Finite Group Databasebeta

← #27542235

Subgroups — #27542235

Order 13568.

466 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 127 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#508×D53C212yesmaximal, complemented
2#20277034C212yesmaximal, complemented
3#20277031C212yesmaximal, complemented
4#20277029C212yesmaximal, complemented
5#20277026C212yesmaximal, complemented
6#20277024C211yesmaximal, characteristic, complemented
7#20277022C212yesmaximal, complemented
8C2×#16810456C211yesmaximal, characteristic, complemented
9C53×#2262C211yesmaximal, characteristic, complemented
10#135×D53—22no—
11#16811056C2212yes—
12#16811055C2212yescomplemented
13#16811054—22no—
14#16811053—22no—
15#16811052C2212yes—
16Q32×D53C2212yescomplemented
17SD32×D53—22no—
18D16×D53C2212yescomplemented
19C2×C16×D53C2211yescharacteristic
20Dic848C2211yescharacteristic, complemented
21#16810463C2212yescomplemented
22#16810462C2212yescomplemented
23#16810461C2212yescomplemented
24#16810460C2212yescomplemented
25D1696C2211yescharacteristic, complemented
26#16810458C2212yescomplemented
27#16810456C2212yescomplemented
28#16810456C2212yescomplemented
29C2×#13133123C2211yescharacteristic, complemented
30C53×#508C2212yescomplemented
31C53×Q64C2211yescharacteristic, complemented
32C53×SD64C2212yescomplemented
33C53×D32C2211yescharacteristic, complemented
34C2×C32×C53C2211yescharacteristic, complemented
35#41×D53—42no—
36#13133240—22no—
37#13133239—22no—
38#13133238—42no—
39#13133237—42no—
40#13133236—22no—
41Q16×D53—22no—
42SD16×D53—42no—
43D8×D53—22no—
44C2×C8×D53D411yescharacteristic
45Dic424C2312yes—
46#13133163—22no—
47#13133162—22no—
48#13133161—22no—
49#13133160—22no—
50D848C2312yes—
51#13133158—22no—
52C2×#11719908D411yescharacteristic
53C16×D53D412yes—
54C16×D53C2312yes—
55#13133123C2312yes—
56#135×C53—22no—
57Q32×C53C2312yes—
58SD32×C53—22no—
59D16×C53C2312yes—
60C2×C16×C53C2311yescharacteristic
61C32×C53C2311yescharacteristic
62C32×C53C2311yescharacteristic
63#41×C53—42no—
64#11719949—42no—
65Q8×D53—42no—
66#11719947—42no—
67#11719947—82no—
68D4×D53—42no—
69D4×D53—82no—
70#11719945—42no—
71#11719945—82no—
72C2×C4×D53—82no—
73C2×C4×D53D811yescharacteristic
74Q16×C53—22no—
75SD16×C53—42no—
76D8×C53—22no—
77C2×C8×C53C2×D411yescharacteristic
78#11719925—42no—
79#11719924—42no—
80#11719923—42no—
81#11719922—42no—
82C2×#88322D811yescharacteristic
83Dic212—22no—
84D424—22no—
85#11719913—42no—
86C8×D53D812yes—
87C8×D53C2×D412yes—
88C16×C53C2×D411yescharacteristic
89C16×C53C2411yescharacteristic
90#11719908C2×D412yes—
91C22×D53—82no—
92C22×D53D1611yescharacteristic
93Q8×C53—42no—
94D4×C53—42no—
95D4×C53—82no—
96C2×C4×C53—82no—
97C2×C4×C53C2×D811yescharacteristic
98C22⋊D53—82no—
99C22⋊D53—82no—
100C22⋊D53—82no—
101C2×Dic53—82no—
102C2×Dic53D1611yescharacteristic
103D212—42no—
104D212—82no—
105C4×D53—82no—
106C4×D53—82no—
107C4×D53—82no—
108C4×D53D1612yes—
109C4×D53C2×D812yes—
110Dic106—42no—
111Dic106—82no—
112C8×C53C2×D811yescharacteristic
113C8×C53C22×D411yescharacteristic
114#88322C2×D812yes—
115#33981—531nomaximal, Sylow
116C22×C53—82no—
117C22×C53C2×D1611yescharacteristic
118C2×D53—41no—
119C2×D53—82no—
120C2×D53—164no—
121C2×D53C2×D1612yes—
122C4×C53—82no—
123C4×C53C2×D1611yescharacteristic
124C4×C53C22×D811yescharacteristic
125Dic53—82no—
126Dic53C2×D1612yes—
127C2×#508—532no—
128#2264—532no—
129#2264—532no—
130#2263—532no—
131#2263—532no—
132#2262—532no—
133#2262—531no—
134#2262D5311yescharacteristic, complemented
135C2×M64—531no—
136C2×C53—21no—
137C2×C53—162no—
138C2×C53C22×D1611yescharacteristic
139D53—22no—
140D53—162no—
141C2×#135—1062no—
142#508—532no—
143#508—532no—
144#508—532no—
145#508—1062no—
146#508—1062no—
147#508C2×D5312yescomplemented
148C2×Q32—532no—
149C2×SD32—1062no—
150C2×D16—532no—
151C22×C16—531no—
152Q64—532no—
153Q64—531no—
154Q64C2×D5311yescharacteristic, complemented
155SD64—532no—
156SD64—532no—
157SD64—532no—
158SD64C2×D5312yescomplemented
159D32—532no—
160D32—531no—
161D32C2×D5311yescharacteristic, complemented
162M64—532no—
163M64—532no—
164C2×C32—531no—
165C2×C32C2×D5311yescharacteristic, complemented
166C53#3398111yescharacteristic, complemented, Sylow
167C2×#41—2122no—
168#135—22no—
169#135—1062no—
170#135—1062no—
171#135—1062no—
172#135—2122no—
173#135—2122no—
174C2×Q16—1062no—
175C2×SD16—2122no—
176C2×D8—1062no—
177C22×C8—531no—
178Q32—532no—
179Q32—1062no—
180Q32C22×D5312yes—
181SD32—22no—
182SD32—1062no—
183SD32—1062no—
184SD32—1062no—
185D16—532no—
186D16—1062no—
187D16C22×D5312yes—
188C2×C16—531no—
189C2×C16—532no—
190C2×C16—532no—
191C2×C16C22×D5311yescharacteristic
192C32—532no—
193C32C22×D5311yescharacteristic
194C32C22×D5311yescharacteristic
195#41—42no—
196#41—4242no—
197#41—4242no—
198#41—2122no—
199#41—2122no—
200#41—2122no—
201C2×Q8—2122no—
202C2×D4—4242no—
203C2×D4—2122no—
204C22×C4—531no—
205C22×C4—4242no—
206Q16—22no—
207Q16—1062no—
208Q16—2122no—
209SD16—42no—
210SD16—2122no—
211SD16—2122no—
212SD16—2122no—
213D8—22no—
214D8—1062no—
215D8—2122no—
216C2×C8—531no—
217C2×C8—532no—
218C2×C8—532no—
219C2×C8D4×D5311yescharacteristic
220C16—532no—
221C16D4×D5311yescharacteristic
222C16C23×D5311yescharacteristic
223C23—531no—
224C23—4242no—
225Q8—42no—
226Q8—4242no—
227Q8—2122no—
228D4—42no—
229D4—82no—
230D4—4242no—
231D4—4242no—
232D4—4242no—
233D4—4242no—
234D4—2122no—
235C2×C4—82no—
236C2×C4—531no—
237C2×C4—532no—
238C2×C4—532no—
239C2×C4—4242no—
240C2×C4—4242no—
241C2×C4—4242no—
242C2×C4—4242no—
243C2×C4D8×D5311yescharacteristic
244C8—532no—
245C8D8×D5311yescharacteristic
246C8C2×D4×D5311yescharacteristic
247C22—82no—
248C22—532no—
249C22—8484no—
250C22—4242no—
251C22—2121no—
252C22D16×D5311yescharacteristic
253C4—82no—
254C4—532no—
255C4—4242no—
256C4D16×D5311yescharacteristic
257C4C2×D8×D5311yescharacteristic
258C2—21no—
259C2—162no—
260C2—8482no—
261C2—1062no—
262C2C2×D16×D5311yescharacteristic
subgroupG/H
centerC4D16×D53
commutator, cosocleC16×C53C24
FrattiniC16C23×D53
FittingC53×#2262C2
socleC2×C53C22×D16
hypercenter#2262D53
hypocenterC53#33981
LCSG ▷ C16×C53 ▷ C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ #2262

