Finite Group Databasebeta

← #28637076

Subgroups — #28637076

Order 15104.

900 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837723C211yesmaximal, characteristic
2C2×#16837646C211yesmaximal, characteristic
3#21428549C211yesmaximal, characteristic, complemented
4#21425567C211yesmaximal, characteristic, complemented
5#21425509C211yesmaximal, characteristic, complemented
6#21425474C211yesmaximal, characteristic, complemented
7#21425413C211yesmaximal, characteristic, complemented
8#21424864C211yesmaximal, characteristic, complemented
9#21424737C211yesmaximal, characteristic, complemented
10#21424430C211yesmaximal, characteristic, complemented
11#21424403C211yesmaximal, characteristic, complemented
12#21424281C211yesmaximal, characteristic, complemented
13#21424096C211yesmaximal, characteristic, complemented
14#21423815C211yesmaximal, characteristic, complemented
15C59×#2361C211yesmaximal, characteristic
16C22×#11745670C2211yescharacteristic
17C22×#11745664C2211yescharacteristic
18C2×#14273706C2211yescharacteristic
19C2×#14273686C2211yescharacteristic
20#16837723—24no—
21#16837646—24no—
22C2×#14273660C2211yescharacteristic
23C2×#14273660C2211yescharacteristic
24C2×#14273658C2211yescharacteristic
25C2×C4⋊(C59⋊C8)C2211yescharacteristic
26C2×C4⋊(C59⋊C8)C2211yescharacteristic
27C2×#14273650C2211yescharacteristic
28C2×#14273650C2211yescharacteristic
29C2×#14273650C2211yescharacteristic
30C2×#14273648C2211yescharacteristic
31C2×#14273648C2211yescharacteristic
32C2×#14273648C2211yescharacteristic
33C2×C22⋊(C59⋊C8)C2211yescharacteristic
34#16837327—21no—
35#16837327—21no—
36#16837308C2211yescharacteristic
37#16837308C2211yescharacteristic
38#16837287—21no—
39#16837287—21no—
40#16837272C2211yescharacteristic
41#16837268C2211yescharacteristic
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837157C2211yescharacteristic
49#16837157C2211yescharacteristic
50C2×#121×C59C2211yescharacteristic
51C2×C4×D4×C59—21no—
52C2×C4×D4×C59C2211yescharacteristic
53C22×C4⋊C4×C59C2211yescharacteristic
54C59×#390C2211yescharacteristic
55C59×#389C2211yescharacteristic
56C59×#388—21no—
57C59×#386—21no—
58C59×#385—21no—
59C4×C4⋊C4×C59C2211yescharacteristic
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#14273706—22no—
74#14273686—22no—
75#14273686—22no—
76C2×C4⋊Dic59—21no—
77C2×C4⋊Dic59—21no—
78C2×C4⋊Dic59C2311yescharacteristic
79C2×C4⋊Dic59C2311yescharacteristic
80C2×C4⋊Dic59C2311yescharacteristic
81C2×Dic59⋊C4—21no—
82C2×C22⋊Dic59—21no—
83C2×C22⋊Dic59—21no—
84C2×C4×Dic59—21no—
85#14273660—22no—
86#14273660—22no—
87#14273658—22no—
88C4⋊(C59⋊C8)—22no—
89C4⋊(C59⋊C8)—41no—
90#14273650—22no—
91#14273650—22no—
92#14273650—41no—
93#14273648—22no—
94#14273648—22no—
95#14273648—22no—
96C22⋊(C59⋊C8)—22no—
97C59⋊#95—21no—
98C59⋊#95—21no—
99C59⋊#95—21no—
100C59⋊#95—21no—
101C59⋊#95—21no—
102C59⋊#95—21no—
103C22×D4×C59—21no—
104C22×D4×C59C2311yescharacteristic
105C23×C4×C59—21no—
106C23×C4×C59—21no—
107C23×C4×C59C2311yescharacteristic
108#121×C59—22no—
