Finite Group Databasebeta

← #27276512

Subgroups — #27276512

Order 13568.

672 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20273420C211yesmaximal, characteristic, complemented
2#20273346C211yesmaximal, characteristic, complemented
3#20273292C211yesmaximal, characteristic, complemented
4#20273266C211yesmaximal, characteristic, complemented
5#20273262C211yesmaximal, characteristic, complemented
6#20273188C211yesmaximal, characteristic, complemented
7C53×#3547C211yesmaximal, characteristic
8#16810680C2211yescharacteristic
9#16810680C2211yescharacteristic
10#16810676C2211yescharacteristic
11#16810491C2211yescharacteristic
12#16810369—21no—
13#16810367—21no—
14#16810365—21no—
15#16810363—21no—
16#16810360—22no—
17#16810360—21no—
18#16810357—21no—
19#16810357—21no—
20#16810357—21no—
21Q8⋊#88322—22no—
22C53×#562—22no—
23C53×#555—21no—
24C53×#555C411yescharacteristic, complemented
25C53×#554C411yescharacteristic, complemented
26C53×#553—21no—
27C53×#552—21no—
28D4×Q8×C53C411yescharacteristic, complemented
29C53×#548—21no—
30C53×#547C411yescharacteristic, complemented
31C53×#544—21no—
32C53×#542—22no—
33C53×#540—21no—
34C53×#537—21no—
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—22no—
55C4⋊Q8×C53—21no—
56C4⋊Q8×C53C2×C411yescharacteristic
57C4⋊Q8×C53C2×C411yescharacteristic
58C4⋊Q8×C53D411yescharacteristic
59#127×C53D411yescharacteristic
60#126×C53—22no—
61#126×C53—21no—
62#125×C53—22no—
63#125×C53—21no—
64#125×C53—21no—
65#125×C53D411yescharacteristic
66#124×C53—22no—
67#124×C53—22no—
68#124×C53—21no—
69#124×C53—22no—
70#124×C53D411yescharacteristic
71#123×C53—22no—
72#123×C53—22no—
73#123×C53—22no—
74#123×C53—22no—
75#123×C53—21no—
76C22⋊Q8×C53—22no—
77C22⋊Q8×C53—22no—
78C22⋊Q8×C53—21no—
79C22⋊Q8×C53—21no—
80C22⋊Q8×C53—22no—
81C22⋊Q8×C53—21no—
82C22⋊Q8×C53—22no—
83C22⋊Q8×C53—21no—
84C22⋊Q8×C53D411yescharacteristic
85C22⋊Q8×C53D412yes—
86#121×C53—22no—
87#121×C53—22no—
88#121×C53—22no—
89#121×C53—22no—
90#121×C53—21no—
91#121×C53—21no—
92#121×C53—21no—
93#121×C53—21no—
94#121×C53D411yescharacteristic
95#121×C53D411yescharacteristic
96#121×C53D411yescharacteristic
97C4×Q8×C53—21no—
98C4×Q8×C53C2×C411yescharacteristic
99C4×Q8×C53D412yes—
100C4×D4×C53—22no—
101C4×D4×C53—21no—
102C4×D4×C53—21no—
103C4×D4×C53—21no—
104C4×D4×C53—21no—
105C4×D4×C53C2×C411yescharacteristic
106C4×D4×C53C2×C411yescharacteristic
107C4×D4×C53C2×C411yescharacteristic
108#117×C53—22no—
109#117×C53—21no—
110#117×C53—21no—
111#117×C53—21no—
112#117×C53—21no—
113C2×C4⋊C4×C53—21no—
114C2×C4⋊C4×C53—21no—
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⋊C412yes—
124C2×D4×C53—22no—
125C2×D4×C53—22no—
126C2×D4×C53—21no—
127C2×D4×C53—21no—
128C2×D4×C53—21no—
129C2×D4×C53—21no—
130C2×D4×C53—21no—
131C2×D4×C53C22⋊C411yescharacteristic
132C2×D4×C53C22⋊C411yescharacteristic
