Finite Group Databasebeta

← #27369322

Subgroups — #27369322

Order 13568.

696 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285840C211yesmaximal, characteristic, complemented
2C2×#16810914C211yesmaximal, characteristic, complemented
3#20284175C211yesmaximal, characteristic
4#20283738C211yesmaximal, characteristic, complemented
5#20274123C214yesmaximal, complemented
6#20274089C214yesmaximal, complemented
7#20273485C211yesmaximal, characteristic, complemented
8#20273395C211yesmaximal, characteristic, complemented
9C53×#1540C211yesmaximal, characteristic, complemented
10C2×#13133242—21no—
11C2×#13133217C2211yescharacteristic
12C2×C4×D212C2211yescharacteristic
13#16811019C2212yescomplemented
14#16810969C2212yescomplemented
15#16810939—22no—
16#16810914—22no—
17#16810914C2214yes—
18#16810913—22no—
19#16810835—22no—
20#16810825C2212yes—
21#16810815C2212yes—
22C8×D212—22no—
23#16810688C2214yes—
24#16810676C2211yescharacteristic
25C2×#13133149—21no—
26#16810647C2211yescharacteristic
27#16810624—21no—
28C2×#13133139—21no—
29C22⋊C8×D53—22no—
30#16810373C2214yescomplemented
31#16810371C2214yescomplemented
32C53×#532C2211yescharacteristic
33C53×#431C2211yescharacteristic
34C2×C4⋊C8×C53C2211yescharacteristic
35C53×#332C2212yescomplemented
36C53×#331C2212yescomplemented
37C22×Dic106—21no—
38C22×D212D411yescharacteristic
39C22×C4×D53—21no—
40#13133242—44no—
41C2×#11719924—22no—
42C2×#11719922—22no—
43C2×#11719917—21no—
44C2×C8×D53—24no—
45#13133217—21no—
46#13133217—21no—
47#13133217—21no—
48#13133217C2312yes—
49C4×D212—21no—
50C4×D212—21no—
51C4×D212D412yes—
52C4×D212C2312yes—
53C2×C4⋊Dic53—21no—
54C2×C4⋊Dic53—21no—
55C2×C4⋊Dic53D411yescharacteristic
56C2×#11719921—21no—
57#13133149—24no—
58#13133148—24no—
59C4⋊#88322—21no—
60C4⋊#88322D412yes—
61C4⋊#88322C2312yes—
62#13133141—22no—
63#13133139—24no—
64#13133138—22no—
65#13133138—24no—
66C22⋊#88322D412yes—
67#13133129—24no—
68C4×#88322C2312yes—
69C2×#41×C53—21no—
70C22×C8×C53—21no—
71C4⋊Q8×C53C2311yescharacteristic
72#127×C53C2311yescharacteristic
73#124×C53—21no—
74#121×C53—21no—
75#121×C53—21no—
76C4×D4×C53—21no—
77C4×D4×C53—21no—
78C2×C42×C53C2311yescharacteristic
79C4⋊C8×C53—21no—
80C4⋊C8×C53C2312yes—
81C4⋊C8×C53C2312yes—
82C22⋊C8×C53—21no—
83C4×C8×C53—21no—
84C23×D53—21no—
85#41×C53—42no—
86#41×C53—42no—
87C2×Q8×C53—21no—
88C2×D4×C53—21no—
89C2×D4×C53—21no—
90C2×D4×C53—21no—
91C2×D4×C53—41no—
92C22×C4×C53—41no—
93C22×C4×C53C2×D411yescharacteristic
94C22×C4×C53C2×D411yescharacteristic
95C22×C4×C53#4111yescharacteristic
96C22×Dic53—21no—
97C22×Dic53—41no—
98C2×D212—21no—
99C2×D212—22no—
100C2×D212SD1614yes—
101C2×D212C2×D412yes—
102C2×C4×D53—22no—
103C2×C4×D53—22no—
104C2×C4×D53—82no—
105C2×Dic106—22no—
106C2×Dic106—41no—
107C2×Dic106—82no—
108C2×C8×C53—22no—
109C2×C8×C53—22no—
110C2×C8×C53—42no—
111C4⋊C4×C53—21no—
112C4⋊C4×C53—21no—
113C22⋊C4×C53—41no—
114C22⋊C4×C53—41no—
115C42×C53C2×D411yescharacteristic
116C42×C53C2×D411yescharacteristic
117C42×C53#4111yescharacteristic
118C42×C53C2411yescharacteristic
119#11719924—44no—
120#11719922—44no—
121#11719921—22no—
122#11719921—22no—
123C4⋊Dic53—21no—
124C4⋊Dic53—22no—
125C4⋊Dic53—22no—
126C4⋊Dic53—41no—
