Finite Group Databasebeta

← #28647551

Subgroups — #28647551

Order 15104.

818 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×#16837812C211yesmaximal, characteristic, complemented
2C2×#16837445C211yesmaximal, characteristic
3#21427798C211yesmaximal, characteristic
4#21425573C211yesmaximal, characteristic, complemented
5#21425567C211yesmaximal, characteristic, complemented
6#21425431C211yesmaximal, characteristic, complemented
7#21425429C211yesmaximal, characteristic, complemented
8#21424831C211yesmaximal, characteristic, complemented
9#21424826C211yesmaximal, characteristic, complemented
10#21424295C211yesmaximal, characteristic
11#21424288C211yesmaximal, characteristic
12#21424252C211yesmaximal, characteristic, complemented
13#21424127C211yesmaximal, characteristic
14#21423880C211yesmaximal, characteristic
15C59×#2441C211yesmaximal, characteristic
16C22×#11745664C2211yescharacteristic
17C2×#14273727C2211yescharacteristic
18C2×#14273704—21no—
19C22×C4⋊Dic59C2211yescharacteristic
20#16837812—24no—
21#16837445—24no—
22C2×#14273660C2211yescharacteristic
23C2×#14273660C2211yescharacteristic
24C2×#14273660C2211yescharacteristic
25C2×#14273660C2211yescharacteristic
26C2×#14273658C2211yescharacteristic
27C2×#14273658C2211yescharacteristic
28C2×#14273658C2211yescharacteristic
29C2×#14273658C2211yescharacteristic
30C2×#14273658C2211yescharacteristic
31C2×#14273658C2211yescharacteristic
32C2×C22⋊(C59⋊C8)C2211yescharacteristic
33C2×#14273638C2211yescharacteristic
34C2×C4×C59⋊C8C2211yescharacteristic
35#16837334—21no—
36#16837323—21no—
37#16837300C2211yescharacteristic
38#16837300C2211yescharacteristic
39#16837277—21no—
40#16837272C2211yescharacteristic
41#16837262C2211yescharacteristic
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837157C2211yescharacteristic
49#16837157C2211yescharacteristic
50C2×#125×C59C2211yescharacteristic
51C2×#117×C59C2211yescharacteristic
52C59×#399—21no—
53C59×#396—21no—
54C59×#389—21no—
55C59×#389C2211yescharacteristic
56C59×#389C2211yescharacteristic
57C59×#387—21no—
58C59×#384C2211yescharacteristic
59C59×#380C2211yescharacteristic
60C4×C22⋊C4×C59C2211yescharacteristic
61C22×Dic118—21no—
62C23×Dic59—21no—
63C2×#11745664—22no—
64C2×#11745664—22no—
65C2×#11745664Q814yescomplemented
66C22×C59⋊C8C2311yescharacteristic
67C22×C59⋊C8C2311yescharacteristic
68C22×C59⋊C8C2311yescharacteristic
69C22×C59⋊C8C2311yescharacteristic
70#14273727—22no—
71#14273727—41no—
72#14273704—44no—
73C2×C4⋊Dic59—22no—
74C2×C4⋊Dic59—21no—
75C2×C4⋊Dic59—21no—
76C2×C4⋊Dic59—21no—
77C2×C4⋊Dic59—21no—
78C2×C4⋊Dic59—41no—
79C2×C4⋊Dic59C2311yescharacteristic
80C2×C4⋊Dic59C2311yescharacteristic
81C2×C4⋊Dic59C2311yescharacteristic
82C2×C4⋊Dic59C2311yescharacteristic
83C2×Dic59⋊C4—21no—
84C2×Dic59⋊C4—21no—
85C2×C22⋊Dic59—21no—
