Finite Group Databasebeta

← #28588486

Subgroups — #28588486

Order 15104.

1774 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4⋊C4×D59C211yesmaximal, characteristic
2C2×#16837365C211yesmaximal, characteristic
3#21426475C211yesmaximal, characteristic
4#21424185C214yesmaximal, complemented
5#21424181C214yesmaximal, complemented
6#21423817C211yesmaximal, characteristic, complemented
7#21423816C211yesmaximal, characteristic, complemented
8#21423815C211yesmaximal, characteristic, complemented
9C59×#1736C211yesmaximal, characteristic, complemented
10C23×C4×D59—21no—
11C23×C4×D59C2211yescharacteristic
12C22×#11745664C2211yescharacteristic
13C2×#14273686C2211yescharacteristic
14C22×C4⋊Dic59C2211yescharacteristic
15C22×Dic59⋊C4—21no—
16C2×C4⋊C4×D59—21no—
17C2×C4⋊C4×D59—21no—
18C2×C4⋊C4×D59—24no—
19C2×C4⋊C4×D59C418yescomplemented
20C2×C4⋊C4×D59C2212yes—
21C2×C4⋊C4×D59C2212yes—
22#16837365—24no—
23C2×(C2×D59)⋊C8C2212yes—
24C2×(C2×D59)⋊C8C2212yes—
25#16837263—21no—
26#16837259C2212yes—
27#16837259C2212yes—
28#16837257—21no—
29C4×(C2×D59)⋊C4—21no—
30C4×(C2×D59)⋊C4—21no—
31#16837165C2214yescomplemented
32#16837164C2214yescomplemented
33#16837164C2214yescomplemented
34C22×M16×C59C2211yescharacteristic
35C2×#117×C59C2211yescharacteristic
36C22×C4⋊C4×C59C2211yescharacteristic
37C59×#340C2214yescomplemented
38C24×D59D411yescharacteristic
39C23×Dic59—21no—
40C23×Dic59D411yescharacteristic
41C22×C4×D59—22no—
42C22×C4×D59—22no—
43C22×C4×D59—22no—
44C22×C4×D59—22no—
45C22×C4×D59—28no—
46C22×C4×D59—22no—
47C22×C4×D59—22no—
48C22×C4×D59—48no—
49C22×C4×D59C2×C414yes—
50C22×C4×D59D411yescharacteristic
51C22×C4×D59D411yescharacteristic
52C22×C4×D59D414yes—
53C22×C4×D59C2311yescharacteristic
54C22×C4×D59C2311yescharacteristic
55C2×#11745664—22no—
56C2×#11745664—22no—
57C2×#11745664—22no—
58C22×C59⋊C8C2312yes—
59#14273686—22no—
60#14273686—22no—
61C2×C4⋊Dic59—21no—
62C2×C4⋊Dic59—22no—
63C2×C4⋊Dic59C2×C414yes—
64C2×C4⋊Dic59C2312yes—
65C2×C4⋊Dic59C2312yes—
66C2×Dic59⋊C4—28no—
67C2×Dic59⋊C4—22no—
68C2×Dic59⋊C4—22no—
69C4⋊C4×D59—28no—
70C4⋊C4×D59—28no—
71C4⋊C4×D59—24no—
72C4⋊C4×D59—24no—
73C4⋊C4×D59C2×C418yes—
74C4⋊C4×D59C2×C418yes—
75C2×C22⋊Dic59—21no—
76C2×(C2×D59)⋊C4—22no—
77C2×(C2×D59)⋊C4—22no—
78C2×C4×Dic59—21no—
79(C2×D59)⋊C8—24no—
80(C2×D59)⋊C8—24no—
81C59⋊#95—22no—
82C59⋊#95—22no—
83C23×C4×C59—21no—
84C23×C4×C59C2311yescharacteristic
85C2×M16×C59—22no—
86C2×M16×C59—22no—
87C2×M16×C59—22no—
88C22×C8×C59C2312yes—
89#117×C59—22no—
90#117×C59—22no—
91C2×C4⋊C4×C59—21no—
92C2×C4⋊C4×C59—22no—
93C2×C4⋊C4×C59C2×C414yes—
94C2×C4⋊C4×C59C2312yes—
95C2×C4⋊C4×C59C2312yes—
96C2×C22⋊C4×C59—21no—
