Finite Group Databasebeta

← #27369315

Subgroups — #27369315

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
1C2×#16810910C211yesmaximal, characteristic, complemented
2#20284175C211yesmaximal, characteristic
3#20283738C211yesmaximal, characteristic, complemented
4#20274121C214yesmaximal, complemented
5#20274087C214yesmaximal, complemented
6#20273486C211yesmaximal, characteristic, complemented
7#20273396C211yesmaximal, characteristic, complemented
8#20273364C211yesmaximal, characteristic, complemented
9C53×#3227C211yesmaximal, characteristic, complemented
10C2×#13133242—21no—
11C2×C4×D212C2211yescharacteristic
12#16810910—22no—
13#16810910C2214yes—
14#16810835—22no—
15#16810825C2212yes—
16#16810815C2212yes—
17C8×D212C2214yes—
18#16810688—22no—
19#16810676C2211yescharacteristic
20C2×#13133148—21no—
21#16810647C2211yescharacteristic
22C2×C4⋊#88322C2211yescharacteristic
23#16810624—21no—
24C2×#13133137—21no—
25C22⋊C8×D53—22no—
26#16810374C2214yescomplemented
27#16810372C2214yescomplemented
28#16810369C2212yescomplemented
29#16810367C2212yescomplemented
30C53×#532C2211yescharacteristic
31C2×C4⋊Q8×C53C2211yescharacteristic
32C53×#498C2212yescomplemented
33C53×#486C2212yescomplemented
34C53×#481—22no—
35C53×#476—22no—
36C53×#431C2211yescharacteristic
37C22×D212D411yescharacteristic
38C22×C4×D53—21no—
39#13133242—44no—
40C2×#11719924—22no—
41C2×#11719922—22no—
42C2×#11719917—21no—
43C22×#88322—21no—
44C2×C8×D53—24no—
45C4×D212—21no—
46C4×D212—21no—
47C4×D212D412yes—
48C4×D212C2312yes—
49C2×C4⋊Dic53D411yescharacteristic
50C2×#11719921—21no—
51#13133148—24no—
52C4⋊#88322—21no—
53C4⋊#88322—21no—
54C4⋊#88322C2312yes—
55C4⋊#88322C2312yes—
56C4⋊#88322C2312yes—
57#13133141—22no—
58#13133138—22no—
59#13133137—24no—
60C22⋊#88322—21no—
61#13133129—24no—
62C4×#88322—21no—
63C2×#41×C53—21no—
64C22×Q8×C53—21no—
65C4⋊Q8×C53—21no—
66C4⋊Q8×C53—21no—
67C4⋊Q8×C53—21no—
68C4⋊Q8×C53C2312yes—
69C4⋊Q8×C53C2311yescharacteristic
70#127×C53C2311yescharacteristic
71#124×C53—21no—
72#121×C53—21no—
73#121×C53—21no—
74C4×D4×C53—21no—
75C4×D4×C53—21no—
76C2×C4⋊C4×C53—21no—
77C2×C4⋊C4×C53—21no—
78C2×C42×C53C2311yescharacteristic
79#106×C53—24no—
80C4⋊C8×C53D412yes—
81#102×C53—24no—
82C22⋊C8×C53D412yes—
83C4×C8×C53C2312yes—
84C23×D53—21no—
85#41×C53—42no—
86#41×C53—42no—
87C2×Q8×C53—21no—
88C2×Q8×C53—22no—
89C2×Q8×C53—41no—
90C2×Q8×C53—82no—
91C2×D4×C53—21no—
92C2×D4×C53—21no—
93C2×D4×C53—21no—
94C2×D4×C53—41no—
95C22×C4×C53—41no—
96C22×C4×C53—41no—
97C22×C4×C53C2×D411yescharacteristic
98C22×C4×C53C2×D411yescharacteristic
99C22×C4×C53#4111yescharacteristic
100C22×Dic53—21no—
101C2×D212—21no—
102C2×D212—22no—
103C2×D212SD1614yes—
104C2×D212C2×D412yes—
105C2×C4×D53—22no—
106C2×C4×D53—22no—
107C2×C4×D53—82no—
108C2×C8×C53C2×D414yes—
109C4⋊C4×C53—22no—
110C4⋊C4×C53—21no—