86#14273660—22no—
87#14273660—22no—
88#14273660—22no—
89#14273660D414yes—
90#14273658—22no—
91#14273658—22no—
92#14273658—22no—
93#14273658—22no—
94#14273658—41no—
95#14273658—41no—
96C22⋊(C59⋊C8)Q814yescomplemented
97#14273638—22no—
98C4×C59⋊C8—22no—
99C59⋊#95—21no—
100C59⋊#95—21no—
101C59⋊#95—21no—
102C23×C4×C59C2311yescharacteristic
103#125×C59—22no—
104#125×C59—41no—
105#117×C59—22no—
106#117×C59—41no—
107C2×C4⋊C4×C59—21no—
108C2×C4⋊C4×C59—21no—
109C2×C4⋊C4×C59—21no—
110C2×C4⋊C4×C59—21no—
111C2×C4⋊C4×C59—21no—
112C2×C4⋊C4×C59—21no—
113C2×C4⋊C4×C59C2311yescharacteristic
114C2×C4⋊C4×C59C2311yescharacteristic
115C2×C4⋊C4×C59C2311yescharacteristic
116C2×C4⋊C4×C59C2311yescharacteristic
117C2×C22⋊C4×C59—21no—
118C2×C22⋊C4×C59—21no—
119C2×C22⋊C4×C59D411yescharacteristic
120C2×C22⋊C4×C59D411yescharacteristic
121C2×C42×C59—21no—
122C2×C42×C59C2311yescharacteristic
123C2×C42×C59C2311yescharacteristic
124#95×C59—21no—
125#95×C59—21no—
126#95×C59—21no—
127#95×C59D411yescharacteristic
128#95×C59D411yescharacteristic
129C24×C59C2×D411yescharacteristic
130C22×C4×C59—21no—
131C22×C4×C59—21no—
132C22×C4×C59—21no—
133C22×C4×C59—21no—
134C22×C4×C59—21no—
135C22×C4×C59—21no—
136C22×C4×C59—21no—
137C22×C4×C59—21no—
138C22×C4×C59C2×D411yescharacteristic
139C22×C4×C59C2×D411yescharacteristic
140C22×C4×C59C2×D411yescharacteristic
141C22×C4×C59C2×D411yescharacteristic
142C22×C4×C59C2×D411yescharacteristic
143C22×C4×C59C2×Q812yes—
144C22×C4×C59C2×Q812yes—
145C22×C4×C59C2411yescharacteristic
146C22×Dic59—21no—
147C22×Dic59—21no—
148C22×Dic59—21no—
149C22×Dic59—21no—
150C22×Dic59—42no—
151C22×Dic59—42no—
152C22×Dic59—81no—
153C2×Dic118—42no—
154C2×Dic118—42no—
155C2×Dic118—81no—
156C4⋊C4×C59—21no—
157C4⋊C4×C59—21no—
158C4⋊C4×C59—21no—
159C4⋊C4×C59—21no—
160C4⋊C4×C59—21no—
161C4⋊C4×C59—21no—
162C4⋊C4×C59—21no—
163C4⋊C4×C59—42no—
164C4⋊C4×C59—42no—
165C4⋊C4×C59—42no—
166C4⋊C4×C59—42no—
167C4⋊C4×C59—81no—
168C4⋊C4×C59—81no—
169C4⋊C4×C59C2×D412yes—
170C22⋊C4×C59—22no—
171C22⋊C4×C59—22no—
172C22⋊C4×C59—42no—
173C22⋊C4×C59—81no—
174C42×C59—21no—
175C42×C59—21no—
176C42×C59—21no—
177C42×C59—21no—
178C42×C59—42no—
179C22⋊Dic59—42no—
180C4⋊Dic59—21no—
181C4⋊Dic59—21no—
182C4⋊Dic59—21no—
183C4⋊Dic59—21no—
184C4⋊Dic59—21no—
185C4⋊Dic59—21no—
186C4⋊Dic59—21no—
187C4⋊Dic59—42no—
188C4⋊Dic59—42no—
189C4⋊Dic59—42no—
190C4⋊Dic59—81no—
191C4⋊Dic59C2×D412yes—
192Dic59⋊C4—42no—
193Dic59⋊C4—42no—
194#11745664—28no—
195C2×C59⋊C8—21no—
196C2×C59⋊C8—21no—
197C2×C59⋊C8—21no—
198C2×C59⋊C8—21no—
199C2×C59⋊C8—21no—
200C2×C59⋊C8—21no—
201C2×C59⋊C8C2×D412yes—
202C2×C59⋊C8C2×Q812yes—
203C2×C59⋊C8C2×Q812yes—
204C2×C59⋊C8C2×Q812yes—
205C2×C59⋊C8C2×Q812yes—
206C2×C59⋊C8#4112yes—
207C23×C59—21no—
208C23×C59—21no—
209C23×C59—21no—
210C23×C59—21no—
211C23×C59—21no—
212C23×C59—21no—