Complements

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

NH ≅ G/Ntype
1. #508×D53C2semidirect
2. #20277034C2semidirect
3. #20277031C2semidirect
4. #20277029C2semidirect
5. #20277026C2semidirect
6. #20277024C2semidirect
7. #20277022C2semidirect
8. C2×#16810456C2semidirect
9. C53×#2262C2semidirect
12. #16811055C22semidirect
16. Q32×D53C22semidirect
18. D16×D53C22semidirect
20. Dic848C22semidirect
21. #16810463C22semidirect
22. #16810462C22semidirect
23. #16810461C22semidirect
24. #16810460C22semidirect
25. D1696C22semidirect
26. #16810458C22semidirect
27. #16810456C22semidirect
28. #16810456C22semidirect
29. C2×#13133123C22semidirect
30. C53×#508C22semidirect
31. C53×Q64C22semidirect
32. C53×SD64C22semidirect
33. C53×D32C22semidirect
34. C2×C32×C53C22semidirect
134. #2262D53semidirect
147. #508C2×D53semidirect
154. Q64C2×D53semidirect
158. SD64C2×D53semidirect
161. D32C2×D53semidirect
165. C2×C32C2×D53semidirect
166. C53#33981semidirect

Maximal subgroups

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

subgroupindex
#508×D532
#202770342
#202770312
#202770292
#202770262
#202770242
#202770222
C2×#168104562
C53×#22622
#3398153