Finite Group Databasebeta

← #28637303

Subgroups — #28637303

Order 15104.

850 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837732C211yesmaximal, characteristic, complemented
2C2×#16837646C211yesmaximal, characteristic
3#21427121C211yesmaximal, characteristic, complemented
4#21425578C211yesmaximal, characteristic, complemented
5#21425515C211yesmaximal, characteristic, complemented
6#21425470C211yesmaximal, characteristic, complemented
7#21425413C211yesmaximal, characteristic, complemented
8#21424862C211yesmaximal, characteristic, complemented
9#21424720C211yesmaximal, characteristic, complemented
10#21424454C211yesmaximal, characteristic, complemented
11#21424419C211yesmaximal, characteristic
12#21424281C211yesmaximal, characteristic, complemented
13#21424108C211yesmaximal, characteristic, complemented
14#21423815C211yesmaximal, characteristic, complemented
15C59×#2504C211yesmaximal, characteristic
16C22×#11745670C2211yescharacteristic
17C22×#11745664C2211yescharacteristic
18C2×#14273706C2211yescharacteristic
19C2×C4×Dic118C2211yescharacteristic
20C2×#14273686C2211yescharacteristic
21#16837732—24no—
22#16837646—24no—
23C2×#14273660C2211yescharacteristic
24C2×#14273658C2211yescharacteristic
25C2×#14273658C2211yescharacteristic
26C2×C4⋊(C59⋊C8)C2211yescharacteristic
27C2×C4⋊(C59⋊C8)C2211yescharacteristic
28C2×#14273650C2211yescharacteristic
29C2×#14273650C2211yescharacteristic
30C2×#14273650C2211yescharacteristic
31C2×#14273648C2211yescharacteristic
32C2×#14273648C2211yescharacteristic
33C2×#14273648C2211yescharacteristic
34C2×C22⋊(C59⋊C8)C2211yescharacteristic
35#16837308C2211yescharacteristic
36#16837300C2211yescharacteristic
37#16837296—21no—
38#16837290—21no—
39#16837284—21no—
40C4×C4⋊Dic59C2211yescharacteristic
41#16837253—21no—
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837157C2211yescharacteristic
49#16837157C2211yescharacteristic
50C2×#121×C59C2211yescharacteristic
51C22×C4⋊C4×C59C2211yescharacteristic
52C59×#397—21no—
53C59×#397—21no—
54C59×#390C2211yescharacteristic
55C59×#390C2211yescharacteristic
56C59×#386—21no—
57C59×#386—21no—
58C59×#382C2211yescharacteristic
59C59×#382C2211yescharacteristic
60C4×C22⋊C4×C59C2211yescharacteristic
61C22×Dic118C2311yescharacteristic
62C2×#11745670—22no—
63C2×#11745670—22no—
64C2×#11745670—41no—
65C2×#11745664—22no—
66C2×#11745664—22no—
67C2×#11745664—41no—
68C22×C59⋊C8C2311yescharacteristic
69C22×C59⋊C8C2311yescharacteristic
70C22×C59⋊C8C2311yescharacteristic
71C22×C59⋊C8C2311yescharacteristic
72#14273706—22no—
73#14273706D414yes—
74C4×Dic118—22no—
75C4×Dic118—41no—
76#14273686—22no—
77#14273686—41no—
78C2×C4⋊Dic59C2311yescharacteristic
79C2×C4⋊Dic59C2311yescharacteristic
80C2×C4⋊Dic59C2311yescharacteristic
81C2×Dic59⋊C4—21no—
82C2×Dic59⋊C4—21no—
83C2×Dic59⋊C4—21no—
84C2×C22⋊Dic59—21no—
85C2×C22⋊Dic59—21no—
86C2×C4×Dic59—21no—
87C2×C4×Dic59—21no—
88C2×C4×Dic59—21no—
89#14273660—22no—
90#14273658—22no—
91#14273658—22no—
92C4⋊(C59⋊C8)—22no—
93C4⋊(C59⋊C8)—41no—