97C2×C42×C59—21no—
98C24×C59C2×D411yescharacteristic
99C23×D59—22no—
100C23×D59—22no—
101C23×D59—24no—
102C23×D59—24no—
103C23×D59C22⋊C414yes—
104C23×D59C2×D411yescharacteristic
105C23×D59C2×D411yescharacteristic
106C22×C4×C59—24no—
107C22×C4×C59—22no—
108C22×C4×C59—22no—
109C22×C4×C59—44no—
110C22×C4×C59C22×C412yes—
111C22×C4×C59C2×D411yescharacteristic
112C22×C4×C59C2×D412yes—
113C22×C4×C59#4112yes—
114C22×C4×C59#4112yes—
115C22×C4×C59#4112yes—
116C22×C4×C59#4112yes—
117C22×C4×C59C2411yescharacteristic
118C22×Dic59—22no—
119C22×Dic59—22no—
120C22×Dic59—24no—
121C22×Dic59—22no—
122C22×Dic59—22no—
123C22×Dic59—44no—
124C22×Dic59C22⋊C414yes—
125C22×Dic59C2×D411yescharacteristic
126C22×Dic59C2×D411yescharacteristic
127C2×C4×D59—28no—
128C2×C4×D59—28no—
129C2×C4×D59—28no—
130C2×C4×D59—28no—
131C2×C4×D59—22no—
132C2×C4×D59—22no—
133C2×C4×D59—22no—
134C2×C4×D59—22no—
135C2×C4×D59—21no—
136C2×C4×D59—21no—
137C2×C4×D59—21no—
138C2×C4×D59—21no—
139C2×C4×D59—21no—
140C2×C4×D59—21no—
141C2×C4×D59—21no—
142C2×C4×D59—21no—
143C2×C4×D59—28no—
144C2×C4×D59—28no—
145C2×C4×D59—48no—
146C2×C4×D59—432no—
147C2×C4×D59—48no—
148C2×C4×D59C22⋊C414yes—
149C2×C4×D59C22⋊C414yes—
150C2×C4×D59C22×C412yes—
151C2×C4×D59C22×C412yes—
152C2×C4×D59C2×D412yes—
153C2×C4×D59C2×D412yes—
154M16×C59—44no—
155C2×C8×C59—22no—
156C2×C8×C59—22no—
157C2×C8×C59—22no—
158C4⋊C4×C59—24no—
159C4⋊C4×C59—22no—
160C4⋊C4×C59—22no—
161C4⋊C4×C59—22no—
162C4⋊C4×C59C22×C414yes—
163C22⋊C4×C59—42no—
164C42×C59—42no—
165C22⋊Dic59—42no—
166(C2×D59)⋊C4—44no—
167(C2×D59)⋊C4—44no—
168C4⋊Dic59—24no—
169C4⋊Dic59—22no—
170C4⋊Dic59—22no—
171C4⋊Dic59—22no—
172C4⋊Dic59C22×C414yes—
173Dic59⋊C4—28no—
174Dic59⋊C4—28no—
175C4×Dic59—42no—
176#11745664—44no—
177C2×C59⋊C8—22no—
178C2×C59⋊C8—22no—
179C2×C59⋊C8—22no—
180C23×C59—22no—
181C23×C59—22no—
182C23×C59C2×C22⋊C412yes—
183C23×C59#12312yes—
184C23×C59#12312yes—
185C23×C59C22×D411yescharacteristic
186C22×D59—216no—
187C22×D59—216no—
188C22×D59—21no—
189C22×D59—21no—
190C22×D59—21no—
191C22×D59—21no—
192C22×D59—22no—
193C22×D59—22no—
194C22×D59—28no—
195C22×D59—28no—
196C22×D59—28no—
197C22×D59C2×C22⋊C412yes—
198C22×D59C2×C22⋊C412yes—
199C22×D59AGL(1,ℤ/8)14yes—
200C22×D59#13714yes—
201C2×C4×C59—24no—
202C2×C4×C59—48no—
203C2×C4×C59—44no—
204C2×C4×C59—42no—
205C2×C4×C59—42no—
206C2×C4×C59—42no—
207C2×C4×C59C2×C22⋊C412yes—
208C2×C4×C59#11714yes—
209C2×C4×C59#11714yes—
210C2×C4×C59#12314yes—
211C2×C4×C59#12312yes—
212C2×C4×C59#12314yes—
213C2×C4×C59#12312yes—
214C2×C4×C59C23×C411yescharacteristic
215C2×C4×C59C22×D411yescharacteristic
216C2×C4×C59C2×#4111yescharacteristic
