Finite Group Databasebeta

← #28612029

Subgroups — #28612029

Order 15104.

778 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×#16837682C211yesmaximal, characteristic
2#21427185C211yesmaximal, characteristic, complemented
3#21425068C211yesmaximal, characteristic, complemented
4#21424862C211yesmaximal, characteristic, complemented
5#21424796C211yesmaximal, characteristic
6#21424782C211yesmaximal, characteristic
7#21424720C211yesmaximal, characteristic
8#21424713C211yesmaximal, characteristic
9#21424688C211yesmaximal, characteristic, complemented
10#21424667C211yesmaximal, characteristic, complemented
11#21423994C211yesmaximal, characteristic, complemented
12C4×#14273660C211yesmaximal, characteristic, complemented
13#21423971C211yesmaximal, characteristic, complemented
14#21423921C211yesmaximal, characteristic
15C59×#2357C211yesmaximal, characteristic
16C2×#14273727C2211yescharacteristic
17#16837682C418yescomplemented
18C2×#14273660C2211yescharacteristic
19C2×#14273660C2211yescharacteristic
20C2×#14273660C2211yescharacteristic
21C2×#14273658C2211yescharacteristic
22C2×C4⋊(C59⋊C8)C2211yescharacteristic
23C2×#14273648C2211yescharacteristic
24C2×#14273648C2211yescharacteristic
25C2×#14273648C2211yescharacteristic
26C2×#14273648C2211yescharacteristic
27C2×#14273638C2211yescharacteristic
28C2×#14273638C2211yescharacteristic
29C2×C4×C59⋊C8C2211yescharacteristic
30#16837306—21no—
31#16837306—21no—
32#16837300C2211yescharacteristic
33#16837300C2211yescharacteristic
34#16837298—21no—
35C4⋊C4×Dic59—21no—
36C4×C4⋊Dic59C2211yescharacteristic
37C4×C4⋊Dic59C2211yescharacteristic
38C4×C4⋊Dic59C2211yescharacteristic
39#16837164C2211yescharacteristic
40#16837164C2211yescharacteristic
41#16837164C2211yescharacteristic
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837157C2211yescharacteristic
48#16837157C2211yescharacteristic
49C2×C4×D4×C59C2211yescharacteristic
50C59×#390C2211yescharacteristic
51C59×#390C2211yescharacteristic
52C59×#388—21no—
53C59×#388—21no—
54C59×#382C2211yescharacteristic
55C59×#381—21no—
56C4×C4⋊C4×C59C2211yescharacteristic
57C4×C4⋊C4×C59C2211yescharacteristic
58C4×C4⋊C4×C59C2211yescharacteristic
59C4×C22⋊C4×C59—21no—
60C22×Dic118—21no—
61C22×C59⋊C8C2311yescharacteristic
62C22×C59⋊C8C2311yescharacteristic
63C22×C59⋊C8C2311yescharacteristic
64C22×C59⋊C8C2311yescharacteristic
65#14273727—22no—
66#14273727C2×C414yes—
67C2×C4⋊Dic59—21no—
68C2×C4⋊Dic59C2311yescharacteristic
69C2×C4⋊Dic59C2311yescharacteristic
70C2×C4⋊Dic59C2311yescharacteristic
71C2×C4⋊Dic59C2311yescharacteristic
72C2×Dic59⋊C4—21no—
73C2×Dic59⋊C4—21no—
74C2×C4×Dic59—21no—
75C2×C4×Dic59—21no—
76C2×C4×Dic59—21no—
77#14273660—22no—
78#14273660C2×C414yes—
79#14273660C2×C414yes—
80#14273658—22no—
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—
98C2×C4⋊C4×C59D411yescharacteristic
99C2×C4⋊C4×C59D411yescharacteristic
100C2×C4⋊C4×C59C2311yescharacteristic
101C2×C4⋊C4×C59C2311yescharacteristic
102C2×C4⋊C4×C59C2311yescharacteristic
103C2×C22⋊C4×C59—21no—
104C2×C22⋊C4×C59—21no—
105C2×C22⋊C4×C59—21no—
106C2×C22⋊C4×C59—21no—
107C2×C22⋊C4×C59—21no—
108C2×C42×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×C59—21no—
117#95×C59D411yescharacteristic
118#95×C59D411yescharacteristic
119C24×C59—21no—
120C2×D4×C59—21no—
121C2×D4×C59—21no—
122C2×D4×C59—21no—
123C2×D4×C59—21no—
124C2×D4×C59—21no—
125C2×D4×C59C22×C412yes—
126C22×C4×C59—21no—
127C22×C4×C59—24no—
128C22×C4×C59—21no—
129C22×C4×C59—21no—
130C22×C4×C59—81no—
131C22×C4×C59—81no—
132C22×C4×C59C2×D411yescharacteristic
133C22×C4×C59C2×D411yescharacteristic
134C22×C4×C59C2×D411yescharacteristic
135C22×C4×C59C2×D411yescharacteristic
136C22×C4×C59C2×D411yescharacteristic
