Finite Group Databasebeta

← #27539639

Subgroups — #27539639

Order 13568.

526 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285285C211yesmaximal, characteristic, complemented
2#20276991C211yesmaximal, characteristic, complemented
3#20276974C211yesmaximal, characteristic, complemented
4#20276949C211yesmaximal, characteristic, complemented
5#20276946C211yesmaximal, characteristic, complemented
6#20276813C211yesmaximal, characteristic, complemented
7#20276809C211yesmaximal, characteristic, complemented
8#20276804C211yesmaximal, characteristic, complemented
9#20276799C211yesmaximal, characteristic
10C4×#13133163C211yesmaximal, characteristic, complemented
11C4×#13133160C211yesmaximal, characteristic, complemented
12C4×#13133161C211yesmaximal, characteristic, complemented
13C4×#13133158C211yesmaximal, characteristic, complemented
14C2×C4×#11719908C211yesmaximal, characteristic, complemented
15C53×#3168C211yesmaximal, characteristic, complemented
16#16811157—21no—
17C2×#13133163C2211yescharacteristic
18C2×#13133161C2211yescharacteristic
19C2×#13133160C2211yescharacteristic
20C2×#13133158C2211yescharacteristic
21C22×#11719908C2211yescharacteristic
22C22×#11719908C2211yescharacteristic
23#16811025C2211yescharacteristic
24#16811007C2211yescharacteristic
25#16810978—21no—
26#16810864C2211yescharacteristic
27#16810859C2211yescharacteristic
28#16810852—21no—
29C4×Dic212C2211yescharacteristic
30C4×#11719913—21no—
31C4×D424C2211yescharacteristic
32#16810450C2211yescharacteristic
33#16810450C2211yescharacteristic
34#16810448C2211yescharacteristic
35#16810448C2211yescharacteristic
36#16810427C2211yescharacteristic
37#16810427C2211yescharacteristic
38#16810426C2211yescharacteristic
39#16810426C2211yescharacteristic
40#16810424C2211yescharacteristic
41#16810424C2211yescharacteristic
42#16810423C2211yescharacteristic
43#16810423C2211yescharacteristic
44C4×#11719908C2211yescharacteristic
45C4×#11719908C2211yescharacteristic
46C4×#11719908C2211yescharacteristic
47C4×#11719908C2211yescharacteristic
48C53×#529—21no—
49C53×#499C2211yescharacteristic
50C53×#495C2211yescharacteristic
51C53×#488—21no—
52C53×#467C2211yescharacteristic
53C53×#466C2211yescharacteristic
54C53×#465—21no—
55C4×Q16×C53C2211yescharacteristic
56C4×SD16×C53—21no—
57C4×D8×C53C2211yescharacteristic
58C2×C4×C8×C53C2211yescharacteristic
59C2×Dic212C2311yescharacteristic
60C2×#11719913—21no—
61C2×D424C2311yescharacteristic
62#13133211—41no—
63#13133209—21no—
64#13133205—21no—
65#13133200—41no—
66#13133196—21no—
67#13133191—21no—
68C4×Dic106—21no—
69C4×D212—21no—
70C4×C22⋊D53—41no—
71#13133174—41no—
72#13133163—22no—
73#13133161—22no—
74#13133160—22no—
75#13133158—22no—
76C2×#11719908—22no—
77C2×#11719908—22no—
78C2×#11719908C2311yescharacteristic
79C2×#11719908C2311yescharacteristic
80C2×#11719908C2311yescharacteristic
81C2×#11719908C2311yescharacteristic
82#13133151C2311yescharacteristic
83#13133151C2311yescharacteristic
84#13133149—21no—
85#13133142—21no—
86#13133142—21no—
87#13133139—21no—
