Finite Group Databasebeta

← #28656656

Subgroups — #28656656

Order 15104.

772 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#21427190C211yesmaximal, characteristic, complemented
2#21425597C211yesmaximal, characteristic, complemented
3#21425327C211yesmaximal, characteristic
4#21425242C211yesmaximal, characteristic, complemented
5#21425242C211yesmaximal, characteristic, complemented
6#21425024C211yesmaximal, characteristic
7#21424796C211yesmaximal, characteristic, complemented
8#21424720C211yesmaximal, characteristic, complemented
9#21424713C211yesmaximal, characteristic, complemented
10#21424713C211yesmaximal, characteristic, complemented
11#21424664C211yesmaximal, characteristic, complemented
12#21424662C211yesmaximal, characteristic, complemented
13#21424357C211yesmaximal, characteristic, complemented
14#21423921C211yesmaximal, characteristic, complemented
15C59×#2445C211yesmaximal, characteristic
16C2×#14273719C2211yescharacteristic
17C2×#14273660C2211yescharacteristic
18C2×#14273658C2211yescharacteristic
19C2×C4⋊(C59⋊C8)C2211yescharacteristic
20C2×#14273648C2211yescharacteristic
21C2×#14273648C2211yescharacteristic
22C2×#14273648C2211yescharacteristic
23C2×#14273648C2211yescharacteristic
24C2×#14273648C2211yescharacteristic
25C2×#14273648C2211yescharacteristic
26C2×#14273648C2211yescharacteristic
27C2×#14273648C2211yescharacteristic
28C2×#14273638C2211yescharacteristic
29#16837300C2211yescharacteristic
30#16837300C2211yescharacteristic
31#16837297—21no—
32#16837284—21no—
33#16837283—21no—
34C4×C4⋊Dic59C2211yescharacteristic
35C4×C4⋊Dic59C2211yescharacteristic
36C4×C4⋊Dic59C2211yescharacteristic
37#16837253—21no—
38#16837164C2211yescharacteristic
39#16837164C2211yescharacteristic
40#16837164C2211yescharacteristic
41#16837164C2211yescharacteristic
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837157C2211yescharacteristic
45#16837157C2211yescharacteristic
46#16837157C2211yescharacteristic
47#16837157C2211yescharacteristic
48C2×#127×C59C2211yescharacteristic
49C2×#123×C59—21no—
50C2×C4×D4×C59C2211yescharacteristic
51C59×#394—21no—
52C59×#394—21no—
53C59×#390C2211yescharacteristic
54C59×#390C2211yescharacteristic
55C59×#390C2211yescharacteristic
56C59×#389C2211yescharacteristic
57C59×#386—21no—
58C4×C4⋊C4×C59C2211yescharacteristic
59C22×C59⋊C8C2311yescharacteristic
60C22×C59⋊C8C2311yescharacteristic
61C22×C59⋊C8C2311yescharacteristic
62C22×C59⋊C8C2311yescharacteristic
63#14273719—22no—
64#14273719—22no—
65C2×C4⋊Dic59C2311yescharacteristic
66C2×C4⋊Dic59C2311yescharacteristic
67C2×C4⋊Dic59C2311yescharacteristic
68C2×C4⋊Dic59C2311yescharacteristic
69C2×Dic59⋊C4—21no—
70C2×Dic59⋊C4—21no—
71C2×Dic59⋊C4—21no—
72C2×Dic59⋊C4—21no—
73C2×C4×Dic59—21no—
74C2×C4×Dic59—21no—
75C2×C4×Dic59—21no—
76#14273660—22no—
77#14273658—41no—
78C4⋊(C59⋊C8)—41no—
79#14273648—22no—
80#14273648—22no—