127C4⋊Dic53—41no—
128C4⋊Dic53C2×D412yes—
129#11719917—42no—
130C2×#88322—22no—
131C2×#88322C2×D414yes—
132C8×D53—84no—
133C23×C53—41no—
134C23×C53#12111yescharacteristic
135C22×D53—22no—
136C22×D53—24no—
137C22×D53—81no—
138C22×D53—81no—
139Q8×C53—41no—
140Q8×C53—41no—
141D4×C53—41no—
142D4×C53—41no—
143D4×C53—42no—
144D4×C53—81no—
145D4×C53—82no—
146C2×C4×C53—22no—
147C2×C4×C53—41no—
148C2×C4×C53—41no—
149C2×C4×C53—82no—
150C2×C4×C53#12111yescharacteristic
151C2×C4×C53#12111yescharacteristic
152C2×C4×C53#12111yescharacteristic
153C2×C4×C53C2×SD1612yes—
154C2×C4×C53#13512yes—
155C2×C4×C53C22×D411yescharacteristic
156C2×C4×C53C22×D411yescharacteristic
157C2×C4×C53C2×#4111yescharacteristic
158C2×Dic53—22no—
159C2×Dic53—42no—
160C2×Dic53—81no—
161C2×Dic53—82no—
162D212—22no—
163D212—24no—
164D212C2×SD1614yes—
165C4×D53—44no—
166C4×D53—82no—
167Dic106—82no—
168Dic106—82no—
169C8×C53—42no—
170C8×C53—42no—
171#88322—24no—
172#88322—42no—
173#19391—531nomaximal, Sylow
174C22×C53—21no—
175C22×C53—21no—
176C22×C53—81no—
177C22×C53—81no—
178C22×C53—81no—
179C22×C53#46512yes—
180C22×C53C2×#12111yescharacteristic
181C2×D53—24no—
182C2×D53—42no—
183C2×D53—42no—
184C2×D53—84no—
185C4×C53—21no—
186C4×C53—21no—
187C4×C53—21no—
188C4×C53—81no—
189C4×C53#46512yes—
190C4×C53C22×SD1611yescharacteristic
191C4×C53C2×#13511yescharacteristic
192C4×C53#57712yes—
193Dic53—42no—
194Dic53—82no—
195#3227—531no—
196C2×#476—531no—
197#3036—531no—
198#2972—531no—
199#1677—534no—
200#1671—534no—
201#1570—531no—
202#1548—531no—
203#1540D5311yescharacteristic, complemented
204C2×C53—22no—
205C2×C53—161no—
206C2×C53#304211yescharacteristic
207C2×C53C2×#46511yescharacteristic
208C2×C53#306811yescharacteristic
209D53—44no—
210C2×AGL(1,ℤ/8)—1061no—
211#532C2×D5311yescharacteristic
212C2×C4⋊Q8—531no—
213C2×C4×D4—531no—
214#498—532no—
215#486—532no—
216#481—1062no—
217#476—534no—
218#476—1062no—
219#476—1062no—
220#463—1062no—
221#461—532no—
222#459—532no—
223C8×D4—534no—
224C8×D4—1062no—
225#431—531no—
226#431C2×D5311yescharacteristic
227C2×#106—1061no—
228#423—531no—
229C2×C4⋊C8C2×D5311yescharacteristic
230#417—1061no—
231C2×#102—1061no—
232C2×C22⋊C8—1062no—
233#334—534no—
234#333—534no—
235#332C2×D5312yescomplemented
236#331C2×D5312yescomplemented
237C53#1939111yescharacteristic, complemented, Sylow
238C2×#41—21no—
239C22×Q8—1061no—
240C22×D4—531no—
241C23×C4—1061no—
242AGL(1,ℤ/8)—2124no—
243C2×SD16—1062no—
244C2×D8—1062no—
245C2×M16—1061no—
246C22×C8—21no—
247C22×C8—1064no—
248C4⋊Q8—532no—
249C4⋊Q8—1061no—
250C4⋊Q8—1061no—
251C4⋊Q8—1061no—
252C4⋊Q8C22×D5311yescharacteristic
253#127C22×D5311yescharacteristic
254#124—21no—
255#121—21no—
256#121—21no—
257C4×D4—21no—
258C4×D4—21no—
259C4×D4—532no—
260C4×D4—532no—
261C4×D4—1061no—
262C4×D4—1061no—
263C2×C4⋊C4—531no—
264C2×C4⋊C4—1061no—
265C2×C4⋊C4—1061no—
266C2×C22⋊C4—1061no—
267C2×C42C22×D5311yescharacteristic
268#106—1064no—
269#106—1064no—
270C4⋊C8—21no—
271C4⋊C8—532no—
272C4⋊C8—532no—
273C4⋊C8—1061no—
274C4⋊C8C22×D5312yes—
275C4⋊C8C22×D5312yes—
276#103—1062no—
277#102—1062no—
278#102—1064no—
279#102—1064no—
280C22⋊C8—21no—
281C22⋊C8—532no—
282C22⋊C8—1064no—
283C4×C8—21no—
284C4×C8—532no—
285C24—1061no—
