Finite Group Databasebeta

← #27418437

Subgroups — #27418437

Order 13568.

1394 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810905C211yesmaximal, characteristic, complemented
2C2×#16810820C211yesmaximal, characteristic, complemented
3C4×#13133196C211yesmaximal, characteristic, complemented
4#20275901C212yesmaximal, complemented
5#20275718C212yesmaximal, complemented
6#20275668C211yesmaximal, characteristic, complemented
7#20275643C211yesmaximal, characteristic, complemented
8#20275049C212yesmaximal, complemented
9#20274976C212yesmaximal, complemented
10#20274695C211yesmaximal, characteristic, complemented
11C2×C53×#532C211yesmaximal, characteristic
12C22×#11719923C2212yes—
13C2×#13133196C2211yescharacteristic
14C2×C4×Dic106C2211yescharacteristic
15#16810905—24no—
16#16810820—24no—
17C2×#13133148C2212yes—
18C2×C4⋊#88322C2211yescharacteristic
19C2×C4⋊#88322C2211yescharacteristic
20C2×#13133140C2212yes—
21C2×#13133138C2212yes—
22C2×#13133138C2212yes—
23C2×#13133138C2212yes—
24C2×C22⋊#88322C2212yes—
25#16810529C2211yescharacteristic
26#16810521C2211yescharacteristic
27#16810507—21no—
28#16810496—21no—
29#16810483C2211yescharacteristic
30C4×#11719919—21no—
31C4×C4⋊Dic53C2211yescharacteristic
32C4×C22⋊Dic53—21no—
33#16810385C2212yes—
34#16810385C2212yes—
35#16810378C2212yes—
36C22×#41×C53—21no—
37C53×#532—24no—
38C53×#532—24no—
39C2×C4⋊Q8×C53C2211yescharacteristic
40C2×#127×C53C2211yescharacteristic
41C2×#124×C53—21no—
42C2×#121×C53—21no—
43C2×#121×C53C2212yes—
44C2×C4×D4×C53—21no—
45C2×C4×D4×C53C2212yes—
46C22×C42×C53C2211yescharacteristic
47C22×Dic106C2311yescharacteristic
48C2×#11719923—24no—
49C2×#11719923—24no—
50C2×#11719923—42no—
51C22×#88322C2312yes—
52C22×#88322C2312yes—
53#13133196—22no—
54#13133196—22no—
55C4×Dic106—22no—
56C4×Dic106—22no—
57C2×C4⋊Dic53C2311yescharacteristic
58C2×C4⋊Dic53C2311yescharacteristic
59C2×C4⋊Dic53C2311yescharacteristic
60C2×#11719919—21no—
61C2×#11719919—21no—
62C2×#11719919—21no—
63C2×C22⋊Dic53—21no—
64C2×C22⋊Dic53—21no—
65C2×C4×Dic53—21no—
66C2×C4×Dic53—21no—
67#13133148—24no—
68C4⋊#88322—22no—
69C4⋊#88322—22no—
70#13133140—24no—
71#13133138—24no—
72#13133138—24no—
73#13133138—42no—
74C22⋊#88322—42no—
75#13133124—21no—
76#13133124—21no—
77#13133124—21no—
78#13133124—21no—
79#13133124—21no—
80C2×#41×C53—28no—
81C2×#41×C53—44no—
82C2×#41×C53—44no—
83C22×Q8×C53—21no—
84C22×D4×C53—22no—
85C22×D4×C53—21no—
86C22×D4×C53—21no—
87C22×D4×C53C2312yes—
88C23×C4×C53—22no—
89C23×C4×C53D411yescharacteristic
90C23×C4×C53D411yescharacteristic
91C23×C4×C53C2311yescharacteristic
92C4⋊Q8×C53—22no—
93C4⋊Q8×C53D414yes—
94#127×C53—22no—
95#127×C53D414yes—
96#124×C53—28no—
97#121×C53—28no—
98#121×C53—24no—
99#121×C53—24no—
100C4×D4×C53—28no—
101C4×D4×C53—24no—
102C4×D4×C53—24no—
103C2×C4⋊C4×C53—21no—
104C2×C4⋊C4×C53C2312yes—