109#121×C59D414yes—
110C4×D4×C59—22no—
111C4×D4×C59—41no—
112C4×D4×C59—44no—
113C2×C4⋊C4×C59—22no—
114C2×C4⋊C4×C59—21no—
115C2×C4⋊C4×C59—21no—
116C2×C4⋊C4×C59—41no—
117C2×C4⋊C4×C59C2311yescharacteristic
118C2×C4⋊C4×C59C2311yescharacteristic
119C2×C4⋊C4×C59C2311yescharacteristic
120C2×C22⋊C4×C59—21no—
121C2×C22⋊C4×C59—21no—
122C2×C22⋊C4×C59—21no—
123C2×C22⋊C4×C59—21no—
124C2×C22⋊C4×C59D411yescharacteristic
125C2×C22⋊C4×C59D411yescharacteristic
126C2×C42×C59—21no—
127C2×C42×C59C2311yescharacteristic
128C2×C42×C59C2311yescharacteristic
129#95×C59—21no—
130#95×C59—21no—
131#95×C59D411yescharacteristic
132#95×C59D411yescharacteristic
133C24×C59—21no—
134C24×C59C2×D411yescharacteristic
135C2×D4×C59—24no—
136C2×D4×C59—21no—
137C2×D4×C59—21no—
138C2×D4×C59—21no—
139C2×D4×C59—21no—
140C2×D4×C59—21no—
141C2×D4×C59—42no—
142C2×D4×C59—42no—
143C2×D4×C59C2×D412yes—
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—21no—
151C22×C4×C59—42no—
152C22×C4×C59—42no—
153C22×C4×C59—42no—
154C22×C4×C59—42no—
155C22×C4×C59—81no—
156C22×C4×C59—81no—
157C22×C4×C59C2×D411yescharacteristic
158C22×C4×C59C2×D411yescharacteristic
159C22×C4×C59C2×D411yescharacteristic
160C22×C4×C59C2×D411yescharacteristic
161C22×C4×C59C2×D412yes—
162C22×C4×C59C2×D411yescharacteristic
163C22×C4×C59#4112yes—
164C22×C4×C59C2411yescharacteristic
165C22×Dic59—21no—
166C22×Dic59—21no—
167C22×Dic59—21no—
168C22×Dic59—21no—
169C2×Dic118—21no—
170C2×Dic118—21no—
171C2×Dic118—21no—
172C2×Dic118—21no—
173C2×Dic118—21no—
174C2×Dic118—21no—
175C4⋊C4×C59—21no—
176C4⋊C4×C59—21no—
177C4⋊C4×C59—21no—
178C4⋊C4×C59—21no—
179C4⋊C4×C59—42no—
180C4⋊C4×C59—42no—
181C4⋊C4×C59C2×D412yes—
182C4⋊C4×C59#4112yes—
183C22⋊C4×C59—24no—
184C22⋊C4×C59—22no—
185C22⋊C4×C59—41no—
186C22⋊C4×C59—42no—
187C22⋊C4×C59—42no—
188C22⋊C4×C59—81no—
189C42×C59—21no—
190C42×C59—21no—
191C42×C59—21no—
192C42×C59—21no—
193C42×C59—42no—
194C22⋊Dic59—42no—
195C22⋊Dic59—42no—
196#11745670—44no—
197C4⋊Dic59—21no—
198C4⋊Dic59—21no—
199C4⋊Dic59—21no—
200C4⋊Dic59—21no—
201C4⋊Dic59—21no—
202C4⋊Dic59—21no—
203C4⋊Dic59—42no—
204C4⋊Dic59—42no—
205Dic59⋊C4—42no—
206C4×Dic59—42no—
207#11745664—44no—
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—21no—
217C2×C59⋊C8—21no—
218C2×C59⋊C8—21no—
219C2×C59⋊C8#4112yes—
220C23×C59—21no—
221C23×C59—21no—
222C23×C59—21no—
223C23×C59—21no—
224C23×C59—21no—
225C23×C59—21no—
226C23×C59—22no—
227C23×C59—41no—
228C23×C59—41no—
229C23×C59—41no—
230C23×C59—41no—
231C23×C59—41no—
232C23×C59—41no—
233C23×C59#12112yes—
234C23×C59C22×D411yescharacteristic
235D4×C59—22no—
236D4×C59—22no—
237D4×C59—48no—
238D4×C59—41no—
239D4×C59—41no—
240C2×C4×C59—21no—
241C2×C4×C59—21no—
242C2×C4×C59—21no—
243C2×C4×C59—21no—
244C2×C4×C59—22no—
245C2×C4×C59—22no—
246C2×C4×C59—22no—
247C2×C4×C59—22no—
248C2×C4×C59—22no—
249C2×C4×C59—21no—
250C2×C4×C59—22no—
251C2×C4×C59—21no—
252C2×C4×C59—21no—
253C2×C4×C59—21no—
254C2×C4×C59—21no—
255C2×C4×C59—21no—
256C2×C4×C59—21no—
257C2×C4×C59—21no—
258C2×C4×C59—21no—