88#13133139—21no—
89C2×Q16×C53C2311yescharacteristic
90C2×SD16×C53—21no—
91C2×D8×C53C2311yescharacteristic
92C22×C8×C53D411yescharacteristic
93C22×C8×C53C2311yescharacteristic
94#126×C53—41no—
95#125×C53—21no—
96#124×C53—21no—
97#123×C53—41no—
98C22⋊Q8×C53—21no—
99#121×C53—21no—
100C4×Q8×C53—21no—
101C4×D4×C53—21no—
102C4×D4×C53—41no—
103#117×C53—41no—
104C2×C42×C53D411yescharacteristic
105#107×C53C2311yescharacteristic
106#107×C53C2311yescharacteristic
107#106×C53—21no—
108#103×C53—21no—
109#103×C53—21no—
110#102×C53—21no—
111#102×C53—21no—
112C4×C8×C53D411yescharacteristic
113C4×C8×C53D411yescharacteristic
114C4×C8×C53C2311yescharacteristic
115C4×C8×C53C2311yescharacteristic
116C2×Q8×C53—21no—
117C2×D4×C53—21no—
118C2×D4×C53—41no—
119C22×C4×C53—41no—
120C22×C4×C53D811yescharacteristic
121C22×C4×C53D811yescharacteristic
122C22×C4×C53C2×D411yescharacteristic
123C2×C22⋊D53—41no—
124C2×D212—21no—
125C2×C4×D53—41no—
126C2×Dic106—21no—
127Q16×C53—21no—
128Q16×C53—21no—
129SD16×C53—42no—
130D8×C53—21no—
131D8×C53—21no—
132C2×C8×C53—22no—
133C2×C8×C53—21no—
134C2×C8×C53—21no—
135C2×C8×C53C2×D411yescharacteristic
136C2×C8×C53C2×D411yescharacteristic
137C2×C8×C53C2×D411yescharacteristic
138C2×C8×C53C2411yescharacteristic
139C4⋊C4×C53—21no—
140C4⋊C4×C53—21no—
141C4⋊C4×C53—41no—
142C4⋊C4×C53—41no—
143C4⋊C4×C53—41no—
144C4⋊C4×C53—41no—
145C22⋊C4×C53—41no—
146C22⋊C4×C53—41no—
147C22⋊C4×C53—41no—
148C22⋊C4×C53—41no—
149C22⋊C4×C53—41no—
150C42×C53—41no—
151C42×C53D811yescharacteristic
152C42×C53D811yescharacteristic
153C42×C53C2×D411yescharacteristic
154C42×C53C2×D411yescharacteristic
155C22⋊Dic53—41no—
156#11719921—41no—
157#11719921—41no—
158#11719921—41no—
159#11719921—41no—
160C4⋊Dic53—21no—
161C4⋊Dic53—21no—
162#11719919—41no—
163#11719919—41no—
164#11719919—41no—
165#11719919—41no—
166C4×Dic53—41no—
167Dic212—21no—
168Dic212—21no—
169D424—21no—
170D424—21no—
171#11719913—42no—
172#11719908—21no—
173#11719908—21no—
174#11719908#4112yes—
175#11719908#4112yes—
176C23×C53—41no—
177C23×C53C2×D811yescharacteristic
178C22×D53—41no—
179Q8×C53—41no—
180Q8×C53—41no—
181D4×C53—41no—
182D4×C53—41no—
183D4×C53—82no—
184C2×C4×C53—22no—
185C2×C4×C53—21no—
186C2×C4×C53—21no—
187C2×C4×C53—22no—
188C2×C4×C53—41no—
189C2×C4×C53—41no—
190C2×C4×C53—41no—
191C2×C4×C53—82no—
192C2×C4×C53C2×D811yescharacteristic
193C2×C4×C53C2×D811yescharacteristic
194C2×C4×C53C2×D811yescharacteristic
195C2×C4×C53C2×D811yescharacteristic
196C2×C4×C53C2×D811yescharacteristic
197C2×C4×C53C22×D411yescharacteristic
198C22⋊D53—82no—
199C2×Dic53—41no—
200C2×Dic53—41no—
201C2×Dic53—41no—
202D212—41no—
203D212—41no—
204C4×D53—82no—
205Dic106—41no—
206Dic106—41no—
207C8×C53—21no—
208C8×C53—21no—
209C8×C53#13512yes—
210C8×C53C2×#4111yescharacteristic
211C8×C53C2×#4111yescharacteristic
212#33735—531nomaximal, Sylow
213C22×C53—21no—
214C22×C53—21no—
215C22×C53—81no—
216C22×C53—81no—
217C22×C53—81no—
218C22×C53#50812yes—
219C22×C53C22×D811yescharacteristic
220C2×D53—81no—
221C2×D53—81no—
222C2×D53—81no—