105C2×C22⋊C4×C53—22no—
106C2×C22⋊C4×C53—22no—
107C2×C42×C53—22no—
108C2×C42×C53—22no—
109C2×C42×C53D411yescharacteristic
110C2×C42×C53D411yescharacteristic
111C2×C42×C53C2311yescharacteristic
112C2×C42×C53C2311yescharacteristic
113C24×C53—22no—
114C24×C53C2×D411yescharacteristic
115#41×C53—432no—
116C2×Q8×C53—24no—
117C2×Q8×C53—42no—
118C2×Q8×C53—42no—
119C2×D4×C53—28no—
120C2×D4×C53—24no—
121C2×D4×C53—24no—
122C2×D4×C53—22no—
123C2×D4×C53—22no—
124C2×D4×C53—22no—
125C2×D4×C53—22no—
126C2×D4×C53—22no—
127C2×D4×C53—42no—
128C2×D4×C53—42no—
129C2×D4×C53—44no—
130C2×D4×C53—44no—
131C2×D4×C53—42no—
132C2×D4×C53—42no—
133C2×D4×C53C2×D414yes—
134C22×C4×C53—22no—
135C22×C4×C53—22no—
136C22×C4×C53—28no—
137C22×C4×C53—22no—
138C22×C4×C53—22no—
139C22×C4×C53—21no—
140C22×C4×C53—21no—
141C22×C4×C53—21no—
142C22×C4×C53—21no—
143C22×C4×C53—21no—
144C22×C4×C53—21no—
145C22×C4×C53—41no—
146C22×C4×C53—41no—
147C22×C4×C53—44no—
148C22×C4×C53—44no—
149C22×C4×C53SD1614yescomplemented
150C22×C4×C53C2×D411yescharacteristic
151C22×C4×C53C2×D411yescharacteristic
152C22×C4×C53C2×D411yescharacteristic
153C22×C4×C53C2×D411yescharacteristic
154C22×C4×C53C2×D411yescharacteristic
155C22×C4×C53C2411yescharacteristic
156C22×Dic53—21no—
157C22×Dic53—21no—
158C22×Dic53—21no—
159C22×Dic53—21no—
160C2×Dic106—21no—
161C2×Dic106—21no—
162C2×Dic106—21no—
163C2×Dic106—21no—
164C2×Dic106—21no—
165C2×Dic106#4112yescomplemented
166C4⋊C4×C53—24no—
167C4⋊C4×C53—22no—
168C4⋊C4×C53C2×D414yes—
169C22⋊C4×C53—28no—
170C22⋊C4×C53—28no—
171C42×C53—22no—
172C42×C53—22no—
173C42×C53—21no—
174C42×C53—21no—
175C42×C53C2×D412yes—
176C42×C53C2×D412yes—
177C22⋊Dic53—24no—
178C22⋊Dic53—42no—
179#11719923—48no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53#4112yescomplemented
186#11719919—24no—
187#11719919—42no—
188#11719919—42no—
189C4×Dic53—42no—
190C4×Dic53—42no—
191C2×#88322—22no—
192C2×#88322—22no—
193C2×#88322—22no—
194C2×#88322—22no—
195C2×#88322—22no—
196C2×#88322—22no—
197C23×C53—21no—
198C23×C53—21no—
199C23×C53—21no—
200C23×C53—21no—
201C23×C53—21no—
202C23×C53—21no—
203C23×C53—24no—
204C23×C53—42no—
205C23×C53—42no—
206C23×C53—42no—
207C23×C53—42no—
208C23×C53—42no—
209C23×C53—42no—
210C23×C53C2×SD1612yes—
211C23×C53C22×D411yescharacteristic
212Q8×C53—48no—
213D4×C53—24no—
214D4×C53—24no—
215D4×C53—48no—
216D4×C53—416no—
217D4×C53—42no—
218D4×C53—42no—
219D4×C53—48no—
220C2×C4×C53—21no—
221C2×C4×C53—21no—
222C2×C4×C53—21no—
223C2×C4×C53—21no—
224C2×C4×C53—28no—
225C2×C4×C53—24no—
226C2×C4×C53—21no—
227C2×C4×C53—21no—
228C2×C4×C53—21no—
229C2×C4×C53—21no—
230C2×C4×C53—21no—
231C2×C4×C53—21no—
232C2×C4×C53—44no—
233C2×C4×C53—416no—
234C2×C4×C53—42no—
235C2×C4×C53—42no—
236C2×C4×C53—41no—
237C2×C4×C53—41no—
238C2×C4×C53—41no—
239C2×C4×C53—41no—
240C2×C4×C53#12012yes—
241C2×C4×C53#12012yes—