94#14273650—22no—
95#14273650—22no—
96#14273650—22no—
97#14273648—22no—
98#14273648—22no—
99#14273648—41no—
100C22⋊(C59⋊C8)—22no—
101C59⋊#95—21no—
102C59⋊#95—21no—
103C59⋊#95—21no—
104C59⋊#95—21no—
105C22×D4×C59—21no—
106C22×D4×C59C2311yescharacteristic
107C23×C4×C59—21no—
108C23×C4×C59C2311yescharacteristic
109#121×C59—22no—
110#121×C59—22no—
111C2×C4⋊C4×C59—22no—
112C2×C4⋊C4×C59—22no—
113C2×C4⋊C4×C59—21no—
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×C59—21no—
120C2×C22⋊C4×C59—21no—
121C2×C22⋊C4×C59—21no—
122C2×C22⋊C4×C59D411yescharacteristic
123C2×C22⋊C4×C59D411yescharacteristic
124C2×C42×C59C2311yescharacteristic
125C2×C42×C59C2311yescharacteristic
126#95×C59—21no—
127#95×C59—21no—
128#95×C59—21no—
129#95×C59—21no—
130#95×C59D411yescharacteristic
131#95×C59D411yescharacteristic
132C24×C59—21no—
133C24×C59C2×D411yescharacteristic
134C2×D4×C59—21no—
135C2×D4×C59—21no—
136C2×D4×C59—21no—
137C2×D4×C59—21no—
138C2×D4×C59—21no—
139C2×D4×C59—21no—
140C2×D4×C59—42no—
141C2×D4×C59—42no—
142C2×D4×C59—42no—
143C22×C4×C59—21no—
144C22×C4×C59—21no—
145C22×C4×C59—21no—
146C22×C4×C59—21no—
147C22×C4×C59—21no—
148C22×C4×C59—21no—
149C22×C4×C59—21no—
150C22×C4×C59—42no—
151C22×C4×C59—81no—
152C22×C4×C59—81no—
153C22×C4×C59C2×D411yescharacteristic
154C22×C4×C59C2×D411yescharacteristic
155C22×C4×C59C2×D411yescharacteristic
156C22×C4×C59C2×D411yescharacteristic
157C22×C4×C59C2×D411yescharacteristic
158C22×C4×C59C2×D412yes—
159C22×C4×C59#4112yes—
160C22×C4×C59C2411yescharacteristic
161C22×Dic59—21no—
162C22×Dic59—21no—
163C22×Dic59—21no—
164C22×Dic59—21no—
165C2×Dic118—21no—
166C2×Dic118—21no—
167C2×Dic118—21no—
168C2×Dic118—21no—
169C2×Dic118—21no—
170C2×Dic118C2×D412yes—
171C4⋊C4×C59—21no—
172C4⋊C4×C59—21no—
173C4⋊C4×C59—21no—
174C4⋊C4×C59—21no—
175C4⋊C4×C59—21no—
176C4⋊C4×C59—21no—
177C4⋊C4×C59—42no—
178C22⋊C4×C59—41no—
179C22⋊C4×C59—41no—
180C22⋊C4×C59—42no—
181C22⋊C4×C59—81no—
182C22⋊C4×C59—81no—
183C22⋊C4×C59—81no—
184C22⋊C4×C59—81no—
185C42×C59—21no—
186C42×C59—21no—
187C42×C59—21no—
188C42×C59—21no—
189C22⋊Dic59—24no—
190C22⋊Dic59—42no—
191#11745670—44no—
192C4⋊Dic59—21no—
193C4⋊Dic59—21no—
194C4⋊Dic59—21no—
195C4⋊Dic59—21no—
196C4⋊Dic59C2×D412yes—
197C4⋊Dic59#4112yes—
198Dic59⋊C4—24no—
199Dic59⋊C4—42no—
200Dic59⋊C4—42no—
201C4×Dic59—42no—
202C4×Dic59—42no—
203C4×Dic59—42no—
204#11745664—44no—
205C2×C59⋊C8—21no—
206C2×C59⋊C8—21no—
207C2×C59⋊C8—21no—
208C2×C59⋊C8—21no—
209C2×C59⋊C8—21no—
210C2×C59⋊C8—21no—
211C2×C59⋊C8—21no—
212C2×C59⋊C8—21no—
213C2×C59⋊C8—21no—
214C2×C59⋊C8—21no—
215C2×C59⋊C8—21no—
216C2×C59⋊C8#4112yes—
217C23×C59—21no—
218C23×C59—21no—
219C23×C59—21no—
220C23×C59—21no—
221C23×C59—21no—
222C23×C59—21no—
223C23×C59—41no—
224C23×C59—41no—
225C23×C59—41no—
226C23×C59—41no—
227C23×C59—41no—
228C23×C59—41no—
229C23×C59—41no—
230C23×C59#12112yes—
231C23×C59C22×D411yescharacteristic
232D4×C59—41no—
233D4×C59—41no—
234D4×C59—41no—
235D4×C59—41no—
236D4×C59—84no—
237C2×C4×C59—21no—
238C2×C4×C59—21no—
