Finite Group Databasebeta

← #27276519

Subgroups — #27276519

Order 13568.

672 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20273425C211yesmaximal, characteristic, complemented
2#20273346C211yesmaximal, characteristic, complemented
3#20273292C211yesmaximal, characteristic, complemented
4#20273266C211yesmaximal, characteristic, complemented
5#20273262C211yesmaximal, characteristic, complemented
6#20273184C211yesmaximal, characteristic, complemented
7C53×#3554C211yesmaximal, characteristic
8#16810680C2211yescharacteristic
9#16810680C2211yescharacteristic
10#16810676C2211yescharacteristic
11#16810491C2211yescharacteristic
12#16810372—21no—
13#16810369—21no—
14#16810365—21no—
15#16810363—21no—
16#16810360—21no—
17#16810360—21no—
18#16810360—21no—
19#16810357—22no—
20#16810357—21no—
21D4⋊#88322—22no—
22C53×#562—22no—
23C53×#559—21no—
24C53×#559C411yescharacteristic, complemented
25C53×#557C411yescharacteristic, complemented
26C53×#554—21no—
27C53×#550C411yescharacteristic, complemented
28D4×Q8×C53C411yescharacteristic, complemented
29C53×#548—21no—
30C53×#547—21no—
31C53×#542—21no—
32C53×#542—21no—
33C53×#540—21no—
34C53×#539—22no—
35C53×#536—21no—
36C53×#533C2211yescharacteristic
37C53×#532C2211yescharacteristic
38C53×#529C2211yescharacteristic
39C4⋊#88322—21no—
40C4⋊#88322—21no—
41C4⋊#88322—21no—
42C22⋊#88322—21no—
43C22⋊#88322—21no—
44C22⋊#88322—21no—
45#13133128—21no—
46#13133128—21no—
47C4×#88322—21no—
48#13133124—21no—
49#13133124—21no—
50#13133124—21no—
51C2×#41×C53—21no—
52C2×#41×C53—21no—
53C22×Q8×C53—21no—
54C4⋊Q8×C53—21no—
55C4⋊Q8×C53—21no—
56C4⋊Q8×C53C2×C411yescharacteristic
57C4⋊Q8×C53D411yescharacteristic
58C4⋊Q8×C53D411yescharacteristic
59#127×C53—21no—
60#127×C53C2×C411yescharacteristic
61#126×C53—22no—
62#126×C53—21no—
63#125×C53—22no—
64#125×C53—21no—
65#125×C53—21no—
66#125×C53D411yescharacteristic
67#124×C53—22no—
68#124×C53—22no—
69#124×C53—21no—
70#124×C53—21no—
71#124×C53—21no—
72#124×C53D411yescharacteristic
73#123×C53—22no—
74#123×C53—22no—
75#123×C53—22no—
76#123×C53—22no—
77#123×C53—22no—
78#123×C53—22no—
79#123×C53—21no—
80C22⋊Q8×C53—22no—
81C22⋊Q8×C53—22no—
82C22⋊Q8×C53—22no—
83C22⋊Q8×C53—22no—
84C22⋊Q8×C53—22no—
85C22⋊Q8×C53—21no—
86C22⋊Q8×C53—21no—
87C22⋊Q8×C53D411yescharacteristic
88C22⋊Q8×C53D411yescharacteristic
89C22⋊Q8×C53D411yescharacteristic
90#121×C53—22no—
91#121×C53—22no—
92#121×C53—21no—
93#121×C53—22no—
94#121×C53—21no—
95#121×C53D411yescharacteristic
96#121×C53D412yes—
97C4×Q8×C53—21no—
98C4×Q8×C53—21no—
99C4×Q8×C53—21no—
100C4×Q8×C53C2×C411yescharacteristic
101C4×Q8×C53C2×C411yescharacteristic
102C4×Q8×C53C2×C411yescharacteristic
103C4×D4×C53—22no—
104C4×D4×C53—21no—
105C4×D4×C53—21no—
106C4×D4×C53—21no—
107C4×D4×C53—21no—
108C4×D4×C53—21no—
109C4×D4×C53—21no—
110C4×D4×C53C2×C411yescharacteristic
111C4×D4×C53D412yes—
112#117×C53—21no—
113#117×C53—21no—
114C2×C4⋊C4×C53—22no—
115C2×C42×C53C2311yescharacteristic
116#41×C53—82no—
