Finite Group Databasebeta

← #44558359

Subgroups — #44558359

Order 32128.

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

normal subgroups only

This group has 712 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
1C2×#22548683C212yesmaximal, complemented
2#30631616C212yesmaximal, complemented
3#30631612C211yesmaximal, characteristic, complemented
4#30631603C212yesmaximal, complemented
5#30631602C212yesmaximal, complemented
6#30631601C212yesmaximal, complemented
7#30631595C212yesmaximal, complemented
8#30631542C211yesmaximal, characteristic, complemented
9#392×C251C211yesmaximal, characteristic, complemented
10C24×D251C2211yescharacteristic
11C22×C22⋊D251C2212yes—
12C22×C22⋊D251C2212yes—
13#22548683—28no—
14C2×C4⋊Dic251C2211yescharacteristic, complemented
15C2×Dic251⋊C4C2212yescomplemented
16C2×C22⋊Dic251C2212yes—
17C2×C22⋊Dic251C2211yescharacteristic
18C2×C22⋊Dic251C2212yescomplemented
19C2×C22⋊Dic251C2212yescomplemented
20C2×(C2×D251)⋊C4C2212yes—
21C2×(C2×D251)⋊C4C2212yes—
22C2×(C2×D251)⋊C4C2212yes—
23C2×(C2×D251)⋊C4C2212yes—
24C251⋊#95C2212yescomplemented
25C251⋊#95C2212yescomplemented
26C251⋊#95C2211yescharacteristic, complemented
27C22×D4×C251C2212yes—
28C22×D4×C251C2211yescharacteristic, complemented
29C2×C22⋊C4×C251C2211yescharacteristic
30C2×C22⋊C4×C251C2212yescomplemented
31#95×C251C2211yescharacteristic, complemented
32C23×D251—41no—
33C23×D251—42no—
34C23×D251—42no—
35C23×D251—41no—
36C23×D251D414yescomplemented
37C23×D251C2312yes—
38C2×C22⋊D251—24no—
39C2×C22⋊D251—24no—
40C2×C22⋊D251—42no—
41C2×C22⋊D251—42no—
42C2×C22⋊D251—42no—
43C2×C22⋊D251—42no—
44C22×Dic251C2312yes—
45C22×Dic251C2312yes—
46C22×Dic251C2312yes—
47C4⋊Dic251—41no—
48Dic251⋊C4—42no—
49C22⋊Dic251—24no—
50C22⋊Dic251—41no—
51C22⋊Dic251—42no—
52C22⋊Dic251—42no—
53(C2×D251)⋊C4—24no—
54(C2×D251)⋊C4—24no—
55(C2×D251)⋊C4—42no—
56(C2×D251)⋊C4—42no—
57C24×C251C2312yes—
58C24×C251C2311yescharacteristic
59C2×D4×C251—24no—
60C2×D4×C251—42no—
61C2×D4×C251—42no—
62C2×D4×C251—41no—
63C2×D4×C251—42no—
64C22×C4×C251C2311yescharacteristic
65C22×C4×C251C2312yes—
66C22×C4×C251C2311yescharacteristic
67C22⋊C4×C251—41no—
68C22⋊C4×C251—42no—
69C22×D251—22no—
70C22×D251—22no—
71C22×D251—22no—
72C22×D251—22no—
73C22×D251—22no—
74C22×D251—22no—
75C22×D251—44no—
76C22×D251—48no—
77C22×D251—44no—
78C22×D251—48no—
79C22×D251—44no—
80C22×D251C2×D414yes—
81C22⋊D251—48no—
82C22⋊D251—48no—
83C2×Dic251—22no—
84C2×Dic251—22no—
85C2×Dic251—22no—
86C2×Dic251—22no—
87C2×Dic251—22no—
88C2×Dic251—22no—
89C2×Dic251—22no—
90C2×Dic251—22no—
91C2×Dic251—22no—
92C23×C251—21no—
93C23×C251—22no—
94C23×C251—22no—
95C23×C251—21no—
96C23×C251—22no—
97C23×C251—22no—
98C23×C251—22no—
99C23×C251—22no—
100C23×C251—22no—
101C23×C251—22no—
102C23×C251—22no—
103C23×C251C2×D412yes—
104C23×C251C2411yescharacteristic
105D4×C251—48no—
106D4×C251—44no—
107C2×C4×C251—22no—
108C2×C4×C251—22no—
109C2×C4×C251—21no—
110C2×C4×C251—22no—
111C2×C4×C251—21no—
112C2×C4×C251—22no—
113C2×C4×C251—22no—
114C22×C251—24no—
115C22×C251—22no—
116C22×C251—41no—
117C22×C251—42no—
118C22×C251—41no—
119C22×C251—42no—
120C22×C251—42no—
121C22×C251—42no—
122C22×C251—42no—
123C22×C251—42no—
124C22×C251—42no—
125C22×C251—42no—
126C22×C251C22×D411yescharacteristic
127C22×C251H(5,𝔽2)12yes—
128C22×C251H(5,𝔽2)11yescharacteristic
129C22×C251H(5,𝔽2)12yes—
130C22×C251H(5,𝔽2)11yescharacteristic
131C2×D251—24no—
132C2×D251—24no—
133C2×D251—24no—
134C2×D251—416no—
135C2×D251—42no—
136C2×D251—42no—
137C2×D251—42no—
138C2×D251—42no—
139C4×C251—42no—
140C4×C251—41no—
141C4×C251—41no—
142Dic251—42no—
143Dic251—42no—
144Dic251—42no—
145C2×C251—42no—
146C2×C251—82no—
147C2×C251#53411yescharacteristic
148C2×C251#53412yes—
149C2×C251#56011yescharacteristic
150C2×C251#56012yes—
151C2×C251H(3,𝔽4)11yescharacteristic
152D251—44no—