137C22×C4×C59C2×D411yescharacteristic
138C22×C4×C59C2411yescharacteristic
139C22×Dic59—21no—
140C22×Dic59—21no—
141C22×Dic59—21no—
142C22×Dic59—21no—
143C2×Dic118—24no—
144C2×Dic118—81no—
145C2×Dic118—81no—
146C4⋊C4×C59—22no—
147C4⋊C4×C59—21no—
148C4⋊C4×C59—21no—
149C4⋊C4×C59—21no—
150C4⋊C4×C59—21no—
151C4⋊C4×C59—21no—
152C4⋊C4×C59—41no—
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—81no—
164C42×C59—81no—
165C42×C59C22×C412yes—
166C42×C59#4112yes—
167C4⋊Dic59—24no—
168C4⋊Dic59—21no—
169C4⋊Dic59—21no—
170C4⋊Dic59—21no—
171C4⋊Dic59—21no—
172C4⋊Dic59—21no—
173C4⋊Dic59C22×C412yes—
174C4⋊Dic59C22×C412yes—
175C4⋊Dic59#4112yes—
176Dic59⋊C4—81no—
177Dic59⋊C4—81no—
178C4×Dic59—42no—
179C4×Dic59—81no—
180C4×Dic59—81no—
181C2×C59⋊C8—21no—
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⋊C8C22×C412yes—
191C2×C59⋊C8C22×C412yes—
192C2×C59⋊C8#4112yes—
193C23×C59—22no—
194C23×C59—41no—
195C23×C59—41no—
196C23×C59—41no—
197C23×C59—41no—
198C23×C59—41no—
199C23×C59—41no—
200C23×C59C22×D411yescharacteristic
201D4×C59—22no—
202D4×C59—22no—
203D4×C59—41no—
204D4×C59—41no—
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—22no—
212C2×C4×C59—21no—
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—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—84no—
230C2×C4×C59C4×D412yes—
231C2×C4×C59C4×D412yes—
232C2×C4×C59#13712yes—
233C2×C4×C59C23×C411yescharacteristic
234C2×C4×C59C2×#4111yescharacteristic
235C2×C4×C59C2×#4111yescharacteristic
236C2×C4×C59C2×#4111yescharacteristic
237C2×C4×C592−1+411yescharacteristic
238C2×C4×C592−1+411yescharacteristic
239C2×Dic59—22no—
240C2×Dic59—22no—
241C2×Dic59—22no—
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—
251Dic118—84no—
252C59⋊C8—22no—
253C59⋊C8—22no—
254C59⋊C8—22no—
255C59⋊C8—41no—
256#23079—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×#13711yescharacteristic
268C22×C59#57511yescharacteristic
269C22×C59#57611yescharacteristic
270C22×C59#57611yescharacteristic
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×#474—591no—
289#2369—591no—
290#2357D5911yescharacteristic
291#1993—591no—
292#1949—591no—
293#1936—591no—
294#1934—591no—
295#1926—591no—
296#1925—591no—
297#1921—591no—
298#1918—591no—
299#1776—591no—
300C4×#107—591no—
301#1771—591no—
302#1762—591no—
303C2×C59—82no—
304C2×C59#236211yescharacteristic
305C2×C59C4×#13711yescharacteristic
306C2×C59#294811yescharacteristic
307C2×C59#334611yescharacteristic
308C2×C59#334711yescharacteristic
309C2×C59#337011yescharacteristic
310C2×C59#337511yescharacteristic
311C2×C4⋊Q8—591no—
312C2×C4×D4C2×D5911yescharacteristic
313#474—598no—
314C2×#107—591no—
315C2×#107—591no—
316C2×#107—591no—
317C2×#106—591no—
318C2×C4⋊C8—591no—
319C2×#102—591no—
320C2×#102—591no—
321C2×#102—591no—
322C2×#102—591no—
323C2×#97—591no—
324C2×#97—591no—
325C2×C4×C8—591no—
326#391—1181no—
327#391—1181no—
328#390C2×D5911yescharacteristic
329#390C2×D5911yescharacteristic
330#389—591no—
331#389—591no—
332#389—1181no—
333#388—21no—
334#388—21no—
335#382C2×D5911yescharacteristic
336#381—21no—
337C4×C4⋊C4—591no—
338C4×C4⋊C4—591no—
339C4×C4⋊C4—591no—
340C4×C4⋊C4—1181no—
341C4×C4⋊C4C2×D5911yescharacteristic
342C4×C4⋊C4C2×D5911yescharacteristic
343C4×C4⋊C4C2×D5911yescharacteristic
344C4×C22⋊C4—21no—
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#2307911yescharacteristic, complemented, Sylow
356C22×Q8—1181no—
357C22×D4C22×D5911yescharacteristic
358C23×C4—21no—
359C22×C8—591no—
360C22×C8—591no—
361C22×C8—591no—
362C22×C8—591no—
363C4⋊Q8—594no—
364C4⋊Q8—1182no—
365C4×D4—22no—
366C4×D4C4×D5914yes—
367C2×C4⋊C4—591no—
368C2×C4⋊C4—591no—
369C2×C4⋊C4—591no—
370C2×C4⋊C4—591no—
371C2×C4⋊C4—1181no—
372C2×C4⋊C4—1181no—
373C2×C4⋊C4—1181no—
