Finite Group Databasebeta

← #27385063

Subgroups — #27385063

Order 13568.

1550 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811091C211yesmaximal, characteristic
2C2×#16810625C211yesmaximal, characteristic
3C2×#16810622C211yesmaximal, characteristic
4C2×#16810519C211yesmaximal, characteristic, complemented
5#20274980C212yesmaximal, complemented
6#20274973C212yesmaximal, complemented
7#20274966C212yesmaximal, complemented
8#20274959C212yesmaximal, complemented
9#20274436C211yesmaximal, characteristic, complemented
10#20274408C211yesmaximal, characteristic, complemented
11C2×C53×#428C211yesmaximal, characteristic, complemented
12C22×#11719945C2211yescharacteristic
13C22×#11719917C2211yescharacteristic
14#16811091—24no—
15#16811091—24no—
16C2×#13133182C2211yescharacteristic
17C2×#13133182C2211yescharacteristic
18C2×#13133179C2211yescharacteristic
19C2×#13133179C2211yescharacteristic
20C2×#13133177—21no—
21C2×#13133177—21no—
22C2×#13133174—21no—
23C22×C4⋊Dic53C2211yescharacteristic
24C22×#11719919—21no—
25C22×C4×Dic53C2211yescharacteristic
26C2×C42×D53—21no—
27#16810625C418yescomplemented
28#16810622—24no—
29C2×#13133142C2212yes—
30C2×#13133140C2212yes—
31C2×#13133139C2212yes—
32C2×#13133137C2212yes—
33#16810519—21no—
34#16810519—21no—
35#16810519C2212yes—
36#16810519C2212yes—
37C2×#13133124—21no—
38#16810385C2212yes—
39#16810385C2212yes—
40#16810377C2212yes—
41#16810377C2212yes—
42C22×M16×C53C2211yescharacteristic
43C2×#117×C53C2211yescharacteristic
44C22×C4⋊C4×C53C2211yescharacteristic
45C53×#428—24no—
46C2×#107×C53C2212yes—
47C2×#106×C53C2212yes—
48C2×#11719945—22no—
49C2×#11719945—22no—
50C2×#11719945—22no—
51C2×#11719945C2×C414yes—
52C2×#11719945D414yes—
53C2×#11719945Q814yes—
54C22×Dic106C2311yescharacteristic
55C22×D212C2311yescharacteristic
56C22×C22⋊D53—21no—
57C23×Dic53—21no—
58C23×Dic53D411yescharacteristic
59C23×Dic53D411yescharacteristic
60C22×C4×D53—21no—
61C22×C4×D53—21no—
62C22×C4×D53—21no—
63C2×#11719917—22no—
64C2×#11719917—22no—
65C2×#11719917—22no—
66C22×#88322C2312yes—
67#13133182—22no—
68#13133182—22no—
69#13133182—22no—
70#13133182—22no—
71#13133179—22no—
72#13133179—22no—
73#13133179—22no—
74#13133179—22no—
75#13133177—28no—
76#13133177—28no—
77#13133174—28no—
78C2×C4⋊Dic53—21no—
79C2×C4⋊Dic53—22no—
80C2×C4⋊Dic53C2×C414yescomplemented
81C2×C4⋊Dic53C2312yes—
82C2×#11719919—21no—
83C2×#11719919—21no—
84C2×#11719919—22no—
85C2×#11719919—22no—
86C2×#11719919—44no—
87C2×#11719921—21no—
88C2×#11719921—21no—
89C2×C4×Dic53—22no—
90C2×C4×Dic53—22no—
