Finite Group Databasebeta

← #27401329

Subgroups — #27401329

Order 13568.

982 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810622C211yesmaximal, characteristic
2#20277592C211yesmaximal, characteristic, complemented
3#20277552C211yesmaximal, characteristic
4#20275413C211yesmaximal, characteristic, complemented
5#20275399C211yesmaximal, characteristic, complemented
6#20275340C211yesmaximal, characteristic, complemented
7#20275333C211yesmaximal, characteristic, complemented
8#20274984C211yesmaximal, characteristic, complemented
9#20274970C211yesmaximal, characteristic, complemented
10#20274746C211yesmaximal, characteristic, complemented
11#20274656C211yesmaximal, characteristic
12#20274544C211yesmaximal, characteristic, complemented
13#20274534C211yesmaximal, characteristic, complemented
14#20274501C211yesmaximal, characteristic, complemented
15C53×#1823C211yesmaximal, characteristic, complemented
16C22×#11719945C2211yescharacteristic
17C22×#11719917C2211yescharacteristic
18C2×#13133182C2211yescharacteristic
19C2×#13133182C2211yescharacteristic
20C2×#13133150C2211yescharacteristic
21C2×#13133148C2211yescharacteristic
22#16810622C418yescomplemented
23C2×#13133142C2211yescharacteristic
24C2×#13133142C2211yescharacteristic
25C2×#13133140C2211yescharacteristic
26C2×#13133140C2211yescharacteristic
27C2×#13133139C2211yescharacteristic
28C2×#13133139C2211yescharacteristic
29C2×#13133137C2211yescharacteristic
30C2×#13133137C2211yescharacteristic
31C2×#13133127C2211yescharacteristic
32C2×C8×Dic53C2211yescharacteristic
33#16810524C2211yescharacteristic
34#16810524C2211yescharacteristic
35#16810523—21no—
36#16810523—21no—
37#16810522C2211yescharacteristic
38#16810522C2211yescharacteristic
39#16810509—21no—
40#16810509—21no—
41#16810483—21no—
42#16810483C2211yescharacteristic
43#16810482—21no—
44C4⋊C4×Dic53C2211yescharacteristic
45C4⋊C4×Dic53C2211yescharacteristic
46C4×C22⋊Dic53—21no—
47C4×C22⋊Dic53C2211yescharacteristic
48#16810470—21no—
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810377C2211yescharacteristic
52#16810377C2211yescharacteristic
53C22×C4⋊C4×C53C2211yescharacteristic
54C2×C22⋊C8×C53C2211yescharacteristic
55C53×#380C2211yescharacteristic
56C53×#340C2211yescharacteristic
57C53×#340C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C53×#340C2211yescharacteristic
60C2×#11719945—22no—
61C2×#11719945—22no—
62C2×#11719945—22no—
63C2×#11719945—41no—
64C2×#11719945—41no—
65C2×#11719945C2×C414yes—
66C22×Dic106C2311yescharacteristic
67C22×D212C2311yescharacteristic
68C22×C22⋊D53—21no—
69C22×C4×D53—21no—
70C2×#11719917—41no—
71C2×#11719917—41no—
72C2×#11719917C2×C414yes—
73C22×#88322C2311yescharacteristic
74C22×#88322C2311yescharacteristic
75#13133182—22no—
76#13133182—22no—
77#13133182—41no—
78#13133182—41no—
79C2×C4⋊Dic53—21no—
80C2×C4⋊Dic53—21no—
81C2×C4⋊Dic53C2311yescharacteristic
82C2×C4⋊Dic53C2311yescharacteristic
83C2×#11719919—21no—
84C2×#11719919—21no—
85C2×C22⋊Dic53—21no—
86C2×C22⋊Dic53D411yescharacteristic
87C2×C22⋊Dic53D411yescharacteristic
88C2×#11719921—21no—
89C2×#11719921—21no—
90C2×#11719921—21no—
91C2×#11719921—21no—
