Finite Group Databasebeta

← #27401332

Subgroups — #27401332

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×#16810624C211yesmaximal, characteristic
2#20277590C211yesmaximal, characteristic, complemented
3#20275415C211yesmaximal, characteristic, complemented
4#20275401C211yesmaximal, characteristic, complemented
5#20275339C211yesmaximal, characteristic, complemented
6#20275332C211yesmaximal, characteristic, complemented
7#20274983C211yesmaximal, characteristic, complemented
8#20274969C211yesmaximal, characteristic, complemented
9#20274799C211yesmaximal, characteristic, complemented
10#20274746C211yesmaximal, characteristic, complemented
11#20274655C211yesmaximal, characteristic
12#20274545C211yesmaximal, characteristic, complemented
13C4×#13133141C211yesmaximal, characteristic, complemented
14#20274499C211yesmaximal, characteristic, complemented
15C53×#2341C211yesmaximal, characteristic
16C22×#11719917C2211yescharacteristic
17C22×C4⋊Dic53C2211yescharacteristic
18C2×#13133150C2211yescharacteristic
19C2×#13133148C2211yescharacteristic
20#16810624C418yescomplemented
21C2×#13133141C2211yescharacteristic
22C2×#13133141C2211yescharacteristic
23C2×#13133141C2211yescharacteristic
24C2×#13133141C2211yescharacteristic
25C2×#13133138C2211yescharacteristic
26C2×#13133138C2211yescharacteristic
27C2×#13133138C2211yescharacteristic
28C2×#13133138C2211yescharacteristic
29C2×C22⋊#88322C2211yescharacteristic
30C2×#13133128C2211yescharacteristic
31C2×C4×#88322C2211yescharacteristic
32#16810521C2211yescharacteristic
33#16810521C2211yescharacteristic
34#16810506—21no—
35#16810506—21no—
36#16810483C2211yescharacteristic
37#16810481—21no—
38C4×C4⋊Dic53C2211yescharacteristic
39C4×C4⋊Dic53C2211yescharacteristic
40C22⋊C4×Dic53—21no—
41#16810385C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810378C2211yescharacteristic
48#16810378C2211yescharacteristic
49C22×#41×C53C2211yescharacteristic
50C2×C4×Q8×C53C2211yescharacteristic
51C2×C4×Q8×C53C2211yescharacteristic
52C53×#390—21no—
53C53×#390—21no—
54C53×#390C2211yescharacteristic
55C53×#390C2211yescharacteristic
56C53×#380—21no—
57C53×#380C2211yescharacteristic
58C4×C22⋊C4×C53—21no—
59C4×C22⋊C4×C53C2211yescharacteristic
60C23×Dic53—21no—
61C2×#11719917—41no—
62C2×#11719917—41no—
63C2×#11719917C2×C414yes—
64C22×#88322C2311yescharacteristic
65C22×#88322C2311yescharacteristic
66C22×#88322C2311yescharacteristic
67C22×#88322C2311yescharacteristic
68C2×C4⋊Dic53—21no—
69C2×C4⋊Dic53—22no—
70C2×C4⋊Dic53C2×C414yes—
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×#11719919—21no—
76C2×#11719919—21no—
77C2×C22⋊Dic53—21no—
78C2×C4×Dic53—21no—
79C2×C4×Dic53—21no—
80#13133150—41no—
81#13133148—41no—
82#13133141—22no—
83#13133141—22no—
84#13133141C2×C414yes—
85#13133141C2×C414yes—
86#13133138—41no—
87#13133138—41no—
88#13133138C2×C414yes—
89#13133138C2×C414yes—
90C22⋊#88322—22no—
91#13133128—41no—