117#41×C53—82no—
118C2×Q8×C53—22no—
119C2×Q8×C53—21no—
120C2×Q8×C53—21no—
121C2×Q8×C53—82no—
122C2×Q8×C53C22⋊C411yescharacteristic
123C2×Q8×C53C22⋊C411yescharacteristic
124C2×Q8×C53C22⋊C411yescharacteristic
125C2×D4×C53—22no—
126C2×D4×C53—22no—
127C2×D4×C53—21no—
128C2×D4×C53—22no—
129C2×D4×C53—21no—
130C2×D4×C53—21no—
131C2×D4×C53C22⋊C411yescharacteristic
132C2×D4×C53C22⋊C412yes—
133C22×C4×C53—22no—
134C22×C4×C53—22no—
135C22×C4×C53—22no—
136C22×C4×C53—21no—
137C22×C4×C53—22no—
138C22×C4×C53C2×D411yescharacteristic
139C22×C4×C53C2×D411yescharacteristic
140C22×C4×C53C2×D411yescharacteristic
141C22×Dic53—41no—
142C4⋊C4×C53—22no—
143C4⋊C4×C53—22no—
144C4⋊C4×C53—22no—
145C4⋊C4×C53—22no—
146C4⋊C4×C53—21no—
147C4⋊C4×C53—21no—
148C4⋊C4×C53—21no—
149C4⋊C4×C53—21no—
150C4⋊C4×C53—21no—
151C4⋊C4×C53—21no—
152C4⋊C4×C53—21no—
153C4⋊C4×C53—21no—
154C4⋊C4×C53—21no—
155C4⋊C4×C53—21no—
156C4⋊C4×C53—21no—
157C4⋊C4×C53—21no—
158C4⋊C4×C53C22⋊C411yescharacteristic
159C4⋊C4×C53C22⋊C411yescharacteristic
160C4⋊C4×C53C22⋊C411yescharacteristic
161C4⋊C4×C53C22⋊C412yes—
162C4⋊C4×C53C22⋊C411yescharacteristic
163C22⋊C4×C53—22no—
164C22⋊C4×C53—22no—
165C22⋊C4×C53—22no—
166C22⋊C4×C53—22no—
167C22⋊C4×C53—22no—
168C22⋊C4×C53—21no—
169C22⋊C4×C53—21no—
170C22⋊C4×C53—21no—
171C22⋊C4×C53—21no—
172C22⋊C4×C53—21no—
173C22⋊C4×C53—21no—
174C22⋊C4×C53—21no—
175C22⋊C4×C53—22no—
176C22⋊C4×C53—22no—
177C42×C53—22no—
178C42×C53—21no—
179C42×C53—21no—
180C42×C53—21no—
181C42×C53C22×C411yescharacteristic
182C42×C53C2×D411yescharacteristic
183C42×C53C2×D411yescharacteristic
184C42×C53C2×D411yescharacteristic
185C2×#88322—41no—
186C2×#88322—41no—
187C2×#88322—41no—
188C23×C53—22no—
189C23×C53—21no—
190C23×C53#12011yescharacteristic
191Q8×C53—42no—
192Q8×C53—41no—
193Q8×C53—41no—
194Q8×C53—41no—
195Q8×C53—81no—
196Q8×C53—82no—
197D4×C53—42no—
198D4×C53—41no—
199D4×C53—42no—
200D4×C53—42no—
201D4×C53—81no—
202D4×C53—82no—
203D4×C53—82no—
204D4×C53—82no—
205C2×C4×C53—21no—
206C2×C4×C53—21no—
207C2×C4×C53—22no—
208C2×C4×C53—21no—
209C2×C4×C53—21no—
210C2×C4×C53—21no—
211C2×C4×C53—21no—
212C2×C4×C53—21no—
213C2×C4×C53—41no—
214C2×C4×C53—41no—
215C2×C4×C53—41no—
216C2×C4×C53—42no—
217C2×C4×C53—82no—
218C2×C4×C53—82no—
219C2×C4×C53—82no—
220C2×C4×C53—82no—
221C2×C4×C53C2×C22⋊C411yescharacteristic
222C2×C4×C53C2×C22⋊C411yescharacteristic
223C2×C4×C53C2×C22⋊C411yescharacteristic
224C2×C4×C53#12011yescharacteristic
225C2×C4×C53#12011yescharacteristic
226C2×C4×C53#12011yescharacteristic
227C2×Dic53—81no—
228C2×Dic53—81no—
229C2×Dic53—81no—
230#88322—81no—
231#88322—81no—
232#88322—81no—
233#8503—531nomaximal, Sylow
234C22×C53—21no—
235C22×C53—21no—
236C22×C53—21no—
237C22×C53—42no—
238C22×C53—41no—
239C22×C53—41no—
240C22×C53—42no—
241C22×C53—42no—
242C22×C53—41no—
243C22×C53#37911yescharacteristic
244C4×C53—21no—
245C4×C53—21no—
246C4×C53—21no—
247C4×C53—21no—
248C4×C53—21no—
249C4×C53—21no—
250C4×C53—41no—
251C4×C53—41no—
252C4×C53—41no—
253C4×C53—41no—
