Finite Group Databasebeta

← #44559640

Subgroups — #44559640

Order 32128.

752 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 752 conjugacy classes of subgroups — too many to draw. Only its 149 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×#22548761C211yesmaximal, characteristic
2C2×#22548746C211yesmaximal, characteristic, complemented
3C22×#16860797C211yesmaximal, characteristic, complemented
4#30631847C214yesmaximal, complemented
5#30631837C214yesmaximal, complemented
6#30631691C211yesmaximal, characteristic, complemented
7#30631681C211yesmaximal, characteristic, complemented
8#30631654C211yesmaximal, characteristic, complemented
9#536×C251C211yesmaximal, characteristic, complemented
10#22548761—24no—
11#22548761—24no—
12C2×#16860811—21no—
13C2×#16860810C2212yes—
14C2×#16860810C2211yescharacteristic
15C22×D1004C2212yes—
16C22×D1004C2211yescharacteristic
17C2×D4×D251—22no—
18C2×D4×D251—21no—
19#22548746—24no—
20C2×#16860800C2212yescomplemented
21C2×#16860797—24no—
22C2×#16860797C2214yescomplemented
23C2×#16860797C2212yescomplemented
24C2×#16860794C2211yescharacteristic, complemented
25C22×C251⋊C8C2211yescharacteristic, complemented
26#22548638C2212yescomplemented
27#22548635C2212yescomplemented
28#22548635C2214yescomplemented
29C22⋊(C251⋊C8)C2214yescomplemented
30C2×#41×C251C2211yescharacteristic
31C22×Q8×C251C2211yescharacteristic, complemented
32C4⋊Q8×C251—21no—
33#124×C251—21no—
34#123×C251—22no—
35C22⋊Q8×C251C2214yescomplemented
36#117×C251C2211yescharacteristic, complemented
37C23×D251—22no—
38C23×D251—21no—
39#16860811—28no—
40#16860810—24no—
41#16860810—24no—
42#16860810—22no—
43#16860810D414yes—
44C2×Dic502C2311yescharacteristic
45C2×D1004—24no—
46C2×D1004—24no—
47C2×D1004—22no—
48C2×D1004—21no—
49C2×D1004—22no—
50C2×D1004D414yes—
51C2×D1004C2311yescharacteristic
52C2×D1004C2312yes—
53C2×C22⋊D251—22no—
54C2×C22⋊D251—21no—
55D4×D251—216no—
56D4×D251—28no—
57C2×C4×D251—22no—
58C2×C4×D251—21no—
59#16860800—24no—
60#16860797—24no—
61#16860797—28no—
62#16860794—22no—
63C2×C251⋊C8—22no—
64C2×C251⋊C8C2312yes—
65C2×C251⋊C8C2312yes—
66#41×C251—22no—
67#41×C251—22no—
68C2×Q8×C251—21no—
69C2×Q8×C251—22no—
70C2×Q8×C251—22no—
71C2×Q8×C251D411yescharacteristic
72C2×Q8×C251C2312yes—
73C2×D4×C251D412yes—
74C2×D4×C251D411yescharacteristic
75C22×C4×C251—21no—
76C22×C4×C251C2312yes—
77C22×C4×C251C2311yescharacteristic
78C4⋊C4×C251—24no—
79C4⋊C4×C251C2312yes—
80C22⋊C4×C251—21no—
81C22⋊C4×C251—24no—
82C42×C251—21no—
83C22×D251—22no—
84C22×D251—21no—
85C22×D251—28no—
86C22×D251—24no—
87C22×D251—82no—
88C22×D251—84no—
89Dic502—21no—
90Dic502C2×D412yes—
91D1004—28no—
92D1004—24no—
93D1004—22no—
94D1004—21no—
95D1004C2×D412yes—
96D1004C2×D414yes—
97C22⋊D251—28no—
98C22⋊D251—24no—
99C2×Dic251—21no—
100C4×D251—28no—
101C4×D251—24no—
102C251⋊C8—22no—
103C251⋊C8—22no—
104C23×C251C2×D412yes—
105C23×C251C2×D411yescharacteristic
106Q8×C251—22no—
107Q8×C251—22no—
108Q8×C251—22no—
109Q8×C251—42no—
110D4×C251—24no—
111D4×C251—22no—
112C2×C4×C251—22no—
113C2×C4×C251—22no—
114C2×C4×C251—22no—
115C2×C4×C251—22no—
116C2×C4×C251—42no—
117C2×C4×C251C2×D412yes—
118C2×C4×C251C2×D412yes—
119C2×C4×C251C2×D412yes—
120C2×C4×C251C2×D411yescharacteristic
121C2×C4×C251C2411yescharacteristic
122C22×C251—22no—
123C22×C251—22no—
124C22×C251—24no—
125C22×C251#12012yes—
126C22×C251C22×D411yescharacteristic
127C2×D251—24no—
128C2×D251—22no—
129C2×D251—44no—
130C2×D251—42no—
131C2×D251—84no—
132C2×D251—88no—
133C4×C251—22no—
134C4×C251—42no—
135C4×C251—42no—
136C4×C251#12012yes—
137C4×C251C22×D411yescharacteristic
138C4×C251C22×D411yescharacteristic
139Dic251—22no—
140C2×C251—22no—
141C2×C251—42no—
142C2×C251C2×#12011yescharacteristic
143C2×C251#57712yes—
144D251—44no—
145D251—42no—
146C251#300911yescharacteristic, complemented, Sylow
147#3009—2511nomaximal, Sylow
148C2×H(5,𝔽2)—2511no—
