Finite Group Databasebeta

← #28645851

Subgroups — #28645851

Order 15104.

946 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837709C211yesmaximal, characteristic, complemented
2C22×#14273651C211yesmaximal, characteristic
3#21427820C211yesmaximal, characteristic
4#21425644C211yesmaximal, characteristic, complemented
5#21425491C211yesmaximal, characteristic, complemented
6#21425477C211yesmaximal, characteristic, complemented
7#21425247C211yesmaximal, characteristic, complemented
8#21424364C211yesmaximal, characteristic, complemented
9#21424350C211yesmaximal, characteristic, complemented
10#21424283C211yesmaximal, characteristic
11#21424276C211yesmaximal, characteristic
12#21424119C211yesmaximal, characteristic
13#21424111C211yesmaximal, characteristic, complemented
14#21424031C211yesmaximal, characteristic
15Q8×C22⋊C4×C59C211yesmaximal, characteristic
16C23×C59⋊C8C2211yescharacteristic
17C2×#14273719C2211yescharacteristic
18C22×C4⋊Dic59C2211yescharacteristic
19#16837709—24no—
20C2×#14273660C2211yescharacteristic
21C2×#14273660C2211yescharacteristic
22C2×#14273658C2211yescharacteristic
23C2×#14273658C2211yescharacteristic
24C2×C4⋊(C59⋊C8)C2211yescharacteristic
25C2×C4⋊(C59⋊C8)C2211yescharacteristic
26C2×#14273651—24no—
27C2×#14273651C2211yescharacteristic
28C2×#14273651C2211yescharacteristic
29C2×#14273651C2211yescharacteristic
30C2×#14273651C2211yescharacteristic
31C2×#14273651C2211yescharacteristic
32C2×#14273651C2211yescharacteristic
33C2×#14273651C2211yescharacteristic
34C2×#14273651C2211yescharacteristic
35C2×C22⋊(C59⋊C8)C2211yescharacteristic
36#16837344—21no—
37#16837336—21no—
38#16837300C2211yescharacteristic
39#16837300C2211yescharacteristic
40#16837274—21no—
41#16837268C2211yescharacteristic
42#16837262C2211yescharacteristic
43#16837260—21no—
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837157C2211yescharacteristic
49#16837157C2211yescharacteristic
50C23×Q8×C59C2211yescharacteristic
51C2×C4×Q8×C59C2211yescharacteristic
52C2×C4×Q8×C59C2211yescharacteristic
53C59×#391—21no—
54C59×#391—21no—
55C59×#391C2211yescharacteristic
56C59×#391C2211yescharacteristic
57C59×#380—21no—
58C59×#380C2211yescharacteristic
59C4×C22⋊C4×C59—21no—
60C4×C22⋊C4×C59C2211yescharacteristic
61C23×Dic59—21no—
62C22×C59⋊C8—22no—
63C22×C59⋊C8—22no—
64C22×C59⋊C8—41no—
65C22×C59⋊C8C2311yescharacteristic
66C22×C59⋊C8C2311yescharacteristic
67C22×C59⋊C8C2311yescharacteristic
68C22×C59⋊C8C2311yescharacteristic
69#14273719—22no—
70#14273719—41no—
71C2×C4⋊Dic59—22no—
72C2×C4⋊Dic59—21no—
73C2×C4⋊Dic59—41no—
74C2×C4⋊Dic59C2311yescharacteristic
75C2×C4⋊Dic59C2311yescharacteristic
76C2×C4⋊Dic59C2311yescharacteristic
77C2×C4⋊Dic59C2311yescharacteristic
78C2×Dic59⋊C4—21no—
79C2×Dic59⋊C4—21no—
80C2×Dic59⋊C4—21no—
81C2×Dic59⋊C4—21no—
82C2×C22⋊Dic59—21no—
83#14273660—22no—
84#14273660—41no—
85#14273658—22no—
86#14273658—41no—
87C4⋊(C59⋊C8)—22no—
88C4⋊(C59⋊C8)—41no—
89#14273651—22no—
