Finite Group Databasebeta

← #28661449

Subgroups — #28661449

Order 15104.

700 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#21428496C211yesmaximal, characteristic
2#21427099C211yesmaximal, characteristic
3#21425516C211yesmaximal, characteristic, complemented
4#21425476C211yesmaximal, characteristic, complemented
5#21425300C211yesmaximal, characteristic
6#21425293C211yesmaximal, characteristic
7#21424866C211yesmaximal, characteristic, complemented
8#21424824C211yesmaximal, characteristic, complemented
9#21424689C211yesmaximal, characteristic, complemented
10#21424682C211yesmaximal, characteristic, complemented
11#21424367C211yesmaximal, characteristic, complemented
12#21424291C211yesmaximal, characteristic
13#21424183C211yesmaximal, characteristic
14C4×#14273652C211yesmaximal, characteristic, complemented
15C59×#1923C211yesmaximal, characteristic
16C2×#14273706C2211yescharacteristic
17C2×C4×Dic118C2211yescharacteristic
18C22×C4⋊Dic59C2211yescharacteristic
19C2×#14273661C2211yescharacteristic
20C2×#14273659C2211yescharacteristic
21C2×#14273659C2211yescharacteristic
22C2×#14273659C2211yescharacteristic
23C2×#14273652C2211yescharacteristic
24C2×#14273652C2211yescharacteristic
25C2×#14273652C2211yescharacteristic
26C2×#14273652C2211yescharacteristic
27C59⋊#401—21no—
28#16837330—21no—
29#16837308C2211yescharacteristic
30#16837300C2211yescharacteristic
31#16837300C2211yescharacteristic
32#16837300C2211yescharacteristic
33#16837299—21no—
34#16837296—21no—
35#16837289—21no—
36#16837284—21no—
37#16837274—21no—
38#16837268C2211yescharacteristic
39#16837262C2211yescharacteristic
40#16837262C2211yescharacteristic
41C4×C4⋊Dic59C2211yescharacteristic
42C4×C4⋊Dic59C2211yescharacteristic
43C22⋊C4×Dic59—21no—
44#16837165C2211yescharacteristic
45#16837165C2211yescharacteristic
46#16837165C2211yescharacteristic
47#16837165C2211yescharacteristic
48#16837165C2211yescharacteristic
49#16837165C2211yescharacteristic
50#16837165C2211yescharacteristic
51#16837165C2211yescharacteristic
52C2×C4⋊C8×C59C2211yescharacteristic
53C2×C22⋊C8×C59C2211yescharacteristic
54C2×C22⋊C8×C59C2211yescharacteristic
55C2×C4×C8×C59C2211yescharacteristic
56C4×C22⋊C4×C59C2211yescharacteristic
57C59×#336C2211yescharacteristic
58C59×#336C2211yescharacteristic
59C22×Dic118C2311yescharacteristic
60C23×Dic59—21no—
61#14273706—22no—
62#14273706—22no—
63C4×Dic118—22no—
64C4×Dic118—41no—
65C2×C4⋊Dic59—22no—
66C2×C4⋊Dic59—21no—
67C2×C4⋊Dic59—41no—
68C2×C4⋊Dic59C2311yescharacteristic
69C2×C4⋊Dic59C2311yescharacteristic
70C2×C4⋊Dic59C2311yescharacteristic
71C2×C4⋊Dic59C2311yescharacteristic
72C2×C4⋊Dic59C2311yescharacteristic
73C2×C4⋊Dic59C2311yescharacteristic
74C2×C4⋊Dic59C2311yescharacteristic
75C2×Dic59⋊C4—21no—
