Finite Group Databasebeta

← #28618720

Subgroups — #28618720

Order 15104.

836 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#121×Dic59C211yesmaximal, characteristic
2#21426491C211yesmaximal, characteristic, complemented
3#21425667C211yesmaximal, characteristic, complemented
4#21425615C211yesmaximal, characteristic, complemented
5#21425580C211yesmaximal, characteristic, complemented
6#21425532C211yesmaximal, characteristic, complemented
7#21425524C211yesmaximal, characteristic, complemented
8#21425512C211yesmaximal, characteristic, complemented
9#21425282C211yesmaximal, characteristic
10#107×Dic59C211yesmaximal, characteristic, complemented
11#21423984C211yesmaximal, characteristic, complemented
12#102×Dic59C211yesmaximal, characteristic, complemented
13#102×Dic59C211yesmaximal, characteristic, complemented
14C22⋊C8×Dic59C211yesmaximal, characteristic, complemented
15C59×#2033C211yesmaximal, characteristic
16C2×D4×Dic59—21no—
17C2×D4×Dic59C2211yescharacteristic
18C22×C4×Dic59C2211yescharacteristic
19C2×#14273661C2211yescharacteristic
20C2×#14273658C2211yescharacteristic
21C2×#14273658C2211yescharacteristic
22C2×#14273648C2211yescharacteristic
23C2×#14273648C2211yescharacteristic
24C2×C22⋊(C59⋊C8)C2211yescharacteristic
25C2×C8×Dic59C2211yescharacteristic
26C2×C8×Dic59C2211yescharacteristic
27#16837305C2211yescharacteristic
28#16837301C2211yescharacteristic
29#16837295—21no—
30#16837282—21no—
31#16837269—21no—
32#16837267C2211yescharacteristic
33#16837267C2211yescharacteristic
34#16837263—21no—
35#16837262C2211yescharacteristic
36#16837262C2211yescharacteristic
37C4⋊C4×Dic59C2211yescharacteristic
38C4⋊C4×Dic59C2211yescharacteristic
39C4⋊C4×Dic59C2211yescharacteristic
40C22⋊C4×Dic59—21no—
41C22⋊C4×Dic59—21no—
42#16837247—21no—
43#16837165C2211yescharacteristic
44#16837165C2211yescharacteristic
45#16837165C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837164C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837156C2211yescharacteristic
51#16837156C2211yescharacteristic
52C2×#121×C59C2211yescharacteristic
53C2×#107×C59C2211yescharacteristic
54C2×#102×C59C2211yescharacteristic
55C2×#102×C59C2211yescharacteristic
56C2×C22⋊C8×C59C2211yescharacteristic
57C59×#380C2211yescharacteristic
58C59×#340C2211yescharacteristic
59C23×Dic59—21no—
60C23×Dic59D411yescharacteristic
61C23×Dic59D411yescharacteristic
62C22×C59⋊C8C2311yescharacteristic
63C22×C59⋊C8C2311yescharacteristic
64D4×Dic59—22no—
65D4×Dic59—22no—
66D4×Dic59—44no—
67C2×C4⋊Dic59—21no—
68C2×C4⋊Dic59C2311yescharacteristic
69C2×C4⋊Dic59C2311yescharacteristic
70C2×C4⋊Dic59C2311yescharacteristic
71C2×C4⋊Dic59C2311yescharacteristic
72C2×C22⋊Dic59—21no—
73C2×C22⋊Dic59—21no—
74C2×C22⋊Dic59—21no—
75C2×C22⋊Dic59—21no—
76C2×C22⋊Dic59—21no—
77C2×C4×Dic59—22no—
78C2×C4×Dic59—22no—
79C2×C4×Dic59—21no—
80C2×C4×Dic59—21no—
81C2×C4×Dic59—21no—
82C2×C4×Dic59D411yescharacteristic