223C4×C53—21no—
224C4×C53—21no—
225C4×C53—21no—
226C4×C53—21no—
227C4×C53—81no—
228C4×C53—81no—
229C4×C53—81no—
230C4×C53#50812yes—
231C4×C53#53211yescharacteristic
232C4×C53C2×#13511yescharacteristic
233Dic53—81no—
234Dic53—81no—
235Dic53—81no—
236#3168—531no—
237#3168D5311yescharacteristic, complemented
238#2253—531no—
239#2249—531no—
240#2242—531no—
241#2242—531no—
242#2217—531no—
243#2217—531no—
244#2216—531no—
245#2215—531no—
246C4×Q32—531no—
247C4×SD32—531no—
248C4×SD32—531no—
249C4×D16—531no—
250C2×C4×C16—531no—
251C2×C53—22no—
252C2×C53—161no—
253C2×C53#316911yescharacteristic
254C2×C53C2×#50811yescharacteristic
255C2×C53C2×#50811yescharacteristic
256D53—161no—
257#529—21no—
258#529—1061no—
259C2×Q32—531no—
260C2×SD32—531no—
261C2×SD32—531no—
262C2×D16—531no—
263C22×C16—531no—
264C22×C16—531no—
265#499—531no—
266#499C2×D5311yescharacteristic
267#495—531no—
268#495C2×D5311yescharacteristic
269#488—21no—
270#488—1061no—
271#467—531no—
272#467C2×D5311yescharacteristic
273#466—531no—
274#466C2×D5311yescharacteristic
275#465—21no—
276#465—1061no—
277C4×Q16—531no—
278C4×Q16C2×D5311yescharacteristic
279C4×SD16—21no—
280C4×SD16—1061no—
281C4×D8—531no—
282C4×D8C2×D5311yescharacteristic
283C2×C4×C8C2×D5311yescharacteristic
284#367—531no—
285#367—531no—
286#366—531no—
287#366—531no—
288#358—531no—
289#358—531no—
290#358—531no—
291#358—531no—
292#357—531no—
293#357—531no—
294#357—531no—
295#357—531no—
296C4×C16—531no—
297C4×C16—531no—
298C4×C16—531no—
299C4×C16—531no—
300C53#3373511yescharacteristic, complemented, Sylow
301C2×Q16—531no—
302C2×Q16C22×D5311yescharacteristic
303C2×SD16—21no—
304C2×SD16—1061no—
305C2×D8—531no—
306C2×D8C22×D5311yescharacteristic
307C22×C8C22⋊D5311yescharacteristic
308C22×C8C22×D5311yescharacteristic
309#126—41no—
310#126—2121no—
311#125—21no—
312#125—1061no—
313#124—21no—
314#124—1061no—
315#123—41no—
316#123—2121no—
317C22⋊Q8—21no—
318C22⋊Q8—1061no—
319#121—21no—
320#121—1061no—
321C4×Q8—21no—
322C4×Q8—1061no—
323C4×D4—21no—
324C4×D4—41no—
325C4×D4—1061no—
326C4×D4—2121no—
327#117—41no—
328#117—2121no—
329C2×C42C22⋊D5311yescharacteristic
330Q32—1062no—
331SD32—1062no—
332SD32—1062no—
333D16—1062no—
334C2×C16—531no—
335C2×C16—531no—
336C2×C16—531no—
337C2×C16—531no—
338C2×C16—1062no—
339C2×C16—1062no—
340#107—531no—
341#107—531no—
342#107C22×D5311yescharacteristic
343#107C22×D5311yescharacteristic
344#106—21no—
345#106—1061no—
346#103—21no—
347#103—21no—
348#103—1061no—
349#103—1061no—
350#102—21no—
351#102—21no—
352#102—1061no—
353#102—1061no—
354C4×C8C22⋊D5311yescharacteristic
355C4×C8C22⋊D5311yescharacteristic
356C4×C8C22×D5311yescharacteristic
357C4×C8C22×D5311yescharacteristic
358C2×Q8—21no—
359C2×Q8—1061no—
360C2×D4—21no—
361C2×D4—41no—
362C2×D4—1061no—
363C2×D4—2121no—
364C22×C4—41no—
365C22×C4—2121no—
366C22×C4#1171992211yescharacteristic
367C22×C4#1171992211yescharacteristic
368C22×C4C2×C22⋊D5311yescharacteristic
369Q16—21no—
370Q16—21no—
371Q16—1061no—
372Q16—1061no—
373SD16—42no—
374SD16—2122no—
375D8—21no—
376D8—21no—
