Finite Group Databasebeta

← #44562159

Subgroups — #44562159

Order 32128.

460 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 125 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#30632499C212yesmaximal, complemented
2#30632494C212yesmaximal, complemented
3#30632060C211yesmaximal, characteristic, complemented
4#30631951C212yesmaximal, complemented
5#30631950C212yesmaximal, complemented
6#30631871C212yesmaximal, complemented
7#30631856C212yesmaximal, complemented
8#30631805C211yesmaximal, characteristic, complemented
9#443×C251C211yesmaximal, characteristic, complemented
10#22548762C2212yes—
11#22548761C2212yescomplemented
12#22548748—22no—
13#22548748C2212yescomplemented
14#22548747—22no—
15#22548745C2212yescomplemented
16#22548743—22no—
17#22548743C2212yes—
18#22548740C2212yes—
19#22548739—22no—
20#22548737—22no—
21#22548737C2211yescharacteristic
22#22548736—22no—
23C2×#16860799—22no—
24C2×#16860799C2212yescomplemented
25C2×#16860799C2212yescomplemented
26SD16×D251—22no—
27#22548721C2212yescomplemented
28#22548715C2211yescharacteristic, complemented
29#22548712C2211yescharacteristic
30#22548651C2211yescharacteristic, complemented
31#22548643C2212yescomplemented
32#22548633C2212yescomplemented
33#22548631C2212yescomplemented
34#135×C251C2211yescharacteristic
35#131×C251C2212yescomplemented
36#108×C251C2211yescharacteristic, complemented
37#104×C251C2212yescomplemented
38C4×C8×C251C2211yescharacteristic, complemented
39#16860811—22no—
40#16860810—22no—
41#16860810—22no—
42#16860810C2312yes—
43#16860809—22no—
44#16860809—42no—
45C2×Dic502—22no—
46C2×Dic502C2312yes—
47Q8×D251—22no—
48C2×D1004—22no—
49C2×D1004—41no—
50C2×D1004C2312yes—
51D4×D251—22no—
52D4×D251—42no—
53Dic1004—22no—
54Dic1004D411yescharacteristic, complemented
55#16860800—22no—
56#16860799—22no—
57#16860799—22no—
58#16860799—22no—
59#16860799—22no—
60#16860799—22no—
61#16860799—22no—
62#16860799D412yescomplemented
63#16860798—22no—
64#16860797—22no—
65D2008—22no—
66D2008D411yescharacteristic, complemented
67#16860795—22no—
68#16860794C2312yes—
69#16860793—22no—
70C8×D251—22no—
71C4⋊Dic251—41no—
72#41×C251C2312yes—
73Q16×C251D411yescharacteristic, complemented
74SD16×C251D412yescomplemented
75D8×C251D411yescharacteristic, complemented
76M16×C251—22no—
77M16×C251C2312yes—
78C2×C8×C251—22no—
79C2×C8×C251C2311yescharacteristic
80C2×C8×C251C2311yescharacteristic
81C42×C251C2311yescharacteristic
82C22×D251—22no—
83C22×D251—42no—
84Dic502—22no—
85Dic502—22no—
86Dic502—42no—
87Dic502C2×D412yes—
88D1004—22no—
89D1004—22no—
90D1004—42no—
91D1004—82no—
92D1004C2×D412yes—
93C22⋊D251—22no—
94C22⋊D251—42no—
95C22⋊D251—42no—
96C2×Dic251—22no—
97C2×Dic251—42no—
98C4×D251—22no—
99C4×D251—42no—
100C4×D251—42no—
101C251⋊C8—22no—
102Q8×C251C2×D412yes—
103D4×C251—22no—
104D4×C251C2×D412yes—
105C2×C4×C251—21no—
106C2×C4×C251—22no—
107C2×C4×C251C2411yescharacteristic
108C8×C251—22no—
109C8×C251C2×D411yescharacteristic
110C8×C251C2×D411yescharacteristic
111C8×C251C2×D411yescharacteristic
112C8×C251C2×D411yescharacteristic
113C22×C251—22no—
114C22×C251H(5,𝔽2)11yescharacteristic
115C2×D251—22no—
116C2×D251—42no—
117C2×D251—82no—
118C2×D251—162no—
119C4×C251—22no—
120C4×C251—22no—
121C4×C251C22×D411yescharacteristic
122C4×C251C22×D411yescharacteristic
123Dic251—22no—
124Dic251—42no—
125C2×C251—21no—
126C2×C251—42no—
127C2×C251D4211yescharacteristic
128D251—42no—
129D251—82no—
130C251#329111yescharacteristic, complemented, Sylow
131#3291—2511nomaximal, Sylow
132#577—2512no—
133#577—2512no—
134#492—2511no—
135#471—2512no—
136#471—2512no—
137#456—2512no—
138#453—2512no—
139#443—2511no—
140#443D25111yescharacteristic, complemented
1412−1+4—2512no—
142H(5,𝔽2)—2512no—
143#137—5022no—
144#137—5022no—
145#137—2512no—
146#137—2512no—
147AGL(1,ℤ/8)—5022no—
148AGL(1,ℤ/8)—5022no—
149AGL(1,ℤ/8)—2512no—
150AGL(1,ℤ/8)—2512no—
151#135—5022no—
152#135—5022no—
153#135—2511no—
154#135C2×D25111yescharacteristic
155C2×SD16—5022no—
156C2×SD16—5022no—
157C2×SD16—2512no—
158C2×SD16—2512no—
159#131—2512no—
160#131C2×D25112yescomplemented
161C4⋊Q8—2511no—
162#127—2511no—
163#108—2511no—
164#108C2×D25111yescharacteristic, complemented
165#104—2512no—
166#104C2×D25112yescomplemented
167#101—2512no—
168#100—2512no—
169C4×C8C2×D25111yescharacteristic, complemented
170#41—10042no—
171#41—5022no—
172#41—5022no—
173#41—5022no—
174#41—5022no—
175#41—2512no—
176#41C22×D25112yes—
177C2×Q8—5022no—
178C2×Q8—5022no—
179C2×Q8—2512no—
180C2×D4—10041no—
181C2×D4—10042no—
182C2×D4—5022no—
183C2×D4—5022no—
184C2×D4—2512no—
185Q16—5022no—
186Q16—5022no—
187Q16—2511no—
188Q16D100411yescharacteristic, complemented
189SD16—5022no—
190SD16—5022no—
191SD16—5022no—
192SD16—5022no—
193SD16—5022no—
194SD16—5022no—
195SD16—5022no—
196SD16—5022no—
197SD16—2512no—
198SD16D100412yescomplemented
199D8—5022no—
200D8—5022no—
201D8—2511no—
202D8D100411yescharacteristic, complemented
203M16—22no—
204M16—5022no—
205M16—2512no—
206M16C22×D25112yes—
207C2×C8—22no—
208C2×C8—5022no—
209C2×C8C22×D25111yescharacteristic
210C2×C8C22×D25111yescharacteristic
211C4⋊C4—10041no—
212C42C22×D25111yescharacteristic
213C23—10042no—
214C23—5022no—
215Q8—10042no—
216Q8—5022no—
217Q8—5022no—
218Q8—2512no—
219Q8C2×D100412yes—
220D4—22no—
221D4—20082no—
222D4—10042no—
223D4—10042no—
224D4—10042no—
225D4—5022no—
226D4—5022no—
227D4—5022no—
228D4—2512no—
229D4C2×D100412yes—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—10042no—
233C2×C4—10042no—
234C2×C4—10042no—
235C2×C4—5022no—
236C2×C4—5022no—
237C2×C4C23×D25111yescharacteristic
238C8—22no—
239C8—5022no—
240C8D4×D25111yescharacteristic
241C8D4×D25111yescharacteristic
242C8C2×D100411yescharacteristic
243C8C2×D100411yescharacteristic
244C22—22no—
245C22—40162no—
246C22—20082no—
247C22—10042no—
248C22—5022no—
249C22#2254876111yescharacteristic
250C4—22no—
251C4—22no—
252C4—10042no—
253C4—5022no—
254C4C2×D4×D25111yescharacteristic
255C4C22×D100411yescharacteristic
256C2—21no—
257C2—42no—
258C2—20082no—
259C2—10042no—
260C2D4×D100411yescharacteristic
subgroupG/H
centerC2D4×D1004
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#443×C251C2
socleC2×C251D42
hypercenter#443D251
hypocenterC251#3291
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C2 ◁ C2×C4 ◁ #443

