Finite Group Databasebeta

← #28599083

Subgroups — #28599083

Order 15104.

830 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837424C211yesmaximal, characteristic
2#21426500C211yesmaximal, characteristic
3#21426489C211yesmaximal, characteristic
4#21424866C211yesmaximal, characteristic, complemented
5#21424864C211yesmaximal, characteristic, complemented
6#21424862C211yesmaximal, characteristic, complemented
7#21424858C211yesmaximal, characteristic, complemented
8#21424856C211yesmaximal, characteristic, complemented
9#21424853C211yesmaximal, characteristic, complemented
10#21424851C211yesmaximal, characteristic, complemented
11#21424130C211yesmaximal, characteristic
12#21424128C211yesmaximal, characteristic
13#21424128C211yesmaximal, characteristic
14#21424126C211yesmaximal, characteristic
15C59×#1949C211yesmaximal, characteristic, complemented
16C22×#11745659C2211yescharacteristic
17C22×#11745659C2211yescharacteristic
18C2×#14273684—21no—
19C2×C42×D59C2211yescharacteristic
20C2×#14273661C2211yescharacteristic
21C2×#14273660C2211yescharacteristic
22C2×#14273660C2211yescharacteristic
23C2×#14273659C2211yescharacteristic
24C2×#14273658C2211yescharacteristic
25C2×#14273658C2211yescharacteristic
26#16837424C418yescomplemented
27C2×C4⋊(C59⋊C8)C2211yescharacteristic
28C2×Dic59⋊C8C2211yescharacteristic
29C2×Dic59⋊C8C2211yescharacteristic
30#16837271C2211yescharacteristic
31#16837271C2211yescharacteristic
32#16837269—21no—
33#16837268C2211yescharacteristic
34#16837266—21no—
35#16837264—21no—
36#16837261—21no—
37#16837259C2211yescharacteristic
38#16837259C2211yescharacteristic
39#16837259C2211yescharacteristic
40#16837259C2211yescharacteristic
41C4⋊C4×Dic59C2211yescharacteristic
42C4⋊C4×Dic59C2211yescharacteristic
43C4×Dic59⋊C4—21no—
44C4×C4⋊Dic59C2211yescharacteristic
45#16837249—21no—
46#16837247—21no—
47#16837165C2211yescharacteristic
48#16837165C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837164C2211yescharacteristic
51#16837164C2211yescharacteristic
52#16837164C2211yescharacteristic
53C2×#107×C59C2211yescharacteristic
54C2×#106×C59C2211yescharacteristic
55C2×C4⋊C8×C59C2211yescharacteristic
56C59×#382C2211yescharacteristic
57C4×C4⋊C4×C59C2211yescharacteristic
58C59×#340C2211yescharacteristic
59C59×#340C2211yescharacteristic
60C22×C4×D59D411yescharacteristic
61C22×C4×D59D411yescharacteristic
62C22×C4×D59C2311yescharacteristic
63C2×#11745659—41no—
64C2×#11745659—41no—
65C2×#11745659—41no—
66C2×#11745659—41no—
67C2×#11745659C2×C414yes—
68C2×#11745659C2×C414yes—
69C22×C59⋊C8C2311yescharacteristic
70C22×C59⋊C8C2311yescharacteristic
71#14273684—44no—
72C2×C4⋊Dic59C2311yescharacteristic
73C2×C4⋊Dic59C2311yescharacteristic
74C2×C4⋊Dic59C2311yescharacteristic
75C2×C4⋊Dic59C2311yescharacteristic
76C2×Dic59⋊C4—21no—
