Finite Group Databasebeta

← #28586102

Subgroups — #28586102

Order 15104.

2158 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
1C22×#14273735C211yesmaximal, characteristic
2C2×#16837883C211yesmaximal, characteristic
3C2×#16837882C211yesmaximal, characteristic
4#21423816C211yesmaximal, characteristic, complemented
5#21423814C212yesmaximal, complemented
6#21423814C212yesmaximal, complemented
7#21423812C211yesmaximal, characteristic, complemented
8#21423811C212yesmaximal, complemented
9C2×#16837164C211yesmaximal, characteristic, complemented
10#340×D59C212yesmaximal, complemented
11C59×#1736C211yesmaximal, characteristic, complemented
12C22×#11745685C2211yescharacteristic
13C2×#14273735—24no—
14C2×#14273735—24no—
15C2×#14273735C418yescomplemented
16C22×#11745659C2212yes—
17C23×C59⋊C8C2211yescharacteristic
18C22×C8×D59C2212yes—
19#16837883—24no—
20#16837883C418yescomplemented
21#16837882—24no—
22#16837882C418yescomplemented
23C2×C4×Dic118—21no—
24C2×#14273692—21no—
25C2×C4×D236—21no—
26C2×#14273689—21no—
27C2×C4×C22⋊D59—22no—
28C2×(C22⋊D59)⋊C4—22no—
29C2×#14273686C2211yescharacteristic
30C2×#14273685C2212yes—
31C2×#14273685C2212yes—
32C22×C4⋊Dic59C2211yescharacteristic
33C2×C4⋊C4×D59C2212yes—
34C2×C4⋊C4×D59C2212yes—
35#16837165C2212yes—
36#16837165C2212yes—
37#16837164C2211yescharacteristic
38#16837164C2212yes—
39#16837164C2212yes—
40#16837164C2211yescharacteristic
41#16837164C2211yescharacteristic
42#16837164C2211yescharacteristic
43C22×M16×C59C2211yescharacteristic
44C2×#117×C59C2211yescharacteristic
45C22×C4⋊C4×C59C2211yescharacteristic
46C59×#340C2212yes—
47C59×#340C2212yes—
48C2×#11745685C2×C414yes—
49C2×#11745685C2×C414yes—
50C2×#11745685C2×C414yes—
51C2×#11745685D414yes—
52C2×#11745685D414yes—
53C2×#11745685Q814yes—
54C22×Dic118D411yescharacteristic
55C22×D236D411yescharacteristic
56C22×C22⋊D59D412yes—
57C23×Dic59—21no—
58C22×C4×D59—22no—
59C22×C4×D59—22no—
60C22×C4×D59C2312yes—
61#14273735—232no—
62C2×#11745659—24no—
63C2×#11745659—24no—
64C2×#11745659C2×C418yescomplemented
65C22×C59⋊C8—22no—
66C22×C59⋊C8—22no—
67C22×C59⋊C8C2×C414yescomplemented
68C22×C59⋊C8C2311yescharacteristic
69C22×C59⋊C8C2311yescharacteristic
70C2×C8×D59—24no—
71C2×C8×D59—24no—
72C2×C8×D59C2×C418yescomplemented
73C4×Dic118—28no—
74#14273692—28no—
75C4×D236—28no—
76#14273689—28no—
77C4×C22⋊D59—216no—
78(C22⋊D59)⋊C4—216no—
79#14273686—22no—
80#14273686C2×C414yescomplemented
81#14273685—24no—
82#14273685—24no—
83#14273685C2×C418yescomplemented
84#14273685C2×C418yescomplemented
85C2×C4⋊Dic59—21no—
