Finite Group Databasebeta

← #44553503

Subgroups — #44553503

Order 32128.

688 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 688 conjugacy classes of subgroups — too many to draw. Only its 173 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
1C2×#22548683C211yesmaximal, characteristic
2#30631590C211yesmaximal, characteristic, complemented
3#30631587C212yesmaximal, complemented
4#30631581C212yesmaximal, complemented
5#30631579C212yesmaximal, complemented
6#30631576C212yesmaximal, complemented
7#30631566C212yesmaximal, complemented
8#30631542C212yesmaximal, complemented
9#386×C251C211yesmaximal, characteristic, complemented
10C24×D251C2211yescharacteristic
11C22×C22⋊D251C2212yes—
12#22548683C418yescomplemented
13C2×C4⋊Dic251C2211yescharacteristic, complemented
14C2×Dic251⋊C4C2212yescomplemented
15C2×C22⋊Dic251C2212yes—
16C2×C22⋊Dic251C2211yescharacteristic
17C2×C22⋊Dic251C2212yescomplemented
18C2×(C2×D251)⋊C4C2212yes—
19C2×(C2×D251)⋊C4C2212yes—
20C2×(C2×D251)⋊C4C2212yes—
21C2×(C2×D251)⋊C4C2212yes—
22C2×(C2×D251)⋊C4C2212yes—
23C2×C4×Dic251C2211yescharacteristic, complemented
24C2×C4×Dic251C2212yescomplemented
25C251⋊#95C2212yescomplemented
26C251⋊#95C2212yescomplemented
27C22×D4×C251C2211yescharacteristic
28C23×C4×C251C2211yescharacteristic, complemented
29C2×C22⋊C4×C251C2212yes—
30C2×C22⋊C4×C251C2211yescharacteristic, complemented
31#95×C251C2212yescomplemented
32C23×D251—42no—
33C23×D251—41no—
34C23×D251—41no—
35C23×D251—42no—
36C23×D251C2×C414yescomplemented
37C23×D251C2312yes—
38C2×C22⋊D251—42no—
39C2×C22⋊D251—42no—
40C2×C22⋊D251C2×C418yes—
41C22×Dic251C2312yes—
42C22×Dic251C2312yes—
43C22×Dic251C2312yes—
44C4⋊Dic251—41no—
45Dic251⋊C4—42no—
46C22⋊Dic251—42no—
47C22⋊Dic251—42no—
48C22⋊Dic251C2×C414yescomplemented
49(C2×D251)⋊C4—42no—
50(C2×D251)⋊C4—42no—
51(C2×D251)⋊C4—42no—
52(C2×D251)⋊C4—42no—
53(C2×D251)⋊C4C2×C418yescomplemented
54C4×Dic251—41no—
55C4×Dic251—42no—
56C24×C251C2311yescharacteristic
57C24×C251C2311yescharacteristic
58C2×D4×C251—41no—
59C2×D4×C251—41no—
60C2×D4×C251C2×C414yes—
61C22×C4×C251—42no—
62C22×C4×C251—41no—
63C22×C4×C251C2312yes—
64C22×C4×C251C2312yes—
65C22×C4×C251C2311yescharacteristic
66C22⋊C4×C251—42no—
67C22⋊C4×C251—41no—
68C22×D251—22no—
69C22×D251—22no—
70C22×D251—22no—
71C22×D251—22no—
72C22×D251—22no—
73C22×D251—22no—
74C22×D251—48no—
75C22×D251—44no—
76C22×D251—48no—
77C22×D251—44no—
78C22×D251—44no—
79C22×D251C22×C414yes—
80C22⋊D251—48no—
81C2×Dic251—22no—
82C2×Dic251—22no—
83C2×Dic251—22no—
84C2×Dic251—22no—
85C2×Dic251—22no—
86C2×Dic251—22no—
87C2×Dic251—22no—
88C2×Dic251—22no—
89C2×Dic251C22×C414yes—
90C23×C251—21no—
91C23×C251—22no—
92C23×C251—21no—
93C23×C251—22no—
94C23×C251—21no—
95C23×C251—21no—
96C23×C251—22no—
97C23×C251—22no—
98C23×C251C22×C412yes—
99C23×C251C22×C412yes—
100C23×C251C2411yescharacteristic
101D4×C251—44no—
102C2×C4×C251—21no—
103C2×C4×C251—21no—
104C2×C4×C251—22no—
105C2×C4×C251—22no—
106C2×C4×C251—22no—
107C2×C4×C251—22no—
108C2×C4×C251—22no—
109C2×C4×C251—22no—
110C2×C4×C251—44no—
111C2×C4×C251C22×C412yes—
112C22×C251—22no—
113C22×C251—22no—
114C22×C251—24no—
115C22×C251—24no—
116C22×C251—42no—
117C22×C251—41no—
118C22×C251—42no—
119C22×C251—41no—
120C22×C251—41no—
121C22×C251—41no—
122C22×C251C23×C411yescharacteristic
123C22×C251H(5,𝔽2)11yescharacteristic
124C22×C251H(5,𝔽2)12yes—
125C22×C251H(5,𝔽2)11yescharacteristic
126C22×C251H(5,𝔽2)12yes—
127C2×D251—24no—
128C2×D251—24no—
129C2×D251—24no—
130C2×D251—416no—
131C2×D251—42no—
132C2×D251—42no—
133C2×D251—42no—
134C2×D251—42no—
135C4×C251—22no—
136C4×C251—42no—
137C4×C251—42no—
138Dic251—24no—
139Dic251—42no—
140Dic251—42no—
141C2×C251—42no—
142C2×C251—42no—
143C2×C251#51812yes—
144C2×C251#51811yescharacteristic
145C2×C251#56012yes—
146C2×C251#56011yescharacteristic
147C2×C251H(3,𝔽4)11yescharacteristic
148D251—44no—
149C251#237511yescharacteristic, complemented, Sylow
