Finite Group Databasebeta

← #27537409

Subgroups — #27537409

Order 13568.

738 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811061C211yesmaximal, characteristic
2#20283486C211yesmaximal, characteristic, complemented
3C2×#16810705C211yesmaximal, characteristic, complemented
4#20276590C212yesmaximal, complemented
5#20276578C212yesmaximal, complemented
6#20276566C212yesmaximal, complemented
7#20276561C212yesmaximal, complemented
8#20276430C211yesmaximal, characteristic, complemented
9#20276425C211yesmaximal, characteristic, complemented
10M32×Dic53C211yesmaximal, characteristic, complemented
11C53×#2159C211yesmaximal, characteristic, complemented
12C2×#13133239C2211yescharacteristic
13C22×D424C2211yescharacteristic
14#16811091—21no—
15C2×#13133179—21no—
16#16811061C418yescomplemented
17C2×#13133162C2212yes—
18C2×D848C2212yes—
19C2×#13133157C2211yescharacteristic
20#16810717C2211yescharacteristic, complemented
21#16810717C2211yescharacteristic, complemented
22#16810710—21no—
23#16810710—21no—
24#16810705—21no—
25#16810705—21no—
26#16810705C2212yes—
27#16810705C2212yes—
28#16810705C2211yescharacteristic, complemented
29#16810705C2211yescharacteristic, complemented
30#16810617—21no—
31C2×#13133137—21no—
32C2×C8×Dic53C2211yescharacteristic
33#16810426C2212yescomplemented
34#16810423C2212yes—
35#16810423C2212yes—
36#16810423C2212yescomplemented
37#16810398C2212yescomplemented
38C16×Dic53C2212yescomplemented
39C2×M32×C53C2211yescharacteristic
40C53×#427C2211yescharacteristic
41C2×#107×C53C2211yescharacteristic
42C53×#367C2212yescomplemented
43C53×#366C2212yescomplemented
44C2×#11719945—21no—
45C22×D212—21no—
46C22×C4×D53—41no—
47#13133239—22no—
48#13133239C2×C414yescomplemented
49C2×Dic212C2311yescharacteristic
50C2×#11719913—21no—
51C2×D424—21no—
52C2×D424—22no—
53C2×D424C2×C414yescomplemented
54C2×D424C2312yes—
55C2×D424C2311yescharacteristic
56C22×#88322D411yescharacteristic
57#13133182—21no—
58#13133182—21no—
59#13133179—22no—
60#13133179—22no—
61#13133179—21no—
62#13133179—21no—
63#13133179—41no—
64#13133179—41no—
65#13133177—41no—
66#13133177—41no—
67#13133174—41no—
68C2×#11719921—41no—
69C2×C4×Dic53D411yescharacteristic
70C42×D53—41no—
71#13133162C2×C418yescomplemented
72D848C2×C418yescomplemented
73#13133157—22no—
74C2×#11719908C2312yes—
75#13133140—21no—
76#13133140—21no—
77#13133137—22no—
78#13133137—22no—
79#13133137—21no—
80#13133137—21no—
81C8×Dic53D411yescharacteristic
82C8×Dic53D411yescharacteristic
83C8×Dic53C2311yescharacteristic
84C8×Dic53C2311yescharacteristic
85C22×C8×C53C2311yescharacteristic
86#117×C53—21no—
87C2×C4⋊C4×C53—21no—
88M32×C53C2×C414yescomplemented
89C2×C16×C53C2312yes—
90#107×C53—21no—
91#107×C53C2311yescharacteristic
92#107×C53C2312yes—
93#107×C53C2311yescharacteristic
94#106×C53—21no—
95C23×D53—41no—
96C22×C4×C53—41no—
97C22×C4×C53C2×D411yescharacteristic
98C2×C22⋊D53—41no—
99C22×Dic53D811yescharacteristic
100C22×Dic53D811yescharacteristic
101#11719945—24no—
102#11719945—42no—
103C2×D212—24no—
104C2×D212—22no—
105C2×D212—21no—
106C2×D212—41no—
107C2×D212—42no—
108C2×C4×D53—41no—
109C2×C4×D53—41no—
110C2×C4×D53—41no—
111C2×C4×D53—42no—
112C2×C4×D53—42no—
113C2×C4×D53—84no—
114C2×Dic106—21no—
115C2×C8×C53C22×C412yes—
116C2×C8×C53C2×D411yescharacteristic
117C2×C8×C53#4112yes—
