Finite Group Databasebeta

← #28606447

Subgroups — #28606447

Order 15104.

782 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837444C211yesmaximal, characteristic
2#21426987C211yesmaximal, characteristic
3#21426971C211yesmaximal, characteristic
4#21424839C211yesmaximal, characteristic, complemented
5#21424824C211yesmaximal, characteristic, complemented
6#21424689C211yesmaximal, characteristic, complemented
7#21424682C211yesmaximal, characteristic, complemented
8#21424675C211yesmaximal, characteristic, complemented
9#21424668C211yesmaximal, characteristic, complemented
10#21424192C211yesmaximal, characteristic
11#21424183C211yesmaximal, characteristic
12#21424129C211yesmaximal, characteristic
13#21423993C211yesmaximal, characteristic, complemented
14C4×#14273661C211yesmaximal, characteristic, complemented
15C59×#1753C211yesmaximal, characteristic
16C2×#14273686C2211yescharacteristic
17C22×C4⋊Dic59C2211yescharacteristic
18#16837444C418yescomplemented
19C2×#14273661C2211yescharacteristic
20C2×#14273661C2211yescharacteristic
21C2×#14273661C2211yescharacteristic
22C2×#14273659C2211yescharacteristic
23C2×#14273659C2211yescharacteristic
24C2×#14273659C2211yescharacteristic
25#16837300C2211yescharacteristic
26#16837300C2211yescharacteristic
27#16837300C2211yescharacteristic
28#16837300C2211yescharacteristic
29#16837296—21no—
30#16837296—21no—
31#16837285—21no—
32#16837285—21no—
33#16837269—21no—
34#16837262C2211yescharacteristic
35#16837262C2211yescharacteristic
36#16837260—21no—
37C4×C4⋊Dic59C2211yescharacteristic
38C4×C4⋊Dic59C2211yescharacteristic
39C4×C4⋊Dic59C2211yescharacteristic
40C4×C4⋊Dic59C2211yescharacteristic
41#16837253—21no—
42C22⋊C4×Dic59—21no—
43#16837165C2211yescharacteristic
44#16837165C2211yescharacteristic
45#16837165C2211yescharacteristic
46#16837165C2211yescharacteristic
47#16837165C2211yescharacteristic
48#16837165C2211yescharacteristic
49#16837165C2211yescharacteristic
50#16837165C2211yescharacteristic
51#16837165C2211yescharacteristic
52#16837165C2211yescharacteristic
53C22×M16×C59C2211yescharacteristic
54C2×C22⋊C8×C59C2211yescharacteristic
55C2×#97×C59C2211yescharacteristic
56C2×C4×C8×C59C2211yescharacteristic
57C4×C22⋊C4×C59C2211yescharacteristic
58C59×#336C2211yescharacteristic
59C59×#336C2211yescharacteristic
60C23×Dic59—21no—
61#14273686—22no—
62#14273686C2×C414yes—
63C2×C4⋊Dic59—21no—
64C2×C4⋊Dic59—22no—
65C2×C4⋊Dic59C2×C414yes—
66C2×C4⋊Dic59C2311yescharacteristic
67C2×C4⋊Dic59C2311yescharacteristic
68C2×C4⋊Dic59C2311yescharacteristic
69C2×C4⋊Dic59C2311yescharacteristic
70C2×C4⋊Dic59C2311yescharacteristic
71C2×C4⋊Dic59C2311yescharacteristic
72C2×C4⋊Dic59C2311yescharacteristic
73C2×C4⋊Dic59C2311yescharacteristic
74C2×Dic59⋊C4—21no—
75C2×Dic59⋊C4—21no—
