Finite Group Databasebeta

← #27369318

Subgroups — #27369318

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#20285843C211yesmaximal, characteristic, complemented
2C2×#16810910C211yesmaximal, characteristic, complemented
3#20284181C211yesmaximal, characteristic
4#20283738C211yesmaximal, characteristic, complemented
5#20274127C214yesmaximal, complemented
6#20274093C214yesmaximal, complemented
7#20273487C211yesmaximal, characteristic, complemented
8#20273362C211yesmaximal, characteristic, complemented
9C53×#1548C211yesmaximal, characteristic, complemented
10C2×#13133245—21no—
11#16811173C2211yescharacteristic
12C2×C4×D212C2211yescharacteristic
13#16811019C2212yescomplemented
14#16810971C2212yescomplemented
15#16810941—22no—
16#16810910—22no—
17#16810910—22no—
18#16810910C2214yes—
19#16810839—22no—
20#16810829C2212yes—
21#16810819C2212yes—
22C8×D212—22no—
23#16810688C2214yes—
24#16810676C2211yescharacteristic
25C2×#13133148—21no—
26C2×C4⋊#88322C2211yescharacteristic
27#16810625—21no—
28C2×#13133137—21no—
29C22⋊C8×D53—22no—
30#16810373C2214yescomplemented
31#16810368C2212yescomplemented
32#16810366C2212yescomplemented
33C2×C4⋊Q8×C53C2211yescharacteristic
34C53×#431C2211yescharacteristic
35C53×#423C2211yescharacteristic
36C53×#333C2214yescomplemented
37C2×#11719945—21no—
38C22×D212D411yescharacteristic
39C22×C4×D53—21no—
40#13133245—44no—
41C2×#11719913—22no—
42C2×D424—22no—
43C22×#88322—21no—
44C2×C8×D53—24no—
45#13133217C2311yescharacteristic
46#13133214C2311yescharacteristic
47#13133205—21no—
48#13133191—21no—
49#13133191—21no—
50C4×D212—21no—
51C4×D212—21no—
52C4×D212—21no—
53C4×D212—21no—
54C4×D212D412yes—
55C4×D212C2312yes—
56C2×C4⋊Dic53D411yescharacteristic
57C2×#11719921—21no—
58#13133148—24no—
59#13133148—24no—
60C4⋊#88322—21no—
61C4⋊#88322D412yes—
62C4⋊#88322C2312yes—
63#13133142—22no—
64#13133139—22no—
65#13133137—24no—
66#13133137—24no—
67C22⋊#88322D412yes—
68#13133129—24no—
69C4×#88322C2312yes—
70C22×Q8×C53—21no—
71C2×M16×C53—21no—
72C4⋊Q8×C53—21no—
73C4⋊Q8×C53—21no—
74C4⋊Q8×C53—21no—
75C4⋊Q8×C53C2312yes—
76C2×C4⋊C4×C53—21no—
77C2×C4⋊C4×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—
85C2×Q8×C53—22no—
86C2×Q8×C53—41no—
87C2×Q8×C53—82no—
88C22×C4×C53—41no—
89C22×C4×C53C2×D411yescharacteristic
90C22×C4×C53C2×D411yescharacteristic
91C22×C4×C53#4111yescharacteristic
92C2×C22⋊D53—41no—
93C22×Dic53—21no—
94#11719945—42no—
95#11719945—42no—
96C2×D212—21no—
97C2×D212—22no—
98C2×D212—21no—
99C2×D212—21no—
100C2×D212—21no—
101C2×D212SD1614yes—
102C2×D212C2×D412yes—
103C2×C4×D53—22no—
104C2×C4×D53—22no—
105C2×C4×D53—41no—
106C2×C4×D53—82no—
107C2×Dic106—21no—
108M16×C53—42no—
109C2×C8×C53—22no—
110C2×C8×C53—22no—
111C4⋊C4×C53—22no—
112C4⋊C4×C53—22no—
113C4⋊C4×C53—41no—
114C4⋊C4×C53—41no—
115C42×C53C2×D411yescharacteristic
116C42×C53C2×D411yescharacteristic
117C42×C53#4111yescharacteristic
118C42×C53C2411yescharacteristic
119#11719921—22no—