76C2×Dic59⋊C4—21no—
77C2×Dic59⋊C4—21no—
78C2×Dic59⋊C4—21no—
79C2×Dic59⋊C4—21no—
80C2×C22⋊Dic59—21no—
81C2×C22⋊Dic59—21no—
82C2×C22⋊Dic59—21no—
83C2×C4×Dic59—21no—
84C2×C4×Dic59—21no—
85C2×C4×Dic59—21no—
86#14273661—22no—
87#14273659—22no—
88#14273659—22no—
89#14273659—41no—
90#14273652—22no—
91#14273652—22no—
92#14273652—22no—
93#14273652—41no—
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—
104C59⋊#95—21no—
105C23×C4×C59C2311yescharacteristic
106C22×C8×C59C2311yescharacteristic
107C22×C8×C59C2311yescharacteristic
108C22×C8×C59C2311yescharacteristic
109C22×C8×C59C2311yescharacteristic
110C2×C22⋊C4×C59D411yescharacteristic
111C2×C22⋊C4×C59D411yescharacteristic
112C2×C42×C59C2311yescharacteristic
113C2×C42×C59C2311yescharacteristic
114C4⋊C8×C59—22no—
115C22⋊C8×C59—22no—
116C22⋊C8×C59—41no—
117C4×C8×C59—41no—
118#95×C59D411yescharacteristic
119#95×C59D411yescharacteristic
120C24×C59C2×D411yescharacteristic
121C22×C4×C59—21no—
122C22×C4×C59—21no—
123C22×C4×C59—21no—
124C22×C4×C59—21no—
125C22×C4×C59—21no—
126C22×C4×C59—21no—
127C22×C4×C59C2×D411yescharacteristic
128C22×C4×C59C2×D411yescharacteristic
129C22×C4×C59C2×D411yescharacteristic
130C22×C4×C59C2×D411yescharacteristic
131C22×C4×C59C2×D411yescharacteristic
132C22×C4×C59C2411yescharacteristic
133C22×Dic59—21no—
134C22×Dic59—21no—
135C22×Dic59—21no—
136C22×Dic59—21no—
137C22×Dic59—21no—
138C22×Dic59—21no—
139C22×Dic59—21no—
140C22×Dic59—21no—
141C22×Dic59—42no—
142C22×Dic59—42no—
143C22×Dic59—81no—
144C2×Dic118—21no—
145C2×Dic118—21no—
146C2×Dic118—21no—
147C2×Dic118—21no—
148C2×Dic118—21no—
149C2×Dic118#4112yes—
150C2×C8×C59—21no—
151C2×C8×C59—21no—
152C2×C8×C59—21no—
153C2×C8×C59—21no—
154C2×C8×C59—21no—
155C2×C8×C59—21no—
156C2×C8×C59—21no—
157C2×C8×C59—21no—
158C2×C8×C59—21no—
159C2×C8×C59—21no—
160C2×C8×C59—21no—
161C2×C8×C59#4112yes—
162C22⋊C4×C59—22no—
163C22⋊C4×C59—22no—
164C42×C59—21no—
165C42×C59—21no—
166C42×C59—21no—
167C42×C59—21no—
168C22⋊Dic59—24no—
169C22⋊Dic59—42no—
170C22⋊Dic59—81no—
171C4⋊Dic59—21no—
172C4⋊Dic59—21no—
173C4⋊Dic59—21no—
174C4⋊Dic59—21no—
175C4⋊Dic59—21no—
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⋊Dic59#4112yes—
185C4⋊Dic59#4112yes—
186Dic59⋊C4—24no—
187Dic59⋊C4—42no—
188Dic59⋊C4—42no—
189Dic59⋊C4—81no—
190Dic59⋊C4—81no—
191C4×Dic59—42no—
192C4×Dic59—81no—
193C4×Dic59—81no—
194C23×C59—21no—
195C23×C59—21no—
196C23×C59—21no—
197C23×C59—21no—
198C23×C59—21no—
199C23×C59—21no—
200C23×C59#13512yes—
201C23×C59C22×D411yescharacteristic
202C2×C4×C59—21no—
203C2×C4×C59—21no—
204C2×C4×C59—21no—
205C2×C4×C59—21no—
206C2×C4×C59—21no—
207C2×C4×C59—21no—
208C2×C4×C59—21no—
209C2×C4×C59—21no—
210C2×C4×C59—21no—
211C2×C4×C59—21no—
212C2×C4×C59—21no—
213C2×C4×C59—21no—
