Finite Group Databasebeta

← #27435878

Subgroups — #27435878

Order 13568.

812 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811029C211yesmaximal, characteristic, complemented
2C2×#16810924C211yesmaximal, characteristic
3#20279669C211yesmaximal, characteristic
4#20277811C211yesmaximal, characteristic, complemented
5#20276195C211yesmaximal, characteristic, complemented
6#20276190C211yesmaximal, characteristic, complemented
7#20276104C211yesmaximal, characteristic
8#20276090C211yesmaximal, characteristic
9#20275477C211yesmaximal, characteristic, complemented
10#20275472C211yesmaximal, characteristic, complemented
11#20275436C211yesmaximal, characteristic, complemented
12#20275340C211yesmaximal, characteristic
13#20275333C211yesmaximal, characteristic
14#20274543C211yesmaximal, characteristic
15C53×#1987C211yesmaximal, characteristic, complemented
16C2×#13133209C2211yescharacteristic
17C2×#13133193—21no—
18C2×#13133193—21no—
19C2×C4×D212C2211yescharacteristic
20#16811029—24no—
21#16810924—24no—
22C2×#13133151C2211yescharacteristic
23C2×#13133150C2211yescharacteristic
24C2×#13133150C2211yescharacteristic
25C2×#13133149C2211yescharacteristic
26C2×#13133148C2211yescharacteristic
27C2×#13133148C2211yescharacteristic
28C2×#13133146C2211yescharacteristic
29C2×#13133146C2211yescharacteristic
30C2×#13133139C2211yescharacteristic
31C2×#13133139C2211yescharacteristic
32C2×#13133137C2211yescharacteristic
33C2×#13133137C2211yescharacteristic
34C2×#13133128C2211yescharacteristic
35#16810558—21no—
36#16810558—21no—
37#16810524C2211yescharacteristic
38#16810524C2211yescharacteristic
39#16810520—21no—
40#16810519C2211yescharacteristic
41#16810519C2211yescharacteristic
42#16810510—21no—
43#16810507—21no—
44#16810492C2211yescharacteristic
45#16810492C2211yescharacteristic
46C4⋊C4×Dic53C2211yescharacteristic
47C4⋊C4×Dic53C2211yescharacteristic
48C4×#11719919—21no—
49C4×C4⋊Dic53C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810385C2211yescharacteristic
52#16810377C2211yescharacteristic
53#16810377C2211yescharacteristic
54C2×C4⋊Q8×C53C2211yescharacteristic
55C2×C4⋊C8×C53C2211yescharacteristic
56C4×C4⋊C4×C53C2211yescharacteristic
57C53×#340C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C53×#340C2211yescharacteristic
60C53×#340C2211yescharacteristic
61C22×D212C2311yescharacteristic
62C22×C4×D53—21no—
63C22×#88322C2311yescharacteristic
64C22×#88322C2311yescharacteristic
65#13133209—22no—
66#13133209D414yes—
67#13133193—44no—
68#13133193—44no—
69C4×D212—22no—
70C4×D212—22no—
71C2×C4⋊Dic53D411yescharacteristic
72C2×C4⋊Dic53D411yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×#11719919—21no—
77C2×#11719919—21no—
78C2×#11719919—21no—
79C2×#11719919—21no—
80C2×#11719919—21no—
81C2×#11719919—21no—
82C2×#11719919—21no—
83C2×#11719919—21no—
84C2×#11719921—21no—
85C2×#11719921—21no—