90#14273651—22no—
91#14273651—22no—
92#14273651—22no—
93#14273651—22no—
94#14273651—41no—
95#14273651—41no—
96#14273651D414yes—
97C22⋊(C59⋊C8)—22no—
98C59⋊#95—21no—
99C59⋊#95—21no—
100C59⋊#95—21no—
101C59⋊#95—21no—
102C59⋊#95—21no—
103C22×Q8×C59—21no—
104C22×Q8×C59—22no—
105C22×Q8×C59—22no—
106C22×Q8×C59—22no—
107C22×Q8×C59—41no—
108C22×Q8×C59—41no—
109C22×Q8×C59Q814yes—
110C22×Q8×C59C2311yescharacteristic
111C22×Q8×C59C2311yescharacteristic
112C23×C4×C59—21no—
113C23×C4×C59C2311yescharacteristic
114C4×Q8×C59—22no—
115C4×Q8×C59—22no—
116C4×Q8×C59—41no—
117C4×Q8×C59—41no—
118C2×C4⋊C4×C59—21no—
119C2×C4⋊C4×C59—21no—
120C2×C4⋊C4×C59C2311yescharacteristic
121C2×C4⋊C4×C59C2311yescharacteristic
122C2×C22⋊C4×C59—21no—
123C2×C22⋊C4×C59D411yescharacteristic
124C2×C22⋊C4×C59D411yescharacteristic
125C2×C42×C59—21no—
126C2×C42×C59—21no—
127C2×C42×C59C2311yescharacteristic
128C2×C42×C59C2311yescharacteristic
129#95×C59—21no—
130#95×C59—21no—
131#95×C59—21no—
132#95×C59—21no—
133#95×C59—21no—
134#95×C59D411yescharacteristic
135#95×C59D411yescharacteristic
136C24×C59C2×D411yescharacteristic
137C2×Q8×C59—24no—
138C2×Q8×C59—21no—
139C2×Q8×C59—21no—
140C2×Q8×C59—21no—
141C2×Q8×C59—21no—
142C2×Q8×C59—28no—
143C2×Q8×C59—21no—
144C2×Q8×C59—21no—
145C2×Q8×C59—21no—
146C2×Q8×C59—21no—
147C2×Q8×C59—42no—
148C2×Q8×C59—42no—
149C2×Q8×C59—44no—
150C2×Q8×C59—44no—
151C2×Q8×C59—44no—
152C2×Q8×C59C2×D412yes—
153C2×Q8×C59C2×Q812yes—
154C2×Q8×C59C2×Q812yes—
155C2×Q8×C59#4112yes—
156C22×C4×C59—21no—
157C22×C4×C59—21no—
158C22×C4×C59—21no—
159C22×C4×C59—21no—
160C22×C4×C59—21no—
161C22×C4×C59—24no—
162C22×C4×C59—21no—
163C22×C4×C59—21no—
164C22×C4×C59—21no—
165C22×C4×C59—42no—
166C22×C4×C59—42no—
167C22×C4×C59C2×D411yescharacteristic
168C22×C4×C59C2×D411yescharacteristic
169C22×C4×C59C2×D411yescharacteristic
170C22×C4×C59C2×D411yescharacteristic
171C22×C4×C59C2×D411yescharacteristic
172C22×C4×C59C2×Q812yes—
173C22×C4×C59#4112yes—
174C22×C4×C59C2411yescharacteristic
175C22×Dic59—21no—
176C22×Dic59—21no—
177C22×Dic59—21no—
178C22×Dic59—21no—
179C22×Dic59—42no—
180C22×Dic59—42no—
181C22×Dic59—81no—
182C4⋊C4×C59—21no—
183C4⋊C4×C59—21no—
184C4⋊C4×C59—21no—
185C4⋊C4×C59—21no—
186C4⋊C4×C59—42no—
187C4⋊C4×C59—42no—
188C22⋊C4×C59—22no—
189C22⋊C4×C59—22no—
190C22⋊C4×C59—42no—
191C42×C59—21no—
192C42×C59—21no—
193C42×C59—21no—
194C42×C59—21no—
195C42×C59—42no—
196C42×C59—42no—
197C22⋊Dic59—42no—
198C4⋊Dic59—21no—
199C4⋊Dic59—21no—
200C4⋊Dic59—21no—
201C4⋊Dic59—21no—
202C4⋊Dic59—21no—
203C4⋊Dic59—21no—
204C4⋊Dic59—21no—
205C4⋊Dic59—42no—
206C4⋊Dic59C2×D412yes—
207Dic59⋊C4—42no—
208Dic59⋊C4—42no—
209Dic59⋊C4—81no—
210Dic59⋊C4—81no—
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—21no—
220C2×C59⋊C8—21no—
221C2×C59⋊C8—21no—
222C2×C59⋊C8—44no—
223C2×C59⋊C8C2×D412yes—
224C23×C59—21no—
225C23×C59—21no—
226C23×C59—21no—
227C23×C59—21no—
228C23×C59C22⋊Q812yes—
229C23×C59#13512yes—
