Finite Group Databasebeta

← #27369321

Subgroups — #27369321

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×#16810913C211yesmaximal, characteristic, complemented
3#20283739C211yesmaximal, characteristic, complemented
4#20274122C214yesmaximal, complemented
5#20274088C214yesmaximal, complemented
6#20273487C211yesmaximal, characteristic, complemented
7#20273397C211yesmaximal, characteristic, complemented
8#20273364C211yesmaximal, characteristic, complemented
9C53×#3036C211yesmaximal, characteristic
10C2×#13133217C2211yescharacteristic
11#16811019C2212yescomplemented
12#16810969C2212yescomplemented
13#16810939—22no—
14#16810913—22no—
15#16810913—22no—
16#16810913C2214yes—
17D4×#88322—22no—
18D4×#88322C2214yes—
19#16810676C2211yescharacteristic
20#16810676C2211yescharacteristic
21C2×#13133148—21no—
22C2×C4⋊#88322C2211yescharacteristic
23C2×#13133138—21no—
24C2×C22⋊#88322—22no—
25#16810373C2214yescomplemented
26#16810371C2214yescomplemented
27#16810369C2212yescomplemented
28#16810367C2212yescomplemented
29C2×AGL(1,ℤ/8)×C53—21no—
30C53×#532C2211yescharacteristic
31C2×C4×D4×C53C2211yescharacteristic
32C53×#463—22no—
33C53×#461C2212yes—
34C53×#459C2212yes—
35C53×#423C2211yescharacteristic
36C53×#417—21no—
37C22×Dic106—21no—
38C22×#88322—24no—
39C22×#88322—21no—
40#13133217—21no—
41#13133217—21no—
42#13133217—21no—
43#13133217C2312yes—
44C2×C4⋊Dic53—21no—
45C2×C4⋊Dic53—21no—
46#13133148—24no—
47#13133148—24no—
48C4⋊#88322—21no—
49C4⋊#88322D412yes—
50C4⋊#88322C2312yes—
51C4⋊#88322C2312yes—
52#13133138—24no—
53#13133138—24no—
54C22⋊#88322—24no—
55C22⋊#88322—21no—
56C22⋊#88322D412yes—
57C4×#88322—21no—
58C4×#88322C2312yes—
59C2×#41×C53—21no—
60C22×D4×C53D411yescharacteristic
61C23×C4×C53—21no—
62AGL(1,ℤ/8)×C53—44no—
63C2×SD16×C53—22no—
64C2×D8×C53—22no—
65C2×M16×C53—21no—
66C4⋊Q8×C53C2311yescharacteristic
67#127×C53C2311yescharacteristic
68#124×C53—21no—
69#121×C53—21no—
70#121×C53—21no—
71C4×D4×C53—21no—
72C4×D4×C53—21no—
73C4×D4×C53—21no—
74C4×D4×C53—21no—
75C4×D4×C53D412yes—
76C4×D4×C53C2312yes—
77C2×C4⋊C4×C53D411yescharacteristic
78C2×C22⋊C4×C53—21no—
79C2×C42×C53C2311yescharacteristic
80C4⋊C8×C53—21no—
81C4⋊C8×C53C2312yes—
82#103×C53—22no—
83#102×C53—22no—
84C24×C53—21no—
85#41×C53—42no—
86#41×C53—42no—
87C2×Q8×C53—21no—
88C2×D4×C53—21no—
89C2×D4×C53—22no—
90C2×D4×C53—21no—
91C2×D4×C53—21no—
92C2×D4×C53—21no—
93C2×D4×C53—41no—
94C2×D4×C53SD1614yes—
95C2×D4×C53C2×D412yes—
96C22×C4×C53—21no—
97C22×C4×C53—22no—
98C22×C4×C53—22no—
99C22×C4×C53—41no—
100C22×C4×C53—82no—
101C22×C4×C53C2×D411yescharacteristic
102C22×C4×C53C2×D411yescharacteristic
103C22×C4×C53#4111yescharacteristic
104C22×Dic53—41no—
105C2×Dic106—22no—
106C2×Dic106—41no—
107C2×Dic106—82no—
108SD16×C53—44no—
109D8×C53—44no—
110M16×C53—42no—
111C2×C8×C53—22no—
112C4⋊C4×C53—21no—
113C4⋊C4×C53—21no—
114C4⋊C4×C53—21no—
115C4⋊C4×C53C2×D412yes—
116C22⋊C4×C53—22no—
117C22⋊C4×C53—22no—
118C22⋊C4×C53—41no—
119C22⋊C4×C53—41no—
120C42×C53C2×D411yescharacteristic
121C42×C53C2×D411yescharacteristic
122C42×C53#4111yescharacteristic