86C2×C4⋊Dic59—22no—
87C2×C4⋊Dic59C2×C414yescomplemented
88C2×C4⋊Dic59C2311yescharacteristic
89C2×C4⋊Dic59C2312yes—
90C2×C4⋊Dic59C2311yescharacteristic
91C2×Dic59⋊C4—22no—
92C2×Dic59⋊C4—22no—
93C4⋊C4×D59—24no—
94C4⋊C4×D59—24no—
95C4⋊C4×D59C2×C418yescomplemented
96C4⋊C4×D59C2×C418yescomplemented
97C2×C22⋊Dic59—21no—
98C2×(C2×D59)⋊C4—22no—
99C2×(C2×D59)⋊C4—22no—
100C2×C4×Dic59—21no—
101C2×C4×Dic59—22no—
102C2×C4×Dic59—22no—
103C23×C4×C59—21no—
104C23×C4×C59C2311yescharacteristic
105C2×M16×C59—22no—
106C2×M16×C59—22no—
107C2×M16×C59C2×C414yescomplemented
108C22×C8×C59C2312yes—
109#117×C59—22no—
110#117×C59C2×C414yescomplemented
111C2×C4⋊C4×C59—21no—
112C2×C4⋊C4×C59—22no—
113C2×C4⋊C4×C59C2×C414yescomplemented
114C2×C4⋊C4×C59C2312yes—
115C2×C4⋊C4×C59C2311yescharacteristic
116C2×C4⋊C4×C59C2311yescharacteristic
117C2×C22⋊C4×C59—21no—
118C2×C42×C59—21no—
119C24×C59C2×D411yescharacteristic
120C23×D59C2×D412yes—
121C22×C4×C59—21no—
122C22×C4×C59—21no—
123C22×C4×C59—22no—
124C22×C4×C59—24no—
125C22×C4×C59—81no—
126C22×C4×C59—81no—
127C22×C4×C59C22×C412yes—
128C22×C4×C59C22×C412yes—
129C22×C4×C59C22×C412yes—
130C22×C4×C59C2×D411yescharacteristic
131C22×C4×C59C2×D412yes—
132C22×C4×C59C2×D412yes—
133C22×C4×C59C2×Q812yes—
134C22×C4×C59C2411yescharacteristic
135C2×C22⋊D59C22⋊C418yes—
136C2×C22⋊D59C22⋊C418yes—
137C2×C22⋊D59C4⋊C418yes—
138C22×Dic59—21no—
139C22×Dic59—22no—
140C22×Dic59—21no—
141C22×Dic59—24no—
142C22×Dic59—81no—
143C22×Dic59—81no—
144C22×Dic59C2×D412yes—
145#11745685C42116yes—
146#11745685C22⋊C4116yes—
147#11745685C4⋊C4116yes—
148#11745685C4⋊C4116yes—
149C2×D236C22⋊C414yes—
150C2×D236C22⋊C414yes—
151C2×D236C4⋊C414yes—
152C2×C4×D59—28no—
153C2×C4×D59—28no—
154C2×C4×D59—82no—
155C2×C4×D59—82no—
156C2×C4×D59—82no—
157C2×C4×D59—82no—
158C2×C4×D59C22×C414yes—
159C2×C4×D59C22×C414yes—
160C2×C4×D59C22×C414yes—
161C2×C4×D59C2×D414yes—
162C2×C4×D59C2×D414yes—
163C2×C4×D59C2×Q814yes—
164C2×Dic118C22⋊C414yes—
165C2×Dic118C22⋊C414yes—
166C2×Dic118C4⋊C414yes—
167M16×C59—28no—
168C2×C8×C59—22no—
169C2×C8×C59—22no—
170C2×C8×C59C22×C414yes—
171C4⋊C4×C59—24no—
172C4⋊C4×C59—22no—
173C4⋊C4×C59—21no—
174C4⋊C4×C59—21no—
175C4⋊C4×C59C22×C414yes—
176C4⋊C4×C59C22×C412yes—
177C4⋊C4×C59C22×C412yes—
178C22⋊C4×C59—24no—
179C42×C59—24no—
180C22⋊Dic59—24no—
181(C2×D59)⋊C4—28no—
182(C2×D59)⋊C4—28no—
183C4⋊Dic59—24no—
184C4⋊Dic59—21no—
185C4⋊Dic59—22no—
186C4⋊Dic59—21no—
187C4⋊Dic59C22×C412yes—
188C4⋊Dic59C22×C414yes—
189C4⋊Dic59C22×C412yes—
190Dic59⋊C4—28no—
191Dic59⋊C4—28no—
192C4×Dic59—24no—
