Finite Group Databasebeta

← #28656701

Subgroups — #28656701

Order 15104.

680 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21427190C211yesmaximal, characteristic, complemented
2#21425587C211yesmaximal, characteristic, complemented
3#21425332C211yesmaximal, characteristic
4#21425247C211yesmaximal, characteristic, complemented
5#21425242C211yesmaximal, characteristic
6#21425029C211yesmaximal, characteristic
7#21424796C211yesmaximal, characteristic
8#21424734C211yesmaximal, characteristic, complemented
9#21424713C211yesmaximal, characteristic
10#21424713C211yesmaximal, characteristic
11#21424685C211yesmaximal, characteristic, complemented
12#21424662C211yesmaximal, characteristic, complemented
13#21424350C211yesmaximal, characteristic, complemented
14#21423926C211yesmaximal, characteristic, complemented
15C59×#2448C211yesmaximal, characteristic
16C2×#14273719C2211yescharacteristic
17C2×#14273660C2211yescharacteristic
18C2×#14273660C2211yescharacteristic
19C2×C4⋊(C59⋊C8)C2211yescharacteristic
20C2×#14273651C2211yescharacteristic
21C2×#14273651C2211yescharacteristic
22C2×#14273651C2211yescharacteristic
23C2×#14273651C2211yescharacteristic
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×#124×C59C2211yescharacteristic
49C2×C4×Q8×C59C2211yescharacteristic
50C59×#399—21no—
51C59×#397—21no—
52C59×#391C2211yescharacteristic
53C59×#390C2211yescharacteristic
54C59×#390C2211yescharacteristic
55C59×#389C2211yescharacteristic
56C59×#388—21no—
57C59×#388—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#14273660—41no—
78C4⋊(C59⋊C8)—41no—
79#14273651—22no—
80#14273651—22no—
81#14273651—22no—
82#14273651—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×Q8×C59C2311yescharacteristic
94C22×D4×C59C2311yescharacteristic
95#124×C59—22no—
96#124×C59—41no—
97C4×Q8×C59—22no—
98C4×Q8×C59—41no—
99C2×C4⋊C4×C59—21no—
100C2×C4⋊C4×C59—21no—
101C2×C4⋊C4×C59D411yescharacteristic
102C2×C4⋊C4×C59D411yescharacteristic
103C2×C4⋊C4×C59C2311yescharacteristic
104C2×C4⋊C4×C59C2311yescharacteristic
105C2×C22⋊C4×C59—21no—
106C2×C22⋊C4×C59—21no—
107C2×C22⋊C4×C59—21no—
108C2×C22⋊C4×C59—21no—
109C2×C22⋊C4×C59—21no—
110C2×C22⋊C4×C59—21no—
111C2×C42×C59—21no—
112C2×C42×C59C2311yescharacteristic
113C2×C42×C59C2311yescharacteristic
114C2×C42×C59C2311yescharacteristic
115#95×C59—21no—
116#95×C59—21no—
117#95×C59—21no—
118#95×C59D411yescharacteristic
119#95×C59D411yescharacteristic
120C24×C59—21no—
121C2×Q8×C59—21no—
122C2×Q8×C59—21no—
123C2×Q8×C59—21no—
124C2×Q8×C59—21no—
125C2×Q8×C59—21no—
126C2×Q8×C59#4112yes—
127C2×D4×C59—21no—
128C2×D4×C59—21no—
129C2×D4×C59—21no—
130C2×D4×C59—21no—
131C2×D4×C59—21no—
132C2×D4×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—
146C22×Dic59—21no—
147C4⋊C4×C59—21no—
148C4⋊C4×C59—21no—
149C4⋊C4×C59—21no—
150C4⋊C4×C59—21no—
151C4⋊C4×C59—41no—
152C4⋊C4×C59—41no—
153C4⋊C4×C59—42no—
154C4⋊C4×C59—81no—
