Finite Group Databasebeta

← #28608463

Subgroups — #28608463

Order 15104.

846 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837489C211yesmaximal, characteristic
2#21427049C211yesmaximal, characteristic, complemented
3#21427041C211yesmaximal, characteristic
4#21424977C211yesmaximal, characteristic
5#21424975C211yesmaximal, characteristic
6#21424737C211yesmaximal, characteristic, complemented
7#21424735C211yesmaximal, characteristic, complemented
8#21424716C211yesmaximal, characteristic
9#21424714C211yesmaximal, characteristic
10#21424254C211yesmaximal, characteristic, complemented
11#21424100C211yesmaximal, characteristic, complemented
12#21424095C211yesmaximal, characteristic
13#21423996C211yesmaximal, characteristic, complemented
14#21423964C211yesmaximal, characteristic, complemented
15C59×#1776C211yesmaximal, characteristic, complemented
16C22×#11745660C2211yescharacteristic
17C2×#14273692C2211yescharacteristic
18C2×#14273689C2211yescharacteristic
19C2×#14273684—21no—
20#16837489C418yescomplemented
21C2×#14273661C2211yescharacteristic
22C2×#14273660C2211yescharacteristic
23C2×#14273660C2211yescharacteristic
24C2×Dic59⋊C8C2211yescharacteristic
25C2×#14273652C2211yescharacteristic
26C2×#14273650C2211yescharacteristic
27C2×#14273650C2211yescharacteristic
28C2×#14273649C2211yescharacteristic
29C2×#14273647C2211yescharacteristic
30C2×#14273647C2211yescharacteristic
31C2×#14273637C2211yescharacteristic
32C2×C8×Dic59C2211yescharacteristic
33#16837308C2211yescharacteristic
34#16837307C2211yescharacteristic
35#16837304C2211yescharacteristic
36#16837303C2211yescharacteristic
37#16837293—21no—
38#16837291—21no—
39#16837286—21no—
40#16837284—21no—
41#16837271C2211yescharacteristic
42#16837271C2211yescharacteristic
43#16837266—21no—
44#16837261—21no—
45C4⋊C4×Dic59C2211yescharacteristic
46C4⋊C4×Dic59C2211yescharacteristic
47C4×Dic59⋊C4—21no—
48C4×Dic59⋊C4C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837164C2211yescharacteristic
51#16837157C2211yescharacteristic
52#16837156C2211yescharacteristic
53C2×#107×C59C2211yescharacteristic
54C2×#106×C59C2211yescharacteristic
55C2×#97×C59C2211yescharacteristic
56C4×C4⋊C4×C59C2211yescharacteristic
57C4×C4⋊C4×C59C2211yescharacteristic
58C59×#340C2211yescharacteristic
59C59×#340C2211yescharacteristic
60C22×Dic118C2311yescharacteristic
61C22×D236C2311yescharacteristic
62C22×C4×D59—21no—
63C2×#11745660—22no—
64C2×#11745660—22no—
65C2×#11745660C2×C414yes—
66C22×C59⋊C8C2311yescharacteristic
67C22×C59⋊C8C2311yescharacteristic
68#14273692—22no—
69#14273692C2×C414yes—
70#14273689—22no—
71#14273689C2×C414yes—
72#14273684—44no—
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×Dic59⋊C4D411yescharacteristic
81C2×Dic59⋊C4D411yescharacteristic
82C2×(C2×D59)⋊C4—21no—
83C2×(C2×D59)⋊C4—21no—
84C2×(C2×D59)⋊C4—21no—
85C2×(C2×D59)⋊C4—21no—