242C2×C4×C53C2×SD1612yes—
243C2×C4×C53C2×SD1612yes—
244C2×C4×C53AGL(1,ℤ/8)12yes—
245C2×C4×C53C22×D411yescharacteristic
246C2×C4×C53C22×D411yescharacteristic
247C2×C4×C53C2×#4111yescharacteristic
248C2×C4×C53C2×#4111yescharacteristic
249C2×C4×C53H(5,𝔽2)11yescharacteristic
250C2×C4×C532−1+411yescharacteristic
251C2×Dic53—22no—
252C2×Dic53—22no—
253C2×Dic53—41no—
254C2×Dic53—41no—
255C2×Dic53—41no—
256C2×Dic53—41no—
257C2×Dic53—41no—
258C2×Dic53—41no—
259C2×Dic53—41no—
260C2×Dic53—41no—
261C2×Dic53—41no—
262C2×Dic53—41no—
263Dic106—22no—
264Dic106—22no—
265Dic106—41no—
266Dic106—41no—
267#88322—42no—
268#88322—42no—
269#24154—531nomaximal, Sylow
270C22×C53—22no—
271C22×C53—22no—
272C22×C53—22no—
273C22×C53—41no—
274C22×C53—41no—
275C22×C53—41no—
276C22×C53—41no—
277C22×C53—44no—
278C22×C53—44no—
279C22×C53—44no—
280C22×C53—82no—
281C22×C53—82no—
282C22×C53—82no—
283C22×C53—82no—
284C22×C53C2×#12011yescharacteristic
285C22×C53#52511yescharacteristic
286C22×C53#53711yescharacteristic
287C22×C53C22×SD1611yescharacteristic
288C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
289C22×C53#57511yescharacteristic
290C22×C53#57511yescharacteristic
291C4×C53—22no—
292C4×C53—22no—
293C4×C53—41no—
294C4×C53—44no—
295C4×C53—41no—
296C4×C53#45014yes—
297C4×C53#54611yescharacteristic
298C4×C53#54711yescharacteristic
299C4×C53#54811yescharacteristic
300C4×C53#55011yescharacteristic
301Dic53—42no—
302Dic53—42no—
303Dic53—81no—
304Dic53—81no—
305C2×#532D5311yescharacteristic
306C2×#475—531no—
307C2×#460—531no—
308C4×C22⋊Q8—531no—
309#2034—532no—
310#2000—532no—
311#1988—531no—
312#1982—531no—
313#1877—532no—
314#1864—532no—
315#1797—531no—
316C2×C53—42no—
317C2×C53—84no—
318C2×C53#240711yescharacteristic
319C2×C53C2×#45011yescharacteristic
320C2×C53#300511yescharacteristic
321C2×C53#310311yescharacteristic
322C2×C53#311511yescharacteristic
323C2×C53#324611yescharacteristic
324C2×C53#325111yescharacteristic
325C22×#41—21no—
326C22×SD16—532no—
327#532—24no—
328#532—24no—
329C2×C4⋊Q8C2×D5311yescharacteristic
330C2×#127C2×D5311yescharacteristic
331C2×#124—21no—
332C2×C22⋊Q8—531no—
333C2×#121—21no—
334C2×#121C2×D5312yes—
335C2×C4×Q8—531no—
336C2×C4×D4—21no—
337C2×C4×D4C2×D5312yes—
338C22×C42C2×D5311yescharacteristic
339#475—1064no—
340#460—1064no—
341C2×#106—532no—
342C2×C4⋊C8—531no—
343C2×C4⋊C8—531no—
344C2×#103—532no—
345C2×#102—532no—
346C2×#102—532no—
347C2×#102—532no—
348C2×C22⋊C8—532no—
349#391—531no—
350#389—531no—
351#387—1061no—
352#385—1061no—
353#380—531no—
354C4×C4⋊C4—531no—
355C4×C4⋊C4—1061no—
356C4×C22⋊C4—1061no—
357#340—532no—
358#340—532no—
359#336—532no—
360C53#2415411yescharacteristic, complemented, Sylow
361C2×#41—28no—
362C2×#41—44no—
363C2×#41—44no—
364C22×Q8—21no—
365C22×Q8—531no—
366C22×D4—22no—
367C22×D4—21no—
368C22×D4—21no—
369C22×D4C22×D5312yes—
370C23×C4—22no—