123C42×C53C2411yescharacteristic
124C4⋊Dic53—22no—
125C4⋊Dic53—22no—
126C4⋊Dic53—41no—
127C4⋊Dic53—41no—
128C2×#88322—22no—
129C2×#88322—22no—
130C2×#88322—42no—
131C2×#88322—84no—
132C2×#88322C2×D414yes—
133C23×C53—22no—
134C23×C53—24no—
135C23×C53—41no—
136C23×C53—81no—
137C23×C53—81no—
138C23×C53#12111yescharacteristic
139Q8×C53—41no—
140Q8×C53—41no—
141D4×C53—22no—
142D4×C53—24no—
143D4×C53—41no—
144D4×C53—41no—
145D4×C53—42no—
146D4×C53—81no—
147D4×C53—82no—
148D4×C53C2×SD1614yes—
149C2×C4×C53—22no—
150C2×C4×C53—22no—
151C2×C4×C53—41no—
152C2×C4×C53—44no—
153C2×C4×C53—41no—
154C2×C4×C53—82no—
155C2×C4×C53—81no—
156C2×C4×C53—82no—
157C2×C4×C53#12111yescharacteristic
158C2×C4×C53#12111yescharacteristic
159C2×C4×C53#12111yescharacteristic
160C2×C4×C53C2×SD1612yes—
161C2×C4×C53#13512yes—
162C2×C4×C53C22×D411yescharacteristic
163C2×C4×C53C22×D411yescharacteristic
164C2×C4×C53C2×#4111yescharacteristic
165C2×Dic53—42no—
166C2×Dic53—82no—
167Dic106—82no—
168Dic106—82no—
169C8×C53—42no—
170#88322—24no—
171#88322—42no—
172#88322—42no—
173#19391—531nomaximal, Sylow
174C22×C53—21no—
175C22×C53—21no—
176C22×C53—24no—
177C22×C53—42no—
178C22×C53—42no—
179C22×C53—84no—
180C22×C53—81no—
181C22×C53—81no—
182C22×C53—81no—
183C22×C53#46512yes—
184C22×C53C2×#12111yescharacteristic
185C4×C53—21no—
186C4×C53—21no—
187C4×C53—21no—
188C4×C53—42no—
189C4×C53—81no—
190C4×C53#46512yes—
191C4×C53C22×SD1611yescharacteristic
192C4×C53C2×#13511yescharacteristic
193C4×C53#57712yes—
194Dic53—82no—
195#3227—531no—
196C2×#476—531no—
197#3036D5311yescharacteristic
198#2972—531no—
199#1677—534no—
200#1671—534no—
201#1570—531no—
202#1548—531no—
203#1540—531no—
204C2×C53—22no—
205C2×C53—44no—
206C2×C53—161no—
207C2×C53#304211yescharacteristic
208C2×C53C2×#46511yescharacteristic
209C2×C53#306811yescharacteristic
210C2×AGL(1,ℤ/8)—21no—
211#532C2×D5311yescharacteristic
212C2×C4⋊Q8—531no—
213C2×C4×D4C2×D5311yescharacteristic
214#498—532no—
215#486—532no—
216#481—1062no—
217#476—534no—
218#476—1062no—
219#476—1062no—
220#463—22no—
221#461C2×D5312yes—
222#459C2×D5312yes—
223C8×D4—534no—
224C8×D4—1062no—
225#431—531no—
226#431—531no—
227C2×#106—1061no—
228#423C2×D5311yescharacteristic
229C2×C4⋊C8—531no—
230#417—21no—
231C2×#102—1061no—
232C2×C22⋊C8—1062no—
233#334—534no—
234#333—534no—
235#332—532no—
236#331—532no—
237C53#1939111yescharacteristic, complemented, Sylow
238C2×#41—21no—
239C22×Q8—1061no—
240C22×D4C22⋊D5311yescharacteristic
241C23×C4—21no—
242AGL(1,ℤ/8)—44no—
243C2×SD16—22no—
244C2×D8—22no—
245C2×M16—21no—
246C22×C8—1064no—
247C22×C8—1061no—
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—21no—
260C4×D4—21no—
261C4×D4C22⋊D5312yes—
262C4×D4C22×D5312yes—
263C2×C4⋊C4—1061no—
264C2×C4⋊C4—1061no—
265C2×C4⋊C4C22⋊D5311yescharacteristic
266C2×C22⋊C4—21no—
267C2×C42C22×D5311yescharacteristic
268#106—1064no—
269#106—1064no—
270C4⋊C8—21no—
271C4⋊C8—532no—
272C4⋊C8—532no—
273C4⋊C8—532no—
274C4⋊C8—1061no—
275C4⋊C8C22×D5312yes—
276#103—22no—
277#102—22no—
278#102—1064no—
279#102—1064no—
280C22⋊C8—532no—
281C22⋊C8—1064no—