111C4⋊C4×C53—22no—
112C4⋊C4×C53—21no—
113C4⋊C4×C53—41no—
114C4⋊C4×C53—41no—
115C22⋊C4×C53—41no—
116C22⋊C4×C53—41no—
117C42×C53C2×D411yescharacteristic
118C42×C53C2×D411yescharacteristic
119C42×C53#4111yescharacteristic
120C42×C53C2411yescharacteristic
121#11719924—44no—
122#11719922—44no—
123#11719921—22no—
124#11719921—22no—
125C4⋊Dic53—21no—
126C4⋊Dic53C2×D412yes—
127#11719917—42no—
128C2×#88322—22no—
129C2×#88322—22no—
130C2×#88322—22no—
131C2×#88322—42no—
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—
141Q8×C53—82no—
142Q8×C53—82no—
143D4×C53—41no—
144D4×C53—41no—
145D4×C53—42no—
146D4×C53—81no—
147D4×C53—82no—
148C2×C4×C53—22no—
149C2×C4×C53—41no—
150C2×C4×C53—41no—
151C2×C4×C53—42no—
152C2×C4×C53—82no—
153C2×C4×C53—82no—
154C2×C4×C53#12111yescharacteristic
155C2×C4×C53#12111yescharacteristic
156C2×C4×C53#12111yescharacteristic
157C2×C4×C53C2×SD1612yes—
158C2×C4×C53#13512yes—
159C2×C4×C53C22×D411yescharacteristic
160C2×C4×C53C22×D411yescharacteristic
161C2×C4×C53C2×#4111yescharacteristic
162C2×Dic53—22no—
163C2×Dic53—81no—
164D212—22no—
165D212—24no—
166D212C2×SD1614yes—
167C4×D53—44no—
168C4×D53—82no—
169C8×C53—24no—
170#88322—42no—
171#88322—42no—
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—82no—
190C4×C53#46512yes—
191C4×C53C22×SD1611yescharacteristic
192C4×C53C2×#13511yescharacteristic
193C4×C53#57712yes—
194Dic53—42no—
195#3227D5311yescharacteristic, complemented
196C2×#476—531no—
197#3036—531no—
198#2972—531no—
199#1677—534no—
200#1671—534no—
201#1570—531no—
202#1548—531no—
203#1540—531no—
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⋊Q8C2×D5311yescharacteristic
213C2×C4×D4—531no—
214#498C2×D5312yescomplemented
215#486C2×D5312yescomplemented
216#481—22no—
217#476—22no—
218#476—534no—
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⋊C8—531no—
230#417—1061no—
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—21no—
240C22×D4—531no—
241C23×C4—1061no—
242AGL(1,ℤ/8)—2124no—
243C2×SD16—1062no—
244C2×D8—1062no—
245C2×M16—1061no—
246C22×C8—1061no—
247C22×C8—1064no—
248C4⋊Q8—21no—
249C4⋊Q8—21no—
250C4⋊Q8—21no—
251C4⋊Q8C22×D5312yes—
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—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—531no—
266C2×C22⋊C4—1061no—
267C2×C42C22×D5311yescharacteristic
268#106—24no—
269#106—1064no—
270C4⋊C8—532no—
271C4⋊C8—532no—
272C4⋊C8—532no—
273C4⋊C8—1061no—
274C4⋊C8—1061no—
275C4⋊C8C22⋊D5312yes—
276#103—1062no—
277#102—24no—
278#102—1064no—
279#102—1062no—
280C22⋊C8—1061no—
281C22⋊C8—1064no—
282C22⋊C8C22⋊D5312yes—
283C4×C8—1061no—
284C4×C8C22×D5312yes—
285C24—1061no—
286#41—42no—
287#41—42no—
288C2×Q8—21no—
289C2×Q8—22no—
290C2×Q8—41no—
291C2×Q8—82no—
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—41no—
302C22×C4—1061no—
303C22×C4—1062no—
304C22×C4—1062no—