92C2×C4×Dic53—21no—
93C2×C4×Dic53—21no—
94C2×C4×Dic53—21no—
95C2×C4×Dic53—21no—
96C2×C4×Dic53C2311yescharacteristic
97C2×C4×Dic53C2311yescharacteristic
98#13133150—41no—
99#13133148—41no—
100#13133142—22no—
101#13133142—22no—
102#13133140C2×C414yes—
103#13133140C2×C414yes—
104#13133139—41no—
105#13133139—41no—
106#13133137C2×C414yes—
107#13133137C2×C414yes—
108#13133127—41no—
109C8×Dic53—41no—
110#13133124—21no—
111#13133124—21no—
112#13133124—21no—
113#13133124—21no—
114#13133124—21no—
115#13133124D411yescharacteristic
116#13133124D411yescharacteristic
117C23×C4×C53—21no—
118C23×C4×C53C2311yescharacteristic
119C22×C8×C53C2311yescharacteristic
120C22×C8×C53C2311yescharacteristic
121C2×C4⋊C4×C53—21no—
122C2×C4⋊C4×C53—22no—
123C2×C4⋊C4×C53C2×C414yes—
124C2×C4⋊C4×C53C2311yescharacteristic
125C2×C4⋊C4×C53C2311yescharacteristic
126C2×C4⋊C4×C53C2311yescharacteristic
127C2×C4⋊C4×C53C2311yescharacteristic
128C2×C22⋊C4×C53—21no—
129C22⋊C8×C53—22no—
130#95×C53—21no—
131C24×C53C2×D411yescharacteristic
132C23×D53—21no—
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53—21no—
137C22×C4×C53—21no—
138C22×C4×C53—21no—
139C22×C4×C53—24no—
140C22×C4×C53—21no—
141C22×C4×C53—21no—
142C22×C4×C53—81no—
143C22×C4×C53—81no—
144C22×C4×C53C22×C412yes—
145C22×C4×C53C2×D411yescharacteristic
146C22×C4×C53#4112yes—
147C22×C4×C53C2411yescharacteristic
148C2×C22⋊D53—24no—
149C2×C22⋊D53—42no—
150C2×C22⋊D53—42no—
151C22×Dic53—21no—
152C22×Dic53—21no—
153C22×Dic53—21no—
154C22×Dic53C2×D411yescharacteristic
155C22×Dic53C2×D411yescharacteristic
156C22×Dic53C2×D411yescharacteristic
157C22×Dic53C2×D411yescharacteristic
158#11719945—28no—
159#11719945—44no—
160#11719945—44no—
161#11719945—44no—
162C2×D212—21no—
163C2×D212—21no—
164C2×D212—21no—
165C2×D212—21no—
166C2×D212—21no—
167C2×D212C22×C412yes—
168C2×C4×D53—24no—
169C2×C4×D53—42no—
170C2×C4×D53—42no—
171C2×Dic106—21no—
172C2×Dic106—21no—
173C2×Dic106—21no—
174C2×Dic106C22×C412yes—
175C2×Dic106#4112yes—
176C2×Dic106#4112yes—
177C2×C8×C53—21no—
178C2×C8×C53—21no—
179C2×C8×C53—21no—
180C2×C8×C53—21no—
181C2×C8×C53—21no—
182C2×C8×C53—21no—
183C4⋊C4×C53—24no—
184C4⋊C4×C53—21no—
185C4⋊C4×C53—21no—
186C4⋊C4×C53—21no—
187C4⋊C4×C53—21no—
188C4⋊C4×C53—21no—
189C4⋊C4×C53—21no—
190C4⋊C4×C53C22×C412yes—
191C4⋊C4×C53C22×C412yes—
192C22⋊C4×C53—81no—
193C22⋊Dic53—22no—
194C22⋊Dic53—41no—
195C22⋊Dic53—42no—
196#11719921—42no—
197#11719921—42no—
198#11719921—81no—
199#11719921—81no—
200C4⋊Dic53—21no—
201C4⋊Dic53—21no—
202C4⋊Dic53—21no—
203C4⋊Dic53—21no—
204C4⋊Dic53—81no—
205C4⋊Dic53—81no—
206#11719919—42no—
207#11719919—42no—
208C4×Dic53—21no—
209C4×Dic53—21no—
210C4×Dic53—21no—
211C4×Dic53—21no—
212C4×Dic53—42no—
213C4×Dic53—42no—
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—21no—
224C23×C53—21no—
225C23×C53—21no—
226C23×C53—21no—
227C23×C53—21no—
228C23×C53—21no—
229C23×C53C4×D412yes—
230C23×C53C22×D411yescharacteristic
231C22×D53—22no—
232C22×D53—41no—
233C22×D53—41no—
234C22×D53—41no—
235C22×D53—41no—
236C22×D53—41no—
237C22×D53—41no—
238C2×C4×C53—21no—
239C2×C4×C53—21no—
240C2×C4×C53—22no—
241C2×C4×C53—22no—
242C2×C4×C53—22no—
243C2×C4×C53—22no—