77C2×Dic59⋊C4—21no—
78C2×Dic59⋊C4—21no—
79C2×Dic59⋊C4—21no—
80C2×(C2×D59)⋊C4—21no—
81C2×(C2×D59)⋊C4—21no—
82C2×(C2×D59)⋊C4—21no—
83C2×(C2×D59)⋊C4—21no—
84C2×C4×Dic59—21no—
85C2×C4×Dic59—21no—
86C2×C4×Dic59—21no—
87C2×C4×Dic59D411yescharacteristic
88C2×C4×Dic59D411yescharacteristic
89C2×C4×Dic59C2311yescharacteristic
90C42×D59C2×C414yes—
91C42×D59Q814yes—
92#14273661—41no—
93#14273660—41no—
94#14273660—41no—
95#14273659—41no—
96#14273658—41no—
97#14273658—41no—
98C4⋊(C59⋊C8)C2×C414yes—
99Dic59⋊C8C2×C414yes—
100Dic59⋊C8C2×C414yes—
101C59⋊#95—21no—
102C59⋊#95—21no—
103C59⋊#95—21no—
104C59⋊#95—21no—
105C59⋊#95—21no—
106C59⋊#95—21no—
107C59⋊#95—21no—
108C59⋊#95—21no—
109C59⋊#95—21no—
110C22×C8×C59C2311yescharacteristic
111C22×C8×C59C2311yescharacteristic
112C2×C4⋊C4×C59C2311yescharacteristic
113C2×C4⋊C4×C59C2311yescharacteristic
114C2×C4⋊C4×C59C2311yescharacteristic
115C2×C4⋊C4×C59C2311yescharacteristic
116C2×C42×C59—21no—
117C2×C42×C59C2311yescharacteristic
118#107×C59—41no—
119#106×C59—41no—
120C4⋊C8×C59C2×C414yes—
121#95×C59—21no—
122#95×C59—21no—
123#95×C59—21no—
124C23×D59C2×D411yescharacteristic
125C22×C4×C59—21no—
126C22×C4×C59—21no—
127C22×C4×C59—21no—
128C22×C4×C59—21no—
129C22×C4×C59C2×D411yescharacteristic
130C22×C4×C59C2×D411yescharacteristic
131C22×C4×C59C2411yescharacteristic
132C22×Dic59—21no—
133C22×Dic59—21no—
134C22×Dic59—21no—
135C22×Dic59—21no—
136C22×Dic59C2×D411yescharacteristic
137C22×Dic59C2×D411yescharacteristic
138C22×Dic59C2×D411yescharacteristic
139C2×C4×D59—21no—
140C2×C4×D59—21no—
141C2×C4×D59—21no—
142C2×C4×D59—21no—
143C2×C4×D59—22no—
144C2×C4×D59—22no—
145C2×C4×D59—41no—
146C2×C4×D59—41no—
147C2×C4×D59C4⋊C414yes—
148C2×C4×D59C4⋊C414yes—
149C2×C4×D59C22×C412yes—
150C2×C4×D59C2×Q812yes—
151C2×C8×C59—21no—
152C2×C8×C59—21no—
153C2×C8×C59—21no—
154C2×C8×C59—21no—
155C2×C8×C59C22×C412yes—
156C2×C8×C59C22×C412yes—
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—21no—
164C4⋊C4×C59—21no—
165C42×C59—42no—
166C42×C59C22×C412yes—
167C42×C59C2×Q812yes—
168(C2×D59)⋊C4—42no—
169(C2×D59)⋊C4—42no—
170(C2×D59)⋊C4—42no—
171(C2×D59)⋊C4—42no—
172C4⋊Dic59—21no—
173C4⋊Dic59—21no—
174C4⋊Dic59—21no—
175C4⋊Dic59—21no—
176C4⋊Dic59—21no—
177C4⋊Dic59—21no—
178C4⋊Dic59—21no—
179C4⋊Dic59—21no—
180Dic59⋊C4—42no—
181Dic59⋊C4—42no—
182Dic59⋊C4—42no—
183Dic59⋊C4—42no—
184C4×Dic59—42no—
185C4×Dic59—42no—
186C4×Dic59—42no—
187C4×Dic59C4⋊C414yes—
188C4×Dic59C4⋊C414yes—
189C4×Dic59C22×C412yes—
190C4×Dic59C2×Q812yes—
191C2×C59⋊C8—21no—
192C2×C59⋊C8—21no—
193C2×C59⋊C8—21no—
194C2×C59⋊C8—21no—
195C2×C59⋊C8C22×C412yes—
196C2×C59⋊C8C22×C412yes—
197#11745659—44no—
198#11745659—44no—
199C23×C59C22×D411yescharacteristic
200C22×D59—21no—
201C22×D59—21no—
202C22×D59—21no—
203C22×D59—21no—
204C22×D59—21no—
205C22×D59—21no—
