Finite Group Databasebeta

← #27401325

Subgroups — #27401325

Order 13568.

1010 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810624C211yesmaximal, characteristic
2#20277594C211yesmaximal, characteristic, complemented
3#20277545C211yesmaximal, characteristic
4#20275402C211yesmaximal, characteristic, complemented
5#20275388C211yesmaximal, characteristic, complemented
6#20275359C211yesmaximal, characteristic, complemented
7#20275352C211yesmaximal, characteristic, complemented
8#20274974C211yesmaximal, characteristic, complemented
9#20274960C211yesmaximal, characteristic, complemented
10#20274802C211yesmaximal, characteristic, complemented
11#20274746C211yesmaximal, characteristic, complemented
12#20274542C211yesmaximal, characteristic, complemented
13#102×Dic53C211yesmaximal, characteristic, complemented
14#20274501C211yesmaximal, characteristic, complemented
15C53×#1788C211yesmaximal, characteristic
16C22×#11719917C2211yescharacteristic
17C2×D4×Dic53C2211yescharacteristic
18C2×D4×Dic53C2211yescharacteristic
19C22×C4⋊Dic53C2211yescharacteristic
20C2×#13133150C2211yescharacteristic
21C2×#13133148C2211yescharacteristic
22#16810624C418yescomplemented
23C2×#13133141C2211yescharacteristic
24C2×#13133141C2211yescharacteristic
25C2×#13133138C2211yescharacteristic
26C2×#13133138C2211yescharacteristic
27C2×#13133127C2211yescharacteristic
28C2×C8×Dic53C2211yescharacteristic
29#16810530C2211yescharacteristic
30#16810530C2211yescharacteristic
31#16810519C2211yescharacteristic
32#16810519C2211yescharacteristic
33#16810508—21no—
34#16810508—21no—
35#16810503—21no—
36#16810503—21no—
37#16810486—21no—
38#16810483C2211yescharacteristic
39#16810481—21no—
40C4⋊C4×Dic53C2211yescharacteristic
41C4⋊C4×Dic53C2211yescharacteristic
42C22⋊C4×Dic53—21no—
43C4×C22⋊Dic53C2211yescharacteristic
44C2×#13133124—21no—
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810386C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810377C2211yescharacteristic
52#16810377C2211yescharacteristic
53C22×#41×C53C2211yescharacteristic
54C2×#103×C53C2211yescharacteristic
55C2×#103×C53C2211yescharacteristic
56C2×#102×C53C2211yescharacteristic
57C2×#102×C53C2211yescharacteristic
58C2×C22⋊C8×C53C2211yescharacteristic
59C53×#380C2211yescharacteristic
60C23×Dic53—21no—
61C23×Dic53—21no—
62C23×Dic53—21no—
63C2×#11719917—41no—
64C2×#11719917—41no—
65C2×#11719917C2×C414yescomplemented
66C22×#88322C2311yescharacteristic
67C22×#88322C2311yescharacteristic
68D4×Dic53—22no—
69D4×Dic53—22no—
70D4×Dic53—41no—
71D4×Dic53—41no—
72C2×C4⋊Dic53—21no—
73C2×C4⋊Dic53—22no—
74C2×C4⋊Dic53C2×C414yescomplemented
75C2×C4⋊Dic53C2311yescharacteristic