120#11719921—22no—
121#11719921—41no—
122#11719921—41no—
123C4⋊Dic53—21no—
124C4⋊Dic53—21no—
125C4⋊Dic53—21no—
126C4⋊Dic53C2×D412yes—
127C2×#88322—22no—
128C2×#88322—42no—
129C2×#88322C2×D414yes—
130D424—44no—
131#11719913—44no—
132C8×D53—84no—
133C23×C53#12111yescharacteristic
134C22×D53—22no—
135C22×D53—24no—
136C22×D53—41no—
137C22×D53—81no—
138C22×D53—81no—
139Q8×C53—82no—
140Q8×C53—82no—
141C2×C4×C53—22no—
142C2×C4×C53—41no—
143C2×C4×C53—42no—
144C2×C4×C53—82no—
145C2×C4×C53#12111yescharacteristic
146C2×C4×C53#12111yescharacteristic
147C2×C4×C53#12111yescharacteristic
148C2×C4×C53C2×SD1612yes—
149C2×C4×C53#13512yes—
150C2×C4×C53C22×D411yescharacteristic
151C2×C4×C53C22×D411yescharacteristic
152C2×C4×C53C2×#4111yescharacteristic
153C22⋊D53—82no—
154C2×Dic53—22no—
155C2×Dic53—41no—
156C2×Dic53—81no—
157D212—22no—
158D212—24no—
159D212—41no—
160D212—41no—
161D212—42no—
162D212—81no—
163D212C2×SD1614yes—
164C4×D53—44no—
165C4×D53—82no—
166C4×D53—82no—
167Dic106—41no—
168Dic106—41no—
169C8×C53—42no—
170C8×C53—42no—
171#88322—24no—
172#88322—42no—
173#19391—531nomaximal, Sylow
174C22×C53—21no—
175C22×C53—21no—
176C22×C53#46512yes—
177C22×C53C2×#12111yescharacteristic
178C2×D53—24no—
179C2×D53—42no—
180C2×D53—42no—
181C2×D53—84no—
182C2×D53—81no—
183C2×D53—81no—
184C2×D53—81no—
185C4×C53—21no—
186C4×C53—21no—
187C4×C53—21no—
188C4×C53—82no—
189C4×C53#46512yes—
190C4×C53C22×SD1611yescharacteristic
191C4×C53C2×#13511yescharacteristic
192C4×C53#57712yes—
193Dic53—42no—
194Dic53—81no—
195#3227—531no—
196C2×#476—531no—
197#3036—531no—
198#2972—531no—
199#1677—534no—
200#1671—534no—
201#1570—531no—
202#1548D5311yescharacteristic, complemented
203#1540—531no—
204C2×C53—22no—
205C2×C53#304211yescharacteristic
206C2×C53C2×#46511yescharacteristic
207C2×C53#306811yescharacteristic
208D53—44no—
209D53—161no—
210C2×AGL(1,ℤ/8)—1061no—
211#532—531no—
212C2×C4⋊Q8C2×D5311yescharacteristic
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#423C2×D5311yescharacteristic
229C2×C4⋊C8—531no—
230#417—1061no—
231C2×#102—1061no—
232C2×C22⋊C8—1062no—
233#334—534no—
234#333C2×D5314yescomplemented
235#332—532no—
236#331—532no—
237C53#1939111yescharacteristic, complemented, Sylow
238C2×#41—1061no—
239C22×Q8—21no—
240C22×D4—531no—
241C23×C4—1061no—
242AGL(1,ℤ/8)—2124no—
243C2×SD16—1062no—
244C2×D8—1062no—
245C2×M16—21no—
246C22×C8—1064no—
247C22×C8—1061no—
248C4⋊Q8—21no—
249C4⋊Q8—21no—
250C4⋊Q8—21no—
251C4⋊Q8—531no—
252C4⋊Q8C22×D5312yes—
253#127—531no—
254#124—1061no—
255#121—1061no—
256#121—1061no—
257C4×D4—532no—
258C4×D4—532no—
259C4×D4—1061no—
260C4×D4—1061no—
261C4×D4—1061no—
262C4×D4—1061no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—531no—
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—1064no—
278#102—1064no—
279#102—1062no—
280C22⋊C8—21no—
281C22⋊C8—532no—
282C22⋊C8—1064no—
283C4×C8—21no—
284C4×C8—532no—
285C24—1061no—
286#41—2122no—
287#41—2122no—
288C2×Q8—22no—
