Finite Group Databasebeta

← #28611924

Subgroups — #28611924

Order 15104.

798 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
1C2×#16837702C211yesmaximal, characteristic
2#21427190C211yesmaximal, characteristic, complemented
3#21425064C211yesmaximal, characteristic, complemented
4#21424857C211yesmaximal, characteristic, complemented
5#21424796C211yesmaximal, characteristic
6#21424782C211yesmaximal, characteristic
7#21424720C211yesmaximal, characteristic
8#21424720C211yesmaximal, characteristic
9#21424674C211yesmaximal, characteristic, complemented
10#21424667C211yesmaximal, characteristic, complemented
11#21423994C211yesmaximal, characteristic, complemented
12C4×#14273658C211yesmaximal, characteristic, complemented
13#21423971C211yesmaximal, characteristic, complemented
14#21423921C211yesmaximal, characteristic
15C59×#2354C211yesmaximal, characteristic
16C2×#14273719C2211yescharacteristic
17#16837702C418yescomplemented
18C2×#14273660C2211yescharacteristic
19C2×#14273658C2211yescharacteristic
20C2×#14273658C2211yescharacteristic
21C2×#14273658C2211yescharacteristic
22C2×C4⋊(C59⋊C8)C2211yescharacteristic
23C2×#14273648C2211yescharacteristic
24C2×#14273648C2211yescharacteristic
25C2×#14273648C2211yescharacteristic
26C2×#14273648C2211yescharacteristic
27C2×#14273638C2211yescharacteristic
28C2×#14273638C2211yescharacteristic
29C2×C4×C59⋊C8C2211yescharacteristic
30#16837300C2211yescharacteristic
31#16837300C2211yescharacteristic
32#16837297—21no—
33#16837284—21no—
34#16837283—21no—
35C4×C4⋊Dic59C2211yescharacteristic
36C4×C4⋊Dic59C2211yescharacteristic
37C4×C4⋊Dic59C2211yescharacteristic
38#16837253—21no—
39#16837164C2211yescharacteristic
40#16837164C2211yescharacteristic
41#16837164C2211yescharacteristic
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837157C2211yescharacteristic
48#16837157C2211yescharacteristic
49C2×C4×D4×C59C2211yescharacteristic
50C2×#117×C59—21no—
51C59×#390C2211yescharacteristic
52C59×#390C2211yescharacteristic
53C59×#388—21no—
54C59×#388—21no—
55C59×#384C2211yescharacteristic
56C59×#380—21no—
57C4×C4⋊C4×C59C2211yescharacteristic
58C4×C4⋊C4×C59C2211yescharacteristic
59C4×C4⋊C4×C59C2211yescharacteristic
60C22×C59⋊C8C2311yescharacteristic
61C22×C59⋊C8C2311yescharacteristic
62C22×C59⋊C8C2311yescharacteristic
63C22×C59⋊C8C2311yescharacteristic
64#14273719—22no—
65#14273719C2×C414yes—
66C2×C4⋊Dic59C2311yescharacteristic
67C2×C4⋊Dic59C2311yescharacteristic
68C2×C4⋊Dic59C2311yescharacteristic
69C2×C4⋊Dic59C2311yescharacteristic
70C2×Dic59⋊C4—21no—
71C2×Dic59⋊C4—21no—
72C2×Dic59⋊C4—21no—
73C2×Dic59⋊C4—21no—
74C2×C4×Dic59—21no—
75C2×C4×Dic59—21no—
76C2×C4×Dic59—21no—
77#14273660C2×C414yes—
78#14273658—22no—
79#14273658—22no—
80#14273658C2×C414yes—
81C4⋊(C59⋊C8)C2×C414yes—
82#14273648—22no—
83#14273648—22no—
84#14273648C2×C414yes—
85#14273648C2×C414yes—
86#14273638—22no—
87#14273638—22no—
88C4×C59⋊C8—22no—
89C59⋊#95—21no—
90C59⋊#95—21no—
91C59⋊#95—21no—
92C59⋊#95—21no—
93C59⋊#95—21no—
94C22×D4×C59C2311yescharacteristic
95C23×C4×C59—21no—
96C4×D4×C59—22no—
97C4×D4×C59C2×C414yes—