86C2×(C2×D59)⋊C4—21no—
87C2×C4×Dic59—21no—
88C2×C4×Dic59—21no—
89C2×C4×Dic59—21no—
90C2×C4×Dic59C2311yescharacteristic
91C2×C4×Dic59C2311yescharacteristic
92#14273661—22no—
93#14273660—41no—
94#14273660—41no—
95Dic59⋊C8—22no—
96#14273652—41no—
97#14273650—41no—
98#14273650C2×C414yes—
99#14273649—41no—
100#14273647—41no—
101#14273647C2×C414yes—
102#14273637—22no—
103C8×Dic59C2×C414yes—
104C59⋊#95—21no—
105C59⋊#95—21no—
106C59⋊#95—21no—
107C59⋊#95—21no—
108C59⋊#95—21no—
109C59⋊#95—21no—
110C59⋊#95D411yescharacteristic
111C59⋊#95D411yescharacteristic
112C22×C8×C59C2311yescharacteristic
113C22×C8×C59C2311yescharacteristic
114C2×C4⋊C4×C59C2311yescharacteristic
115C2×C4⋊C4×C59C2311yescharacteristic
116C2×C4⋊C4×C59C2311yescharacteristic
117C2×C4⋊C4×C59C2311yescharacteristic
118C2×C42×C59—21no—
119C2×C42×C59—21no—
120C2×C42×C59C2311yescharacteristic
121#107×C59—22no—
122#106×C59C2×C414yes—
123#97×C59—22no—
124#95×C59—21no—
125#95×C59—21no—
126C23×D59—21no—
127C22×C4×C59—21no—
128C22×C4×C59—21no—
129C22×C4×C59—21no—
130C22×C4×C59—21no—
131C22×C4×C59C2×D411yescharacteristic
132C22×C4×C59C2×D411yescharacteristic
133C22×C4×C59C2411yescharacteristic
134C22×Dic59—21no—
135C22×Dic59—21no—
136C22×Dic59—21no—
137C22×Dic59C2×D411yescharacteristic
138C22×Dic59C2×D411yescharacteristic
139C22×Dic59C2×D411yescharacteristic
140C22×Dic59C2×D411yescharacteristic
141C2×D236—21no—
142C2×D236—21no—
143C2×D236—21no—
144C2×D236—21no—
145C2×D236—21no—
146C2×D236C22×C412yes—
147C2×C4×D59—24no—
148C2×C4×D59—42no—
149C2×C4×D59—42no—
150C2×Dic118—21no—
151C2×Dic118—21no—
152C2×Dic118—21no—
153C2×Dic118—21no—
154C2×Dic118—21no—
155C2×Dic118C22×C412yes—
156C2×C8×C59—21no—
157C2×C8×C59—21no—
158C2×C8×C59—21no—
159C2×C8×C59C22×C412yes—
160C2×C8×C59#4112yes—
161C2×C8×C59#4112yes—
162C4⋊C4×C59—21no—
163C4⋊C4×C59—21no—
164C4⋊C4×C59—21no—
165C4⋊C4×C59—21no—
166C4⋊C4×C59—21no—
167C4⋊C4×C59—21no—
168C4⋊C4×C59C22×C412yes—
169C4⋊C4×C59C22×C412yes—
170C42×C59—21no—
171C42×C59—21no—
172C42×C59—42no—
173C42×C59—81no—
174(C2×D59)⋊C4—24no—
175(C2×D59)⋊C4—42no—
176(C2×D59)⋊C4—42no—
177(C2×D59)⋊C4—81no—
178(C2×D59)⋊C4—81no—
179C4⋊Dic59—21no—
180C4⋊Dic59—21no—
181C4⋊Dic59—21no—
182C4⋊Dic59—21no—
183Dic59⋊C4—24no—
184Dic59⋊C4—22no—
185Dic59⋊C4—22no—
186Dic59⋊C4—42no—
187Dic59⋊C4—42no—
188Dic59⋊C4—81no—
189Dic59⋊C4—81no—
190C4×Dic59—21no—
191C4×Dic59—21no—
192C4×Dic59—24no—
193C4×Dic59—42no—
194C4×Dic59—81no—
195C4×Dic59C22×C412yes—
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⋊C8C22×C412yes—
203#11745660—28no—
204C23×C59C22×D411yescharacteristic
205C22×D59—22no—
206C22×D59—41no—
207C22×D59—41no—
208C22×D59—41no—
209C22×D59—41no—
210C22×D59—41no—
211C22×D59—41no—
212C2×C4×C59—22no—
213C2×C4×C59—22no—
214C2×C4×C59—21no—
215C2×C4×C59—21no—
216C2×C4×C59—21no—
217C2×C4×C59—21no—
218C2×C4×C59—21no—
219C2×C4×C59—21no—