83C2×C4×Dic59D411yescharacteristic
84C2×C4×Dic59C2311yescharacteristic
85C2×C4×Dic59C2311yescharacteristic
86#14273661—22no—
87#14273658—22no—
88#14273658—22no—
89#14273648—22no—
90#14273648—22no—
91C22⋊(C59⋊C8)—22no—
92C8×Dic59—22no—
93C8×Dic59—22no—
94C59⋊#95—21no—
95C59⋊#95—21no—
96C59⋊#95—21no—
97C59⋊#95—21no—
98C59⋊#95—21no—
99C59⋊#95—21no—
100C59⋊#95—21no—
101C59⋊#95—21no—
102C59⋊#95—21no—
103C59⋊#95—21no—
104C22×D4×C59—21no—
105C22×D4×C59C2311yescharacteristic
106C23×C4×C59C2311yescharacteristic
107C22×C8×C59C2311yescharacteristic
108C22×C8×C59C2311yescharacteristic
109#121×C59—22no—
110#121×C59—22no—
111C2×C4⋊C4×C59C2311yescharacteristic
112C2×C4⋊C4×C59C2311yescharacteristic
113C2×C4⋊C4×C59C2311yescharacteristic
114C2×C22⋊C4×C59—21no—
115C2×C22⋊C4×C59—21no—
116#107×C59—22no—
117#102×C59—22no—
118#102×C59—22no—
119C22⋊C8×C59—22no—
120#95×C59—21no—
121C24×C59—21no—
122C24×C59C2×D411yescharacteristic
123C2×D4×C59—24no—
124C2×D4×C59—21no—
125C2×D4×C59—21no—
126C2×D4×C59—21no—
127C2×D4×C59—21no—
128C2×D4×C59—21no—
129C2×D4×C59—42no—
130C2×D4×C59—42no—
131C2×D4×C59#4112yes—
132C22×C4×C59—21no—
133C22×C4×C59—21no—
134C22×C4×C59—21no—
135C22×C4×C59—21no—
136C22×C4×C59—21no—
137C22×C4×C59—21no—
138C22×C4×C59—21no—
139C22×C4×C59—21no—
140C22×C4×C59—21no—
141C22×C4×C59C2×D411yescharacteristic
142C22×C4×C59C2411yescharacteristic
143C22×Dic59—22no—
144C22×Dic59—22no—
145C22×Dic59—22no—
146C22×Dic59—22no—
147C22×Dic59—22no—
148C22×Dic59—21no—
149C22×Dic59—21no—
150C22×Dic59—21no—
151C22×Dic59—21no—
152C22×Dic59—41no—
153C22×Dic59—42no—
154C22×Dic59—42no—
155C22×Dic59—42no—
156C22×Dic59C2×D411yescharacteristic
157C22×Dic59C2×D411yescharacteristic
158C22×Dic59C2×D411yescharacteristic
159C22×Dic59C2×D411yescharacteristic
160C2×C8×C59—21no—
161C2×C8×C59—21no—
162C2×C8×C59—21no—
163C2×C8×C59—21no—
164C2×C8×C59—21no—
165C2×C8×C59#4112yes—
166C4⋊C4×C59—21no—
167C4⋊C4×C59—21no—
168C4⋊C4×C59—21no—
169C4⋊C4×C59—21no—
170C4⋊C4×C59—21no—
171C4⋊C4×C59#4112yes—
172C22⋊C4×C59—24no—
173C22⋊C4×C59—42no—
174C22⋊Dic59—42no—
175C22⋊Dic59—42no—
176C22⋊Dic59—42no—
177C22⋊Dic59—42no—
178C22⋊Dic59—42no—
179C4⋊Dic59—21no—
180C4⋊Dic59—21no—
181C4⋊Dic59—21no—
182C4⋊Dic59—21no—
183C4⋊Dic59—21no—
184C4⋊Dic59—21no—
185C4⋊Dic59—21no—
186C4⋊Dic59—21no—
187C4⋊Dic59—42no—
188C4×Dic59—21no—
189C4×Dic59—21no—
190C4×Dic59—22no—
191C4×Dic59—22no—
192C4×Dic59—42no—
193C4×Dic59—42no—
194C4×Dic59—42no—
195C4×Dic59#4112yes—
196C4×Dic59#4112yes—
197C2×C59⋊C8—21no—
198C2×C59⋊C8—21no—
199C2×C59⋊C8—21no—
200C2×C59⋊C8—21no—
201C2×C59⋊C8—21no—
202C2×C59⋊C8#4112yes—
203C23×C59—21no—
204C23×C59—21no—
205C23×C59—21no—
206C23×C59—21no—
207C23×C59—21no—
208C23×C59—21no—
209C23×C59—22no—
210C23×C59—41no—
211C23×C59—41no—
212C23×C59—41no—
213C23×C59—41no—
214C23×C59—41no—
215C23×C59—41no—
216C23×C59#13512yes—
217C23×C59C22×D411yescharacteristic
218D4×C59—22no—