118C2×C8×C53C2411yescharacteristic
119C4⋊C4×C53—21no—
120C4⋊C4×C53—21no—
121C4⋊C4×C53—22no—
122C4⋊C4×C53—41no—
123C22⋊C4×C53—41no—
124C42×C53—41no—
125#11719921—42no—
126#11719921—42no—
127#11719921—41no—
128#11719921—41no—
129#11719919—41no—
130#11719919—41no—
131C4×Dic53—41no—
132C4×Dic53—41no—
133C4×Dic53D811yescharacteristic
134C4×Dic53D811yescharacteristic
135C4×Dic53C2×D411yescharacteristic
136C4×Dic53C2×D411yescharacteristic
137C2×#88322—22no—
138C2×#88322C2×D411yescharacteristic
139C2×#88322C2×D411yescharacteristic
140Dic212—21no—
141Dic212C22×C412yescomplemented
142D424—24no—
143D424—22no—
144D424—21no—
145D424C22×C414yes—
146D424C22×C412yescomplemented
147#11719913—24no—
148C16×C53C22×C414yescomplemented
149#11719908—22no—
150C23×C53C2×D811yescharacteristic
151C22×D53—41no—
152C22×D53—42no—
153C22×D53—44no—
154C22×D53—82no—
155C22×D53—82no—
156C2×C4×C53—42no—
157C2×C4×C53—41no—
158C2×C4×C53—41no—
159C2×C4×C53—82no—
160C2×C4×C53C4×D412yes—
161C2×C4×C53C2×D811yescharacteristic
162C2×C4×C53#13512yes—
163C2×C4×C53C22×D411yescharacteristic
164C22⋊D53—44no—
165C2×Dic53—22no—
166C2×Dic53—41no—
167C2×Dic53—41no—
168C2×Dic53C2×D811yescharacteristic
169C2×Dic53C2×D811yescharacteristic
170C2×Dic53C2×D811yescharacteristic
171C2×Dic53C2×D811yescharacteristic
172D212—24no—
173D212—22no—
174D212—42no—
175D212—41no—
176D212—44no—
177C4×D53—44no—
178C4×D53—82no—
179C4×D53—84no—
180C4×D53—82no—
181C4×D53—84no—
182Dic106—22no—
183Dic106—41no—
184C8×C53C4×D412yes—
185C8×C53C23×C411yescharacteristic
186C8×C53C2×#4111yescharacteristic
187#88322—21no—
188#88322—21no—
189C4×#509—531nomaximal, Sylow
190C22×C53—21no—
191C22×C53—21no—
192C22×C53C4×D812yes—
193C22×C53C22×D811yescharacteristic
194C2×D53—42no—
195C2×D53—44no—
196C2×D53—88no—
197C2×D53—82no—
198C2×D53—81no—
199C2×D53—81no—
200C2×D53—82no—
201C4×C53—82no—
202C4×C53—81no—
203C4×C53—81no—
204C4×C53C4×D812yes—
205C4×C53C2×C4×D411yescharacteristic
206C4×C53C2×#13511yescharacteristic
207Dic53—21no—
208Dic53—21no—
209Dic53—21no—
210Dic53—42no—
211Dic53#50912yescomplemented
212C2×#509—531no—
213C4×#135—531no—
214C2×C4×D8—531no—
215#2177—532no—
216#2175—532no—
217C4×SD32—532no—
218C4×D16—532no—
219#2159D5311yescharacteristic, complemented
220#2140—531no—
221#2139—531no—
222C4×M32—531no—
223C2×C53—22no—
224C2×C53C2×C4×D811yescharacteristic
225C2×C53C2×#50911yescharacteristic
226C2×C53#341411yescharacteristic
227D53—82no—
228D53—84no—
229C2×#135—531no—
230C22×D8—531no—
231C4×#41—1061no—
232C2×C4×D4—1061no—
233#509—538no—
234C2×SD32—532no—
235C2×D16—532no—
236C2×M32—531no—
237C2×M32C2×D5311yescharacteristic
238C4×Q16—531no—
239C4×Q16—531no—
240C4×SD16—1061no—
241C4×SD16—1061no—
242C4×D8—532no—
243C4×D8—532no—
244C4×D8—531no—
245C4×D8—531no—
246C4×D8—1061no—
247C4×D8—1061no—
248#427C2×D5311yescharacteristic
249C2×#107C2×D5311yescharacteristic
250#416—1061no—
251C2×#102—1061no—
252C2×C4×C8—531no—
253#367C2×D5312yescomplemented
254#366C2×D5312yescomplemented
255#358—532no—
256#357—532no—
257#357—532no—
258#357—532no—
259#346—532no—
260C4×C16—532no—
261C53C4×#50911yescharacteristic, complemented, Sylow
262C2×#41—1061no—
263C22×D4—1061no—
264C23×C4—2121no—
265#135—534no—
266#135—1062no—
267C2×Q16—531no—
268C2×SD16—1061no—
269C2×D8—534no—
270C2×D8—532no—
271C2×D8—531no—