150#2375—2511nomaximal, Sylow
151C2×#120—2511no—
152#390—2511no—
153#390—2512no—
154#388—2512no—
155#388—2512no—
156#388—2512no—
157#386—2512no—
158#386D25111yescharacteristic, complemented
159#379—2512no—
160C25—2511no—
161C22×D4—2512no—
162C22×D4C2×D25111yescharacteristic
163C23×C4C2×D25111yescharacteristic, complemented
164#120—2518no—
165C2×C4⋊C4—2512no—
166C2×C4⋊C4—2511no—
167C2×C22⋊C4—2512no—
168C2×C22⋊C4—2512no—
169C2×C22⋊C4—2512no—
170C2×C22⋊C4—2512no—
171C2×C22⋊C4—2512no—
172C2×C22⋊C4—2512no—
173C2×C22⋊C4—2511no—
174C2×C22⋊C4—2512no—
175C2×C22⋊C4C2×D25112yes—
176C2×C22⋊C4C2×D25111yescharacteristic, complemented
177C2×C42—2511no—
178C2×C42—2512no—
179#95—2512no—
180#95—2512no—
181#95C2×D25112yescomplemented
182C24—10042no—
183C24—10041no—
184C24—10041no—
185C24—10042no—
186C24—2514no—
187C24—2512no—
188C24C22×D25111yescharacteristic
189C24C22×D25111yescharacteristic
190C2×D4—41no—
191C2×D4—41no—
192C2×D4—10042no—
193C2×D4—10042no—
194C2×D4—2518no—
195C2×D4C4×D25114yes—
196C22×C4—42no—
197C22×C4—41no—
198C22×C4—2512no—
199C22×C4—2512no—
200C22×C4—2512no—
201C22×C4C22×D25112yes—
202C22×C4C22×D25112yes—
203C22×C4C22×D25111yescharacteristic
204C4⋊C4—10042no—
205C4⋊C4—10041no—
206C22⋊C4—42no—
207C22⋊C4—41no—
208C22⋊C4—10042no—
209C22⋊C4—10042no—
210C22⋊C4—10042no—
211C22⋊C4—10042no—
212C22⋊C4—10042no—
213C22⋊C4—10042no—
214C22⋊C4—2514no—
215C22⋊C4—2518no—
216C42—10041no—
217C42—10042no—
218C23—21no—
219C23—22no—
220C23—21no—
221C23—22no—
222C23—21no—
223C23—21no—
224C23—22no—
225C23—22no—
226C23—10048no—
227C23—10044no—
228C23—10048no—
229C23—10044no—
230C23—10044no—
231C23—5022no—
232C23—5022no—
233C23—5022no—
234C23—5022no—
235C23—5022no—
236C23—5022no—
237C23—2514no—
238C23C2×C4×D25112yes—
239C23C2×C4×D25112yes—
240C23C23×D25111yescharacteristic
241D4—44no—
242D4—10048no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—44no—
252C2×C4—5022no—
253C2×C4—5022no—
254C2×C4—5022no—
255C2×C4—5022no—
256C2×C4—5022no—
257C2×C4—5022no—
258C2×C4—5022no—
259C2×C4—5022no—
260C2×C4—2514no—
261C2×C4C2×C4×D25112yes—
262C22—24no—
263C22—24no—
264C22—22no—
265C22—22no—
266C22—42no—
267C22—42no—
268C22—41no—
269C22—41no—
270C22—41no—
271C22—41no—
272C22—100416no—
273C22—10042no—
274C22—10042no—
275C22—10042no—
276C22—10042no—
277C22—5024no—
278C22—5024no—
279C22—5024no—
280C22C22×C4×D25111yescharacteristic
281C22#2254876012yes—
282C22#2254876011yescharacteristic
283C22#2254876011yescharacteristic
284C22#2254876112yes—
285C4—22no—
286C4—42no—
287C4—42no—
288C4—10042no—
289C4—10042no—
290C4—5024no—
291C2—42no—
292C2—42no—
293C2—10044no—
294C2#3063215912yes—
295C2#3063216011yescharacteristic
296C2#3063242512yes—
297C2#3063242711yescharacteristic
298C2#3063243011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D251
commutatorC22×C251C23×C4
Fitting#386×C251C2
socle, cosocleC23×C251C24
prime radicalC2×#22548683C2
hypercenter#386D251
hypocenterC251#2375
LCSG ▷ C22×C251 ▷ C251
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. #30631590C2semidirect
3. #30631587C2semidirect
4. #30631581C2semidirect
5. #30631579C2semidirect
6. #30631576C2semidirect
7. #30631566C2semidirect
8. #30631542C2semidirect
9. #386×C251C2semidirect
12. #22548683C4semidirect
13. C2×C4⋊Dic251C22semidirect
14. C2×Dic251⋊C4C22semidirect
17. C2×C22⋊Dic251C22semidirect
23. C2×C4×Dic251C22semidirect
24. C2×C4×Dic251C22semidirect
25. C251⋊#95C22semidirect
26. C251⋊#95C22semidirect
28. C23×C4×C251C22semidirect
30. C2×C22⋊C4×C251C22semidirect
31. #95×C251C22semidirect
36. C23×D251C2×C4semidirect
48. C22⋊Dic251C2×C4semidirect
53. (C2×D251)⋊C4C2×C4semidirect
149. C251#2375semidirect
158. #386D251semidirect
163. C23×C4C2×D251semidirect
176. C2×C22⋊C4C2×D251semidirect
181. #95C2×D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225486832
#306315902
#306315872
#306315812
#306315792
#306315762
#306315662
#306315422
#386×C2512
#2375251