214C2×C4×C59—21no—
215C2×C4×C59—21no—
216C2×C4×C59—41no—
217C2×C4×C59—41no—
218C2×C4×C59—41no—
219C2×C4×C59—41no—
220C2×C4×C59#13712yes—
221C2×C4×C59C2×#4111yescharacteristic
222C2×C4×C59C2×#4111yescharacteristic
223C2×C4×C59C2×#4111yescharacteristic
224C2×C4×C59C2×#4111yescharacteristic
225C2×C4×C59H(5,𝔽2)11yescharacteristic
226C2×C4×C592−1+411yescharacteristic
227C2×Dic59—22no—
228C2×Dic59—22no—
229C2×Dic59—22no—
230C2×Dic59—41no—
231C2×Dic59—41no—
232C2×Dic59—41no—
233C2×Dic59—41no—
234C2×Dic59—41no—
235C2×Dic59—41no—
236C2×Dic59—41no—
237C2×Dic59—41no—
238C2×Dic59—41no—
239C2×Dic59—41no—
240C2×Dic59—41no—
241C2×Dic59—41no—
242C2×Dic59—41no—
243C2×Dic59—41no—
244C2×Dic59—41no—
245C2×Dic59—41no—
246C2×Dic59—41no—
247C2×Dic59—41no—
248C2×Dic59—41no—
249C2×Dic59—41no—
250C2×Dic59—41no—
251C2×Dic59—84no—
252Dic118—22no—
253Dic118—22no—
254Dic118—41no—
255Dic118—41no—
256C8×C59—22no—
257C8×C59—41no—
258C8×C59—41no—
259C8×C59—41no—
260#27037—591nomaximal, Sylow
261C22×C59—22no—
262C22×C59—22no—
263C22×C59—22no—
264C22×C59—41no—
265C22×C59—41no—
266C22×C59—41no—
267C22×C59—41no—
268C22×C59#52511yescharacteristic
269C22×C59#54611yescharacteristic
270C22×C59#54711yescharacteristic
271C22×C59C2×#13511yescharacteristic
272C22×C59C2×#13711yescharacteristic
273C22×C59#57711yescharacteristic
274C22×C59#57711yescharacteristic
275C4×C59—22no—
276C4×C59—41no—
277C4×C59—41no—
278C4×C59—41no—
279C4×C59—41no—
280C4×C59#52911yescharacteristic
281C4×C59#53911yescharacteristic
282C4×C59#55311yescharacteristic
283C4×C59#55611yescharacteristic
284Dic59—42no—
285Dic59—42no—
286Dic59—42no—
287Dic59—81no—
288Dic59—81no—
289Dic59—81no—
290Dic59—81no—
291Dic59—81no—
292#2494—591no—
293#2360—591no—
294#2074—591no—
295#2066—591no—
296#2036—591no—
297#2035—591no—
298#1949—591no—
299#1941—591no—
300#1923D5911yescharacteristic
301#1921—591no—
302#1920—591no—
303#1865—591no—
304#1853—591no—
305#1823—591no—
306C4×#103—591no—
307C2×C59—42no—
308C2×C59#246611yescharacteristic
309C2×C59#310811yescharacteristic
310C2×C59#312511yescharacteristic
311C2×C59#329611yescharacteristic
312C2×C59#332011yescharacteristic
313C2×C59#333711yescharacteristic
314C2×C59#335611yescharacteristic
315C2×C22⋊Q8—591no—
316C2×C4×Q8—591no—
317C22×C4⋊C4—591no—
318C2×#107—591no—
319C2×#106—591no—
320C2×#106—591no—
321C2×#106—591no—
322C2×C4⋊C8C2×D5911yescharacteristic
323C2×#103—591no—
324C2×#103—591no—
325C2×#103—591no—
326C2×#103—591no—
327C2×C22⋊C8C2×D5911yescharacteristic
328C2×C22⋊C8C2×D5911yescharacteristic
329C2×C4×C8C2×D5911yescharacteristic
330#401—1181no—
331#397—1181no—
332#391—591no—
333#389—591no—
334#389—591no—
335#389—591no—
336#389—1181no—
337#388—1181no—
338#387—1181no—
339#387—1181no—
340#385—1181no—