133C2×D4×C53C22⋊C411yescharacteristic
134C22×C4×C53—22no—
135C22×C4×C53—22no—
136C22×C4×C53—22no—
137C22×C4×C53—21no—
138C22×C4×C53—21no—
139C22×C4×C53—21no—
140C22×C4×C53C2×D411yescharacteristic
141C22×C4×C53C2×D411yescharacteristic
142C22×C4×C53C2×D411yescharacteristic
143C22×Dic53—41no—
144C4⋊C4×C53—22no—
145C4⋊C4×C53—22no—
146C4⋊C4×C53—22no—
147C4⋊C4×C53—22no—
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—22no—
157C4⋊C4×C53—22no—
158C4⋊C4×C53C22⋊C411yescharacteristic
159C4⋊C4×C53C22⋊C411yescharacteristic
160C4⋊C4×C53C22⋊C411yescharacteristic
161C4⋊C4×C53C22⋊C411yescharacteristic
162C4⋊C4×C53C22⋊C412yes—
163C22⋊C4×C53—22no—
164C22⋊C4×C53—22no—
165C22⋊C4×C53—22no—
166C22⋊C4×C53—22no—
167C22⋊C4×C53—22no—
168C22⋊C4×C53—22no—
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—21no—
176C22⋊C4×C53—21no—
177C22⋊C4×C53—21no—
178C42×C53—22no—
179C42×C53—21no—
180C42×C53—22no—
181C42×C53C22×C411yescharacteristic
182C42×C53C2×D411yescharacteristic
183C42×C53C2×D411yescharacteristic
184C42×C53C2×D411yescharacteristic
185C2×#88322—41no—
186C2×#88322—41no—
187C2×#88322—41no—
188C23×C53—21no—
189C23×C53—21no—
190C23×C53—21no—
191C23×C53#12011yescharacteristic
192Q8×C53—42no—
193Q8×C53—41no—
194Q8×C53—42no—
195Q8×C53—81no—
196Q8×C53—82no—
197D4×C53—41no—
198D4×C53—41no—
199D4×C53—41no—
200D4×C53—81no—
201D4×C53—81no—
202D4×C53—81no—
203D4×C53—81no—
204D4×C53—81no—
205D4×C53—82no—
206D4×C53—82no—
207C2×C4×C53—21no—
208C2×C4×C53—22no—
209C2×C4×C53—21no—
210C2×C4×C53—21no—
211C2×C4×C53—21no—
212C2×C4×C53—22no—
213C2×C4×C53—21no—
214C2×C4×C53—41no—
215C2×C4×C53—41no—
216C2×C4×C53—41no—
217C2×C4×C53—42no—
218C2×C4×C53—42no—
219C2×C4×C53—42no—
220C2×C4×C53—82no—
221C2×C4×C53—82no—
222C2×C4×C53—82no—
223C2×C4×C53C2×C22⋊C411yescharacteristic
224C2×C4×C53C2×C22⋊C411yescharacteristic
225C2×C4×C53C2×C22⋊C411yescharacteristic
226C2×C4×C53#12011yescharacteristic
227C2×C4×C53#12011yescharacteristic
228C2×C4×C53#12011yescharacteristic
229C2×Dic53—81no—
230C2×Dic53—81no—
231C2×Dic53—81no—
232#88322—81no—
233#88322—81no—
234#88322—81no—
235#8502—531nomaximal, Sylow
236C22×C53—21no—
237C22×C53—21no—
238C22×C53—21no—
239C22×C53—41no—
240C22×C53—41no—
241C22×C53—41no—
242C22×C53—41no—
243C22×C53—41no—
244C22×C53—41no—
245C22×C53—41no—
246C22×C53—41no—
247C22×C53—41no—
248C22×C53#37911yescharacteristic
249C4×C53—21no—
250C4×C53—21no—
251C4×C53—21no—
252C4×C53—21no—
253C4×C53—21no—
254C4×C53—21no—
255C4×C53—41no—
256C4×C53—42no—
257C4×C53—41no—
258C4×C53—41no—
259C4×C53—42no—
260C4×C53—41no—
261C4×C53—41no—
262Dic53—161no—
263#3547D5311yescharacteristic
264#1553—531no—
265#1535—531no—
266#1521—531no—
267#1515—531no—
268#1514—531no—
269#1497—531no—
270C2×C53—41no—
271C2×C53—81no—
272C2×C53—81no—
273C2×C53—81no—
274C2×C53#179411yescharacteristic
275C2×C53#179511yescharacteristic
