Finite Group Databasebeta

← #27275532

Subgroups — #27275532

Order 13568.

1620 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133191C211yesmaximal, characteristic
2#20274625C211yesmaximal, characteristic, complemented
3C2×#16810356C214yesmaximal, complemented
4C53×C22⋊#95C211yesmaximal, characteristic, complemented
5C23×D212C411yescharacteristic, complemented
6C23×C22⋊D53—21no—
7C2×#13133191—28no—
8C2×#13133191—21no—
9C2×#13133191—21no—
10C2×#13133191C412yescomplemented
11C2×#13133191C2212yes—
12C22×C4⋊Dic53C411yescharacteristic, complemented
13C22×#11719921—21no—
14C2×C22⋊#88322—21no—
15C2×C22⋊#88322—21no—
16C2×C22⋊#88322C2212yescomplemented
17#16810356—28no—
18C24×C4×C53C2211yescharacteristic
19C22×C22⋊C4×C53—21no—
20C2×#95×C53—21no—
21C2×#95×C53C2212yescomplemented
22C24×D53—21no—
23C22×D212—21no—
24C22×D212—24no—
25C22×D212—42no—
26C22×D212—42no—
27C22×D212C2×C412yes—
28C22×C22⋊D53—28no—
29C22×C22⋊D53—22no—
30C22×C22⋊D53—22no—
31C22×C22⋊D53—44no—
32C22×C22⋊D53—42no—
33C22×C22⋊D53—42no—
34C23×Dic53—21no—
35C22×#88322—22no—
36#13133191—28no—
37#13133191—28no—
38#13133191—24no—
39#13133191—24no—
40#13133191—24no—
41#13133191D418yes—
42C2×C4⋊Dic53—24no—
43C2×C4⋊Dic53—21no—
44C2×C4⋊Dic53C2×C412yes—
45C2×#11719921—28no—
46C2×#11719921—22no—
47C2×#11719921—22no—
48C22⋊#88322—24no—
49C22⋊#88322—42no—
50C22⋊#88322—42no—
51C25×C53D411yescharacteristic
52C23×C4×C53—24no—
53C23×C4×C53—22no—
54C23×C4×C53—42no—
55C23×C4×C53—42no—
56C23×C4×C53C2×C411yescharacteristic
57C23×C4×C53C2×C411yescharacteristic
58C23×C4×C53D411yescharacteristic
59C23×C4×C53D411yescharacteristic
60C23×C4×C53D411yescharacteristic
61C23×C4×C53C2311yescharacteristic
62C2×C22⋊C4×C53—24no—
63C2×C22⋊C4×C53—42no—
64C2×C22⋊C4×C53—42no—
65#95×C53—24no—
66#95×C53—42no—
67C24×C53—22no—
68C24×C53—22no—
69C24×C53—22no—
70C24×C53—22no—
71C24×C53—41no—
72C24×C53—41no—
73C24×C53C22⋊C411yescharacteristic
74C24×C53C22⋊C411yescharacteristic
75C24×C53D812yes—
76C24×C53SD1612yes—
77C24×C53C2×D411yescharacteristic
78C23×D53—24no—
79C23×D53—22no—
80C23×D53—42no—
81C23×D53—42no—
82C23×D53—42no—
83C23×D53—42no—
84C23×D53—42no—
85C23×D53—42no—
86C22×C4×C53—22no—
87C22×C4×C53—22no—
88C22×C4×C53—22no—
89C22×C4×C53—22no—
90C22×C4×C53—22no—
91C22×C4×C53—22no—
92C22×C4×C53—22no—
93C22×C4×C53—22no—
94C22×C4×C53—22no—
95C22×C4×C53—22no—
96C22×C4×C53—24no—
97C22×C4×C53—41no—
98C22×C4×C53—41no—
99C22×C4×C53—41no—
100C22×C4×C53—41no—
101C22×C4×C53—41no—
102C22×C4×C53—41no—
103C22×C4×C53—41no—
104C22×C4×C53—416no—
105C22×C4×C53—42no—
106C22×C4×C53—42no—
107C22×C4×C53—42no—
108C22×C4×C53—42no—
109C22×C4×C53—42no—
110C22×C4×C53—42no—
111C22×C4×C53C22⋊C411yescharacteristic
112C22×C4×C53C22⋊C411yescharacteristic
113C22×C4×C53C22×C411yescharacteristic
114C22×C4×C53C2×D411yescharacteristic
115C22×C4×C53C2×D414yes—
116C2×C22⋊D53—28no—
117C2×C22⋊D53—28no—
118C2×C22⋊D53—416no—
119C2×C22⋊D53—416no—
120C2×C22⋊D53—44no—
121C2×C22⋊D53—44no—
122C2×C22⋊D53—42no—
123C2×C22⋊D53—42no—
124C2×C22⋊D53—44no—
125C22×Dic53—24no—
