Finite Group Databasebeta

← #27401840

Subgroups — #27401840

Order 13568.

810 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810663C211yesmaximal, characteristic
2#20277606C211yesmaximal, characteristic
3#20275450C211yesmaximal, characteristic, complemented
4#20275445C211yesmaximal, characteristic, complemented
5#20275307C211yesmaximal, characteristic, complemented
6#20275297C211yesmaximal, characteristic, complemented
7#20275290C211yesmaximal, characteristic, complemented
8#20275286C211yesmaximal, characteristic, complemented
9#20274807C211yesmaximal, characteristic
10#20274805C211yesmaximal, characteristic
11#20274746C211yesmaximal, characteristic
12#20274613C211yesmaximal, characteristic, complemented
13C4×#13133148C211yesmaximal, characteristic, complemented
14#20274499C211yesmaximal, characteristic
15C53×#2344C211yesmaximal, characteristic
16C22×#11719917C2211yescharacteristic
17C2×#13133176C2211yescharacteristic
18#16810663C418yescomplemented
19C2×#13133150C2211yescharacteristic
20C2×#13133150C2211yescharacteristic
21C2×#13133150C2211yescharacteristic
22C2×#13133148C2211yescharacteristic
23C2×#13133148C2211yescharacteristic
24C2×#13133148C2211yescharacteristic
25C2×C22⋊#88322C2211yescharacteristic
26C2×#13133128C2211yescharacteristic
27C2×C4×#88322C2211yescharacteristic
28#16810521C2211yescharacteristic
29#16810521C2211yescharacteristic
30#16810517—21no—
31#16810517—21no—
32#16810490—21no—
33#16810483C2211yescharacteristic
34C4×C4⋊Dic53C2211yescharacteristic
35C4×C4⋊Dic53C2211yescharacteristic
36#16810474—21no—
37#16810385C2211yescharacteristic
38#16810385C2211yescharacteristic
39#16810385C2211yescharacteristic
40#16810385C2211yescharacteristic
41#16810385C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810378C2211yescharacteristic
48#16810378C2211yescharacteristic
49C2×#117×C53—21no—
50C22×C4⋊C4×C53C2211yescharacteristic
51C53×#389C2211yescharacteristic
52C53×#389C2211yescharacteristic
53C53×#387—21no—
54C53×#387—21no—
55C53×#381—21no—
56C53×#380C2211yescharacteristic
57C4×C4⋊C4×C53C2211yescharacteristic
58C4×C4⋊C4×C53C2211yescharacteristic
59C4×C22⋊C4×C53C2211yescharacteristic
60C2×#11719917—22no—
61C2×#11719917—22no—
62C2×#11719917C2×C414yes—
63C22×#88322C2311yescharacteristic
64C22×#88322C2311yescharacteristic
65C22×#88322C2311yescharacteristic
66C22×#88322C2311yescharacteristic
67#13133176—22no—
68#13133176C2×C414yes—
69C2×C4⋊Dic53C2311yescharacteristic
70C2×C4⋊Dic53C2311yescharacteristic
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×#11719919—21no—
74C2×#11719919—21no—
75C2×C22⋊Dic53—21no—
76C2×C22⋊Dic53—21no—
77C2×C4×Dic53—21no—
78C2×C4×Dic53—21no—
79C2×C4×Dic53—21no—
80#13133150—22no—
81#13133150C2×C414yes—
82#13133150C2×C414yes—
83#13133148—22no—
84#13133148C2×C414yes—
85#13133148C2×C414yes—
86C22⋊#88322—22no—
87#13133128—22no—
88C4×#88322—22no—
89#13133124—21no—
90#13133124—21no—
91#13133124—21no—
92#13133124—21no—
93#13133124—21no—
94C23×C4×C53—21no—
95C23×C4×C53C2311yescharacteristic
96#117×C53—44no—
97C2×C4⋊C4×C53—21no—
98C2×C4⋊C4×C53—21no—
99C2×C4⋊C4×C53—21no—
100C2×C4⋊C4×C53—22no—
101C2×C4⋊C4×C53C2×C414yes—
102C2×C4⋊C4×C53C2311yescharacteristic
103C2×C4⋊C4×C53C2311yescharacteristic
104C2×C4⋊C4×C53C2311yescharacteristic
105C2×C4⋊C4×C53C2311yescharacteristic
106C2×C22⋊C4×C53—21no—
107C2×C22⋊C4×C53D411yescharacteristic
108C2×C22⋊C4×C53D411yescharacteristic