289C2×Q8—41no—
290C2×Q8—82no—
291C2×Q8—1061no—
292C2×D4—534no—
293C2×D4—532no—
294C2×D4—1061no—
295C2×D4—1062no—
296C2×D4—1061no—
297C2×D4—1061no—
298C2×D4—1061no—
299C2×D4—2121no—
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—42no—
312C2×C8—22no—
313C2×C8—22no—
314C2×C8—534no—
315C2×C8—1062no—
316C2×C8—4244no—
317C2×C8—2122no—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—532no—
323C4⋊C4—1061no—
324C4⋊C4—1061no—
325C4⋊C4—1061no—
326C22⋊C4—1062no—
327C22⋊C4—1062no—
328C22⋊C4—2121no—
329C22⋊C4—2121no—
330C42#1171994511yescharacteristic
331C42D4×D5311yescharacteristic
332C42D4×D5311yescharacteristic
333C42C23×D5311yescharacteristic
334C23—1064no—
335C23—1062no—
336C23—4241no—
337C23—4241no—
338C23—2121no—
339C23#1313319011yescharacteristic
340Q8—82no—
341Q8—82no—
342Q8—2121no—
343Q8—2121no—
344D4—534no—
345D4—1064no—
346D4—1062no—
347D4—4241no—
348D4—4242no—
349D4—2121no—
350D4—2121no—
351D4—2122no—
352C2×C4—22no—
353C2×C4—41no—
354C2×C4—42no—
355C2×C4—82no—
356C2×C4—1062no—
357C2×C4—4242no—
358C2×C4—4242no—
359C2×C4—4241no—
360C2×C4—2121no—
361C2×C4—2124no—
362C2×C4#1313319011yescharacteristic
363C2×C4#1313319011yescharacteristic
364C2×C4#1313319011yescharacteristic
365C2×C4SD16×D5312yes—
366C2×C4#1313324012yes—
367C2×C4C2×D4×D5311yescharacteristic
368C2×C4C2×D4×D5311yescharacteristic
369C2×C4C2×#1171994511yescharacteristic
370C8—42no—
371C8—42no—
372C8—1064no—
373C8—2122no—
374C22—21no—
375C22—21no—
376C22—1064no—
377C22—4244no—
378C22—4241no—
379C22—4241no—
380C22—4241no—
381C22—2122no—
382C22—2122no—
383C22#1681085512yes—
384C22C2×#1313319011yescharacteristic
385C4—21no—
386C4—21no—
387C4—21no—
388C4—82no—
389C4—4241no—
390C4—2122no—
391C4#1681085512yes—
392C4C2×SD16×D5311yescharacteristic
393C4C2×#1313324011yescharacteristic
394C4#1681143212yes—
395C2—22no—
396C2—8481no—
397C2—2124no—
398C2#2028422611yescharacteristic
399C2C2×#1681085511yescharacteristic
400C2#2028444211yescharacteristic
subgroupG/H
centerC22C2×#13133190
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#1548C2
socleC22×C53C2×#121
prime radical#20284181C2
hypercenter#1548D53
hypocenterC53#19391
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1548

Complements

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

NH ≅ G/Ntype
1. #20285843C2semidirect
2. C2×#16810910C2semidirect
4. #20283738C2semidirect
5. #20274127C2semidirect
6. #20274093C2semidirect
7. #20273487C2semidirect
8. #20273362C2semidirect
9. C53×#1548C2semidirect
13. #16811019C22semidirect
14. #16810971C22semidirect
30. #16810373C22semidirect
31. #16810368C22semidirect
32. #16810366C22semidirect
36. C53×#333C22semidirect
202. #1548D53semidirect
234. #333C2×D53semidirect
237. C53#19391semidirect

Maximal subgroups

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

subgroupindex
#202858432
C2×#168109102
#202841812
#202837382
#202741272
#202740932
#202734872
#202733622
C53×#15482
#1939153