239C2×C4×C59—22no—
240C2×C4×C59—22no—
241C2×C4×C59—22no—
242C2×C4×C59—22no—
243C2×C4×C59—21no—
244C2×C4×C59—21no—
245C2×C4×C59—21no—
246C2×C4×C59—21no—
247C2×C4×C59—21no—
248C2×C4×C59—21no—
249C2×C4×C59—21no—
250C2×C4×C59—21no—
251C2×C4×C59—21no—
252C2×C4×C59—21no—
253C2×C4×C59—21no—
254C2×C4×C59—21no—
255C2×C4×C59—41no—
256C2×C4×C59—41no—
257C2×C4×C59—41no—
258C2×C4×C59—41no—
259C2×C4×C59—41no—
260C2×C4×C59—41no—
261C2×C4×C59—41no—
262C2×C4×C59—41no—
263C2×C4×C59—41no—
264C2×C4×C59—84no—
265C2×C4×C59#12112yes—
266C2×C4×C59C22×D411yescharacteristic
267C2×C4×C59C2×#4111yescharacteristic
268C2×C4×C59C2×#4111yescharacteristic
269C2×C4×C59C2×#4111yescharacteristic
270C2×C4×C59H(5,𝔽2)11yescharacteristic
271C2×C4×C592−1+411yescharacteristic
272C2×Dic59—22no—
273C2×Dic59—22no—
274C2×Dic59—22no—
275C2×Dic59—41no—
276C2×Dic59—41no—
277C2×Dic59—41no—
278C2×Dic59—41no—
279C2×Dic59—41no—
280C2×Dic59—41no—
281C2×Dic59—41no—
282C2×Dic59—41no—
283C2×Dic59—41no—
284Dic118—22no—
285Dic118—22no—
286Dic118—41no—
287Dic118—41no—
288C59⋊C8—22no—
289C59⋊C8—41no—
290C59⋊C8—41no—
291C59⋊C8—41no—
292#25231—591nomaximal, Sylow
293C22×C59—22no—
294C22×C59—22no—
295C22×C59—22no—
296C22×C59—41no—
297C22×C59—41no—
298C22×C59—41no—
299C22×C59—41no—
300C22×C59—81no—
301C22×C59—81no—
302C22×C59—81no—
303C22×C59—81no—
304C22×C59—81no—
305C22×C59—81no—
306C22×C59—81no—
307C22×C59C2×#12111yescharacteristic
308C22×C59#54611yescharacteristic
309C22×C59#54711yescharacteristic
310C22×C59#57511yescharacteristic
311C22×C59#57511yescharacteristic
312C22×C59#57711yescharacteristic
313C22×C59#57711yescharacteristic
314C4×C59—22no—
315C4×C59—41no—
316C4×C59—41no—
317C4×C59—41no—
318C4×C59—41no—
319C4×C59—81no—
320C4×C59—81no—
321C4×C59—81no—
322C4×C59#52911yescharacteristic
323C4×C59#53911yescharacteristic
324C4×C59#54611yescharacteristic
325C4×C59#54711yescharacteristic
326Dic59—42no—
327Dic59—42no—
328Dic59—42no—
329Dic59—81no—
330C2×#483—591no—
331C2×#468—591no—
332#2504D5911yescharacteristic
333#2362—591no—
334#2085—591no—
335#2074—591no—
336#2065—591no—
337#2055—591no—
338#1949—591no—
339#1926—591no—
340#1880—591no—
341#1875—591no—
342#1851—591no—
343#1805—591no—
344#1736—591no—
345C2×C59—42no—
346C2×C59—161no—
347C2×C59#242511yescharacteristic
348C2×C59#304011yescharacteristic
349C2×C59#304311yescharacteristic
350C2×C59#330711yescharacteristic
351C2×C59#332611yescharacteristic
352C2×C59#335011yescharacteristic
353C2×C59#335111yescharacteristic
354C22×SD16—591no—
355C22×M16—591no—
356C2×C22⋊Q8—591no—
357C2×#121C2×D5911yescharacteristic
358C2×C4×Q8—591no—
359C2×#117—591no—
360C22×C4⋊C4C2×D5911yescharacteristic
361#483—1184no—
362#468—1184no—
363C2×#107—591no—
364C2×#106—591no—
365C2×#106—591no—
366C2×C4⋊C8—591no—
367C2×C4⋊C8—591no—
368C2×#103—591no—
369C2×#103—591no—
370C2×#103—591no—
371C2×#102—591no—
372C2×#102—591no—
373C2×#102—591no—
374C2×C22⋊C8—591no—
375#397—21no—
376#397—21no—
377#391—591no—
378#390C2×D5911yescharacteristic
379#390C2×D5911yescharacteristic
