Finite Group Databasebeta

← #27537501

Subgroups — #27537501

Order 13568.

652 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
1#20287849C211yesmaximal, characteristic
2C2×#16810740C211yesmaximal, characteristic, complemented
3C2×#16810739C211yesmaximal, characteristic, complemented
4#20276598C214yesmaximal, complemented
5#20276593C214yesmaximal, complemented
6#20276509C211yesmaximal, characteristic, complemented
7#20276444C211yesmaximal, characteristic, complemented
8M32×Dic53C211yesmaximal, characteristic, complemented
9C53×#2143C211yesmaximal, characteristic, complemented
10C2×#13133239C2211yescharacteristic
11C2×#13133225—21no—
12C2×#13133224—21no—
13#16811064C412yescomplemented
14#16811063C412yescomplemented
15#16811059C412yescomplemented
16#16811058C412yescomplemented
17#16811053C2214yes—
18C2×#13133157C2211yescharacteristic
19#16810740—21no—
20#16810740—21no—
21#16810740C2212yescomplemented
22#16810740C2212yescomplemented
23#16810739—21no—
24#16810739—21no—
25#16810739C2212yescomplemented
26#16810739C2212yescomplemented
27C2×#13133153C2211yescharacteristic
28C2×#13133144—21no—
29C2×#13133143—21no—
30C2×C8×Dic53C2211yescharacteristic
31#16810452C2214yescomplemented
32#16810431C2214yescomplemented
33#16810398C2212yescomplemented
34C16×Dic53C2212yescomplemented
35C2×#135×C53C2211yescharacteristic
36C2×M32×C53C2211yescharacteristic
37C2×#108×C53C2211yescharacteristic
38C53×#359C2214yescomplemented
39C2×#11719945—21no—
40#13133239—21no—
41#13133239—21no—
42#13133239C2×C412yes—
43#13133239C2312yes—
44C2×Dic212C2×C411yescharacteristic
45C2×#11719913—21no—
46C2×D424C2×C411yescharacteristic
47#13133225—22no—
48#13133225—22no—
49#13133225—41no—
50#13133225—41no—
51#13133224—22no—
52#13133224—22no—
53#13133224—41no—
54#13133224—41no—
55C2×#11719917—21no—
56C2×#11719917—41no—
57C2×#11719912—41no—
58C22×#88322—41no—
59C22×#88322D411yescharacteristic
60C2×C8×D53—41no—
61C2×C4×Dic53D411yescharacteristic
62#13133163C2×C414yescomplemented
63#13133161C2×C414yescomplemented
64#13133160C2×C414yescomplemented
65#13133158C2×C414yescomplemented
66#13133157C2×C412yes—
67#13133157C2×C412yes—
68C2×#11719908C2312yes—
69#13133153—21no—
70#13133153C2312yes—
71#13133144—22no—
72#13133144—22no—
73#13133143—22no—
74#13133143—22no—
75C8×Dic53D411yescharacteristic
76C8×Dic53D411yescharacteristic
77C8×Dic53C2311yescharacteristic
78C8×Dic53C2311yescharacteristic
79C2×#41×C53—21no—
80#135×C53—21no—
81#135×C53—21no—
82#135×C53C2×C412yes—
83#135×C53C2312yes—
84C2×Q16×C53C2×C411yescharacteristic
85C2×SD16×C53—21no—
86C2×D8×C53C2×C411yescharacteristic
87C2×M16×C53—21no—
88C22×C8×C53C2311yescharacteristic
89M32×C53—21no—
90M32×C53—21no—
91C2×C16×C53C2312yes—
92#108×C53—21no—
93#108×C53C2312yes—
94#41×C53—22no—
95#41×C53—22no—
