Finite Group Databasebeta

← #28319265

Subgroups — #28319265

Order 13568.

1318 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20291192C212yesmaximal, complemented
2#20290613C212yesmaximal, complemented
3#20290577C212yesmaximal, complemented
4#20290576C212yesmaximal, complemented
5#20290544C212yesmaximal
6#20290462C212yesmaximal, complemented
7C4×#13133276C212yesmaximal, complemented
8C2×#16811103C211yesmaximal, characteristic
9C2×#13133276C2214yes—
10C2×#13133275C2214yes—
11#16811572—28no—
12#16811566C2214yes—
13#16811565C2214yes—
14C4×C22⋊#6157C2214yes—
15#16811501C2214yes—
16#16811501C2214yes—
17C22×#11719945C2211yescharacteristic
18#16811103—28no—
19C2×C4×Dic106C411yescharacteristic, complemented
20C2×C4×Dic106C411yescharacteristic, complemented
21C2×C4×D212C2211yescharacteristic
22C2×C4×D212C2211yescharacteristic
23C2×#13133177C411yescharacteristic, complemented
24C2×#13133177C411yescharacteristic, complemented
25C2×#13133177C2211yescharacteristic
26C2×#13133177C2211yescharacteristic
27C2×#13133175C411yescharacteristic, complemented
28C2×#13133175C2211yescharacteristic
29C22×C4⋊Dic53C411yescharacteristic, complemented
30C2×C4⋊C4×D53C411yescharacteristic, complemented
31C2×C4⋊C4×D53C2211yescharacteristic
32C2×#117×C53C411yescharacteristic, complemented
33C2×C4⋊#6157—24no—
34C2×C22⋊#6157C2318yes—
35C2×C4×#6157—24no—
36#13133276—28no—
37#13133275—28no—
38(C2×#6157)⋊C4—24no—
39(C2×#6157)⋊C4—24no—
40(C2×#6157)⋊C4—28no—
41C2×#11719945—24no—
42C2×#11719945—24no—
43C2×#11719945—24no—
44C22×Dic106C2×C411yescharacteristic
45C22×D212C2311yescharacteristic
46C22×C22⋊D53C2×C411yescharacteristic
47C22×C22⋊D53C2311yescharacteristic
48C23×Dic53C2×C411yescharacteristic
49C23×Dic53C2×C411yescharacteristic
50C22×C4×D53C2×C411yescharacteristic
51C22×C4×D53C2×C411yescharacteristic
52C22×C4×D53C2×C411yescharacteristic
53C22×C4×D53C2311yescharacteristic
54C22×C4×D53C2311yescharacteristic
55C22×C4×D53C2311yescharacteristic
56C4×Dic106—24no—
57C4×Dic106—24no—
58C4×D212—24no—
59C4×D212—24no—
60#13133177—24no—
61#13133177—24no—
62#13133177—24no—
63#13133177—24no—
64#13133175—24no—
65#13133175—24no—
66C2×C4⋊Dic53—24no—
67C2×C4⋊Dic53C2×C411yescharacteristic
68C2×C4⋊Dic53C2×C411yescharacteristic
69C2×C4⋊Dic53C2×C411yescharacteristic
70C2×C4⋊Dic53C2×C411yescharacteristic
71C2×#11719919C2×C411yescharacteristic
72C2×#11719919C2×C411yescharacteristic
73C2×#11719919C2×C411yescharacteristic
74C2×#11719919C2×C411yescharacteristic
75C4⋊C4×D53—24no—
76C4⋊C4×D53D418yes—
77C2×#11719921C2×C411yescharacteristic
