Finite Group Databasebeta

← #28617673

Subgroups — #28617673

Order 15104.

898 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837439C211yesmaximal, characteristic
2C2×#16837437C211yesmaximal, characteristic
3#21426577C211yesmaximal, characteristic
4#21426502C211yesmaximal, characteristic
5#21425579C211yesmaximal, characteristic, complemented
6#21425578C211yesmaximal, characteristic, complemented
7#21425577C211yesmaximal, characteristic, complemented
8#21425576C211yesmaximal, characteristic, complemented
9#21425575C211yesmaximal, characteristic, complemented
10#21425571C211yesmaximal, characteristic, complemented
11#21425570C211yesmaximal, characteristic, complemented
12#21423811C211yesmaximal, characteristic
13#21423811C211yesmaximal, characteristic
14#336×D59C211yesmaximal, characteristic
15C59×#2085C211yesmaximal, characteristic, complemented
16C22×C8×D59C2211yescharacteristic
17C22×C8×D59C2211yescharacteristic
18C2×#14273685C2211yescharacteristic
19C2×#14273685C2211yescharacteristic
20C2×#14273685C2211yescharacteristic
21C2×#14273685C2211yescharacteristic
22C2×C42×D59C2211yescharacteristic
23#16837439—24no—
24#16837437—24no—
25C2×#14273661C2211yescharacteristic
26C2×#14273660C2211yescharacteristic
27C2×#14273660C2211yescharacteristic
28C2×#14273659C2211yescharacteristic
29C2×#14273658C2211yescharacteristic
30C2×#14273658C2211yescharacteristic
31#16837301C2211yescharacteristic
32#16837300C2211yescharacteristic
33#16837298C2211yescharacteristic
34#16837276—21no—
35#16837273—21no—
36#16837271C2211yescharacteristic
37#16837271C2211yescharacteristic
38#16837268C2211yescharacteristic
39#16837261—21no—
40#16837261—21no—
41C4⋊C4×Dic59—21no—
42#16837253—21no—
43#16837253—21no—
44#16837253—21no—
45#16837165C2211yescharacteristic
46#16837165C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837164C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837164C2211yescharacteristic
51#16837157C2211yescharacteristic
52#16837156C2211yescharacteristic
53#16837156C2211yescharacteristic
54C2×#107×C59C2211yescharacteristic
55C2×#106×C59C2211yescharacteristic
56C59×#389C2211yescharacteristic
57C59×#382C2211yescharacteristic
58C59×#340C2211yescharacteristic
59C59×#340C2211yescharacteristic
60C59×#336C2211yescharacteristic
61C22×C4×D59D411yescharacteristic
62C22×C4×D59D411yescharacteristic
63C22×C4×D59C2311yescharacteristic
64C22×C59⋊C8C2311yescharacteristic
65C22×C59⋊C8C2311yescharacteristic
66C2×C8×D59—22no—
67C2×C8×D59—22no—
68C2×C8×D59—22no—
69C2×C8×D59—22no—
70C2×C8×D59Q814yes—
71C2×C8×D59Q814yes—
72#14273685—22no—
73#14273685—22no—
74#14273685—22no—
75#14273685—22no—
76#14273685—22no—
77#14273685—22no—
78#14273685—22no—
79#14273685D414yes—
80C2×C4⋊Dic59—21no—
81C2×C4⋊Dic59C2311yescharacteristic
82C2×C4⋊Dic59C2311yescharacteristic