96#41×C53—41no—
97#41×C53—41no—
98C2×Q8×C53—21no—
99C2×D4×C53—21no—
100C2×D4×C53—41no—
101C22×C4×C53—41no—
102C22×C4×C53C2×D411yescharacteristic
103C2×C22⋊D53—41no—
104C22×Dic53—21no—
105#11719945—22no—
106#11719945—22no—
107#11719945—41no—
108#11719945—41no—
109C2×D212—21no—
110C2×C4×D53—41no—
111C2×Dic106—21no—
112Q16×C53—21no—
113Q16×C53C22×C412yes—
114SD16×C53—22no—
115SD16×C53—22no—
116D8×C53—21no—
117D8×C53C22×C412yes—
118M16×C53—22no—
119M16×C53—41no—
120C2×C8×C53—41no—
121C2×C8×C53C22×C411yescharacteristic
122C2×C8×C53C22×C411yescharacteristic
123C2×C8×C53C2×D411yescharacteristic
124C2×C8×C53#4111yescharacteristic
125C2×C8×C53#4111yescharacteristic
126C2×C8×C53C2411yescharacteristic
127C4×Dic53—21no—
128C4×Dic53C2×D411yescharacteristic
129C4×Dic53C2×D411yescharacteristic
130#11719917—22no—
131#11719917—41no—
132#11719917—42no—
133#11719917—42no—
134C2×#88322—41no—
135C2×#88322—42no—
136C2×#88322—42no—
137C2×#88322D811yescharacteristic
138C2×#88322D811yescharacteristic
139C2×#88322D811yescharacteristic
140C2×#88322D811yescharacteristic
141C2×#88322C2×D411yescharacteristic
142C2×#88322C2×D411yescharacteristic
143Dic212—21no—
144Dic212C22×C412yes—
145D424—21no—
146D424C22×C412yes—
147#11719913—22no—
148#11719913—22no—
149#11719912—42no—
150#11719912—42no—
151C8×D53—42no—
152C8×D53—42no—
153C16×C53—22no—
154#11719908C22×C414yes—
155C23×C53—41no—
156C23×C53#13511yescharacteristic
157C22×D53—41no—
158Q8×C53—22no—
159Q8×C53—41no—
160D4×C53—22no—
161D4×C53—42no—
162D4×C53—42no—
163D4×C53—41no—
164C2×C4×C53—41no—
165C2×C4×C53—42no—
166C2×C4×C53—42no—
167C2×C4×C53C4×D411yescharacteristic
168C2×C4×C53C4×D411yescharacteristic
169C2×C4×C53C2×D811yescharacteristic
170C2×C4×C53C2×D811yescharacteristic
171C2×C4×C53#13511yescharacteristic
172C2×C4×C53C22×D411yescharacteristic
173C22⋊D53—42no—
174C22⋊D53—42no—
175C2×Dic53—21no—
176C2×Dic53—21no—
177C2×Dic53—42no—
178C2×Dic53—41no—
179D212—22no—
180D212—41no—
181C4×D53—42no—
182C4×D53—42no—
183Dic106—22no—
184Dic106—41no—
185C8×C53—42no—
186C8×C53C4×D411yescharacteristic
187C8×C53C4×D411yescharacteristic
188C8×C53C23×C411yescharacteristic
189C8×C53C2×#4111yescharacteristic
190#88322—42no—
191#88322C2×D811yescharacteristic
192#88322C2×D811yescharacteristic
193#88322C2×D811yescharacteristic
194#88322C2×D811yescharacteristic
195#33463—531nomaximal, Sylow
196C22×C53—41no—
197C22×C53—42no—
198C22×C53—82no—
199C22×C53C4×D811yescharacteristic
200C22×C53C4×D811yescharacteristic
201C22×C53C2×#13511yescharacteristic
202C2×D53—42no—
203C2×D53—82no—
204C4×C53—42no—
205C4×C53C4×D811yescharacteristic
206C4×C53C4×D811yescharacteristic
207C4×C53C2×C4×D411yescharacteristic
208C4×C53C22×D811yescharacteristic
209Dic53—22no—
210Dic53—41no—
211Dic53—42no—
212#3414—531no—