282C22⋊C8—1061no—
283C4×C8—532no—
284C4×C8—1061no—
285C24—21no—
286#41—42no—
287#41—42no—
288C2×Q8—21no—
289C2×Q8—1062no—
290C2×Q8—4242no—
291C2×Q8—2121no—
292C2×D4—21no—
293C2×D4—22no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—41no—
298C2×D4#1171992314yes—
299C2×D4C2×C22⋊D5312yes—
300C22×C4—21no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—41no—
304C22×C4—82no—
305C22×C4—2121no—
306C22×C4D4×D5311yescharacteristic
307C22×C4#1171994711yescharacteristic
308C22×C4C2×C22⋊D5311yescharacteristic
309SD16—44no—
310D8—44no—
311M16—42no—
312C2×C8—22no—
313C2×C8—534no—
314C2×C8—1062no—
315C2×C8—1062no—
316C2×C8—4244no—
317C2×C8—2122no—
318C4⋊C4—21no—
319C4⋊C4—21no—
320C4⋊C4—21no—
321C4⋊C4—1062no—
322C4⋊C4—1062no—
323C4⋊C4—2121no—
324C4⋊C4—2121no—
325C4⋊C4C2×C22⋊D5312yes—
326C22⋊C4—22no—
327C22⋊C4—22no—
328C22⋊C4—41no—
329C22⋊C4—41no—
330C42D4×D5311yescharacteristic
331C42#1171994711yescharacteristic
332C42C2×C22⋊D5311yescharacteristic
333C42C23×D5311yescharacteristic
334C23—22no—
335C23—24no—
336C23—41no—
337C23—81no—
338C23—81no—
339C23#1313318911yescharacteristic
340Q8—41no—
341Q8—41no—
342Q8—4242no—
343Q8—4242no—
344D4—24no—
345D4—22no—
346D4—41no—
347D4—41no—
348D4—42no—
349D4—81no—
350D4—82no—
351D4C2×#1171992314yes—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—41no—
355C2×C4—44no—
356C2×C4—41no—
357C2×C4—82no—
358C2×C4—81no—
359C2×C4—82no—
360C2×C4—4242no—
361C2×C4—2122no—
362C2×C4#1313318911yescharacteristic
363C2×C4#1313318911yescharacteristic
364C2×C4#1313318911yescharacteristic
365C2×C4C2×#1171992312yes—
366C2×C4#1313323712yes—
367C2×C4C2×D4×D5311yescharacteristic
368C2×C4C22×C22⋊D5311yescharacteristic
369C2×C4C2×#1171994711yescharacteristic
370C8—42no—
371C8—1064no—
372C8—2122no—
373C8—2122no—
374C22—21no—
375C22—21no—
376C22—24no—
377C22—42no—
378C22—42no—
379C22—84no—
380C22—81no—
381C22—81no—
382C22—81no—
383C22#1681085012yes—
384C22C2×#1313318911yescharacteristic
385C4—21no—
386C4—21no—
387C4—21no—
388C4—42no—
389C4—81no—
390C4—4242no—
391C4#1681085012yes—
392C4C22×#1171992311yescharacteristic
393C4C2×#1313323711yescharacteristic
394C4#1681143312yes—
395C2—22no—
396C2—44no—
397C2—161no—
398C2#2028422911yescharacteristic
399C2C2×#1681085011yescharacteristic
400C2#2028444511yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
Fitting, prime radicalC53×#3036C2
socleC22×C53C2×#121
hypercenter#3036D53
hypocenterC53#19391
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3036

Complements

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

NH ≅ G/Ntype
1. #20285840C2semidirect
2. C2×#16810913C2semidirect
3. #20283739C2semidirect
4. #20274122C2semidirect
5. #20274088C2semidirect
6. #20273487C2semidirect
7. #20273397C2semidirect
8. #20273364C2semidirect
11. #16811019C22semidirect
12. #16810969C22semidirect
25. #16810373C22semidirect
26. #16810371C22semidirect
27. #16810369C22semidirect
28. #16810367C22semidirect
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×#168109132
#202837392
#202741222
#202740882
#202734872
#202733972
#202733642
C53×#30362
#1939153