219D4×C59—22no—
220D4×C59—48no—
221D4×C59—41no—
222D4×C59—41no—
223C2×C4×C59—21no—
224C2×C4×C59—21no—
225C2×C4×C59—22no—
226C2×C4×C59—21no—
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×C59—41no—
234C2×C4×C59—41no—
235C2×C4×C59—41no—
236C2×C4×C59—41no—
237C2×C4×C59—41no—
238C2×C4×C59—41no—
239C2×C4×C59C2×#4111yescharacteristic
240C2×C4×C59C2×#4111yescharacteristic
241C2×C4×C59C2×#4111yescharacteristic
242C2×C4×C59C2×#4111yescharacteristic
243C2×C4×C59C2×#4111yescharacteristic
244C2×C4×C59C2×#4111yescharacteristic
245C2×Dic59—21no—
246C2×Dic59—21no—
247C2×Dic59—21no—
248C2×Dic59—21no—
249C2×Dic59—21no—
250C2×Dic59—21no—
251C2×Dic59—21no—
252C2×Dic59—21no—
253C2×Dic59—44no—
254C2×Dic59—44no—
255C2×Dic59—41no—
256C2×Dic59—41no—
257C2×Dic59—41no—
258C2×Dic59—41no—
259C2×Dic59—41no—
260C2×Dic59—41no—
261C2×Dic59—41no—
262C2×Dic59—41no—
263C2×Dic59—41no—
264C2×Dic59—41no—
265C2×Dic59—41no—
266C2×Dic59—41no—
267C2×Dic59—84no—
268C2×Dic59#12312yes—
269C2×Dic59#12312yes—
270C2×Dic59#13512yes—
271C2×Dic59AGL(1,ℤ/8)12yes—
272C8×C59—22no—
273C8×C59—41no—
274C59⋊C8—22no—
275C59⋊C8—41no—
276#23758—591nomaximal, Sylow
277C22×C59—22no—
278C22×C59—22no—
279C22×C59—22no—
280C22×C59—41no—
281C22×C59—41no—
282C22×C59—41no—
283C22×C59—41no—
284C22×C59—42no—
285C22×C59—42no—
286C22×C59—42no—
287C22×C59—81no—
288C22×C59—81no—
289C22×C59—81no—
290C22×C59—81no—
291C22×C59C2×#12311yescharacteristic
292C22×C59#52511yescharacteristic
293C22×C59#53211yescharacteristic
294C22×C59C2×#13511yescharacteristic
295C22×C59C2×#13511yescharacteristic
296C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
297C22×C59#57511yescharacteristic
298C4×C59—42no—
299C4×C59—41no—
300C4×C59—81no—
301C4×C59—81no—
302C4×C59#52911yescharacteristic
303C4×C59#52911yescharacteristic
304C4×C59#52911yescharacteristic
305C4×C59#52911yescharacteristic
306Dic59—22no—
307Dic59—41no—
308Dic59—41no—
309Dic59—81no—
310Dic59—81no—
311Dic59—81no—
312Dic59—81no—
313Dic59#48214yes—
314C4×#121—591no—
315#2283—591no—
316#2101—591no—
317#2092—591no—
318#2085—591no—
319#2077—591no—
320#2076—591no—
321#2073—591no—
322#2034—591no—
323#2033D5911yescharacteristic
324C4×#107—591no—
325C4×#106—591no—
326C4×#102—591no—
327C4×#102—591no—
328C4×C22⋊C8—591no—
329C2×C59—42no—
330C2×C59—82no—
331C2×C59#239511yescharacteristic
332C2×C59C2×#48211yescharacteristic
333C2×C59#309111yescharacteristic
334C2×C59#310211yescharacteristic
335C2×C59#310511yescharacteristic
336C2×C59#316811yescharacteristic
337C2×C59#317111yescharacteristic
338C2×#121C2×D5911yescharacteristic
339C2×C4×D4—591no—
340C2×C4×D4—1181no—
341C22×C42—591no—
342C2×#107—591no—
343C2×#107C2×D5911yescharacteristic
344C2×#106—591no—
345C2×#106—591no—
346C2×#102—591no—
347C2×#102—591no—
348C2×#102C2×D5911yescharacteristic
349C2×#102C2×D5911yescharacteristic
350C2×C22⋊C8—591no—
351C2×C22⋊C8C2×D5911yescharacteristic
352C2×C4×C8—591no—
