Finite Group Databasebeta

← #28691843

Subgroups — #28691843

Order 15104.

708 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21430056C211yesmaximal, characteristic, complemented
2#21425587C211yesmaximal, characteristic, complemented
3#21425470C211yesmaximal, characteristic, complemented
4#21425467C211yesmaximal, characteristic, complemented
5#21425406C211yesmaximal, characteristic
6#21424961C211yesmaximal, characteristic
7#21424821C211yesmaximal, characteristic, complemented
8#21424796C211yesmaximal, characteristic, complemented
9#21424792C211yesmaximal, characteristic, complemented
10#21424723C211yesmaximal, characteristic, complemented
11#21424667C211yesmaximal, characteristic, complemented
12#21424444C211yesmaximal, characteristic
13#21424357C211yesmaximal, characteristic, complemented
14#21423953C211yesmaximal, characteristic, complemented
15C59×#2319C211yesmaximal, characteristic, complemented
16C22×#11745669C2211yescharacteristic
17C2×#14273660C2211yescharacteristic
18C2×#14273658C2211yescharacteristic
19C2×#14273658C2211yescharacteristic
20C2×C4⋊(C59⋊C8)C2211yescharacteristic
21C2×C4⋊(C59⋊C8)C2211yescharacteristic
22C2×#14273648C2211yescharacteristic
23C2×#14273648C2211yescharacteristic
24C2×#14273648C2211yescharacteristic
25C2×#14273647C2211yescharacteristic
26C2×#14273647C2211yescharacteristic
27C2×#14273647C2211yescharacteristic
28C2×#14273638C2211yescharacteristic
29C2×C4×C59⋊C8C2211yescharacteristic
30#16837331—21no—
31#16837331—21no—
32#16837331—21no—
33#16837326—21no—
34#16837304C2211yescharacteristic
35#16837304C2211yescharacteristic
36#16837304C2211yescharacteristic
37#16837300C2211yescharacteristic
38#16837300C2211yescharacteristic
39#16837300C2211yescharacteristic
40#16837164C2211yescharacteristic
41#16837164C2211yescharacteristic
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837157C2211yescharacteristic
47#16837157C2211yescharacteristic
48C2×C4×D4×C59C2211yescharacteristic
49C59×#390C2211yescharacteristic
50C59×#390C2211yescharacteristic
51C59×#389C2211yescharacteristic
52C59×#389C2211yescharacteristic
53C59×#388—21no—
54C59×#388—21no—
55C59×#381—21no—
56C59×#380—21no—
57C4×C4⋊C4×C59C2211yescharacteristic
58C59×#376C2211yescharacteristic
59C22×D236C2311yescharacteristic
60C2×#11745669—22no—
61C2×#11745669—22no—
62C2×#11745669—41no—
63C22×C59⋊C8C2311yescharacteristic
64C22×C59⋊C8C2311yescharacteristic
65C22×C59⋊C8C2311yescharacteristic
66C22×C59⋊C8C2311yescharacteristic
67C2×C4⋊Dic59C2311yescharacteristic
68C2×C4⋊Dic59C2311yescharacteristic
69C2×C4⋊Dic59C2311yescharacteristic
70C2×Dic59⋊C4—21no—
71C2×Dic59⋊C4—21no—
72C2×Dic59⋊C4—21no—
73C2×(C2×D59)⋊C4—21no—
74C2×(C2×D59)⋊C4—21no—
75C2×(C2×D59)⋊C4—21no—
76C2×(C2×D59)⋊C4—21no—
77C2×(C2×D59)⋊C4—21no—
78C2×(C2×D59)⋊C4—21no—
79#14273660—22no—
80#14273658—22no—
