Finite Group Databasebeta

← #44562160

Subgroups — #44562160

Order 32128.

460 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 117 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#30632498C212yesmaximal, complemented
2#30632497C212yesmaximal, complemented
3#30632059C211yesmaximal, characteristic, complemented
4#30631949C212yesmaximal, complemented
5#30631948C212yesmaximal, complemented
6#30631871C212yesmaximal, complemented
7#30631854C211yesmaximal, characteristic, complemented
8#30631801C212yesmaximal, complemented
9#453×C251C211yesmaximal, characteristic, complemented
10#22548762C2212yes—
11#22548760C2211yescharacteristic, complemented
12#22548748C2212yescomplemented
13#22548745—22no—
14#22548745C2212yescomplemented
15#22548744—22no—
16#22548741—22no—
17#22548741C2212yes—
18#22548739—22no—
19#22548735—22no—
20#22548735C2212yes—
21#22548734—22no—
22C2×#16860798—22no—
23C2×#16860798C2212yescomplemented
24C2×#16860798C2212yescomplemented
25SD16×D251—22no—
26#22548723C2212yescomplemented
27#22548722C2212yescomplemented
28#22548715C2211yescharacteristic, complemented
29#22548650C2212yescomplemented
30#22548643C2212yescomplemented
31#22548633C2212yescomplemented
32#22548632C2211yescharacteristic, complemented
33C4×C251⋊C8C2211yescharacteristic, complemented
34H(5,𝔽2)×C251C2211yescharacteristic, complemented
35AGL(1,ℤ/8)×C251C2212yes—
36#127×C251C2211yescharacteristic
37#104×C251C2212yescomplemented
38#100×C251C2211yescharacteristic, complemented
39#16860810—22no—
40#16860810C2312yes—
41#16860809—22no—
42#16860809—22no—
43#16860809—42no—
44C2×Dic502—22no—
45C2×Dic502C2312yes—
46Q8×D251—22no—
47C2×C22⋊D251—41no—
48D4×D251—22no—
49Dic1004—22no—
50#16860800—22no—
51#16860800D412yescomplemented
52#16860799—22no—
53#16860798—22no—
54#16860798—22no—
55#16860798—22no—
56#16860798—22no—
57#16860798—22no—
58#16860798—22no—
59#16860798D412yescomplemented
60#16860797D412yescomplemented
61#16860795—22no—
62#16860795D412yescomplemented
63#16860794—22no—
64#16860794C2312yes—
65#16860793—22no—
66C2×C251⋊C8—22no—
67C2×C251⋊C8C2312yes—
68C8×D251—22no—
69C4⋊Dic251—41no—
70#41×C251—22no—
71#41×C251C2312yes—
72C2×D4×C251—22no—
73C2×D4×C251—41no—
74C2×D4×C251—41no—
75C2×D4×C251C2311yescharacteristic
76C2×D4×C251C2311yescharacteristic
77SD16×C251—22no—
78D8×C251—22no—
79M16×C251C2312yes—
80C42×C251C2311yescharacteristic
81C22×D251—41no—
82Dic502—22no—
83Dic502—22no—
84Dic502—42no—
85Dic502C2×D412yes—
86D1004C2×D412yes—
87C22⋊D251—22no—
88C22⋊D251—42no—
89C22⋊D251—42no—
90C2×Dic251—22no—
91C2×Dic251—41no—
92C2×Dic251—42no—
93C4×D251—22no—
94C4×D251—42no—
95C251⋊C8—22no—
96C251⋊C8C2×D412yes—
97C251⋊C8C2×D412yes—
98C23×C251—21no—
99C23×C251—21no—
100C23×C251—41no—
101Q8×C251C2×D412yes—
102D4×C251—22no—
103D4×C251—22no—
104D4×C251—22no—
105D4×C251—42no—
106D4×C251—82no—
107D4×C251C2×D412yes—
108C2×C4×C251—21no—
109C2×C4×C251—22no—
110C2×C4×C251—41no—
111C2×C4×C251C2411yescharacteristic
112C8×C251—22no—
113C22×C251—22no—
114C22×C251—41no—
115C22×C251—41no—
116C22×C251—81no—
117C22×C251—81no—
118C22×C251—161no—
119C22×C251H(5,𝔽2)11yescharacteristic
120C2×D251—22no—
121C2×D251—161no—
122C4×C251—22no—
123C4×C251—22no—
124C4×C251C22×D412yes—
125Dic251—22no—
126Dic251—42no—
127C2×C251—21no—
128C2×C251—42no—
129C2×C251—81no—
130C2×C251—81no—
131C2×C251D4211yescharacteristic
132D251—42no—
133C251#329111yescharacteristic, complemented, Sylow
134#3291—2511nomaximal, Sylow
135#577—2512no—
136#577—2512no—
137#492—2511no—
138#471—2512no—
139#471—2512no—
140#456—2512no—
141#453—2511no—
142#453D25111yescharacteristic, complemented
143#443—2512no—
1442−1+4—2512no—
145H(5,𝔽2)—2511no—
146H(5,𝔽2)C2×D25111yescharacteristic, complemented
147#137—5022no—
148#137—5022no—
149#137—2512no—
150#137—2512no—
151AGL(1,ℤ/8)—5022no—
152AGL(1,ℤ/8)—5022no—
153AGL(1,ℤ/8)—2512no—
154AGL(1,ℤ/8)C2×D25112yes—
155#135—5022no—
156#135—5022no—
157#135—2512no—
158C2×SD16—5022no—
159C2×SD16—5022no—
160C2×SD16—2512no—
161C2×SD16—2512no—
162#131—2512no—
163#131—2512no—
164C4⋊Q8—2511no—
165#127C2×D25111yescharacteristic
166#108—2512no—
167#104—2512no—
168#104C2×D25112yescomplemented
169#101—2512no—
170#100—2511no—
171#100C2×D25111yescharacteristic, complemented
172C4×C8—2511no—
173#41—22no—
174#41—10042no—
175#41—5022no—
176#41—5022no—
177#41—5022no—
178#41—2512no—
179#41C22×D25112yes—
180C2×Q8—5022no—
181C2×Q8—5022no—
182C2×Q8—2512no—
183C2×D4—22no—
184C2×D4—41no—
185C2×D4—41no—
186C2×D4—10041no—
187C2×D4—5022no—
188C2×D4C22×D25111yescharacteristic
189C2×D4C22×D25111yescharacteristic
190Q16—5022no—
191Q16—5022no—
192Q16—2512no—
193SD16—22no—
194SD16—5022no—
195SD16—5022no—
196SD16—5022no—
197SD16—5022no—
198SD16—5022no—
199SD16—5022no—
200SD16—5022no—
201SD16—2512no—
202SD16—2512no—
203D8—22no—
204D8—5022no—
205D8—2512no—
206M16—5022no—
207M16—5022no—
208M16—2512no—
209M16C22×D25112yes—
210C2×C8—5022no—
211C2×C8—5022no—
212C2×C8—2512no—
213C4⋊C4—10041no—
214C42C22×D25111yescharacteristic
215C23—21no—
216C23—21no—
217C23—41no—
218C23—10041no—
219Q8—10042no—
220Q8—5022no—
221Q8—5022no—
222Q8—2512no—
223Q8D4×D25112yes—
224D4—22no—
225D4—22no—
226D4—22no—
227D4—42no—
228D4—82no—
229D4—10042no—
230D4—10042no—
231D4—5022no—
232D4—2512no—
233D4D4×D25112yes—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—41no—
237C2×C4—10042no—
238C2×C4—10042no—
239C2×C4—10041no—
240C2×C4—5022no—
241C2×C4—5022no—
242C2×C4C23×D25111yescharacteristic
243C8—22no—
244C8—5022no—
245C8—2512no—
246C8—2512no—
247C22—22no—
248C22—41no—
249C22—41no—
250C22—81no—
251C22—81no—
252C22—161no—
253C22—40161no—
254C22—5022no—
255C22#2254876011yescharacteristic
256C4—22no—
257C4—22no—
258C4—10042no—
259C4—5022no—
260C4C2×D4×D25112yes—
261C2—21no—
262C2—42no—
263C2—81no—
264C2—81no—
265C2—10042no—
266C2#3063231911yescharacteristic
subgroupG/H
centerC2#30632319
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#453×C251C2
socleC2×C251D42
hypercenter#453D251
hypocenterC251#3291
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C2 ◁ C2×C4 ◁ #453