286#41—42no—
287#41—42no—
288C2×Q8—21no—
289C2×Q8—1062no—
290C2×Q8—4242no—
291C2×Q8—2121no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—41no—
296C2×D4—534no—
297C2×D4—532no—
298C2×D4—1061no—
299C2×D4—1062no—
300C22×C4—41no—
301C22×C4—1061no—
302C22×C4—1062no—
303C22×C4—1062no—
304C22×C4—4242no—
305C22×C4—2121no—
306C22×C4#1171994511yescharacteristic
307C22×C4D4×D5311yescharacteristic
308C22×C4D4×D5311yescharacteristic
309SD16—2124no—
310D8—2124no—
311M16—2122no—
312C2×C8—22no—
313C2×C8—22no—
314C2×C8—42no—
315C2×C8—534no—
316C2×C8—1062no—
317C2×C8—4244no—
318C4⋊C4—21no—
319C4⋊C4—21no—
320C4⋊C4—532no—
321C4⋊C4—1061no—
322C4⋊C4—1062no—
323C4⋊C4—1062no—
324C4⋊C4—2121no—
325C4⋊C4—2121no—
326C22⋊C4—41no—
327C22⋊C4—41no—
328C22⋊C4—1062no—
329C22⋊C4—1062no—
330C42#1171994511yescharacteristic
331C42D4×D5311yescharacteristic
332C42D4×D5311yescharacteristic
333C42C23×D5311yescharacteristic
334C23—41no—
335C23—1064no—
336C23—1062no—
337C23—4241no—
338C23—4241no—
339C23#1313319011yescharacteristic
340Q8—41no—
341Q8—41no—
342Q8—4242no—
343Q8—4242no—
344D4—41no—
345D4—41no—
346D4—42no—
347D4—81no—
348D4—82no—
349D4—534no—
350D4—1064no—
351D4—1062no—
352C2×C4—22no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—82no—
356C2×C4—1062no—
357C2×C4—4242no—
358C2×C4—4242no—
359C2×C4—4241no—
360C2×C4—2122no—
361C2×C4—2124no—
362C2×C4#1313319011yescharacteristic
363C2×C4#1313319011yescharacteristic
364C2×C4#1313319011yescharacteristic
365C2×C4SD16×D5312yes—
366C2×C4#1313323612yes—
367C2×C4C2×D4×D5311yescharacteristic
368C2×C4C2×D4×D5311yescharacteristic
369C2×C4C2×#1171994511yescharacteristic
370C8—42no—
371C8—42no—
372C8—1064no—
373C8—2122no—
374C22—21no—
375C22—21no—
376C22—81no—
377C22—81no—
378C22—81no—
379C22—1064no—
380C22—4244no—
381C22—2122no—
382C22—2122no—
383C22#1681085112yes—
384C22C2×#1313319011yescharacteristic
385C4—21no—
386C4—21no—
387C4—21no—
388C4—81no—
389C4—4242no—
390C4—2122no—
391C4#1681085112yes—
392C4C2×SD16×D5311yescharacteristic
393C4C2×#1313323611yescharacteristic
394C4#1681143012yes—
395C2—22no—
396C2—161no—
397C2—2124no—
398C2#2028423011yescharacteristic
399C2C2×#1681085111yescharacteristic
400C2#2028444611yescharacteristic
subgroupG/H
centerC22C2×#13133190
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#1540C2
socleC22×C53C2×#121
prime radical#20284175C2
hypercenter#1540D53
hypocenterC53#19391
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1540

Complements

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

NH ≅ G/Ntype
1. #20285840C2semidirect
2. C2×#16810914C2semidirect
4. #20283738C2semidirect
5. #20274123C2semidirect
6. #20274089C2semidirect
7. #20273485C2semidirect
8. #20273395C2semidirect
9. C53×#1540C2semidirect
13. #16811019C22semidirect
14. #16810969C22semidirect
30. #16810373C22semidirect
31. #16810371C22semidirect
35. C53×#332C22semidirect
36. C53×#331C22semidirect
203. #1540D53semidirect
235. #332C2×D53semidirect
236. #331C2×D53semidirect
237. C53#19391semidirect

Maximal subgroups

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

subgroupindex
#202858402
C2×#168109142
#202841752
#202837382
#202741232
#202740892
#202734852
#202733952
C53×#15402
#1939153