109C2×C42×C53—21no—
110C2×C42×C53—21no—
111C2×C42×C53C2311yescharacteristic
112C2×C42×C53C2311yescharacteristic
113#95×C53—21no—
114#95×C53—21no—
115#95×C53—21no—
116#95×C53—21no—
117#95×C53—21no—
118#95×C53D411yescharacteristic
119#95×C53D411yescharacteristic
120C24×C53C2×D411yescharacteristic
121C22×C4×C53—21no—
122C22×C4×C53—21no—
123C22×C4×C53—21no—
124C22×C4×C53—21no—
125C22×C4×C53—24no—
126C22×C4×C53—21no—
127C22×C4×C53—21no—
128C22×C4×C53—21no—
129C22×C4×C53—21no—
130C22×C4×C53—42no—
131C22×C4×C53—42no—
132C22×C4×C53C22×C412yes—
133C22×C4×C53C2×D411yescharacteristic
134C22×C4×C53C2×D411yescharacteristic
135C22×C4×C53C2×D411yescharacteristic
136C22×C4×C53C2×D411yescharacteristic
137C22×C4×C53C2×D411yescharacteristic
138C22×C4×C53#4112yes—
139C22×C4×C53C2411yescharacteristic
140C22×Dic53—21no—
141C22×Dic53—21no—
142C22×Dic53—21no—
143C22×Dic53—21no—
144C4⋊C4×C53—24no—
145C4⋊C4×C53—21no—
146C4⋊C4×C53—21no—
147C4⋊C4×C53—21no—
148C4⋊C4×C53—21no—
149C4⋊C4×C53—21no—
150C4⋊C4×C53—21no—
151C4⋊C4×C53—42no—
152C4⋊C4×C53—42no—
153C4⋊C4×C53C22×C412yes—
154C4⋊C4×C53C22×C412yes—
155C22⋊C4×C53—22no—
156C22⋊C4×C53—41no—
157C22⋊C4×C53—42no—
158C42×C53—21no—
159C42×C53—21no—
160C42×C53—21no—
161C42×C53—21no—
162C42×C53—42no—
163C42×C53—42no—
164C22⋊Dic53—24no—
165C22⋊Dic53—81no—
166C4⋊Dic53—21no—
167C4⋊Dic53—21no—
168C4⋊Dic53—21no—
169C4⋊Dic53—21no—
170C4⋊Dic53—21no—
171C4⋊Dic53—21no—
172C4⋊Dic53C22×C412yes—
173C4⋊Dic53C22×C412yes—
174#11719919—42no—
175#11719919—42no—
176C4×Dic53—24no—
177C4×Dic53—81no—
178C4×Dic53—81no—
179#11719917—28no—
180C2×#88322—21no—
181C2×#88322—21no—
182C2×#88322—21no—
183C2×#88322—21no—
184C2×#88322—21no—
185C2×#88322—21no—
186C2×#88322—21no—
187C2×#88322—21no—
188C2×#88322C22×C412yes—
189C2×#88322C22×C412yes—
190C2×#88322#4112yes—
191C2×#88322#4112yes—
192C23×C53—21no—
193C23×C53—21no—
194C23×C53—21no—
195C23×C53—21no—
196C23×C53—21no—
197C23×C53—21no—
198C23×C53C4×D412yes—
199C23×C53C22×D411yescharacteristic
200C2×C4×C53—21no—
201C2×C4×C53—21no—
202C2×C4×C53—21no—
203C2×C4×C53—21no—
204C2×C4×C53—22no—
205C2×C4×C53—22no—
206C2×C4×C53—22no—
207C2×C4×C53—22no—
208C2×C4×C53—22no—
209C2×C4×C53—22no—
210C2×C4×C53—21no—
211C2×C4×C53—21no—
212C2×C4×C53—21no—
213C2×C4×C53—21no—
214C2×C4×C53—21no—
215C2×C4×C53—21no—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—48no—
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—41no—
230C2×C4×C53C4×D412yes—
231C2×C4×C53AGL(1,ℤ/8)12yes—
232C2×C4×C53C23×C411yescharacteristic
233C2×C4×C53C2×#4111yescharacteristic
234C2×C4×C53C2×#4111yescharacteristic
235C2×C4×C53C2×#4111yescharacteristic
236C2×C4×C532−1+411yescharacteristic
237C2×C4×C532−1+411yescharacteristic
238C2×Dic53—22no—
239C2×Dic53—22no—
240C2×Dic53—41no—
241C2×Dic53—41no—
242C2×Dic53—41no—
243C2×Dic53—41no—
244C2×Dic53—41no—
245C2×Dic53—41no—
246C2×Dic53—41no—
247C2×Dic53—41no—
248C2×Dic53—41no—
249C2×Dic53—41no—
250#88322—22no—
251#88322—41no—
252#88322—41no—
253#88322#11714yescomplemented
254#22622—531nomaximal, Sylow
255C22×C53—22no—
256C22×C53—22no—
257C22×C53—22no—
258C22×C53—41no—
259C22×C53—41no—
260C22×C53—41no—
261C22×C53—41no—
262C22×C53C2×C4×D411yescharacteristic
263C22×C53#54711yescharacteristic
264C22×C53#54711yescharacteristic
265C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
266C22×C53#57511yescharacteristic
267C22×C53#57711yescharacteristic