81#14273648—22no—
82#14273648—22no—
83#14273648—22no—
84#14273648—22no—
85#14273648—22no—
86#14273648—41no—
87#14273638—41no—
88C59⋊#95—21no—
89C59⋊#95—21no—
90C59⋊#95—21no—
91C59⋊#95—21no—
92C59⋊#95—21no—
93C22×D4×C59—21no—
94C22×D4×C59—21no—
95C22×D4×C59C2311yescharacteristic
96C22×D4×C59C2311yescharacteristic
97C23×C4×C59—21no—
98#127×C59—22no—
99#127×C59—41no—
100#123×C59—44no—
101C4×D4×C59—22no—
102C4×D4×C59—41no—
103C2×C4⋊C4×C59—21no—
104C2×C4⋊C4×C59D411yescharacteristic
105C2×C4⋊C4×C59D411yescharacteristic
106C2×C4⋊C4×C59C2311yescharacteristic
107C2×C4⋊C4×C59C2311yescharacteristic
108C2×C22⋊C4×C59—21no—
109C2×C22⋊C4×C59—21no—
110C2×C22⋊C4×C59—21no—
111C2×C22⋊C4×C59—21no—
112C2×C22⋊C4×C59—21no—
113C2×C22⋊C4×C59—21no—
114C2×C22⋊C4×C59—21no—
115C2×C42×C59C2311yescharacteristic
116C2×C42×C59C2311yescharacteristic
117C2×C42×C59C2311yescharacteristic
118#95×C59—21no—
119#95×C59D411yescharacteristic
120#95×C59D411yescharacteristic
121C24×C59—21no—
122C24×C59—21no—
123C2×D4×C59—21no—
124C2×D4×C59—21no—
125C2×D4×C59—21no—
126C2×D4×C59—21no—
127C2×D4×C59—21no—
128C2×D4×C59—21no—
129C2×D4×C59—21no—
130C2×D4×C59—21no—
131C2×D4×C59—21no—
132C2×D4×C59—21no—
133C2×D4×C59—21no—
134C2×D4×C59—42no—
135C2×D4×C59—42no—
136C2×D4×C59—42no—
137C2×D4×C59—42no—
138C2×D4×C59—81no—
139C2×D4×C59—81no—
140C2×D4×C59#4112yes—
141C22×C4×C59—21no—
142C22×C4×C59—21no—
143C22×C4×C59—42no—
144C22×C4×C59—42no—
145C22×C4×C59—81no—
146C22×C4×C59C2×D411yescharacteristic
147C22×C4×C59C2×D411yescharacteristic
148C22×C4×C59C2×D411yescharacteristic
149C22×C4×C59C2×D411yescharacteristic
150C22×C4×C59C2×D411yescharacteristic
151C22×C4×C59C2×D411yescharacteristic
152C22×C4×C59C2411yescharacteristic
153C22×Dic59—21no—
154C22×Dic59—21no—
155C22×Dic59—21no—
156C22×Dic59—21no—
157C4⋊C4×C59—22no—
158C4⋊C4×C59—21no—
159C4⋊C4×C59—21no—
160C4⋊C4×C59—21no—
161C4⋊C4×C59—21no—
162C4⋊C4×C59—41no—
163C4⋊C4×C59—42no—
164C22⋊C4×C59—42no—
165C22⋊C4×C59—42no—
166C22⋊C4×C59—42no—
167C22⋊C4×C59—42no—
168C22⋊C4×C59—81no—
169C22⋊C4×C59—81no—
170C22⋊C4×C59—81no—
171C42×C59—21no—
172C42×C59—21no—
173C42×C59—21no—
174C42×C59—21no—
175C42×C59—21no—
176C42×C59—21no—
177C4⋊Dic59—21no—
178C4⋊Dic59—21no—
179C4⋊Dic59—21no—
180C4⋊Dic59—21no—
181C4⋊Dic59—21no—
182C4⋊Dic59#4112yes—
183C4⋊Dic59#4112yes—
184C4⋊Dic59#4112yes—
185Dic59⋊C4—24no—
186Dic59⋊C4—24no—
187Dic59⋊C4—42no—
188Dic59⋊C4—81no—
189C4×Dic59—42no—
190C4×Dic59—81no—
191C4×Dic59—81no—
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⋊C8—21no—
201C2×C59⋊C8—21no—
202C2×C59⋊C8—21no—
203C2×C59⋊C8—21no—
204C23×C59—22no—
205C23×C59—41no—
206C23×C59—41no—
207C23×C59—41no—
208C23×C59—41no—
209C23×C59—41no—
210C23×C59—41no—
211C23×C59—41no—
212C23×C59—41no—
213C23×C59—41no—
214C23×C59—41no—
215C23×C59—41no—
216C23×C59—41no—
217C23×C59—41no—
218C23×C59C22×D411yescharacteristic
219D4×C59—22no—