213C23×C59C4⋊Q812yes—
214C23×C59C22×D411yescharacteristic
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—22no—
222C2×C4×C59—22no—
223C2×C4×C59—22no—
224C2×C4×C59—22no—
225C2×C4×C59—21no—
226C2×C4×C59—21no—
227C2×C4×C59—22no—
228C2×C4×C59—21no—
229C2×C4×C59—21no—
230C2×C4×C59—21no—
231C2×C4×C59—21no—
232C2×C4×C59—41no—
233C2×C4×C59—41no—
234C2×C4×C59—41no—
235C2×C4×C59—41no—
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×C59C4⋊Q812yes—
244C2×C4×C59AGL(1,ℤ/8)12yes—
245C2×C4×C59#13712yes—
246C2×C4×C59C22×D411yescharacteristic
247C2×C4×C59C22×Q811yescharacteristic
248C2×C4×C59C22×Q811yescharacteristic
249C2×C4×C59C2×#4111yescharacteristic
250C2×C4×C59H(5,𝔽2)11yescharacteristic
251C2×C4×C592−1+411yescharacteristic
252C2×Dic59—22no—
253C2×Dic59—41no—
254C2×Dic59—41no—
255C2×Dic59—41no—
256C2×Dic59—41no—
257C2×Dic59—41no—
258C2×Dic59—41no—
259C2×Dic59—41no—
260C2×Dic59—41no—
261C2×Dic59—41no—
262C2×Dic59—41no—
263C2×Dic59—41no—
264C2×Dic59—84no—
265Dic118—84no—
266C59⋊C8—22no—
267C59⋊C8—22no—
268C59⋊C8—22no—
269C59⋊C8C22⋊Q814yescomplemented
270#26078—591nomaximal, Sylow
271C22×C59—22no—
272C22×C59—22no—
273C22×C59—22no—
274C22×C59—41no—
275C22×C59—41no—
276C22×C59—41no—
277C22×C59—41no—
278C22×C59C2×C4⋊Q811yescharacteristic
279C22×C59D4211yescharacteristic
280C22×C59#54711yescharacteristic
281C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
282C22×C59C2×#13711yescharacteristic
283C22×C59#57711yescharacteristic
284C22×C59#57711yescharacteristic
285C4×C59—22no—
286C4×C59—22no—
287C4×C59—22no—
288C4×C59—41no—
289C4×C59—42no—
290C4×C59—41no—
291C4×C59—81no—
292C4×C59—81no—
293C4×C59—81no—
294C4×C59C2×C22⋊Q811yescharacteristic
295C4×C59#54411yescharacteristic
296C4×C59D4×Q811yescharacteristic
297C4×C59#55411yescharacteristic
298Dic59—42no—
299Dic59—81no—
300Dic59—81no—
301Dic59—81no—
302C2×#501—591no—
303C2×#428—591no—
304#2441D5911yescharacteristic
305#2440—591no—
306#2084—591no—
307#2083—591no—
308#2059—591no—
309#2058—591no—
310#1943—591no—
311#1942—591no—
312#1854—591no—
313#1853—591no—
314#1844—591no—
315#1809—591no—
316#1753—591no—
317C2×C59—42no—
318C2×C59#244011yescharacteristic
319C2×C59#316311yescharacteristic
320C2×C59#316511yescharacteristic
321C2×C59#327211yescharacteristic
322C2×C59#327811yescharacteristic
323C2×C59#334411yescharacteristic
324C2×C59#334511yescharacteristic
325C22×M16—591no—
326C2×C4⋊Q8—591no—
327C2×#125C2×D5911yescharacteristic
328C2×C22⋊Q8—1181no—
329C2×#117C2×D5911yescharacteristic
330C22×C4⋊C4—591no—
331#501—1184no—
332#428—1184no—
333C2×#107—591no—
334C2×#107—591no—
335C2×#107—591no—
336C2×#107—591no—
337C2×#106—591no—
338C2×#106—591no—
339C2×#106—591no—
340C2×#106—591no—
341C2×#106—591no—
342C2×#106—591no—
343C2×C22⋊C8—591no—
344C2×#97—591no—
345C2×C4×C8—591no—