230C23×C59#13712yes—
231C23×C59C22×D411yescharacteristic
232Q8×C59—22no—
233Q8×C59—22no—
234Q8×C59—22no—
235Q8×C59—22no—
236Q8×C59—22no—
237Q8×C59—48no—
238Q8×C59—41no—
239Q8×C59—41no—
240Q8×C59C22⋊Q814yes—
241C2×C4×C59—22no—
242C2×C4×C59—21no—
243C2×C4×C59—22no—
244C2×C4×C59—22no—
245C2×C4×C59—21no—
246C2×C4×C59—22no—
247C2×C4×C59—22no—
248C2×C4×C59—22no—
249C2×C4×C59—22no—
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—21no—
260C2×C4×C59—21no—
261C2×C4×C59—21no—
262C2×C4×C59—21no—
263C2×C4×C59—48no—
264C2×C4×C59—41no—
265C2×C4×C59—41no—
266C2×C4×C59—41no—
267C2×C4×C59—41no—
268C2×C4×C59—41no—
269C2×C4×C59—41no—
270C2×C4×C59—41no—
271C2×C4×C59—41no—
272C2×C4×C59C22⋊Q812yes—
273C2×C4×C59C22×D411yescharacteristic
274C2×C4×C59C22×Q811yescharacteristic
275C2×C4×C59C2×#4111yescharacteristic
276C2×C4×C59C2×#4111yescharacteristic
277C2×C4×C59H(5,𝔽2)11yescharacteristic
278C2×C4×C59H(5,𝔽2)11yescharacteristic
279C2×Dic59—22no—
280C2×Dic59—41no—
281C2×Dic59—41no—
282C2×Dic59—41no—
283C2×Dic59—41no—
284C2×Dic59—41no—
285C2×Dic59—41no—
286C2×Dic59—41no—
287C2×Dic59—41no—
288C2×Dic59—41no—
289C2×Dic59—41no—
290C2×Dic59—41no—
291C2×Dic59—84no—
292C59⋊C8—22no—
293C59⋊C8—41no—
294C59⋊C8—41no—
295C59⋊C8—41no—
296#25957—591nomaximal, Sylow
297C22×C59—22no—
298C22×C59—22no—
299C22×C59—22no—
300C22×C59—22no—
301C22×C59—22no—
302C22×C59—22no—
303C22×C59#47914yescomplemented
304C22×C59C2×C22⋊Q811yescharacteristic
305C22×C59D4211yescharacteristic
306C22×C59#54611yescharacteristic
307C22×C59C2×#13511yescharacteristic
308C22×C59C2×#13711yescharacteristic
309C22×C59#57711yescharacteristic
310C22×C59#57811yescharacteristic
311C4×C59—22no—
312C4×C59—24no—
313C4×C59—42no—
314C4×C59—41no—
315C4×C59—41no—
316C4×C59—41no—
317C4×C59—41no—
318C4×C59—81no—
319C4×C59—81no—
320C4×C59C2×C22⋊Q811yescharacteristic
321C4×C59#54311yescharacteristic
322C4×C59#54611yescharacteristic
323C4×C59#55111yescharacteristic
324Dic59—42no—
325Dic59—81no—
326Dic59—81no—
327Dic59—81no—
328C2×#479—591no—
329C22×#103—591no—
330#2442—591no—
331Q8×C22⋊C4D5911yescharacteristic
332#2097—591no—
333#2068—591no—
334#2066—591no—
335#2026—591no—
336#1865—591no—
337#1863—591no—
338#1852—591no—
339#1850—591no—
340#1807—591no—
341#1805—591no—
342#1787—591no—
343C2×C59—24no—
344C2×C59#243711yescharacteristic
345C2×C59C2×#47911yescharacteristic
346C2×C59#308211yescharacteristic
347C2×C59#327011yescharacteristic
348C2×C59#328211yescharacteristic
349C2×C59#330011yescharacteristic
350C2×C59#332011yescharacteristic
351C23×Q8C2×D5911yescharacteristic
352C23×C8—591no—
353C2×#125—591no—
354C2×C4×Q8C2×D5911yescharacteristic
355C2×C4×Q8C2×D5911yescharacteristic
356C22×C4⋊C4—591no—
357#479—1184no—
358C2×#107—591no—
359C2×#107—591no—
360C2×#106—591no—
361C2×#106—591no—
362C2×C4⋊C8—591no—
363C2×C4⋊C8—591no—
364C2×#103—591no—
365C2×#103—591no—
366C2×#103—591no—
367C2×#103—591no—
368C2×#103—591no—
369C2×#103—591no—
370C2×#103—591no—