98#117×C59—44no—
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×C4⋊C4×C59C2311yescharacteristic
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×C59—21no—
113C2×C42×C59C2311yescharacteristic
114C2×C42×C59C2311yescharacteristic
115C2×C42×C59C2311yescharacteristic
116#95×C59—21no—
117#95×C59—21no—
118#95×C59D411yescharacteristic
119#95×C59D411yescharacteristic
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—24no—
129C22×C4×C59—21no—
130C22×C4×C59—21no—
131C22×C4×C59—42no—
132C22×C4×C59—42no—
133C22×C4×C59C2×D411yescharacteristic
134C22×C4×C59C2×D411yescharacteristic
135C22×C4×C59C2×D411yescharacteristic
136C22×C4×C59C2×D411yescharacteristic
137C22×C4×C59C2×D411yescharacteristic
138C22×C4×C59C2×D411yescharacteristic
139C22×C4×C59C2411yescharacteristic
140C22×Dic59—21no—
141C22×Dic59—21no—
142C22×Dic59—21no—
143C22×Dic59—21no—
144C4⋊C4×C59—22no—
145C4⋊C4×C59—21no—
146C4⋊C4×C59—21no—
147C4⋊C4×C59—21no—
148C4⋊C4×C59—21no—
149C4⋊C4×C59—21no—
150C4⋊C4×C59—41no—
151C4⋊C4×C59—42no—
152C4⋊C4×C59—81no—
153C4⋊C4×C59C22×C412yes—
154C22⋊C4×C59—24no—
155C22⋊C4×C59—42no—
156C22⋊C4×C59—42no—
157C22⋊C4×C59—81no—
158C22⋊C4×C59—81no—
159C42×C59—21no—
160C42×C59—21no—
161C42×C59—21no—
162C42×C59—21no—
163C42×C59—42no—
164C42×C59—42no—
165C42×C59C22×C412yes—
166C42×C59#4112yes—
167C4⋊Dic59—21no—
168C4⋊Dic59—21no—
169C4⋊Dic59—21no—
170C4⋊Dic59—21no—
171C4⋊Dic59—21no—
172C4⋊Dic59C22×C412yes—
173C4⋊Dic59C22×C412yes—
174C4⋊Dic59#4112yes—
175Dic59⋊C4—24no—
176Dic59⋊C4—24no—
177Dic59⋊C4—42no—
178Dic59⋊C4—81no—
179C4×Dic59—42no—
180C4×Dic59—81no—
181C4×Dic59—81no—
182C2×C59⋊C8—21no—
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⋊C8C22×C412yes—
192C2×C59⋊C8C22×C412yes—
193C2×C59⋊C8#4112yes—
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—21no—
211C2×C4×C59—21no—
212C2×C4×C59—22no—
213C2×C4×C59—21no—
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—21no—
221C2×C4×C59—21no—
222C2×C4×C59—48no—
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×C59—41no—
231C2×C4×C59C4×D412yes—
232C2×C4×C59C4×D412yes—
233C2×C4×C59AGL(1,ℤ/8)12yes—
234C2×C4×C59C23×C411yescharacteristic
235C2×C4×C59C2×#4111yescharacteristic
236C2×C4×C59C2×#4111yescharacteristic
237C2×C4×C59C2×#4111yescharacteristic
238C2×C4×C592−1+411yescharacteristic
239C2×C4×C592−1+411yescharacteristic
240C2×Dic59—22no—
241C2×Dic59—22no—
242C2×Dic59—22no—
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—41no—
252C59⋊C8—22no—
253C59⋊C8—22no—
254C59⋊C8—22no—
255C59⋊C8—41no—
256#23072—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#54711yescharacteristic
266C22×C59#54811yescharacteristic
267C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
268C22×C59#57511yescharacteristic
269C22×C59#57711yescharacteristic
270C22×C59#57711yescharacteristic
271C4×C59—22no—
272C4×C59—22no—
273C4×C59—22no—
274C4×C59—41no—
275C4×C59—41no—
276C4×C59—41no—
277C4×C59—42no—
278C4×C59—81no—
279C4×C59—81no—
280C4×C59C4×#4111yescharacteristic
281C4×C59#51911yescharacteristic
282C4×C59#55211yescharacteristic
283C4×C59#55611yescharacteristic
284Dic59—42no—