217C2×C4×C59C2×#4111yescharacteristic
218C2×C4×C59C2×#4111yescharacteristic
219C2×C4×C59C2×#4111yescharacteristic
220C2×Dic59—21no—
221C2×Dic59—21no—
222C2×Dic59—21no—
223C2×Dic59—21no—
224C2×Dic59—28no—
225C2×Dic59—24no—
226C2×Dic59—48no—
227C2×Dic59—44no—
228C2×Dic59—42no—
229C2×Dic59—42no—
230C2×Dic59—42no—
231C2×Dic59C2×C22⋊C412yes—
232C2×Dic59C2×C22⋊C412yes—
233C4×D59—22no—
234C4×D59—22no—
235C4×D59—22no—
236C4×D59—22no—
237C4×D59—22no—
238C4×D59—22no—
239C4×D59—22no—
240C4×D59—22no—
241C4×D59—28no—
242C4×D59—28no—
243C4×D59—416no—
244C4×D59—416no—
245C8×C59—42no—
246C59⋊C8—42no—
247#20700—591nomaximal, Sylow
248C22×C59—24no—
249C22×C59—24no—
250C22×C59—22no—
251C22×C59—22no—
252C22×C59#38114yes—
253C22×C59C22×C22⋊C411yescharacteristic
254C22×C59C2×#12311yescharacteristic
255C22×C59C2×#12311yescharacteristic
256C22×C59C2×AGL(1,ℤ/8)12yes—
257C22×C59C2×#13712yes—
258C2×D59—232no—
259C2×D59—24no—
260C2×D59—24no—
261C2×D59—24no—
262C2×D59—24no—
263C2×D59—22no—
264C2×D59—22no—
265C2×D59—22no—
266C2×D59—22no—
267C2×D59#41814yes—
268C2×D59#41914yes—
269C4×C59—44no—
270C4×C59—82no—
271C4×C59#38114yes—
272C4×C59C2×#11711yescharacteristic
273C4×C59C2×#11711yescharacteristic
274C4×C59C2×#12311yescharacteristic
275C4×C59C2×#12311yescharacteristic
276Dic59—22no—
277Dic59—22no—
278Dic59—44no—
279Dic59—82no—
280C23×C4⋊C4—591no—
281C2×#407—591no—
282#2280—591no—
283#1824—594no—
284#1823—594no—
285#1736—591no—
286#1736—591no—
287#1736—591no—
288#1736D5911yescharacteristic, complemented
289C2×C59—24no—
290C2×C59C2×#38111yescharacteristic
291C2×C59C2×#41811yescharacteristic
292C2×C59C2×#41911yescharacteristic
293C2×C59#309212yes—
294C2×C59#309212yes—
295D59—24no—
296D59—24no—
297C24×C4—591no—
298C24×C4—1181no—
299C22×M16—591no—
300C22×M16C2×D5911yescharacteristic
301C2×#117—591no—
302C2×#117C2×D5911yescharacteristic
303C22×C4⋊C4—591no—
304C22×C4⋊C4—598no—
305C22×C4⋊C4—592no—
306C22×C4⋊C4—592no—
307C22×C4⋊C4—1181no—
308C22×C4⋊C4—1181no—
309C22×C4⋊C4—1184no—
310C22×C4⋊C4—1181no—
311C22×C4⋊C4C2×D5911yescharacteristic
312#407—1184no—
313C2×C22⋊C8—592no—
314C2×C22⋊C8—592no—
315#381—1181no—
316#380—592no—
317#380—592no—
318#379—1181no—
319C4×C22⋊C4—1181no—
320C4×C22⋊C4—1181no—
321#340—594no—
322#340—594no—
323#340—594no—
324#340C2×D5914yescomplemented
325C59#2070011yescharacteristic, complemented, Sylow
326C25—591no—
327C23×C4—21no—
328C23×C4—591no—
329C23×C4—594no—
330C23×C4—591no—
331C23×C4—591no—
332C23×C4—594no—
333C23×C4—591no—
334C23×C4—591no—
335C23×C4—1188no—
336C23×C4—1182no—
337C23×C4—1182no—
338C23×C4—1182no—
339C23×C4—1182no—
340C23×C4—1182no—
341C23×C4—1182no—