149C2×#137—2511no—
150C22×SD16—2511no—
151#536D25111yescharacteristic, complemented
152#452—2514no—
153#450—2514no—
154#418—2511no—
155#416—2511no—
156#408—2511no—
157H(5,𝔽2)—5024no—
158H(5,𝔽2)—5024no—
159C2×#41—5021no—
160C2×#41—2512no—
161C2×#41—2511no—
162C2×#41C2×D25111yescharacteristic
163C22×Q8C2×D25111yescharacteristic, complemented
164C22×D4—5022no—
165C22×D4—5021no—
166C22×D4—2512no—
167C22×D4—2511no—
168#137—5024no—
169C2×Q16—2512no—
170C2×SD16—5024no—
171C2×SD16—2514no—
172C2×SD16—2512no—
173C2×M16—2511no—
174C22×C8—2511no—
175C4⋊Q8—21no—
176#124—21no—
177#123—22no—
178C22⋊Q8C2×D25114yescomplemented
179#117C2×D25111yescharacteristic, complemented
180#103—2512no—
181#102—2512no—
182#102—2514no—
183C22⋊C8—2514no—
184C24—5022no—
185C24—5021no—
186#41—22no—
187#41—22no—
188#41—5024no—
189#41—5024no—
190#41—5028no—
191#41—5022no—
192#41—2514no—
193C2×Q8—21no—
194C2×Q8—22no—
195C2×Q8—22no—
196C2×Q8—2511no—
197C2×Q8C22⋊D25111yescharacteristic
198C2×Q8C22×D25112yes—
199C2×D4—5024no—
200C2×D4—5024no—
201C2×D4—50216no—
202C2×D4—5028no—
203C2×D4—5022no—
204C2×D4—5021no—
205C2×D4—5022no—
206C2×D4—5022no—
207C2×D4—5021no—
208C2×D4—2514no—
209C2×D4—2511no—
210C2×D4—2512no—
211C2×D4C22⋊D25112yes—
212C2×D4C22⋊D25111yescharacteristic
213C22×C4—21no—
214C22×C4—5022no—
215C22×C4—5021no—
216C22×C4C22×D25112yes—
217C22×C4C22×D25111yescharacteristic
218Q16—5024no—
219SD16—5028no—
220SD16—5024no—
221M16—5022no—
222C2×C8—5022no—
223C2×C8—2512no—
224C2×C8—2512no—
225C4⋊C4—24no—
226C4⋊C4C22×D25112yes—
227C22⋊C4—21no—
228C22⋊C4—24no—
229C42—21no—
230C23—20082no—
231C23—20084no—
232C23—5028no—
233C23—5024no—
234C23—5022no—
235C23—5021no—
236C23C2×C22⋊D25112yes—
237C23C2×C22⋊D25111yescharacteristic
238Q8—22no—
239Q8—22no—
240Q8—22no—
241Q8—42no—
242Q8—5021no—
243Q8—2512no—
244D4—24no—
245D4—22no—
246D4—5028no—
247D4—5028no—
248D4—5024no—
249D4—5022no—
250D4—5021no—
251D4—5024no—
252D4—2512no—
253D4—2514no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—42no—
259C2×C4—5028no—
260C2×C4—5021no—
261C2×C4—5024no—
262C2×C4D4×D25112yes—
263C2×C4D4×D25112yes—
264C2×C4C2×C22⋊D25112yes—
265C2×C4C2×C22⋊D25111yescharacteristic
266C2×C4C23×D25111yescharacteristic
267C8—5022no—
268C8—5022no—
269C22—22no—
270C22—24no—
271C22—22no—
272C22—20084no—
273C22—20088no—
274C22—10044no—
275C22—10042no—
276C22—5024no—
277C22—5022no—
278C22#2254868312yes—
279C22C22×C22⋊D25111yescharacteristic
280C4—22no—
281C4—42no—
282C4—42no—
283C4—5022no—
284C4#2254868312yes—
285C4C2×D4×D25111yescharacteristic
286C4C2×D4×D25111yescharacteristic
287C2—22no—
288C2—42no—
289C2—10044no—
290C2—10042no—
291C2C2×#2254868311yescharacteristic
292C2#3063249512yes—
subgroupG/H
centerC22C22×C22⋊D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#536×C251C2
socleC22×C251C22×D4
prime radicalC2×#22548761C2
hypercenter#536D251
hypocenterC251#3009
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C2×#41 ◁ #536

Complements

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

NH ≅ G/Ntype
2. C2×#22548746C2semidirect
3. C22×#16860797C2semidirect
4. #30631847C2semidirect
5. #30631837C2semidirect
6. #30631691C2semidirect
7. #30631681C2semidirect
8. #30631654C2semidirect
9. #536×C251C2semidirect
20. C2×#16860800C22semidirect
22. C2×#16860797C22semidirect
23. C2×#16860797C22semidirect
24. C2×#16860794C22semidirect
25. C22×C251⋊C8C22semidirect
26. #22548638C22semidirect
27. #22548635C22semidirect
28. #22548635C22semidirect
29. C22⋊(C251⋊C8)C22semidirect
31. C22×Q8×C251C22semidirect
35. C22⋊Q8×C251C22semidirect
36. #117×C251C22semidirect
146. C251#3009semidirect
151. #536D251semidirect
163. C22×Q8C2×D251semidirect
178. C22⋊Q8C2×D251semidirect
179. #117C2×D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487612
C2×#225487462
C22×#168607972
#306318472
#306318372
#306316912
#306316812
#306316542
#536×C2512
#3009251