Finite Group Databasebeta

← #28600776

Subgroups — #28600776

Order 15104.

796 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#125×Dic59C211yesmaximal, characteristic, complemented
2#21426561C211yesmaximal, characteristic
3#21425054C211yesmaximal, characteristic, complemented
4#21424866C211yesmaximal, characteristic, complemented
5#21424830C211yesmaximal, characteristic, complemented
6#21424825C211yesmaximal, characteristic, complemented
7#21424781C211yesmaximal, characteristic
8#21424781C211yesmaximal, characteristic
9#21424725C211yesmaximal, characteristic
10#21424718C211yesmaximal, characteristic
11#21424690C211yesmaximal, characteristic, complemented
12#21424683C211yesmaximal, characteristic, complemented
13#21424106C211yesmaximal, characteristic
14C4⋊C8×Dic59C211yesmaximal, characteristic, complemented
15C2×C59×#478C211yesmaximal, characteristic
16C2×#14273686—21no—
17C2×#14273661C2211yescharacteristic
18C2×#14273659C2211yescharacteristic
19C2×C4⋊(C59⋊C8)C2211yescharacteristic
20C2×#14273648C2211yescharacteristic
21C2×#14273648C2211yescharacteristic
22C2×C8×Dic59C2211yescharacteristic
23C2×C8×Dic59C2211yescharacteristic
24#16837305C2211yescharacteristic
25#16837305C2211yescharacteristic
26#16837301C2211yescharacteristic
27#16837301C2211yescharacteristic
28#16837301C2211yescharacteristic
29#16837301C2211yescharacteristic
30#16837295—21no—
31#16837295—21no—
32#16837288—21no—
33#16837288—21no—
34#16837268C2211yescharacteristic
35C4⋊C4×Dic59—21no—
36C4⋊C4×Dic59—21no—
37C4⋊C4×Dic59C2211yescharacteristic
38C4⋊C4×Dic59C2211yescharacteristic
39C4×C4⋊Dic59C2211yescharacteristic
40C4×C22⋊Dic59—21no—
41C42×Dic59C2211yescharacteristic
42#16837165C2211yescharacteristic
43#16837165C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837164C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837156C2211yescharacteristic
51#16837156C2211yescharacteristic
52C2×#125×C59C2211yescharacteristic
53C2×C4×D4×C59C2211yescharacteristic
54C59×#478C418yescomplemented
55C2×#107×C59C2211yescharacteristic
56C2×#106×C59C2211yescharacteristic
57C2×C4⋊C8×C59C2211yescharacteristic
58C2×#102×C59C2211yescharacteristic
59C2×#102×C59C2211yescharacteristic
60C22×C59⋊C8C2311yescharacteristic
61C22×C59⋊C8C2311yescharacteristic
62#14273686—44no—
63C2×C4⋊Dic59—21no—
64C2×C4⋊Dic59—21no—
65C2×C4⋊Dic59—21no—
66C2×C4⋊Dic59—21no—
67C2×C4⋊Dic59C2311yescharacteristic
68C2×C4⋊Dic59C2311yescharacteristic
69C2×C4⋊Dic59C2311yescharacteristic
70C2×C4⋊Dic59C2311yescharacteristic
71C2×C22⋊Dic59—21no—
72C2×C22⋊Dic59—21no—
73C2×C22⋊Dic59—21no—
74C2×C22⋊Dic59—21no—
75C2×C4×Dic59—21no—
76C2×C4×Dic59—21no—
77C2×C4×Dic59—21no—
78C2×C4×Dic59D411yescharacteristic
79C2×C4×Dic59D411yescharacteristic