254C4×C53—41no—
255C4×C53—42no—
256C4×C53—41no—
257C4×C53—41no—
258Dic53—161no—
259#3554D5311yescharacteristic
260#1554—531no—
261#1535—531no—
262#1521—531no—
263#1515—531no—
264#1514—531no—
265#1496—531no—
266C2×C53—41no—
267C2×C53—82no—
268C2×C53—81no—
269C2×C53#179411yescharacteristic
270C2×C53#179511yescharacteristic
271C2×C53#179611yescharacteristic
272#562—22no—
273#559—21no—
274#559Dic5311yescharacteristic, complemented
275#557Dic5311yescharacteristic, complemented
276#554—21no—
277#550Dic5311yescharacteristic, complemented
278D4×Q8Dic5311yescharacteristic, complemented
279#548—21no—
280#547—21no—
281#542—21no—
282#542—21no—
283#540—21no—
284#539—22no—
285#536—21no—
286#533C2×D5311yescharacteristic
287#532C2×D5311yescharacteristic
288#529C2×D5311yescharacteristic
289#432—531no—
290#432—531no—
291#431—531no—
292#383—531no—
293#333—1061no—
294#332—1061no—
295#330—1061no—
296#329—1061no—
297#328—1061no—
298#328—1061no—
299#328—1061no—
300#327—1062no—
301#327—1061no—
302D4⋊C8—1062no—
303C53#850311yescharacteristic, complemented, Sylow
304C2×#41—21no—
305C2×#41—21no—
306C22×Q8—21no—
307C4⋊Q8—21no—
308C4⋊Q8—21no—
309C4⋊Q8C2×Dic5311yescharacteristic
310C4⋊Q8C22⋊D5311yescharacteristic
311C4⋊Q8C22⋊D5311yescharacteristic
312#127—21no—
313#127C2×Dic5311yescharacteristic
314#126—22no—
315#126—21no—
316#125—22no—
317#125—21no—
318#125—21no—
319#125C22⋊D5311yescharacteristic
320#124—22no—
321#124—22no—
322#124—21no—
323#124—21no—
324#124—21no—
325#124C22⋊D5311yescharacteristic
326#123—22no—
327#123—22no—
328#123—22no—
329#123—22no—
330#123—22no—
331#123—22no—
332#123—21no—
333C22⋊Q8—22no—
334C22⋊Q8—22no—
335C22⋊Q8—22no—
336C22⋊Q8—22no—
337C22⋊Q8—21no—
338C22⋊Q8—22no—
339C22⋊Q8—21no—
340C22⋊Q8C22⋊D5311yescharacteristic
341C22⋊Q8C22⋊D5311yescharacteristic
342C22⋊Q8C22⋊D5311yescharacteristic
343#121—22no—
344#121—22no—
345#121—22no—
346#121—21no—
347#121—21no—
348#121C22⋊D5311yescharacteristic
349#121C22⋊D5312yes—
350C4×Q8—21no—
351C4×Q8—21no—
352C4×Q8—21no—
353C4×Q8C2×Dic5311yescharacteristic
354C4×Q8C2×Dic5311yescharacteristic
355C4×Q8C2×Dic5311yescharacteristic
356C4×D4—22no—
357C4×D4—21no—
358C4×D4—21no—
359C4×D4—21no—
360C4×D4—21no—
361C4×D4—21no—
362C4×D4—21no—
363C4×D4C2×Dic5311yescharacteristic
364C4×D4C22⋊D5312yes—
365#117—21no—
366#117—21no—
367C2×C4⋊C4—22no—
368C2×C42C22×D5311yescharacteristic
369C4⋊C8—1061no—
370C4⋊C8—1061no—
371C4⋊C8—1061no—
372C22⋊C8—1061no—
373C22⋊C8—1061no—
374C22⋊C8—1061no—
375#97—1061no—
376#97—1061no—
377C4×C8—1061no—
378#95—1061no—
379#95—1061no—
380#95—1061no—
381#41—82no—
382#41—82no—
383C2×Q8—22no—
384C2×Q8—21no—
385C2×Q8—21no—
386C2×Q8—82no—
387C2×Q8C22⋊Dic5311yescharacteristic
388C2×Q8C22⋊Dic5311yescharacteristic
389C2×Q8C22⋊Dic5311yescharacteristic
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—21no—
393C2×D4—22no—
394C2×D4—21no—
395C2×D4—21no—
396C2×D4C22⋊Dic5311yescharacteristic
397C2×D4C22⋊Dic5312yes—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—2121no—