272C2×D8—1061no—
273C2×D8—1062no—
274C22×C8—531no—
275C22×C8C22×D5311yescharacteristic
276C4×Q8—1061no—
277C4×Q8—1061no—
278C4×D4—1062no—
279C4×D4—1062no—
280C4×D4—1061no—
281C4×D4—1061no—
282C4×D4—2121no—
283C4×D4—2121no—
284C4×D4—2121no—
285C4×D4—2121no—
286#117—21no—
287#117—2121no—
288C2×C4⋊C4—21no—
289C2×C22⋊C4—2121no—
290C2×C42—531no—
291C2×C42—2121no—
292SD32—538no—
293D16—538no—
294M32—1062no—
295M32C4×D5314yescomplemented
296C2×C16—532no—
297C2×C16C22×D5312yes—
298#107—21no—
299#107C22×D5311yescharacteristic
300#107C22×D5311yescharacteristic
301#107C22×D5312yes—
302#106—21no—
303#103—1061no—
304#103—1061no—
305#102—1062no—
306#102—1062no—
307#102—1061no—
308#102—1061no—
309C4×C8—531no—
310C4×C8—531no—
311C4×C8—531no—
312C4×C8—531no—
313C24—2121no—
314#41—1064no—
315#41—2122no—
316C2×Q8—1061no—
317C2×D4—1064no—
318C2×D4—1062no—
319C2×D4—1061no—
320C2×D4—2121no—
321C2×D4—2121no—
322C2×D4—2122no—
323C22×C4—41no—
324C22×C4—531no—
325C22×C4—531no—
326C22×C4—4244no—
327C22×C4—2121no—
328C22×C4—2121no—
329C22×C4—2122no—
330C22×C4—2122no—
331C22×C4—2121no—
332C22×C4D4×D5311yescharacteristic
333Q16—532no—
334Q16—1061no—
335SD16—1064no—
336D8—532no—
337D8—534no—
338D8—1064no—
339D8—1062no—
340D8—1061no—
341C2×C8—531no—
342C2×C8—531no—
343C2×C8—1062no—
344C2×C8C2×C4×D5312yes—
345C2×C8D4×D5311yescharacteristic
346C2×C8#1171994912yes—
347C2×C8C23×D5311yescharacteristic
348C4⋊C4—22no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—41no—
352C4⋊C4—2121no—
353C4⋊C4—2121no—
354C22⋊C4—41no—
355C22⋊C4—2122no—
356C22⋊C4—2122no—
357C22⋊C4—2121no—
358C22⋊C4—2121no—
359C42—41no—
360C42—531no—
361C42—531no—
362C42—531no—
363C42—531no—
364C42—2121no—
365C42—2121no—
366C16—1062no—
367C16C2×C4×D5314yescomplemented
368C23—4242no—
369C23—4242no—
370C23—2121no—
371C23—2122no—
372C23—2124no—
373C23D8×D5311yescharacteristic
374Q8—1062no—
375Q8—2121no—
376D4—1064no—
377D4—1062no—
378D4—2124no—
379D4—2122no—
380D4—2121no—
381D4—2124no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—42no—
385C2×C4—82no—
386C2×C4—531no—
387C2×C4—531no—
388C2×C4—531no—
389C2×C4—531no—
390C2×C4—1062no—
391C2×C4—4244no—
392C2×C4—4242no—
393C2×C4—4244no—
394C2×C4—4242no—
395C2×C4—2121no—
396C2×C4—2121no—
397C2×C4—2124no—
398C2×C4#1313317912yes—
399C2×C4D8×D5311yescharacteristic
400C2×C4#1313324012yes—
401C2×C4C2×D4×D5311yescharacteristic
402C8—1061no—
403C8—1061no—
404C8#1313317912yes—
405C8C22×C4×D5311yescharacteristic
406C8C2×#1171994911yescharacteristic
407C22—21no—
408C22—21no—
409C22—4248no—
410C22—4242no—
411C22—4241no—
412C22—4241no—
413C22—4242no—
414C22—2122no—
415C22—2124no—
416C22#1681070512yes—
417C22C2×D8×D5311yescharacteristic
418C4—82no—
419C4—81no—
420C4—81no—
421C4—532no—
422C4—1061no—
423C4—1061no—
424C4—1061no—
425C4—2122no—
426C4#1681070512yes—
427C4C2×#1313317911yescharacteristic
428C4C2×#1313324011yescharacteristic
429C2—22no—
430C2—4244no—
431C2—4242no—
432C2C2×#1681070511yescharacteristic
433C2#509×D5311yescharacteristic
434C2#2028785011yescharacteristic
subgroupG/H
centerC22C2×D8×D53
commutatorC8×C53C23×C4
FrattiniC2×C8C23×D53
FittingC53×#2159C2
socleC22×C53C22×D8
prime radicalC2×#16811061C2
cosocleC2×C8×C53C24
hypercenter#2159D53
hypocenterC53C4×#509
LCSG ▷ C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2159