80C2×C4×Dic59D411yescharacteristic
81C2×C4×Dic59D411yescharacteristic
82C2×C4×Dic59C2311yescharacteristic
83C2×C4×Dic59C2311yescharacteristic
84#14273661—41no—
85#14273659—41no—
86C4⋊(C59⋊C8)—22no—
87#14273648—41no—
88#14273648—41no—
89C8×Dic59—41no—
90C8×Dic59—41no—
91C59⋊#95—21no—
92C59⋊#95—21no—
93C59⋊#95—21no—
94C59⋊#95—21no—
95C59⋊#95—21no—
96C59⋊#95—21no—
97C59⋊#95—21no—
98C59⋊#95—21no—
99C59⋊#95—21no—
100C22×D4×C59C2311yescharacteristic
101C23×C4×C59—21no—
102C22×C8×C59C2311yescharacteristic
103C22×C8×C59C2311yescharacteristic
104#125×C59—22no—
105#125×C59C2×C414yes—
106C4×D4×C59—22no—
107C4×D4×C59C2×C414yes—
108C2×C4⋊C4×C59—21no—
109C2×C4⋊C4×C59—21no—
110C2×C4⋊C4×C59C2311yescharacteristic
111C2×C4⋊C4×C59C2311yescharacteristic
112C2×C4⋊C4×C59C2311yescharacteristic
113C2×C22⋊C4×C59—21no—
114C2×C42×C59C2311yescharacteristic
115#107×C59C2×C414yes—
116#106×C59C2×C414yes—
117C4⋊C8×C59C2×C414yes—
118#102×C59C2×C414yes—
119#102×C59C2×C414yes—
120C24×C59—21no—
121C2×D4×C59—21no—
122C2×D4×C59—21no—
123C2×D4×C59—21no—
124C2×D4×C59—21no—
125C2×D4×C59—21no—
126C2×D4×C59C22×C412yes—
127C22×C4×C59—21no—
128C22×C4×C59—21no—
129C22×C4×C59—21no—
130C22×C4×C59—24no—
131C22×C4×C59—42no—
132C22×C4×C59—42no—
133C22×C4×C59C2×D411yescharacteristic
134C22×C4×C59C2×D411yescharacteristic
135C22×C4×C59C2411yescharacteristic
136C22×Dic59—21no—
137C22×Dic59—21no—
138C22×Dic59—21no—
139C22×Dic59—21no—
140C22×Dic59C2×D411yescharacteristic
141C22×Dic59C2×D411yescharacteristic
142C22×Dic59C2×D411yescharacteristic
143C22×Dic59C2×D411yescharacteristic
144C2×C8×C59—21no—
145C2×C8×C59—21no—
146C2×C8×C59—21no—
147C2×C8×C59—21no—
148C2×C8×C59C22×C412yes—
149C2×C8×C59C22×C412yes—
150C4⋊C4×C59—24no—
151C4⋊C4×C59—24no—
152C4⋊C4×C59—21no—
153C4⋊C4×C59—21no—
154C4⋊C4×C59—21no—
155C4⋊C4×C59C22×C412yes—
156C4⋊C4×C59C22×C412yes—
157C4⋊C4×C59C22×C412yes—
158C22⋊C4×C59—24no—
159C42×C59C22×C412yes—
160C42×C59#4112yes—
161C22⋊Dic59—42no—
162C22⋊Dic59—42no—
163C22⋊Dic59—42no—
164C22⋊Dic59—42no—
165C4⋊Dic59—21no—
166C4⋊Dic59—21no—
167C4⋊Dic59—21no—
168C4⋊Dic59—21no—
169C4⋊Dic59—21no—
170C4⋊Dic59—21no—
171C4⋊Dic59—21no—
172C4⋊Dic59—21no—
173C4⋊Dic59—42no—
174C4⋊Dic59—42no—
175C4⋊Dic59—42no—
176C4⋊Dic59—42no—
177C4×Dic59—21no—
178C4×Dic59—21no—
179C4×Dic59—21no—
180C4×Dic59—21no—
181C4×Dic59—22no—
182C4×Dic59—22no—
183C4×Dic59—22no—
184C4×Dic59—41no—
185C4×Dic59—42no—
186C4×Dic59—42no—
187C4×Dic59—42no—
188C2×C59⋊C8—21no—
189C2×C59⋊C8—21no—
190C2×C59⋊C8—21no—
191C2×C59⋊C8—21no—
192C2×C59⋊C8—21no—
193C2×C59⋊C8—21no—
194C23×C59—22no—
195C23×C59—41no—
196C23×C59—41no—
197C23×C59—41no—
198C23×C59—41no—
199C23×C59—41no—
200C23×C59—41no—
201C23×C59C22×D411yescharacteristic
202D4×C59—22no—
203D4×C59—22no—