213C2×#443—531no—
214C2×#443—531no—
215#2179—534no—
216#2178—534no—
217#2161—531no—
218#2143—531no—
219#2143D5311yescharacteristic, complemented
220C4×M32—531no—
221C2×C53—21no—
222C2×C53—21no—
223C2×C53—21no—
224C2×C53—82no—
225C2×C53C2×C4×D811yescharacteristic
226D53—82no—
227C2×#135—531no—
228C2×#135C2×D5311yescharacteristic
229C2×#131—1061no—
230C2×#131—1061no—
231#510—532no—
232#510—532no—
233#509—532no—
234#509—532no—
235#508—534no—
236C2×M32—531no—
237C2×M32C2×D5311yescharacteristic
238#443—532no—
239#443—532no—
240#443—532no—
241#443—532no—
242#443—1061no—
243#443—1061no—
244#443—1061no—
245#443—1061no—
246C2×#108—531no—
247C2×#108C2×D5311yescharacteristic
248C2×#104—1061no—
249C2×#104—1061no—
250C2×C4×C8—531no—
251#368—534no—
252#359—534no—
253#359C2×D5314yescomplemented
254#346—532no—
255C4×C16—532no—
256C53#3346311yescharacteristic, complemented, Sylow
257C2×#41—21no—
258C2×#41—1061no—
259#135—21no—
260#135—21no—
261#135—532no—
262#135—532no—
263#135—1061no—
264#135—1061no—
265#135C4×D5312yes—
266#135C22×D5312yes—
267C2×Q16—531no—
268C2×Q16C4×D5311yescharacteristic
269C2×SD16—21no—
270C2×SD16—1061no—
271C2×D8—531no—
272C2×D8C4×D5311yescharacteristic
273#131—1062no—
274#131—1062no—
275#131—1062no—
276#131—1062no—
277#131—2121no—
278#131—2121no—
279#131—2121no—
280#131—2121no—
281C2×M16—21no—
282C2×M16—1061no—
283C2×M16—2121no—
284C2×M16—2121no—
285C22×C8—531no—
286C22×C8—2121no—
287C22×C8—2121no—
288C22×C8C22×D5311yescharacteristic
289C2×C42—531no—
290Q32—534no—
291SD32—534no—
292SD32—534no—
293D16—534no—
294M32—21no—
295M32—21no—
296M32—532no—
297M32—532no—
298C2×C16—532no—
299C2×C16C22×D5312yes—
300#108—21no—
301#108—532no—
302#108—1061no—
303#108C22×D5312yes—
304#104—1062no—
305#104—1062no—
306#104—1062no—
307#104—1062no—
308C4×C8—531no—
309C4×C8—531no—
310C4×C8—531no—
311C4×C8—531no—
312#41—22no—
313#41—22no—
314#41—41no—
315#41—41no—
316#41—1062no—
317#41—1062no—
318#41—2121no—
319#41—2121no—
320C2×Q8—21no—
321C2×Q8—1061no—
322C2×D4—21no—
323C2×D4—41no—
324C2×D4—1061no—
325C2×D4—2121no—
326C22×C4—41no—
327C22×C4—1061no—
328C22×C4—2121no—
329C22×C4D4×D5311yescharacteristic
330Q16—21no—
331Q16—532no—
332Q16—1061no—
333Q16C2×C4×D5312yes—
334SD16—22no—
335SD16—22no—
336SD16—1062no—
337SD16—1062no—
338D8—21no—
339D8—532no—
340D8—1061no—
341D8C2×C4×D5312yes—
342M16—22no—
343M16—41no—
344M16—1062no—
345M16—2122no—
346M16—2122no—
347M16—2122no—
348M16—2122no—
349M16—2121no—
350C2×C8—41no—
351C2×C8—531no—
352C2×C8—531no—
353C2×C8—531no—
354C2×C8—531no—
355C2×C8—531no—
356C2×C8—531no—
357C2×C8—2122no—
358C2×C8—2122no—
359C2×C8—2122no—
360C2×C8—2122no—
361C2×C8—2121no—
362C2×C8C2×C4×D5311yescharacteristic
363C2×C8C2×C4×D5311yescharacteristic
364C2×C8D4×D5311yescharacteristic
365C2×C8#1171994711yescharacteristic