86C2×#11719921—21no—
87C2×#11719921—21no—
88C2×#11719921—21no—
89C2×C4×Dic53—21no—
90C2×C4×Dic53—21no—
91C2×C4×Dic53C2311yescharacteristic
92C2×C4×Dic53C2311yescharacteristic
93#13133151—22no—
94#13133150—22no—
95#13133150—22no—
96#13133149—22no—
97#13133148—22no—
98#13133148—22no—
99#13133146—22no—
100#13133146—22no—
101#13133139—22no—
102#13133139—22no—
103#13133137—22no—
104#13133137Q814yescomplemented
105#13133128—22no—
106#13133124—21no—
107#13133124—21no—
108#13133124—21no—
109#13133124—21no—
110#13133124D411yescharacteristic
111#13133124D411yescharacteristic
112C22×Q8×C53—21no—
113C22×C8×C53C2311yescharacteristic
114C22×C8×C53C2311yescharacteristic
115C4⋊Q8×C53—22no—
116C4⋊Q8×C53—22no—
117C2×C4⋊C4×C53—21no—
118C2×C4⋊C4×C53C2311yescharacteristic
119C2×C4⋊C4×C53C2311yescharacteristic
120C2×C4⋊C4×C53C2311yescharacteristic
121C2×C4⋊C4×C53C2311yescharacteristic
122C2×C42×C53—21no—
123C2×C42×C53C2311yescharacteristic
124C4⋊C8×C53—22no—
125#95×C53—21no—
126C23×D53—21no—
127C2×Q8×C53—42no—
128C2×Q8×C53—42no—
129C2×Q8×C53—42no—
130C22×C4×C53—21no—
131C22×C4×C53—21no—
132C22×C4×C53—21no—
133C22×C4×C53—21no—
134C22×C4×C53C2×D411yescharacteristic
135C22×C4×C53C2×D411yescharacteristic
136C22×C4×C53C2411yescharacteristic
137C22×Dic53—21no—
138C22×Dic53—21no—
139C22×Dic53—21no—
140C22×Dic53C2×D411yescharacteristic
141C22×Dic53C2×D411yescharacteristic
142C22×Dic53C2×D411yescharacteristic
143C22×Dic53C2×D411yescharacteristic
144C2×D212—21no—
145C2×D212—21no—
146C2×D212—21no—
147C2×D212—21no—
148C2×D212—21no—
149C2×D212C2×Q812yes—
150C2×C4×D53—42no—
151C2×C4×D53—42no—
152C2×C4×D53—42no—
153C2×C8×C53—21no—
154C2×C8×C53—21no—
155C2×C8×C53—21no—
156C2×C8×C53—21no—
157C2×C8×C53—21no—
158C2×C8×C53—21no—
159C4⋊C4×C53—21no—
160C4⋊C4×C53—21no—
161C4⋊C4×C53—21no—
162C4⋊C4×C53—21no—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—42no—
167C4⋊C4×C53C2×Q812yes—
168C42×C53—81no—
169C42×C53C2×D412yes—
170C42×C53#4112yes—
171#11719921—42no—
172#11719921—42no—
173#11719921—42no—
174#11719921—42no—
175#11719921—42no—
176C4⋊Dic53—21no—
177C4⋊Dic53—21no—
178C4⋊Dic53—22no—
179C4⋊Dic53—22no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53C2×D412yes—
183C4⋊Dic53C2×D412yes—
184#11719919—24no—
185#11719919—24no—
186#11719919—42no—
187#11719919—42no—
188#11719919—42no—
189#11719919—42no—
190#11719919—42no—
191#11719919—81no—
192C4×Dic53—21no—
193C4×Dic53—21no—
194C4×Dic53—21no—
195C4×Dic53—21no—
196C4×Dic53—42no—
197C4×Dic53—81no—
198C2×#88322—21no—
199C2×#88322—21no—
200C2×#88322—21no—
201C2×#88322—21no—
202C2×#88322C2×Q812yes—
203C2×#88322#4112yes—
204C23×C53C22×D411yescharacteristic
205C22×D53—22no—
206C22×D53—41no—
207C22×D53—41no—
208C22×D53—41no—
209C22×D53—41no—
210C22×D53—41no—
211C22×D53—41no—
212Q8×C53—84no—
213C2×C4×C53—21no—
214C2×C4×C53—21no—
215C2×C4×C53—21no—
216C2×C4×C53—21no—