244C2×C4×C53—22no—
245C2×C4×C53—22no—
246C2×C4×C53—41no—
247C2×C4×C53—41no—
248C2×C4×C53—41no—
249C2×C4×C53—41no—
250C2×C4×C53—41no—
251C2×C4×C53—41no—
252C2×C4×C53—41no—
253C2×C4×C53—41no—
254C2×C4×C53—41no—
255C2×C4×C53—41no—
256C2×C4×C53—84no—
257C2×C4×C53C4×D412yes—
258C2×C4×C53C23×C411yescharacteristic
259C2×C4×C53C2×#4111yescharacteristic
260C2×C4×C53C2×#4111yescharacteristic
261C2×C4×C53C2×#4111yescharacteristic
262C2×C4×C53H(5,𝔽2)11yescharacteristic
263C2×C4×C53H(5,𝔽2)11yescharacteristic
264C22⋊D53—48no—
265C2×Dic53—22no—
266C2×Dic53—22no—
267C2×Dic53—22no—
268C2×Dic53—21no—
269C2×Dic53—21no—
270C2×Dic53—21no—
271C2×Dic53—21no—
272C2×Dic53—21no—
273C2×Dic53—21no—
274C2×Dic53—21no—
275C2×Dic53—21no—
276C2×Dic53—21no—
277C2×Dic53—21no—
278C2×Dic53—21no—
279C2×Dic53—41no—
280C2×Dic53—41no—
281C2×Dic53—41no—
282C2×Dic53—41no—
283C2×Dic53—41no—
284C2×Dic53—41no—
285C2×Dic53#13712yes—
286D212—22no—
287D212—22no—
288D212—41no—
289D212—41no—
290C4×D53—48no—
291Dic106—22no—
292Dic106—22no—
293Dic106—22no—
294Dic106#11714yes—
295C8×C53—41no—
296C8×C53—41no—
297#88322—22no—
298#88322—22no—
299#22579—531nomaximal, Sylow
300C22×C53—22no—
301C22×C53—22no—
302C22×C53—22no—
303C22×C53—41no—
304C22×C53—41no—
305C22×C53—41no—
306C22×C53—41no—
307C22×C53C2×C4×D411yescharacteristic
308C22×C53#54611yescharacteristic
309C22×C53#54611yescharacteristic
310C22×C53C2×#13711yescharacteristic
311C22×C53#57511yescharacteristic
312C22×C53#57611yescharacteristic
313C22×C53#57711yescharacteristic
314C2×D53—42no—
315C2×D53—42no—
316C2×D53—42no—
317C2×D53—81no—
318C2×D53—81no—
319C2×D53—81no—
320C2×D53—81no—
321C4×C53—22no—
322C4×C53—42no—
323C4×C53—42no—
324C4×C53—81no—
325C4×C53—81no—
326C4×C53C2×#11711yescharacteristic
327C4×C53#51811yescharacteristic
328C4×C53#55111yescharacteristic
329C4×C53#55111yescharacteristic
330Dic53—24no—
331Dic53—22no—
332Dic53—41no—
333Dic53—41no—
334Dic53—41no—
335Dic53—81no—
336Dic53—81no—
337C2×#417—531no—
338#2345—531no—
339#2341—531no—
340#1936—531no—
341#1934—531no—
342#1926—531no—
343#1925—531no—
344#1865—531no—
345#1863—531no—
346#1823D5311yescharacteristic, complemented
347#1809—531no—
348#1788—531no—
349#1763—531no—
350C4×#103—531no—
351#1753—531no—
352C2×C53—42no—
353C2×C53#234211yescharacteristic
354C2×C53C4×#13711yescharacteristic
355C2×C53#294611yescharacteristic
356C2×C53#330911yescharacteristic
357C2×C53#331611yescharacteristic
358C2×C53#332611yescharacteristic
359C2×C53#333311yescharacteristic
360D53—82no—
361C22×#41—531no—
362C22×M16—531no—
363C2×C4×Q8—531no—
364C2×C4×Q8—531no—
365C22×C4⋊C4C2×D5311yescharacteristic
366C2×#107—531no—
367C2×#106—531no—
368#417—538no—
369C2×#103—531no—
370C2×#103—531no—
371C2×#103—531no—
372C2×#103—531no—
373C2×#102—531no—
374C2×#102—531no—
375C2×#102—531no—
376C2×#102—531no—
377C2×C22⋊C8C2×D5311yescharacteristic
378C2×#97—531no—
379C2×C4×C8—531no—
380#390—531no—
381#390—531no—
382#390—1061no—
383#390—1061no—
384#389—531no—
385#389—531no—
386#387—1061no—
387#387—1061no—
388#380—531no—
389#380—1061no—
390#380—1061no—
391#380C2×D5311yescharacteristic
392C4×C4⋊C4—531no—
393C4×C4⋊C4—531no—