193C4×Dic59—28no—
194C4×Dic59—28no—
195C2×C59⋊C8—28no—
196C2×C59⋊C8—21no—
197C2×C59⋊C8—21no—
198C2×C59⋊C8—21no—
199C2×C59⋊C8—21no—
200C2×C59⋊C8C22×C412yes—
201C2×C59⋊C8C22×C412yes—
202#11745659—216no—
203C8×D59—216no—
204C23×C59—21no—
205C23×C59—21no—
206C23×C59—21no—
207C23×C59—21no—
208C23×C59C2×C22⋊C412yes—
209C23×C59C2×C22⋊C412yes—
210C23×C59C2×C4⋊C412yes—
211C23×C59C22×D411yescharacteristic
212C22×D59—22no—
213C22×D59—22no—
214C22×D59—22no—
215C22×D59—22no—
216C22×D59C2×C22⋊C414yes—
217C22×D59C2×C22⋊C414yes—
218C22×D59C2×C4⋊C414yes—
219C2×C4×C59—22no—
220C2×C4×C59—24no—
221C2×C4×C59—22no—
222C2×C4×C59—41no—
223C2×C4×C59—42no—
224C2×C4×C59—41no—
225C2×C4×C59—42no—
226C2×C4×C59—41no—
227C2×C4×C59—41no—
228C2×C4×C59—84no—
229C2×C4×C59C2×C4214yes—
230C2×C4×C59C2×C22⋊C414yes—
231C2×C4×C59C2×C22⋊C412yes—
232C2×C4×C59C2×C22⋊C412yes—
233C2×C4×C59C2×C4⋊C414yes—
234C2×C4×C59C2×C4⋊C414yes—
235C2×C4×C59C2×C4⋊C412yes—
236C2×C4×C59C23×C411yescharacteristic
237C2×C4×C59C23×C411yescharacteristic
238C2×C4×C59C23×C411yescharacteristic
239C2×C4×C59C22×D411yescharacteristic
240C2×C4×C59C22×D411yescharacteristic
241C2×C4×C59C22×Q811yescharacteristic
242C22⋊D59#95132yes—
243C2×Dic59—22no—
244C2×Dic59—22no—
245C2×Dic59—24no—
246C2×Dic59—41no—
247C2×Dic59—42no—
248C2×Dic59—41no—
249C2×Dic59—42no—
250C2×Dic59—41no—
251C2×Dic59—41no—
252C2×Dic59—84no—
253C2×Dic59C2×C22⋊C414yes—
254C2×Dic59C2×C22⋊C414yes—
255C2×Dic59C2×C4⋊C414yes—
256D236#95116yes—
257C4×D59—88no—
258C4×D59—88no—
259C4×D59C2×C4218yes—
260C4×D59C2×C22⋊C418yes—
261C4×D59C2×C4⋊C418yes—
262C4×D59C2×C4⋊C418yes—
263Dic118#95116yes—
264C8×C59—24no—
265C59⋊C8—22no—
266C59⋊C8—22no—
267#20348—591nomaximal, Sylow
268C22×C59—22no—
269C22×C59—22no—
270C22×C59—22no—
271C22×C59—22no—
272C22×C59—22no—
273C22×C59—22no—
274C22×C59C2×#9514yes—
275C22×C59C22×C22⋊C411yescharacteristic
276C22×C59C22×C22⋊C411yescharacteristic
277C22×C59C22×C4⋊C411yescharacteristic
278C22×C59#57611yescharacteristic
279C22×C59#57711yescharacteristic
280C22×C59#57711yescharacteristic
281C22×C59#57811yescharacteristic
282C2×D59—24no—
283C2×D59—24no—
284C2×D59—24no—
285C2×D59—24no—
286C2×D59—24no—
287C2×D59—24no—
288C2×D59C2×#9518yes—
289C4×C59—42no—
290C4×C59—44no—
291C4×C59—42no—
292C4×C59C2×#9514yes—
293C4×C59C22×C4211yescharacteristic
294C4×C59C22×C22⋊C411yescharacteristic
295C4×C59C22×C4⋊C411yescharacteristic
296C4×C59C22×C4⋊C411yescharacteristic
297Dic59—42no—
298Dic59—44no—
299Dic59—42no—
300Dic59C2×#9518yes—
301C22×#131—591no—
302C2×#520—591no—
303C2×#520—591no—