220D4×C59—22no—
221D4×C59—41no—
222D4×C59—41no—
223D4×C59—41no—
224D4×C59—41no—
225D4×C59—41no—
226D4×C59—41no—
227D4×C59—84no—
228D4×C59—84no—
229C2×C4×C59—21no—
230C2×C4×C59—21no—
231C2×C4×C59—21no—
232C2×C4×C59—21no—
233C2×C4×C59—21no—
234C2×C4×C59—21no—
235C2×C4×C59—21no—
236C2×C4×C59—21no—
237C2×C4×C59—21no—
238C2×C4×C59—21no—
239C2×C4×C59—21no—
240C2×C4×C59—21no—
241C2×C4×C59—21no—
242C2×C4×C59—21no—
243C2×C4×C59—21no—
244C2×C4×C59—21no—
245C2×C4×C59—21no—
246C2×C4×C59—41no—
247C2×C4×C59—41no—
248C2×C4×C59—41no—
249C2×C4×C59—41no—
250C2×C4×C59—41no—
251C2×C4×C59—41no—
252C2×C4×C59—84no—
253C2×C4×C59AGL(1,ℤ/8)12yes—
254C2×C4×C59C2×#4111yescharacteristic
255C2×C4×C59C2×#4111yescharacteristic
256C2×C4×C59C2×#4111yescharacteristic
257C2×C4×C59C2×#4111yescharacteristic
258C2×C4×C59H(5,𝔽2)11yescharacteristic
259C2×C4×C592−1+411yescharacteristic
260C2×Dic59—22no—
261C2×Dic59—22no—
262C2×Dic59—22no—
263C2×Dic59—41no—
264C2×Dic59—41no—
265C2×Dic59—41no—
266C2×Dic59—41no—
267C2×Dic59—41no—
268C2×Dic59—41no—
269C2×Dic59—41no—
270C2×Dic59—41no—
271C2×Dic59—41no—
272C59⋊C8—41no—
273C59⋊C8—41no—
274C59⋊C8—41no—
275C59⋊C8—41no—
276#26716—591nomaximal, Sylow
277C22×C59—42no—
278C22×C59—42no—
279C22×C59—42no—
280C22×C59—81no—
281C22×C59—81no—
282C22×C59—81no—
283C22×C59—81no—
284C22×C59—81no—
285C22×C59—81no—
286C22×C59—81no—
287C22×C59—81no—
288C22×C59—81no—
289C22×C59—81no—
290C22×C59—81no—
291C22×C59#52511yescharacteristic
292C22×C59#54611yescharacteristic
293C22×C59#54811yescharacteristic
294C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
295C22×C59#57511yescharacteristic
296C22×C59#57611yescharacteristic
297C22×C59#57711yescharacteristic
298C4×C59—22no—
299C4×C59—41no—
300C4×C59—41no—
301C4×C59—41no—
302C4×C59—41no—
303C4×C59—41no—
304C4×C59—81no—
305C4×C59—81no—
306C4×C59#52911yescharacteristic
307C4×C59#54211yescharacteristic
308C4×C59#55211yescharacteristic
309C4×C59#55911yescharacteristic
310Dic59—42no—
311Dic59—42no—
312Dic59—42no—
313Dic59—81no—
314#2445D5911yescharacteristic
315#2370—591no—
316#2088—591no—
317#2041—591no—
318#2025—591no—
319#2025—591no—
320#1982—591no—
321#1936—591no—
322#1926—591no—
323#1925—591no—
324#1925—591no—
325#1918—591no—
326#1917—591no—
327#1864—591no—
328#1762—591no—
329C2×C59—82no—
330C2×C59—161no—
331C2×C59#245811yescharacteristic
332C2×C59#311511yescharacteristic
333C2×C59#311911yescharacteristic
334C2×C59#331011yescharacteristic
335C2×C59#332211yescharacteristic
336C2×C59#336611yescharacteristic
337C2×C59#337111yescharacteristic
338C2×#127C2×D5911yescharacteristic
339C2×#125—591no—
340C2×#123—21no—
341C2×C4×D4C2×D5911yescharacteristic
342C2×#107—591no—
343C2×#106—591no—
344C2×C4⋊C8—591no—
345C2×#102—591no—
346C2×#102—591no—
347C2×#102—591no—
348C2×#102—591no—
349C2×#102—591no—
350C2×#102—591no—
351C2×#102—591no—
352C2×#102—591no—