83C2×C4⋊Dic59C2311yescharacteristic
84C2×C4⋊Dic59C2311yescharacteristic
85C2×Dic59⋊C4—21no—
86C2×Dic59⋊C4—21no—
87C2×Dic59⋊C4—21no—
88C2×Dic59⋊C4—21no—
89C2×Dic59⋊C4—21no—
90C2×Dic59⋊C4—21no—
91C2×(C2×D59)⋊C4—21no—
92C2×(C2×D59)⋊C4—21no—
93C2×(C2×D59)⋊C4—21no—
94C2×(C2×D59)⋊C4—21no—
95C2×C4×Dic59—21no—
96C2×C4×Dic59—21no—
97C2×C4×Dic59—21no—
98C2×C4×Dic59—21no—
99C2×C4×Dic59D411yescharacteristic
100C2×C4×Dic59D411yescharacteristic
101C2×C4×Dic59C2311yescharacteristic
102C42×D59—22no—
103C42×D59—22no—
104#14273661—22no—
105#14273660—22no—
106#14273660—22no—
107#14273659—22no—
108#14273658—22no—
109#14273658—22no—
110C59⋊#95—21no—
111C59⋊#95—21no—
112C59⋊#95—21no—
113C59⋊#95—21no—
114C59⋊#95—21no—
115C22×C8×C59C2311yescharacteristic
116C22×C8×C59C2311yescharacteristic
117C2×C4⋊C4×C59—21no—
118C2×C4⋊C4×C59C2311yescharacteristic
119C2×C4⋊C4×C59C2311yescharacteristic
120C2×C4⋊C4×C59C2311yescharacteristic
121C2×C4⋊C4×C59C2311yescharacteristic
122C2×C42×C59C2311yescharacteristic
123#107×C59—22no—
124#106×C59—22no—
125#95×C59—21no—
126#95×C59—21no—
127#95×C59—21no—
128C23×D59C2×D411yescharacteristic
129C22×C4×C59—21no—
130C22×C4×C59—21no—
131C22×C4×C59—21no—
132C22×C4×C59—21no—
133C22×C4×C59C2×D411yescharacteristic
134C22×C4×C59C2×D411yescharacteristic
135C22×C4×C59C2411yescharacteristic
136C22×Dic59—21no—
137C22×Dic59—21no—
138C22×Dic59—21no—
139C22×Dic59—21no—
140C22×Dic59C2×D411yescharacteristic
141C22×Dic59C2×D411yescharacteristic
142C22×Dic59C2×D411yescharacteristic
143C2×C4×D59—22no—
144C2×C4×D59—22no—
145C2×C4×D59—22no—
146C2×C4×D59—22no—
147C2×C4×D59—41no—
148C2×C4×D59—41no—
149C2×C4×D59C2×D412yes—
150C2×C4×D59C2×Q812yes—
151C2×C4×D59C2×Q812yes—
152C2×C4×D59#4112yes—
153C2×C4×D59#4112yes—
154C2×C4×D59#4112yes—
155C2×C8×C59—21no—
156C2×C8×C59—21no—
157C2×C8×C59—21no—
158C2×C8×C59—21no—
159C2×C8×C59C2×Q812yes—
160C2×C8×C59C2×Q812yes—
161C4⋊C4×C59—21no—
162C4⋊C4×C59—21no—
163C4⋊C4×C59—21no—
164C4⋊C4×C59—21no—
165C4⋊C4×C59—21no—
166C4⋊C4×C59—21no—
167C4⋊C4×C59—21no—
168C4⋊C4×C59—42no—
169C4⋊C4×C59C2×D412yes—
170C42×C59—21no—
171C42×C59—21no—
172(C2×D59)⋊C4—24no—
173(C2×D59)⋊C4—42no—
174(C2×D59)⋊C4—42no—
175(C2×D59)⋊C4—42no—
176C4⋊Dic59—21no—
177C4⋊Dic59—21no—
178C4⋊Dic59—21no—
179C4⋊Dic59—21no—
180C4⋊Dic59—21no—
181C4⋊Dic59—21no—
182C4⋊Dic59—21no—
183C4⋊Dic59—42no—
184C4⋊Dic59C2×D412yes—
185Dic59⋊C4—42no—
186Dic59⋊C4—42no—
187Dic59⋊C4—42no—
188Dic59⋊C4—42no—
189Dic59⋊C4—42no—
190Dic59⋊C4—42no—
191C4×Dic59—21no—
192C4×Dic59—21no—
193C4×Dic59—24no—
194C4×Dic59—22no—
195C4×Dic59—22no—
196C4×Dic59—42no—
197C4×Dic59—42no—
198C4×Dic59—42no—
199C2×C59⋊C8—21no—
200C2×C59⋊C8—21no—
201C2×C59⋊C8—21no—
202C2×C59⋊C8—21no—
203C2×C59⋊C8C2×Q812yes—
204C2×C59⋊C8C2×Q812yes—
205C8×D59—28no—
206C8×D59—28no—
207C23×C59C22×D411yescharacteristic
208C22×D59—21no—