206C22×D59C2×C4⋊C412yes—
207C2×C4×C59—21no—
208C2×C4×C59—21no—
209C2×C4×C59—21no—
210C2×C4×C59—21no—
211C2×C4×C59—41no—
212C2×C4×C59—41no—
213C2×C4×C59—41no—
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×C59—41no—
221C2×C4×C59—41no—
222C2×C4×C59—41no—
223C2×C4×C59C2×C4⋊C412yes—
224C2×C4×C59C2×C4⋊C412yes—
225C2×C4×C59C23×C411yescharacteristic
226C2×C4×C59C22×Q811yescharacteristic
227C2×C4×C59H(5,𝔽2)11yescharacteristic
228C2×C4×C59H(5,𝔽2)11yescharacteristic
229C2×C4×C592−1+411yescharacteristic
230C2×C4×C592−1+411yescharacteristic
231C2×Dic59—21no—
232C2×Dic59—21no—
233C2×Dic59—21no—
234C2×Dic59—21no—
235C2×Dic59—41no—
236C2×Dic59—41no—
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×Dic59C2×C4⋊C412yes—
248C2×Dic59C2×C4⋊C412yes—
249C2×Dic59C2×C4⋊C412yes—
250C2×Dic59AGL(1,ℤ/8)12yes—
251C2×Dic59#13712yes—
252C4×D59—22no—
253C4×D59—22no—
254C4×D59—22no—
255C4×D59—22no—
256C4×D59—28no—
257C4×D59—44no—
258C8×C59—22no—
259C8×C59—22no—
260C59⋊C8—22no—
261C59⋊C8—22no—
262#21963—591nomaximal, Sylow
263C22×C59C22×C4⋊C411yescharacteristic
264C22×C59#53711yescharacteristic
265C22×C59#53711yescharacteristic
266C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
267C22×C59C2×#13711yescharacteristic
268C22×C59#57511yescharacteristic
269C22×C59#57511yescharacteristic
270C2×D59—22no—
271C2×D59—22no—
272C2×D59—22no—
273C2×D59—41no—
274C2×D59—41no—
275C2×D59—41no—
276C2×D59—41no—
277C4×C59—22no—
278C4×C59—22no—
279C4×C59—81no—
280C4×C59—81no—
281C4×C59—81no—
282C4×C59—81no—
283C4×C59#51811yescharacteristic
284C4×C59#51911yescharacteristic
285C4×C59#54411yescharacteristic
286C4×C59#54411yescharacteristic
287Dic59—22no—
288Dic59—22no—
289Dic59—81no—
290Dic59—81no—
291Dic59—81no—
292Dic59—81no—
293Dic59#42814yes—
294C2×#423—591no—
295#2285—591no—
296#2283—591no—
297#1949—591no—
298#1949—591no—
299#1949—591no—
300#1949D5911yescharacteristic, complemented
301#1948—591no—
302#1948—591no—
303#1947—591no—
304#1947—591no—
305#1809—591no—
306#1809—591no—
307#1809—591no—
308#1809—591no—
309C2×C59#231611yescharacteristic
310C2×C59C2×#42811yescharacteristic
311C2×C59#291111yescharacteristic
312C2×C59#325111yescharacteristic
313C2×C59#325111yescharacteristic
314C2×C59#325211yescharacteristic
315C2×C59#325211yescharacteristic
316D59—42no—
317C22×M16—591no—
318C22×M16—591no—
319C2×#117—1181no—
320C22×C42—591no—
321C2×#107—591no—
322C2×#107—591no—
323C2×#107—591no—
324C2×#107C2×D5911yescharacteristic
325C2×#106—591no—
326C2×#106—591no—
327C2×#106—591no—
328C2×#106C2×D5911yescharacteristic
329#423—598no—
330C2×C4⋊C8—591no—
331C2×C4⋊C8—591no—
332C2×C4⋊C8—591no—
333C2×C4⋊C8C2×D5911yescharacteristic
334#384—591no—
335#384—591no—
336#383—1181no—
337#382—591no—
338#382—1181no—
339#382C2×D5911yescharacteristic
340#381—1181no—
341#380—591no—
342#380—591no—