276C2×C53#179611yescharacteristic
277#562—22no—
278#555—21no—
279#555Dic5311yescharacteristic, complemented
280#554Dic5311yescharacteristic, complemented
281#553—21no—
282#552—21no—
283D4×Q8Dic5311yescharacteristic, complemented
284#548—21no—
285#547Dic5311yescharacteristic, complemented
286#544—21no—
287#542—22no—
288#540—21no—
289#537—21no—
290#536—21no—
291#533C2×D5311yescharacteristic
292#532C2×D5311yescharacteristic
293#529C2×D5311yescharacteristic
294#432—531no—
295#432—531no—
296#431—531no—
297#383—531no—
298#332—1061no—
299#331—1061no—
300#330—1061no—
301#329—1061no—
302#328—1062no—
303#328—1061no—
304#327—1061no—
305#327—1061no—
306#327—1061no—
307Q8⋊C8—1062no—
308C53#850211yescharacteristic, complemented, Sylow
309C2×#41—21no—
310C2×#41—21no—
311C22×Q8—21no—
312C4⋊Q8—22no—
313C4⋊Q8—21no—
314C4⋊Q8C2×Dic5311yescharacteristic
315C4⋊Q8C2×Dic5311yescharacteristic
316C4⋊Q8C22⋊D5311yescharacteristic
317#127C22⋊D5311yescharacteristic
318#126—22no—
319#126—21no—
320#125—22no—
321#125—21no—
322#125—21no—
323#125C22⋊D5311yescharacteristic
324#124—22no—
325#124—22no—
326#124—22no—
327#124—21no—
328#124C22⋊D5311yescharacteristic
329#123—22no—
330#123—22no—
331#123—22no—
332#123—21no—
333#123—22no—
334C22⋊Q8—22no—
335C22⋊Q8—22no—
336C22⋊Q8—21no—
337C22⋊Q8—22no—
338C22⋊Q8—21no—
339C22⋊Q8—21no—
340C22⋊Q8—21no—
341C22⋊Q8—22no—
342C22⋊Q8C22⋊D5311yescharacteristic
343C22⋊Q8C22⋊D5312yes—
344#121—22no—
345#121—22no—
346#121—22no—
347#121—22no—
348#121—21no—
349#121—21no—
350#121—21no—
351#121—21no—
352#121C22⋊D5311yescharacteristic
353#121C22⋊D5311yescharacteristic
354#121C22⋊D5311yescharacteristic
355C4×Q8—21no—
356C4×Q8C2×Dic5311yescharacteristic
357C4×Q8C22⋊D5312yes—
358C4×D4—22no—
359C4×D4—21no—
360C4×D4—21no—
361C4×D4—21no—
362C4×D4—21no—
363C4×D4C2×Dic5311yescharacteristic
364C4×D4C2×Dic5311yescharacteristic
365C4×D4C2×Dic5311yescharacteristic
366#117—22no—
367#117—21no—
368#117—21no—
369#117—21no—
370#117—21no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—21no—
373C2×C42C22×D5311yescharacteristic
374C4⋊C8—1061no—
375C4⋊C8—1061no—
376C4⋊C8—1061no—
377C22⋊C8—1061no—
378C22⋊C8—1061no—
379C22⋊C8—1061no—
380#97—1061no—
381#97—1061no—
382C4×C8—1061no—
383#95—1061no—
384#95—1061no—
385#95—1061no—
386#41—82no—
387#41—82no—
388C2×Q8—22no—
389C2×Q8—21no—
390C2×Q8—21no—
391C2×Q8—82no—
392C2×Q8C22⋊Dic5311yescharacteristic
393C2×Q8C22⋊Dic5312yes—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—21no—
397C2×D4—21no—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4—21no—
401C2×D4C22⋊Dic5311yescharacteristic
402C2×D4C22⋊Dic5311yescharacteristic
403C2×D4C22⋊Dic5311yescharacteristic
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—2121no—
411C22×C4C2×C22⋊D5311yescharacteristic
412C22×C4C2×C22⋊D5311yescharacteristic
413C22×C4C2×C22⋊D5311yescharacteristic
414C2×C8—2121no—