209C22×D59—21no—
210C22×D59—21no—
211C22×D59—21no—
212C22×D59#12112yes—
213C22×D59#12312yes—
214C22×D59C4⋊Q812yes—
215C2×C4×C59—22no—
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×C59C22×D411yescharacteristic
234C2×C4×C59C22×Q811yescharacteristic
235C2×C4×C59C22×Q811yescharacteristic
236C2×C4×C59C2×#4111yescharacteristic
237C2×C4×C59C2×#4111yescharacteristic
238C2×C4×C59C2×#4111yescharacteristic
239C2×Dic59—21no—
240C2×Dic59—21no—
241C2×Dic59—21no—
242C2×Dic59—21no—
243C2×Dic59—22no—
244C2×Dic59—41no—
245C2×Dic59—41no—
246C2×Dic59—41no—
247C2×Dic59—41no—
248C2×Dic59—41no—
249C2×Dic59—41no—
250C2×Dic59—41no—
251C2×Dic59—41no—
252C2×Dic59—41no—
253C2×Dic59—41no—
254C2×Dic59—41no—
255C2×Dic59#12112yes—
256C2×Dic59#12312yes—
257C2×Dic59C4⋊Q812yes—
258C2×Dic59#13512yes—
259C2×Dic59#13512yes—
260C4×D59—44no—
261C4×D59—44no—
262C4×D59C22⋊Q814yes—
263C4×D59C22⋊Q814yes—
264C4×D59#12514yes—
265C4×D59#12514yes—
266C8×C59—22no—
267C8×C59—22no—
268C59⋊C8—22no—
269C59⋊C8—22no—
270#23656—591nomaximal, Sylow
271C22×C59C2×#12111yescharacteristic
272C22×C59C2×#12311yescharacteristic
273C22×C59C2×C4⋊Q811yescharacteristic
274C22×C59C2×#13511yescharacteristic
275C22×C59C2×#13511yescharacteristic
276C22×C59#57511yescharacteristic
277C22×C59#57511yescharacteristic
278C2×D59—22no—
279C2×D59—22no—
280C2×D59—22no—
281C2×D59—22no—
282C2×D59—22no—
283C2×D59—22no—
284C2×D59#39814yes—
285C4×C59—41no—
286C4×C59—41no—
287C4×C59—42no—
288C4×C59—81no—
289C4×C59—81no—
290C4×C59—81no—
291C4×C59C2×C22⋊Q811yescharacteristic
292C4×C59C2×C22⋊Q811yescharacteristic
293C4×C59C2×#12511yescharacteristic
294C4×C59C2×#12511yescharacteristic
295Dic59—22no—
296Dic59—22no—
297Dic59—42no—
298Dic59—81no—
299Dic59—81no—
300Dic59—81no—
301Dic59#39814yes—
302C2×#427—591no—
303C2×#427—591no—
304#2296—591no—
305#2285—591no—
306#2085—591no—
307#2085—591no—
308#2085—591no—
309#2085D5911yescharacteristic, complemented
310#2084—591no—
311#2084—591no—
312#2083—591no—
313#2083—591no—
314#1735—591no—
315#1735—591no—
316C2×#336—591no—
317C2×C59C2×#39811yescharacteristic
318C2×C59#303511yescharacteristic
319C2×C59#303511yescharacteristic
320C2×C59#309111yescharacteristic
321C2×C59#309111yescharacteristic
322C2×C59#316011yescharacteristic
323C2×C59#316211yescharacteristic
324D59—24no—
325C23×C8—591no—
326C23×C8—591no—
327C2×#117—591no—
328C2×#117—591no—
329C2×#117—591no—
330C2×#117—591no—
331C22×C42—591no—
332#427—1184no—
333#427—1184no—
334C2×#107—591no—
335C2×#107—591no—
336C2×#107—591no—
337C2×#107C2×D5911yescharacteristic
338C2×#106—591no—
339C2×#106—591no—
340C2×#106—591no—
341C2×#106C2×D5911yescharacteristic
342#389—591no—
343#389—591no—
344#389—591no—
345#389C2×D5911yescharacteristic
346#385—1181no—
347#385—1181no—
348#384—591no—
349#384—591no—
350#382—591no—
351#382C2×D5911yescharacteristic