394C4×C22⋊C4—531no—
395C4×C22⋊C4—1061no—
396C4×C22⋊C4—1061no—
397#340—531no—
398#340—531no—
399#340C2×D5311yescharacteristic
400#340C2×D5311yescharacteristic
401#340C2×D5311yescharacteristic
402#340C2×D5311yescharacteristic
403#336—531no—
404#336—531no—
405C53#2257911yescharacteristic, complemented, Sylow
406C2×#41—534no—
407C2×#41—1062no—
408C2×#41—1062no—
409C2×#41—1062no—
410C2×#41—2121no—
411C2×#41—2121no—
412C22×Q8—531no—
413C22×D4—531no—
414C22×D4—1061no—
415C23×C4—21no—
416C23×C4—1061no—
417C23×C4C22×D5311yescharacteristic
418C2×M16—534no—
419C2×M16—2121no—
420C2×M16—2121no—
421C22×C8—531no—
422C22×C8—531no—
423C22×C8C22×D5311yescharacteristic
424C22×C8C22×D5311yescharacteristic
425C4×Q8—1062no—
426C4×Q8—1062no—
427C4×Q8—2121no—
428C4×Q8—2121no—
429C2×C4⋊C4—21no—
430C2×C4⋊C4—22no—
431C2×C4⋊C4—531no—
432C2×C4⋊C4—531no—
433C2×C4⋊C4—1061no—
434C2×C4⋊C4—1061no—
435C2×C4⋊C4—1061no—
436C2×C4⋊C4—1061no—
437C2×C4⋊C4C4×D5314yes—
438C2×C4⋊C4C22×D5311yescharacteristic
439C2×C4⋊C4C22×D5311yescharacteristic
440C2×C4⋊C4C22×D5311yescharacteristic
441C2×C4⋊C4C22×D5311yescharacteristic
442C2×C22⋊C4—21no—
443C2×C22⋊C4—531no—
444C2×C22⋊C4—531no—
445C2×C22⋊C4—1061no—
446C2×C22⋊C4—1061no—
447C2×C22⋊C4—1061no—
448C2×C22⋊C4—1061no—
449C2×C22⋊C4—1061no—
450C2×C42—531no—
451C2×C42—531no—
452C2×C42—1061no—
453C2×C42—1061no—
454C2×C42—1061no—
455C2×C42—1061no—
456#107—2121no—
457#106—2121no—
458#103—534no—
459#103—534no—
460#103—1062no—
461#103—1062no—
462#102—534no—
463#102—534no—
464#102—2121no—
465#102—2121no—
466C22⋊C8—22no—
467#97—2121no—
468C4×C8—2121no—
469#95—21no—
470#95—531no—
471#95—531no—
472#95—1061no—
473#95—1061no—
474#95—1061no—
475#95—1061no—
476#95—1061no—
477C24—1061no—
478C24D4×D5311yescharacteristic
479#41—1068no—
480#41—2124no—
481#41—2124no—
482#41—2124no—
483C2×Q8—532no—
484C2×Q8—532no—
485C2×Q8—532no—
486C2×Q8—1061no—
487C2×Q8—1061no—
488C2×Q8—1061no—
489C2×D4—532no—
490C2×D4—1064no—
491C2×D4—1061no—
492C2×D4—1061no—
493C2×D4—1061no—
494C2×D4—1061no—
495C2×D4—1061no—
496C2×D4—2122no—
497C2×D4—2122no—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—24no—
505C22×C4—21no—
506C22×C4—21no—
507C22×C4—81no—
508C22×C4—81no—
509C22×C4—531no—
510C22×C4—531no—
511C22×C4—531no—
512C22×C4—531no—
513C22×C4—1061no—
514C22×C4—1064no—
515C22×C4—1061no—
516C22×C4—1061no—
517C22×C4—2122no—
518C22×C4—2122no—
519C22×C4C2×C4×D5312yes—
520C22×C4D4×D5311yescharacteristic
521C22×C4#1171994712yes—
522C22×C4C23×D5311yescharacteristic
523M16—2124no—
524C2×C8—21no—
525C2×C8—21no—
526C2×C8—21no—
527C2×C8—21no—
528C2×C8—21no—
529C2×C8—21no—
530C2×C8—532no—
531C2×C8—532no—
532C2×C8—1061no—
533C2×C8—1061no—
534C2×C8—1061no—
535C2×C8—1061no—
536C4⋊C4—24no—
537C4⋊C4—21no—
538C4⋊C4—21no—
539C4⋊C4—21no—
540C4⋊C4—21no—
541C4⋊C4—21no—
542C4⋊C4—21no—
543C4⋊C4—1061no—
544C4⋊C4—1061no—
545C4⋊C4—1061no—
546C4⋊C4—1061no—
547C4⋊C4—4241no—
548C4⋊C4—4241no—
549C4⋊C4—2122no—
550C4⋊C4—2122no—
551C4⋊C4C2×C4×D5312yes—
552C4⋊C4C2×C4×D5312yes—
553C22⋊C4—81no—
554C22⋊C4—1062no—
555C22⋊C4—4241no—
556C22⋊C4—4241no—
557C22⋊C4—2122no—
558C22⋊C4—2122no—
559C22⋊C4—2121no—
560C22⋊C4—2122no—
561C42—1061no—