305C22×C4—4242no—
306C22×C4D4×D5311yescharacteristic
307C22×C4#1171994711yescharacteristic
308C22×C4C2×C22⋊D5311yescharacteristic
309SD16—2124no—
310D8—2124no—
311M16—2122no—
312C2×C8—1062no—
313C2×C8—1062no—
314C2×C8—1062no—
315C2×C8—4244no—
316C2×C8—2122no—
317C2×C8C2×C22⋊D5314yes—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—21no—
321C4⋊C4—21no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—532no—
325C4⋊C4—1061no—
326C22⋊C4—41no—
327C22⋊C4—41no—
328C22⋊C4—1062no—
329C22⋊C4—1062no—
330C42D4×D5311yescharacteristic
331C42#1171994711yescharacteristic
332C42C2×C22⋊D5311yescharacteristic
333C42C23×D5311yescharacteristic
334C23—41no—
335C23—1064no—
336C23—1062no—
337C23—4241no—
338C23—4241no—
339C23#1313318811yescharacteristic
340Q8—41no—
341Q8—41no—
342Q8—82no—
343Q8—82no—
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—42no—
356C2×C4—82no—
357C2×C4—82no—
358C2×C4—1062no—
359C2×C4—4242no—
360C2×C4—4241no—
361C2×C4—2124no—
362C2×C4#1313318811yescharacteristic
363C2×C4#1313318811yescharacteristic
364C2×C4#1313318811yescharacteristic
365C2×C4SD16×D5312yes—
366C2×C4#1313323812yes—
367C2×C4C2×D4×D5311yescharacteristic
368C2×C4C22×C22⋊D5311yescharacteristic
369C2×C4C2×#1171994711yescharacteristic
370C8—24no—
371C8—2122no—
372C8—2122no—
373C8—2122no—
374C22—21no—
375C22—21no—
376C22—81no—
377C22—81no—
378C22—81no—
379C22—1064no—
380C22—4244no—
381C22—2122no—
382C22—2122no—
383C22#1681084912yes—
384C22C2×#1313318811yescharacteristic
385C4—21no—
386C4—21no—
387C4—21no—
388C4—82no—
389C4—81no—
390C4—2122no—
391C4#1681084912yes—
392C4C2×SD16×D5311yescharacteristic
393C4C2×#1313323811yescharacteristic
394C4#1681143112yes—
395C2—22no—
396C2—161no—
397C2—2124no—
398C2#2028422311yescharacteristic
399C2C2×#1681084911yescharacteristic
400C2#2028443911yescharacteristic
subgroupG/H
centerC22C2×#13133188
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3227C2
socleC22×C53C2×#121
prime radical#20284175C2
hypercenter#3227D53
hypocenterC53#19391
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3227

Complements

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

NH ≅ G/Ntype
1. C2×#16810910C2semidirect
3. #20283738C2semidirect
4. #20274121C2semidirect
5. #20274087C2semidirect
6. #20273486C2semidirect
7. #20273396C2semidirect
8. #20273364C2semidirect
9. C53×#3227C2semidirect
26. #16810374C22semidirect
27. #16810372C22semidirect
28. #16810369C22semidirect
29. #16810367C22semidirect
32. C53×#498C22semidirect
33. C53×#486C22semidirect
195. #3227D53semidirect
214. #498C2×D53semidirect
215. #486C2×D53semidirect
237. C53#19391semidirect

Maximal subgroups

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

subgroupindex
C2×#168109102
#202841752
#202837382
#202741212
#202740872
#202734862
#202733962
#202733642
C53×#32272
#1939153