76C2×Dic59⋊C4—21no—
77C2×Dic59⋊C4—21no—
78C2×C22⋊Dic59—21no—
79C2×C22⋊Dic59—21no—
80C2×C22⋊Dic59—21no—
81C2×C4×Dic59—21no—
82C2×C4×Dic59—21no—
83C2×C4×Dic59—21no—
84C2×C4×Dic59—21no—
85C2×C4×Dic59—21no—
86#14273661—22no—
87#14273661C2×C414yes—
88#14273661C2×C414yes—
89#14273659—22no—
90#14273659C2×C414yes—
91#14273659C2×C414yes—
92C59⋊#95—21no—
93C59⋊#95—21no—
94C59⋊#95—21no—
95C59⋊#95—21no—
96C59⋊#95—21no—
97C59⋊#95—21no—
98C59⋊#95—21no—
99C59⋊#95—21no—
100C59⋊#95—21no—
101C59⋊#95—21no—
102C23×C4×C59C2311yescharacteristic
103C2×M16×C59—22no—
104C2×M16×C59—22no—
105C2×M16×C59C2×C414yes—
106C22×C8×C59C2311yescharacteristic
107C22×C8×C59C2311yescharacteristic
108C22×C8×C59C2311yescharacteristic
109C22×C8×C59C2311yescharacteristic
110C2×C22⋊C4×C59D411yescharacteristic
111C2×C22⋊C4×C59D411yescharacteristic
112C2×C42×C59C2311yescharacteristic
113C2×C42×C59C2311yescharacteristic
114C22⋊C8×C59—22no—
115#97×C59—22no—
116C4×C8×C59—22no—
117#95×C59D411yescharacteristic
118#95×C59D411yescharacteristic
119C24×C59C2×D411yescharacteristic
120C22×C4×C59—21no—
121C22×C4×C59—21no—
122C22×C4×C59—21no—
123C22×C4×C59—21no—
124C22×C4×C59C22×C412yes—
125C22×C4×C59C2×D411yescharacteristic
126C22×C4×C59C2×D411yescharacteristic
127C22×C4×C59C2×D411yescharacteristic
128C22×C4×C59C2×D411yescharacteristic
129C22×C4×C59C2×D411yescharacteristic
130C22×C4×C59#4112yes—
131C22×C4×C59C2411yescharacteristic
132C22×Dic59—21no—
133C22×Dic59—21no—
134C22×Dic59—21no—
135C22×Dic59—21no—
136C22×Dic59—24no—
137C22×Dic59—21no—
138C22×Dic59—21no—
139C22×Dic59—21no—
140C22×Dic59—21no—
141C22×Dic59—81no—
142C22×Dic59—81no—
143M16×C59—28no—
144C2×C8×C59—21no—
145C2×C8×C59—21no—
146C2×C8×C59—21no—
147C2×C8×C59—21no—
148C2×C8×C59—21no—
149C2×C8×C59—21no—
150C2×C8×C59—21no—
151C2×C8×C59—21no—
152C2×C8×C59C22×C412yes—
153C2×C8×C59C22×C412yes—
154C2×C8×C59#4112yes—
155C2×C8×C59#4112yes—
156C22⋊C4×C59—22no—
157C22⋊C4×C59—41no—
158C42×C59—21no—
159C42×C59—21no—
160C42×C59—21no—
161C42×C59—21no—
162C22⋊Dic59—24no—
163C22⋊Dic59—42no—
164C22⋊Dic59—81no—
165C4⋊Dic59—24no—
166C4⋊Dic59—21no—
167C4⋊Dic59—21no—
168C4⋊Dic59—21no—
169C4⋊Dic59—21no—
170C4⋊Dic59—21no—
171C4⋊Dic59—21no—
172C4⋊Dic59—21no—
173C4⋊Dic59—21no—
174C4⋊Dic59—21no—
175C4⋊Dic59—21no—
176C4⋊Dic59—21no—
177C4⋊Dic59—21no—
178C4⋊Dic59C22×C412yes—
179C4⋊Dic59C22×C412yes—
180C4⋊Dic59C22×C412yes—
181C4⋊Dic59C22×C412yes—
182Dic59⋊C4—42no—
183Dic59⋊C4—42no—
184Dic59⋊C4—81no—
185Dic59⋊C4—81no—
186C4×Dic59—24no—
187C4×Dic59—81no—
188C4×Dic59—81no—
189C4×Dic59—81no—
190C4×Dic59—81no—
191C23×C59—21no—
192C23×C59—21no—
193C23×C59—21no—
194C23×C59—21no—
195C23×C59—21no—
196C23×C59—21no—
197C23×C59C4×D412yes—
198C23×C59C22×D411yescharacteristic
199C2×C4×C59—21no—
200C2×C4×C59—21no—
201C2×C4×C59—21no—