220C2×C4×C59—41no—
221C2×C4×C59—41no—
222C2×C4×C59—41no—
223C2×C4×C59—41no—
224C2×C4×C59—41no—
225C2×C4×C59—41no—
226C2×C4×C59—41no—
227C2×C4×C59—41no—
228C2×C4×C59—41no—
229C2×C4×C59—41no—
230C2×C4×C59C23×C411yescharacteristic
231C2×C4×C59C2×#4111yescharacteristic
232C2×C4×C59C2×#4111yescharacteristic
233C2×C4×C59C2×#4111yescharacteristic
234C2×C4×C59H(5,𝔽2)11yescharacteristic
235C2×C4×C592−1+411yescharacteristic
236C2×Dic59—21no—
237C2×Dic59—21no—
238C2×Dic59—22no—
239C2×Dic59—21no—
240C2×Dic59—21no—
241C2×Dic59—21no—
242C2×Dic59—21no—
243C2×Dic59—21no—
244C2×Dic59—21no—
245C2×Dic59—41no—
246C2×Dic59—41no—
247C2×Dic59—41no—
248C2×Dic59—41no—
249C2×Dic59—41no—
250C2×Dic59—41no—
251C2×Dic59—41no—
252C2×Dic59—41no—
253C2×Dic59C4×D412yes—
254C2×Dic59C4×D412yes—
255C2×Dic59AGL(1,ℤ/8)12yes—
256C2×Dic59#13712yes—
257D236—22no—
258D236—22no—
259D236—41no—
260D236—41no—
261C4×D59—48no—
262Dic118—22no—
263Dic118—22no—
264Dic118—41no—
265Dic118—41no—
266C8×C59—41no—
267C8×C59#11714yescomplemented
268C59⋊C8—22no—
269C59⋊C8—41no—
270#22811—591nomaximal, Sylow
271C22×C59C2×C4×D411yescharacteristic
272C22×C59#54711yescharacteristic
273C22×C59#55011yescharacteristic
274C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
275C22×C59C2×#13711yescharacteristic
276C22×C59#57611yescharacteristic
277C22×C59#57811yescharacteristic
278C2×D59—42no—
279C2×D59—42no—
280C2×D59—42no—
281C2×D59—81no—
282C2×D59—81no—
283C2×D59—81no—
284C2×D59—81no—
285C4×C59—41no—
286C4×C59—41no—
287C4×C59—42no—
288C4×C59—42no—
289C4×C59—81no—
290C4×C59—81no—
291C4×C59C2×#11711yescharacteristic
292C4×C59#52011yescharacteristic
293C4×C59#55511yescharacteristic
294C4×C59#55611yescharacteristic
295Dic59—22no—
296Dic59—41no—
297Dic59—41no—
298Dic59—42no—
299Dic59—81no—
300Dic59—81no—
301Dic59#44014yes—
302C2×C4×SD16—591no—
303#2355—591no—
304#2354—591no—
305#1973—591no—
306#1973—591no—
307#1928—591no—
308#1928—591no—
309#1925—591no—
310#1925—591no—
311#1845—591no—
312#1803—591no—
313#1802—591no—
314#1776—591no—
315#1776D5911yescharacteristic, complemented
316#1770—591no—
317C2×C59#235411yescharacteristic
318C2×C59C2×#44011yescharacteristic
319C2×C59#295111yescharacteristic
320C2×C59#335311yescharacteristic
321C2×C59#335711yescharacteristic
322C2×C59#339811yescharacteristic
323C2×C59#339911yescharacteristic
324D59—82no—
325C22×SD16—591no—
326C2×C4×Q8—591no—
327C2×C4×D4—591no—
328C2×#117—1181no—
329C4×SD16—598no—
330C2×#107—591no—
331C2×#107—591no—
332C2×#107—591no—
333C2×#107C2×D5911yescharacteristic
334C2×#106C2×D5911yescharacteristic
335C2×C4⋊C8—591no—
336C2×#103—591no—
337C2×#103—591no—
338C2×#103—591no—
339C2×#102—591no—
340C2×#102—591no—
341C2×#102—591no—
342C2×#97—591no—
343C2×#97C2×D5911yescharacteristic
344C2×C4×C8—591no—
345#391—591no—
346#391—591no—
347#390—591no—
348#390—591no—
349#388—1181no—