285Dic59—42no—
286Dic59—42no—
287Dic59—81no—
288C2×#478—591no—
289#2370—591no—
290#2354D5911yescharacteristic
291#1992—591no—
292#1948—591no—
293#1936—591no—
294#1934—591no—
295#1926—591no—
296#1926—591no—
297#1919—591no—
298#1918—591no—
299#1776—591no—
300C4×#106—591no—
301#1771—591no—
302#1762—591no—
303C2×C59—82no—
304C2×C59#236211yescharacteristic
305C2×C59C4×AGL(1,ℤ/8)11yescharacteristic
306C2×C59#295211yescharacteristic
307C2×C59#334511yescharacteristic
308C2×C59#334811yescharacteristic
309C2×C59#336811yescharacteristic
310C2×C59#337611yescharacteristic
311C2×#125—591no—
312C2×C4×D4C2×D5911yescharacteristic
313C2×#117—21no—
314#478—598no—
315C2×#107—591no—
316C2×#106—591no—
317C2×#106—591no—
318C2×#106—591no—
319C2×C4⋊C8—591no—
320C2×#102—591no—
321C2×#102—591no—
322C2×#102—591no—
323C2×#102—591no—
324C2×#97—591no—
325C2×#97—591no—
326C2×C4×C8—591no—
327#390C2×D5911yescharacteristic
328#390C2×D5911yescharacteristic
329#389—591no—
330#389—591no—
331#389—1181no—
332#388—21no—
333#388—21no—
334#387—1181no—
335#387—1181no—
336#384C2×D5911yescharacteristic
337#380—21no—
338C4×C4⋊C4—591no—
339C4×C4⋊C4—591no—
340C4×C4⋊C4—591no—
341C4×C4⋊C4—1181no—
342C4×C4⋊C4C2×D5911yescharacteristic
343C4×C4⋊C4C2×D5911yescharacteristic
344C4×C4⋊C4C2×D5911yescharacteristic
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#2307211yescharacteristic, complemented, Sylow
356C22×D4C22×D5911yescharacteristic
357C23×C4—21no—
358C22×C8—591no—
359C22×C8—591no—
360C22×C8—591no—
361C22×C8—591no—
362#125—594no—
363#125—1182no—
364C4×D4—22no—
365C4×D4C4×D5914yes—
366#117—44no—
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×C4⋊C4C22×D5911yescharacteristic
381C2×C4⋊C4C22×D5911yescharacteristic
382C2×C22⋊C4—21no—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—21no—
385C2×C22⋊C4—21no—
386C2×C22⋊C4—21no—
387C2×C42—21no—
388C2×C42—21no—
389C2×C42—1181no—
390C2×C42—1181no—
391C2×C42—1181no—
392C2×C42C22×D5911yescharacteristic
393C2×C42C22×D5911yescharacteristic
394C2×C42C22×D5911yescharacteristic
395#107—594no—
396#106—594no—
397#106—1182no—
398#106—1182no—
399C4⋊C8—594no—
400#102—594no—
401#102—594no—
402#102—1182no—
403#102—1182no—
404#97—1182no—
405#97—1182no—
406C4×C8—1182no—
407#95—21no—
408#95—21no—
409#95—1181no—
410#95—1181no—
411#95—1181no—
412#95—1181no—
413#95—1181no—
414#95C22⋊D5911yescharacteristic
415#95C22⋊D5911yescharacteristic
416C24—21no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4C2×C4×D5912yes—
423C22×C4—21no—
424C22×C4—24no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—1181no—
430C22×C4—1181no—
431C22×C4—1181no—
432C22×C4—1181no—
433C22×C4C2×C22⋊D5911yescharacteristic
434C22×C4C2×C22⋊D5911yescharacteristic
435C22×C4C2×C22⋊D5911yescharacteristic
436C22×C4C2×C22⋊D5911yescharacteristic
437C22×C4C2×C22⋊D5911yescharacteristic
438C22×C4C2×C22⋊D5911yescharacteristic
439C22×C4C23×D5911yescharacteristic
440C2×C8—592no—
441C2×C8—592no—
442C2×C8—592no—
443C2×C8—1181no—
444C2×C8—1181no—
445C2×C8—1181no—
446C2×C8—1181no—
447C2×C8—1181no—
448C2×C8—1181no—
449C2×C8—1181no—
450C2×C8—1181no—
451C2×C8—1181no—