91C2×C4×Dic53—21no—
92C2×C4×Dic53—21no—
93C2×C4×Dic53D411yescharacteristic
94C2×C4×Dic53D411yescharacteristic
95C2×C4×Dic53C2311yescharacteristic
96C2×C4×Dic53C2311yescharacteristic
97C42×D53—28no—
98#13133142C2×C418yescomplemented
99#13133140—24no—
100#13133139C2×C418yescomplemented
101#13133137—24no—
102#13133124—22no—
103#13133124—22no—
104C23×C4×C53—21no—
105C23×C4×C53C2311yescharacteristic
106C2×M16×C53—22no—
107C2×M16×C53—22no—
108C2×M16×C53C2×C414yescomplemented
109C22×C8×C53C2312yes—
110#117×C53—22no—
111#117×C53D414yes—
112C2×C4⋊C4×C53—21no—
113C2×C4⋊C4×C53—22no—
114C2×C4⋊C4×C53—22no—
115C2×C4⋊C4×C53C2312yes—
116C2×C4⋊C4×C53C2311yescharacteristic
117C2×C4⋊C4×C53C2311yescharacteristic
118C2×C22⋊C4×C53—21no—
119C2×C42×C53—21no—
120#107×C53—24no—
121#106×C53—24no—
122C24×C53C2×D411yescharacteristic
123C23×D53—21no—
124C22×C4×C53—21no—
125C22×C4×C53—21no—
126C22×C4×C53—22no—
127C22×C4×C53—42no—
128C22×C4×C53—42no—
129C22×C4×C53—42no—
130C22×C4×C53C22×C412yes—
131C22×C4×C53C2×D412yes—
132C22×C4×C53C2×D411yescharacteristic
133C22×C4×C53C2×D412yes—
134C22×C4×C53C2×Q812yes—
135C22×C4×C53#4112yes—
136C22×C4×C53#4112yes—
137C22×C4×C53C2411yescharacteristic
138C2×C22⋊D53—24no—
139C2×C22⋊D53—24no—
140C2×C22⋊D53—24no—
141C22×Dic53—21no—
142C22×Dic53—24no—
143C22×Dic53—22no—
144C22×Dic53—22no—
145C22×Dic53—22no—
146C22×Dic53—22no—
147C22×Dic53—22no—
148C22×Dic53—41no—
149C22×Dic53—41no—
150C22×Dic53—42no—
151C22×Dic53—42no—
152C22×Dic53C2×D411yescharacteristic
153C22×Dic53C2×D411yescharacteristic
154C22×Dic53C2×D411yescharacteristic
155C22×Dic53C2×D411yescharacteristic
156#11719945—28no—
157#11719945—28no—
158#11719945—28no—
159#11719945C4⋊C4116yescomplemented
160C2×D212—21no—
161C2×D212—21no—
162C2×D212—21no—
163C2×D212C22×C412yes—
164C2×D212C2×D412yes—
165C2×D212C2×Q812yes—
166C2×C4×D53—24no—
167C2×C4×D53—24no—
168C2×C4×D53—24no—
169C2×C4×D53—24no—
170C2×C4×D53—24no—
171C2×C4×D53—42no—
172C2×C4×D53—42no—
173C2×C4×D53—42no—
174C2×C4×D53—42no—
175C2×Dic106—21no—
176C2×Dic106—21no—
177C2×Dic106—21no—
178C2×Dic106C22×C412yes—
179C2×Dic106C2×D412yes—
180C2×Dic106C2×Q812yes—
181M16×C53—28no—
182C2×C8×C53—22no—
183C2×C8×C53—22no—
184C2×C8×C53C22×C414yes—
185C4⋊C4×C53—21no—
186C4⋊C4×C53—21no—
187C4⋊C4×C53—22no—
188C4⋊C4×C53—22no—
189C4⋊C4×C53—42no—
190C4⋊C4×C53C2×D412yes—
191C4⋊C4×C53C2×D412yes—
192C22⋊C4×C53—24no—
193C42×C53—24no—
194#11719921—24no—
195#11719921—24no—
196C4⋊Dic53—24no—
197C4⋊Dic53—22no—
198C4⋊Dic53C22×C414yes—
199#11719919—24no—
200#11719919—24no—
201#11719919—44no—