204D4×C59—41no—
205D4×C59—41no—
206C2×C4×C59—21no—
207C2×C4×C59—21no—
208C2×C4×C59—21no—
209C2×C4×C59—21no—
210C2×C4×C59—22no—
211C2×C4×C59—22no—
212C2×C4×C59—22no—
213C2×C4×C59—48no—
214C2×C4×C59—41no—
215C2×C4×C59—41no—
216C2×C4×C59—41no—
217C2×C4×C59—41no—
218C2×C4×C59—41no—
219C2×C4×C59—41no—
220C2×C4×C59C4×D412yes—
221C2×C4×C59C4×D412yes—
222C2×C4×C59C23×C411yescharacteristic
223C2×C4×C59C2×#4111yescharacteristic
224C2×C4×C59H(5,𝔽2)11yescharacteristic
225C2×C4×C592−1+411yescharacteristic
226C2×C4×C592−1+411yescharacteristic
227C2×C4×C592−1+411yescharacteristic
228C2×Dic59—21no—
229C2×Dic59—21no—
230C2×Dic59—21no—
231C2×Dic59—21no—
232C2×Dic59—21no—
233C2×Dic59—21no—
234C2×Dic59—21no—
235C2×Dic59—21no—
236C2×Dic59—21no—
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#13512yes—
250C2×Dic59#13512yes—
251C2×Dic59AGL(1,ℤ/8)12yes—
252C8×C59—22no—
253C8×C59—22no—
254C59⋊C8—41no—
255C59⋊C8—41no—
256#22114—591nomaximal, Sylow
257C22×C59—42no—
258C22×C59—42no—
259C22×C59—42no—
260C22×C59—81no—
261C22×C59—81no—
262C22×C59—81no—
263C22×C59—81no—
264C22×C59C2×C4×D411yescharacteristic
265C22×C59#53611yescharacteristic
266C22×C59#53711yescharacteristic
267C22×C59C2×#13511yescharacteristic
268C22×C59C2×#13511yescharacteristic
269C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
270C22×C59#57511yescharacteristic
271C4×C59—22no—
272C4×C59—22no—
273C4×C59—42no—
274C4×C59—42no—
275C4×C59—42no—
276C4×C59#51911yescharacteristic
277C4×C59#52011yescharacteristic
278C4×C59#54111yescharacteristic
279C4×C59#54211yescharacteristic
280Dic59—22no—
281Dic59—22no—
282Dic59—22no—
283Dic59—41no—
284Dic59—81no—
285Dic59—81no—
286Dic59—81no—
287Dic59—81no—
288C2×#478D5911yescharacteristic
289C4×#125—591no—
290#2293—591no—
291#1989—591no—
292#1949—591no—
293#1942—591no—
294#1941—591no—
295#1934—591no—
296#1934—591no—
297#1926—591no—
298#1925—591no—
299#1921—591no—
300#1920—591no—
301#1804—591no—
302C4×C4⋊C8—591no—
303C2×C59—82no—
304C2×C59#232311yescharacteristic
305C2×C59C4×#13511yescharacteristic
306C2×C59C4×AGL(1,ℤ/8)11yescharacteristic
307C2×C59#322811yescharacteristic
308C2×C59#323211yescharacteristic
309C2×C59#324711yescharacteristic
310C2×C59#325011yescharacteristic
311C2×#125C2×D5911yescharacteristic
312C2×C4×D4C2×D5911yescharacteristic
313C2×#117—1181no—
314#478Dic5918yescomplemented
315C2×#107—591no—
316C2×#107C2×D5911yescharacteristic
317C2×#106—591no—
318C2×#106C2×D5911yescharacteristic
319C2×C4⋊C8—591no—
320C2×C4⋊C8C2×D5911yescharacteristic
321C2×#102—591no—
322C2×#102—591no—
323C2×#102C2×D5911yescharacteristic
324C2×#102C2×D5911yescharacteristic
325C2×C4×C8—591no—
326C2×C4×C8—591no—
327#390—591no—
328#390—591no—
329#389—591no—
330#389—591no—
331#389—591no—