126C22×Dic53—22no—
127C22×Dic53—42no—
128C22×Dic53—42no—
129C2×D212—24no—
130C2×D212—22no—
131C2×D212—416no—
132C2×D212—42no—
133C2×D212—42no—
134C2×D212—42no—
135C2×D212—42no—
136C2×D212C22⋊C414yescomplemented
137C22⋊C4×C53—48no—
138#11719921—28no—
139#11719921—28no—
140C4⋊Dic53—24no—
141C4⋊Dic53—22no—
142C4⋊Dic53C22⋊C414yescomplemented
143C2×#88322—42no—
144C2×#88322—42no—
145C2×#88322—42no—
146C23×C53—21no—
147C23×C53—21no—
148C23×C53—21no—
149C23×C53—21no—
150C23×C53—21no—
151C23×C53—21no—
152C23×C53—21no—
153C23×C53—21no—
154C23×C53—24no—
155C23×C53—24no—
156C23×C53—21no—
157C23×C53—21no—
158C23×C53—21no—
159C23×C53—21no—
160C23×C53—21no—
161C23×C53—21no—
162C23×C53—24no—
163C23×C53—24no—
164C23×C53—42no—
165C23×C53—42no—
166C23×C53—44no—
167C23×C53—44no—
168C23×C53—44no—
169C23×C53—44no—
170C23×C53#9912yes—
171C23×C53#10212yes—
172C23×C53#10212yes—
173C23×C53#10412yes—
174C23×C53C2×C22⋊C411yescharacteristic
175C23×C53#12012yes—
176C23×C53C2×D811yescharacteristic
177C23×C53C2×SD1611yescharacteristic
178C23×C53AGL(1,ℤ/8)12yes—
179C22×D53—24no—
180C22×D53—48no—
181C22×D53—48no—
182C22×D53—48no—
183C22×D53—42no—
184C22×D53—42no—
185C22×D53—42no—
186C22×D53—42no—
187C22×D53—42no—
188C22×D53—42no—
189C22×D53—84no—
190C22×D53—84no—
191C22×D53—84no—
192C22×D53—84no—
193C2×C4×C53—28no—
194C2×C4×C53—21no—
195C2×C4×C53—21no—
196C2×C4×C53—21no—
197C2×C4×C53—21no—
198C2×C4×C53—21no—
199C2×C4×C53—21no—
200C2×C4×C53—21no—
201C2×C4×C53—21no—
202C2×C4×C53—21no—
203C2×C4×C53—44no—
204C2×C4×C53—44no—
205C2×C4×C53—44no—
206C2×C4×C53—44no—
207C2×C4×C53—44no—
208C2×C4×C53—44no—
209C2×C4×C53—44no—
210C2×C4×C53—44no—
211C2×C4×C53—82no—
212C2×C4×C53—82no—
213C2×C4×C53—82no—
214C2×C4×C53—82no—
215C2×C4×C53#9912yes—
216C2×C4×C53C2×C22⋊C412yes—
217C2×C4×C53#12012yes—
218C22⋊D53—416no—
219C22⋊D53—48no—
220C22⋊D53—48no—
221C2×Dic53—24no—
222C2×Dic53—48no—
223C2×Dic53—42no—
224C2×Dic53—42no—
225D212—44no—
226D212—48no—
227D212—44no—
228#88322—82no—
229#8328—531nomaximal, Sylow
230C22×C53—21no—
231C22×C53—21no—
232C22×C53—22no—
233C22×C53—22no—
234C22×C53—22no—
235C22×C53—22no—
236C22×C53—21no—
237C22×C53—22no—
238C22×C53—21no—
239C22×C53—22no—
240C22×C53—22no—
241C22×C53—22no—
242C22×C53—22no—
243C22×C53—22no—
244C22×C53—22no—
245C22×C53—22no—
246C22×C53—22no—
247C22×C53—48no—
248C22×C53—48no—
249C22×C53—41no—
250C22×C53—41no—
251C22×C53—41no—
252C22×C53—41no—
253C22×C53#32714yescomplemented
254C22×C53#37911yescharacteristic
255C22×C53C2×#9911yescharacteristic
256C22×C53C2×#9911yescharacteristic
257C22×C53C2×#10211yescharacteristic
258C22×C53#41811yescharacteristic
259C22×C53C2×#10411yescharacteristic
260C22×C53#42111yescharacteristic
261C22×C53#44712yes—
262C22×C53#45012yes—
263C2×D53—44no—
264C2×D53—44no—
265C2×D53—44no—
266C2×D53—816no—
267C2×D53—82no—
268C2×D53—82no—
269C2×D53—82no—
270C2×D53—82no—
271C4×C53—22no—
272C4×C53—22no—
273C4×C53—22no—
274C4×C53—41no—
275C4×C53—84no—
276Dic53—44no—
277C22×#121—531no—
278#1779—531no—
279C2×#327—534no—