259C2×C4×C59—41no—
260C2×C4×C59—41no—
261C2×C4×C59—41no—
262C2×C4×C59—41no—
263C2×C4×C59—41no—
264C2×C4×C59—41no—
265C2×C4×C59—41no—
266C2×C4×C59—84no—
267C2×C4×C59—84no—
268C2×C4×C59#12112yes—
269C2×C4×C59AGL(1,ℤ/8)12yes—
270C2×C4×C59C22×D411yescharacteristic
271C2×C4×C59C2×#4111yescharacteristic
272C2×C4×C59C2×#4111yescharacteristic
273C2×C4×C59C2×#4111yescharacteristic
274C2×C4×C59H(5,𝔽2)11yescharacteristic
275C2×C4×C592−1+411yescharacteristic
276C2×Dic59—41no—
277C2×Dic59—41no—
278C2×Dic59—41no—
279C2×Dic59—41no—
280C2×Dic59—41no—
281C2×Dic59—41no—
282C2×Dic59—41no—
283C2×Dic59—41no—
284C2×Dic59—41no—
285C2×Dic59—41no—
286C2×Dic59—41no—
287C2×Dic59—41no—
288Dic118—41no—
289Dic118—41no—
290Dic118—41no—
291Dic118—41no—
292C59⋊C8—22no—
293C59⋊C8—41no—
294C59⋊C8—41no—
295C59⋊C8—41no—
296#25216—591nomaximal, Sylow
297C22×C59—22no—
298C22×C59—22no—
299C22×C59—22no—
300C22×C59—41no—
301C22×C59—41no—
302C22×C59—41no—
303C22×C59—41no—
304C22×C59—42no—
305C22×C59—42no—
306C22×C59—42no—
307C22×C59—81no—
308C22×C59—81no—
309C22×C59—81no—
310C22×C59—81no—
311C22×C59C2×#12111yescharacteristic
312C22×C59#54611yescharacteristic
313C22×C59#54711yescharacteristic
314C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
315C22×C59#57511yescharacteristic
316C22×C59#57711yescharacteristic
317C22×C59#57811yescharacteristic
318C4×C59—22no—
319C4×C59—22no—
320C4×C59—41no—
321C4×C59—41no—
322C4×C59—42no—
323C4×C59—42no—
324C4×C59—41no—
325C4×C59—81no—
326C4×C59#52911yescharacteristic
327C4×C59#53911yescharacteristic
328C4×C59#54611yescharacteristic
329C4×C59#54711yescharacteristic
330Dic59—81no—
331Dic59—81no—
332Dic59—81no—
333Dic59—81no—
334C2×#482—591no—
335C2×#468—591no—
336#2499—591no—
337#2361D5911yescharacteristic
338#2083—591no—
339#2073—591no—
340#2066—591no—
341#2055—591no—
342#1949—591no—
343#1928—591no—
344#1877—591no—
345#1872—591no—
346#1851—591no—
347#1802—591no—
348#1736—591no—
349C2×C59—42no—
350C2×C59—82no—
351C2×C59#242511yescharacteristic
352C2×C59#303611yescharacteristic
353C2×C59#304611yescharacteristic
354C2×C59#331011yescharacteristic
355C2×C59#332711yescharacteristic
356C2×C59#335011yescharacteristic
357C2×C59#335311yescharacteristic
358C22×SD16—591no—
359C22×M16—591no—
360C2×C22⋊Q8—591no—
361C2×#121C2×D5911yescharacteristic
362C2×C4×D4—21no—
363C2×C4×D4C2×D5911yescharacteristic
364C2×#117—591no—
365C22×C4⋊C4C2×D5911yescharacteristic
366#482—1184no—
367#468—1184no—
368C2×#107—591no—
369C2×#107—591no—
370C2×#106—591no—
371C2×C4⋊C8—591no—
372C2×C4⋊C8—591no—
373C2×#103—591no—
374C2×#103—591no—
375C2×#103—591no—
376C2×#102—591no—
377C2×#102—591no—
378C2×#102—591no—
379C2×C22⋊C8—591no—
380#397—1181no—
381#397—1181no—
382#391—591no—
383#391—591no—
384#390C2×D5911yescharacteristic
385#389C2×D5911yescharacteristic
386#388—21no—
387#387—1181no—
388#387—1181no—
389#386—21no—
390#385—21no—
391#384—591no—
392#382—591no—
393C4×C4⋊C4C2×D5911yescharacteristic
394C4×C22⋊C4C2×D5911yescharacteristic
395#340—591no—
396#340—591no—
397#340—591no—