346#399—21no—
347#398—1181no—
348#396—21no—
349#395—1181no—
350#389—21no—
351#389—591no—
352#389—591no—
353#389C2×D5911yescharacteristic
354#389C2×D5911yescharacteristic
355#387—21no—
356#385—1181no—
357#384—591no—
358#384C2×D5911yescharacteristic
359#380—591no—
360#380C2×D5911yescharacteristic
361C4×C22⋊C4C2×D5911yescharacteristic
362#340—591no—
363#340—591no—
364#340—591no—
365#340—591no—
366#340—591no—
367#340—591no—
368#336—591no—
369#336—591no—
370C59#2607811yescharacteristic, complemented, Sylow
371C22×Q8—1181no—
372C23×C4—1181no—
373C23×C4C22×D5911yescharacteristic
374C2×M16—594no—
375C2×M16—1182no—
376C2×M16—1182no—
377C22×C8—591no—
378C22×C8—591no—
379C22×C8—591no—
380C22×C8—591no—
381C4⋊Q8—1182no—
382C4⋊Q8—2361no—
383#125—22no—
384#125—41no—
385C22⋊Q8—2364no—
386#117—22no—
387#117—41no—
388C2×C4⋊C4—21no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—591no—
395C2×C4⋊C4—591no—
396C2×C4⋊C4—591no—
397C2×C4⋊C4—591no—
398C2×C4⋊C4—1182no—
399C2×C4⋊C4—1181no—
400C2×C4⋊C4—1181no—
401C2×C4⋊C4—1181no—
402C2×C4⋊C4—1181no—
403C2×C4⋊C4—1181no—
404C2×C4⋊C4—1181no—
405C2×C4⋊C4—2361no—
406C2×C4⋊C4C22×D5911yescharacteristic
407C2×C4⋊C4C22×D5911yescharacteristic
408C2×C4⋊C4C22×D5911yescharacteristic
409C2×C4⋊C4C22×D5911yescharacteristic
410C2×C22⋊C4—21no—
411C2×C22⋊C4—21no—
412C2×C22⋊C4—1181no—
413C2×C22⋊C4C22⋊D5911yescharacteristic
414C2×C22⋊C4C22⋊D5911yescharacteristic
415C2×C42—21no—
416C2×C42C22×D5911yescharacteristic
417C2×C42C22×D5911yescharacteristic
418#107—594no—
419#107—1182no—
420#107—1182no—
421#107—1182no—
422#106—1182no—
423#106—1182no—
424#106—1182no—
425#106—1182no—
426#106—2361no—
427#106—2361no—
428C22⋊C8—594no—
429#97—1182no—
430C4×C8—1182no—
431#95—21no—
432#95—21no—
433#95—21no—
434#95—1181no—
435#95—1181no—
436#95—1181no—
437#95C22⋊D5911yescharacteristic
438#95C22⋊D5911yescharacteristic
439C24C2×C22⋊D5911yescharacteristic
440C2×Q8—4721no—
441C2×Q8—2362no—
442C2×Q8—2362no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—1181no—
452C22×C4—1181no—
453C22×C4—1181no—
454C22×C4—1181no—
455C22×C4—4721no—
456C22×C4—2362no—
457C22×C4—2362no—
458C22×C4Q8×D5912yes—
459C22×C4Q8×D5912yes—
460C22×C4C2×C22⋊D5911yescharacteristic
461C22×C4C2×C22⋊D5911yescharacteristic
462C22×C4C2×C22⋊D5911yescharacteristic
463C22×C4C2×C22⋊D5911yescharacteristic
464C22×C4C2×C22⋊D5911yescharacteristic
465C22×C4C23×D5911yescharacteristic
466M16—1188no—
467C2×C8—592no—
468C2×C8—592no—
469C2×C8—592no—
470C2×C8—592no—
471C2×C8—592no—
472C2×C8—592no—
473C2×C8—1181no—
474C2×C8—1181no—
475C2×C8—1181no—
476C2×C8—1181no—
477C2×C8—1181no—
478C2×C8—1181no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—81no—
491C4⋊C4—81no—
492C4⋊C4—592no—
493C4⋊C4—1181no—
494C4⋊C4—1181no—
495C4⋊C4—1181no—
496C4⋊C4—1181no—
497C4⋊C4—1181no—
498C4⋊C4—1181no—
