Finite Group Databasebeta

← #27538832

Subgroups — #27538832

Order 13568.

524 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285227C211yesmaximal, characteristic, complemented
2C2×#16810740C211yesmaximal, characteristic
3#20276889C214yesmaximal, complemented
4#20276884C214yesmaximal, complemented
5#20276488C211yesmaximal, characteristic, complemented
6#20276487C211yesmaximal, characteristic, complemented
7C2×#16810443C211yesmaximal, characteristic, complemented
8#20276451C211yesmaximal, characteristic
9C53×#2145C211yesmaximal, characteristic
10C2×#13133224—21no—
11#16811176—21no—
12C2×#13133157C2211yescharacteristic
13#16811027—21no—
14#16811024—21no—
15#16811023C2212yescomplemented
16#16811020C2212yescomplemented
17#16810740—21no—
18#16810740—21no—
19#16810740C2212yes—
20#16810740C2212yes—
21C2×#13133153C2211yescharacteristic
22#16810661C2211yescharacteristic
23#16810659—21no—
24C2×#13133144—21no—
25C2×C8×Dic53C2211yescharacteristic
26#16810446C2214yescomplemented
27#16810445C2214yescomplemented
28#16810443C2212yescomplemented
29#16810443C2212yescomplemented
30#16810433C2214yes—
31C2×#135×C53C2211yescharacteristic
32C2×M32×C53C2211yescharacteristic
33C53×#427C2211yescharacteristic
34C53×#360C2214yes—
35#13133224—22no—
36#13133224—22no—
37#13133224—41no—
38#13133224—41no—
39C2×#11719917—21no—
40C2×#11719917—41no—
41C22×#88322—41no—
42C22×#88322D411yescharacteristic
43#13133216—22no—
44#13133208—22no—
45#13133194—42no—
46C4×Dic106—42no—
47#13133176—21no—
48C2×C4×Dic53D411yescharacteristic
49#13133157—21no—
50#13133157C2312yes—
51C2×#11719908—21no—
52#13133153—21no—
53#13133153C2312yes—
54#13133151—21no—
55#13133150—22no—
56#13133149C2312yes—
57#13133148—22no—
58#13133144—22no—
59#13133144—22no—
60C8×Dic53D411yescharacteristic
61C8×Dic53D411yescharacteristic
62C8×Dic53C2311yescharacteristic
63C8×Dic53C2311yescharacteristic
64C2×#41×C53—21no—
65#135×C53—21no—
66#135×C53—21no—
67#135×C53Q812yescomplemented
68#135×C53C2312yes—
69C2×Q16×C53Q811yescharacteristic, complemented
70C2×SD16×C53—21no—
71C2×D8×C53Q811yescharacteristic, complemented
72C22×C8×C53C2311yescharacteristic
73#117×C53—21no—
74M32×C53—21no—
75M32×C53C2312yes—
76C2×C16×C53—21no—
77#107×C53—21no—
78#106×C53C2312yes—
79#41×C53—22no—
80#41×C53—22no—
81#41×C53—41no—
82#41×C53—41no—
83C2×Q8×C53—21no—
84C2×D4×C53—21no—
85C2×D4×C53—41no—
86C22×C4×C53—41no—
87C22×C4×C53C2×D411yescharacteristic
88C22×Dic53—21no—
89C2×Dic106—42no—
90Q16×C53—21no—
91Q16×C53C2×Q812yes—
92SD16×C53—22no—
93SD16×C53—22no—
94D8×C53—21no—
95D8×C53C2×Q812yes—
96C2×C8×C53C2×D411yescharacteristic
97C2×C8×C53C2×Q811yescharacteristic
98C2×C8×C53C2×Q811yescharacteristic
99C2×C8×C53#4111yescharacteristic
100C2×C8×C53#4111yescharacteristic
101C2×C8×C53C2411yescharacteristic
102C4⋊C4×C53—22no—
103C22⋊C4×C53—41no—
104C42×C53—41no—
105C22⋊Dic53—41no—
106C4⋊Dic53—22no—
107C4⋊Dic53—42no—
108#11719919—42no—
109#11719919—42no—
110C4×Dic53—21no—
111C4×Dic53—41no—
112C4×Dic53C2×D411yescharacteristic
113C4×Dic53C2×D411yescharacteristic
114#11719917—22no—
115#11719917—41no—
116#11719917—42no—
117#11719917—42no—
118C2×#88322—41no—
119C2×#88322—42no—