415C2×C8—2121no—
416C2×C8—2121no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4C22⋊Dic5311yescharacteristic
432C4⋊C4C22⋊Dic5311yescharacteristic
433C4⋊C4C22⋊Dic5311yescharacteristic
434C4⋊C4C22⋊Dic5311yescharacteristic
435C4⋊C4C22⋊Dic5312yes—
436C22⋊C4—22no—
437C22⋊C4—22no—
438C22⋊C4—22no—
439C22⋊C4—22no—
440C22⋊C4—22no—
441C22⋊C4—22no—
442C22⋊C4—21no—
443C22⋊C4—21no—
444C22⋊C4—21no—
445C22⋊C4—21no—
446C22⋊C4—21no—
447C22⋊C4—21no—
448C22⋊C4—21no—
449C22⋊C4—21no—
450C22⋊C4—21no—
451C42—22no—
452C42—21no—
453C42—22no—
454C42C22×Dic5311yescharacteristic
455C42C2×C22⋊D5311yescharacteristic
456C42C2×C22⋊D5311yescharacteristic
457C42C2×C22⋊D5311yescharacteristic
458C23—21no—
459C23—21no—
460C23—21no—
461C23#1313318611yescharacteristic
462Q8—41no—
463Q8—42no—
464Q8—42no—
465Q8—81no—
466Q8—82no—
467D4—41no—
468D4—41no—
469D4—41no—
470D4—81no—
471D4—81no—
472D4—81no—
473D4—81no—
474D4—81no—
475D4—82no—
476D4—82no—
477C2×C4—21no—
478C2×C4—22no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—22no—
483C2×C4—21no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—4241no—
494C2×C4—4241no—
495C2×C4—4241no—
496C2×C4C2×C22⋊Dic5311yescharacteristic
497C2×C4C2×C22⋊Dic5311yescharacteristic
498C2×C4C2×C22⋊Dic5311yescharacteristic
499C2×C4#1313318611yescharacteristic
500C2×C4#1313318611yescharacteristic
501C2×C4#1313318611yescharacteristic
502C8—4241no—
503C8—4241no—
504C8—4241no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—41no—
509C22—41no—
510C22—41no—
511C22—41no—
512C22—41no—
513C22—41no—
514C22—41no—
515C22—41no—
516C22—41no—
517C22#1681047911yescharacteristic
518C4—21no—
519C4—21no—
520C4—21no—
521C4—21no—
522C4—21no—
523C4—21no—
524C4—41no—
525C4—41no—
526C4—42no—
527C4—41no—
528C4—41no—
529C4—42no—
530C4—41no—
531C4—8481no—
532C2—41no—
533C2—81no—
534C2—81no—
535C2—81no—
536C2#2027467911yescharacteristic
537C2#2027468511yescharacteristic
538C2#2027469011yescharacteristic
subgroupG/H
centerC22#16810479
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
Fitting, prime radicalC53×#3547C2
socleC22×C53#379
cosocleC2×C42×C53C23
hypercenter#3547D53
hypocenterC53#8502
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3547

Complements

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

NH ≅ G/Ntype
1. #20273420C2semidirect
2. #20273346C2semidirect
3. #20273292C2semidirect
4. #20273266C2semidirect
5. #20273262C2semidirect
6. #20273188C2semidirect
24. C53×#555C4semidirect
25. C53×#554C4semidirect
28. D4×Q8×C53C4semidirect
30. C53×#547C4semidirect
279. #555Dic53semidirect
280. #554Dic53semidirect
283. D4×Q8Dic53semidirect
285. #547Dic53semidirect
308. C53#8502semidirect

Maximal subgroups

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

subgroupindex
#202734202
#202733462
#202732922
#202732662
#202732622
#202731882
C53×#35472
#850253