78C2×#11719921C2×C411yescharacteristic
79C2×#11719921C2311yescharacteristic
80C2×#11719921C2311yescharacteristic
81C2×C4×Dic53C2×C411yescharacteristic
82C2×C4×Dic53C2×C411yescharacteristic
83C2×C4×Dic53C2×C411yescharacteristic
84C2×C4×Dic53C2×C411yescharacteristic
85C23×C4×C53C2×C411yescharacteristic
86#117×C53—24no—
87C2×C4⋊C4×C53C2×C411yescharacteristic
88C2×C4⋊C4×C53C2×C411yescharacteristic
89C2×C22⋊C4×C53C2×C411yescharacteristic
90C2×C22⋊C4×C53C2×C411yescharacteristic
91C2×C42×C53C2×C411yescharacteristic
92C2×C42×C53C2×C411yescharacteristic
93C24×C53C22×C411yescharacteristic
94C23×D53C22×C411yescharacteristic
95C23×D53C2411yescharacteristic
96C22×#6157—28no—
97C22×C4×C53—22no—
98C22×C4×C53—22no—
99C22×C4×C53—22no—
100C22×C4×C53C22×C411yescharacteristic
101C22×C4×C53C22×C411yescharacteristic
102C22×C4×C53C22×C411yescharacteristic
103C22×C4×C53C22×C411yescharacteristic
104C22×C4×C53C22×C411yescharacteristic
105C22×C4×C53C22×C411yescharacteristic
106C2×C22⋊D53—22no—
107C2×C22⋊D53—22no—
108C2×C22⋊D53—22no—
109C2×C22⋊D53—22no—
110C2×C22⋊D53—22no—
111C2×C22⋊D53#4114yes—
112C22×Dic53—22no—
113C22×Dic53—22no—
114C22×Dic53—81no—
115C22×Dic53—81no—
116C22×Dic53C22×C411yescharacteristic
117C22×Dic53C22×C411yescharacteristic
118C22×Dic53C22×C411yescharacteristic
119C22×Dic53C22×C411yescharacteristic
120C22×Dic53C22×C411yescharacteristic
121C22×Dic53C22×C411yescharacteristic
122#11719945—232no—
123C2×D212—22no—
124C2×D212—22no—
125C2×D212#4114yes—
126C2×C4×D53—22no—
127C2×C4×D53—22no—
128C2×C4×D53—22no—
129C2×C4×D53—22no—
130C2×C4×D53—22no—
131C2×C4×D53—22no—
132C2×C4×D53—22no—
133C2×C4×D53—81no—
134C2×C4×D53—81no—
135C2×C4×D53—81no—
136C2×C4×D53—81no—
137C2×C4×D53C2×D414yes—
138C2×C4×D53C2×D414yes—
139C2×C4×D53C2×D414yes—
140C2×Dic106—22no—
141C2×Dic106—22no—
142C2×Dic106—22no—
143C22⋊#6157—28no—
144C22⋊#6157—28no—
145C4⋊#6157—84no—
146C4×#6157—84no—
147C4⋊C4×C53—22no—
148C4⋊C4×C53C22⋊C414yes—
149C22⋊C4×C53—22no—
150C22⋊C4×C53—22no—
151C42×C53—22no—
152C42×C53—22no—
153#11719921—22no—
154#11719921—22no—
155#11719921—22no—
156#11719921—22no—
157C4⋊Dic53—22no—
158C4⋊Dic53—22no—
159C4⋊Dic53—22no—
160C4⋊Dic53C22⋊C414yes—
161#11719919—22no—
162#11719919—22no—
163#11719919C22⋊C414yes—
164#11719919C22⋊C414yes—
165C4×Dic53—22no—
166C4×Dic53—22no—
167C4×Dic53—22no—
168C4×Dic53—22no—
169C23×C53—21no—
170C23×C53—21no—
171C23×C53—41no—
172C23×C53—41no—
173C23×C53#11712yes—
174C23×C53C23×C411yescharacteristic
175C22×D53—21no—
176C22×D53—21no—
177C22×D53—41no—
178C22×D53—41no—