81#14273658—41no—
82C4⋊(C59⋊C8)—22no—
83C4⋊(C59⋊C8)—22no—
84#14273648—22no—
85#14273648—22no—
86#14273648—41no—
87#14273647—22no—
88#14273647—22no—
89#14273647—41no—
90#14273638—41no—
91C4×C59⋊C8—41no—
92C59⋊#95—21no—
93C59⋊#95—21no—
94C59⋊#95—21no—
95C22×D4×C59C2311yescharacteristic
96C23×C4×C59—21no—
97C4×D4×C59—22no—
98C4×D4×C59—22no—
99C2×C4⋊C4×C59—21no—
100C2×C4⋊C4×C59—21no—
101C2×C4⋊C4×C59C2311yescharacteristic
102C2×C4⋊C4×C59C2311yescharacteristic
103C2×C4⋊C4×C59C2311yescharacteristic
104C2×C22⋊C4×C59—21no—
105C2×C22⋊C4×C59—21no—
106C2×C22⋊C4×C59—21no—
107C2×C22⋊C4×C59—21no—
108C2×C22⋊C4×C59—21no—
109C2×C42×C59—21no—
110C2×C42×C59C2311yescharacteristic
111C2×C42×C59C2311yescharacteristic
112C2×C42×C59C2311yescharacteristic
113#95×C59—21no—
114#95×C59—21no—
115#95×C59—21no—
116#95×C59D411yescharacteristic
117#95×C59D411yescharacteristic
118#95×C59D411yescharacteristic
119#95×C59D411yescharacteristic
120C24×C59—21no—
121C23×D59—21no—
122C2×D4×C59—21no—
123C2×D4×C59—21no—
124C2×D4×C59—21no—
125C2×D4×C59—21no—
126C2×D4×C59—21no—
127C2×D4×C59#4112yes—
128C22×C4×C59—21no—
129C22×C4×C59—24no—
130C22×C4×C59—21no—
131C22×C4×C59—21no—
132C22×C4×C59—81no—
133C22×C4×C59—81no—
134C22×C4×C59C2×D411yescharacteristic
135C22×C4×C59C2×D411yescharacteristic
136C22×C4×C59C2×D411yescharacteristic
137C22×C4×C59C2×D411yescharacteristic
138C22×C4×C59C2×D411yescharacteristic
139C22×C4×C59C2×D411yescharacteristic
140C22×C4×C59C2411yescharacteristic
141C22×Dic59—21no—
142C22×Dic59—21no—
143C22×Dic59—21no—
144C2×D236—21no—
145C2×D236—21no—
146C2×D236—21no—
147C2×D236—21no—
148C2×D236—21no—
149C2×D236—21no—
150C4⋊C4×C59—21no—
151C4⋊C4×C59—21no—
152C4⋊C4×C59—21no—
153C4⋊C4×C59—21no—
154C4⋊C4×C59—21no—
155C4⋊C4×C59—81no—
156C4⋊C4×C59—81no—
157C4⋊C4×C59#4112yes—
158C22⋊C4×C59—24no—
159C22⋊C4×C59—81no—
160C22⋊C4×C59—81no—
161C22⋊C4×C59—81no—
162C22⋊C4×C59—81no—
163C42×C59—21no—
164C42×C59—21no—
165C42×C59—21no—
166C42×C59—21no—
167C42×C59—21no—
168C42×C59—21no—
169C42×C59—81no—
170#11745669—44no—
171(C2×D59)⋊C4—81no—
172(C2×D59)⋊C4—81no—
173(C2×D59)⋊C4—81no—
174(C2×D59)⋊C4—81no—
175(C2×D59)⋊C4—81no—
176(C2×D59)⋊C4—81no—
177C4⋊Dic59—21no—
178C4⋊Dic59—21no—
179C4⋊Dic59—21no—
180C4⋊Dic59—21no—
181C4⋊Dic59—21no—
182C4⋊Dic59—21no—
183Dic59⋊C4—81no—
184Dic59⋊C4—81no—
185Dic59⋊C4—81no—
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—
195C2×C59⋊C8—21no—
196C2×C59⋊C8#4112yes—
197C2×C59⋊C8#4112yes—
198C23×C59—22no—
199C23×C59—41no—
200C23×C59—41no—
201C23×C59—41no—
202C23×C59—41no—
203C23×C59—41no—
204C23×C59—41no—
205C23×C59C22×D411yescharacteristic
206C22×D59—41no—
207C22×D59—41no—
208C22×D59—41no—
209C22×D59—41no—
210C22×D59—41no—
211C22×D59—41no—
212C22×D59—41no—
213D4×C59—22no—
214D4×C59—22no—
215D4×C59—41no—
216D4×C59—41no—
217C2×C4×C59—21no—
218C2×C4×C59—21no—
219C2×C4×C59—22no—
