Finite Group Databasebeta

← #44565145

Subgroups — #44565145

Order 32128.

1464 conjugacy classes of subgroups; only the 545 normal subgroups are shown.

normal subgroups only

This group has 1464 conjugacy classes of subgroups — too many to draw. Only its 545 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#30632523C211yesmaximal, characteristic, complemented
2#30632498C218yesmaximal, complemented
3#30632490C218yesmaximal, complemented
4#30632485C216yesmaximal, complemented
5C2×#22548741C216yesmaximal, complemented
6#30632454C211yesmaximal, characteristic, complemented
7C2×H(5,𝔽2)×C251C211yesmaximal, characteristic, complemented
8#22548763—21no—
9#22548762C2214yescomplemented
10#22548761C2214yescomplemented
11#22548760—23no—
12C2×#16860810C2216yescomplemented
13#41×D251—24no—
14#22548746C2214yescomplemented
15#22548745C22112yescomplemented
16#22548741C22112yescomplemented
17#22548741C22124yescomplemented
18#22548740C22112yescomplemented
19#22548735C22124yescomplemented
20C2×#16860798C22112yescomplemented
21C2×#16860795C22112yescomplemented
22#22548722C2218yescomplemented
23C2×#16860794C2216yescomplemented
24H(5,𝔽2)×C251—24no—
25H(5,𝔽2)×C251C2218yescomplemented
26C2×#41×C251—21no—
27C2×#41×C251C2211yescharacteristic
28C2×#41×C251C2213yescomplemented
29C22×D4×C251—23no—
30C22×D4×C251C2213yescomplemented
31#16860811—24no—
32#16860810—26no—
33#16860810—26no—
34#16860810C2314yes—
35#16860810C23112yescomplemented
36#16860809—212no—
37C2×Dic502C2316yescomplemented
38Q8×D251—24no—
39C2×D1004C2316yescomplemented
40C2×C22⋊D251—26no—
41D4×D251—212no—
42C2×C4×D251—26no—
43#16860800C2318yescomplemented
44#16860798C23124yescomplemented
45#16860797C2318yescomplemented
46#16860795C23124yescomplemented
47#16860794C2314yes—
48#16860794C23112yescomplemented
49C2×C251⋊C8C23112yescomplemented
50C24×C251—23no—
51#41×C251—28no—
52#41×C251—26no—
53#41×C251D414yes—
54#41×C251C23112yes—
55#41×C251C2314yes—
56C2×Q8×C251D411yescharacteristic
57C2×Q8×C251C2311yescharacteristic
58C2×D4×C251—23no—
59C2×D4×C251—224no—
60C2×D4×C251—23no—
61C2×D4×C251—26no—
62C2×D4×C251D413yes—
63C2×D4×C251C2313yes—
64C2×D4×C251C23112yes—
65C22×C4×C251—23no—
66C22×C4×C251C2313yes—
67C22×D251—26no—
68Dic502—26no—
69Dic502C2414yes—
70D1004—26no—
71D1004C2414yes—
72C22⋊D251—212no—
73C22⋊D251—24no—
74C2×Dic251—26no—
75C4×D251—24no—
76C4×D251—212no—
77C251⋊C8C2418yes—
78C23×C251—23no—
79C23×C251—212no—
80C23×C251—86no—
81C23×C251C2×D413yes—
82Q8×C251—21no—
83Q8×C251C2×D414yes—
84Q8×C251C2412yes—
85D4×C251—212no—
86D4×C251—212no—
87D4×C251—23no—
88D4×C251C2×D4112yes—
89D4×C251C2416yes—
90C2×C4×C251—21no—
91C2×C4×C251—212no—
92C2×C4×C251C2×D413yes—
93C2×C4×C251C2×D416yes—
94C2×C4×C251C2411yescharacteristic
95C2×C4×C251C2416yes—
96C22×C251—26no—
97C22×C251—43no—
98C22×C251—46no—
99C22×C251—812no—
100C22×C251C22×D411yescharacteristic
101C22×C251C22×D416yes—
102C2×D251—24no—
103C2×D251—86no—
104C4×C251—22no—
105C4×C251C22×D416yes—
106C4×C251C22×D411yescharacteristic
107C4×C251C2511yescharacteristic
108Dic251—24no—
109C2×C251—26no—
110C2×C251—21no—
111C2×C251—46no—
112C2×C251C23×D411yescharacteristic
113D251—44no—
114C251#358511yescharacteristic, complemented, Sylow
115#3585—2511nomaximal, Sylow
116#585—2511no—
117C2×H(5,𝔽2)D25111yescharacteristic, complemented
118#577—2518no—
119#576—2518no—
120#575—2516no—
121C2×AGL(1,ℤ/8)—2516no—
122#568—2511no—
1232−1+4—5021no—
1242−1+4—2514no—
125H(5,𝔽2)—24no—
126H(5,𝔽2)—5023no—