179C22×D53—41no—
180C22×D53—41no—
181C22×D53#11712yes—
182C22×D53C22×D412yes—
183C22×D53C2×#4112yes—
184C22×D53C2×#4112yes—
185C2×#6157—48no—
186C2×#6157—48no—
187C2×#6157—48no—
188C2×C4×C53—28no—
189C2×C4×C53—21no—
190C2×C4×C53—21no—
191C2×C4×C53—21no—
192C2×C4×C53—21no—
193C2×C4×C53—21no—
194C2×C4×C53—21no—
195C2×C4×C53—41no—
196C2×C4×C53—41no—
197C2×C4×C53—41no—
198C2×C4×C53—41no—
199C2×C4×C53C2×C22⋊C412yes—
200C2×C4×C53C2×C22⋊C412yes—
201C2×C4×C53C2×C22⋊C412yes—
202C2×C4×C53#11712yes—
203C22⋊D53—28no—
204C22⋊D53—28no—
205C2×Dic53—21no—
206C2×Dic53—21no—
207C2×Dic53—21no—
208C2×Dic53—21no—
209C2×Dic53—21no—
210C2×Dic53—21no—
211C2×Dic53—41no—
212C2×Dic53—41no—
213C2×Dic53—41no—
214C2×Dic53—41no—
215C2×Dic53—82no—
216C2×Dic53—82no—
217C2×Dic53C2×C22⋊C412yes—
218C2×Dic53C2×C22⋊C412yes—
219C2×Dic53C2×C22⋊C412yes—
220C2×Dic53#11712yes—
221D212—28no—
222C4×D53—28no—
223C4×D53—28no—
224C4×D53—44no—
225C4×D53—44no—
226C4×D53—82no—
227C4×D53—82no—
228Dic106—28no—
229#20725—531nomaximal, Sylow
230C22×C53—21no—
231C22×C53—21no—
232C22×C53—22no—
233C22×C53—42no—
234C22×C53—42no—
235C22×C53—41no—
236C22×C53—41no—
237C22×C53C22×C22⋊C411yescharacteristic
238C22×C53C2×#11711yescharacteristic
239C22×C53C2×#11711yescharacteristic
240C2×D53—22no—
241C2×D53—22no—
242C2×D53—22no—
243C2×D53—42no—
244C2×D53—42no—
245C2×D53—42no—
246C2×D53—41no—
247C2×D53—41no—
248C2×D53—42no—
249C2×D53#52514yes—
250#6157—88no—
251C4×C53—22no—
252C4×C53—22no—
253C4×C53—22no—
254C4×C53—22no—
255C4×C53—41no—
256C4×C53—41no—
257Dic53—22no—
258Dic53—22no—
259Dic53—22no—
260Dic53—22no—
261Dic53—41no—
262Dic53—41no—
263C2×#520—531no—
264#2313—532no—
265#1820—532no—
266#1799—532no—
267#1799—532no—
268#1782—532no—
269C4×#99—532no—
270C4×#99—532no—
271C2×C53—21no—
272C2×C53—21no—
273C2×C53—42no—
274C2×C53#228011yescharacteristic
275C2×C53#288311yescharacteristic
276C2×C53#288411yescharacteristic
277D53—24no—
278D53—42no—
279C22×#41—531no—
280#520—1068no—
281C2×C4×Q8—531no—
282C2×C4×Q8—531no—
283C2×C4×D4—531no—
284C2×C4×D4—531no—
285C2×C4×D4—531no—
286C2×C4×D4—531no—
287C2×C4×D4—531no—
288C2×C4×D4—531no—
289C2×#117—531no—
290C2×#117—531no—
291C2×#117#615711yescharacteristic, complemented
292C22×C4⋊C4—531no—
293C22×C4⋊C4—531no—
294C22×C4⋊C4—531no—
295C2×#99—534no—
296C2×#99—534no—
297#387—1068no—
298#381—534no—
299#380—534no—
300C4×C22⋊C4—534no—
301#342—534no—
302#342—534no—
303C53#2072511yescharacteristic, complemented, Sylow
304C2×#41—1064no—
305C2×#41—1064no—
306C2×#41—1064no—
307C22×Q8—531no—
308C22×D4—531no—