217C2×C4×C53—22no—
218C2×C4×C53—41no—
219C2×C4×C53—41no—
220C2×C4×C53—41no—
221C2×C4×C53—41no—
222C2×C4×C53—41no—
223C2×C4×C53—41no—
224C2×C4×C53—41no—
225C2×C4×C53—41no—
226C2×C4×C53—41no—
227C2×C4×C53—41no—
228C2×C4×C53—41no—
229C2×C4×C53#12112yes—
230C2×C4×C53#12112yes—
231C2×C4×C53C22×D411yescharacteristic
232C2×C4×C53C22×Q811yescharacteristic
233C2×C4×C53C2×#4111yescharacteristic
234C2×C4×C53C2×#4111yescharacteristic
235C2×C4×C532−1+411yescharacteristic
236C2×C4×C532−1+411yescharacteristic
237C2×Dic53—21no—
238C2×Dic53—21no—
239C2×Dic53—21no—
240C2×Dic53—21no—
241C2×Dic53—22no—
242C2×Dic53—22no—
243C2×Dic53—21no—
244C2×Dic53—21no—
245C2×Dic53—21no—
246C2×Dic53—21no—
247C2×Dic53—21no—
248C2×Dic53—21no—
249C2×Dic53—41no—
250C2×Dic53—41no—
251C2×Dic53—41no—
252C2×Dic53—41no—
253C2×Dic53—41no—
254C2×Dic53—41no—
255C2×Dic53—41no—
256C2×Dic53AGL(1,ℤ/8)12yes—
257C2×Dic53AGL(1,ℤ/8)12yes—
258D212—22no—
259D212—22no—
260D212—41no—
261D212—41no—
262C4×D53—84no—
263C8×C53—41no—
264C8×C53—41no—
265#88322—22no—
266#88322—22no—
267#25465—531nomaximal, Sylow
268C22×C53C2×#12111yescharacteristic
269C22×C53#54711yescharacteristic
270C22×C53D4×Q811yescharacteristic
271C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
272C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
273C22×C53#57711yescharacteristic
274C22×C53#57811yescharacteristic
275C2×D53—42no—
276C2×D53—42no—
277C2×D53—42no—
278C2×D53—81no—
279C2×D53—81no—
280C2×D53—81no—
281C2×D53—81no—
282C4×C53—22no—
283C4×C53—22no—
284C4×C53—42no—
285C4×C53—81no—
286C4×C53—81no—
287C4×C53—81no—
288C4×C53#53311yescharacteristic
289C4×C53#54111yescharacteristic
290C4×C53#54811yescharacteristic
291C4×C53D4×Q811yescharacteristic
292Dic53—22no—
293Dic53—22no—
294Dic53—41no—
295Dic53—41no—
296Dic53—42no—
297Dic53—42no—
298Dic53—81no—
299C2×#501—531no—
300C2×#478—531no—
301#2542—531no—
302#2370—531no—
303#2084—531no—
304#2083—531no—
305#2067—531no—
306#2065—531no—
307#1987D5311yescharacteristic, complemented
308#1948—531no—
309#1947—531no—
310#1940—531no—
311#1926—531no—
312#1925—531no—
313#1762—531no—
314C2×C53#242911yescharacteristic
315C2×C53#303811yescharacteristic
316C2×C53#304611yescharacteristic
317C2×C53#334311yescharacteristic
318C2×C53#334511yescharacteristic
319C2×C53#338411yescharacteristic
320C2×C53#338611yescharacteristic
321D53—82no—
322C2×C4⋊Q8C2×D5311yescharacteristic
323C2×#125—531no—
324C2×C22⋊Q8—1061no—
325C2×C22⋊Q8—1061no—
326C2×C4×D4—531no—
327#501—1064no—
328#478—1064no—
329C2×#107—531no—
330C2×#107—531no—
331C2×#107—531no—
332C2×#106—531no—
333C2×#106—531no—
334C2×#106—531no—
335C2×C4⋊C8—531no—
336C2×C4⋊C8—531no—
337C2×C4⋊C8C2×D5311yescharacteristic
338C2×#102—531no—
339C2×#102—531no—
340C2×#102—531no—
341C2×#102—531no—
342C2×#97—531no—
343#399—1061no—
344#399—1061no—
345#390—531no—
346#390—531no—
347#389—531no—