280C22⋊#95D5311yescharacteristic, complemented
281C2×C53—22no—
282C2×C53—24no—
283C2×C53—22no—
284C2×C53—42no—
285C2×C53C2×#32711yescharacteristic
286C2×C53#150711yescharacteristic
287C2×C53#151011yescharacteristic
288C2×C53#151911yescharacteristic
289C2×C53#178111yescharacteristic
290C2×C53#178611yescharacteristic
291C2×C53#178911yescharacteristic
292D53—84no—
293C23×D4—531no—
294C23×D4—1061no—
295C24×C4C2×D5311yescharacteristic
296C2×#121—532no—
297C2×#121—532no—
298C2×#121—1068no—
299C2×#121—1061no—
300C2×#121—1061no—
301C22×C4⋊C4—531no—
302C22×C22⋊C4—21no—
303C22×C22⋊C4—1061no—
304C2×C22⋊C8—532no—
305C2×C22⋊C8—1061no—
306C2×C22⋊C8—1061no—
307C2×#95—21no—
308C2×#95C2×D5312yescomplemented
309#327—1068no—
310C53#832811yescharacteristic, complemented, Sylow
311C25—1061no—
312C25C22⋊D5311yescharacteristic
313C22×D4—532no—
314C22×D4—1064no—
315C22×D4—1068no—
316C22×D4—1062no—
317C22×D4—1062no—
318C22×D4—1061no—
319C22×D4—2122no—
320C22×D4—2122no—
321C22×D4—2124no—
322C22×D4—2122no—
323C22×D4—2122no—
324C23×C4—24no—
325C23×C4—22no—
326C23×C4—42no—
327C23×C4—42no—
328C23×C4—1061no—
329C23×C4C4×D5311yescharacteristic
330C23×C4C4×D5311yescharacteristic
331C23×C4D21211yescharacteristic
332C23×C4D21211yescharacteristic
333C23×C4C22⋊D5311yescharacteristic
334C23×C4C22×D5311yescharacteristic
335C22×C8—1062no—
336#121—538no—
337#121—1064no—
338#121—1064no—
339#121—1068no—
340#121—1068no—
341#121—1064no—
342C2×C4⋊C4—532no—
343C2×C4⋊C4—1064no—
344C2×C4⋊C4—1061no—
345C2×C22⋊C4—24no—
346C2×C22⋊C4—42no—
347C2×C22⋊C4—42no—
348C2×C22⋊C4—1068no—
349C2×C22⋊C4—1062no—
350C2×C22⋊C4—1062no—
351C22⋊C8—1064no—
352C22⋊C8—2122no—
353C22⋊C8—2122no—
354#95—24no—
355#95—42no—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—22no—
360C24—41no—
361C24—41no—
362C24—1064no—
363C24—1062no—
364C24—2122no—
365C24—2122no—
366C24—2122no—
367C24—2122no—
368C24—2122no—
369C24—2122no—
370C24#1171992111yescharacteristic
371C24#1171992111yescharacteristic
372C24#1171992212yes—
373C24#1171992412yes—
374C24C2×C22⋊D5311yescharacteristic
375C2×D4—534no—
376C2×D4—1064no—
377C2×D4—1068no—
378C2×D4—1068no—
379C2×D4—1062no—
380C2×D4—2122no—
381C2×D4—2122no—
382C2×D4—21216no—
383C2×D4—2122no—
384C2×D4—21216no—
385C2×D4—21216no—
386C2×D4—2124no—
387C2×D4—2124no—
388C2×D4—2122no—
389C2×D4—2122no—
390C2×D4—2124no—
391C2×D4—2122no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—416no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—1064no—
418C22×C4—1062no—
419C22×C4—2122no—
420C22×C4—2122no—
421C22×C4#1171992111yescharacteristic
422C22×C4#1171992111yescharacteristic
423C22×C4C2×C4×D5311yescharacteristic
424C22×C4C2×D21211yescharacteristic
425C22×C4D4×D5314yes—
426C2×C8—2122no—
427C2×C8—2122no—
428C2×C8—2122no—
429C4⋊C4—534no—
430C4⋊C4—1064no—
431C4⋊C4—1062no—
432C22⋊C4—48no—
433C22⋊C4—1068no—
434C22⋊C4—1068no—
435C23—21no—
436C23—21no—
437C23—21no—
438C23—21no—
439C23—21no—
440C23—21no—
441C23—21no—
442C23—21no—
443C23—24no—
444C23—24no—
445C23—21no—
446C23—21no—
447C23—24no—