341#382—591no—
342#380—591no—
343#380—591no—
344C4×C4⋊C4—591no—
345C4×C4⋊C4—591no—
346C4×C22⋊C4—1181no—
347C4×C22⋊C4C2×D5911yescharacteristic
348#340—591no—
349#340—591no—
350#340—591no—
351#340—591no—
352#340—591no—
353#340—591no—
354#340—591no—
355#340—591no—
356#336C2×D5911yescharacteristic
357#336C2×D5911yescharacteristic
358C59#2703711yescharacteristic, complemented, Sylow
359C22×Q8—591no—
360C23×C4—1181no—
361C23×C4C22×D5911yescharacteristic
362C22×C8C22×D5911yescharacteristic
363C22×C8C22×D5911yescharacteristic
364C22×C8C22×D5911yescharacteristic
365C22×C8C22×D5911yescharacteristic
366C22⋊Q8—1182no—
367C22⋊Q8—1182no—
368C4×Q8—1182no—
369C4×Q8—2361no—
370C2×C4⋊C4—591no—
371C2×C4⋊C4—591no—
372C2×C4⋊C4—591no—
373C2×C4⋊C4—591no—
374C2×C4⋊C4—591no—
375C2×C4⋊C4—591no—
376C2×C4⋊C4—591no—
377C2×C4⋊C4—1182no—
378C2×C4⋊C4—1181no—
379C2×C4⋊C4—1181no—
380C2×C4⋊C4—1181no—
381C2×C4⋊C4—1181no—
382C2×C4⋊C4—1181no—
383C2×C4⋊C4—1181no—
384C2×C4⋊C4—2361no—
385C2×C22⋊C4—1181no—
386C2×C22⋊C4—1181no—
387C2×C22⋊C4—1181no—
388C2×C22⋊C4D23611yescharacteristic
389C2×C22⋊C4D23611yescharacteristic
390C2×C42—1181no—
391C2×C42—1181no—
392C2×C42—1181no—
393C2×C42C22×D5911yescharacteristic
394C2×C42C22×D5911yescharacteristic
395#107—1182no—
396#106—1182no—
397#106—1182no—
398#106—2361no—
399C4⋊C8—22no—
400#103—1182no—
401#103—1182no—
402#103—1182no—
403#103—2361no—
404C22⋊C8—22no—
405C22⋊C8—41no—
406C4×C8—41no—
407#95—1181no—
408#95—1181no—
409#95—1181no—
410#95—1181no—
411#95—1181no—
412#95—1181no—
413#95—1181no—
414#95—1181no—
415#95—1181no—
416#95—1181no—
417#95—1181no—
418#95D23611yescharacteristic
419#95D23611yescharacteristic
420C24C2×D23611yescharacteristic
421C2×Q8—592no—
422C2×Q8—1181no—
423C2×Q8—1181no—
424C2×Q8—1181no—
425C2×Q8—1181no—
426C2×Q8—1181no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—1181no—
434C22×C4—1181no—
435C22×C4—1181no—
436C22×C4—1181no—
437C22×C4—1181no—
438C22×C4—1181no—
439C22×C4—1181no—
440C22×C4—1181no—
441C22×C4—4721no—
442C22×C4—2362no—
443C22×C4—2362no—
444C22×C4C2×D23611yescharacteristic
445C22×C4C2×D23611yescharacteristic
446C22×C4C2×D23611yescharacteristic
447C22×C4C2×D23611yescharacteristic
448C22×C4C2×D23611yescharacteristic
449C22×C4C23×D5911yescharacteristic
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8#1174568712yes—
462C4⋊C4—592no—
463C4⋊C4—592no—
464C4⋊C4—1184no—
465C4⋊C4—1181no—
466C4⋊C4—1181no—
467C4⋊C4—1181no—
468C4⋊C4—1181no—
469C4⋊C4—1181no—
470C4⋊C4—1181no—
471C4⋊C4—1181no—
472C4⋊C4—1181no—
473C4⋊C4—1181no—
474C4⋊C4—1181no—
475C4⋊C4—1181no—
476C4⋊C4—1181no—
477C4⋊C4—4721no—
478C4⋊C4—4721no—
479C4⋊C4—2362no—
480C4⋊C4—2362no—
481C4⋊C4—2362no—
482C22⋊C4—22no—
483C22⋊C4—22no—
484C22⋊C4—1184no—
485C22⋊C4—4721no—
486C22⋊C4—2362no—