348#389—531no—
349#389—1061no—
350#387—1061no—
351#387—1061no—
352#384—531no—
353#384—531no—
354C4×C4⋊C4—531no—
355C4×C4⋊C4—531no—
356C4×C4⋊C4—531no—
357C4×C4⋊C4—1061no—
358C4×C4⋊C4C2×D5311yescharacteristic
359#340—531no—
360#340—531no—
361#340C2×D5311yescharacteristic
362#340C2×D5311yescharacteristic
363#340C2×D5311yescharacteristic
364#340C2×D5311yescharacteristic
365#336—531no—
366#336—531no—
367C53#2546511yescharacteristic, complemented, Sylow
368C22×Q8—21no—
369C22×D4—531no—
370C23×C4—1061no—
371C22×C8—531no—
372C22×C8—531no—
373C22×C8C22×D5311yescharacteristic
374C22×C8C22×D5311yescharacteristic
375C4⋊Q8—22no—
376C4⋊Q8—22no—
377#125—534no—
378#125—1062no—
379C22⋊Q8—2124no—
380C22⋊Q8—2124no—
381C4×D4—1062no—
382C4×D4—1062no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—531no—
385C2×C4⋊C4—531no—
386C2×C4⋊C4—531no—
387C2×C4⋊C4—531no—
388C2×C4⋊C4—531no—
389C2×C4⋊C4—1061no—
390C2×C4⋊C4—1061no—
391C2×C4⋊C4—1061no—
392C2×C4⋊C4—1061no—
393C2×C4⋊C4—1061no—
394C2×C4⋊C4—1061no—
395C2×C4⋊C4—1061no—
396C2×C4⋊C4—1061no—
397C2×C4⋊C4C22×D5311yescharacteristic
398C2×C4⋊C4C22×D5311yescharacteristic
399C2×C4⋊C4C22×D5311yescharacteristic
400C2×C4⋊C4C22×D5311yescharacteristic
401C2×C22⋊C4—1061no—
402C2×C22⋊C4—1061no—
403C2×C22⋊C4—1061no—
404C2×C22⋊C4—1061no—
405C2×C22⋊C4—1061no—
406C2×C42—21no—
407C2×C42—531no—
408C2×C42—531no—
409C2×C42—1061no—
410C2×C42—1061no—
411C2×C42C22×D5311yescharacteristic
412#107—1062no—
413#107—1062no—
414#107—1062no—
415#106—1062no—
416#106—1062no—
417#106—1062no—
418C4⋊C8—22no—
419C4⋊C8—1062no—
420C4⋊C8—1062no—
421#102—534no—
422#102—1062no—
423#102—1062no—
424#102—1062no—
425#97—1062no—
426#95—21no—
427#95—531no—
428#95—531no—
429#95—1061no—
430#95—1061no—
431#95—1061no—
432#95—1061no—
433C24—1061no—
434C2×Q8—42no—
435C2×Q8—42no—
436C2×Q8—42no—
437C2×D4—532no—
438C2×D4—1061no—
439C2×D4—1061no—
440C2×D4—1061no—
441C2×D4—1061no—
442C2×D4—1061no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—531no—
448C22×C4—531no—
449C22×C4—531no—
450C22×C4—531no—
451C22×C4—1061no—
452C22×C4—1061no—
453C22×C4—1061no—
454C22×C4—2122no—
455C22×C4—2122no—
456C22×C4—2122no—
457C22×C4D4×D5311yescharacteristic
458C22×C4D4×D5311yescharacteristic
459C22×C4C23×D5311yescharacteristic
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—532no—
467C2×C8—532no—
468C2×C8—1061no—
469C2×C8—1061no—
470C2×C8—1061no—
471C2×C8—1061no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—42no—
480C4⋊C4—532no—
481C4⋊C4—532no—
482C4⋊C4—1062no—
483C4⋊C4—1062no—
484C4⋊C4—1064no—
485C4⋊C4—1064no—
486C4⋊C4—1061no—
487C4⋊C4—1061no—
488C4⋊C4—1061no—
489C4⋊C4—1061no—
490C4⋊C4—4241no—
491C4⋊C4—2122no—
492C4⋊C4—2122no—
493C4⋊C4—2122no—
494C4⋊C4—2122no—
495C4⋊C4—2122no—
496C4⋊C4Q8×D5312yes—
497C22⋊C4—2122no—
498C22⋊C4—2122no—
499C22⋊C4—2122no—
500C22⋊C4—2122no—
501C22⋊C4—2122no—
502C42—81no—