202C2×C4×C59—21no—
203C2×C4×C59—22no—
204C2×C4×C59—22no—
205C2×C4×C59—22no—
206C2×C4×C59—22no—
207C2×C4×C59—21no—
208C2×C4×C59—21no—
209C2×C4×C59—21no—
210C2×C4×C59—21no—
211C2×C4×C59—21no—
212C2×C4×C59—21no—
213C2×C4×C59—21no—
214C2×C4×C59—21no—
215C2×C4×C59—21no—
216C2×C4×C59C4×D412yes—
217C2×C4×C59#13712yes—
218C2×C4×C59C23×C411yescharacteristic
219C2×C4×C59C2×#4111yescharacteristic
220C2×C4×C59C2×#4111yescharacteristic
221C2×C4×C59C2×#4111yescharacteristic
222C2×C4×C592−1+411yescharacteristic
223C2×C4×C592−1+411yescharacteristic
224C2×Dic59—22no—
225C2×Dic59—22no—
226C2×Dic59—22no—
227C2×Dic59—22no—
228C2×Dic59—41no—
229C2×Dic59—41no—
230C2×Dic59—41no—
231C2×Dic59—41no—
232C2×Dic59—41no—
233C2×Dic59—41no—
234C2×Dic59—41no—
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×Dic59—41no—
248C2×Dic59—84no—
249C8×C59—22no—
250C8×C59—41no—
251C8×C59—41no—
252C8×C59#11714yes—
253#22623—591nomaximal, Sylow
254C22×C59—22no—
255C22×C59—22no—
256C22×C59—22no—
257C22×C59—41no—
258C22×C59—41no—
259C22×C59—41no—
260C22×C59—41no—
261C22×C59C2×C4×D411yescharacteristic
262C22×C59#54711yescharacteristic
263C22×C59#54711yescharacteristic
264C22×C59C2×#13711yescharacteristic
265C22×C59#57511yescharacteristic
266C22×C59#57611yescharacteristic
267C22×C59#57711yescharacteristic
268C4×C59—22no—
269C4×C59—22no—
270C4×C59—41no—
271C4×C59—41no—
272C4×C59—41no—
273C4×C59C2×#11711yescharacteristic
274C4×C59#51911yescharacteristic
275C4×C59#55211yescharacteristic
276C4×C59#55211yescharacteristic
277Dic59—42no—
278Dic59—42no—
279Dic59—42no—
280Dic59—42no—
281Dic59—81no—
282Dic59—81no—
283Dic59—81no—
284Dic59—81no—
285C2×#428—591no—
286#2347—591no—
287#2345—591no—
288#1944—591no—
289#1941—591no—
290#1921—591no—
291#1920—591no—
292#1919—591no—
293#1918—591no—
294#1825—591no—
295#1823—591no—
296#1809—591no—
297#1776—591no—
298C4×#107—591no—
299#1753D5911yescharacteristic
300C2×C59—42no—
301C2×C59#234411yescharacteristic
302C2×C59C4×#13711yescharacteristic
303C2×C59#294611yescharacteristic
304C2×C59#334611yescharacteristic
305C2×C59#335011yescharacteristic
306C2×C59#335511yescharacteristic
307C2×C59#335611yescharacteristic
308C22×M16C2×D5911yescharacteristic
309C2×#117—591no—
310C22×C4⋊C4—591no—
311#428—598no—
312C2×#107—591no—
313C2×#107—591no—
314C2×#107—591no—
315C2×#106—591no—
316C2×#106—591no—
317C2×#106—591no—
318C2×C22⋊C8C2×D5911yescharacteristic
319C2×#97C2×D5911yescharacteristic
320C2×C4×C8C2×D5911yescharacteristic
321#389—591no—
322#389—591no—
323#389—591no—
324#389—591no—
325#388—1181no—
326#388—1181no—
327#387—1181no—
328#387—1181no—
329#383—1181no—
330#380—591no—
331#380—591no—
332#380—1181no—
333C4×C4⋊C4—591no—
334C4×C4⋊C4—591no—
335C4×C4⋊C4—591no—
336C4×C4⋊C4—591no—
337C4×C4⋊C4—1181no—