352#380—1181no—
353#380—1181no—
354C4×C4⋊C4—1181no—
355C4×C4⋊C4—1181no—
356C4×C4⋊C4—1181no—
357C4×C4⋊C4—1181no—
358#340—591no—
359#340—591no—
360#340—591no—
361#340—591no—
362#340—591no—
363#340—591no—
364#340C2×D5911yescharacteristic
365#340C2×D5911yescharacteristic
366#336—591no—
367#336—591no—
368#336—591no—
369#336C2×D5911yescharacteristic
370C59#2365611yescharacteristic, complemented, Sylow
371C23×C4—591no—
372C23×C4—591no—
373C23×C4—591no—
374C22×C8—591no—
375C22×C8—591no—
376C22×C8—594no—
377C22×C8—594no—
378C22×C8—1182no—
379C22×C8—1182no—
380C22×C8—1182no—
381C22×C8—1182no—
382C22×C8C22×D5911yescharacteristic
383C22×C8C22×D5911yescharacteristic
384#117—594no—
385#117—1182no—
386#117—1182no—
387#117—1182no—
388#117—1182no—
389#117—1182no—
390#117—1182no—
391#117—1182no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—591no—
394C2×C4⋊C4—591no—
395C2×C4⋊C4—591no—
396C2×C4⋊C4—591no—
397C2×C4⋊C4—1181no—
398C2×C4⋊C4—1181no—
399C2×C4⋊C4—1181no—
400C2×C4⋊C4—1181no—
401C2×C4⋊C4—1181no—
402C2×C4⋊C4—1181no—
403C2×C4⋊C4—1181no—
404C2×C4⋊C4C22×D5911yescharacteristic
405C2×C4⋊C4C22×D5911yescharacteristic
406C2×C4⋊C4C22×D5911yescharacteristic
407C2×C4⋊C4C22×D5911yescharacteristic
408C2×C22⋊C4—1181no—
409C2×C22⋊C4—1181no—
410C2×C22⋊C4—1181no—
411C2×C22⋊C4—1181no—
412C2×C42—591no—
413C2×C42—591no—
414C2×C42—591no—
415C2×C42—1182no—
416C2×C42—1182no—
417C2×C42—1181no—
418C2×C42—1181no—
419C2×C42—1181no—
420C2×C42—1181no—
421C2×C42C22×D5911yescharacteristic
422#107—22no—
423#107—1182no—
424#107—1182no—
425#107—1182no—
426#106—22no—
427#106—1182no—
428#106—1182no—
429#106—1182no—
430#95—21no—
431#95—21no—
432#95—21no—
433#95—1181no—
434#95—1181no—
435#95—1181no—
436#95—1181no—
437#95—1181no—
438C24—591no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—591no—
444C22×C4—592no—
445C22×C4—592no—
446C22×C4—592no—
447C22×C4—592no—
448C22×C4—592no—
449C22×C4—592no—
450C22×C4—591no—
451C22×C4—591no—
452C22×C4—1182no—
453C22×C4—1182no—
454C22×C4—1182no—
455C22×C4—1182no—
456C22×C4—1181no—
457C22×C4—1181no—
458C22×C4—1181no—
459C22×C4—1181no—
460C22×C4—2361no—
461C22×C4—2361no—
462C22×C4D4×D5911yescharacteristic
463C22×C4D4×D5911yescharacteristic
464C22×C4C23×D5911yescharacteristic
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—592no—
470C2×C8—592no—
471C2×C8—1188no—
472C2×C8—1188no—
473C2×C8—1181no—
474C2×C8—1181no—
475C2×C8—1181no—
476C2×C8—1181no—
477C2×C8Q8×D5912yes—
478C2×C8Q8×D5912yes—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—42no—
487C4⋊C4—592no—
488C4⋊C4—1181no—
489C4⋊C4—1181no—
490C4⋊C4—1181no—
491C4⋊C4—1181no—
492C4⋊C4—1181no—
493C4⋊C4—1181no—
494C4⋊C4—1181no—
495C4⋊C4—2362no—
496C4⋊C4—2362no—
497C4⋊C4—2362no—
498C4⋊C4—2362no—
499C4⋊C4—2362no—
500C4⋊C4—2362no—
501C4⋊C4—2362no—
502C4⋊C4D4×D5912yes—
503C22⋊C4—1184no—