127H(5,𝔽2)—2514no—
128H(5,𝔽2)C2×D25118yescomplemented
129C2×#41—21no—
130C2×#41—5024no—
131C2×#41—2516no—
132C2×#41C2×D25111yescharacteristic
133C2×#41C2×D25113yescomplemented
134C22×D4—23no—
135C22×D4C2×D25113yescomplemented
136#137—25112no—
137#137—2514no—
138AGL(1,ℤ/8)—25112no—
139AGL(1,ℤ/8)—25112no—
140AGL(1,ℤ/8)—25124no—
141#135—25124no—
142C2×SD16—25112no—
143C2×D8—25112no—
144#131—2518no—
145C2×M16—2516no—
146C24—23no—
147#41—28no—
148#41—26no—
149#41—5024no—
150#41—50212no—
151#41—5026no—
152#41—5026no—
153#41—25112no—
154#41—2514no—
155#41C22⋊D25114yes—
156#41C22×D251112yes—
157#41C22×D25114yes—
158C2×Q8—5024no—
159C2×Q8—2516no—
160C2×Q8C22⋊D25111yescharacteristic
161C2×Q8C22×D25111yescharacteristic
162C2×D4—23no—
163C2×D4—224no—
164C2×D4—23no—
165C2×D4—26no—
166C2×D4—5026no—
167C2×D4—50212no—
168C2×D4—2516no—
169C2×D4C22⋊D25113yes—
170C2×D4C22×D251112yes—
171C2×D4C22×D25113yes—
172C22×C4—23no—
173C22×C4—5026no—
174C22×C4C22×D25113yes—
175Q16—2518no—
176SD16—2518no—
177SD16—25124no—
178D8—25124no—
179M16—2514no—
180M16—25112no—
181C2×C8—25112no—
182C23—23no—
183C23—212no—
184C23—86no—
185C23—5026no—
186C23C2×C22⋊D25113yes—
187Q8—21no—
188Q8—5026no—
189Q8—2514no—
190Q8C2×C22⋊D25114yes—
191Q8C23×D25112yes—
192D4—212no—
193D4—212no—
194D4—23no—
195D4—5024no—
196D4—50212no—
197D4—5026no—
198D4—2514no—
199D4C2×C22⋊D251112yes—
200D4C23×D25116yes—
201C2×C4—21no—
202C2×C4—212no—
203C2×C4—5026no—
204C2×C4—5024no—
205C2×C4—50212no—
206C2×C4C2×C22⋊D25113yes—
207C2×C4C2×C22⋊D25116yes—
208C2×C4C23×D25116yes—
209C2×C4C23×D25111yescharacteristic
210C8—2518no—
211C22—26no—
212C22—43no—
213C22—46no—
214C22—812no—
215C22—20086no—
216C22—5024no—
217C22C22×C22⋊D25111yescharacteristic
218C22C22×C22⋊D25116yes—
219C4—22no—
220C4—5024no—
221C4C22×C22⋊D25116yes—
222C4C22×C22⋊D25111yescharacteristic
223C4C24×D25111yescharacteristic
224C2—21no—
225C2—26no—
226C2—46no—
227C2—10044no—
228C2C23×C22⋊D25111yescharacteristic
subgroupG/H
centerC2C23×C22⋊D251
commutator, cosocleC4×C251C25
FrattiniC4C24×D251
FittingC2×H(5,𝔽2)×C251C2
socleC2×C251C23×D4
hypercenterC2×H(5,𝔽2)D251
hypocenterC251#3585
LCSG ▷ C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C2 ◁ C2×#41 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #30632523C2semidirect
2. #30632498C2semidirect
3. #30632490C2semidirect
4. #30632485C2semidirect
5. C2×#22548741C2semidirect
6. #30632454C2semidirect
7. C2×H(5,𝔽2)×C251C2semidirect
9. #22548762C22semidirect
10. #22548761C22semidirect
12. C2×#16860810C22semidirect
14. #22548746C22semidirect
15. #22548745C22semidirect
16. #22548741C22semidirect
17. #22548741C22semidirect
18. #22548740C22semidirect
19. #22548735C22semidirect
20. C2×#16860798C22semidirect
21. C2×#16860795C22semidirect
22. #22548722C22semidirect
23. C2×#16860794C22semidirect
25. H(5,𝔽2)×C251C22semidirect
28. C2×#41×C251C22semidirect
30. C22×D4×C251C22semidirect
35. #16860810C23semidirect
37. C2×Dic502C23semidirect
39. C2×D1004C23semidirect
43. #16860800C23semidirect
44. #16860798C23semidirect
45. #16860797C23semidirect
46. #16860795C23semidirect
48. #16860794C23semidirect
49. C2×C251⋊C8C23semidirect
114. C251#3585semidirect
117. C2×H(5,𝔽2)D251semidirect
128. H(5,𝔽2)C2×D251semidirect
133. C2×#41C2×D251semidirect
135. C22×D4C2×D251semidirect

Maximal subgroups

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

subgroupindex
#306325232
#306324982
#306324902
#306324852
C2×#225487412
#306324542
C2×H(5,𝔽2)×C2512
#3585251