350#388—1181no—
351#387—1181no—
352#387—1181no—
353#384—591no—
354#384—591no—
355#382—1181no—
356#380—1181no—
357C4×C4⋊C4—591no—
358C4×C4⋊C4—591no—
359C4×C4⋊C4—591no—
360C4×C4⋊C4—1181no—
361C4×C4⋊C4C2×D5911yescharacteristic
362C4×C4⋊C4C2×D5911yescharacteristic
363#340—591no—
364#340—591no—
365#340C2×D5911yescharacteristic
366#340C2×D5911yescharacteristic
367#336—591no—
368#336—591no—
369C59#2281111yescharacteristic, complemented, Sylow
370C22×Q8—591no—
371C22×D4—591no—
372C23×C4—1181no—
373C2×SD16—594no—
374C2×SD16—1182no—
375C2×SD16—1182no—
376C22×C8—591no—
377C22×C8—591no—
378C22×C8C22×D5911yescharacteristic
379C22×C8C22×D5911yescharacteristic
380C4×Q8—594no—
381C4×Q8—1182no—
382C4×D4—594no—
383C4×D4—1182no—
384#117—2364no—
385C2×C4⋊C4—591no—
386C2×C4⋊C4—591no—
387C2×C4⋊C4—591no—
388C2×C4⋊C4—591no—
389C2×C4⋊C4—1181no—
390C2×C4⋊C4—1181no—
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×C4⋊C4C22×D5911yescharacteristic
398C2×C22⋊C4—1181no—
399C2×C22⋊C4—1181no—
400C2×C22⋊C4—1181no—
401C2×C22⋊C4—1181no—
402C2×C22⋊C4—1181no—
403C2×C42—21no—
404C2×C42—21no—
405C2×C42—591no—
406C2×C42—591no—
407C2×C42—1181no—
408C2×C42—1181no—
409C2×C42—1181no—
410C2×C42C22×D5911yescharacteristic
411#107—22no—
412#107—1182no—
413#107—2361no—
414#107—2361no—
415#106C4×D5914yes—
416C4⋊C8—1182no—
417#103—594no—
418#103—2361no—
419#103—2361no—
420#102—594no—
421#102—2361no—
422#102—2361no—
423#97—22no—
424#97—1182no—
425C4×C8—594no—
426#95—21no—
427#95—21no—
428#95—591no—
429#95—591no—
430#95—1181no—
431#95—1181no—
432#95—1181no—
433#95—1181no—
434#95—1181no—
435#95—1181no—
436C24—1181no—
437C2×Q8—592no—
438C2×Q8—1181no—
439C2×Q8—1181no—
440C2×Q8—1181no—
441C2×Q8—1181no—
442C2×Q8—1181no—
443C2×D4—592no—
444C2×D4—1181no—
445C2×D4—1181no—
446C2×D4—1181no—
447C2×D4—1181no—
448C2×D4—1181no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—591no—
454C22×C4—591no—
455C22×C4—591no—
456C22×C4—591no—
457C22×C4—1184no—
458C22×C4—1181no—
459C22×C4—1181no—
460C22×C4—1181no—
461C22×C4—2362no—
462C22×C4—2362no—
463C22×C4D4×D5911yescharacteristic
464C22×C4D4×D5911yescharacteristic
465C22×C4C23×D5911yescharacteristic
466SD16—1188no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—592no—
471C2×C8—1181no—
472C2×C8—1181no—
473C2×C8—1181no—
474C2×C8—1181no—
475C2×C8—1181no—
476C2×C8C2×C4×D5912yes—
477C2×C8#1174568512yes—
478C2×C8#1174568512yes—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—1182no—
486C4⋊C4—1182no—
487C4⋊C4—1184no—
488C4⋊C4—1181no—
489C4⋊C4—1181no—
490C4⋊C4—1181no—
491C4⋊C4—1181no—
492C4⋊C4—4721no—
493C4⋊C4—4721no—
494C4⋊C4—2362no—
495C4⋊C4—2362no—
496C4⋊C4C2×C4×D5912yes—
497C4⋊C4C2×C4×D5912yes—
498C22⋊C4—1184no—
499C22⋊C4—4721no—
500C22⋊C4—4721no—
501C22⋊C4—2362no—
502C22⋊C4—2362no—
503C42—21no—
504C42—21no—
505C42—42no—
506C42—81no—
507C42—592no—
508C42—592no—
509C42—1184no—