338C4×C22⋊C4—1181no—
339C4×C22⋊C4C2×D5911yescharacteristic
340#340—591no—
341#340—591no—
342#340—591no—
343#340—591no—
344#340—591no—
345#340—591no—
346#340—591no—
347#340—591no—
348#340—591no—
349#340—591no—
350#336C2×D5911yescharacteristic
351#336C2×D5911yescharacteristic
352C59#2262311yescharacteristic, complemented, Sylow
353C23×C4—1181no—
354C23×C4C22×D5911yescharacteristic
355C2×M16—22no—
356C2×M16—22no—
357C2×M16C4×D5914yes—
358C22×C8C22×D5911yescharacteristic
359C22×C8C22×D5911yescharacteristic
360C22×C8C22×D5911yescharacteristic
361C22×C8C22×D5911yescharacteristic
362#117—594no—
363#117—1182no—
364C2×C4⋊C4—594no—
365C2×C4⋊C4—591no—
366C2×C4⋊C4—591no—
367C2×C4⋊C4—591no—
368C2×C4⋊C4—591no—
369C2×C4⋊C4—591no—
370C2×C4⋊C4—591no—
371C2×C4⋊C4—591no—
372C2×C4⋊C4—591no—
373C2×C4⋊C4—1181no—
374C2×C4⋊C4—1182no—
375C2×C4⋊C4—1181no—
376C2×C4⋊C4—1181no—
377C2×C4⋊C4—1181no—
378C2×C4⋊C4—1181no—
379C2×C22⋊C4—1181no—
380C2×C22⋊C4—1181no—
381C2×C22⋊C4—1181no—
382C2×C22⋊C4D23611yescharacteristic
383C2×C22⋊C4D23611yescharacteristic
384C2×C42—1181no—
385C2×C42—1181no—
386C2×C42—1181no—
387C2×C42—1181no—
388C2×C42—1181no—
389C2×C42C22×D5911yescharacteristic
390C2×C42C22×D5911yescharacteristic
391#107—594no—
392#107—594no—
393#107—1182no—
394#106—594no—
395#106—594no—
396#106—1182no—
397C22⋊C8—22no—
398#97—22no—
399C4×C8—22no—
400#95—1181no—
401#95—1181no—
402#95—1181no—
403#95—1181no—
404#95—1181no—
405#95—1181no—
406#95—1181no—
407#95—1181no—
408#95—1181no—
409#95—1181no—
410#95D23611yescharacteristic
411#95D23611yescharacteristic
412C24C2×D23611yescharacteristic
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—1184no—
418C22×C4—1181no—
419C22×C4—1181no—
420C22×C4—1181no—
421C22×C4—1181no—
422C22×C4—1181no—
423C22×C4—1181no—
424C22×C4—1181no—
425C22×C4—1181no—
426C22×C4—4721no—
427C22×C4—4721no—
428C22×C4C2×C4×D5912yes—
429C22×C4C2×D23611yescharacteristic
430C22×C4C2×D23611yescharacteristic
431C22×C4C2×D23611yescharacteristic
432C22×C4C2×D23611yescharacteristic
433C22×C4C2×D23611yescharacteristic
434C22×C4#1174568512yes—
435C22×C4C23×D5911yescharacteristic
436M16—28no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8C2×C4×D5912yes—
446C2×C8C2×C4×D5912yes—
447C2×C8#1174568712yes—
448C2×C8#1174568712yes—
449C4⋊C4—592no—
450C4⋊C4—592no—
451C4⋊C4—592no—
452C4⋊C4—592no—
453C4⋊C4—1184no—
454C4⋊C4—1181no—
455C4⋊C4—1181no—
456C4⋊C4—1181no—
457C4⋊C4—1181no—
458C4⋊C4—1181no—
459C4⋊C4—1181no—
460C4⋊C4—1181no—
461C4⋊C4—1181no—
462C4⋊C4—1181no—
463C4⋊C4—1181no—
464C4⋊C4—1181no—
465C4⋊C4—1181no—
466C4⋊C4—4721no—
467C4⋊C4—4721no—
468C4⋊C4—2362no—
469C4⋊C4—2362no—
470C22⋊C4—22no—
471C22⋊C4—41no—
472C22⋊C4—1184no—
473C22⋊C4—4721no—
474C22⋊C4—2362no—
475C42—21no—
476C42—21no—
477C42—21no—
478C42—21no—
479C42—1184no—