76C2×C4⋊Dic53C2311yescharacteristic
77C2×C4⋊Dic53C2311yescharacteristic
78C2×C4⋊Dic53C2311yescharacteristic
79C2×#11719919—21no—
80C2×#11719919—21no—
81C2×C22⋊Dic53—21no—
82C2×C22⋊Dic53—21no—
83C2×C22⋊Dic53—21no—
84C2×C22⋊Dic53D411yescharacteristic
85C2×C22⋊Dic53D411yescharacteristic
86C2×C4×Dic53—21no—
87C2×C4×Dic53—21no—
88C2×C4×Dic53C2311yescharacteristic
89C2×C4×Dic53C2311yescharacteristic
90#13133150—41no—
91#13133148—41no—
92#13133141C2×C414yescomplemented
93#13133141C2×C414yescomplemented
94#13133138C2×C414yes—
95#13133138C2×C414yes—
96#13133127—41no—
97C8×Dic53—41no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105#13133124—21no—
106#13133124—21no—
107#13133124D411yescharacteristic
108#13133124D411yescharacteristic
109C2×#41×C53—22no—
110C2×#41×C53—22no—
111C2×#41×C53—22no—
112C2×#41×C53—41no—
113C2×#41×C53—41no—
114C2×#41×C53C2×C414yes—
115C22×Q8×C53C2311yescharacteristic
116C22×D4×C53—21no—
117C22×D4×C53C2311yescharacteristic
118C23×C4×C53—21no—
119C23×C4×C53C2311yescharacteristic
120C22×C8×C53C2311yescharacteristic
121C22×C8×C53C2311yescharacteristic
122C2×C4⋊C4×C53C2311yescharacteristic
123C2×C4⋊C4×C53C2311yescharacteristic
124C2×C22⋊C4×C53—21no—
125#103×C53—41no—
126#103×C53—41no—
127#102×C53—22no—
128#102×C53—22no—
129C22⋊C8×C53—22no—
130#95×C53—21no—
131C24×C53—21no—
132C24×C53C2×D411yescharacteristic
133#41×C53—28no—
134#41×C53—44no—
135#41×C53—44no—
136#41×C53—44no—
137C2×Q8×C53—21no—
138C2×Q8×C53—21no—
139C2×Q8×C53—21no—
140C2×Q8×C53—21no—
141C2×Q8×C53—21no—
142C2×Q8×C53C22×C412yes—
143C2×D4×C53—24no—
144C2×D4×C53—21no—
145C2×D4×C53—21no—
146C2×D4×C53—21no—
147C2×D4×C53—42no—
148C2×D4×C53—42no—
149C2×D4×C53C22×C412yes—
150C2×D4×C53#4112yes—
151C2×D4×C53#4112yes—
152C22×C4×C53—21no—
153C22×C4×C53—21no—
154C22×C4×C53—21no—
155C22×C4×C53—21no—
156C22×C4×C53—21no—
157C22×C4×C53—24no—
158C22×C4×C53—21no—
159C22×C4×C53—21no—
160C22×C4×C53—42no—
161C22×C4×C53—42no—
162C22×C4×C53C22×C412yes—
163C22×C4×C53C2×D411yescharacteristic
164C22×C4×C53#4112yes—
165C22×C4×C53C2411yescharacteristic
166C22×Dic53—21no—
167C22×Dic53—21no—
168C22×Dic53—24no—
169C22×Dic53—21no—
170C22×Dic53—21no—
171C22×Dic53—42no—
172C22×Dic53—42no—
173C22×Dic53—42no—
174C22×Dic53—42no—
175C22×Dic53—81no—
176C22×Dic53—81no—
177C22×Dic53—81no—
178C22×Dic53—81no—
179C22×Dic53C2×D411yescharacteristic
180C22×Dic53C2×D411yescharacteristic
181C22×Dic53C2×D411yescharacteristic
182C22×Dic53C2×D411yescharacteristic
183C2×C8×C53—21no—
184C2×C8×C53—21no—
185C2×C8×C53—21no—
186C2×C8×C53—21no—
187C2×C8×C53—21no—
188C2×C8×C53—21no—
189C4⋊C4×C53—21no—
190C4⋊C4×C53—21no—
191C4⋊C4×C53—21no—
192C4⋊C4×C53—21no—