309C22×D4—531no—
310C22×D4—531no—
311C23×C4—531no—
312C23×C4—531no—
313C23×C4—531no—
314C23×C4—531no—
315C23×C4—531no—
316C23×C4—531no—
317C23×C4—531no—
318C23×C4—531no—
319C23×C4C2×#615711yescharacteristic
320C4×Q8—1064no—
321C4×Q8—1064no—
322C4×D4—1064no—
323C4×D4—1064no—
324C4×D4—1064no—
325C4×D4—1064no—
326C4×D4—1064no—
327C4×D4—1064no—
328#117—24no—
329#117—1064no—
330#117—1064no—
331C2×C4⋊C4—531no—
332C2×C4⋊C4—531no—
333C2×C4⋊C4—531no—
334C2×C4⋊C4—538no—
335C2×C4⋊C4—531no—
336C2×C4⋊C4—531no—
337C2×C4⋊C4—531no—
338C2×C4⋊C4—531no—
339C2×C4⋊C4—531no—
340C2×C4⋊C4—1064no—
341C2×C4⋊C4—1064no—
342C2×C4⋊C4—1064no—
343C2×C4⋊C4C2×#615711yescharacteristic
344C2×C4⋊C4C2×#615711yescharacteristic
345C2×C22⋊C4—531no—
346C2×C22⋊C4—531no—
347C2×C22⋊C4—531no—
348C2×C22⋊C4—531no—
349C2×C22⋊C4—538no—
350C2×C22⋊C4C2×#615711yescharacteristic
351C2×C22⋊C4C2×#615711yescharacteristic
352C2×C42—531no—
353C2×C42—531no—
354C2×C42—531no—
355C2×C42—531no—
356C2×C42—1064no—
357C2×C42C2×#615711yescharacteristic
358C2×C42C2×#615711yescharacteristic
359#99—1068no—
360#99—1068no—
361#95—1064no—
362#95—1064no—
363#95—1068no—
364C24—531no—
365C24—531no—
366C24C22×#615711yescharacteristic
367#41—10632no—
368C2×Q8—1062no—
369C2×Q8—1062no—
370C2×Q8—1062no—
371C2×D4—534no—
372C2×D4—534no—
373C2×D4—1062no—
374C2×D4—1062no—
375C2×D4—1062no—
376C2×D4—1062no—
377C2×D4—1062no—
378C2×D4—1062no—
379C2×D4—1062no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—531no—
384C22×C4—531no—
385C22×C4—531no—
386C22×C4—531no—
387C22×C4—534no—
388C22×C4—534no—
389C22×C4—534no—
390C22×C4—531no—
391C22×C4—531no—
392C22×C4—1062no—
393C22×C4—1062no—
394C22×C4—1062no—
395C22×C4—1062no—
396C22×C4—1062no—
397C22×C4—1062no—
398C22×C4—1062no—
399C22×C4—1062no—
400C22×C4—1062no—
401C22×C4—1068no—
402C22×C4—4241no—
403C22×C4—4241no—
404C22×C4—4241no—
405C22×C4—4241no—
406C22×C4—4241no—
407C22×C4—4241no—
408C22×C4C22×#615711yescharacteristic
409C22×C4C22×#615711yescharacteristic
410C22×C4C22×#615711yescharacteristic
411C22×C4C22×#615711yescharacteristic
412C22×C4C22×#615711yescharacteristic
413C22×C4C22×#615711yescharacteristic
414C4⋊C4—22no—
415C4⋊C4—534no—
416C4⋊C4—534no—
417C4⋊C4—534no—
418C4⋊C4—1062no—
419C4⋊C4—1062no—
420C4⋊C4—1062no—
421C4⋊C4—1062no—
422C4⋊C4—1062no—
423C4⋊C4—4244no—
424C4⋊C4C22⋊#615714yes—
425C22⋊C4—22no—
426C22⋊C4—22no—
427C22⋊C4—1062no—
428C22⋊C4—1062no—
429C22⋊C4—1062no—
430C22⋊C4—1062no—
431C22⋊C4—1068no—
432C22⋊C4—1068no—
433C42—22no—
434C42—22no—
435C42—1062no—
436C42—1062no—
437C42—1062no—