155C22⋊C4×C59—42no—
156C22⋊C4×C59—42no—
157C22⋊C4×C59—42no—
158C22⋊C4×C59—42no—
159C22⋊C4×C59—81no—
160C22⋊C4×C59—81no—
161C42×C59—21no—
162C42×C59—21no—
163C42×C59—21no—
164C42×C59—21no—
165C42×C59—21no—
166C42×C59—21no—
167C42×C59—42no—
168C4⋊Dic59—21no—
169C4⋊Dic59—21no—
170C4⋊Dic59—21no—
171C4⋊Dic59—21no—
172C4⋊Dic59—21no—
173C4⋊Dic59#4112yes—
174C4⋊Dic59#4112yes—
175C4⋊Dic59#4112yes—
176Dic59⋊C4—24no—
177Dic59⋊C4—24no—
178Dic59⋊C4—42no—
179Dic59⋊C4—81no—
180C4×Dic59—42no—
181C4×Dic59—81no—
182C4×Dic59—81no—
183C2×C59⋊C8—21no—
184C2×C59⋊C8—21no—
185C2×C59⋊C8—21no—
186C2×C59⋊C8—21no—
187C2×C59⋊C8—21no—
188C2×C59⋊C8—21no—
189C2×C59⋊C8—21no—
190C2×C59⋊C8—21no—
191C2×C59⋊C8—21no—
192C2×C59⋊C8—21no—
193C2×C59⋊C8—21no—
194C2×C59⋊C8—21no—
195C23×C59—41no—
196C23×C59—41no—
197C23×C59—41no—
198C23×C59—41no—
199C23×C59—41no—
200C23×C59—41no—
201C23×C59—41no—
202C23×C59C22×D411yescharacteristic
203Q8×C59—22no—
204Q8×C59—22no—
205Q8×C59—41no—
206Q8×C59—41no—
207D4×C59—41no—
208D4×C59—41no—
209D4×C59—41no—
210D4×C59—41no—
211C2×C4×C59—21no—
212C2×C4×C59—21no—
213C2×C4×C59—21no—
214C2×C4×C59—21no—
215C2×C4×C59—21no—
216C2×C4×C59—21no—
217C2×C4×C59—22no—
218C2×C4×C59—21no—
219C2×C4×C59—21no—
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—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×C59C2×#4111yescharacteristic
239C2×C4×C59C2×#4111yescharacteristic
240C2×C4×C59C2×#4111yescharacteristic
241C2×C4×C59C2×#4111yescharacteristic
242C2×C4×C59H(5,𝔽2)11yescharacteristic
243C2×C4×C592−1+411yescharacteristic
244C2×Dic59—22no—
245C2×Dic59—22no—
246C2×Dic59—22no—
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—41no—
256C59⋊C8—41no—
257C59⋊C8—41no—
258C59⋊C8—41no—
259C59⋊C8—41no—
260#26719—591nomaximal, Sylow
261C22×C59—81no—
262C22×C59—81no—
263C22×C59—81no—
264C22×C59—81no—
265C22×C59—81no—
266C22×C59—81no—
267C22×C59—81no—
268C22×C59#52511yescharacteristic
269C22×C59#54611yescharacteristic
270C22×C59#54811yescharacteristic
271C22×C59#57511yescharacteristic
272C22×C59#57511yescharacteristic
273C22×C59#57611yescharacteristic
274C22×C59#57811yescharacteristic
275C4×C59—41no—
276C4×C59—41no—
277C4×C59—42no—
278C4×C59—41no—
279C4×C59—41no—
280C4×C59—41no—
281C4×C59—41no—
282C4×C59—81no—
283C4×C59—81no—
284C4×C59#52911yescharacteristic
285C4×C59#54211yescharacteristic
286C4×C59#55211yescharacteristic
287C4×C59#55911yescharacteristic
288Dic59—42no—
289Dic59—42no—
290Dic59—42no—
291Dic59—81no—
292#2448D5911yescharacteristic
293#2370—591no—
294#2086—591no—
295#2042—591no—
296#2026—591no—
297#2025—591no—
298#1983—591no—
299#1936—591no—
300#1928—591no—
301#1925—591no—
302#1925—591no—
303#1921—591no—
304#1917—591no—
305#1863—591no—
306#1763—591no—
307C2×C59—161no—
308C2×C59#245811yescharacteristic
309C2×C59#311111yescharacteristic
310C2×C59#312411yescharacteristic
311C2×C59#330911yescharacteristic
312C2×C59#332711yescharacteristic