202#11719919—44no—
203C4×Dic53—24no—
204C4×Dic53—24no—
205C4×Dic53—22no—
206C4×Dic53—22no—
207C4×Dic53—21no—
208C4×Dic53—21no—
209C4×Dic53—21no—
210C4×Dic53—21no—
211#11719917—44no—
212C2×#88322—22no—
213C2×#88322—22no—
214C2×#88322—22no—
215C23×C53—21no—
216C23×C53—21no—
217C23×C53—21no—
218C23×C53—21no—
219C23×C53C4×D412yes—
220C23×C53#12012yes—
221C23×C53C22⋊Q812yes—
222C23×C53C22×D411yescharacteristic
223C22×D53—22no—
224C22×D53—22no—
225C22×D53—22no—
226C22×D53—41no—
227C22×D53—41no—
228C22×D53—41no—
229C22×D53—41no—
230C2×C4×C53—22no—
231C2×C4×C53—22no—
232C2×C4×C53—41no—
233C2×C4×C53—41no—
234C2×C4×C53—41no—
235C2×C4×C53—41no—
236C2×C4×C53—42no—
237C2×C4×C53—42no—
238C2×C4×C53—42no—
239C2×C4×C53—84no—
240C2×C4×C53C2×C4⋊C414yes—
241C2×C4×C53C4×D414yes—
242C2×C4×C53C4×D412yes—
243C2×C4×C53#12012yes—
244C2×C4×C53C22⋊Q814yes—
245C2×C4×C53C22⋊Q812yes—
246C2×C4×C53#12314yes—
247C2×C4×C53C23×C411yescharacteristic
248C2×C4×C53C22×D411yescharacteristic
249C2×C4×C53C22×D411yescharacteristic
250C2×C4×C53C22×Q811yescharacteristic
251C2×C4×C53C2×#4111yescharacteristic
252C2×C4×C53C2×#4111yescharacteristic
253C22⋊D53—216no—
254C2×Dic53—22no—
255C2×Dic53—22no—
256C2×Dic53—21no—
257C2×Dic53—21no—
258C2×Dic53—22no—
259C2×Dic53—21no—
260C2×Dic53—21no—
261C2×Dic53—24no—
262C2×Dic53—21no—
263C2×Dic53—21no—
264C2×Dic53—21no—
265C2×Dic53—21no—
266C2×Dic53—21no—
267C2×Dic53—21no—
268C2×Dic53—44no—
269C2×Dic53—44no—
270C2×Dic53—48no—
271C2×Dic53—42no—
272C2×Dic53—42no—
273C2×Dic53AGL(1,ℤ/8)12yes—
274C2×Dic53#13712yes—
275D212—22no—
276D212—22no—
277D212—22no—
278D212C2×C4⋊C414yescomplemented
279C4×D53—216no—
280C4×D53—48no—
281C4×D53—48no—
282Dic106—22no—
283Dic106—22no—
284Dic106—22no—
285Dic106C2×C4⋊C414yescomplemented
286C8×C53—24no—
287#88322—42no—
288#20875—531nomaximal, Sylow
289C22×C53—22no—
290C22×C53—22no—
291C22×C53—22no—
292C22×C53—22no—
293C22×C53—22no—
294C22×C53—22no—
295C22×C53#38514yes—
296C22×C53C2×C4×D411yescharacteristic
297C22×C53C2×#12011yescharacteristic
298C22×C53C2×C22⋊Q811yescharacteristic
299C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
300C22×C53C2×#13711yescharacteristic
301C22×C53#57511yescharacteristic
302C22×C53#57511yescharacteristic
303C2×D53—24no—
304C2×D53—42no—
305C2×D53—42no—
306C2×D53—42no—
307C2×D53—42no—
308C2×D53—42no—
309C2×D53—42no—
310C4×C53—42no—
311C4×C53—42no—
312C4×C53—82no—
313C4×C53#38514yes—
314C4×C53C22×C4⋊C411yescharacteristic
315C4×C53C2×C4×D411yescharacteristic
316C4×C53C2×C22⋊Q811yescharacteristic
317C4×C53C2×#12311yescharacteristic