120C2×#88322—42no—
121C2×#88322SD1611yescharacteristic
122C2×#88322SD1611yescharacteristic
123C2×#88322SD1611yescharacteristic
124C2×#88322SD1611yescharacteristic
125C2×#88322C2×D411yescharacteristic
126C2×#88322C2×D411yescharacteristic
127C16×C53—22no—
128#11719908—22no—
129C23×C53—41no—
130C23×C53AGL(1,ℤ/8)11yescharacteristic
131Q8×C53—22no—
132Q8×C53—41no—
133D4×C53—22no—
134D4×C53—42no—
135D4×C53—42no—
136D4×C53—41no—
137C2×C4×C53—41no—
138C2×C4×C53—42no—
139C2×C4×C53—42no—
140C2×C4×C53—42no—
141C2×C4×C53C22⋊Q811yescharacteristic
142C2×C4×C53C22⋊Q811yescharacteristic
143C2×C4×C53C2×SD1611yescharacteristic
144C2×C4×C53C2×SD1611yescharacteristic
145C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
146C2×C4×C53C22×D411yescharacteristic
147C2×Dic53—21no—
148C2×Dic53—21no—
149C2×Dic53—42no—
150C2×Dic53—42no—
151Dic106—84no—
152C8×C53C22⋊Q811yescharacteristic
153C8×C53C22⋊Q811yescharacteristic
154C8×C53C22×Q811yescharacteristic
155C8×C53C2×#4111yescharacteristic
156#88322—42no—
157#88322C2×SD1611yescharacteristic
158#88322C2×SD1611yescharacteristic
159#88322C2×SD1611yescharacteristic
160#88322C2×SD1611yescharacteristic
161#33637—531nomaximal, Sylow
162C22×C53—41no—
163C22×C53—42no—
164C22×C53—82no—
165C22×C53#47611yescharacteristic
166C22×C53#47611yescharacteristic
167C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
168C4×C53—42no—
169C4×C53—82no—
170C4×C53#47611yescharacteristic
171C4×C53#47611yescharacteristic
172C4×C53C2×C22⋊Q811yescharacteristic
173C4×C53C22×SD1611yescharacteristic
174Dic53—22no—
175Dic53—41no—
176Dic53—82no—
177#3160—531no—
178C2×#443—531no—
179#2231—534no—
180#2230—534no—
181#2155—531no—
182#2155—531no—
183C2×#364—531no—
184#2145—531no—
185#2145D5311yescharacteristic
186C2×C53—21no—
187C2×C53—21no—
188C2×C53—21no—
189C2×C53—82no—
190C2×C53C2×#47611yescharacteristic
191C2×#135C2×D5311yescharacteristic
192C2×#131—1061no—
193#533—1061no—
194C2×M32—531no—
195C2×M32C2×D5311yescharacteristic
196#500—1061no—
197#499—532no—
198#499—1061no—
199#498—532no—
200#443—532no—
201#443—532no—
202#443—1061no—
203#443—1061no—
204C2×#108—531no—
205#427—531no—
206#427—1061no—
207#427C2×D5311yescharacteristic
208C2×#104—1061no—
209C2×C4×C8—531no—
210#365—534no—
211#365—534no—
212#364—532no—
213#364—532no—
214#360—534no—
215#360C2×D5314yes—
216C53#3363711yescharacteristic, complemented, Sylow
217C2×#41—21no—
218#135—21no—
219#135—21no—
220#135Dic10612yescomplemented
221#135C22×D5312yes—
222C2×Q16Dic10611yescharacteristic, complemented
223C2×SD16—21no—
224C2×D8Dic10611yescharacteristic, complemented
225#131—1062no—
226#131—1062no—
227#131—2121no—
228#131—2121no—
229C2×M16—1061no—
230C2×M16—2121no—
231C22×C8—531no—
232C22×C8—2121no—
233C22×C8C22×D5311yescharacteristic
234C4⋊Q8—1062no—
235#125—1062no—
236C22⋊Q8—2122no—
237C4×Q8—2122no—
238#117—21no—
239#117—1061no—
240C2×C42—531no—
241M32—21no—
242M32—532no—
243M32—1061no—
244M32C22×D5312yes—
245C2×C16—21no—
246C2×C16—1061no—
247#108—532no—
248#108—1061no—
249#107—21no—
250#107—1062no—
251#107—1061no—
252#106—532no—
253#106—1062no—
254#106C22×D5312yes—
255#104—1062no—
256#104—1062no—
257C4×C8—531no—
258C4×C8—531no—