342C23×C4—1181no—
343C23×C4—2368no—
344C23×C4C22×D5911yescharacteristic
345C2×M16—22no—
346C2×M16—22no—
347C2×M16—22no—
348C2×M16—1182no—
349C2×M16—1182no—
350C2×M16—1182no—
351C22×C8—592no—
352C22×C8C22×D5912yes—
353#117—22no—
354#117—22no—
355#117—1182no—
356#117—1182no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—22no—
359C2×C4⋊C4—594no—
360C2×C4⋊C4—592no—
361C2×C4⋊C4—598no—
362C2×C4⋊C4—598no—
363C2×C4⋊C4—592no—
364C2×C4⋊C4—1188no—
365C2×C4⋊C4—1188no—
366C2×C4⋊C4—1184no—
367C2×C4⋊C4—1184no—
368C2×C4⋊C4—1188no—
369C2×C4⋊C4—1182no—
370C2×C4⋊C4—1182no—
371C2×C4⋊C4—1181no—
372C2×C4⋊C4—1182no—
373C2×C4⋊C4C4×D5914yes—
374C2×C4⋊C4C22×D5912yes—
375C2×C4⋊C4C22×D5912yes—
376C2×C22⋊C4—21no—
377C2×C22⋊C4—1182no—
378C2×C22⋊C4—1182no—
379C2×C22⋊C4—1181no—
380C2×C42—21no—
381C2×C42—1181no—
382C22⋊C8—1184no—
383C22⋊C8—1184no—
384#95—1182no—
385#95—1182no—
386C24—591no—
387C24—591no—
388C24—594no—
389C24—1184no—
390C24—1184no—
391C24—1182no—
392C24—1182no—
393C24D4×D5911yescharacteristic
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—44no—
398C22×C4—594no—
399C22×C4—591no—
400C22×C4—591no—
401C22×C4—594no—
402C22×C4—594no—
403C22×C4—592no—
404C22×C4—592no—
405C22×C4—592no—
406C22×C4—592no—
407C22×C4—1188no—
408C22×C4—1188no—
409C22×C4—1182no—
410C22×C4—1182no—
411C22×C4—1188no—
412C22×C4—1188no—
413C22×C4—1182no—
414C22×C4—1182no—
415C22×C4—1188no—
416C22×C4—1182no—
417C22×C4—1182no—
418C22×C4—1188no—
419C22×C4—1181no—
420C22×C4—1181no—
421C22×C4—1181no—
422C22×C4—1181no—
423C22×C4—1181no—
424C22×C4—1181no—
425C22×C4—1181no—
426C22×C4—1181no—
427C22×C4—1184no—
428C22×C4—1182no—
429C22×C4—1182no—
430C22×C4—2368no—
431C22×C4—23632no—
432C22×C4—2368no—
433C22×C4—2364no—
434C22×C4C2×C4×D5912yes—
435C22×C4#1174568512yes—
436C22×C4#1174568512yes—
437C22×C4D4×D5911yescharacteristic
438C22×C4D4×D5912yes—
439C22×C4#1174568712yes—
440C22×C4#1174568912yes—
441C22×C4C23×D5911yescharacteristic
442M16—44no—
443M16—2364no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—1182no—
448C2×C8—1182no—
449C2×C8—1182no—
450C4⋊C4—24no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—594no—
455C4⋊C4—1188no—
456C4⋊C4—1188no—
457C4⋊C4—1184no—
458C4⋊C4—1182no—
459C4⋊C4—1182no—
460C4⋊C4—1182no—
461C4⋊C4C2×C4×D5914yes—
462C22⋊C4—42no—
463C22⋊C4—2364no—
464C22⋊C4—2364no—
465C22⋊C4—2362no—
466C42—42no—
467C42—2362no—
468C23—22no—
469C23—22no—
470C23—594no—
471C23—594no—
472C23—592no—
473C23—592no—
474C23—1188no—
475C23—1188no—
476C23—11816no—
477C23—11816no—
478C23—1181no—
479C23—1181no—
480C23—1181no—
481C23—1181no—
482C23—1182no—
483C23—1182no—
484C23—1188no—
485C23C22⋊C4×D5912yes—