318Dic53—22no—
319Dic53—22no—
320Dic53—24no—
321Dic53—44no—
322Dic53—41no—
323Dic53—41no—
324C2×C4×#41—531no—
325C2×#428D5311yescharacteristic, complemented
326C2×#417—531no—
327C2×#417—531no—
328C2×#389—531no—
329#1865—532no—
330#1864—532no—
331#1863—532no—
332#1862—532no—
333#1736—531no—
334#1728—531no—
335C2×C53—24no—
336C2×C53C2×#38511yescharacteristic
337C2×C53C4×AGL(1,ℤ/8)11yescharacteristic
338C2×C53C4×#13711yescharacteristic
339C2×C53#300611yescharacteristic
340C2×C53#300711yescharacteristic
341C2×C53#307911yescharacteristic
342C2×C53#308011yescharacteristic
343D53—44no—
344C22×#41—531no—
345C22×M16—531no—
346C22×M16C2×D5311yescharacteristic
347C4×#41—1064no—
348C4×#41—1064no—
349C2×C4×Q8—531no—
350C2×C4×Q8—531no—
351C2×C4×D4—531no—
352C2×C4×D4—531no—
353C2×C4×D4—1061no—
354C2×C4×D4—1061no—
355C2×#117—1061no—
356C2×#117C2×D5311yescharacteristic
357C22×C4⋊C4—531no—
358C22×C4⋊C4—1061no—
359C22×C4⋊C4C2×D5311yescharacteristic
360C22×C42—531no—
361C22×C42—1061no—
362#428—24no—
363C2×#107C2×D5312yes—
364C2×#106C2×D5312yes—
365#417—538no—
366#417—1064no—
367C2×#103—532no—
368C2×#103—532no—
369C2×#102—532no—
370C2×#102—532no—
371#389—532no—
372#389—532no—
373#389—1061no—
374#389—1061no—
375C2×#95—1061no—
376#340—532no—
377#340—532no—
378#336—532no—
379#336—532no—
380C53#2087511yescharacteristic, complemented, Sylow
381C2×#41—534no—
382C2×#41—534no—
383C2×#41—534no—
384C2×#41—1062no—
385C2×#41—1062no—
386C2×#41—1062no—
387C22×Q8—531no—
388C22×D4—531no—
389C22×D4—1061no—
390C23×C4—21no—
391C23×C4—531no—
392C23×C4—531no—
393C23×C4—1061no—
394C23×C4—1061no—
395C23×C4—1061no—
396C23×C4—1061no—
397C23×C4C22×D5311yescharacteristic
398C2×M16—22no—
399C2×M16—22no—
400C2×M16—1062no—
401C2×M16—1062no—
402C2×M16—1062no—
403C2×M16C4×D5314yescomplemented
404C22×C8—532no—
405C22×C8C22×D5312yes—
406C4×Q8—1062no—
407C4×Q8—1062no—
408C4×Q8—1062no—
409C4×Q8—1062no—
410C4×D4—1068no—
411C4×D4—1068no—
412C4×D4—1062no—
413C4×D4—1062no—
414C4×D4—1062no—
415C4×D4—1062no—
416#117—22no—
417#117—1068no—
418#117C22⋊D5314yes—
419C2×C4⋊C4—22no—
420C2×C4⋊C4—22no—
421C2×C4⋊C4—21no—
422C2×C4⋊C4—534no—
423C2×C4⋊C4—532no—
424C2×C4⋊C4—1061no—
425C2×C4⋊C4—1061no—
426C2×C4⋊C4—1061no—
427C2×C4⋊C4—1062no—
428C2×C4⋊C4—1062no—
429C2×C4⋊C4—1062no—
430C2×C4⋊C4—2124no—
431C2×C4⋊C4C22×D5312yes—
432C2×C4⋊C4C22×D5311yescharacteristic
433C2×C4⋊C4C22×D5311yescharacteristic
434C2×C22⋊C4—21no—
435C2×C22⋊C4—1061no—
436C2×C22⋊C4—1061no—
437C2×C42—21no—
438C2×C42—531no—
439C2×C42—531no—
440C2×C42—531no—
441C2×C42—531no—
442C2×C42—1068no—