398#340—591no—
399#340—591no—
400#340—591no—
401#336—591no—
402#336—591no—
403C59#2521611yescharacteristic, complemented, Sylow
404C22×Q8—591no—
405C22×D4—21no—
406C22×D4C22×D5911yescharacteristic
407C23×C4—21no—
408C23×C4—21no—
409C23×C4C22×D5911yescharacteristic
410C2×SD16—1182no—
411C2×SD16—1182no—
412C2×SD16—2361no—
413C2×M16—1182no—
414C2×M16—1182no—
415C2×M16—2361no—
416C22×C8—591no—
417C22×C8—591no—
418C22×C8—591no—
419C22×C8—591no—
420C22⋊Q8—1182no—
421C22⋊Q8—1182no—
422#121—22no—
423#121D23614yes—
424C4×D4—22no—
425C4×D4—41no—
426C4×D4—44no—
427#117—1182no—
428#117—1182no—
429C2×C4⋊C4—22no—
430C2×C4⋊C4—21no—
431C2×C4⋊C4—21no—
432C2×C4⋊C4—41no—
433C2×C4⋊C4—591no—
434C2×C4⋊C4—591no—
435C2×C4⋊C4—591no—
436C2×C4⋊C4—1181no—
437C2×C4⋊C4—1181no—
438C2×C4⋊C4—1181no—
439C2×C4⋊C4C22×D5911yescharacteristic
440C2×C4⋊C4C22×D5911yescharacteristic
441C2×C4⋊C4C22×D5911yescharacteristic
442C2×C22⋊C4—21no—
443C2×C22⋊C4—21no—
444C2×C22⋊C4—21no—
445C2×C22⋊C4—21no—
446C2×C22⋊C4—1181no—
447C2×C22⋊C4—1181no—
448C2×C22⋊C4C22⋊D5911yescharacteristic
449C2×C22⋊C4C22⋊D5911yescharacteristic
450C2×C42—21no—
451C2×C42—1181no—
452C2×C42C22×D5911yescharacteristic
453C2×C42C22×D5911yescharacteristic
454#107—1182no—
455#107—1182no—
456#106—1182no—
457C4⋊C8—1182no—
458C4⋊C8—2361no—
459#103—1182no—
460#103—1182no—
461#103—2361no—
462#102—1182no—
463#102—1182no—
464#102—1182no—
465C22⋊C8—1182no—
466#95—21no—
467#95—21no—
468#95—1181no—
469#95—1181no—
470#95—1181no—
471#95—1181no—
472#95—1181no—
473#95—1181no—
474#95C22⋊D5911yescharacteristic
475#95C22⋊D5911yescharacteristic
476C24—21no—
477C24C2×C22⋊D5911yescharacteristic
478C2×Q8—1181no—
479C2×Q8—1181no—
480C2×Q8—1181no—
481C2×Q8—1181no—
482C2×Q8—1181no—
483C2×Q8—1181no—
484C2×D4—24no—
485C2×D4—21no—
486C2×D4—21no—
487C2×D4—21no—
488C2×D4—21no—
489C2×D4—21no—
490C2×D4—42no—
491C2×D4—42no—
492C2×D4C2×D23612yes—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—42no—
501C22×C4—42no—
502C22×C4—42no—
503C22×C4—42no—
504C22×C4—81no—
505C22×C4—81no—
506C22×C4—1181no—
507C22×C4—1181no—
508C22×C4—1181no—
509C22×C4—1181no—
510C22×C4C2×D23612yes—
511C22×C4#1174568512yes—
512C22×C4C2×C22⋊D5911yescharacteristic
513C22×C4C2×C22⋊D5911yescharacteristic
514C22×C4C2×C22⋊D5911yescharacteristic
515C22×C4C2×C22⋊D5911yescharacteristic
516C22×C4C2×C22⋊D5911yescharacteristic
517C22×C4C23×D5911yescharacteristic
518SD16—2364no—
519M16—2364no—
520C2×C8—592no—
521C2×C8—1181no—
522C2×C8—1181no—
523C2×C8—1181no—
524C2×C8—1181no—
525C2×C8—1181no—
526C2×C8—1181no—
527C2×C8—1181no—
528C2×C8—1181no—
529C2×C8—1181no—
530C2×C8—1181no—
531C2×C8—1181no—
532C4⋊C4—21no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C4⋊C4—21no—
536C4⋊C4—42no—
537C4⋊C4—42no—
538C4⋊C4—1181no—
539C4⋊C4—1181no—
540C4⋊C4—1181no—
541C4⋊C4—1181no—
542C4⋊C4—1181no—
543C4⋊C4—1181no—
544C4⋊C4—2362no—
545C4⋊C4—2362no—
546C4⋊C4—2362no—
547C4⋊C4C2×D23612yes—
548C4⋊C4#1174568712yes—
549C22⋊C4—24no—
550C22⋊C4—22no—
551C22⋊C4—41no—
552C22⋊C4—42no—
553C22⋊C4—42no—