Complements

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

NH ≅ G/Ntype
1. #30632499C2semidirect
2. #30632494C2semidirect
3. #30632060C2semidirect
4. #30631951C2semidirect
5. #30631950C2semidirect
6. #30631871C2semidirect
7. #30631856C2semidirect
8. #30631805C2semidirect
9. #443×C251C2semidirect
11. #22548761C22semidirect
13. #22548748C22semidirect
15. #22548745C22semidirect
24. C2×#16860799C22semidirect
25. C2×#16860799C22semidirect
27. #22548721C22semidirect
28. #22548715C22semidirect
30. #22548651C22semidirect
31. #22548643C22semidirect
32. #22548633C22semidirect
33. #22548631C22semidirect
35. #131×C251C22semidirect
36. #108×C251C22semidirect
37. #104×C251C22semidirect
38. C4×C8×C251C22semidirect
54. Dic1004D4semidirect
62. #16860799D4semidirect
66. D2008D4semidirect
73. Q16×C251D4semidirect
74. SD16×C251D4semidirect
75. D8×C251D4semidirect
130. C251#3291semidirect
140. #443D251semidirect
160. #131C2×D251semidirect
164. #108C2×D251semidirect
166. #104C2×D251semidirect
169. C4×C8C2×D251semidirect
188. Q16D1004semidirect
198. SD16D1004semidirect
202. D8D1004semidirect

Maximal subgroups

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

subgroupindex
#306324992
#306324942
#306320602
#306319512
#306319502
#306318712
#306318562
#306318052
#443×C2512
#3291251