438C42—1062no—
439C42—4244no—
440C23—21no—
441C23—21no—
442C23—41no—
443C23—41no—
444C23—532no—
445C23—532no—
446C23—532no—
447C23—532no—
448C23—1061no—
449C23—1061no—
450C23—2121no—
451C23—2121no—
452C23—2121no—
453C23—2121no—
454C23#1313329612yes—
455C23C23×#615711yescharacteristic
456Q8—1068no—
457D4—1068no—
458D4—1068no—
459D4—1068no—
460C2×C4—28no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—532no—
472C2×C4—532no—
473C2×C4—532no—
474C2×C4—532no—
475C2×C4—1068no—
476C2×C4—1068no—
477C2×C4—1061no—
478C2×C4—1061no—
479C2×C4—1061no—
480C2×C4—1061no—
481C2×C4—1061no—
482C2×C4—1061no—
483C2×C4—4242no—
484C2×C4—4242no—
485C2×C4—4242no—
486C2×C4—4242no—
487C2×C4—2124no—
488C2×C4—2124no—
489C2×C4—2121no—
490C2×C4—2121no—
491C2×C4—2128no—
492C2×C4—2128no—
493C2×C4—2128no—
494C2×C4—2121no—
495C2×C4—2121no—
496C2×C4C2×C22⋊#615712yes—
497C2×C4C2×C22⋊#615712yes—
498C2×C4C2×C22⋊#615712yes—
499C2×C4#1313329612yes—
500C22—21no—
501C22—21no—
502C22—22no—
503C22—42no—
504C22—42no—
505C22—41no—
506C22—41no—
507C22—534no—
508C22—1062no—
509C22—1062no—
510C22—1062no—
511C22—2122no—
512C22—2122no—
513C22—2122no—
514C22—2122no—
515C22—2121no—
516C22—2121no—
517C22C22×C22⋊#615711yescharacteristic
518C22C2×#1313329611yescharacteristic
519C22C2×#1313329611yescharacteristic
520C4—22no—
521C4—22no—
522C4—22no—
523C4—22no—
524C4—41no—
525C4—41no—
526C4—1062no—
527C4—1062no—
528C4—1062no—
529C4—1062no—
530C4—4248no—
531C4—2121no—
532C4—2121no—
533C2—21no—
534C2—21no—
535C2—42no—
536C2—1064no—
537C2—2122no—
538C2#2029115911yescharacteristic
539C2#2029131511yescharacteristic
540C2#2029131711yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6157
commutatorC23×C53C23×C4
FrattiniC23C23×#6157
FittingC2×#117×C53C4
socleC22×C53C22×C22⋊C4
prime radicalC22×#11719945C22
cosocleC23×D53C24
hypercenterC2×#117#6157
hypocenterC53#20725
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
1. #20291192C2semidirect
2. #20290613C2semidirect
3. #20290577C2semidirect
4. #20290576C2semidirect
6. #20290462C2semidirect
7. C4×#13133276C2semidirect
19. C2×C4×Dic106C4semidirect
20. C2×C4×Dic106C4semidirect
23. C2×#13133177C4semidirect
24. C2×#13133177C4semidirect
27. C2×#13133175C4semidirect
29. C22×C4⋊Dic53C4semidirect
30. C2×C4⋊C4×D53C4semidirect
32. C2×#117×C53C4semidirect
291. C2×#117#6157semidirect
303. C53#20725semidirect

Maximal subgroups

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

subgroupindex
#202911922
#202906132
#202905772
#202905762
#202905442
#202904622
C4×#131332762
C2×#168111032
#2072553