259C4×C8—531no—
260C4×C8—531no—
261#41—22no—
262#41—22no—
263#41—41no—
264#41—41no—
265C2×Q8—21no—
266C2×Q8—2122no—
267C2×D4—21no—
268C2×D4—41no—
269C22×C4—41no—
270C22×C4—1061no—
271C22×C4D4×D5311yescharacteristic
272Q16—21no—
273Q16C2×Dic10612yes—
274SD16—22no—
275SD16—22no—
276D8—21no—
277D8C2×Dic10612yes—
278M16—1062no—
279M16—2122no—
280M16—2122no—
281M16—2121no—
282C2×C8—531no—
283C2×C8—531no—
284C2×C8—531no—
285C2×C8—531no—
286C2×C8—531no—
287C2×C8—531no—
288C2×C8—2122no—
289C2×C8—2122no—
290C2×C8—2121no—
291C2×C8C2×Dic10611yescharacteristic
292C2×C8C2×Dic10611yescharacteristic
293C2×C8D4×D5311yescharacteristic
294C2×C8#1171994711yescharacteristic
295C2×C8#1171994711yescharacteristic
296C2×C8C23×D5311yescharacteristic
297C4⋊C4—22no—
298C4⋊C4—1062no—
299C4⋊C4—2122no—
300C4⋊C4—2122no—
301C4⋊C4—2122no—
302C22⋊C4—41no—
303C22⋊C4—2121no—
304C42—41no—
305C42—531no—
306C42—531no—
307C42—1061no—
308C42—2121no—
309C16—22no—
310C16—1062no—
311C23—41no—
312C23#1313324111yescharacteristic
313Q8—22no—
314Q8—41no—
315Q8—4244no—
316D4—22no—
317D4—42no—
318D4—42no—
319D4—41no—
320C2×C4—41no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—1061no—
325C2×C4—1061no—
326C2×C4—2122no—
327C2×C4—2122no—
328C2×C4#1313319411yescharacteristic
329C2×C4#1313319411yescharacteristic
330C2×C4SD16×D5311yescharacteristic
331C2×C4SD16×D5311yescharacteristic
332C2×C4#1313324111yescharacteristic
333C2×C4C2×D4×D5311yescharacteristic
334C8—531no—
335C8—531no—
336C8—531no—
337C8—531no—
338C8—2122no—
339C8#1313319411yescharacteristic
340C8#1313319411yescharacteristic
341C8C22×Dic10611yescharacteristic
342C8C2×#1171994711yescharacteristic
343C22—41no—
344C22—42no—
345C22—82no—
346C22#1681091111yescharacteristic
347C22#1681091111yescharacteristic
348C22C2×#1313324111yescharacteristic
349C4—42no—
350C4—82no—
351C4—1062no—
352C4—4242no—
353C4—2121no—
354C4#1681091111yescharacteristic
355C4#1681091111yescharacteristic
356C4C2×#1313319411yescharacteristic
357C4C2×SD16×D5311yescharacteristic
358C2—21no—
359C2—21no—
360C2—21no—
361C2—82no—
362C2C2×#1681091111yescharacteristic
subgroupG/H
centerC4#16810911
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#2145C2
socleC2×C53C2×#476
prime radicalC2×#135×C53C22
hypercenter#2145D53
hypocenterC53#33637
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C22×C4 ◁ C22×C8 ◁ #2145

Complements

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

NH ≅ G/Ntype
1. #20285227C2semidirect
3. #20276889C2semidirect
4. #20276884C2semidirect
5. #20276488C2semidirect
6. #20276487C2semidirect
7. C2×#16810443C2semidirect
15. #16811023C22semidirect
16. #16811020C22semidirect
26. #16810446C22semidirect
27. #16810445C22semidirect
28. #16810443C22semidirect
29. #16810443C22semidirect
67. #135×C53Q8semidirect
69. C2×Q16×C53Q8semidirect
71. C2×D8×C53Q8semidirect
216. C53#33637semidirect
220. #135Dic106semidirect
222. C2×Q16Dic106semidirect
224. C2×D8Dic106semidirect

Maximal subgroups

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

subgroupindex
#202852272
C2×#168107402
#202768892
#202768842
#202764882
#202764872
C2×#168104432
#202764512
C53×#21452
#3363753