Finite Group Databasebeta

← #3288×C251 #44547875

Subgroups — #3288×C251 #44547875

Order 32128.

464 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#577×C251C212yesmaximal, complemented
2#576×C251C212yesmaximal, complemented
3#496×C251C211yesmaximal, characteristic, complemented
4#472×C251C212yesmaximal, complemented
5#471×C251C212yesmaximal, complemented
6#455×C251C211yesmaximal, characteristic, complemented
7#454×C251C211yesmaximal, characteristic, complemented
8#453×C251C212yesmaximal, complemented
9#444×C251C212yesmaximal, complemented
102−1+4×C251C2211yescharacteristic, complemented
11H(5,𝔽2)×C251C2211yescharacteristic
12H(5,𝔽2)×C251C2212yescomplemented
13#137×C251—22no—
14#137×C251C2212yescomplemented
15AGL(1,ℤ/8)×C251—22no—
16AGL(1,ℤ/8)×C251—22no—
17AGL(1,ℤ/8)×C251—22no—
18AGL(1,ℤ/8)×C251C2212yes—
19AGL(1,ℤ/8)×C251C2212yescomplemented
20AGL(1,ℤ/8)×C251C2212yescomplemented
21#135×C251—22no—
22#135×C251—22no—
23#135×C251C2212yes—
24C2×SD16×C251—22no—
25C2×SD16×C251C2212yescomplemented
26C2×D8×C251—22no—
27C2×D8×C251C2212yescomplemented
28#131×C251C2212yescomplemented
29#131×C251C2212yescomplemented
30#127×C251C2211yescharacteristic, complemented
31#124×C251C2211yescharacteristic, complemented
32#108×C251C2212yescomplemented
33#104×C251C2212yescomplemented
34#104×C251C2212yescomplemented
35#101×C251C2211yescharacteristic, complemented
36#100×C251C2212yescomplemented
37#100×C251C2211yescharacteristic, complemented
38#97×C251C2211yescharacteristic, complemented
39#41×C251—22no—
40#41×C251—22no—
41#41×C251—22no—
42#41×C251—22no—
43#41×C251—41no—
44#41×C251C2312yes—
45#41×C251C2312yes—
46C2×Q8×C251—22no—
47C2×Q8×C251C2311yescharacteristic
48C2×D4×C251—22no—
49C2×D4×C251—22no—
50C2×D4×C251—22no—
51C2×D4×C251—41no—
52C2×D4×C251—41no—
53C2×D4×C251—42no—
54C2×D4×C251C2311yescharacteristic
55C2×D4×C251C2312yes—
56Q16×C251—22no—
57Q16×C251D412yescomplemented
58SD16×C251—22no—
59SD16×C251—22no—
60SD16×C251—22no—
61SD16×C251—22no—
62SD16×C251—22no—
63SD16×C251—22no—
64SD16×C251D412yescomplemented
65SD16×C251D412yescomplemented
66D8×C251—22no—
67D8×C251—22no—
68D8×C251—22no—
69D8×C251—22no—
70D8×C251—22no—
71D8×C251D412yescomplemented
72M16×C251—22no—
73M16×C251—22no—
74M16×C251C2312yes—
75M16×C251C2312yes—
76C2×C8×C251—22no—
77C2×C8×C251—22no—
78C2×C8×C251C2312yes—
79C22⋊C4×C251—41no—
80C42×C251C2311yescharacteristic
81C23×C251—21no—
82C23×C251—22no—
83C23×C251—42no—
84C23×C251—41no—
85Q8×C251—22no—
86Q8×C251—41no—
87Q8×C251C2×D412yes—
88Q8×C251C2×D412yes—
89D4×C251—22no—
90D4×C251—22no—
91D4×C251—22no—
92D4×C251—22no—
93D4×C251—22no—
94D4×C251—42no—
95D4×C251—42no—
96D4×C251—42no—
97D4×C251—41no—
98D4×C251—82no—
99D4×C251C2×D412yes—
100D4×C251C2×D412yes—
101C2×C4×C251—22no—
102C2×C4×C251—22no—
103C2×C4×C251—21no—
104C2×C4×C251—21no—
105C2×C4×C251—41no—
106C2×C4×C251—42no—
107C2×C4×C251—42no—
108C2×C4×C251C2411yescharacteristic
109C8×C251—22no—
110C8×C251—22no—
111C8×C251C2×D412yes—
112C8×C251C2×D412yes—
113C22×C251—22no—
114C22×C251—22no—
115C22×C251—41no—
116C22×C251—42no—
117C22×C251—82no—
118C22×C251—81no—
119C22×C251—162no—
120C22×C251—161no—
121C22×C251H(5,𝔽2)11yescharacteristic
122C4×C251—22no—
123C4×C251—22no—
124C4×C251—41no—
125C4×C251—41no—
126C4×C251C22×D412yes—