377D8—1061no—
378D8—1061no—
379C2×C8—22no—
380C2×C8—21no—
381C2×C8—21no—
382C2×C8C2×C22⋊D5311yescharacteristic
383C2×C8C2×C22⋊D5311yescharacteristic
384C2×C8C2×C22⋊D5311yescharacteristic
385C2×C8C23×D5311yescharacteristic
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—1061no—
393C4⋊C4—1061no—
394C4⋊C4—2121no—
395C4⋊C4—2121no—
396C4⋊C4—2121no—
397C4⋊C4—2121no—
398C22⋊C4—41no—
399C22⋊C4—41no—
400C22⋊C4—41no—
401C22⋊C4—41no—
402C22⋊C4—41no—
403C22⋊C4—2121no—
404C22⋊C4—2121no—
405C22⋊C4—2121no—
406C22⋊C4—2121no—
407C22⋊C4—2121no—
408C42—41no—
409C42—2121no—
410C42#1171992211yescharacteristic
411C42#1171992211yescharacteristic
412C42C2×C22⋊D5311yescharacteristic
413C42C2×C22⋊D5311yescharacteristic
414C16—532no—
415C16—532no—
416C16—1061no—
417C16—1061no—
418C23—41no—
419C23—2121no—
420C23C2×#1171992211yescharacteristic
421Q8—41no—
422Q8—41no—
423Q8—2121no—
424Q8—2121no—
425D4—41no—
426D4—41no—
427D4—82no—
428D4—4242no—
429D4—2121no—
430D4—2121no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—82no—
439C2×C4—4242no—
440C2×C4—2121no—
441C2×C4—2121no—
442C2×C4—2121no—
443C2×C4C2×#1171992211yescharacteristic
444C2×C4C2×#1171992211yescharacteristic
445C2×C4C2×#1171992211yescharacteristic
446C2×C4C2×#1171992211yescharacteristic
447C2×C4C2×#1171992211yescharacteristic
448C2×C4C22×C22⋊D5311yescharacteristic
449C8—21no—
450C8—21no—
451C8#1313323712yes—
452C8#41×D5311yescharacteristic
453C8#41×D5311yescharacteristic
454C22—21no—
455C22—21no—
456C22—81no—
457C22—81no—
458C22—81no—
459C22—4241no—
460C22—4241no—
461C22—4241no—
462C22#1681105312yes—
463C22C22×#1171992211yescharacteristic
464C4—21no—
465C4—21no—
466C4—21no—
467C4—21no—
468C4—81no—
469C4—81no—
470C4—81no—
471C4—4241no—
472C4—4241no—
473C4—4241no—
474C4#1681105312yes—
475C4#1681116911yescharacteristic
476C4C2×#1313323711yescharacteristic
477C2—22no—
478C2—161no—
479C2—8481no—
480C2#2028529011yescharacteristic
481C2C2×#1681105311yescharacteristic
482C2C2×#1681105311yescharacteristic
subgroupG/H
centerC2×C4C2×#11719922
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#3168C2
socleC22×C53C22×D8
prime radical#20276799C2
hypercenter#3168D53
hypocenterC53#33735
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ #3168

Complements

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

NH ≅ G/Ntype
1. #20285285C2semidirect
2. #20276991C2semidirect
3. #20276974C2semidirect
4. #20276949C2semidirect
5. #20276946C2semidirect
6. #20276813C2semidirect
7. #20276809C2semidirect
8. #20276804C2semidirect
10. C4×#13133163C2semidirect
11. C4×#13133160C2semidirect
12. C4×#13133161C2semidirect
13. C4×#13133158C2semidirect
14. C2×C4×#11719908C2semidirect
15. C53×#3168C2semidirect
237. #3168D53semidirect
300. C53#33735semidirect

Maximal subgroups

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

subgroupindex
#202852852
#202769912
#202769742
#202769492
#202769462
#202768132
#202768092
#202768042
#202767992
C4×#131331632
C4×#131331602
C4×#131331612
C4×#131331582
C2×C4×#117199082
C53×#31682
#3373553