4C22×D5911yescharacteristic
399C2×C4⋊C4C22×D5911yescharacteristic
400C2×C22⋊C4—1181no—
401C2×C22⋊C4—1181no—
402C2×C22⋊C4—1181no—
403C2×C22⋊C4—1181no—
404C2×C22⋊C4—1181no—
405C2×C42—21no—
406C2×C42C22×D5911yescharacteristic
407C2×C42C22×D5911yescharacteristic
408C2×C42C22×D5911yescharacteristic
409#107—1182no—
410#106—1182no—
411#106—1182no—
412#106—1182no—
413C4⋊C8—594no—
414C4⋊C8—1182no—
415C4⋊C8—1182no—
416#102—594no—
417#102—1182no—
418#102—1182no—
419#102—1182no—
420#97—1182no—
421#97—1182no—
422C4×C8—1182no—
423#95—21no—
424#95—21no—
425#95—21no—
426#95—21no—
427#95—21no—
428#95—1181no—
429#95—1181no—
430#95—1181no—
431#95—1181no—
432C24—1181no—
433C2×Q8—22no—
434C2×Q8—22no—
435C2×Q8—41no—
436C2×Q8—42no—
437C2×Q8—42no—
438C2×Q8—42no—
439C2×D4—592no—
440C2×D4—1181no—
441C2×D4—1181no—
442C2×D4—1181no—
443C2×D4—1181no—
444C2×D4—1181no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—1181no—
450C22×C4—1181no—
451C22×C4—1181no—
452C22×C4—4721no—
453C22×C4—4721no—
454C22×C4—2362no—
455C22×C4C2×C22⋊D5911yescharacteristic
456C22×C4C2×C22⋊D5911yescharacteristic
457C22×C4C2×C22⋊D5911yescharacteristic
458C22×C4C2×C22⋊D5911yescharacteristic
459C22×C4C2×C22⋊D5911yescharacteristic
460C22×C4C2×C22⋊D5911yescharacteristic
461C22×C4C23×D5911yescharacteristic
462C2×C8—592no—
463C2×C8—592no—
464C2×C8—592no—
465C2×C8—1181no—
466C2×C8—1181no—
467C2×C8—1181no—
468C2×C8—1181no—
469C2×C8—1181no—
470C2×C8—1181no—
471C2×C8—1181no—
472C2×C8—1181no—
473C2×C8—1181no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—1181no—
489C4⋊C4—1181no—
490C4⋊C4—1181no—
491C4⋊C4—1181no—
492C4⋊C4—1181no—
493C4⋊C4—1181no—
494C4⋊C4—4721no—
495C4⋊C4—4721no—
496C4⋊C4Q8×D5912yes—
497C4⋊C4#1174568912yes—
498C22⋊C4—4721no—
499C22⋊C4—4721no—
500C22⋊C4—2362no—
501C22⋊C4—2362no—
502C22⋊C4—2362no—
503C42—21no—
504C42—21no—
505C42—21no—
506C42—21no—
507C42—42no—
508C42D4×D5912yes—
509C42#1174568712yes—
510C23—1182no—
511C23—2361no—
512C23—2361no—
513C23—2361no—
514C23—2361no—
515C23—2361no—
516C23—2361no—
517C23C22×C22⋊D5911yescharacteristic
518Q8—44no—
519Q8—84no—
520D4—1182no—
521D4—1182no—
522D4—2361no—
523D4—2361no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—4724no—
552C2×C4—2361no—
553C2×C4—2361no—
554C2×C4—2361no—
555C2×C4—2361no—
556C2×C4—2361no—
557C2×C4—2361no—
558C2×C4—2361no—
559C2×C4—2361no—
560C2×C4—2361no—
561C2×C4#1427369912yes—
562C2×C4#1427369912yes—
563C2×C4#1427375312yes—
564C2×C4C2×D4×D5911yescharacteristic
565C2×C4C2×Q8×D5911yescharacteristic
566C2×C4C2×#1174568711yescharacteristic
567C2×C4C2×#1174568911yescharacteristic
568C2×C4#1427377511yescharacteristic
569C2×C4#1427377511yescharacteristic
570C8—1182no—
571C8—1182no—
572C8—1182no—
573C8—2361no—
574C22—4721no—
575C22—4721no—
576C22—4721no—
577C22—4721no—
578C22—2362no—
579C22—2362no—
580C22—2362no—
581C22C2×#1427369911yescharacteristic
582C22#1683806311yescharacteristic
583C22Q8×C22⋊D5911yescharacteristic
584C22C2×#1427375311yescharacteristic
585C22#1683820011yescharacteristic
586C22#1683821011yescharacteristic
587C22#1683821011yescharacteristic
588C4—22no—
589C4—22no—
590C4—22no—
591C4—42no—
592C4—41no—
593C4—41no—
594C4—41no—
595C4—42no—
596C4—81no—
597C4—81no—
598C4—4721no—
599C4—4721no—
600C4—4721no—
601C4#1683795711yescharacteristic
602C4#1683801311yescharacteristic
603C4#1683805511yescharacteristic
604C4#1683806711yescharacteristic
605C2—4722no—
606C2#2142769811yescharacteristic
607C2#2143371111yescharacteristic
608C2#2143373711yescharacteristic
609C2#2143686411yescharacteristic
610C2#2143688811yescharacteristic
611C2#2143699111yescharacteristic
612C2#2143703511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2546C2
socleC23×C59C22×D4
prime radicalC22×D236C23
hypercenter#2546D59
hypocenterC59#25485
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2546

Complements

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

NH ≅ G/Ntype
2. C2×#16837533C2semidirect
4. #21425598C2semidirect
5. #21425597C2semidirect
8. #21425070C2semidirect
9. #21424864C2semidirect
10. #21424831C2semidirect
11. #21424821C2semidirect
15. C59×#2546C2semidirect
89. #14273647Q8semidirect
302. #2546D59semidirect
367. C59#25485semidirect

Maximal subgroups

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

subgroupindex
C2×#168376892
C2×#168375332
#214294542
#214255982
#214255972
#214254812
#214254672
#214250702
#214248642
#214248312
#214248212
#214247922
#214247782
#214239242
C59×#25462
#2548559