268C22×C53#57811yescharacteristic
269C4×C53—22no—
270C4×C53—22no—
271C4×C53—41no—
272C4×C53—41no—
273C4×C53—41no—
274C4×C53—42no—
275C4×C53—42no—
276C4×C53—81no—
277C4×C53—81no—
278C4×C53C2×#11711yescharacteristic
279C4×C53#51911yescharacteristic
280C4×C53#55211yescharacteristic
281C4×C53#55211yescharacteristic
282Dic53—42no—
283Dic53—42no—
284Dic53—81no—
285Dic53—81no—
286C2×#428—531no—
287#2347—531no—
288#2344D5311yescharacteristic
289#1943—531no—
290#1942—531no—
291#1921—531no—
292#1920—531no—
293#1919—531no—
294#1918—531no—
295#1825—531no—
296#1824—531no—
297#1809—531no—
298#1776—531no—
299C4×#106—531no—
300#1753—531no—
301C2×C53—42no—
302C2×C53#234411yescharacteristic
303C2×C53C4×AGL(1,ℤ/8)11yescharacteristic
304C2×C53#294711yescharacteristic
305C2×C53#334511yescharacteristic
306C2×C53#334911yescharacteristic
307C2×C53#335111yescharacteristic
308C2×C53#335311yescharacteristic
309C22×M16—531no—
310C2×#117—21no—
311C2×#117—531no—
312C22×C4⋊C4C2×D5311yescharacteristic
313#428—538no—
314C2×#107—531no—
315C2×#107—531no—
316C2×#107—531no—
317C2×#106—531no—
318C2×#106—531no—
319C2×#106—531no—
320C2×C22⋊C8—531no—
321C2×#97—531no—
322C2×C4×C8—531no—
323#389—531no—
324#389—531no—
325#389C2×D5311yescharacteristic
326#389C2×D5311yescharacteristic
327#388—1061no—
328#388—1061no—
329#387—21no—
330#387—21no—
331#383—1061no—
332#381—21no—
333#380—531no—
334#380C2×D5311yescharacteristic
335C4×C4⋊C4—531no—
336C4×C4⋊C4—531no—
337C4×C4⋊C4—1061no—
338C4×C4⋊C4C2×D5311yescharacteristic
339C4×C4⋊C4C2×D5311yescharacteristic
340C4×C22⋊C4C2×D5311yescharacteristic
341#340—531no—
342#340—531no—
343#340—531no—
344#340—531no—
345#340—531no—
346#340—531no—
347#340—531no—
348#340—531no—
349#340—531no—
350#340—531no—
351#336—531no—
352#336—531no—
353C53#2262211yescharacteristic, complemented, Sylow
354C23×C4—21no—
355C23×C4C22×D5311yescharacteristic
356C2×M16—534no—
357C2×M16—1062no—
358C2×M16—1062no—
359C22×C8—531no—
360C22×C8—531no—
361C22×C8—531no—
362C22×C8—531no—
363#117—44no—
364#117—534no—
365#117—1062no—
366C2×C4⋊C4—22no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—531no—
371C2×C4⋊C4—531no—
372C2×C4⋊C4—531no—
373C2×C4⋊C4—531no—
374C2×C4⋊C4—1061no—
375C2×C4⋊C4—1061no—
376C2×C4⋊C4C4×D5314yes—
377C2×C4⋊C4C22×D5311yescharacteristic
378C2×C4⋊C4C22×D5311yescharacteristic
379C2×C4⋊C4C22×D5311yescharacteristic
380C2×C4⋊C4C22×D5311yescharacteristic
381C2×C22⋊C4—21no—
382C2×C22⋊C4—1061no—
383C2×C22⋊C4—1061no—
384C2×C22⋊C4C22⋊D5311yescharacteristic
385C2×C22⋊C4C22⋊D5311yescharacteristic
386C2×C42—21no—
387C2×C42—21no—
388C2×C42—1061no—
389C2×C42—1061no—
390C2×C42—1061no—
391C2×C42C22×D5311yescharacteristic
392C2×C42C22×D5311yescharacteristic
393#107—534no—
394#107—534no—
395#107—1062no—
396#106—534no—
397#106—534no—
398#106—1062no—
399C22⋊C8—1062no—
400#97—1062no—
401C4×C8—1062no—
402#95—21no—
403#95—21no—
404#95—21no—
405#95—21no—
406#95—21no—
407#95—1061no—
408#95—1061no—
409#95—1061no—
410#95—1061no—
411#95—1061no—
412#95C22⋊D5311yescharacteristic
413#95C22⋊D5311yescharacteristic
414C24C2×C22⋊D5311yescharacteristic
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—24no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—1061no—
427C22×C4—1061no—
428C22×C4—1061no—
429C22×C4—1061no—
430C22×C4C2×C4×D5312yes—
431C22×C4#1171994512yes—
432C22×C4C2×C22⋊D5311yescharacteristic
433C22×C4C2×C22⋊D5311yescharacteristic