193C22⋊C4×C53—42no—
194C22⋊Dic53—22no—
195C22⋊Dic53—41no—
196C22⋊Dic53—42no—
197C22⋊Dic53—42no—
198C22⋊Dic53—81no—
199C4⋊Dic53—24no—
200C4⋊Dic53—21no—
201C4⋊Dic53—21no—
202C4⋊Dic53—21no—
203C4⋊Dic53—21no—
204C4⋊Dic53—21no—
205C4⋊Dic53—21no—
206C4⋊Dic53C22×C412yes—
207C4⋊Dic53C22×C412yes—
208#11719919—81no—
209#11719919—81no—
210C4×Dic53—21no—
211C4×Dic53—21no—
212C4×Dic53—21no—
213C4×Dic53—21no—
214C4×Dic53—81no—
215C4×Dic53—81no—
216#11719917—44no—
217C2×#88322—21no—
218C2×#88322—21no—
219C2×#88322—21no—
220C2×#88322—21no—
221C2×#88322C22×C412yes—
222C2×#88322C22×C412yes—
223C23×C53—22no—
224C23×C53—22no—
225C23×C53—22no—
226C23×C53—21no—
227C23×C53—21no—
228C23×C53—21no—
229C23×C53—21no—
230C23×C53—21no—
231C23×C53—21no—
232C23×C53—41no—
233C23×C53—41no—
234C23×C53—41no—
235C23×C53—41no—
236C23×C53C4×D412yes—
237C23×C53C22×D411yescharacteristic
238Q8×C53—22no—
239Q8×C53—22no—
240Q8×C53—41no—
241Q8×C53—41no—
242D4×C53—22no—
243D4×C53—22no—
244D4×C53—22no—
245D4×C53—48no—
246D4×C53#11714yes—
247C2×C4×C53—21no—
248C2×C4×C53—21no—
249C2×C4×C53—22no—
250C2×C4×C53—22no—
251C2×C4×C53—22no—
252C2×C4×C53—22no—
253C2×C4×C53—22no—
254C2×C4×C53—48no—
255C2×C4×C53—41no—
256C2×C4×C53—41no—
257C2×C4×C53—41no—
258C2×C4×C53—41no—
259C2×C4×C53—41no—
260C2×C4×C53—41no—
261C2×C4×C53—41no—
262C2×C4×C53—41no—
263C2×C4×C53C4×D412yes—
264C2×C4×C53C23×C411yescharacteristic
265C2×C4×C53C2×#4111yescharacteristic
266C2×C4×C53C2×#4111yescharacteristic
267C2×C4×C53C2×#4111yescharacteristic
268C2×C4×C53H(5,𝔽2)11yescharacteristic
269C2×C4×C53H(5,𝔽2)11yescharacteristic
270C2×Dic53—21no—
271C2×Dic53—21no—
272C2×Dic53—22no—
273C2×Dic53—22no—
274C2×Dic53—21no—
275C2×Dic53—21no—
276C2×Dic53—21no—
277C2×Dic53—21no—
278C2×Dic53—21no—
279C2×Dic53—21no—
280C2×Dic53—21no—
281C2×Dic53—21no—
282C2×Dic53—21no—
283C2×Dic53—41no—
284C2×Dic53—41no—
285C2×Dic53—41no—
286C2×Dic53—41no—
287C2×Dic53—41no—
288C2×Dic53—41no—
289C2×Dic53—41no—
290C2×Dic53—41no—
291C2×Dic53—41no—
292C2×Dic53—41no—
293C2×Dic53—84no—
294C2×Dic53—84no—
295C2×Dic53—84no—
296C2×Dic53#13712yes—
297C8×C53—41no—
298C8×C53—41no—
299#88322—22no—
300#88322—22no—
301#22578—531nomaximal, Sylow
302C22×C53—22no—
303C22×C53—24no—
304C22×C53—22no—
305C22×C53—22no—
306C22×C53—41no—
307C22×C53—42no—
308C22×C53—42no—
309C22×C53—42no—
310C22×C53—41no—
311C22×C53—42no—
312C22×C53—42no—
313C22×C53—41no—
314C22×C53—42no—
315C22×C53—41no—
316C22×C53C2×C4×D411yescharacteristic
317C22×C53#54611yescharacteristic
318C22×C53#54611yescharacteristic
319C22×C53C2×#13711yescharacteristic
320C22×C53#57511yescharacteristic
321C22×C53#57611yescharacteristic
322C22×C53#57711yescharacteristic
323C4×C53—22no—