353C2×#97—591no—
354#394—21no—
355#394—21no—
356#390C2×D5911yescharacteristic
357#390C2×D5911yescharacteristic
358#390C2×D5911yescharacteristic
359#389—591no—
360#389—591no—
361#389—1181no—
362#389C2×D5911yescharacteristic
363#387—1181no—
364#387—1181no—
365#386—21no—
366C4×C4⋊C4—591no—
367C4×C4⋊C4—591no—
368C4×C4⋊C4—591no—
369C4×C4⋊C4—1181no—
370C4×C4⋊C4C2×D5911yescharacteristic
371#340—591no—
372#340—591no—
373#340—591no—
374#340—591no—
375#340—591no—
376#340—591no—
377#336—591no—
378#336—591no—
379#336—591no—
380#336—591no—
381C59#2671611yescharacteristic, complemented, Sylow
382C22×D4—21no—
383C22×D4—21no—
384C22×D4C22×D5911yescharacteristic
385C22×D4C22×D5911yescharacteristic
386C23×C4—21no—
387C22×C8—591no—
388C22×C8—591no—
389C22×C8—591no—
390C22×C8—591no—
391#127—22no—
392#127—41no—
393#125—1182no—
394#125—1182no—
395#123—44no—
396C4×D4—22no—
397C4×D4—41no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—591no—
400C2×C4⋊C4—591no—
401C2×C4⋊C4—591no—
402C2×C4⋊C4—591no—
403C2×C4⋊C4—1181no—
404C2×C4⋊C4—1181no—
405C2×C4⋊C4—1181no—
406C2×C4⋊C4—1181no—
407C2×C4⋊C4C22⋊D5911yescharacteristic
408C2×C4⋊C4C22⋊D5911yescharacteristic
409C2×C4⋊C4C22×D5911yescharacteristic
410C2×C4⋊C4C22×D5911yescharacteristic
411C2×C22⋊C4—21no—
412C2×C22⋊C4—21no—
413C2×C22⋊C4—21no—
414C2×C22⋊C4—21no—
415C2×C22⋊C4—21no—
416C2×C22⋊C4—21no—
417C2×C22⋊C4—21no—
418C2×C42—1181no—
419C2×C42—1181no—
420C2×C42—1181no—
421C2×C42C22×D5911yescharacteristic
422C2×C42C22×D5911yescharacteristic
423C2×C42C22×D5911yescharacteristic
424#107—1182no—
425#106—2361no—
426C4⋊C8—2361no—
427#102—1182no—
428#102—1182no—
429#102—1182no—
430#102—1182no—
431#102—1182no—
432#102—1182no—
433#102—1182no—
434#102—2361no—
435#97—2361no—
436#95—21no—
437#95—1181no—
438#95—1181no—
439#95—1181no—
440#95—1181no—
441#95—1181no—
442#95C22⋊D5911yescharacteristic
443#95C22⋊D5911yescharacteristic
444C24—21no—
445C24—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—42no—
458C2×D4—42no—
459C2×D4—42no—
460C2×D4—42no—
461C2×D4—81no—
462C2×D4—81no—
463C2×D4#1174568712yes—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—81no—
469C22×C4—1181no—
470C22×C4—1181no—
471C22×C4—1181no—
472C22×C4—1181no—
473C22×C4C2×C22⋊D5911yescharacteristic
474C22×C4C2×C22⋊D5911yescharacteristic
475C22×C4C2×C22⋊D5911yescharacteristic
476C22×C4C2×C22⋊D5911yescharacteristic
477C22×C4C2×C22⋊D5911yescharacteristic
478C22×C4C2×C22⋊D5911yescharacteristic
479C22×C4C23×D5911yescharacteristic
480C2×C8—1181no—
481C2×C8—1181no—
482C2×C8—1181no—
483C2×C8—1181no—
484C2×C8—1181no—
485C2×C8—1181no—
486C2×C8—1181no—
487C2×C8—1181no—
488C2×C8—1181no—
489C2×C8—1181no—
490C2×C8—1181no—
491C2×C8—1181no—
492C4⋊C4—22no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—41no—
498C4⋊C4—42no—
499C4⋊C4—592no—
500C4⋊C4—592no—
501C4⋊C4—592no—
502C4⋊C4—1184no—
503C4⋊C4—1184no—
504C4⋊C4—1181no—