Complements

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

NH ≅ G/Ntype
2. #20283486C2semidirect
3. C2×#16810705C2semidirect
4. #20276590C2semidirect
5. #20276578C2semidirect
6. #20276566C2semidirect
7. #20276561C2semidirect
8. #20276430C2semidirect
9. #20276425C2semidirect
10. M32×Dic53C2semidirect
11. C53×#2159C2semidirect
16. #16811061C4semidirect
20. #16810717C22semidirect
21. #16810717C22semidirect
28. #16810705C22semidirect
29. #16810705C22semidirect
33. #16810426C22semidirect
36. #16810423C22semidirect
37. #16810398C22semidirect
38. C16×Dic53C22semidirect
42. C53×#367C22semidirect
43. C53×#366C22semidirect
48. #13133239C2×C4semidirect
53. C2×D424C2×C4semidirect
71. #13133162C2×C4semidirect
72. D848C2×C4semidirect
88. M32×C53C2×C4semidirect
141. Dic212C22×C4semidirect
146. D424C22×C4semidirect
148. C16×C53C22×C4semidirect
211. Dic53#509semidirect
219. #2159D53semidirect
253. #367C2×D53semidirect
254. #366C2×D53semidirect
261. C53C4×#509semidirect
295. M32C4×D53semidirect
367. C16C2×C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168110612
#202834862
C2×#168107052
#202765902
#202765782
#202765662
#202765612
#202764302
#202764252
M32×Dic532
C53×#21592
C4×#50953