92C4×#88322—41no—
93#13133124—21no—
94#13133124—21no—
95#13133124—21no—
96#13133124—21no—
97#13133124—21no—
98C2×#41×C53—22no—
99C2×#41×C53—22no—
100C2×#41×C53—22no—
101C2×#41×C53—41no—
102C2×#41×C53—41no—
103C2×#41×C53C2×C414yes—
104C22×Q8×C53C2311yescharacteristic
105C22×D4×C53—21no—
106C22×D4×C53C2311yescharacteristic
107C23×C4×C53—21no—
108C23×C4×C53C2311yescharacteristic
109C4×Q8×C53—22no—
110C4×Q8×C53—22no—
111C4×Q8×C53—41no—
112C4×Q8×C53—41no—
113C2×C4⋊C4×C53—21no—
114C2×C4⋊C4×C53—21no—
115C2×C4⋊C4×C53C2311yescharacteristic
116C2×C4⋊C4×C53C2311yescharacteristic
117C2×C22⋊C4×C53—21no—
118C2×C22⋊C4×C53—21no—
119C2×C22⋊C4×C53—21no—
120C2×C22⋊C4×C53—21no—
121C2×C22⋊C4×C53—21no—
122C2×C22⋊C4×C53D411yescharacteristic
123C2×C22⋊C4×C53D411yescharacteristic
124C2×C42×C53—21no—
125C2×C42×C53—21no—
126C2×C42×C53C2311yescharacteristic
127C2×C42×C53C2311yescharacteristic
128#95×C53—21no—
129#95×C53D411yescharacteristic
130#95×C53D411yescharacteristic
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×C53C22×C412yes—
141C2×Q8×C53#4112yes—
142C2×Q8×C53#4112yes—
143C2×D4×C53—24no—
144C2×D4×C53—21no—
145C2×D4×C53—21no—
146C2×D4×C53—21no—
147C2×D4×C53—21no—
148C2×D4×C53—21no—
149C2×D4×C53—42no—
150C2×D4×C53—42no—
151C2×D4×C53C22×C412yes—
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×C53C2×D411yescharacteristic
165C22×C4×C53C2×D411yescharacteristic
166C22×C4×C53C2×D411yescharacteristic
167C22×C4×C53C2×D411yescharacteristic
168C22×C4×C53#4112yes—
169C22×C4×C53C2411yescharacteristic
170C22×Dic53—21no—
171C22×Dic53—21no—
172C22×Dic53—24no—
173C22×Dic53—21no—
174C22×Dic53—21no—
175C22×Dic53—81no—
176C22×Dic53—81no—
177C4⋊C4×C53—21no—
178C4⋊C4×C53—21no—
179C4⋊C4×C53—21no—
180C4⋊C4×C53—21no—
181C4⋊C4×C53—42no—
182C4⋊C4×C53—42no—
183C22⋊C4×C53—22no—
184C22⋊C4×C53—41no—
185C22⋊C4×C53—42no—
186C22⋊C4×C53—42no—
187C22⋊C4×C53—81no—
188C22⋊C4×C53—81no—
189C22⋊C4×C53—81no—
190C42×C53—21no—
191C42×C53—21no—
192C42×C53—21no—
193C42×C53—21no—
194C42×C53—42no—
195C42×C53—42no—
196C22⋊Dic53—42no—
197C4⋊Dic53—24no—
198C4⋊Dic53—21no—
199C4⋊Dic53—21no—
200C4⋊Dic53—21no—
201C4⋊Dic53—21no—
202C4⋊Dic53—21no—
203C4⋊Dic53—21no—
204C4⋊Dic53C22×C412yes—
205C4⋊Dic53C22×C412yes—
206#11719919—81no—
207#11719919—81no—
208C4×Dic53—81no—
209C4×Dic53—81no—
210#11719917—44no—
211C2×#88322—21no—
212C2×#88322—21no—
213C2×#88322—21no—
214C2×#88322—21no—
215C2×#88322—21no—
216C2×#88322—21no—
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×C53—22no—
230C23×C53—41no—
231C23×C53—41no—
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—22no—
241Q8×C53#11714yes—
242D4×C53—22no—
243D4×C53—22no—
244D4×C53—48no—
245D4×C53—41no—
246D4×C53—41no—
247C2×C4×C53—22no—
248C2×C4×C53—21no—
249C2×C4×C53—21no—
250C2×C4×C53—22no—
251C2×C4×C53—22no—
252C2×C4×C53—21no—
253C2×C4×C53—21no—
254C2×C4×C53—22no—
255C2×C4×C53—22no—
256C2×C4×C53—22no—
257C2×C4×C53—22no—
258C2×C4×C53—21no—