452C4⋊C4—22no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—41no—
459C4⋊C4—42no—
460C4⋊C4—81no—
461C4⋊C4—592no—
462C4⋊C4—592no—
463C4⋊C4—592no—
464C4⋊C4—1184no—
465C4⋊C4—1184no—
466C4⋊C4—1181no—
467C4⋊C4—1181no—
468C4⋊C4—1181no—
469C4⋊C4—1181no—
470C4⋊C4—1181no—
471C4⋊C4—4721no—
472C4⋊C4—2362no—
473C4⋊C4C2×C4×D5912yes—
474C22⋊C4—24no—
475C22⋊C4—42no—
476C22⋊C4—42no—
477C22⋊C4—81no—
478C22⋊C4—81no—
479C42—21no—
480C42—21no—
481C42—21no—
482C42—21no—
483C42—42no—
484C42—42no—
485C42—4721no—
486C42—4721no—
487C42—2362no—
488C42C2×C4×D5912yes—
489C42#1174568512yes—
490C23—22no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23C22×C22⋊D5911yescharacteristic
498D4—22no—
499D4—22no—
500D4—41no—
501D4—41no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
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—22no—
518C2×C4—48no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—1182no—
528C2×C4—1182no—
529C2×C4—1182no—
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×C4C4×C22⋊D5912yes—
540C2×C4C4×C22⋊D5912yes—
541C2×C4#1427375312yes—
542C2×C4C22×C4×D5911yescharacteristic
543C2×C4#41×D5911yescharacteristic
544C2×C4#41×D5911yescharacteristic
545C2×C4C2×#1174568511yescharacteristic
546C2×C4#1427377411yescharacteristic
547C2×C4#1427377411yescharacteristic
548C8—1182no—
549C8—1182no—
550C8—1182no—
551C8—2361no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22—81no—
556C22—81no—
557C22—81no—
558C22—81no—
559C22C2×C4×C22⋊D5911yescharacteristic
560C22#1683805111yescharacteristic
561C22#1683805911yescharacteristic
562C22C2×#1427375311yescharacteristic
563C22#1683820011yescharacteristic
564C22#1683821011yescharacteristic
565C22#1683821311yescharacteristic
566C4—22no—
567C4—22no—
568C4—22no—
569C4—42no—
570C4—41no—
571C4—41no—
572C4—41no—
573C4—81no—
574C4—81no—
575C4—4721no—
576C4—2362no—
577C4—2362no—
578C4—2362no—
579C4#1683787011yescharacteristic
580C4#1683787911yescharacteristic
581C4#1683809111yescharacteristic
582C4#1683812011yescharacteristic
583C2—82no—
584C2#2142712311yescharacteristic
585C2C4×#1427375311yescharacteristic
586C2#2143297711yescharacteristic
587C2#2143656611yescharacteristic
588C2#2143661111yescharacteristic
589C2#2143682111yescharacteristic
590C2#2143691711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#2354C2
socleC23×C59C22×D4
prime radicalC22×D4×C59C23
cosocleC22×C4×C59C24
hypercenter#2354D59
hypocenterC59#23072
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2354

Complements

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

NH ≅ G/Ntype
2. #21427190C2semidirect
3. #21425064C2semidirect
4. #21424857C2semidirect
9. #21424674C2semidirect
10. #21424667C2semidirect
11. #21423994C2semidirect
12. C4×#14273658C2semidirect
13. #21423971C2semidirect
17. #16837702C4semidirect
355. C59#23072semidirect

Maximal subgroups

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

subgroupindex
C2×#168377022
#214271902
#214250642
#214248572
#214247962
#214247822
#214247202
#214246742
#214246672
#214239942
C4×#142736582
#214239712
#214239212
C59×#23542
#2307259