366C2×C8#1171994711yescharacteristic
367C2×C8C23×D5311yescharacteristic
368C42—531no—
369C42—531no—
370C42—1061no—
371C16—22no—
372C16—534no—
373C23—41no—
374C23—2121no—
375C23#1313323811yescharacteristic
376Q8—22no—
377Q8—41no—
378Q8—1062no—
379Q8—2121no—
380D4—22no—
381D4—42no—
382D4—42no—
383D4—41no—
384D4—1062no—
385D4—2122no—
386D4—2122no—
387D4—2121no—
388C2×C4—41no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—1061no—
392C2×C4—1061no—
393C2×C4—2122no—
394C2×C4—2121no—
395C2×C4—2122no—
396C2×C4—2122no—
397C2×C4#1313317711yescharacteristic
398C2×C4#1313317711yescharacteristic
399C2×C4D8×D5311yescharacteristic
400C2×C4D8×D5311yescharacteristic
401C2×C4#1313323811yescharacteristic
402C2×C4C2×D4×D5311yescharacteristic
403C8—42no—
404C8—531no—
405C8—531no—
406C8—531no—
407C8—531no—
408C8—2122no—
409C8#1313317711yescharacteristic
410C8#1313317711yescharacteristic
411C8C22×C4×D5311yescharacteristic
412C8C2×#1171994711yescharacteristic
413C22—41no—
414C22—42no—
415C22—82no—
416C22—4242no—
417C22—2122no—
418C22#1681070311yescharacteristic
419C22#1681070311yescharacteristic
420C22C2×#1313323811yescharacteristic
421C4—42no—
422C4—1062no—
423C4—2121no—
424C4—2122no—
425C4#1681070311yescharacteristic
426C4#1681070311yescharacteristic
427C4C2×#1313317711yescharacteristic
428C4C2×D8×D5311yescharacteristic
429C2—21no—
430C2—21no—
431C2—21no—
432C2—82no—
433C2—4242no—
434C2C2×#1681070311yescharacteristic
subgroupG/H
centerC4C2×D8×D53
commutatorC8×C53C23×C4
FrattiniC2×C8C23×D53
FittingC53×#2143C2
socleC2×C53C2×C4×D8
prime radical#20287849C2
cosocleC2×C8×C53C24
hypercenter#2143D53
hypocenterC53#33463
LCSG ▷ C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C22×C4 ◁ C22×C8 ◁ #2143

Complements

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

NH ≅ G/Ntype
2. C2×#16810740C2semidirect
3. C2×#16810739C2semidirect
4. #20276598C2semidirect
5. #20276593C2semidirect
6. #20276509C2semidirect
7. #20276444C2semidirect
8. M32×Dic53C2semidirect
9. C53×#2143C2semidirect
13. #16811064C4semidirect
14. #16811063C4semidirect
15. #16811059C4semidirect
16. #16811058C4semidirect
21. #16810740C22semidirect
22. #16810740C22semidirect
25. #16810739C22semidirect
26. #16810739C22semidirect
31. #16810452C22semidirect
32. #16810431C22semidirect
33. #16810398C22semidirect
34. C16×Dic53C22semidirect
38. C53×#359C22semidirect
62. #13133163C2×C4semidirect
63. #13133161C2×C4semidirect
64. #13133160C2×C4semidirect
65. #13133158C2×C4semidirect
219. #2143D53semidirect
253. #359C2×D53semidirect
256. C53#33463semidirect

Maximal subgroups

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

subgroupindex
#202878492
C2×#168107402
C2×#168107392
#202765982
#202765932
#202765092
#202764442
M32×Dic532
C53×#21432
#3346353