304#1736—592no—
305#1736—592no—
306#1736—591no—
307#1736D5911yescharacteristic, complemented
308#1735—592no—
309#1735—591no—
310C2×#340—592no—
311C2×#340—591no—
312C2×C59—24no—
313C2×C59C22×#9511yescharacteristic
314C2×C59#289711yescharacteristic
315C2×C59#289711yescharacteristic
316C2×C59#289811yescharacteristic
317C2×C59#289811yescharacteristic
318C2×C59#291211yescharacteristic
319C2×C59#291311yescharacteristic
320D59—28no—
321C22×#41—591no—
322C2×#131—598no—
323C2×#131—1184no—
324C2×#131—1184no—
325C22×M16—592no—
326C22×M16C2×D5911yescharacteristic
327C23×C8—591no—
328C23×C8—592no—
329#520—598no—
330#520—598no—
331#520—1184no—
332#520—1184no—
333C2×C4×Q8—1181no—
334C2×C4×Q8—1181no—
335C2×C4×D4—1181no—
336C2×C4×D4—1182no—
337C2×C4×D4—1181no—
338C2×C4×D4—1182no—
339C2×#117—591no—
340C2×#117—592no—
341C2×#117—592no—
342C2×#117C2×D5911yescharacteristic
343C22×C4⋊C4—591no—
344C22×C4⋊C4—592no—
345C22×C4⋊C4—592no—
346C22×C4⋊C4C2×D5911yescharacteristic
347#340—592no—
348#340—592no—
349#340—591no—
350#340—592no—
351#340—592no—
352#340—591no—
353#340—591no—
354#340—591no—
355#340C2×D5912yes—
356#340C2×D5912yes—
357C59#2034811yescharacteristic, complemented, Sylow
358C2×#41—594no—
359C2×#41—594no—
360C2×#41—594no—
361C2×#41—594no—
362C2×#41—594no—
363C2×#41—594no—
364C22×Q8—591no—
365C22×D4—592no—
366C22×D4—591no—
367C23×C4—21no—
368C23×C4—592no—
369C23×C4—1182no—
370C23×C4—1182no—
371C23×C4—1181no—
372C23×C4C22×D5911yescharacteristic
373#131—11832no—
374C2×M16—22no—
375C2×M16—22no—
376C2×M16—598no—
377C2×M16—1184no—
378C2×M16—1184no—
379C2×M16C4×D5914yescomplemented
380C22×C8—594no—
381C22×C8—591no—
382C22×C8—591no—
383C22×C8—598no—
384C22×C8—1184no—
385C22×C8—1184no—
386C22×C8—1182no—
387C22×C8—1182no—
388C22×C8C22×D5912yes—
389C4×Q8—1188no—
390C4×Q8—1188no—
391C4×D4—1188no—
392C4×D4—11816no—
393C4×D4—1188no—
394C4×D4—11816no—
395#117—22no—
396#117—594no—
397#117—598no—
398#117—598no—
399#117—1184no—
400#117—1184no—
401#117—1182no—
402#117C4×D5914yescomplemented
403C2×C4⋊C4—21no—
404C2×C4⋊C4—22no—
405C2×C4⋊C4—594no—
406C2×C4⋊C4—592no—
407C2×C4⋊C4—591no—
408C2×C4⋊C4—591no—
409C2×C4⋊C4—598no—
410C2×C4⋊C4—598no—
411C2×C4⋊C4—1184no—
412C2×C4⋊C4—1184no—
413C2×C4⋊C4—1182no—
414C2×C4⋊C4—1182no—
415C2×C4⋊C4—1181no—
416C2×C4⋊C4—1182no—
417C2×C4⋊C4C4×D5914yescomplemented
418C2×C4⋊C4C22×D5911yescharacteristic
419C2×C4⋊C4C22×D5912yes—
420C2×C4⋊C4C22×D5911yescharacteristic
421C2×C22⋊C4—21no—
422C2×C22⋊C4—1182no—
423C2×C22⋊C4—1182no—
424C2×C22⋊C4—1181no—
425C2×C42—21no—
426C2×C42—1182no—
427C2×C42—1182no—