448C23—24no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23—21no—
453C23—42no—
454C23—42no—
455C23—44no—
456C23—44no—
457C23—44no—
458C23—44no—
459C23—1064no—
460C23—4244no—
461C23—4244no—
462C23—4244no—
463C23—4244no—
464C23—2128no—
465C23—2128no—
466C23—2128no—
467C23—2122no—
468C23—2122no—
469C23—2122no—
470C23—2122no—
471C23—2122no—
472C23—2122no—
473C23#1313313112yes—
474C23#1313313712yes—
475C23#1313313712yes—
476C23#1313314512yes—
477C23C2×#1171992111yescharacteristic
478C23#1313318512yes—
479C23C2×#1171992211yescharacteristic
480C23C2×#1171992411yescharacteristic
481C23#1313324212yes—
482D4—2124no—
483D4—2128no—
484D4—2124no—
485D4—21216no—
486D4—2128no—
487D4—2128no—
488C2×C4—28no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4—82no—
509C2×C4—82no—
510C2×C4—1064no—
511C2×C4—2128no—
512C2×C4—2122no—
513C2×C4—2122no—
514C2×C4#1313313112yes—
515C2×C4C22⋊C4×D5312yes—
516C2×C4#1313318712yes—
517C8—4242no—
518C22—21no—
519C22—21no—
520C22—22no—
521C22—22no—
522C22—22no—
523C22—22no—
524C22—21no—
525C22—22no—
526C22—21no—
527C22—22no—
528C22—22no—
529C22—22no—
530C22—22no—
531C22—22no—
532C22—22no—
533C22—22no—
534C22—22no—
535C22—48no—
536C22—48no—
537C22—41no—
538C22—41no—
539C22—41no—
540C22—41no—
541C22—42416no—
542C22—4242no—
543C22—4242no—
544C22—4242no—
545C22—4242no—
546C22—2124no—
547C22—2124no—
548C22—2124no—
549C22#1681035614yescomplemented
550C22#1681047811yescharacteristic
551C22C2×#1313313111yescharacteristic
552C22C2×#1313313111yescharacteristic
553C22C2×#1313313711yescharacteristic
554C22#1681062711yescharacteristic
555C22C2×#1313314511yescharacteristic
556C22#1681064111yescharacteristic
557C22#1681075612yes—
558C22#1681077312yes—
559C4—22no—
560C4—22no—
561C4—22no—
562C4—41no—
563C4—84no—
564C4—2124no—
565C2—22no—
566C2—24no—
567C2—22no—
568C2—42no—
569C2—4244no—
570C2C2×#1681035611yescharacteristic
571C2#2027323011yescharacteristic
572C2#2027324311yescharacteristic
573C2#2027328211yescharacteristic
574C2#2027462911yescharacteristic
575C2#2027464511yescharacteristic
576C2#2027466011yescharacteristic
subgroupG/H
centerC23C2×#11719921
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×C22⋊#95C2
socleC23×C53C2×C22⋊C4
prime radicalC22×#13133191C2
cosocleC23×C4×C53C23
hypercenterC22⋊#95D53
hypocenterC53#8328
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C24×C4 ◁ C22⋊#95

Complements

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

NH ≅ G/Ntype
2. #20274625C2semidirect
3. C2×#16810356C2semidirect
4. C53×C22⋊#95C2semidirect
5. C23×D212C4semidirect
10. C2×#13133191C4semidirect
12. C22×C4⋊Dic53C4semidirect
16. C2×C22⋊#88322C22semidirect
21. C2×#95×C53C22semidirect
136. C2×D212C22⋊C4semidirect
142. C4⋊Dic53C22⋊C4semidirect
253. C22×C53#327semidirect
280. C22⋊#95D53semidirect
308. C2×#95C2×D53semidirect
310. C53#8328semidirect
549. C22#16810356semidirect

Maximal subgroups

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

subgroupindex
C22×#131331912
#202746252
C2×#168103562
C53×C22⋊#952
#832853