486C23#1427370912yes—
487C23#1427371212yes—
488C23C2×D4×D5911yescharacteristic
489C2×C4—24no—
490C2×C4—48no—
491C2×C4—44no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—592no—
496C2×C4—592no—
497C2×C4—1181no—
498C2×C4—1181no—
499C2×C4—1181no—
500C2×C4—1181no—
501C2×C4—1188no—
502C2×C4—1182no—
503C2×C4—1182no—
504C2×C4—1182no—
505C2×C4—1182no—
506C2×C4—1188no—
507C2×C4—1188no—
508C2×C4—1182no—
509C2×C4—1182no—
510C2×C4—1182no—
511C2×C4—1182no—
512C2×C4—1184no—
513C2×C4—23616no—
514C2×C4—23616no—
515C2×C4—2362no—
516C2×C4—2362no—
517C2×C4—2362no—
518C2×C4—2368no—
519C2×C4—2364no—
520C2×C4C22⋊C4×D5912yes—
521C2×C4#1427368414yes—
522C2×C4#1427368514yes—
523C2×C4#1427370914yes—
524C2×C4#1427370912yes—
525C2×C4#1427371214yes—
526C2×C4#1427371212yes—
527C2×C4C22×C4×D5911yescharacteristic
528C2×C4C2×D4×D5911yescharacteristic
529C2×C4C2×#1174568711yescharacteristic
530C2×C4C2×#1174568511yescharacteristic
531C2×C4C2×#1174568511yescharacteristic
532C2×C4C2×#1174568911yescharacteristic
533C8—42no—
534C8—2362no—
535C22—22no—
536C22—22no—
537C22—24no—
538C22—24no—
539C22—594no—
540C22—594no—
541C22—11832no—
542C22—1182no—
543C22—1182no—
544C22—1182no—
545C22—1182no—
546C22—1184no—
547C22—1184no—
548C22—1184no—
549C22—1184no—
550C22#1683726414yes—
551C22C2×C22⋊C4×D5911yescharacteristic
552C22C2×#1427370911yescharacteristic
553C22C2×#1427371211yescharacteristic
554C22AGL(1,ℤ/8)×D5912yes—
555C22#137×D5912yes—
556C4—44no—
557C4—82no—
558C4—1182no—
559C4—1182no—
560C4—4722no—
561C4—2364no—
562C4#1683726414yes—
563C4C2×#1427368411yescharacteristic
564C4C2×#1427368511yescharacteristic
565C4C2×#1427370911yescharacteristic
566C4C2×#1427371211yescharacteristic
567C2—24no—
568C2—1184no—
569C2—1184no—
570C2C2×#1683726411yescharacteristic
571C2#418×D5911yescharacteristic
572C2#419×D5911yescharacteristic
573C2#2143401712yes—
574C2#2143402212yes—
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1736C2
socleC23×C59C22×D4
prime radicalC24×D59D4
cosocleC22×C4×C59C24
hypercenter#1736D59
hypocenterC59#20700
LCSG ▷ C2×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. #21424185C2semidirect
5. #21424181C2semidirect
6. #21423817C2semidirect
7. #21423816C2semidirect
8. #21423815C2semidirect
9. C59×#1736C2semidirect
19. C2×C4⋊C4×D59C4semidirect
31. #16837165C22semidirect
32. #16837164C22semidirect
33. #16837164C22semidirect
37. C59×#340C22semidirect
288. #1736D59semidirect
324. #340C2×D59semidirect
325. C59#20700semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊C4×D592
C2×#168373652
#214264752
#214241852
#214241812
#214238172
#214238162
#214238152
C59×#17362
#2070059