343#380—591no—
344#380—591no—
345#380—1181no—
346C4×C4⋊C4—591no—
347C4×C4⋊C4—591no—
348C4×C4⋊C4—591no—
349C4×C4⋊C4—1181no—
350C4×C4⋊C4C2×D5911yescharacteristic
351C4×C22⋊C4—1181no—
352#376—1181no—
353#340—591no—
354#340—591no—
355#340—591no—
356#340—591no—
357#340—591no—
358#340—591no—
359#340C2×D5911yescharacteristic
360#340C2×D5911yescharacteristic
361C59#2196311yescharacteristic, complemented, Sylow
362C23×C4—591no—
363C23×C4—591no—
364C23×C4—591no—
365C2×M16—594no—
366C2×M16—594no—
367C2×M16—2361no—
368C2×M16—2361no—
369C2×M16—2361no—
370C2×M16—2361no—
371C22×C8—591no—
372C22×C8—591no—
373C22×C8C22×D5911yescharacteristic
374C22×C8C22×D5911yescharacteristic
375#117—2364no—
376C2×C4⋊C4—591no—
377C2×C4⋊C4—591no—
378C2×C4⋊C4—591no—
379C2×C4⋊C4—591no—
380C2×C4⋊C4—1181no—
381C2×C4⋊C4—1181no—
382C2×C4⋊C4—1181no—
383C2×C4⋊C4—1181no—
384C2×C4⋊C4C22×D5911yescharacteristic
385C2×C4⋊C4C22×D5911yescharacteristic
386C2×C4⋊C4C22×D5911yescharacteristic
387C2×C4⋊C4C22×D5911yescharacteristic
388C2×C22⋊C4—1181no—
389C2×C22⋊C4—1181no—
390C2×C22⋊C4—1181no—
391C2×C22⋊C4—1181no—
392C2×C42—21no—
393C2×C42—591no—
394C2×C42—591no—
395C2×C42—591no—
396C2×C42—594no—
397C2×C42—594no—
398C2×C42—1181no—
399C2×C42—1181no—
400C2×C42—1181no—
401C2×C42C22×D5911yescharacteristic
402#107—41no—
403#107—2361no—
404#107—2361no—
405#107—2361no—
406#106—41no—
407#106—2361no—
408#106—2361no—
409#106—2361no—
410C4⋊C8—594no—
411C4⋊C8—594no—
412C4⋊C8—594no—
413C4⋊C8C4×D5914yes—
414#95—21no—
415#95—21no—
416#95—21no—
417#95—1181no—
418#95—1181no—
419#95—1181no—
420#95—1181no—
421#95—1181no—
422#95—1181no—
423#95—1181no—
424#95—1181no—
425#95—1181no—
426C24—591no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—591no—
432C22×C4—591no—
433C22×C4—591no—
434C22×C4—594no—
435C22×C4—592no—
436C22×C4—592no—
437C22×C4—594no—
438C22×C4—1182no—
439C22×C4—1182no—
440C22×C4—1181no—
441C22×C4—1181no—
442C22×C4—1181no—
443C22×C4—1181no—
444C22×C4—1181no—
445C22×C4—1181no—
446C22×C4—1181no—
447C22×C4—1181no—
448C22×C4—2361no—
449C22×C4—2361no—
450C22×C4D4×D5911yescharacteristic
451C22×C4D4×D5911yescharacteristic
452C22×C4C23×D5911yescharacteristic
453M16—2364no—
454M16—2364no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—592no—
460C2×C8—592no—
461C2×C8—1181no—
462C2×C8—1181no—
463C2×C8—1181no—
464C2×C8—1181no—
465C2×C8C2×C4×D5912yes—
466C2×C8C2×C4×D5912yes—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—1181no—
476C4⋊C4—1181no—
477C4⋊C4—1181no—
478C4⋊C4—1181no—
479C4⋊C4—1181no—
480C4⋊C4—1181no—
481C4⋊C4—1181no—
482C4⋊C4—1181no—
483C4⋊C4—2362no—
484C4⋊C4—2362no—
485C4⋊C4—2362no—
486C4⋊C4—2362no—
487C22⋊C4—2362no—
488C22⋊C4—2362no—
489C22⋊C4—2362no—
490C22⋊C4—2362no—
491C42—42no—
492C42—594no—
493C42—594no—
494C42—592no—
495C42—592no—
496C42—2362no—