554C22⋊C4—81no—
555C22⋊C4—2362no—
556C22⋊C4—2362no—
557C42—21no—
558C42—21no—
559C42—21no—
560C42—21no—
561C42—42no—
562C42—2362no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—21no—
569C23—22no—
570C23—41no—
571C23—41no—
572C23—41no—
573C23—41no—
574C23—41no—
575C23—41no—
576C23#1427370112yes—
577C23C22×C22⋊D5911yescharacteristic
578Q8—2361no—
579Q8—2361no—
580Q8—2361no—
581Q8—2361no—
582D4—22no—
583D4—22no—
584D4—48no—
585D4—41no—
586D4—41no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—22no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—84no—
614C2×C4—84no—
615C2×C4—2361no—
616C2×C4—2361no—
617C2×C4—2361no—
618C2×C4—2361no—
619C2×C4—2361no—
620C2×C4—2361no—
621C2×C4—2361no—
622C2×C4—2361no—
623C2×C4—2361no—
624C2×C4—2361no—
625C2×C4—2361no—
626C2×C4—2361no—
627C2×C4#1427370112yes—
628C2×C4#1427375312yes—
629C2×C4C22×D23611yescharacteristic
630C2×C4#41×D5911yescharacteristic
631C2×C4C2×#1174568711yescharacteristic
632C2×C4C2×#1174568511yescharacteristic
633C2×C4#1427377211yescharacteristic
634C2×C4#1427377411yescharacteristic
635C8—1182no—
636C8—2361no—
637C8—2361no—
638C8—2361no—
639C22—22no—
640C22—22no—
641C22—22no—
642C22—41no—
643C22—41no—
644C22—41no—
645C22—41no—
646C22—42no—
647C22—42no—
648C22—42no—
649C22—81no—
650C22—81no—
651C22—81no—
652C22—81no—
653C22C2×#1427370111yescharacteristic
654C22#1683804611yescharacteristic
655C22#1683805111yescharacteristic
656C22C2×#1427375311yescharacteristic
657C22#1683820011yescharacteristic
658C22#1683821311yescharacteristic
659C22#1683821711yescharacteristic
660C4—22no—
661C4—22no—
662C4—41no—
663C4—41no—
664C4—42no—
665C4—42no—
666C4—41no—
667C4—81no—
668C4—4721no—
669C4—4721no—
670C4—4721no—
671C4—4721no—
672C4#1683793211yescharacteristic
673C4#1683799411yescharacteristic
674C4#1683804711yescharacteristic
675C4#1683805611yescharacteristic
676C2—42no—
677C2—82no—
678C2#2142764111yescharacteristic
679C2#2143356011yescharacteristic
680C2#2143364611yescharacteristic
681C2#2143611711yescharacteristic
682C2#2143636411yescharacteristic
683C2#2143663511yescharacteristic
684C2#2143668011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2361C2
socleC23×C59C22×D4
prime radicalC2×#121×C59C22
hypercenter#2361D59
hypocenterC59#25216
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2361

Complements

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

NH ≅ G/Ntype
3. #21428549C2semidirect
4. #21425567C2semidirect
5. #21425509C2semidirect
6. #21425474C2semidirect
7. #21425413C2semidirect
8. #21424864C2semidirect
9. #21424737C2semidirect
10. #21424430C2semidirect
11. #21424403C2semidirect
12. #21424281C2semidirect
13. #21424096C2semidirect
14. #21423815C2semidirect
403. C59#25216semidirect

Maximal subgroups

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

subgroupindex
C2×#168377232
C2×#168376462
#214285492
#214255672
#214255092
#214254742
#214254132
#214248642
#214247372
#214244302
#214244032
#214242812
#214240962
#214238152
C59×#23612
#2521659