371C23×C4C22⋊D5311yescharacteristic
372C23×C4C22⋊D5311yescharacteristic
373C23×C4C22×D5311yescharacteristic
374C2×SD16—1064no—
375C2×SD16—1064no—
376C2×SD16—2122no—
377C22×C8—532no—
378C22×C8—532no—
379C4⋊Q8—22no—
380C4⋊Q8C22⋊D5314yes—
381#127—22no—
382#127C22⋊D5314yes—
383#124—28no—
384C22⋊Q8—1062no—
385C22⋊Q8—1062no—
386#121—28no—
387#121—24no—
388#121—24no—
389C4×Q8—1062no—
390C4×Q8—1062no—
391C4×D4—28no—
392C4×D4—24no—
393C4×D4—24no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—531no—
396C2×C4⋊C4—531no—
397C2×C4⋊C4—531no—
398C2×C4⋊C4—1061no—
399C2×C4⋊C4—1061no—
400C2×C4⋊C4—1061no—
401C2×C4⋊C4C22×D5312yes—
402C2×C22⋊C4—22no—
403C2×C22⋊C4—22no—
404C2×C22⋊C4—1061no—
405C2×C22⋊C4—1061no—
406C2×C42—22no—
407C2×C42—22no—
408C2×C42—1061no—
409C2×C42—1061no—
410C2×C42C22⋊D5311yescharacteristic
411C2×C42C22⋊D5311yescharacteristic
412C2×C42C22×D5311yescharacteristic
413C2×C42C22×D5311yescharacteristic
414#106—1064no—
415C4⋊C8—1062no—
416C4⋊C8—1062no—
417#103—1064no—
418#102—1064no—
419#102—1064no—
420#102—2122no—
421C22⋊C8—2122no—
422#95—1061no—
423#95—1061no—
424#95—1061no—
425#95—1061no—
426#95—1061no—
427C24—22no—
428C24C2×C22⋊D5311yescharacteristic
429#41—432no—
430C2×Q8—24no—
431C2×Q8—42no—
432C2×Q8—42no—
433C2×Q8—532no—
434C2×Q8—1061no—
435C2×Q8—1061no—
436C2×Q8—1061no—
437C2×Q8—1061no—
438C2×Q8—1061no—
439C2×D4—28no—
440C2×D4—24no—
441C2×D4—24no—
442C2×D4—22no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—42no—
448C2×D4—42no—
449C2×D4—44no—
450C2×D4—44no—
451C2×D4—42no—
452C2×D4—42no—
453C2×D4C2×C22⋊D5314yes—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—28no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—41no—
466C22×C4—41no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4—1061no—
470C22×C4—1061no—
471C22×C4—1061no—
472C22×C4—1061no—
473C22×C4#1171992314yescomplemented
474C22×C4C2×C22⋊D5311yescharacteristic
475C22×C4C2×C22⋊D5311yescharacteristic
476C22×C4C2×C22⋊D5311yescharacteristic
477C22×C4C2×C22⋊D5311yescharacteristic
478C22×C4C2×C22⋊D5311yescharacteristic
479C22×C4C23×D5311yescharacteristic
480SD16—2128no—
481C2×C8—1062no—
482C2×C8—1062no—
483C2×C8—1062no—
484C2×C8—1062no—
485C2×C8—1062no—
486C2×C8—1062no—
487C4⋊C4—24no—
488C4⋊C4—22no—
489C4⋊C4—532no—
490C4⋊C4—1064no—
491C4⋊C4—1061no—
492C4⋊C4—1061no—
493C4⋊C4—1061no—
494C4⋊C4—1061no—
495C4⋊C4—1061no—
496C4⋊C4—2122no—
497C4⋊C4—2122no—
498C4⋊C4C2×C22⋊D5314yes—
499C22⋊C4—28no—
500C22⋊C4—28no—
501C22⋊C4—1064no—
502C22⋊C4—2122no—
503C42—22no—
504C42—22no—
505C42—21no—
506C42—21no—
507C42—2122no—
508C42—2122no—
509C42C2×C22⋊D5312yes—
510C42C2×C22⋊D5312yes—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—24no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23C2×#1171992312yes—
525C23C22×C22⋊D5311yescharacteristic
526Q8—48no—
527Q8—1062no—
528Q8—1062no—