380#389—591no—
381#388—1181no—
382#388—1181no—
383#387—1181no—
384#386—21no—
385#386—21no—
386#382C2×D5911yescharacteristic
387#382C2×D5911yescharacteristic
388C4×C4⋊C4—591no—
389C4×C4⋊C4—1181no—
390C4×C22⋊C4C2×D5911yescharacteristic
391#340—591no—
392#340—591no—
393#340—591no—
394#340—591no—
395#340—591no—
396#340—591no—
397#336—591no—
398#336—591no—
399C59#2523111yescharacteristic, complemented, Sylow
400C22×Q8—591no—
401C22×D4—21no—
402C22×D4C22×D5911yescharacteristic
403C23×C4—21no—
404C23×C4C22×D5911yescharacteristic
405C2×SD16—1182no—
406C2×SD16—1182no—
407C2×SD16—2361no—
408C2×M16—1182no—
409C2×M16—1182no—
410C2×M16—2361no—
411C22×C8—591no—
412C22×C8—591no—
413C22×C8—591no—
414C22×C8—591no—
415C22⋊Q8—594no—
416C22⋊Q8—1182no—
417#121—22no—
418#121—22no—
419C4×Q8—1182no—
420C4×Q8—2361no—
421#117—1182no—
422#117—2361no—
423C2×C4⋊C4—22no—
424C2×C4⋊C4—22no—
425C2×C4⋊C4—21no—
426C2×C4⋊C4—591no—
427C2×C4⋊C4—591no—
428C2×C4⋊C4—591no—
429C2×C4⋊C4—1181no—
430C2×C4⋊C4—1181no—
431C2×C4⋊C4—1181no—
432C2×C4⋊C4C22×D5911yescharacteristic
433C2×C4⋊C4C22×D5911yescharacteristic
434C2×C4⋊C4C22×D5911yescharacteristic
435C2×C22⋊C4—21no—
436C2×C22⋊C4—21no—
437C2×C22⋊C4—21no—
438C2×C22⋊C4—21no—
439C2×C22⋊C4—21no—
440C2×C22⋊C4—1181no—
441C2×C22⋊C4—1181no—
442C2×C22⋊C4C22⋊D5911yescharacteristic
443C2×C22⋊C4C22⋊D5911yescharacteristic
444C2×C42—1181no—
445C2×C42—1181no—
446C2×C42—1181no—
447C2×C42C22×D5911yescharacteristic
448C2×C42C22×D5911yescharacteristic
449#107—1182no—
450#106—1182no—
451#106—1182no—
452C4⋊C8—1182no—
453C4⋊C8—2361no—
454#103—1182no—
455#103—1182no—
456#103—1182no—
457#102—1182no—
458#102—1182no—
459#102—2361no—
460C22⋊C8—1182no—
461#95—21no—
462#95—21no—
463#95—21no—
464#95—21no—
465#95—1181no—
466#95—1181no—
467#95—1181no—
468#95—1181no—
469#95C22⋊D5911yescharacteristic
470#95C22⋊D5911yescharacteristic
471C24—21no—
472C24C2×C22⋊D5911yescharacteristic
473C2×Q8—592no—
474C2×Q8—1181no—
475C2×Q8—1181no—
476C2×Q8—1181no—
477C2×Q8—1181no—
478C2×Q8—1181no—
479C2×D4—21no—
480C2×D4—21no—
481C2×D4—21no—
482C2×D4—21no—
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—42no—
486C2×D4—42no—
487C2×D4—42no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—42no—
496C22×C4—81no—
497C22×C4—81no—
498C22×C4—1181no—
499C22×C4—1181no—
500C22×C4—1181no—
501C22×C4—1181no—
502C22×C4D4×D5912yes—
503C22×C4#1174568712yes—
504C22×C4C2×C22⋊D5911yescharacteristic
505C22×C4C2×C22⋊D5911yescharacteristic
506C22×C4C2×C22⋊D5911yescharacteristic
507C22×C4C2×C22⋊D5911yescharacteristic
508C22×C4C2×C22⋊D5911yescharacteristic
509C22×C4C23×D5911yescharacteristic
510SD16—2364no—
511M16—2364no—
512C2×C8—592no—
513C2×C8—1181no—
514C2×C8—1181no—
515C2×C8—1181no—
516C2×C8—1181no—
517C2×C8—1181no—
518C2×C8—1181no—
519C2×C8—1181no—
520C2×C8—1181no—
521C2×C8—1181no—
522C2×C8—1181no—
523C2×C8—1181no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—42no—
531C4⋊C4—592no—
532C4⋊C4—592no—
533C4⋊C4—1184no—
534C4⋊C4—1181no—
535C4⋊C4—1181no—
536C4⋊C4—1181no—
