Finite Group Databasebeta

← #44559642

Subgroups — #44559642

Order 32128.

752 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 752 conjugacy classes of subgroups — too many to draw. Only its 165 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×#22548745C211yesmaximal, characteristic, complemented
2C22×#16860798C211yesmaximal, characteristic, complemented
3#30632258C211yesmaximal, characteristic, complemented
4#30631849C214yesmaximal, complemented
5#30631838C214yesmaximal, complemented
6#30631692C211yesmaximal, characteristic, complemented
7#30631683C211yesmaximal, characteristic, complemented
8#30631654C211yesmaximal, characteristic, complemented
9C2×H(5,𝔽2)×C251C211yesmaximal, characteristic
10C22×Dic502C2211yescharacteristic, complemented
11#22548745—24no—
12C2×#16860800C2212yescomplemented
13C2×#16860798—24no—
14C2×#16860798C2214yescomplemented
15C2×#16860798C2212yescomplemented
16C2×#16860794C2211yescharacteristic, complemented
17C22×C251⋊C8C2211yescharacteristic, complemented
18#22548713—21no—
19#22548702—21no—
20#22548696—22no—
21#22548694C2214yescomplemented
22#22548674C2211yescharacteristic, complemented
23#22548639C2212yescomplemented
24#22548636C2212yescomplemented
25#22548636C2214yescomplemented
26C22⋊(C251⋊C8)C2214yescomplemented
27H(5,𝔽2)×C251—24no—
28H(5,𝔽2)×C251—24no—
29C2×#41×C251—21no—
30C2×#41×C251C2212yes—
31C2×#41×C251C2211yescharacteristic
32C2×#41×C251C2211yescharacteristic
33C22×D4×C251—22no—
34C22×D4×C251—21no—
35C22×D4×C251C2212yes—
36C22×D4×C251C2211yescharacteristic
37C2×Dic502—21no—
38C2×Dic502—22no—
39C2×Dic502—22no—
40C2×Dic502C2312yes—
41C22×Dic251—21no—
42#16860800—24no—
43#16860798—24no—
44#16860798—28no—
45#16860794—22no—
46C2×C251⋊C8—22no—
47C2×C251⋊C8C2312yes—
48C2×C251⋊C8C2312yes—
49C4⋊Dic251C2312yes—
50Dic251⋊C4—24no—
51C22⋊Dic251—21no—
52C22⋊Dic251—24no—
53C4×Dic251—21no—
54C24×C251—22no—
55C24×C251—21no—
56#41×C251—28no—
57#41×C251—24no—
58#41×C251—24no—
59#41×C251—22no—
60#41×C251—22no—
61#41×C251—22no—
62#41×C251D414yes—
63C2×Q8×C251D411yescharacteristic
64C2×Q8×C251C2311yescharacteristic
65C2×D4×C251—216no—
66C2×D4×C251—28no—
67C2×D4×C251—22no—
68C2×D4×C251—21no—
69C2×D4×C251—24no—
70C2×D4×C251—24no—
71C2×D4×C251—22no—
72C2×D4×C251—21no—
73C2×D4×C251—22no—
74C2×D4×C251D412yes—
75C2×D4×C251D411yescharacteristic
76C2×D4×C251D414yes—
77C2×D4×C251C2311yescharacteristic
78C2×D4×C251C2312yes—
79C22×C4×C251—22no—
80C22×C4×C251—21no—
81C22×C4×C251C2312yes—
82C22×C4×C251C2311yescharacteristic
83Dic502—22no—
84Dic502—22no—
85Dic502—42no—
86C2×Dic251—22no—
87C2×Dic251—22no—
88C2×Dic251—42no—
89C251⋊C8—22no—
90C251⋊C8—22no—
91C23×C251—28no—
92C23×C251—22no—
93C23×C251—21no—
94C23×C251—24no—
95C23×C251—82no—
96C23×C251—84no—
97C23×C251C2×D412yes—
98C23×C251C2×D411yescharacteristic
99Q8×C251—22no—
100Q8×C251—21no—
101Q8×C251C2×D412yes—
102D4×C251—28no—
103D4×C251—28no—
104D4×C251—24no—
105D4×C251—24no—
106D4×C251—22no—
107D4×C251—24no—
108D4×C251—22no—
109D4×C251—21no—
110D4×C251C2×D412yes—
111D4×C251C2×D414yes—
112C2×C4×C251—28no—
113C2×C4×C251—21no—
114C2×C4×C251—24no—
115C2×C4×C251—22no—
116C2×C4×C251—22no—
117C2×C4×C251C2×D412yes—
118C2×C4×C251C2×D411yescharacteristic
119C2×C4×C251C2×D412yes—
120C2×C4×C251C2×D412yes—
121C2×C4×C251C2411yescharacteristic
122C22×C251—22no—
123C22×C251—22no—
124C22×C251—24no—
125C22×C251—22no—
126C22×C251—24no—
127C22×C251—44no—
128C22×C251—42no—
129C22×C251—84no—
130C22×C251—88no—
131C22×C251#12012yes—
132C22×C251C22×D411yescharacteristic
133C4×C251—22no—
134C4×C251—22no—
135C4×C251#12012yes—
136C4×C251C22×D411yescharacteristic
137C4×C251C22×D411yescharacteristic
138Dic251—42no—
139Dic251—42no—
140C2×C251—22no—
141C2×C251—42no—
142C2×C251—44no—
143C2×C251—42no—