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—21no—
231C2×C4×C59—21no—
232C2×C4×C59—21no—
233C2×C4×C59—21no—
234C2×C4×C59—21no—
235C2×C4×C59—21no—
236C2×C4×C59—41no—
237C2×C4×C59—41no—
238C2×C4×C59—41no—
239C2×C4×C59—41no—
240C2×C4×C59—41no—
241C2×C4×C59—41no—
242C2×C4×C59—41no—
243C2×C4×C59—41no—
244C2×C4×C59—84no—
245C2×C4×C59C2×#4111yescharacteristic
246C2×C4×C59C2×#4111yescharacteristic
247C2×C4×C59C2×#4111yescharacteristic
248C2×C4×C59C2×#4111yescharacteristic
249C2×C4×C59H(5,𝔽2)11yescharacteristic
250C2×C4×C592−1+411yescharacteristic
251C2×Dic59—41no—
252C2×Dic59—41no—
253C2×Dic59—41no—
254C2×Dic59—41no—
255C2×Dic59—41no—
256C2×Dic59—41no—
257C2×Dic59—41no—
258C2×Dic59—41no—
259C2×Dic59—41no—
260D236—41no—
261D236—41no—
262D236—41no—
263D236—41no—
264C59⋊C8—22no—
265C59⋊C8—22no—
266C59⋊C8—41no—
267C59⋊C8—41no—
268#29437—591nomaximal, Sylow
269C22×C59—42no—
270C22×C59—42no—
271C22×C59—42no—
272C22×C59—81no—
273C22×C59—81no—
274C22×C59—81no—
275C22×C59—81no—
276C22×C59#52511yescharacteristic
277C22×C59#53211yescharacteristic
278C22×C59#53711yescharacteristic
279C22×C59#57611yescharacteristic
280C22×C59#57611yescharacteristic
281C22×C59#57711yescharacteristic
282C22×C59#57711yescharacteristic
283C2×D59—81no—
284C2×D59—81no—
285C2×D59—81no—
286C2×D59—81no—
287C2×D59—81no—
288C2×D59—81no—
289C2×D59—81no—
290C4×C59—41no—
291C4×C59—42no—
292C4×C59—41no—
293C4×C59—41no—
294C4×C59—41no—
295C4×C59—41no—
296C4×C59—41no—
297C4×C59—81no—
298C4×C59—81no—
299C4×C59#55111yescharacteristic
300C4×C59#55211yescharacteristic
301C4×C59#55311yescharacteristic
302C4×C59#55611yescharacteristic
303Dic59—81no—
304Dic59—81no—
305Dic59—81no—
306#2644—591no—
307#2319D5911yescharacteristic, complemented
308#2086—591no—
309#2065—591no—
310#2065—591no—
311#2054—591no—
312#1970—591no—
313#1941—591no—
314#1936—591no—
315#1936—591no—
316#1926—591no—
317#1918—591no—
318#1879—591no—
319#1864—591no—
320#1768—591no—
321C2×C59—82no—
322C2×C59#260111yescharacteristic
323C2×C59#311811yescharacteristic
324C2×C59#311911yescharacteristic
325C2×C59#317311yescharacteristic
326C2×C59#317611yescharacteristic
327C2×C59#325711yescharacteristic
328C2×C59#325711yescharacteristic
329D59—161no—
330C22×D8—591no—
331C2×C4×D4C2×D5911yescharacteristic
332C2×#107—591no—
333C2×#106—591no—
334C2×#106—591no—
335C2×C4⋊C8—591no—
336C2×C4⋊C8—591no—
337C2×#102—591no—
338C2×#102—591no—
339C2×#102—591no—
340C2×#102—591no—
341C2×#102—591no—
342C2×#102—591no—
343C2×#97—591no—
344C2×C4×C8—591no—
345#397—1181no—
346#397—1181no—
347#397—1181no—
348#396—1181no—
349#390—591no—
350#390—591no—
351#390—591no—
352#390C2×D5911yescharacteristic
353#390C2×D5911yescharacteristic
354#389—591no—
355#389—591no—
356#389—591no—
357#389C2×D5911yescharacteristic
358#389C2×D5911yescharacteristic
359#388—21no—
360#388—21no—