332#389—591no—
333#388—1181no—
334#388—1181no—
335#387—1181no—
336#387—1181no—
337#382—591no—
338C4×C4⋊C4—591no—
339C4×C4⋊C4—591no—
340C4×C4⋊C4—591no—
341C4×C4⋊C4—1181no—
342C4×C4⋊C4—1181no—
343C4×C22⋊C4—1181no—
344C43—591no—
345#340—591no—
346#340—591no—
347#340—591no—
348#340—591no—
349#340—591no—
350#340—591no—
351#340—591no—
352#340—591no—
353#336—591no—
354#336—591no—
355C59#2211411yescharacteristic, complemented, Sylow
356C22×D4C22×D5911yescharacteristic
357C23×C4—21no—
358C22×C8—591no—
359C22×C8—591no—
360C22×C8C22×D5911yescharacteristic
361C22×C8C22×D5911yescharacteristic
362#125—22no—
363#125C2×Dic5914yes—
364C4×D4—22no—
365C4×D4C2×Dic5914yes—
366#117—2364no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—591no—
370C2×C4⋊C4—591no—
371C2×C4⋊C4—591no—
372C2×C4⋊C4—591no—
373C2×C4⋊C4—1181no—
374C2×C4⋊C4—1181no—
375C2×C4⋊C4—1181no—
376C2×C4⋊C4—1181no—
377C2×C4⋊C4C22×D5911yescharacteristic
378C2×C4⋊C4C22×D5911yescharacteristic
379C2×C4⋊C4C22×D5911yescharacteristic
380C2×C22⋊C4—21no—
381C2×C22⋊C4—1181no—
382C2×C22⋊C4—1181no—
383C2×C22⋊C4—1181no—
384C2×C22⋊C4—1181no—
385C2×C42—591no—
386C2×C42—591no—
387C2×C42—591no—
388C2×C42—591no—
389C2×C42—591no—
390C2×C42—591no—
391C2×C42—1181no—
392C2×C42—1181no—
393C2×C42—1181no—
394C2×C42C22×D5911yescharacteristic
395#107—2361no—
396#107C2×Dic5914yes—
397#106—2361no—
398#106C2×Dic5914yes—
399C4⋊C8—1182no—
400C4⋊C8C2×Dic5914yes—
401#102—2361no—
402#102—2361no—
403#102C2×Dic5914yes—
404#102C2×Dic5914yes—
405C4×C8—2361no—
406C4×C8—2361no—
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—
416C24—21no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4C22×Dic5912yes—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—24no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—591no—
430C22×C4—591no—
431C22×C4—591no—
432C22×C4—591no—
433C22×C4—1181no—
434C22×C4—1181no—
435C22×C4—1181no—
436C22×C4—1181no—
437C22×C4D4×D5911yescharacteristic
438C22×C4D4×D5911yescharacteristic
439C22×C4C23×D5911yescharacteristic
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—1181no—
445C2×C8—1181no—
446C2×C8—1181no—
447C2×C8—1181no—
448C2×C8—1181no—
449C2×C8—1181no—
450C2×C8C22×Dic5912yes—
451C2×C8C22×Dic5912yes—
452C4⋊C4—24no—
453C4⋊C4—24no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—1181no—
458C4⋊C4—1181no—
459C4⋊C4—1181no—
460C4⋊C4—1181no—
461C4⋊C4—1181no—
462C4⋊C4—1181no—
463C4⋊C4—1181no—
464C4⋊C4—1181no—
465C4⋊C4—2362no—
466C4⋊C4—2362no—
467C4⋊C4—2362no—
468C4⋊C4—2362no—
469C4⋊C4C22×Dic5912yes—
470C4⋊C4C22×Dic5912yes—
471C4⋊C4C22×Dic5912yes—
472C22⋊C4—24no—
473C22⋊C4—2362no—
474C22⋊C4—2362no—
475C22⋊C4—2362no—
476C22⋊C4—2362no—
477C42—1182no—
478C42—1182no—
479C42—1182no—
480C42—1181no—
481C42—1181no—