153C251#281211yescharacteristic, complemented, Sylow
154#2812—2511nomaximal, Sylow
155C2×#120—2512no—
156#397—2511no—
157#397—2512no—
158#394—2512no—
159#394—2512no—
160#394—2512no—
161#392—2512no—
162#392D25111yescharacteristic, complemented
163#379—2511no—
164C25—2511no—
165C22×D4—2512no—
166C22×D4—2512no—
167C22×D4C2×D25112yes—
168C22×D4C2×D25111yescharacteristic, complemented
169#120—5028no—
170C2×C4⋊C4—2512no—
171C2×C4⋊C4—2511no—
172C2×C22⋊C4—2511no—
173C2×C22⋊C4—2512no—
174C2×C22⋊C4—2512no—
175C2×C22⋊C4—2512no—
176C2×C22⋊C4—2512no—
177C2×C22⋊C4—2512no—
178C2×C22⋊C4—2512no—
179C2×C22⋊C4—2512no—
180C2×C22⋊C4C2×D25111yescharacteristic
181C2×C22⋊C4C2×D25112yescomplemented
182#95—2512no—
183#95—2511no—
184#95—2512no—
185#95C2×D25111yescharacteristic, complemented
186C24—10041no—
187C24—10042no—
188C24—10042no—
189C24—10041no—
190C24—2514no—
191C24—2512no—
192C24C22×D25112yes—
193C24C22×D25111yescharacteristic
194C2×D4—24no—
195C2×D4—42no—
196C2×D4—42no—
197C2×D4—41no—
198C2×D4—42no—
199C2×D4—10042no—
200C2×D4—10042no—
201C2×D4—10042no—
202C2×D4—10042no—
203C2×D4—5024no—
204C2×D4—5024no—
205C22×C4—2512no—
206C22×C4—2512no—
207C22×C4—2512no—
208C22×C4C22×D25111yescharacteristic
209C22×C4C22×D25112yes—
210C22×C4C22×D25111yescharacteristic
211C4⋊C4—10042no—
212C4⋊C4—10041no—
213C22⋊C4—41no—
214C22⋊C4—42no—
215C22⋊C4—10042no—
216C22⋊C4—10042no—
217C22⋊C4—10042no—
218C22⋊C4—10041no—
219C22⋊C4—10042no—
220C22⋊C4—5024no—
221C22⋊C4—5024no—
222C22⋊C4—5024no—
223C23—21no—
224C23—22no—
225C23—22no—
226C23—21no—
227C23—22no—
228C23—22no—
229C23—22no—
230C23—22no—
231C23—22no—
232C23—22no—
233C23—22no—
234C23—10044no—
235C23—10048no—
236C23—10044no—
237C23—10048no—
238C23—10044no—
239C23—5022no—
240C23—5022no—
241C23—5022no—
242C23—5022no—
243C23—5022no—
244C23—5022no—
245C23—2514no—
246C23D4×D25112yes—
247C23C23×D25111yescharacteristic
248D4—48no—
249D4—44no—
250D4—10048no—
251D4—10048no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—5022no—
260C2×C4—5022no—
261C2×C4—5022no—
262C2×C4—5022no—
263C2×C4—5022no—
264C2×C4—5022no—
265C2×C4—5022no—
266C2×C4—5022no—
267C2×C4—5022no—
268C22—22no—
269C22—24no—
270C22—41no—
271C22—42no—
272C22—41no—
273C22—42no—
274C22—42no—
275C22—42no—
276C22—42no—
277C22—42no—
278C22—42no—
279C22—42no—
280C22—100416no—
281C22—10042no—
282C22—10042no—
283C22—10042no—
284C22—10042no—
285C22—5024no—
286C22—5024no—
287C22—5024no—
288C22C2×D4×D25111yescharacteristic
289C22#2254876011yescharacteristic
290C22#2254876012yes—
291C22#2254876012yes—
292C22#2254876111yescharacteristic
293C4—42no—
294C4—41no—
295C4—41no—
296C4—10042no—
297C4—10042no—
298C4—10042no—
299C2—42no—
300C2—82no—
301C2—10044no—
302C2#3063224511yescharacteristic
303C2#3063224712yes—
304C2#3063242411yescharacteristic
305C2#3063242512yes—
306C2#3063243011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, cosocleC23×C251C24
Fitting#392×C251C2
hypercenter#392D251
hypocenterC251#2812
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C2×#22548683C2semidirect
2. #30631616C2semidirect
3. #30631612C2semidirect
4. #30631603C2semidirect
5. #30631602C2semidirect
6. #30631601C2semidirect
7. #30631595C2semidirect
8. #30631542C2semidirect
9. #392×C251C2semidirect
14. C2×C4⋊Dic251C22semidirect
15. C2×Dic251⋊C4C22semidirect
18. C2×C22⋊Dic251C22semidirect
19. C2×C22⋊Dic251C22semidirect
24. C251⋊#95C22semidirect
25. C251⋊#95C22semidirect
26. C251⋊#95C22semidirect
28. C22×D4×C251C22semidirect
30. C2×C22⋊C4×C251C22semidirect
31. #95×C251C22semidirect
36. C23×D251D4semidirect
153. C251#2812semidirect
162. #392D251semidirect
168. C22×D4C2×D251semidirect
181. C2×C22⋊C4C2×D251semidirect
185. #95C2×D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225486832
#306316162
#306316122
#306316032
#306316022
#306316012
#306315952
#306315422
#392×C2512
#2812251