324C4×C53—42no—
325C4×C53—81no—
326C4×C53—81no—
327C4×C53C2×#11711yescharacteristic
328C4×C53#51811yescharacteristic
329C4×C53#55111yescharacteristic
330C4×C53#55111yescharacteristic
331Dic53—22no—
332Dic53—41no—
333Dic53—42no—
334Dic53—42no—
335Dic53—41no—
336Dic53—41no—
337Dic53—81no—
338Dic53—81no—
339C2×#417—531no—
340#2345—531no—
341#2339—531no—
342#1935—531no—
343#1933—531no—
344#1928—531no—
345#1927—531no—
346#1864—531no—
347#1862—531no—
348#1823—531no—
349#1809—531no—
350#1788D5311yescharacteristic
351#1762—531no—
352C4×#102—531no—
353#1753—531no—
354C2×C53—42no—
355C2×C53—44no—
356C2×C53#234211yescharacteristic
357C2×C53C4×#13711yescharacteristic
358C2×C53#294611yescharacteristic
359C2×C53#330711yescharacteristic
360C2×C53#331411yescharacteristic
361C2×C53#332211yescharacteristic
362C2×C53#333111yescharacteristic
363C22×#41C2×D5311yescharacteristic
364C22×M16—531no—
365C2×C4×D4—531no—
366C2×C4×D4—531no—
367C22×C4⋊C4—531no—
368C2×#107—531no—
369C2×#106—531no—
370#417—538no—
371C2×#103—531no—
372C2×#103—531no—
373C2×#103C2×D5311yescharacteristic
374C2×#103C2×D5311yescharacteristic
375C2×#102—531no—
376C2×#102—531no—
377C2×#102C2×D5311yescharacteristic
378C2×#102C2×D5311yescharacteristic
379C2×C22⋊C8C2×D5311yescharacteristic
380C2×#97—531no—
381C2×C4×C8—531no—
382#391—531no—
383#391—531no—
384#389—531no—
385#389—531no—
386#387—1061no—
387#387—1061no—
388#386—1061no—
389#386—1061no—
390#381—1061no—
391#380—531no—
392#380—1061no—
393#380C2×D5311yescharacteristic
394C4×C4⋊C4—531no—
395C4×C4⋊C4—531no—
396C4×C22⋊C4—531no—
397C4×C22⋊C4—1061no—
398C2×#95—1061no—
399#340—531no—
400#340—531no—
401#340—531no—
402#340—531no—
403#340—531no—
404#340—531no—
405#336—531no—
406#336—531no—
407C53#2257811yescharacteristic, complemented, Sylow
408C2×#41—22no—
409C2×#41—22no—
410C2×#41—22no—
411C2×#41—41no—
412C2×#41—41no—
413C2×#41C4×D5314yes—
414C22×Q8C22×D5311yescharacteristic
415C22×D4—21no—
416C22×D4C22×D5311yescharacteristic
417C23×C4—21no—
418C23×C4—1061no—
419C23×C4—1061no—
420C23×C4—1061no—
421C23×C4C22×D5311yescharacteristic
422C2×M16—534no—
423C2×M16—2121no—
424C2×M16—2121no—
425C22×C8—531no—
426C22×C8—531no—
427C22×C8C22×D5311yescharacteristic
428C22×C8C22×D5311yescharacteristic
429C4×D4—1062no—
430C4×D4—1062no—
431C4×D4—2121no—
432C4×D4—2121no—
433C2×C4⋊C4—534no—
434C2×C4⋊C4—531no—
435C2×C4⋊C4—531no—
436C2×C4⋊C4—531no—
437C2×C4⋊C4—531no—
438C2×C4⋊C4—1061no—
439C2×C4⋊C4—1062no—
440C2×C4⋊C4—1061no—
441C2×C4⋊C4—1061no—
442C2×C4⋊C4C22×D5311yescharacteristic
443C2×C4⋊C4C22×D5311yescharacteristic
444C2×C22⋊C4—21no—
445C2×C22⋊C4—531no—
446C2×C22⋊C4—531no—
447C2×C22⋊C4—1061no—
448C2×C22⋊C4—1061no—