529Q8—2121no—
530Q8—2121no—
531D4—24no—
532D4—24no—
533D4—48no—
534D4—416no—
535D4—42no—
536D4—48no—
537D4—42no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—28no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—24no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—44no—
551C2×C4—416no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—1062no—
559C2×C4—1062no—
560C2×C4—2121no—
561C2×C4—2121no—
562C2×C4—2121no—
563C2×C4—2121no—
564C2×C4—2121no—
565C2×C4—2121no—
566C2×C4—2121no—
567C2×C4—2121no—
568C2×C4—2121no—
569C2×C4—2121no—
570C2×C4#1313318612yes—
571C2×C4#1313318612yes—
572C2×C4C2×#1171992312yes—
573C2×C4C2×#1171992312yes—
574C2×C4#1313324312yes—
575C2×C4C22×C22⋊D5311yescharacteristic
576C2×C4C22×C22⋊D5311yescharacteristic
577C2×C4#41×D5311yescharacteristic
578C2×C4#41×D5311yescharacteristic
579C2×C4#1313326311yescharacteristic
580C2×C4#1313326411yescharacteristic
581C8—2122no—
582C8—2122no—
583C22—22no—
584C22—22no—
585C22—22no—
586C22—41no—
587C22—41no—
588C22—41no—
589C22—41no—
590C22—44no—
591C22—44no—
592C22—44no—
593C22—82no—
594C22—82no—
595C22—82no—
596C22—82no—
597C22C2×#1313318611yescharacteristic
598C22#1681113011yescharacteristic
599C22#1681120111yescharacteristic
600C22C22×#1171992311yescharacteristic
601C22C2×#1313324311yescharacteristic
602C22#1681142111yescharacteristic
603C22#1681142111yescharacteristic
604C4—22no—
605C4—22no—
606C4—41no—
607C4—41no—
608C4—44no—
609C4—4241no—
610C4—4241no—
611C4—2122no—
612C4—2122no—
613C4#1681077414yes—
614C4#1681126111yescharacteristic
615C4#1681127211yescharacteristic
616C4#1681128011yescharacteristic
617C4#1681129211yescharacteristic
618C2—42no—
619C2—84no—
620C2#2027805411yescharacteristic
621C2C2×#1681077411yescharacteristic
622C2#2028395711yescharacteristic
623C2#2028474211yescharacteristic
624C2#2028486011yescharacteristic
625C2#2028603111yescharacteristic
626C2#2028607111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
Fitting, prime radicalC2×C53×#532C2
socleC23×C53C22×D4
hypercenterC2×#532D53
hypocenterC53#24154
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#532

Complements

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

NH ≅ G/Ntype
1. C2×#16810905C2semidirect
2. C2×#16810820C2semidirect
3. C4×#13133196C2semidirect
4. #20275901C2semidirect
5. #20275718C2semidirect
6. #20275668C2semidirect
7. #20275643C2semidirect
8. #20275049C2semidirect
9. #20274976C2semidirect
10. #20274695C2semidirect
149. C22×C4×C53SD16semidirect
165. C2×Dic106#41semidirect
185. C4⋊Dic53#41semidirect
360. C53#24154semidirect
473. C22×C4#11719923semidirect

Maximal subgroups

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

subgroupindex
C2×#168109052
C2×#168108202
C4×#131331962
#202759012
#202757182
#202756682
#202756432
#202750492
#202749762
#202746952
C2×C53×#5322
#2415453