434C22×C4C2×C22⋊D5311yescharacteristic
435C22×C4C2×C22⋊D5311yescharacteristic
436C22×C4C2×C22⋊D5311yescharacteristic
437C22×C4C23×D5311yescharacteristic
438M16—1068no—
439C2×C8—532no—
440C2×C8—532no—
441C2×C8—532no—
442C2×C8—532no—
443C2×C8—1061no—
444C2×C8—1061no—
445C2×C8—1061no—
446C2×C8—1061no—
447C2×C8—1061no—
448C2×C8—1061no—
449C2×C8—1061no—
450C2×C8—1061no—
451C4⋊C4—24no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—532no—
461C4⋊C4—532no—
462C4⋊C4—1061no—
463C4⋊C4—1061no—
464C4⋊C4—1061no—
465C4⋊C4—1061no—
466C4⋊C4—1061no—
467C4⋊C4—1061no—
468C4⋊C4—2122no—
469C4⋊C4—2122no—
470C4⋊C4C2×C4×D5312yes—
471C4⋊C4C2×C4×D5312yes—
472C22⋊C4—22no—
473C22⋊C4—41no—
474C22⋊C4—42no—
475C22⋊C4—1064no—
476C22⋊C4—4241no—
477C42—21no—
478C42—21no—
479C42—21no—
480C42—21no—
481C42—42no—
482C42—42no—
483C42—1064no—
484C42—4241no—
485C42—4241no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23C4×C22⋊D5312yes—
493C23C22×C22⋊D5311yescharacteristic
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—48no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—1062no—
525C2×C4—1062no—
526C2×C4—2121no—
527C2×C4—2121no—
528C2×C4—2121no—
529C2×C4—2121no—
530C2×C4—2121no—
531C2×C4—2121no—
532C2×C4—2121no—
533C2×C4—2121no—
534C2×C4—2121no—
535C2×C4—2121no—
536C2×C4C4×C22⋊D5312yes—
537C2×C4#1313324312yes—
538C2×C4C22×C4×D5311yescharacteristic
539C2×C4#41×D5311yescharacteristic
540C2×C4#41×D5311yescharacteristic
541C2×C4C2×#1171994511yescharacteristic
542C2×C4#1313326411yescharacteristic
543C2×C4#1313326411yescharacteristic
544C8—534no—
545C8—1062no—
546C8—2121no—
547C8—2121no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—41no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22C2×C4×C22⋊D5311yescharacteristic
556C22#1681127211yescharacteristic
557C22#1681127211yescharacteristic
558C22C2×#1313324311yescharacteristic
559C22#1681142111yescharacteristic
560C22#1681143111yescharacteristic
561C22#1681143811yescharacteristic
562C4—22no—
563C4—22no—
564C4—41no—
565C4—41no—
566C4—41no—
567C4—42no—
568C4—42no—
569C4—81no—
570C4—81no—
571C4—4241no—
572C4—4241no—
573C4—2122no—
574C4—2122no—
575C4#117×D5311yescharacteristic
576C4#1681110011yescharacteristic
577C4#1681130611yescharacteristic
578C4#1681130611yescharacteristic
579C2—42no—
580C2#2027758511yescharacteristic
581C2C4×#1313324311yescharacteristic
582C2#2028355811yescharacteristic
583C2#2028718511yescharacteristic
584C2#2028724511yescharacteristic
585C2#2028727511yescharacteristic
586C2#2028729911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#2344C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
cosocleC22×C4×C53C24
hypercenter#2344D53
hypocenterC53#22622
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2344

Complements

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

NH ≅ G/Ntype
3. #20275450C2semidirect
4. #20275445C2semidirect
5. #20275307C2semidirect
6. #20275297C2semidirect
7. #20275290C2semidirect
8. #20275286C2semidirect
12. #20274613C2semidirect
13. C4×#13133148C2semidirect
18. #16810663C4semidirect
253. #88322#117semidirect
353. C53#22622semidirect

Maximal subgroups

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

subgroupindex
C2×#168106632
#202776062
#202754502
#202754452
#202753072
#202752972
#202752902
#202752862
#202748072
#202748052
#202747462
#202746132
C4×#131331482
#202744992
C53×#23442
#2262253