428C2×C42—1181no—
429C24—592no—
430C24D4×D5911yescharacteristic
431#41—5916no—
432#41—5916no—
433#41—5916no—
434#41—5916no—
435C2×Q8—594no—
436C2×Q8—594no—
437C2×Q8—594no—
438C2×D4—598no—
439C2×D4—598no—
440C2×D4—598no—
441C2×D4—594no—
442C2×D4—594no—
443C2×D4—594no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—81no—
449C22×C4—81no—
450C22×C4—592no—
451C22×C4—594no—
452C22×C4—594no—
453C22×C4—594no—
454C22×C4—594no—
455C22×C4—594no—
456C22×C4—594no—
457C22×C4—1188no—
458C22×C4—1188no—
459C22×C4—1184no—
460C22×C4—1181no—
461C22×C4—1182no—
462C22×C4—1181no—
463C22×C4—4722no—
464C22×C4—4722no—
465C22×C4—4722no—
466C22×C4—4722no—
467C22×C4—4721no—
468C22×C4—4721no—
469C22×C4C2×C4×D5912yes—
470C22×C4C2×C4×D5912yes—
471C22×C4C2×C4×D5912yes—
472C22×C4D4×D5911yescharacteristic
473C22×C4D4×D5912yes—
474C22×C4D4×D5912yes—
475C22×C4Q8×D5912yes—
476C22×C4C23×D5911yescharacteristic
477M16—28no—
478M16—11816no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—592no—
482C2×C8—592no—
483C2×C8—11816no—
484C2×C8—1188no—
485C2×C8—1181no—
486C2×C8—1181no—
487C2×C8—1181no—
488C2×C8—1181no—
489C2×C8C2×C4×D5914yes—
490C4⋊C4—24no—
491C4⋊C4—22no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—594no—
495C4⋊C4—592no—
496C4⋊C4—592no—
497C4⋊C4—1188no—
498C4⋊C4—1188no—
499C4⋊C4—1184no—
500C4⋊C4—1181no—
501C4⋊C4—1182no—
502C4⋊C4—1181no—
503C4⋊C4C2×C4×D5912yes—
504C4⋊C4C2×C4×D5914yes—
505C4⋊C4C2×C4×D5912yes—
506C22⋊C4—24no—
507C22⋊C4—1188no—
508C22⋊C4—1188no—
509C22⋊C4—1184no—
510C42—24no—
511C42—1188no—
512C42—1188no—
513C42—1184no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—594no—
519C23—594no—
520C23—594no—
521C23—1182no—
522C23—1182no—
523C23—1182no—
524C23—1182no—
525C23C22⋊C4×D5912yes—
526C23C22⋊C4×D5912yes—
527C23C4⋊C4×D5912yes—
528C23C2×D4×D5911yescharacteristic
529Q8—5916no—
530D4—5932no—
531D4—5916no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—42no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—41no—
541C2×C4—84no—
542C2×C4—594no—
543C2×C4—594no—
544C2×C4—594no—
545C2×C4—598no—
546C2×C4—598no—
547C2×C4—598no—
548C2×C4—598no—
549C2×C4—1182no—
550C2×C4—1184no—
551C2×C4—1182no—
552C2×C4—4728no—
553C2×C4—4728no—
554C2×C4—4724no—
555C2×C4—2362no—
556C2×C4—2361no—
557C2×C4—2361no—
558C2×C4—2362no—
559C2×C4—2361no—
560C2×C4—2361no—
561C2×C4C42×D5914yes—
562C2×C4C22⋊C4×D5912yes—
563C2×C4C22⋊C4×D5912yes—
564C2×C4C22⋊C4×D5914yes—
565C2×C4C4⋊C4×D5912yes—
566C2×C4C4⋊C4×D5914yes—
567C2×C4C4⋊C4×D5914yes—
568C2×C4C22×C4×D5911yescharacteristic
569C2×C4C22×C4×D5911yescharacteristic