Complements

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

NH ≅ G/Ntype
1. #30632498C2semidirect
2. #30632497C2semidirect
3. #30632059C2semidirect
4. #30631949C2semidirect
5. #30631948C2semidirect
6. #30631871C2semidirect
7. #30631854C2semidirect
8. #30631801C2semidirect
9. #453×C251C2semidirect
11. #22548760C22semidirect
12. #22548748C22semidirect
14. #22548745C22semidirect
23. C2×#16860798C22semidirect
24. C2×#16860798C22semidirect
26. #22548723C22semidirect
27. #22548722C22semidirect
28. #22548715C22semidirect
29. #22548650C22semidirect
30. #22548643C22semidirect
31. #22548633C22semidirect
32. #22548632C22semidirect
33. C4×C251⋊C8C22semidirect
34. H(5,𝔽2)×C251C22semidirect
37. #104×C251C22semidirect
38. #100×C251C22semidirect
51. #16860800D4semidirect
59. #16860798D4semidirect
60. #16860797D4semidirect
62. #16860795D4semidirect
133. C251#3291semidirect
142. #453D251semidirect
146. H(5,𝔽2)C2×D251semidirect
168. #104C2×D251semidirect
171. #100C2×D251semidirect

Maximal subgroups

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

subgroupindex
#306324982
#306324972
#306320592
#306319492
#306319482
#306318712
#306318542
#306318012
#453×C2512
#3291251