374C2×C4⋊C4C22⋊D5911yescharacteristic
375C2×C4⋊C4C22⋊D5911yescharacteristic
376C2×C4⋊C4C22×D5911yescharacteristic
377C2×C4⋊C4C22×D5911yescharacteristic
378C2×C4⋊C4C22×D5911yescharacteristic
379C2×C22⋊C4—21no—
380C2×C22⋊C4—21no—
381C2×C22⋊C4—21no—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—21no—
384C2×C42—21no—
385C2×C42—21no—
386C2×C42—1181no—
387C2×C42—1181no—
388C2×C42—1181no—
389C2×C42C22×D5911yescharacteristic
390C2×C42C22×D5911yescharacteristic
391C2×C42C22×D5911yescharacteristic
392#107—594no—
393#107—594no—
394#107—1182no—
395#106—1182no—
396C4⋊C8—594no—
397#102—594no—
398#102—594no—
399#102—1182no—
400#102—1182no—
401#97—1182no—
402#97—1182no—
403C4×C8—1182no—
404#95—21no—
405#95—21no—
406#95—21no—
407#95—21no—
408#95—1181no—
409#95—1181no—
410#95—1181no—
411#95—1181no—
412#95—1181no—
413#95C22⋊D5911yescharacteristic
414#95C22⋊D5911yescharacteristic
415C24—21no—
416C2×Q8—1184no—
417C2×Q8—4721no—
418C2×Q8—4721no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4C2×C4×D5912yes—
425C22×C4—21no—
426C22×C4—24no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—81no—
430C22×C4—81no—
431C22×C4—1181no—
432C22×C4—1181no—
433C22×C4—1181no—
434C22×C4—1181no—
435C22×C4C2×C22⋊D5911yescharacteristic
436C22×C4C2×C22⋊D5911yescharacteristic
437C22×C4C2×C22⋊D5911yescharacteristic
438C22×C4C2×C22⋊D5911yescharacteristic
439C22×C4C2×C22⋊D5911yescharacteristic
440C22×C4C2×C22⋊D5911yescharacteristic
441C22×C4C23×D5911yescharacteristic
442C2×C8—592no—
443C2×C8—592no—
444C2×C8—592no—
445C2×C8—1181no—
446C2×C8—1181no—
447C2×C8—1181no—
448C2×C8—1181no—
449C2×C8—1181no—
450C2×C8—1181no—
451C2×C8—1181no—
452C2×C8—1181no—
453C2×C8—1181no—
454C4⋊C4—22no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—41no—
461C4⋊C4—592no—
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—4721no—
471C4⋊C4—4721no—
472C4⋊C4C2×C4×D5912yes—
473C22⋊C4—24no—
474C22⋊C4—42no—
475C22⋊C4—42no—
476C22⋊C4—81no—
477C22⋊C4—81no—
478C42—21no—
479C42—21no—
480C42—21no—
481C42—21no—
482C42—81no—
483C42—81no—
484C42—4721no—
485C42—4721no—
486C42—2362no—
487C42C2×C4×D5912yes—
488C42#1174568512yes—
489C23—22no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23C22×C22⋊D5911yescharacteristic
497Q8—4724no—
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—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—84no—
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#1427375712yes—
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×#1427375711yescharacteristic
563C22#1683820011yescharacteristic
564C22#1683820511yescharacteristic
565C22#1683820511yescharacteristic
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×#1427375711yescharacteristic
586C2#2143294611yescharacteristic
587C2#2143658111yescharacteristic
588C2#2143659011yescharacteristic
589C2#2143684311yescharacteristic
590C2#2143690611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#2357C2
socleC23×C59C22×D4
prime radicalC22×D4×C59C23
cosocleC22×C4×C59C24
hypercenter#2357D59
hypocenterC59#23079
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2357

Complements

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

NH ≅ G/Ntype
2. #21427185C2semidirect
3. #21425068C2semidirect
4. #21424862C2semidirect
9. #21424688C2semidirect
10. #21424667C2semidirect
11. #21423994C2semidirect
12. C4×#14273660C2semidirect
13. #21423971C2semidirect
17. #16837682C4semidirect
355. C59#23079semidirect

Maximal subgroups

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

subgroupindex
C2×#168376822
#214271852
#214250682
#214248622
#214247962
#214247822
#214247202
#214247132
#214246882
#214246672
#214239942
C4×#142736602
#214239712
#214239212
C59×#23572
#2307959