443C2×C42—1061no—
444C2×C42—1061no—
445C2×C42—1062no—
446C2×C42—1062no—
447#107—24no—
448#106—24no—
449#103—538no—
450#103—1064no—
451#102—538no—
452#102—1064no—
453#95—1062no—
454#95—1062no—
455C24—1061no—
456C24D4×D5311yescharacteristic
457#41—5316no—
458#41—1068no—
459#41—1068no—
460#41—1068no—
461C2×Q8—532no—
462C2×Q8—532no—
463C2×Q8—532no—
464C2×Q8—1061no—
465C2×Q8—1061no—
466C2×Q8—1061no—
467C2×D4—532no—
468C2×D4—532no—
469C2×D4—532no—
470C2×D4—1064no—
471C2×D4—1064no—
472C2×D4—1064no—
473C2×D4—1061no—
474C2×D4—1061no—
475C2×D4—1061no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—22no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—531no—
483C22×C4—531no—
484C22×C4—531no—
485C22×C4—531no—
486C22×C4—1064no—
487C22×C4—1064no—
488C22×C4—1062no—
489C22×C4—1062no—
490C22×C4—1061no—
491C22×C4—1064no—
492C22×C4—1064no—
493C22×C4—1064no—
494C22×C4—1064no—
495C22×C4—1062no—
496C22×C4—1062no—
497C22×C4—1062no—
498C22×C4—2122no—
499C22×C4—2122no—
500C22×C4—2122no—
501C22×C4—2122no—
502C22×C4—2121no—
503C22×C4—2121no—
504C22×C4—2122no—
505C22×C4—2122no—
506C22×C4C2×C4×D5312yes—
507C22×C4#1171994512yes—
508C22×C4D4×D5312yes—
509C22×C4D4×D5311yescharacteristic
510C22×C4Q8×D5312yes—
511C22×C4#1171994912yes—
512C22×C4C2×C22⋊D5312yes—
513C22×C4C23×D5311yescharacteristic
514M16—28no—
515M16—2124no—
516C2×C8—22no—
517C2×C8—22no—
518C2×C8—1062no—
519C2×C8—1062no—
520C2×C8—1062no—
521C2×C8C2×C4×D5314yes—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—42no—
527C4⋊C4—534no—
528C4⋊C4—1064no—
529C4⋊C4—1064no—
530C4⋊C4—1064no—
531C4⋊C4—1062no—
532C4⋊C4—2124no—
533C4⋊C4—2124no—
534C4⋊C4C2×C22⋊D5312yes—
535C4⋊C4C2×C22⋊D5312yes—
536C22⋊C4—24no—
537C22⋊C4—1064no—
538C22⋊C4—1064no—
539C42—24no—
540C42—1064no—
541C42—1064no—
542C42—1062no—
543C42—1062no—
544C42—1061no—
545C42—1061no—
546C42—1061no—
547C42—1061no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—1062no—
553C23—1062no—
554C23—1062no—
555C23—2121no—
556C23—2121no—
557C23—2121no—
558C23—2121no—
559C23#1313317912yes—
560C23#1313318512yes—
561C23#1313319512yes—
562C23C2×D4×D5311yescharacteristic
563Q8—534no—
564Q8—1062no—
565Q8—1062no—
566Q8—1062no—
567D4—534no—
568D4—10616no—
569D4—1062no—
570D4—1062no—
571D4—1062no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—84no—
582C2×C4—532no—
583C2×C4—532no—
584C2×C4—1062no—
585C2×C4—1061no—
586C2×C4—1061no—
587C2×C4—1062no—
588C2×C4—1062no—
589C2×C4—1061no—
590C2×C4—1061no—
591C2×C4—10616no—
592C2×C4—1064no—
593C2×C4—1061no—
594C2×C4—1061no—
595C2×C4—1061no—