404C22×C4C2×C22⋊D5311yescharacteristic
405C22×C4C2×C22⋊D5311yescharacteristic
406C22×C4C2×C22⋊D5311yescharacteristic
407C2×C8—2121no—
408C2×C8—2121no—
409C2×C8—2121no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—21no—
415C4⋊C4—21no—
416C4⋊C4—21no—
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4C22⋊Dic5311yescharacteristic
427C4⋊C4C22⋊Dic5311yescharacteristic
428C4⋊C4C22⋊Dic5311yescharacteristic
429C4⋊C4C22⋊Dic5312yes—
430C4⋊C4C22⋊Dic5311yescharacteristic
431C22⋊C4—22no—
432C22⋊C4—22no—
433C22⋊C4—22no—
434C22⋊C4—22no—
435C22⋊C4—22no—
436C22⋊C4—21no—
437C22⋊C4—21no—
438C22⋊C4—21no—
439C22⋊C4—21no—
440C22⋊C4—21no—
441C22⋊C4—21no—
442C22⋊C4—21no—
443C22⋊C4—22no—
444C22⋊C4—22no—
445C42—22no—
446C42—21no—
447C42—21no—
448C42—21no—
449C42C22×Dic5311yescharacteristic
450C42C2×C22⋊D5311yescharacteristic
451C42C2×C22⋊D5311yescharacteristic
452C42C2×C22⋊D5311yescharacteristic
453C23—22no—
454C23—21no—
455C23#1313318611yescharacteristic
456Q8—41no—
457Q8—41no—
458Q8—41no—
459Q8—42no—
460Q8—81no—
461Q8—82no—
462D4—41no—
463D4—42no—
464D4—42no—
465D4—42no—
466D4—81no—
467D4—82no—
468D4—82no—
469D4—82no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—22no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—4241no—
487C2×C4—4241no—
488C2×C4—4241no—
489C2×C4C2×C22⋊Dic5311yescharacteristic
490C2×C4C2×C22⋊Dic5311yescharacteristic
491C2×C4C2×C22⋊Dic5311yescharacteristic
492C2×C4#1313318611yescharacteristic
493C2×C4#1313318611yescharacteristic
494C2×C4#1313318611yescharacteristic
495C8—4241no—
496C8—4241no—
497C8—4241no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—42no—
502C22—41no—
503C22—42no—
504C22—41no—
505C22—41no—
506C22—42no—
507C22#1681047911yescharacteristic
508C4—21no—
509C4—21no—
510C4—21no—
511C4—21no—
512C4—21no—
513C4—21no—
514C4—41no—
515C4—41no—
516C4—41no—
517C4—41no—
518C4—41no—
519C4—41no—
520C4—42no—
521C4—41no—
522C4—8481no—
523C2—41no—
524C2—82no—
525C2—81no—
526C2#2027467911yescharacteristic
527C2#2027468511yescharacteristic
528C2#2027469011yescharacteristic
subgroupG/H
centerC22#16810479
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
Fitting, prime radicalC53×#3554C2
socleC22×C53#379
cosocleC2×C42×C53C23
hypercenter#3554D53
hypocenterC53#8503
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3554

Complements

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

NH ≅ G/Ntype
1. #20273425C2semidirect
2. #20273346C2semidirect
3. #20273292C2semidirect
4. #20273266C2semidirect
5. #20273262C2semidirect
6. #20273184C2semidirect
24. C53×#559C4semidirect
25. C53×#557C4semidirect
27. C53×#550C4semidirect
28. D4×Q8×C53C4semidirect
274. #559Dic53semidirect
275. #557Dic53semidirect
277. #550Dic53semidirect
278. D4×Q8Dic53semidirect
303. C53#8503semidirect

Maximal subgroups

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

subgroupindex
#202734252
#202733462
#202732922
#202732662
#202732622
#202731842
C53×#35542
#850353