562C42—1061no—
563C42—1061no—
564C42—1061no—
565C42—4241no—
566C42—4241no—
567C42—2122no—
568C42—2122no—
569C23—21no—
570C23—21no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—1062no—
576C23—2121no—
577C23—2121no—
578C23—2121no—
579C23—2121no—
580C23—2121no—
581C23—2121no—
582C23#1313317712yes—
583C23C2×D4×D5311yescharacteristic
584Q8—534no—
585Q8—1062no—
586Q8—1062no—
587Q8—1062no—
588D4—1062no—
589D4—1062no—
590D4—2128no—
591D4—2121no—
592D4—2121no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—41no—
602C2×C4—41no—
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—84no—
612C2×C4—532no—
613C2×C4—1061no—
614C2×C4—1062no—
615C2×C4—1062no—
616C2×C4—1062no—
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—1061no—
626C2×C4—1061no—
627C2×C4—2128no—
628C2×C4—2121no—
629C2×C4—2121no—
630C2×C4—2121no—
631C2×C4—2121no—
632C2×C4—2121no—
633C2×C4—2121no—
634C2×C4#1313317712yes—
635C2×C4C22×C4×D5311yescharacteristic
636C2×C4#41×D5311yescharacteristic
637C2×C4#41×D5311yescharacteristic
638C2×C4C2×#1171994711yescharacteristic
639C2×C4#1313326311yescharacteristic
640C2×C4#1313326311yescharacteristic
641C8—41no—
642C8—41no—
643C8—1062no—
644C8—1062no—
645C22—22no—
646C22—22no—
647C22—22no—
648C22—41no—
649C22—41no—
650C22—41no—
651C22—41no—
652C22—4241no—
653C22—4241no—
654C22—4241no—
655C22—4241no—
656C22—2122no—
657C22—2122no—
658C22—2122no—
659C22C2×#1313317711yescharacteristic
660C22#1681126511yescharacteristic
661C22#1681126511yescharacteristic
662C22#137×D5311yescharacteristic
663C22#1681142211yescharacteristic
664C22#1681142411yescharacteristic
665C22#1681142911yescharacteristic
666C4—22no—
667C4—42no—
668C4—42no—
669C4—81no—
670C4—81no—
671C4—1064no—
672C4—1062no—
673C4—4241no—
674C4—4241no—
675C4—2121no—
676C4—2121no—
677C4—2121no—
678C4#117×D5311yescharacteristic
679C4#1681109711yescharacteristic
680C4#1681129611yescharacteristic
681C4#1681129611yescharacteristic
682C2—42no—
683C2—4242no—
684C2#2027755711yescharacteristic
685C2#2028353011yescharacteristic
686C2#2028354111yescharacteristic
687C2#2028672411yescharacteristic
688C2#2028682911yescharacteristic
689C2#2028697111yescharacteristic
690C2#2028705111yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1823C2
socleC23×C53C22×D4
prime radicalC22×#11719945C22
cosocleC22×C4×C53C24
hypercenter#1823D53
hypocenterC53#22579
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1823

Complements

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

NH ≅ G/Ntype
2. #20277592C2semidirect
4. #20275413C2semidirect
5. #20275399C2semidirect
6. #20275340C2semidirect
7. #20275333C2semidirect
8. #20274984C2semidirect
9. #20274970C2semidirect
10. #20274746C2semidirect
12. #20274544C2semidirect
13. #20274534C2semidirect
14. #20274501C2semidirect
15. C53×#1823C2semidirect
22. #16810622C4semidirect
346. #1823D53semidirect
405. C53#22579semidirect

Maximal subgroups

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

subgroupindex
C2×#168106222
#202775922
#202775522
#202754132
#202753992
#202753402
#202753332
#202749842
#202749702
#202747462
#202746562
#202745442
#202745342
#202745012
C53×#18232
#2257953