510C42—1181no—
511C42—1181no—
512C42—4721no—
513C42—2362no—
514C23—1182no—
515C23—2361no—
516C23—2361no—
517C23—2361no—
518C23—2361no—
519C23—2361no—
520C23—2361no—
521C23C2×D4×D5911yescharacteristic
522Q8—1182no—
523Q8—1182no—
524Q8—2361no—
525Q8—2361no—
526D4—1182no—
527D4—1182no—
528D4—2361no—
529D4—2361no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—592no—
549C2×C4—592no—
550C2×C4—592no—
551C2×C4—592no—
552C2×C4—1181no—
553C2×C4—1181no—
554C2×C4—1181no—
555C2×C4—1181no—
556C2×C4—1181no—
557C2×C4—1181no—
558C2×C4—1181no—
559C2×C4—1181no—
560C2×C4—1182no—
561C2×C4—2368no—
562C2×C4—2361no—
563C2×C4—2361no—
564C2×C4—2361no—
565C2×C4—2361no—
566C2×C4—2361no—
567C2×C4—2361no—
568C2×C4—2361no—
569C2×C4—2361no—
570C2×C4C22×C4×D5911yescharacteristic
571C2×C4#41×D5911yescharacteristic
572C2×C4C2×#1174568511yescharacteristic
573C2×C4C2×#1174568511yescharacteristic
574C2×C4#1427377311yescharacteristic
575C2×C4#1427377511yescharacteristic
576C8—41no—
577C8—1182no—
578C8—2361no—
579C8#1427368414yescomplemented
580C22—4721no—
581C22—4721no—
582C22—4721no—
583C22—4721no—
584C22—2362no—
585C22—2362no—
586C22—2362no—
587C22C4×D4×D5911yescharacteristic
588C22#1683805311yescharacteristic
589C22#1683807311yescharacteristic
590C22AGL(1,ℤ/8)×D5911yescharacteristic
591C22#137×D5911yescharacteristic
592C22#1683820611yescharacteristic
593C22#1683821811yescharacteristic
594C4—41no—
595C4—42no—
596C4—41no—
597C4—42no—
598C4—81no—
599C4—81no—
600C4—594no—
601C4—1182no—
602C4—4721no—
603C4—4721no—
604C4—2361no—
605C4—2361no—
606C4—2362no—
607C4C2×#1427368411yescharacteristic
608C4#1683788411yescharacteristic
609C4#1683811511yescharacteristic
610C4#1683812411yescharacteristic
611C2—4722no—
612C2#2142704111yescharacteristic
613C2#440×D5911yescharacteristic
614C2#2143296811yescharacteristic
615C2#2143667611yescharacteristic
616C2#2143671811yescharacteristic
617C2#2143712811yescharacteristic
618C2#2143713911yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1776C2
socleC23×C59C22×D4
prime radicalC22×D236C23
cosocleC22×C4×C59C24
hypercenter#1776D59
hypocenterC59#22811
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1776

Complements

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

NH ≅ G/Ntype
2. #21427049C2semidirect
6. #21424737C2semidirect
7. #21424735C2semidirect
10. #21424254C2semidirect
11. #21424100C2semidirect
13. #21423996C2semidirect
14. #21423964C2semidirect
15. C59×#1776C2semidirect
20. #16837489C4semidirect
267. C8×C59#117semidirect
315. #1776D59semidirect
369. C59#22811semidirect
579. C8#14273684semidirect

Maximal subgroups

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

subgroupindex
C2×#168374892
#214270492
#214270412
#214249772
#214249752
#214247372
#214247352
#214247162
#214247142
#214242542
#214241002
#214240952
#214239962
#214239642
C59×#17762
#2281159