371C2×#103—591no—
372C2×#103—1184no—
373C2×C22⋊C8—591no—
374#400—1181no—
375#398—1181no—
376#391—21no—
377#391—21no—
378#391C2×D5911yescharacteristic
379#391C2×D5911yescharacteristic
380#389—591no—
381#389—591no—
382#385—1181no—
383#382—591no—
384#380—21no—
385#380—591no—
386#380—1181no—
387#380C2×D5911yescharacteristic
388C4×C22⋊C4—21no—
389C4×C22⋊C4C2×D5911yescharacteristic
390#340—591no—
391#340—591no—
392#340—591no—
393#340—591no—
394#336—591no—
395#336—591no—
396C59#2595711yescharacteristic, complemented, Sylow
397C22×Q8—22no—
398C22×Q8—21no—
399C22×Q8—22no—
400C22×Q8—22no—
401C22×Q8—41no—
402C22×Q8—41no—
403C22×Q8Dic11814yes—
404C22×Q8C22×D5911yescharacteristic
405C22×Q8C22×D5911yescharacteristic
406C23×C4—21no—
407C23×C4—1181no—
408C23×C4C22×D5911yescharacteristic
409C22×C8—591no—
410C22×C8—591no—
411C22×C8—591no—
412C22×C8—591no—
413C22×C8—1182no—
414C22×C8—1182no—
415C22×C8—2361no—
416#125—1182no—
417#125—2361no—
418C4×Q8—22no—
419C4×Q8—22no—
420C4×Q8—41no—
421C4×Q8—41no—
422C2×C4⋊C4—21no—
423C2×C4⋊C4—21no—
424C2×C4⋊C4—591no—
425C2×C4⋊C4—591no—
426C2×C4⋊C4—591no—
427C2×C4⋊C4—591no—
428C2×C4⋊C4—1182no—
429C2×C4⋊C4—1181no—
430C2×C4⋊C4—1181no—
431C2×C4⋊C4—1181no—
432C2×C4⋊C4—1181no—
433C2×C4⋊C4—1181no—
434C2×C4⋊C4—2361no—
435C2×C4⋊C4C22×D5911yescharacteristic
436C2×C4⋊C4C22×D5911yescharacteristic
437C2×C22⋊C4—21no—
438C2×C22⋊C4—1181no—
439C2×C22⋊C4C22⋊D5911yescharacteristic
440C2×C22⋊C4C22⋊D5911yescharacteristic
441C2×C42—21no—
442C2×C42—21no—
443C2×C42C22×D5911yescharacteristic
444C2×C42C22×D5911yescharacteristic
445#107—1182no—
446#107—2361no—
447#106—1182no—
448#106—2361no—
449C4⋊C8—1182no—
450C4⋊C8—2361no—
451#103—594no—
452#103—1182no—
453#103—1182no—
454#103—1182no—
455#103—1182no—
456#103—1182no—
457#103—2361no—
458#103—2361no—
459C22⋊C8—1182no—
460#95—21no—
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#95—1181no—
470#95C22⋊D5911yescharacteristic
471#95C22⋊D5911yescharacteristic
472C24C2×C22⋊D5911yescharacteristic
473C2×Q8—24no—
474C2×Q8—21no—
475C2×Q8—21no—
476C2×Q8—21no—
477C2×Q8—21no—
478C2×Q8—28no—
479C2×Q8—21no—
480C2×Q8—21no—
481C2×Q8—21no—
482C2×Q8—21no—
483C2×Q8—42no—
484C2×Q8—42no—
485C2×Q8—44no—
486C2×Q8—44no—
487C2×Q8—44no—
488C2×Q8C2×Dic11812yes—
489C2×Q8C2×Dic11812yes—
490C2×Q8D4×D5912yes—
491C2×Q8#1174568712yes—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—24no—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—42no—
502C22×C4—42no—
503C22×C4—1181no—
504C22×C4—1181no—
505C22×C4—1181no—
506C22×C4—1181no—
507C22×C4—4721no—
508C22×C4—2362no—
509C22×C4—2362no—
510C22×C4C2×Dic11812yes—
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
518C2×C8—592no—
519C2×C8—1181no—
520C2×C8—1181no—
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—2364no—
531C4⋊C4—21no—
532C4⋊C4—21no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C4⋊C4—42no—
536C4⋊C4—42no—
537C4⋊C4—592no—
538C4⋊C4—1181no—
539C4⋊C4—1181no—