503C42—1061no—
504C42—1061no—
505C42—1061no—
506C42—1061no—
507C42—4241no—
508C42—2122no—
509C42#1171994512yes—
510C42D4×D5312yes—
511C23—1062no—
512C23—2121no—
513C23—2121no—
514C23—2121no—
515C23—2121no—
516C23—2121no—
517C23—2121no—
518C23C2×D4×D5311yescharacteristic
519Q8—84no—
520D4—1062no—
521D4—1062no—
522D4—2121no—
523D4—2121no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—532no—
541C2×C4—532no—
542C2×C4—1061no—
543C2×C4—1061no—
544C2×C4—1061no—
545C2×C4—1061no—
546C2×C4—1061no—
547C2×C4—1061no—
548C2×C4—1062no—
549C2×C4—1062no—
550C2×C4—1061no—
551C2×C4—1061no—
552C2×C4—1061no—
553C2×C4—1061no—
554C2×C4—4244no—
555C2×C4—2121no—
556C2×C4—2121no—
557C2×C4—2121no—
558C2×C4—2121no—
559C2×C4—2121no—
560C2×C4—2121no—
561C2×C4—2121no—
562C2×C4#1313319012yes—
563C2×C4#1313319012yes—
564C2×C4C2×D4×D5311yescharacteristic
565C2×C4C2×Q8×D5311yescharacteristic
566C2×C4#41×D5311yescharacteristic
567C2×C4C2×#1171994511yescharacteristic
568C2×C4#1313326411yescharacteristic
569C2×C4#1313326511yescharacteristic
570C8—41no—
571C8—41no—
572C8—1062no—
573C8—1062no—
574C22—4241no—
575C22—4241no—
576C22—4241no—
577C22—4241no—
578C22—2122no—
579C22—2122no—
580C22—2122no—
581C22C2×#1313319011yescharacteristic
582C22#1681127511yescharacteristic
583C22#1681128611yescharacteristic
584C22AGL(1,ℤ/8)×D5311yescharacteristic
585C22AGL(1,ℤ/8)×D5311yescharacteristic
586C22#1681142911yescharacteristic
587C22#1681143611yescharacteristic
588C4—22no—
589C4—22no—
590C4—42no—
591C4—81no—
592C4—81no—
593C4—81no—
594C4—1062no—
595C4—1062no—
596C4—4241no—
597C4—2122no—
598C4—2122no—
599C4—2121no—
600C4—2121no—
601C4#1681117711yescharacteristic
602C4#1681123111yescharacteristic
603C4#1681128111yescharacteristic
604C4#1681128811yescharacteristic
605C2—4242no—
606C2#2027831511yescharacteristic
607C2#2028419411yescharacteristic
608C2#2028427011yescharacteristic
609C2#2028715211yescharacteristic
610C2#2028719211yescharacteristic
611C2#2028760911yescharacteristic
612C2#2028763911yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1987C2
socleC23×C53C22×D4
prime radicalC22×D212C23
hypercenter#1987D53
hypocenterC53#25465
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1987

Complements

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

NH ≅ G/Ntype
1. C2×#16811029C2semidirect
4. #20277811C2semidirect
5. #20276195C2semidirect
6. #20276190C2semidirect
9. #20275477C2semidirect
10. #20275472C2semidirect
11. #20275436C2semidirect
15. C53×#1987C2semidirect
104. #13133137Q8semidirect
307. #1987D53semidirect
367. C53#25465semidirect

Maximal subgroups

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

subgroupindex
C2×#168110292
C2×#168109242
#202796692
#202778112
#202761952
#202761902
#202761042
#202760902
#202754772
#202754722
#202754362
#202753402
#202753332
#202745432
C53×#19872
#2546553