353C2×C4×C8—591no—
354#390—591no—
355#389—591no—
356#388—1181no—
357#386—1181no—
358#383—1181no—
359#382—591no—
360#382—591no—
361#381—1181no—
362#380—591no—
363#380—591no—
364#380C2×D5911yescharacteristic
365C4×C4⋊C4—591no—
366C4×C4⋊C4—591no—
367C4×C4⋊C4—591no—
368C4×C22⋊C4—1181no—
369C4×C22⋊C4—1181no—
370#376—1181no—
371#340—591no—
372#340—591no—
373#340—591no—
374#340—591no—
375#340—591no—
376#340—591no—
377#340—591no—
378#340C2×D5911yescharacteristic
379#336—591no—
380#336—591no—
381C59#2375811yescharacteristic, complemented, Sylow
382C22×D4—21no—
383C22×D4C22×D5911yescharacteristic
384C23×C4—591no—
385C23×C4—591no—
386C23×C4—1181no—
387C23×C4C22×D5911yescharacteristic
388C22×C8—591no—
389C22×C8—591no—
390C22×C8C22×D5911yescharacteristic
391C22×C8C22×D5911yescharacteristic
392#121—22no—
393#121—22no—
394C4×D4—1182no—
395C4×D4—1182no—
396C4×D4—2364no—
397C2×C4⋊C4—591no—
398C2×C4⋊C4—591no—
399C2×C4⋊C4—591no—
400C2×C4⋊C4—591no—
401C2×C4⋊C4—1181no—
402C2×C4⋊C4C22×D5911yescharacteristic
403C2×C4⋊C4C22×D5911yescharacteristic
404C2×C4⋊C4C22×D5911yescharacteristic
405C2×C22⋊C4—21no—
406C2×C22⋊C4—21no—
407C2×C22⋊C4—1181no—
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—591no—
416C2×C42—1182no—
417C2×C42—1182no—
418C2×C42—1181no—
419C2×C42—1181no—
420C2×C42—1181no—
421#107—22no—
422#107—1182no—
423#106—1182no—
424#106—1182no—
425#102—22no—
426#102—22no—
427#102—1182no—
428#102—1182no—
429C22⋊C8—22no—
430C22⋊C8—1182no—
431C4×C8—1182no—
432C4×C8—1182no—
433#95—21no—
434#95—1181no—
435#95—1181no—
436#95—1181no—
437#95—1181no—
438#95—1181no—
439#95—1181no—
440#95—1181no—
441#95—1181no—
442#95—1181no—
443#95—1181no—
444C24—21no—
445C24D4×D5911yescharacteristic
446C2×D4—24no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4—42no—
453C2×D4—42no—
454C2×D4#1174568712yes—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—591no—
465C22×C4—591no—
466C22×C4—591no—
467C22×C4—591no—
468C22×C4—1182no—
469C22×C4—1182no—
470C22×C4—1182no—
471C22×C4—1182no—
472C22×C4—1182no—
473C22×C4—1181no—
474C22×C4—1181no—
475C22×C4—1181no—
476C22×C4—1181no—
477C22×C4—2362no—
478C22×C4—2362no—
479C22×C4—2362no—
480C22×C4—2361no—
481C22×C4D4×D5911yescharacteristic
482C22×C4C23×D5911yescharacteristic
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—592no—
489C2×C8—1181no—
490C2×C8—1181no—
491C2×C8—1181no—
492C2×C8—1181no—
493C2×C8—1181no—
494C2×C8#1174568712yes—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—1181no—
501C4⋊C4—1181no—
502C4⋊C4—1181no—
503C4⋊C4—1181no—
504C4⋊C4—1181no—
505C4⋊C4—1181no—
506C4⋊C4—1181no—
507C4⋊C4—1181no—
508C4⋊C4—2362no—
509C4⋊C4#1174568712yes—
510C22⋊C4—24no—
511C22⋊C4—42no—
512C22⋊C4—2362no—
513C22⋊C4—2362no—
514C22⋊C4—2362no—
515C22⋊C4—2362no—
516C22⋊C4—2362no—
517C42—592no—
518C42—592no—
519C42—1182no—
520C42—1182no—
521C42—1181no—
522C42—1181no—
523C42—2362no—
524C42—2362no—