127C2×C251—21no—
128C2×C251—42no—
129C2×C251—42no—
130C2×C251—82no—
131C2×C251—81no—
132C2×C251D4211yescharacteristic
133C251#328811yescharacteristic, direct product factor, Sylow
134#3288C25111yesmaximal, characteristic, direct product factor, Sylow
135#577C2×C25112yescomplemented
136#576C2×C25112yescomplemented
137#496C2×C25111yescharacteristic, complemented
138#472C2×C25112yescomplemented
139#471C2×C25112yescomplemented
140#455C2×C25111yescharacteristic, complemented
141#454C2×C25111yescharacteristic, complemented
142#453C2×C25112yescomplemented
143#444C2×C25112yescomplemented
1442−1+4C22×C25111yescharacteristic, complemented
145H(5,𝔽2)C22×C25111yescharacteristic
146H(5,𝔽2)C22×C25112yescomplemented
147#137—22no—
148#137C22×C25112yescomplemented
149AGL(1,ℤ/8)—22no—
150AGL(1,ℤ/8)—22no—
151AGL(1,ℤ/8)—22no—
152AGL(1,ℤ/8)C22×C25112yes—
153AGL(1,ℤ/8)C22×C25112yescomplemented
154AGL(1,ℤ/8)C22×C25112yescomplemented
155#135—22no—
156#135—22no—
157#135C22×C25112yes—
158C2×SD16—22no—
159C2×SD16C22×C25112yescomplemented
160C2×D8—22no—
161C2×D8C22×C25112yescomplemented
162#131C22×C25112yescomplemented
163#131C22×C25112yescomplemented
164#127C22×C25111yescharacteristic, complemented
165#124C22×C25111yescharacteristic, complemented
166#108C22×C25112yescomplemented
167#104C22×C25112yescomplemented
168#104C22×C25112yescomplemented
169#101C22×C25111yescharacteristic, complemented
170#100C22×C25112yescomplemented
171#100C22×C25111yescharacteristic, complemented
172#97C22×C25111yescharacteristic, complemented
173#41—22no—
174#41—22no—
175#41—22no—
176#41—22no—
177#41—41no—
178#41C23×C25112yes—
179#41C23×C25112yes—
180C2×Q8—22no—
181C2×Q8C23×C25111yescharacteristic
182C2×D4—22no—
183C2×D4—22no—
184C2×D4—22no—
185C2×D4—41no—
186C2×D4—41no—
187C2×D4—42no—
188C2×D4C23×C25111yescharacteristic
189C2×D4C23×C25112yes—
190Q16—22no—
191Q16D4×C25112yescomplemented
192SD16—22no—
193SD16—22no—
194SD16—22no—
195SD16—22no—
196SD16—22no—
197SD16—22no—
198SD16D4×C25112yescomplemented
199SD16D4×C25112yescomplemented
200D8—22no—
201D8—22no—
202D8—22no—
203D8—22no—
204D8—22no—
205D8D4×C25112yescomplemented
206M16—22no—
207M16—22no—
208M16C23×C25112yes—
209M16C23×C25112yes—
210C2×C8—22no—
211C2×C8—22no—
212C2×C8C23×C25112yes—
213C22⋊C4—41no—
214C42C23×C25111yescharacteristic
215C23—21no—
216C23—22no—
217C23—42no—
218C23—41no—
219Q8—22no—
220Q8—41no—
221Q8C2×D4×C25112yes—
222Q8C2×D4×C25112yes—
223D4—22no—
224D4—22no—
225D4—22no—
226D4—22no—
227D4—22no—
228D4—42no—
229D4—42no—
230D4—42no—
231D4—41no—
232D4—82no—
233D4C2×D4×C25112yes—
234D4C2×D4×C25112yes—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—41no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4C24×C25111yescharacteristic
243C8—22no—
244C8—22no—
245C8C2×D4×C25112yes—
246C8C2×D4×C25112yes—
247C22—22no—
248C22—22no—
249C22—41no—
250C22—42no—
251C22—82no—
252C22—81no—
253C22—161no—
254C22—162no—
255C22H(5,𝔽2)×C25111yescharacteristic
256C4—22no—
257C4—22no—
258C4—41no—
259C4—41no—
260C4C22×D4×C25112yes—
261C2—21no—
262C2—42no—
263C2—42no—
264C2—82no—
265C2—81no—
266C2D42×C25111yescharacteristic
subgroupG/H
center, socleC2×C251D42
commutator, Frattini, cosocleC2×C4C24×C251
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C251 ◁ C2×C4×C251 ◁ G