482C42—1181no—
483C42—1181no—
484C42—2361no—
485C42—2362no—
486C42—2362no—
487C42—2362no—
488C42#1174568712yes—
489C42C22×Dic5912yes—
490C23—22no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23C2×D4×D5911yescharacteristic
498D4—22no—
499D4—22no—
500D4—41no—
501D4—41no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—48no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—592no—
517C2×C4—592no—
518C2×C4—592no—
519C2×C4—1181no—
520C2×C4—1181no—
521C2×C4—1181no—
522C2×C4—1181no—
523C2×C4—1181no—
524C2×C4—1181no—
525C2×C4—1181no—
526C2×C4—1181no—
527C2×C4—1181no—
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×C4D4×Dic5912yes—
541C2×C4D4×Dic5912yes—
542C2×C4C23×Dic5911yescharacteristic
543C2×C4C2×#1174568711yescharacteristic
544C2×C4#1427377311yescharacteristic
545C2×C4#1427377411yescharacteristic
546C2×C4#1427377511yescharacteristic
547C2×C4#1427377511yescharacteristic
548C8—22no—
549C8—22no—
550C8—2361no—
551C8—2361no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22—81no—
556C22—81no—
557C22—81no—
558C22—81no—
559C22C2×D4×Dic5911yescharacteristic
560C22#1683797511yescharacteristic
561C22#1683798211yescharacteristic
562C22#135×D5911yescharacteristic
563C22#135×D5911yescharacteristic
564C22AGL(1,ℤ/8)×D5911yescharacteristic
565C22#1683819811yescharacteristic
566C4—22no—
567C4—22no—
568C4—42no—
569C4—42no—
570C4—42no—
571C4—1182no—
572C4—1182no—
573C4—1182no—
574C4—4721no—
575C4—4721no—
576C4—4721no—
577C4—4721no—
578C4—2361no—
579C4#1683788011yescharacteristic
580C4#1683788511yescharacteristic
581C4#1683801211yescharacteristic
582C4#1683802511yescharacteristic
583C2—82no—
584C2#2142678011yescharacteristic
585C2#135×Dic5911yescharacteristic
586C2AGL(1,ℤ/8)×Dic5911yescharacteristic
587C2#2143523111yescharacteristic
588C2#2143526211yescharacteristic
589C2#2143542111yescharacteristic
590C2#2143544711yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC2×C59×#478C2
socleC23×C59C22×D4
prime radicalC22×D4×C59C23
cosocleC22×C4×C59C24
hypercenterC2×#478D59
hypocenterC59#22114
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#478

Complements

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

NH ≅ G/Ntype
1. #125×Dic59C2semidirect
3. #21425054C2semidirect
4. #21424866C2semidirect
5. #21424830C2semidirect
6. #21424825C2semidirect
11. #21424690C2semidirect
12. #21424683C2semidirect
14. C4⋊C8×Dic59C2semidirect
54. C59×#478C4semidirect
314. #478Dic59semidirect
355. C59#22114semidirect

Maximal subgroups

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

subgroupindex
#125×Dic592
#214265612
#214250542
#214248662
#214248302
#214248252
#214247812
#214247252
#214247182
#214246902
#214246832
#214241062
C4⋊C8×Dic592
C2×C59×#4782
#2211459