537C4⋊C4—1181no—
538C4⋊C4—2362no—
539C4⋊C4—2362no—
540C22⋊C4—41no—
541C22⋊C4—41no—
542C22⋊C4—42no—
543C22⋊C4—81no—
544C22⋊C4—81no—
545C22⋊C4—81no—
546C22⋊C4—81no—
547C22⋊C4—1184no—
548C22⋊C4—2362no—
549C42—21no—
550C42—21no—
551C42—21no—
552C42—21no—
553C42—2362no—
554C42—2362no—
555C42—2362no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23—41no—
567C23—41no—
568C23—41no—
569C23#1427369812yes—
570C23C22×C22⋊D5911yescharacteristic
571Q8—1182no—
572Q8—1182no—
573Q8—2361no—
574Q8—2361no—
575D4—41no—
576D4—41no—
577D4—41no—
578D4—41no—
579D4—84no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—84no—
608C2×C4—1182no—
609C2×C4—1182no—
610C2×C4—1182no—
611C2×C4—2361no—
612C2×C4—2361no—
613C2×C4—2361no—
614C2×C4—2361no—
615C2×C4—2361no—
616C2×C4—2361no—
617C2×C4—2361no—
618C2×C4—2361no—
619C2×C4—2361no—
620C2×C4#1427369812yes—
621C2×C4C2×D4×D5911yescharacteristic
622C2×C4#41×D5911yescharacteristic
623C2×C4#41×D5911yescharacteristic
624C2×C4C2×#1174568711yescharacteristic
625C2×C4#1427377311yescharacteristic
626C2×C4#1427377411yescharacteristic
627C8—1182no—
628C8—2361no—
629C8—2361no—
630C8—2361no—
631C22—22no—
632C22—22no—
633C22—22no—
634C22—41no—
635C22—41no—
636C22—41no—
637C22—41no—
638C22—81no—
639C22—81no—
640C22—81no—
641C22—81no—
642C22—81no—
643C22—81no—
644C22—81no—
645C22C2×#1427369811yescharacteristic
646C22#1683804011yescharacteristic
647C22#1683805111yescharacteristic
648C22#1683820011yescharacteristic
649C22#1683820011yescharacteristic
650C22#1683821311yescharacteristic
651C22#1683821311yescharacteristic
652C4—22no—
653C4—41no—
654C4—41no—
655C4—41no—
656C4—41no—
657C4—81no—
658C4—81no—
659C4—81no—
660C4—4721no—
661C4—2362no—
662C4—2362no—
663C4—2362no—
664C4#1683792911yescharacteristic
665C4#1683799611yescharacteristic
666C4#1683804411yescharacteristic
667C4#1683805011yescharacteristic
668C2—42no—
669C2—161no—
670C2#2142764311yescharacteristic
671C2#2143358611yescharacteristic
672C2#2143361311yescharacteristic
673C2#2143606611yescharacteristic
674C2#2143634311yescharacteristic
675C2#2143663511yescharacteristic
676C2#2143665011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2504C2
socleC23×C59C22×D4
prime radicalC2×#121×C59C22
hypercenter#2504D59
hypocenterC59#25231
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2504

Complements

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

NH ≅ G/Ntype
1. C2×#16837732C2semidirect
3. #21427121C2semidirect
4. #21425578C2semidirect
5. #21425515C2semidirect
6. #21425470C2semidirect
7. #21425413C2semidirect
8. #21424862C2semidirect
9. #21424720C2semidirect
10. #21424454C2semidirect
12. #21424281C2semidirect
13. #21424108C2semidirect
14. #21423815C2semidirect
399. C59#25231semidirect

Maximal subgroups

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

subgroupindex
C2×#168377322
C2×#168376462
#214271212
#214255782
#214255152
#214254702
#214254132
#214248622
#214247202
#214244542
#214244192
#214242812
#214241082
#214238152
C59×#25042
#2523159