313C2×C59#337011yescharacteristic
314C2×C59#337711yescharacteristic
315C2×#125—591no—
316C2×#124C2×D5911yescharacteristic
317C2×C4×Q8C2×D5911yescharacteristic
318C2×#107—591no—
319C2×#107—591no—
320C2×C4⋊C8—591no—
321C2×#103—591no—
322C2×#103—591no—
323C2×#103—591no—
324C2×#103—591no—
325C2×#102—591no—
326C2×#102—591no—
327C2×#102—591no—
328C2×#102—591no—
329C2×#97—591no—
330#399—21no—
331#397—21no—
332#391C2×D5911yescharacteristic
333#390C2×D5911yescharacteristic
334#390C2×D5911yescharacteristic
335#389—591no—
336#389—591no—
337#389—1181no—
338#389C2×D5911yescharacteristic
339#388—21no—
340#388—21no—
341#387—1181no—
342#387—1181no—
343C4×C4⋊C4—591no—
344C4×C4⋊C4—591no—
345C4×C4⋊C4—591no—
346C4×C4⋊C4—1181no—
347C4×C4⋊C4C2×D5911yescharacteristic
348#340—591no—
349#340—591no—
350#340—591no—
351#340—591no—
352#340—591no—
353#340—591no—
354#336—591no—
355#336—591no—
356#336—591no—
357#336—591no—
358C59#2671911yescharacteristic, complemented, Sylow
359C22×Q8C22×D5911yescharacteristic
360C22×D4C22×D5911yescharacteristic
361C22×C8—591no—
362C22×C8—591no—
363C22×C8—591no—
364C22×C8—591no—
365#125—1182no—
366#125—1182no—
367#124—22no—
368#124—41no—
369C4×Q8—22no—
370C4×Q8—41no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—591no—
374C2×C4⋊C4—591no—
375C2×C4⋊C4—591no—
376C2×C4⋊C4—591no—
377C2×C4⋊C4—1181no—
378C2×C4⋊C4—1181no—
379C2×C4⋊C4—1181no—
380C2×C4⋊C4—1181no—
381C2×C4⋊C4C22⋊D5911yescharacteristic
382C2×C4⋊C4C22⋊D5911yescharacteristic
383C2×C4⋊C4C22×D5911yescharacteristic
384C2×C4⋊C4C22×D5911yescharacteristic
385C2×C22⋊C4—21no—
386C2×C22⋊C4—21no—
387C2×C22⋊C4—21no—
388C2×C22⋊C4—21no—
389C2×C22⋊C4—21no—
390C2×C22⋊C4—21no—
391C2×C42—21no—
392C2×C42—1181no—
393C2×C42—1181no—
394C2×C42—1181no—
395C2×C42C22×D5911yescharacteristic
396C2×C42C22×D5911yescharacteristic
397C2×C42C22×D5911yescharacteristic
398#107—1182no—
399#107—2361no—
400C4⋊C8—2361no—
401#103—1182no—
402#103—1182no—
403#103—1182no—
404#103—1182no—
405#102—1182no—
406#102—1182no—
407#102—1182no—
408#102—2361no—
409#97—2361no—
410#95—21no—
411#95—21no—
412#95—21no—
413#95—1181no—
414#95—1181no—
415#95—1181no—
416#95—1181no—
417#95—1181no—
418#95C22⋊D5911yescharacteristic
419#95C22⋊D5911yescharacteristic
420C24—21no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8#1174568712yes—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—1181no—
437C22×C4—1181no—
438C22×C4—1181no—
439C22×C4—1181no—
440C22×C4C2×C22⋊D5911yescharacteristic
441C22×C4C2×C22⋊D5911yescharacteristic
442C22×C4C2×C22⋊D5911yescharacteristic
443C22×C4C2×C22⋊D5911yescharacteristic
444C22×C4C2×C22⋊D5911yescharacteristic
445C22×C4C2×C22⋊D5911yescharacteristic
446C22×C4C23×D5911yescharacteristic
447C2×C8—1181no—
448C2×C8—1181no—
449C2×C8—1181no—
450C2×C8—1181no—
451C2×C8—1181no—
452C2×C8—1181no—
453C2×C8—1181no—
454C2×C8—1181no—
455C2×C8—1181no—
456C2×C8—1181no—
457C2×C8—1181no—