259C2×C4×C53—21no—
260C2×C4×C53—21no—
261C2×C4×C53—21no—
262C2×C4×C53—21no—
263C2×C4×C53—21no—
264C2×C4×C53—21no—
265C2×C4×C53—21no—
266C2×C4×C53—21no—
267C2×C4×C53—48no—
268C2×C4×C53—41no—
269C2×C4×C53—41no—
270C2×C4×C53—41no—
271C2×C4×C53—41no—
272C2×C4×C53—41no—
273C2×C4×C53—41no—
274C2×C4×C53C4×D412yes—
275C2×C4×C53#13712yes—
276C2×C4×C53C23×C411yescharacteristic
277C2×C4×C53C2×#4111yescharacteristic
278C2×C4×C53C2×#4111yescharacteristic
279C2×C4×C53C2×#4111yescharacteristic
280C2×C4×C53H(5,𝔽2)11yescharacteristic
281C2×C4×C53H(5,𝔽2)11yescharacteristic
282C2×Dic53—22no—
283C2×Dic53—22no—
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—41no—
294C2×Dic53—84no—
295#88322—22no—
296#88322—22no—
297#88322—41no—
298#88322—41no—
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×C53—42no—
308C22×C53—42no—
309C22×C53—42no—
310C22×C53—81no—
311C22×C53—81no—
312C22×C53—81no—
313C22×C53—81no—
314C22×C53C2×C4×D411yescharacteristic
315C22×C53#54611yescharacteristic
316C22×C53#54611yescharacteristic
317C22×C53C2×#13711yescharacteristic
318C22×C53#57511yescharacteristic
319C22×C53#57611yescharacteristic
320C22×C53#57711yescharacteristic
321C4×C53—22no—
322C4×C53—24no—
323C4×C53—22no—
324C4×C53—41no—
325C4×C53—41no—
326C4×C53—41no—
327C4×C53—81no—
328C4×C53—81no—
329C4×C53C2×#11711yescharacteristic
330C4×C53#51811yescharacteristic
331C4×C53#55111yescharacteristic
332C4×C53#55111yescharacteristic
333Dic53—42no—
334Dic53—42no—
335Dic53—81no—
336Dic53—81no—
337C2×#417—531no—
338#2345—531no—
339#2341D5311yescharacteristic
340#1936—531no—
341#1934—531no—
342#1926—531no—
343#1925—531no—
344#1865—531no—
345#1863—531no—
346#1823—531no—
347#1809—531no—
348#1788—531no—
349#1763—531no—
350C4×#103—531no—
351#1753—531no—
352C2×C53—42no—
353C2×C53—82no—
354C2×C53#234211yescharacteristic
355C2×C53C4×#13711yescharacteristic
356C2×C53#294611yescharacteristic
357C2×C53#330911yescharacteristic
358C2×C53#331611yescharacteristic
359C2×C53#332611yescharacteristic
360C2×C53#333311yescharacteristic
361C22×#41C2×D5311yescharacteristic
362C22×M16—531no—
363C2×C4×Q8C2×D5311yescharacteristic
364C2×C4×Q8C2×D5311yescharacteristic
365C22×C4⋊C4—531no—
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⋊C8—531no—
378C2×#97—531no—
379C2×C4×C8—531no—
380#390—21no—
381#390—21no—
382#390C2×D5311yescharacteristic
383#390C2×D5311yescharacteristic
384#389—531no—
385#389—531no—
386#387—1061no—
387#387—1061no—
388#380—21no—
389#380—531no—
390#380—1061no—
391#380C2×D5311yescharacteristic
392C4×C4⋊C4—531no—
393C4×C4⋊C4—531no—
394C4×C22⋊C4—21no—
395C4×C22⋊C4—1061no—
396C4×C22⋊C4C2×D5311yescharacteristic
397#340—531no—
398#340—531no—
399#340—531no—
400#340—531no—
401#340—531no—
402#340—531no—
403#336—531no—
404#336—531no—
405C53#2257911yescharacteristic, complemented, Sylow
406C2×#41—22no—
407C2×#41—22no—
408C2×#41—22no—
409C2×#41—41no—
410C2×#41—41no—