497C42—2362no—
498C42—2362no—
499C42C2×C4×D5912yes—
500C42Q8×D5912yes—
501C23—592no—
502C23—1181no—
503C23—1181no—
504C23—1181no—
505C23—1181no—
506C23—1181no—
507C23—1181no—
508C23C2×D4×D5911yescharacteristic
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—592no—
526C2×C4—592no—
527C2×C4—592no—
528C2×C4—592no—
529C2×C4—592no—
530C2×C4—1188no—
531C2×C4—1181no—
532C2×C4—1181no—
533C2×C4—1181no—
534C2×C4—1181no—
535C2×C4—1182no—
536C2×C4—1182no—
537C2×C4—1182no—
538C2×C4—1182no—
539C2×C4—2364no—
540C2×C4—2361no—
541C2×C4—2361no—
542C2×C4—2361no—
543C2×C4—2361no—
544C2×C4—2361no—
545C2×C4—2361no—
546C2×C4—2361no—
547C2×C4—2361no—
548C2×C4—2361no—
549C2×C4—2361no—
550C2×C4—2361no—
551C2×C4—2361no—
552C2×C4C4⋊C4×D5912yes—
553C2×C4C4⋊C4×D5912yes—
554C2×C4C22×C4×D5911yescharacteristic
555C2×C4C2×Q8×D5911yescharacteristic
556C2×C4#1427377211yescharacteristic
557C2×C4#1427377311yescharacteristic
558C2×C4#1427377411yescharacteristic
559C2×C4#1427377511yescharacteristic
560C8—22no—
561C8—22no—
562C8—1182no—
563C8—1182no—
564C22—1182no—
565C22—1182no—
566C22—1182no—
567C22—2361no—
568C22—2361no—
569C22—2361no—
570C22—2361no—
571C22C2×C4⋊C4×D5911yescharacteristic
572C22#1683798111yescharacteristic
573C22#1683798211yescharacteristic
574C22AGL(1,ℤ/8)×D5911yescharacteristic
575C22#137×D5911yescharacteristic
576C22#1683819911yescharacteristic
577C22#1683819911yescharacteristic
578C4—22no—
579C4—22no—
580C4—81no—
581C4—81no—
582C4—81no—
583C4—81no—
584C4—594no—
585C4—1182no—
586C4—1182no—
587C4—4721no—
588C4—4721no—
589C4—4721no—
590C4—4721no—
591C4#1683787411yescharacteristic
592C4#1683787811yescharacteristic
593C4#1683802911yescharacteristic
594C4#1683803211yescharacteristic
595C2—2362no—
596C2#2142671111yescharacteristic
597C2#428×D5911yescharacteristic
598C2#2143268411yescharacteristic
599C2#2143545511yescharacteristic
600C2#2143545711yescharacteristic
601C2#2143546611yescharacteristic
602C2#2143546811yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1949C2
socleC23×C59C22×D4
prime radicalC23×D59C2×D4
cosocleC22×C4×C59C24
hypercenter#1949D59
hypocenterC59#21963
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1949

Complements

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

NH ≅ G/Ntype
4. #21424866C2semidirect
5. #21424864C2semidirect
6. #21424862C2semidirect
7. #21424858C2semidirect
8. #21424856C2semidirect
9. #21424853C2semidirect
10. #21424851C2semidirect
15. C59×#1949C2semidirect
26. #16837424C4semidirect
300. #1949D59semidirect
361. C59#21963semidirect

Maximal subgroups

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

subgroupindex
C2×#168374242
#214265002
#214264892
#214248662
#214248642
#214248622
#214248582
#214248562
#214248532
#214248512
#214241302
#214241282
#214241262
C59×#19492
#2196359