Finite Group Databasebeta

← #44559756

Subgroups — #44559756

Order 32128.

766 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 766 conjugacy classes of subgroups — too many to draw. Only its 145 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#30632198C212yesmaximal, complemented
3#30631876C214yesmaximal, complemented
4#30631873C214yesmaximal, complemented
5#30631661C211yesmaximal, characteristic, complemented
6#30631660C212yesmaximal, complemented
7#525×C251C211yesmaximal, characteristic, complemented
8#22548763—21no—
9#22548762—22no—
10#22548761—22no—
11#22548760—21no—
12#22548760C2214yes—
13C2×#16860810—22no—
14C2×#16860810C2212yes—
15#41×D251—24no—
16#22548698C2214yescomplemented
17#22548694—22no—
18#22548689—22no—
19#22548684C2214yescomplemented
20C4×C22⋊D251—24no—
21#22548674C2212yescomplemented
22#22548630C2214yescomplemented
23#22548629C2218yescomplemented
24C2×#41×C251C2211yescharacteristic
25C23×C4×C251C2211yescharacteristic, complemented
26#123×C251C2212yescomplemented
27C22⋊Q8×C251—21no—
28#121×C251—21no—
29#120×C251C2212yescomplemented
30C4×D4×C251—22no—
31#117×C251C2211yescharacteristic, complemented
32#16860811—24no—
33#16860810—24no—
34#16860810—24no—
35#16860810—22no—
36#16860810—22no—
37#16860810—24no—
38#16860810—22no—
39#16860810—22no—
40#16860809—24no—
41#16860809—28no—
42C2×Dic502—22no—
43C2×Dic502D412yescomplemented
44Q8×D251—24no—
45C2×D1004—22no—
46C2×D1004D412yescomplemented
47C2×C22⋊D251—24no—
48C2×C22⋊D251—24no—
49C2×C22⋊D251C2314yes—
50D4×D251—24no—
51D4×D251—28no—
52C2×C4×D251—24no—
53C2×C4×D251D414yescomplemented
54C4⋊Dic251—22no—
55Dic251⋊C4—24no—
56C22⋊Dic251—24no—
57C22⋊Dic251C2314yes—
58(C2×D251)⋊C4—24no—
59C4×Dic251—22no—
60C24×C251C2311yescharacteristic
61#41×C251—22no—
62#41×C251—22no—
63C2×Q8×C251D411yescharacteristic, complemented
64C2×D4×C251—22no—
65C2×D4×C251D411yescharacteristic, complemented
66C2×D4×C251C2312yes—
67C22×C4×C251—21no—
68C22×C4×C251—21no—
69C22×C4×C251—22no—
70C22×C4×C251—22no—
71C22×C4×C251D412yescomplemented
72C22×C4×C251C2311yescharacteristic
73C22×C4×C251C2311yescharacteristic
74C4⋊C4×C251—21no—
75C4⋊C4×C251—22no—
76C22⋊C4×C251—22no—
77C22⋊C4×C251—22no—
78C22⋊C4×C251C2312yes—
79C42×C251—21no—
80C22×D251—24no—
81C22×D251C2×D414yes—
82Dic502—24no—
83Dic502—22no—
84Dic502—22no—
85D1004—24no—
86D1004—22no—
87D1004—22no—
88C22⋊D251—28no—
89C22⋊D251—24no—
90C22⋊D251—24no—
91C22⋊D251—48no—
92C2×Dic251—24no—
93C2×Dic251—24no—
94C2×Dic251C2×D414yes—
95C4×D251—28no—
96C4×D251—24no—
97C4×D251—24no—
98C23×C251—21no—
99C23×C251—22no—
100C23×C251—42no—
101C23×C251C2×D412yes—
102C23×C251C2411yescharacteristic
103Q8×C251—22no—
104D4×C251—24no—
105D4×C251—22no—
106D4×C251—44no—
107C2×C4×C251—22no—
108C2×C4×C251—22no—
109C2×C4×C251—22no—
110C2×C4×C251—21no—
111C2×C4×C251—22no—
112C2×C4×C251—41no—
113C2×C4×C251—42no—
114C2×C4×C251—41no—
115C2×C4×C251C2×D412yes—
116C2×C4×C251C2×D412yes—
117C2×C4×C251C2×D412yes—
118C2×C4×C251C2×D411yescharacteristic
119C2×C4×C251C2×D411yescharacteristic
120C22×C251—22no—
121C22×C251—24no—
122C22×C251—22no—
123C22×C251—41no—
124C22×C251—42no—
125C22×C251—42no—
126C22×C251—41no—
127C22×C251—41no—
128C22×C251C22×D412yes—
129C22×C251C22×D411yescharacteristic
130C2×D251—24no—
131C2×D251—44no—
132C2×D251—84no—
133C4×C251—22no—
134C4×C251—42no—
135C4×C251—41no—
136C4×C251#12011yescharacteristic
137C4×C251#12012yes—
138C4×C251#12011yescharacteristic
139Dic251—24no—
140Dic251—44no—
141C2×C251—22no—
142C2×C251—21no—
143C2×C251—42no—
144C2×C251—42no—
145C2×C251C2×#12011yescharacteristic
146D251—44no—
147C251#302511yescharacteristic, complemented, Sylow
148#3025—2511nomaximal, Sylow
149#585—2511no—
150#525—2512no—
151#525D25111yescharacteristic, complemented
152#458—2514no—
153B(4,𝔽2)—2514no—
154#410—2511no—
155#410—2512no—
1562−1+4—5022no—
1572−1+4—5021no—