540C4⋊C4—1181no—
541C4⋊C4—1181no—
542C4⋊C4—1181no—
543C4⋊C4—1181no—
544C4⋊C4—1181no—
545C4⋊C4—4721no—
546C4⋊C4—4721no—
547C4⋊C4—2362no—
548C4⋊C4—2362no—
549C4⋊C4—2362no—
550C22⋊C4—22no—
551C22⋊C4—22no—
552C22⋊C4—42no—
553C22⋊C4—2362no—
554C42—21no—
555C42—21no—
556C42—21no—
557C42—21no—
558C42—42no—
559C42—42no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23#1427370612yes—
565C23#1427374712yes—
566C23#1427375812yes—
567C23C22×C22⋊D5911yescharacteristic
568Q8—22no—
569Q8—22no—
570Q8—22no—
571Q8—22no—
572Q8—22no—
573Q8—48no—
574Q8—41no—
575Q8—41no—
576Q8#1427370414yes—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
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—21no—
599C2×C4—48no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—1182no—
609C2×C4—4724no—
610C2×C4—2361no—
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—2361no—
621C2×C4#1427370612yes—
622C2×C4C2×D4×D5911yescharacteristic
623C2×C4C22×Dic11811yescharacteristic
624C2×C4C2×#1174568711yescharacteristic
625C2×C4C2×#1174568511yescharacteristic
626C2×C4#1427377211yescharacteristic
627C2×C4#1427377211yescharacteristic
628C8—1182no—
629C8—2361no—
630C8—2361no—
631C8—2361no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—22no—
636C22—22no—
637C22—22no—
638C22#1683770914yescomplemented
639C22C2×#1427370611yescharacteristic
640C22D4×C22⋊D5911yescharacteristic
641C22#1683804611yescharacteristic
642C22C2×#1427374711yescharacteristic
643C22C2×#1427375811yescharacteristic
644C22#1683821011yescharacteristic
645C22#1683821711yescharacteristic
646C4—22no—
647C4—24no—
648C4—41no—
649C4—42no—
650C4—41no—
651C4—41no—
652C4—41no—
653C4—81no—
654C4—81no—
655C4—4721no—
656C4—4721no—
657C4—4721no—
658C4—2362no—
659C4C2×#1427370411yescharacteristic
660C4#1683802811yescharacteristic
661C4#1683804211yescharacteristic
662C4#1683807811yescharacteristic
663C2—24no—
664C2#2142775711yescharacteristic
665C2C2×#1683770911yescharacteristic
666C2#2143393811yescharacteristic
667C2#2143566711yescharacteristic
668C2#2143580411yescharacteristic
669C2#2143598311yescharacteristic
670C2#2143626711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingQ8×C22⋊C4×C59C2
socleC23×C59C22×D4
prime radicalC24×C59C2×D4
hypercenterQ8×C22⋊C4D59
hypocenterC59#25957
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ Q8×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#16837709C2semidirect
4. #21425644C2semidirect
5. #21425491C2semidirect
6. #21425477C2semidirect
7. #21425247C2semidirect
8. #21424364C2semidirect
9. #21424350C2semidirect
13. #21424111C2semidirect
303. C22×C59#479semidirect
396. C59#25957semidirect
638. C22#16837709semidirect

Maximal subgroups

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

subgroupindex
C2×#168377092
C22×#142736512
#214278202
#214256442
#214254912
#214254772
#214252472
#214243642
#214243502
#214242832
#214242762
#214241192
#214241112
#214240312
Q8×C22⋊C4×C592
#2595759