458C2×C8—1181no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—41no—
464C4⋊C4—41no—
465C4⋊C4—42no—
466C4⋊C4—81no—
467C4⋊C4—592no—
468C4⋊C4—592no—
469C4⋊C4—592no—
470C4⋊C4—1184no—
471C4⋊C4—1184no—
472C4⋊C4—1181no—
473C4⋊C4—1181no—
474C4⋊C4—1181no—
475C4⋊C4—1181no—
476C4⋊C4—1181no—
477C4⋊C4—4721no—
478C4⋊C4—2362no—
479C22⋊C4—42no—
480C22⋊C4—42no—
481C22⋊C4—42no—
482C22⋊C4—42no—
483C22⋊C4—81no—
484C22⋊C4—81no—
485C42—21no—
486C42—21no—
487C42—21no—
488C42—21no—
489C42—21no—
490C42—21no—
491C42—42no—
492C42—4721no—
493C42—4721no—
494C42—2362no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23C22×C22⋊D5911yescharacteristic
503Q8—22no—
504Q8—22no—
505Q8—41no—
506Q8—41no—
507D4—41no—
508D4—41no—
509D4—41no—
510D4—41no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—1182no—
539C2×C4—1182no—
540C2×C4—1182no—
541C2×C4—2361no—
542C2×C4—2361no—
543C2×C4—2361no—
544C2×C4—2361no—
545C2×C4—2361no—
546C2×C4—2361no—
547C2×C4—2361no—
548C2×C4—2361no—
549C2×C4—2361no—
550C2×C4#41×D5911yescharacteristic
551C2×C4#41×D5911yescharacteristic
552C2×C4#41×D5911yescharacteristic
553C2×C4C2×#1174568711yescharacteristic
554C2×C4#1427377211yescharacteristic
555C2×C4#1427377411yescharacteristic
556C8—2361no—
557C8—2361no—
558C8—2361no—
559C8—2361no—
560C22—81no—
561C22—81no—
562C22—81no—
563C22—81no—
564C22—81no—
565C22—81no—
566C22—81no—
567C22#1683790911yescharacteristic
568C22#1683804611yescharacteristic
569C22#1683805911yescharacteristic
570C22#1683820011yescharacteristic
571C22#1683820011yescharacteristic
572C22#1683820511yescharacteristic
573C22#1683821711yescharacteristic
574C4—42no—
575C4—41no—
576C4—41no—
577C4—41no—
578C4—41no—
579C4—41no—
580C4—41no—
581C4—81no—
582C4—81no—
583C4—4721no—
584C4—2362no—
585C4—2362no—
586C4—2362no—
587C4#1683792911yescharacteristic
588C4#1683802011yescharacteristic
589C4#1683808911yescharacteristic
590C4#1683813611yescharacteristic
591C2—161no—
592C2#2142801411yescharacteristic
593C2#2143419911yescharacteristic
594C2#2143435211yescharacteristic
595C2#2143610211yescharacteristic
596C2#2143636411yescharacteristic
597C2#2143684311yescharacteristic
598C2#2143693211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2448C2
socleC23×C59C22×D4
prime radicalC22×D4×C59C23
hypercenter#2448D59
hypocenterC59#26719
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2448

Complements

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

NH ≅ G/Ntype
1. #21427190C2semidirect
2. #21425587C2semidirect
4. #21425247C2semidirect
8. #21424734C2semidirect
11. #21424685C2semidirect
12. #21424662C2semidirect
13. #21424350C2semidirect
14. #21423926C2semidirect
358. C59#26719semidirect

Maximal subgroups

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

subgroupindex
#214271902
#214255872
#214253322
#214252472
#214252422
#214250292
#214247962
#214247342
#214247132
#214246852
#214246622
#214243502
#214239262
C59×#24482
#2671959