487C42—21no—
488C42—21no—
489C42—21no—
490C42—21no—
491C42—4721no—
492C42—4721no—
493C42—2362no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23#1427374912yes—
501C23C22×D23611yescharacteristic
502Q8—1182no—
503Q8—1182no—
504Q8—2361no—
505Q8—2361no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
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—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—1182no—
525C2×C4—1182no—
526C2×C4—1182no—
527C2×C4—4724no—
528C2×C4—2361no—
529C2×C4—2361no—
530C2×C4—2361no—
531C2×C4—2361no—
532C2×C4—2361no—
533C2×C4—2361no—
534C2×C4—2361no—
535C2×C4—2361no—
536C2×C4—2361no—
537C2×C4—2361no—
538C2×C4—2361no—
539C2×C4—2361no—
540C2×C4—2361no—
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#1427376012yes—
550C2×C4#41×D5911yescharacteristic
551C2×C4#41×D5911yescharacteristic
552C2×C4#41×D5911yescharacteristic
553C2×C4C2×#1174568711yescharacteristic
554C2×C4#1427377211yescharacteristic
555C2×C4#1427377411yescharacteristic
556C8—22no—
557C8—41no—
558C8—41no—
559C8—41no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22#1683791411yescharacteristic
568C22#1683804711yescharacteristic
569C22#1683805611yescharacteristic
570C22C2×#1427374911yescharacteristic
571C22C2×#1427376011yescharacteristic
572C22#1683821411yescharacteristic
573C22#1683821411yescharacteristic
574C4—22no—
575C4—41no—
576C4—41no—
577C4—41no—
578C4—41no—
579C4—4721no—
580C4—4721no—
581C4—4721no—
582C4—4721no—
583C4—4721no—
584C4—2362no—
585C4—2362no—
586C4—2362no—
587C4#1683793111yescharacteristic
588C4#1683799511yescharacteristic
589C4#1683809511yescharacteristic
590C4#1683811911yescharacteristic
591C2—42no—
592C2#2142813011yescharacteristic
593C2#2143417811yescharacteristic
594C2#2143437511yescharacteristic
595C2#2143593311yescharacteristic
596C2#2143626911yescharacteristic
597C2#2143647711yescharacteristic
598C2#2143671211yescharacteristic
subgroupG/H
centerC23C22×D236
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#1923C2
socleC23×C59C22×D4
prime radicalC24×C59C2×D4
hypercenter#1923D59
hypocenterC59#27037
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1923

Complements

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

NH ≅ G/Ntype
3. #21425516C2semidirect
4. #21425476C2semidirect
7. #21424866C2semidirect
8. #21424824C2semidirect
9. #21424689C2semidirect
10. #21424682C2semidirect
11. #21424367C2semidirect
14. C4×#14273652C2semidirect
358. C59#27037semidirect

Maximal subgroups

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

subgroupindex
#214284962
#214270992
#214255162
#214254762
#214253002
#214252932
#214248662
#214248242
#214246892
#214246822
#214243672
#214242912
#214241832
C4×#142736522
C59×#19232
#2703759