411C2×#41C4×D5314yes—
412C22×Q8C22×D5311yescharacteristic
413C22×D4—21no—
414C22×D4C22×D5311yescharacteristic
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×C8—531no—
424C22×C8—531no—
425C4×Q8—22no—
426C4×Q8—22no—
427C4×Q8—41no—
428C4×Q8—41no—
429C2×C4⋊C4—21no—
430C2×C4⋊C4—21no—
431C2×C4⋊C4—534no—
432C2×C4⋊C4—531no—
433C2×C4⋊C4—531no—
434C2×C4⋊C4—531no—
435C2×C4⋊C4—531no—
436C2×C4⋊C4—1062no—
437C2×C4⋊C4—1061no—
438C2×C4⋊C4—1061no—
439C2×C4⋊C4—1061no—
440C2×C4⋊C4C22×D5311yescharacteristic
441C2×C4⋊C4C22×D5311yescharacteristic
442C2×C22⋊C4—21no—
443C2×C22⋊C4—21no—
444C2×C22⋊C4—21no—
445C2×C22⋊C4—21no—
446C2×C22⋊C4—21no—
447C2×C22⋊C4—1061no—
448C2×C22⋊C4C22⋊D5311yescharacteristic
449C2×C22⋊C4C22⋊D5311yescharacteristic
450C2×C42—21no—
451C2×C42—21no—
452C2×C42—1061no—
453C2×C42—1061no—
454C2×C42C22×D5311yescharacteristic
455C2×C42C22×D5311yescharacteristic
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—1062no—
467#97—2121no—
468C4×C8—2121no—
469#95—21no—
470#95—1061no—
471#95—1061no—
472#95—1061no—
473#95—1061no—
474#95—1061no—
475#95C22⋊D5311yescharacteristic
476#95C22⋊D5311yescharacteristic
477C24—21no—
478C24C2×C22⋊D5311yescharacteristic
479#41—28no—
480#41—44no—
481#41—44no—
482#41—44no—
483C2×Q8—21no—
484C2×Q8—21no—
485C2×Q8—21no—
486C2×Q8C2×C4×D5312yes—
487C2×Q8#1171994712yes—
488C2×Q8#1171994712yes—
489C2×D4—24no—
490C2×D4—21no—
491C2×D4—21no—
492C2×D4—21no—
493C2×D4—21no—
494C2×D4—21no—
495C2×D4—42no—
496C2×D4—42no—
497C2×D4C2×C4×D5312yes—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—21no—
503C22×C4—24no—
504C22×C4—21no—
505C22×C4—21no—
506C22×C4—42no—
507C22×C4—42no—
508C22×C4—1064no—
509C22×C4—1061no—
510C22×C4—1061no—
511C22×C4—1061no—
512C22×C4—1061no—
513C22×C4—4241no—
514C22×C4—4241no—
515C22×C4C2×C4×D5312yes—
516C22×C4#1171994512yes—
517C22×C4C2×C22⋊D5311yescharacteristic
518C22×C4C2×C22⋊D5311yescharacteristic
519C22×C4C2×C22⋊D5311yescharacteristic
520C22×C4C2×C22⋊D5311yescharacteristic
521C22×C4C2×C22⋊D5311yescharacteristic
522C22×C4C23×D5311yescharacteristic
523M16—2124no—
524C2×C8—532no—
525C2×C8—532no—
526C2×C8—1061no—
527C2×C8—1061no—
528C2×C8—1061no—
529C2×C8—1061no—
530C2×C8—1061no—
531C2×C8—1061no—
532C2×C8—1061no—
533C2×C8—1061no—
534C2×C8—1061no—
535C2×C8—1061no—
536C4⋊C4—21no—
537C4⋊C4—21no—
538C4⋊C4—21no—
539C4⋊C4—21no—
540C4⋊C4—42no—
541C4⋊C4—42no—
542C4⋊C4—532no—
543C4⋊C4—532no—
544C4⋊C4—1064no—
545C4⋊C4—1061no—
546C4⋊C4—1061no—
547C4⋊C4—1061no—
548C4⋊C4—1061no—
549C4⋊C4—1061no—
550C4⋊C4—1061no—
551C4⋊C4—4241no—
552C4⋊C4—4241no—
553C22⋊C4—22no—
554C22⋊C4—41no—
555C22⋊C4—42no—
556C22⋊C4—42no—
557C22⋊C4—81no—
558C22⋊C4—81no—
559C22⋊C4—81no—
560C22⋊C4—2122no—
561C42—21no—
562C42—21no—
563C42—21no—
564C42—21no—
565C42—42no—
566C42—42no—
567C42—4241no—
568C42—4241no—
569C23—21no—
570C23—21no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—22no—
576C23—41no—
577C23—41no—
578C23—41no—