570C2×C4C22×C4×D5911yescharacteristic
571C2×C4C2×D4×D5911yescharacteristic
572C2×C4C2×D4×D5911yescharacteristic
573C2×C4C2×Q8×D5911yescharacteristic
574C8—24no—
575C8—1182no—
576C8—1182no—
577C22—22no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—598no—
584C22—1184no—
585C22—1184no—
586C22—1184no—
587C22—1184no—
588C22—1184no—
589C22—1184no—
590C22#95×D5914yes—
591C22C2×C22⋊C4×D5911yescharacteristic
592C22C2×C22⋊C4×D5911yescharacteristic
593C22C2×C4⋊C4×D5911yescharacteristic
594C22#1683820611yescharacteristic
595C22#1683821111yescharacteristic
596C22#1683821211yescharacteristic
597C22#1683821811yescharacteristic
598C4—42no—
599C4—44no—
600C4—42no—
601C4—598no—
602C4—2364no—
603C4—2362no—
604C4—2362no—
605C4#95×D5914yes—
606C4C2×C42×D5911yescharacteristic
607C4C2×C22⋊C4×D5911yescharacteristic
608C4C2×C4⋊C4×D5911yescharacteristic
609C4C2×C4⋊C4×D5911yescharacteristic
610C2—24no—
611C2—1188no—
612C2C2×#95×D5911yescharacteristic
613C2#2143262011yescharacteristic
614C2#2143262111yescharacteristic
615C2#2143262711yescharacteristic
616C2#2143262811yescharacteristic
617C2#2143269011yescharacteristic
618C2#2143269511yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutatorC4×C59C22×C42
FrattiniC22×C4C23×D59
FittingC59×#1736C2
socleC23×C59C22×D4
prime radicalC22×#11745685C22
cosocleC22×C4×C59C24
hypercenter#1736D59
hypocenterC59#20348
LCSG ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1736

Complements

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

NH ≅ G/Ntype
4. #21423816C2semidirect
5. #21423814C2semidirect
6. #21423814C2semidirect
7. #21423812C2semidirect
8. #21423811C2semidirect
9. C2×#16837164C2semidirect
10. #340×D59C2semidirect
11. C59×#1736C2semidirect
15. C2×#14273735C4semidirect
20. #16837883C4semidirect
22. #16837882C4semidirect
64. C2×#11745659C2×C4semidirect
67. C22×C59⋊C8C2×C4semidirect
72. C2×C8×D59C2×C4semidirect
80. #14273686C2×C4semidirect
83. #14273685C2×C4semidirect
84. #14273685C2×C4semidirect
87. C2×C4⋊Dic59C2×C4semidirect
95. C4⋊C4×D59C2×C4semidirect
96. C4⋊C4×D59C2×C4semidirect
107. C2×M16×C59C2×C4semidirect
110. #117×C59C2×C4semidirect
113. C2×C4⋊C4×C59C2×C4semidirect
307. #1736D59semidirect
357. C59#20348semidirect
379. C2×M16C4×D59semidirect
402. #117C4×D59semidirect
417. C2×C4⋊C4C4×D59semidirect

Maximal subgroups

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

subgroupindex
C22×#142737352
C2×#168378832
C2×#168378822
#214238162
#214238142
#214238122
#214238112
C2×#168371642
#340×D592
C59×#17362
#2034859