505C4⋊C4—1181no—
506C4⋊C4—1181no—
507C4⋊C4—1181no—
508C4⋊C4—1181no—
509C4⋊C4—4721no—
510C4⋊C4—2362no—
511C22⋊C4—42no—
512C22⋊C4—42no—
513C22⋊C4—42no—
514C22⋊C4—42no—
515C22⋊C4—81no—
516C22⋊C4—81no—
517C22⋊C4—81no—
518C42—21no—
519C42—21no—
520C42—21no—
521C42—21no—
522C42—21no—
523C42—21no—
524C42—4721no—
525C42—4721no—
526C42—2362no—
527C23—22no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23C22×C22⋊D5911yescharacteristic
542D4—22no—
543D4—22no—
544D4—41no—
545D4—41no—
546D4—41no—
547D4—41no—
548D4—41no—
549D4—41no—
550D4—84no—
551D4—84no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—84no—
576C2×C4—1182no—
577C2×C4—1182no—
578C2×C4—1182no—
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#1427375312yes—
589C2×C4#41×D5911yescharacteristic
590C2×C4#41×D5911yescharacteristic
591C2×C4#41×D5911yescharacteristic
592C2×C4C2×#1174568711yescharacteristic
593C2×C4#1427377211yescharacteristic
594C2×C4#1427377411yescharacteristic
595C8—2361no—
596C8—2361no—
597C8—2361no—
598C8—2361no—
599C22—42no—
600C22—42no—
601C22—42no—
602C22—81no—
603C22—81no—
604C22—81no—
605C22—81no—
606C22—81no—
607C22—81no—
608C22—81no—
609C22—81no—
610C22—81no—
611C22—81no—
612C22—81no—
613C22#1683790911yescharacteristic
614C22#1683804611yescharacteristic
615C22#1683805911yescharacteristic
616C22C2×#1427375311yescharacteristic
617C22#1683820011yescharacteristic
618C22#1683820511yescharacteristic
619C22#1683821311yescharacteristic
620C4—22no—
621C4—41no—
622C4—41no—
623C4—41no—
624C4—41no—
625C4—41no—
626C4—81no—
627C4—81no—
628C4—4721no—
629C4—2362no—
630C4—2362no—
631C4—2362no—
632C4#1683792911yescharacteristic
633C4#1683802011yescharacteristic
634C4#1683808911yescharacteristic
635C4#1683813611yescharacteristic
636C2—82no—
637C2—161no—
638C2#2142801411yescharacteristic
639C2#2143424111yescharacteristic
640C2#2143428911yescharacteristic
641C2#2143611711yescharacteristic
642C2#2143629711yescharacteristic
643C2#2143679911yescharacteristic
644C2#2143685811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2445C2
socleC23×C59C22×D4
prime radicalC2×#127×C59C22
hypercenter#2445D59
hypocenterC59#26716
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2445

Complements

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

NH ≅ G/Ntype
1. #21427190C2semidirect
2. #21425597C2semidirect
4. #21425242C2semidirect
5. #21425242C2semidirect
7. #21424796C2semidirect
8. #21424720C2semidirect
9. #21424713C2semidirect
10. #21424713C2semidirect
11. #21424664C2semidirect
12. #21424662C2semidirect
13. #21424357C2semidirect
14. #21423921C2semidirect
381. C59#26716semidirect

Maximal subgroups

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

subgroupindex
#214271902
#214255972
#214253272
#214252422
#214250242
#214247962
#214247202
#214247132
#214246642
#214246622
#214243572
#214239212
C59×#24452
#2671659