144C2×C251C2×#12011yescharacteristic
145C2×C251#57712yes—
146C251#300911yescharacteristic, complemented, Sylow
147#3009—2511nomaximal, Sylow
148C2×H(5,𝔽2)D25111yescharacteristic
149C2×#137—2511no—
150C22×SD16—2511no—
151#536—2511no—
152#452—2514no—
153#450—2514no—
154#418—2511no—
155#416—2511no—
156#408—2511no—
157H(5,𝔽2)—24no—
158H(5,𝔽2)—24no—
159C2×#41—21no—
160C2×#41C2×D25112yes—
161C2×#41C2×D25111yescharacteristic
162C2×#41C2×D25111yescharacteristic
163C22×Q8—2511no—
164C22×D4—22no—
165C22×D4—21no—
166C22×D4C2×D25112yes—
167C22×D4C2×D25111yescharacteristic
168#137—5024no—
169C2×Q16—2512no—
170C2×SD16—5024no—
171C2×SD16—2514no—
172C2×SD16—2512no—
173C2×M16—2511no—
174C22×C8—2511no—
175C4⋊Q8—5021no—
176#124—5021no—
177#123—5022no—
178C22⋊Q8—2514no—
179#117—2511no—
180#103—2512no—
181#102—2512no—
182#102—2514no—
183C22⋊C8—2514no—
184C24—22no—
185C24—21no—
186#41—28no—
187#41—24no—
188#41—24no—
189#41—22no—
190#41—22no—
191#41—22no—
192#41C22⋊D25114yes—
193C2×Q8—5022no—
194C2×Q8—5022no—
195C2×Q8—5021no—
196C2×Q8—2512no—
197C2×Q8C22⋊D25111yescharacteristic
198C2×Q8C22×D25111yescharacteristic
199C2×D4—216no—
200C2×D4—28no—
201C2×D4—22no—
202C2×D4—21no—
203C2×D4—24no—
204C2×D4—24no—
205C2×D4—22no—
206C2×D4—21no—
207C2×D4—22no—
208C2×D4C22⋊D25112yes—
209C2×D4C22⋊D25111yescharacteristic
210C2×D4C22⋊D25114yes—
211C2×D4C22×D25111yescharacteristic
212C2×D4C22×D25112yes—
213C22×C4—22no—
214C22×C4—21no—
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—5024no—
226C4⋊C4—2512no—
227C22⋊C4—5021no—
228C22⋊C4—5024no—
229C42—5021no—
230C23—28no—
231C23—22no—
232C23—21no—
233C23—24no—
234C23—82no—
235C23—84no—
236C23C2×C22⋊D25112yes—
237C23C2×C22⋊D25111yescharacteristic
238Q8—21no—
239Q8—22no—
240Q8—10042no—
241Q8—5022no—
242Q8—5022no—
243Q8C2×C22⋊D25112yes—
244D4—28no—
245D4—28no—
246D4—24no—
247D4—24no—
248D4—22no—
249D4—21no—
250D4—22no—
251D4—24no—
252D4C2×C22⋊D25112yes—
253D4C2×C22⋊D25114yes—
254C2×C4—28no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—24no—
259C2×C4—10042no—
260C2×C4—5022no—
261C2×C4—5022no—
262C2×C4C2×C22⋊D25112yes—
263C2×C4C2×C22⋊D25112yes—
264C2×C4C2×C22⋊D25112yes—
265C2×C4C2×C22⋊D25111yescharacteristic
266C2×C4C23×D25111yescharacteristic
267C8—5022no—
268C8—5022no—
269C22—22no—
270C22—24no—
271C22—22no—
272C22—24no—
273C22—22no—
274C22—44no—
275C22—42no—
276C22—84no—
277C22—88no—
278C22#2254868412yes—
279C22C22×C22⋊D25111yescharacteristic
280C4—22no—
281C4—22no—
282C4—10042no—
283C4—10042no—
284C4#2254868412yes—
285C4C22×C22⋊D25111yescharacteristic
286C4C22×C22⋊D25111yescharacteristic
287C2—22no—
288C2—42no—
289C2—44no—
290C2—42no—
291C2C2×#2254868411yescharacteristic
292C2#3063249812yes—
subgroupG/H
centerC22C22×C22⋊D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting, prime radicalC2×H(5,𝔽2)×C251C2
socleC22×C251C22×D4
hypercenterC2×H(5,𝔽2)D251
hypocenterC251#3009
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C2×#41 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. C2×#22548745C2semidirect
2. C22×#16860798C2semidirect
3. #30632258C2semidirect
4. #30631849C2semidirect
5. #30631838C2semidirect
6. #30631692C2semidirect
7. #30631683C2semidirect
8. #30631654C2semidirect
10. C22×Dic502C22semidirect
12. C2×#16860800C22semidirect
14. C2×#16860798C22semidirect
15. C2×#16860798C22semidirect
16. C2×#16860794C22semidirect
17. C22×C251⋊C8C22semidirect
21. #22548694C22semidirect
22. #22548674C22semidirect
23. #22548639C22semidirect
24. #22548636C22semidirect
25. #22548636C22semidirect
26. C22⋊(C251⋊C8)C22semidirect
146. C251#3009semidirect

Maximal subgroups

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

subgroupindex
C2×#225487452
C22×#168607982
#306322582
#306318492
#306318382
#306316922
#306316832
#306316542
C2×H(5,𝔽2)×C2512
#3009251