361#381—21no—
362#380—21no—
363C4×C4⋊C4C2×D5911yescharacteristic
364#376C2×D5911yescharacteristic
365#340—591no—
366#340—591no—
367#340—591no—
368#340—591no—
369#340—591no—
370#340—591no—
371#336—591no—
372#336—591no—
373C59#2943711yescharacteristic, complemented, Sylow
374C22×D4—591no—
375C22×D4C22×D5911yescharacteristic
376C23×C4—21no—
377C2×D8—1182no—
378C2×D8—1182no—
379C2×D8—2361no—
380C22×C8—591no—
381C22×C8—591no—
382C22×C8—591no—
383C22×C8—591no—
384C4×D4—22no—
385C4×D4—22no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—591no—
389C2×C4⋊C4—591no—
390C2×C4⋊C4—591no—
391C2×C4⋊C4—1181no—
392C2×C4⋊C4—1181no—
393C2×C4⋊C4—1181no—
394C2×C4⋊C4C22×D5911yescharacteristic
395C2×C4⋊C4C22×D5911yescharacteristic
396C2×C4⋊C4C22×D5911yescharacteristic
397C2×C22⋊C4—21no—
398C2×C22⋊C4—21no—
399C2×C22⋊C4—21no—
400C2×C22⋊C4—21no—
401C2×C22⋊C4—21no—
402C2×C22⋊C4—1181no—
403C2×C22⋊C4—1181no—
404C2×C22⋊C4—1181no—
405C2×C22⋊C4—1181no—
406C2×C22⋊C4—1181no—
407C2×C22⋊C4—1181no—
408C2×C42—21no—
409C2×C42C22×D5911yescharacteristic
410C2×C42C22×D5911yescharacteristic
411C2×C42C22×D5911yescharacteristic
412#107—1182no—
413#106—1182no—
414#106—2361no—
415C4⋊C8—1182no—
416C4⋊C8—1182no—
417#102—1182no—
418#102—1182no—
419#102—1182no—
420#102—1182no—
421#102—2361no—
422#102—2361no—
423#97—2361no—
424C4×C8—2361no—
425#95—21no—
426#95—21no—
427#95—21no—
428#95—1181no—
429#95—1181no—
430#95—1181no—
431#95C22⋊D5911yescharacteristic
432#95C22⋊D5911yescharacteristic
433#95C22⋊D5911yescharacteristic
434#95C22⋊D5911yescharacteristic
435C24—21no—
436C24—1181no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—1181no—
443C2×D4—1181no—
444C2×D4—1181no—
445C2×D4—1181no—
446C2×D4—1181no—
447C2×D4—1181no—
448C2×D4#1174568512yes—
449C22×C4—21no—
450C22×C4—24no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—81no—
454C22×C4—81no—
455C22×C4—1181no—
456C22×C4—1181no—
457C22×C4—1181no—
458C22×C4C2×C22⋊D5911yescharacteristic
459C22×C4C2×C22⋊D5911yescharacteristic
460C22×C4C2×C22⋊D5911yescharacteristic
461C22×C4C2×C22⋊D5911yescharacteristic
462C22×C4C2×C22⋊D5911yescharacteristic
463C22×C4C2×C22⋊D5911yescharacteristic
464C22×C4C23×D5911yescharacteristic
465D8—2364no—
466C2×C8—592no—
467C2×C8—592no—
468C2×C8—1181no—
469C2×C8—1181no—
470C2×C8—1181no—
471C2×C8—1181no—
472C2×C8—1181no—
473C2×C8—1181no—
474C2×C8—1181no—
475C2×C8—1181no—
476C2×C8—1181no—
477C2×C8—1181no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—81no—
484C4⋊C4—81no—
485C4⋊C4—1181no—
486C4⋊C4—1181no—
487C4⋊C4—1181no—
488C4⋊C4—1181no—
489C4⋊C4—1181no—
490C4⋊C4—1181no—
491C4⋊C4—4721no—
492C4⋊C4—4721no—
493C4⋊C4—4721no—
494C4⋊C4#1174568512yes—
495C22⋊C4—24no—
496C22⋊C4—81no—
497C22⋊C4—81no—
498C22⋊C4—81no—
499C22⋊C4—81no—
500C22⋊C4—4721no—
501C22⋊C4—4721no—
502C22⋊C4—4721no—
503C22⋊C4—4721no—
504C22⋊C4—4721no—