504C22⋊C4—2362no—
505C22⋊C4—2362no—
506C22⋊C4—2362no—
507C42—21no—
508C42—21no—
509C42—1182no—
510C42—1182no—
511C42—1184no—
512C42—1181no—
513C42—1181no—
514C42—2362no—
515C42—2362no—
516C42—2362no—
517C23—592no—
518C23—592no—
519C23—592no—
520C23—1181no—
521C23—1181no—
522C23—1181no—
523C23—1181no—
524C23C2×D4×D5911yescharacteristic
525C2×C4—22no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—592no—
544C2×C4—592no—
545C2×C4—592no—
546C2×C4—594no—
547C2×C4—594no—
548C2×C4—594no—
549C2×C4—594no—
550C2×C4—592no—
551C2×C4—592no—
552C2×C4—1181no—
553C2×C4—1181no—
554C2×C4—1181no—
555C2×C4—1181no—
556C2×C4—1182no—
557C2×C4—2364no—
558C2×C4—2364no—
559C2×C4—2361no—
560C2×C4—2361no—
561C2×C4—2361no—
562C2×C4—2361no—
563C2×C4—2361no—
564C2×C4—2361no—
565C2×C4—2361no—
566C2×C4—2361no—
567C2×C4—2361no—
568C2×C4—2361no—
569C2×C4—2361no—
570C2×C4C2×D4×D5911yescharacteristic
571C2×C4C2×Q8×D5911yescharacteristic
572C2×C4C2×Q8×D5911yescharacteristic
573C2×C4#41×D5911yescharacteristic
574C2×C4#41×D5911yescharacteristic
575C2×C4#41×D5911yescharacteristic
576C8—22no—
577C8—22no—
578C8—1182no—
579C8—1182no—
580C22—594no—
581C22—1182no—
582C22—1182no—
583C22—1182no—
584C22—1182no—
585C22—1182no—
586C22—1182no—
587C22#121×D5911yescharacteristic
588C22#123×D5911yescharacteristic
589C22C4⋊Q8×D5911yescharacteristic
590C22#135×D5911yescharacteristic
591C22#135×D5911yescharacteristic
592C22#1683819811yescharacteristic
593C22#1683820111yescharacteristic
594C4—41no—
595C4—41no—
596C4—42no—
597C4—81no—
598C4—81no—
599C4—81no—
600C4—594no—
601C4—1182no—
602C4—1182no—
603C4—4721no—
604C4—4721no—
605C4—4721no—
606C4—2362no—
607C4C22⋊Q8×D5911yescharacteristic
608C4C22⋊Q8×D5911yescharacteristic
609C4#125×D5911yescharacteristic
610C4#125×D5911yescharacteristic
611C2—1184no—
612C2#398×D5911yescharacteristic
613C2#2143354811yescharacteristic
614C2#2143355011yescharacteristic
615C2#2143400811yescharacteristic
616C2#2143401411yescharacteristic
617C2#2143460911yescharacteristic
618C2#2143461711yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2085C2
socleC23×C59C22×D4
prime radicalC23×D59C2×D4
hypercenter#2085D59
hypocenterC59#23656
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2085

Complements

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

NH ≅ G/Ntype
5. #21425579C2semidirect
6. #21425578C2semidirect
7. #21425577C2semidirect
8. #21425576C2semidirect
9. #21425575C2semidirect
10. #21425571C2semidirect
11. #21425570C2semidirect
15. C59×#2085C2semidirect
309. #2085D59semidirect
370. C59#23656semidirect

Maximal subgroups

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

subgroupindex
C2×#168374392
C2×#168374372
#214265772
#214265022
#214255792
#214255782
#214255772
#214255762
#214255752
#214255712
#214255702
#214238112
#336×D592
C59×#20852
#2365659