525C42—2362no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—22no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23#1427374612yes—
540C23C2×D4×D5911yescharacteristic
541D4—22no—
542D4—22no—
543D4—48no—
544D4—41no—
545D4—41no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—592no—
563C2×C4—592no—
564C2×C4—592no—
565C2×C4—592no—
566C2×C4—1181no—
567C2×C4—1181no—
568C2×C4—1181no—
569C2×C4—1181no—
570C2×C4—1181no—
571C2×C4—1181no—
572C2×C4—1181no—
573C2×C4—1181no—
574C2×C4—4724no—
575C2×C4—2364no—
576C2×C4—2364no—
577C2×C4—2361no—
578C2×C4—2361no—
579C2×C4—2361no—
580C2×C4—2361no—
581C2×C4—2361no—
582C2×C4—2361no—
583C2×C4—2361no—
584C2×C4—2361no—
585C2×C4—2361no—
586C2×C4—2361no—
587C2×C4—2361no—
588C2×C4—2361no—
589C2×C4#41×D5911yescharacteristic
590C2×C4#41×D5911yescharacteristic
591C2×C4#41×D5911yescharacteristic
592C2×C4C2×#1174568711yescharacteristic
593C2×C4C2×#1174568711yescharacteristic
594C2×C4C2×#1174568711yescharacteristic
595C8—22no—
596C8—41no—
597C8—1182no—
598C8—2361no—
599C22—22no—
600C22—22no—
601C22—22no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22—41no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22—81no—
610C22—81no—
611C22—81no—
612C22—81no—
613C22#123×D5911yescharacteristic
614C22#1683790811yescharacteristic
615C22#1683794711yescharacteristic
616C22#135×D5911yescharacteristic
617C22C2×#1427374611yescharacteristic
618C22AGL(1,ℤ/8)×D5911yescharacteristic
619C22#1683820111yescharacteristic
620C4—42no—
621C4—41no—
622C4—81no—
623C4—81no—
624C4—594no—
625C4—1182no—
626C4—4721no—
627C4—4721no—
628C4—4721no—
629C4—4721no—
630C4—2361no—
631C4—2361no—
632C4#1683792911yescharacteristic
633C4#1683793111yescharacteristic
634C4#1683793311yescharacteristic
635C4#1683793311yescharacteristic
636C2—42no—
637C2—82no—
638C2#2142735211yescharacteristic
639C2#482×D5911yescharacteristic
640C2#2143400411yescharacteristic
641C2#2143411511yescharacteristic
642C2#2143413811yescharacteristic
643C2#2143466111yescharacteristic
644C2#2143468411yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2033C2
socleC23×C59C22×D4
prime radicalC2×#121×C59C22
hypercenter#2033D59
hypocenterC59#23758
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2033

Complements

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

NH ≅ G/Ntype
2. #21426491C2semidirect
3. #21425667C2semidirect
4. #21425615C2semidirect
5. #21425580C2semidirect
6. #21425532C2semidirect
7. #21425524C2semidirect
8. #21425512C2semidirect
10. #107×Dic59C2semidirect
11. #21423984C2semidirect
12. #102×Dic59C2semidirect
13. #102×Dic59C2semidirect
14. C22⋊C8×Dic59C2semidirect
381. C59#23758semidirect

Maximal subgroups

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

subgroupindex
#121×Dic592
#214264912
#214256672
#214256152
#214255802
#214255322
#214255242
#214255122
#214252822
#107×Dic592
#214239842
#102×Dic592
C22⋊C8×Dic592
C59×#20332
#2375859