505C22⋊C4—4721no—
506C42—21no—
507C42—21no—
508C42—21no—
509C42—21no—
510C42—21no—
511C42—21no—
512C42—81no—
513C23—22no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—2361no—
521C23—2361no—
522C23—2361no—
523C23—2361no—
524C23—2361no—
525C23—2361no—
526C23—2361no—
527C23C22×C22⋊D5911yescharacteristic
528D4—22no—
529D4—22no—
530D4—41no—
531D4—41no—
532D4—2361no—
533D4—2361no—
534D4—2361no—
535D4—2361no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—84no—
564C2×C4—2361no—
565C2×C4—2361no—
566C2×C4—2361no—
567C2×C4—2361no—
568C2×C4—2361no—
569C2×C4—2361no—
570C2×C4—2361no—
571C2×C4—2361no—
572C2×C4—2361no—
573C2×C4#41×D5911yescharacteristic
574C2×C4#41×D5911yescharacteristic
575C2×C4C2×#1174568511yescharacteristic
576C2×C4C2×#1174568511yescharacteristic
577C2×C4#1427377311yescharacteristic
578C2×C4#1427377411yescharacteristic
579C8—1182no—
580C8—1182no—
581C8—2361no—
582C8—2361no—
583C22—42no—
584C22—42no—
585C22—42no—
586C22—81no—
587C22—81no—
588C22—81no—
589C22—81no—
590C22—4721no—
591C22—4721no—
592C22—4721no—
593C22—4721no—
594C22—4721no—
595C22—4721no—
596C22—4721no—
597C22#1683791311yescharacteristic
598C22#1683794811yescharacteristic
599C22#1683798011yescharacteristic
600C22#1683820511yescharacteristic
601C22#1683820511yescharacteristic
602C22#1683821011yescharacteristic
603C22#1683821311yescharacteristic
604C4—41no—
605C4—41no—
606C4—41no—
607C4—41no—
608C4—41no—
609C4—42no—
610C4—41no—
611C4—81no—
612C4—81no—
613C4—4721no—
614C4—4721no—
615C4—4721no—
616C4#1683807611yescharacteristic
617C4#1683808511yescharacteristic
618C4#1683809611yescharacteristic
619C4#1683812011yescharacteristic
620C2—82no—
621C2—9441no—
622C2#2142969311yescharacteristic
623C2#2143428011yescharacteristic
624C2#2143429511yescharacteristic
625C2#2143470711yescharacteristic
626C2#2143474011yescharacteristic
627C2#2143550111yescharacteristic
628C2#2143550711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2319C2
socleC23×C59C22×D4
prime radicalC22×#11745669C22
hypercenter#2319D59
hypocenterC59#29437
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ #376 ◁ #2319

Complements

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

NH ≅ G/Ntype
1. #21430056C2semidirect
2. #21425587C2semidirect
3. #21425470C2semidirect
4. #21425467C2semidirect
7. #21424821C2semidirect
8. #21424796C2semidirect
9. #21424792C2semidirect
10. #21424723C2semidirect
11. #21424667C2semidirect
13. #21424357C2semidirect
14. #21423953C2semidirect
15. C59×#2319C2semidirect
307. #2319D59semidirect
373. C59#29437semidirect

Maximal subgroups

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

subgroupindex
#214300562
#214255872
#214254702
#214254672
#214254062
#214249612
#214248212
#214247962
#214247922
#214247232
#214246672
#214244442
#214243572
#214239532
C59×#23192
#2943759