449C2×C22⋊C4—1061no—
450C2×C42—531no—
451C2×C42—531no—
452C2×C42—1061no—
453C2×C42—1061no—
454#107—2121no—
455#106—2121no—
456#103—41no—
457#103—41no—
458#103—534no—
459#103—534no—
460#102—22no—
461#102—22no—
462#102—534no—
463#102—534no—
464C22⋊C8—22no—
465#97—2121no—
466C4×C8—2121no—
467#95—21no—
468#95—531no—
469#95—531no—
470#95—1061no—
471#95—1061no—
472#95—1061no—
473#95—1061no—
474#95—1061no—
475#95—1061no—
476#95—1061no—
477#95—1061no—
478#95—1061no—
479C24—21no—
480C24D4×D5311yescharacteristic
481#41—28no—
482#41—44no—
483#41—44no—
484#41—44no—
485C2×Q8—21no—
486C2×Q8—21no—
487C2×Q8—21no—
488C2×Q8—21no—
489C2×Q8—21no—
490C2×Q8C2×C4×D5312yes—
491C2×D4—24no—
492C2×D4—21no—
493C2×D4—21no—
494C2×D4—21no—
495C2×D4—42no—
496C2×D4—42no—
497C2×D4C2×C4×D5312yes—
498C2×D4#1171994712yes—
499C2×D4#1171994712yes—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—21no—
505C22×C4—24no—
506C22×C4—21no—
507C22×C4—21no—
508C22×C4—42no—
509C22×C4—42no—
510C22×C4—531no—
511C22×C4—531no—
512C22×C4—531no—
513C22×C4—531no—
514C22×C4—1064no—
515C22×C4—1061no—
516C22×C4—1061no—
517C22×C4—1061no—
518C22×C4—1061no—
519C22×C4—4241no—
520C22×C4—4241no—
521C22×C4—4241no—
522C22×C4—4241no—
523C22×C4—2122no—
524C22×C4—2122no—
525C22×C4—2122no—
526C22×C4—2122no—
527C22×C4C2×C4×D5312yes—
528C22×C4D4×D5311yescharacteristic
529C22×C4#1171994712yes—
530C22×C4C23×D5311yescharacteristic
531M16—2124no—
532C2×C8—21no—
533C2×C8—21no—
534C2×C8—21no—
535C2×C8—21no—
536C2×C8—21no—
537C2×C8—21no—
538C2×C8—532no—
539C2×C8—532no—
540C2×C8—1061no—
541C2×C8—1061no—
542C2×C8—1061no—
543C2×C8—1061no—
544C4⋊C4—21no—
545C4⋊C4—21no—
546C4⋊C4—21no—
547C4⋊C4—21no—
548C4⋊C4—532no—
549C4⋊C4—532no—
550C4⋊C4—1064no—
551C4⋊C4—1061no—
552C4⋊C4—1061no—
553C4⋊C4—1061no—
554C4⋊C4—1061no—
555C4⋊C4—1061no—
556C4⋊C4—1061no—
557C4⋊C4—4241no—
558C4⋊C4—4241no—
559C22⋊C4—42no—
560C22⋊C4—1062no—
561C22⋊C4—4241no—
562C22⋊C4—2121no—
563C22⋊C4—2122no—
564C22⋊C4—2122no—
565C42—1061no—
566C42—1061no—
567C42—1061no—
568C42—1061no—
569C42—4241no—
570C42—4241no—
571C23—22no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—22no—
577C23—22no—
578C23—21no—
579C23—21no—
580C23—41no—
581C23—41no—
582C23—41no—
583C23—41no—
584C23#1313317712yes—
585C23C2×D4×D5311yescharacteristic
586Q8—22no—
587Q8—22no—
588Q8—41no—
589Q8—41no—
590D4—22no—
591D4—22no—
592D4—22no—
593D4—48no—
594D4#1313317314yes—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—48no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—532no—
612C2×C4—1061no—
613C2×C4—1061no—
614C2×C4—1062no—
615C2×C4—1062no—
616C2×C4—1061no—
617C2×C4—1061no—
618C2×C4—1061no—
619C2×C4—1061no—