158H(5,𝔽2)—5022no—
159H(5,𝔽2)—5021no—
160H(5,𝔽2)—2514no—
161C2×#41—5024no—
162C2×#41—5022no—
163C2×#41—2512no—
164C2×#41C2×D25111yescharacteristic
165C23×C4C2×D25111yescharacteristic, complemented
166#123—2514no—
167#123C2×D25112yescomplemented
168C22⋊Q8—21no—
169C22⋊Q8—5022no—
170#121—21no—
171#121—5022no—
172#120—2514no—
173#120C2×D25112yescomplemented
174C4×D4—22no—
175C4×D4—5024no—
176#117—2512no—
177#117C2×D25111yescharacteristic, complemented
178#99—2514no—
179#99—2518no—
180C24C22×D25111yescharacteristic
181#41—22no—
182#41—22no—
183#41—5024no—
184#41—5024no—
185#41—5024no—
186#41—5024no—
187#41—5028no—
188#41—5022no—
189#41—5022no—
190#41—5022no—
191#41—5024no—
192#41—5022no—
193C2×Q8—5022no—
194C2×Q8—5024no—
195C2×Q8—2512no—
196C2×Q8C22⋊D25111yescharacteristic, complemented
197C2×D4—22no—
198C2×D4—5022no—
199C2×D4—5024no—
200C2×D4—5028no—
201C2×D4—5024no—
202C2×D4—5024no—
203C2×D4—2514no—
204C2×D4—2512no—
205C2×D4C22⋊D25111yescharacteristic, complemented
206C2×D4C22×D25112yes—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—5024no—
212C22×C4—2514no—
213C22×C4C22⋊D25112yescomplemented
214C22×C4C22×D25111yescharacteristic
215C22×C4C22×D25111yescharacteristic
216C4⋊C4—21no—
217C4⋊C4—22no—
218C4⋊C4—5024no—
219C4⋊C4—5022no—
220C22⋊C4—22no—
221C22⋊C4—22no—
222C22⋊C4—5024no—
223C22⋊C4—5024no—
224C22⋊C4—2514no—
225C22⋊C4C22×D25112yes—
226C42—21no—
227C42—5022no—
228C23—22no—
229C23—21no—
230C23—42no—
231C23—5024no—
232C23—2514no—
233C23C2×C22⋊D25112yes—
234C23C23×D25111yescharacteristic
235Q8—22no—
236Q8—5024no—
237Q8—5022no—
238Q8—5022no—
239D4—24no—
240D4—22no—
241D4—44no—
242D4—10048no—
243D4—5024no—
244D4—5028no—
245D4—5024no—
246D4—5024no—
247D4—5022no—
248D4—5022no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—41no—
255C2×C4—42no—
256C2×C4—41no—
257C2×C4—5028no—
258C2×C4—5024no—
259C2×C4—5024no—
260C2×C4—5024no—
261C2×C4—5024no—
262C2×C4—2514no—
263C2×C4D4×D25112yes—
264C2×C4D4×D25112yes—
265C2×C4C2×C22⋊D25112yes—
266C2×C4C2×C22⋊D25111yescharacteristic
267C2×C4C2×C22⋊D25111yescharacteristic
268C22—22no—
269C22—24no—
270C22—22no—
271C22—41no—
272C22—42no—
273C22—41no—
274C22—42no—
275C22—41no—
276C22—20084no—
277C22—10044no—
278C22—5024no—
279C22C2×D4×D25112yes—
280C22C22×C22⋊D25111yescharacteristic
281C4—22no—
282C4—42no—
283C4—41no—
284C4—10044no—
285C4—5024no—
286C4#2254868311yescharacteristic
287C4#2254868312yes—
288C4#2254868311yescharacteristic
289C2—21no—
290C2—22no—
291C2—42no—
292C2—42no—
293C2—10044no—
294C2C2×#2254868311yescharacteristic
subgroupG/H
centerC4#22548683
commutator, cosocleC23×C251C24
FrattiniC23C23×D251
Fitting#525×C251C2
socleC2×C251C2×#120
hypercenter#525D251
hypocenterC251#3025
LCSG ▷ C23×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C4 ◁ C22×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. #30632523C2semidirect
2. #30632198C2semidirect
3. #30631876C2semidirect
4. #30631873C2semidirect
5. #30631661C2semidirect
6. #30631660C2semidirect
7. #525×C251C2semidirect
16. #22548698C22semidirect
19. #22548684C22semidirect
21. #22548674C22semidirect
22. #22548630C22semidirect
23. #22548629C22semidirect
25. C23×C4×C251C22semidirect
26. #123×C251C22semidirect
29. #120×C251C22semidirect
31. #117×C251C22semidirect
43. C2×Dic502D4semidirect
46. C2×D1004D4semidirect
53. C2×C4×D251D4semidirect
63. C2×Q8×C251D4semidirect
65. C2×D4×C251D4semidirect
71. C22×C4×C251D4semidirect
147. C251#3025semidirect
151. #525D251semidirect
165. C23×C4C2×D251semidirect
167. #123C2×D251semidirect
173. #120C2×D251semidirect
177. #117C2×D251semidirect
196. C2×Q8C22⋊D251semidirect
205. C2×D4C22⋊D251semidirect
213. C22×C4C22⋊D251semidirect

Maximal subgroups

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

subgroupindex
#306325232
#306321982
#306318762
#306318732
#306316612
#306316602
#525×C2512
#3025251