579C23—41no—
580C23—41no—
581C23—41no—
582C23C4×C22⋊D5312yes—
583C23C22×C22⋊D5311yescharacteristic
584Q8—22no—
585Q8—22no—
586Q8—22no—
587Q8#1313317314yes—
588D4—22no—
589D4—22no—
590D4—48no—
591D4—41no—
592D4—41no—
593C2×C4—21no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—22no—
599C2×C4—21no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—48no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—1062no—
621C2×C4—1062no—
622C2×C4—4244no—
623C2×C4—2121no—
624C2×C4—2121no—
625C2×C4—2121no—
626C2×C4—2121no—
627C2×C4—2121no—
628C2×C4—2121no—
629C2×C4—2121no—
630C2×C4—2121no—
631C2×C4—2121no—
632C2×C4—2121no—
633C2×C4C4×C22⋊D5312yes—
634C2×C4#1313324712yes—
635C2×C4C22×C4×D5311yescharacteristic
636C2×C4C2×#1171994711yescharacteristic
637C2×C4C2×#1171994711yescharacteristic
638C2×C4C2×#1171994511yescharacteristic
639C2×C4#1313326211yescharacteristic
640C2×C4#1313326211yescharacteristic
641C8—1062no—
642C8—1062no—
643C8—2121no—
644C8—2121no—
645C22—22no—
646C22—22no—
647C22—22no—
648C22—41no—
649C22—41no—
650C22—41no—
651C22—41no—
652C22—42no—
653C22—42no—
654C22—42no—
655C22—81no—
656C22—81no—
657C22—81no—
658C22—81no—
659C22C2×C4×C22⋊D5311yescharacteristic
660C22#1681126711yescharacteristic
661C22#1681126711yescharacteristic
662C22C2×#1313324711yescharacteristic
663C22#1681142111yescharacteristic
664C22#1681142611yescharacteristic
665C22#1681143111yescharacteristic
666C4—22no—
667C4—24no—
668C4—22no—
669C4—41no—
670C4—41no—
671C4—41no—
672C4—81no—
673C4—81no—
674C4—4241no—
675C4—4241no—
676C4—2122no—
677C4—2122no—
678C4C2×#1313317311yescharacteristic
679C4#1681109611yescharacteristic
680C4#1681129911yescharacteristic
681C4#1681129911yescharacteristic
682C2—42no—
683C2—82no—
684C2#2027756011yescharacteristic
685C2C4×#1313324711yescharacteristic
686C2#2028354311yescharacteristic
687C2#2028672111yescharacteristic
688C2#2028682611yescharacteristic
689C2#2028696811yescharacteristic
690C2#2028705311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#2341C2
socleC23×C53C22×D4
prime radicalC22×#41×C53C22
cosocleC22×C4×C53C24
hypercenter#2341D53
hypocenterC53#22579
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2341

Complements

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

NH ≅ G/Ntype
2. #20277590C2semidirect
3. #20275415C2semidirect
4. #20275401C2semidirect
5. #20275339C2semidirect
6. #20275332C2semidirect
7. #20274983C2semidirect
8. #20274969C2semidirect
9. #20274799C2semidirect
10. #20274746C2semidirect
12. #20274545C2semidirect
13. C4×#13133141C2semidirect
14. #20274499C2semidirect
20. #16810624C4semidirect
405. C53#22579semidirect

Maximal subgroups

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

subgroupindex
C2×#168106242
#202775902
#202754152
#202754012
#202753392
#202753322
#202749832
#202749692
#202747992
#202747462
#202746552
#202745452
C4×#131331412
#202744992
C53×#23412
#2257953