620C2×C4—1061no—
621C2×C4—1061no—
622C2×C4—1061no—
623C2×C4—1061no—
624C2×C4—1061no—
625C2×C4—4244no—
626C2×C4—4244no—
627C2×C4—4244no—
628C2×C4—2121no—
629C2×C4—2121no—
630C2×C4—2121no—
631C2×C4—2121no—
632C2×C4—2121no—
633C2×C4—2121no—
634C2×C4—2121no—
635C2×C4—2121no—
636C2×C4—2121no—
637C2×C4—2121no—
638C2×C4#1313317712yes—
639C2×C4C22×C4×D5311yescharacteristic
640C2×C4C2×#1171994711yescharacteristic
641C2×C4C2×#1171994711yescharacteristic
642C2×C4C2×#1171994711yescharacteristic
643C2×C4#1313326211yescharacteristic
644C2×C4#1313326211yescharacteristic
645C8—41no—
646C8—41no—
647C8—1062no—
648C8—1062no—
649C22—22no—
650C22—24no—
651C22—22no—
652C22—22no—
653C22—41no—
654C22—42no—
655C22—42no—
656C22—41no—
657C22—42no—
658C22—42no—
659C22—42no—
660C22—41no—
661C22—41no—
662C22—42no—
663C22C2×#1313317711yescharacteristic
664C22#1681126011yescharacteristic
665C22#1681126011yescharacteristic
666C22#137×D5311yescharacteristic
667C22#1681141911yescharacteristic
668C22#1681142511yescharacteristic
669C22#1681143011yescharacteristic
670C4—22no—
671C4—42no—
672C4—81no—
673C4—81no—
674C4—1062no—
675C4—4241no—
676C4—4241no—
677C4—2121no—
678C4—2122no—
679C4—2122no—
680C4—2121no—
681C4—2121no—
682C4C2×#1313317311yescharacteristic
683C4#1681109611yescharacteristic
684C4#1681130411yescharacteristic
685C4#1681130411yescharacteristic
686C2—42no—
687C2—44no—
688C2#2027756611yescharacteristic
689C2#2028353911yescharacteristic
690C2#2028355011yescharacteristic
691C2#2028668411yescharacteristic
692C2#2028678911yescharacteristic
693C2#2028690911yescharacteristic
694C2#2028702411yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1788C2
socleC23×C53C22×D4
prime radicalC22×#41×C53C22
cosocleC22×C4×C53C24
hypercenter#1788D53
hypocenterC53#22578
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1788

Complements

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

NH ≅ G/Ntype
2. #20277594C2semidirect
4. #20275402C2semidirect
5. #20275388C2semidirect
6. #20275359C2semidirect
7. #20275352C2semidirect
8. #20274974C2semidirect
9. #20274960C2semidirect
10. #20274802C2semidirect
11. #20274746C2semidirect
12. #20274542C2semidirect
13. #102×Dic53C2semidirect
14. #20274501C2semidirect
22. #16810624C4semidirect
65. C2×#11719917C2×C4semidirect
74. C2×C4⋊Dic53C2×C4semidirect
92. #13133141C2×C4semidirect
93. #13133141C2×C4semidirect
407. C53#22578semidirect

Maximal subgroups

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

subgroupindex
C2×#168106242
#202775942
#202775452
#202754022
#202753882
#202753592
#202753522
#202749742
#202749602
#202748022
#202747462
#202745422
#102×Dic532
#202745012
C53×#17882
#2257853