596C2×C4—1061no—
597C2×C4—1061no—
598C2×C4—1061no—
599C2×C4—2128no—
600C2×C4—2128no—
601C2×C4—2124no—
602C2×C4—2124no—
603C2×C4—2128no—
604C2×C4—2122no—
605C2×C4—2122no—
606C2×C4C4⋊C4×D5314yes—
607C2×C4C4×C22⋊D5314yes—
608C2×C4#1313317912yes—
609C2×C4#1313318512yes—
610C2×C4#1313319314yes—
611C2×C4#1313319512yes—
612C2×C4#1313320214yes—
613C2×C4C22×C4×D5311yescharacteristic
614C2×C4C2×D4×D5311yescharacteristic
615C2×C4C22×C22⋊D5311yescharacteristic
616C2×C4C2×Q8×D5311yescharacteristic
617C2×C4C2×#1171994511yescharacteristic
618C2×C4C2×#1171994911yescharacteristic
619C8—24no—
620C8—2122no—
621C22—22no—
622C22—22no—
623C22—22no—
624C22—22no—
625C22—22no—
626C22—22no—
627C22—1064no—
628C22—2122no—
629C22—2122no—
630C22—2122no—
631C22—2122no—
632C22—2122no—
633C22—2122no—
634C22#1681049414yes—
635C22C2×#1313317911yescharacteristic
636C22C2×#1313318511yescharacteristic
637C22C2×#1313319511yescharacteristic
638C22AGL(1,ℤ/8)×D5311yescharacteristic
639C22#137×D5311yescharacteristic
640C22#1681141911yescharacteristic
641C22#1681142211yescharacteristic
642C4—42no—
643C4—42no—
644C4—82no—
645C4—1062no—
646C4—1062no—
647C4—1064no—
648C4—2121no—
649C4—2121no—
650C4—2124no—
651C4#1681049414yes—
652C4C2×C4⋊C4×D5311yescharacteristic
653C4C2×C4×C22⋊D5311yescharacteristic
654C4C2×#1313319311yescharacteristic
655C4C2×#1313320211yescharacteristic
656C2—24no—
657C2—2124no—
658C2C2×#1681049411yescharacteristic
659C2#2028352211yescharacteristic
660C2#2028353311yescharacteristic
661C2#2028396311yescharacteristic
662C2#2028397111yescharacteristic
663C2#2028452811yescharacteristic
664C2#2028453611yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC2×C53×#428C2
socleC23×C53C22×D4
prime radicalC22×#11719945C22
cosocleC22×C4×C53C24
hypercenterC2×#428D53
hypocenterC53#20875
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
4. C2×#16810519C2semidirect
5. #20274980C2semidirect
6. #20274973C2semidirect
7. #20274966C2semidirect
8. #20274959C2semidirect
9. #20274436C2semidirect
10. #20274408C2semidirect
11. C2×C53×#428C2semidirect
27. #16810625C4semidirect
80. C2×C4⋊Dic53C2×C4semidirect
98. #13133142C2×C4semidirect
100. #13133139C2×C4semidirect
108. C2×M16×C53C2×C4semidirect
159. #11719945C4⋊C4semidirect
278. D212C2×C4⋊C4semidirect
285. Dic106C2×C4⋊C4semidirect
325. C2×#428D53semidirect
380. C53#20875semidirect
403. C2×M16C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168110912
C2×#168106252
C2×#168106222
C2×#168105192
#202749802
#202749732
#202749662
#202749592
#202744362
#202744082
C2×C53×#4282
#2087553