480C42—4721no—
481C42—4721no—
482C42—4721no—
483C42—4721no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23C4×D23612yes—
491C23C22×D23611yescharacteristic
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—1182no—
510C2×C4—1182no—
511C2×C4—1182no—
512C2×C4—1182no—
513C2×C4—4724no—
514C2×C4—2361no—
515C2×C4—2361no—
516C2×C4—2361no—
517C2×C4—2361no—
518C2×C4—2361no—
519C2×C4—2361no—
520C2×C4—2361no—
521C2×C4—2361no—
522C2×C4—2361no—
523C2×C4—2361no—
524C2×C4—2361no—
525C2×C4—2361no—
526C2×C4—2361no—
527C2×C4—2361no—
528C2×C4—2361no—
529C2×C4—2361no—
530C2×C4—2361no—
531C2×C4—2361no—
532C2×C4—2361no—
533C2×C4—2361no—
534C2×C4C4×D23612yes—
535C2×C4#1427376012yes—
536C2×C4C22×C4×D5911yescharacteristic
537C2×C4C2×#1174568711yescharacteristic
538C2×C4C2×#1174568711yescharacteristic
539C2×C4C2×#1174568511yescharacteristic
540C2×C4#1427377411yescharacteristic
541C2×C4#1427377411yescharacteristic
542C8—22no—
543C8—41no—
544C8—41no—
545C8#1427368314yes—
546C22—22no—
547C22—22no—
548C22—22no—
549C22—41no—
550C22—41no—
551C22—41no—
552C22—41no—
553C22C2×C4×D23611yescharacteristic
554C22#1683805611yescharacteristic
555C22#1683805611yescharacteristic
556C22C2×#1427376011yescharacteristic
557C22#1683820211yescharacteristic
558C22#1683820711yescharacteristic
559C22#1683821411yescharacteristic
560C4—22no—
561C4—22no—
562C4—41no—
563C4—41no—
564C4—41no—
565C4—4721no—
566C4—4721no—
567C4—4721no—
568C4—4721no—
569C4—2362no—
570C4—2362no—
571C4—2362no—
572C4—2362no—
573C4C2×#1427368311yescharacteristic
574C4#1683787911yescharacteristic
575C4#1683809211yescharacteristic
576C4#1683809211yescharacteristic
577C2—42no—
578C2#2142696311yescharacteristic
579C2C4×#1427376011yescharacteristic
580C2#2143293011yescharacteristic
581C2#2143658311yescharacteristic
582C2#2143664311yescharacteristic
583C2#2143670111yescharacteristic
584C2#2143671211yescharacteristic
subgroupG/H
centerC23C22×D236
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1753C2
socleC23×C59C22×D4
prime radicalC24×C59C2×D4
cosocleC22×C4×C59C24
hypercenter#1753D59
hypocenterC59#22623
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1753

Complements

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

NH ≅ G/Ntype
4. #21424839C2semidirect
5. #21424824C2semidirect
6. #21424689C2semidirect
7. #21424682C2semidirect
8. #21424675C2semidirect
9. #21424668C2semidirect
13. #21423993C2semidirect
14. C4×#14273661C2semidirect
18. #16837444C4semidirect
352. C59#22623semidirect

Maximal subgroups

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

subgroupindex
C2×#168374442
#214269872
#214269712
#214248392
#214248242
#214246892
#214246822
#214246752
#214246682
#214241922
#214241832
#214241292
#214239932
C4×#142736612
C59×#17532
#2262359