499C4⋊C4—1181no—
500C4⋊C4—4721no—
501C4⋊C4—2362no—
502C4⋊C4—2362no—
503C4⋊C4—2362no—
504C4⋊C4—2362no—
505C4⋊C4—2362no—
506C4⋊C4D4×D5912yes—
507C22⋊C4—22no—
508C22⋊C4—22no—
509C22⋊C4—42no—
510C22⋊C4—81no—
511C22⋊C4—2362no—
512C42—21no—
513C42—21no—
514C42—21no—
515C42—21no—
516C42—42no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23#1427372512yes—
524C23C22×C22⋊D5911yescharacteristic
525Q8—4724no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—1182no—
555C2×C4—4724no—
556C2×C4—2361no—
557C2×C4—2361no—
558C2×C4—2361no—
559C2×C4—2361no—
560C2×C4—2361no—
561C2×C4—2361no—
562C2×C4—2361no—
563C2×C4—2361no—
564C2×C4—2361no—
565C2×C4—2361no—
566C2×C4—2361no—
567C2×C4#1427372512yes—
568C2×C4#1427375212yes—
569C2×C4#1427375712yes—
570C2×C4C2×D4×D5911yescharacteristic
571C2×C4C2×Q8×D5911yescharacteristic
572C2×C4C2×Q8×D5911yescharacteristic
573C2×C4#41×D5911yescharacteristic
574C2×C4#1427377211yescharacteristic
575C2×C4#1427377411yescharacteristic
576C8—594no—
577C8—1182no—
578C8—1182no—
579C8—1182no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—41no—
584C22—41no—
585C22—41no—
586C22—41no—
587C22C2×#1427372511yescharacteristic
588C22D4×C22⋊D5911yescharacteristic
589C22#1683805111yescharacteristic
590C22C2×#1427375211yescharacteristic
591C22C2×#1427375711yescharacteristic
592C22#1683821011yescharacteristic
593C22#1683821311yescharacteristic
594C4—22no—
595C4—22no—
596C4—22no—
597C4—41no—
598C4—42no—
599C4—41no—
600C4—81no—
601C4—81no—
602C4—81no—
603C4—4721no—
604C4—4721no—
605C4—4721no—
606C4—2362no—
607C4C22⋊Q8×D5911yescharacteristic
608C4#1683803111yescharacteristic
609C4#1683806511yescharacteristic
610C4#1683810611yescharacteristic
611C2—42no—
612C2#2142779511yescharacteristic
613C2#2143462211yescharacteristic
614C2#2143464311yescharacteristic
615C2#2143569611yescharacteristic
616C2#2143576211yescharacteristic
617C2#2143655111yescharacteristic
618C2#2143656011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2441C2
socleC23×C59C22×D4
prime radicalC24×C59C2×D4
hypercenter#2441D59
hypocenterC59#26078
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2441

Complements

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

NH ≅ G/Ntype
1. C2×#16837812C2semidirect
4. #21425573C2semidirect
5. #21425567C2semidirect
6. #21425431C2semidirect
7. #21425429C2semidirect
8. #21424831C2semidirect
9. #21424826C2semidirect
12. #21424252C2semidirect
65. C2×#11745664Q8semidirect
96. C22⋊(C59⋊C8)Q8semidirect
269. C59⋊C8C22⋊Q8semidirect
370. C59#26078semidirect

Maximal subgroups

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

subgroupindex
C2×#168378122
C2×#168374452
#214277982
#214255732
#214255672
#214254312
#214254292
#214248312
#214248262
#214242952
#214242882
#214242522
#214241272
#214238802
C59×#24412
#2607859