Complements

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

NH ≅ G/Ntype
1. #577×C251C2semidirect
2. #576×C251C2semidirect
3. #496×C251C2semidirect
4. #472×C251C2semidirect
5. #471×C251C2semidirect
6. #455×C251C2semidirect
7. #454×C251C2semidirect
8. #453×C251C2semidirect
9. #444×C251C2semidirect
10. 2−1+4×C251C22semidirect
12. H(5,𝔽2)×C251C22semidirect
14. #137×C251C22semidirect
19. AGL(1,ℤ/8)×C251C22semidirect
20. AGL(1,ℤ/8)×C251C22semidirect
25. C2×SD16×C251C22semidirect
27. C2×D8×C251C22semidirect
28. #131×C251C22semidirect
29. #131×C251C22semidirect
30. #127×C251C22semidirect
31. #124×C251C22semidirect
32. #108×C251C22semidirect
33. #104×C251C22semidirect
34. #104×C251C22semidirect
35. #101×C251C22semidirect
36. #100×C251C22semidirect
37. #100×C251C22semidirect
38. #97×C251C22semidirect
57. Q16×C251D4semidirect
64. SD16×C251D4semidirect
65. SD16×C251D4semidirect
71. D8×C251D4semidirect
133. C251#3288direct (N × H)
134. #3288C251direct (N × H)
135. #577C2×C251semidirect
136. #576C2×C251semidirect
137. #496C2×C251semidirect
138. #472C2×C251semidirect
139. #471C2×C251semidirect
140. #455C2×C251semidirect
141. #454C2×C251semidirect
142. #453C2×C251semidirect
143. #444C2×C251semidirect
144. 2−1+4C22×C251semidirect
146. H(5,𝔽2)C22×C251semidirect
148. #137C22×C251semidirect
153. AGL(1,ℤ/8)C22×C251semidirect
154. AGL(1,ℤ/8)C22×C251semidirect
159. C2×SD16C22×C251semidirect
161. C2×D8C22×C251semidirect
162. #131C22×C251semidirect
163. #131C22×C251semidirect
164. #127C22×C251semidirect
165. #124C22×C251semidirect
166. #108C22×C251semidirect
167. #104C22×C251semidirect
168. #104C22×C251semidirect
169. #101C22×C251semidirect
170. #100C22×C251semidirect
171. #100C22×C251semidirect
172. #97C22×C251semidirect
191. Q16D4×C251semidirect
198. SD16D4×C251semidirect
199. SD16D4×C251semidirect
205. D8D4×C251semidirect

Maximal subgroups

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

subgroupindex
#577×C2512
#576×C2512
#496×C2512
#472×C2512
#471×C2512
#455×C2512
#454×C2512
#453×C2512
#444×C2512
#3288251