Finite Group Databasebeta

← #44559943

Subgroups — #44559943

Order 32128.

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

normal subgroups only

This group has 520 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×#22548735C211yesmaximal, characteristic, complemented
2C23×C251⋊C8C211yesmaximal, characteristic, complemented
3#30632198C211yesmaximal, characteristic, complemented
4#30631926C212yesmaximal, complemented
5#30631921C212yesmaximal, complemented
6#30631918C212yesmaximal, complemented
7#30631914C212yesmaximal, complemented
8#30631723C211yesmaximal, characteristic, complemented
9#30631683C211yesmaximal, characteristic, complemented
10#30631681C211yesmaximal, characteristic, complemented
11#525×C251C211yesmaximal, characteristic, complemented
12C2×#16860810C2211yescharacteristic
13#22548735—24no—
14C2×#16860800C2211yescharacteristic, complemented
15C2×#16860798C2211yescharacteristic, complemented
16C2×#16860797C2211yescharacteristic, complemented
17C2×#16860795C2211yescharacteristic, complemented
18C22×C251⋊C8—24no—
19C22×C251⋊C8C2211yescharacteristic
20C22×C251⋊C8C2212yes—
21C22×C251⋊C8C2212yes—
22C22×C251⋊C8C2211yescharacteristic
23#22548698—21no—
24#22548694C2212yes—
25#22548689C2212yes—
26#22548684—21no—
27C4×C22⋊D251—22no—
28#22548674C2211yescharacteristic
29#22548648C2212yescomplemented
30#22548646C2212yescomplemented
31#22548639C2212yescomplemented
32#22548638C2212yescomplemented
33#22548636C2212yescomplemented
34#22548635C2212yescomplemented
35C2×#41×C251C2211yescharacteristic
36C23×C4×C251C2211yescharacteristic
37#123×C251—21no—
38C22⋊Q8×C251C2212yes—
39#121×C251C2212yes—
40#120×C251—21no—
41C4×D4×C251—22no—
42#117×C251C2211yescharacteristic
43#16860810—22no—
44#16860810—22no—
45C2×Dic502C2311yescharacteristic
46C2×D1004C2311yescharacteristic
47C2×C22⋊D251—21no—
48C2×C22⋊D251—22no—
49C2×C4×D251—21no—
50#16860800—22no—
51#16860798—22no—
52#16860797—22no—
53#16860795—22no—
54C2×C251⋊C8—24no—
55C2×C251⋊C8—24no—
56C2×C251⋊C8—22no—
57C2×C251⋊C8D414yescomplemented
58C2×C251⋊C8C2312yes—
59C2×C251⋊C8C2311yescharacteristic
60C2×C251⋊C8C2311yescharacteristic
61C4⋊Dic251C2312yes—
62Dic251⋊C4—22no—
63C22⋊Dic251—21no—
64C22⋊Dic251—22no—
65(C2×D251)⋊C4—22no—
66C4×Dic251—21no—
67C24×C251D411yescharacteristic
68#41×C251—22no—
69#41×C251—22no—
70C2×Q8×C251C2311yescharacteristic
71C2×D4×C251—22no—
72C2×D4×C251—21no—
73C2×D4×C251C2311yescharacteristic
74C22×C4×C251—22no—
75C22×C4×C251—22no—
76C22×C4×C251—21no—
77C22×C4×C251D412yes—
78C22×C4×C251D411yescharacteristic
79C22×C4×C251C2312yes—
80C22×C4×C251C2311yescharacteristic
81C4⋊C4×C251—22no—
82C4⋊C4×C251C2312yes—
83C22⋊C4×C251—21no—
84C22⋊C4×C251—22no—
85C22⋊C4×C251—22no—
86C42×C251—21no—
87C22×D251—21no—
88Dic502—21no—
89Dic502—21no—
90D1004—21no—
91D1004—21no—
92C22⋊D251—42no—
93C22⋊D251—44no—
94C2×Dic251—22no—
95C2×Dic251—21no—
96C4×D251—42no—
97C251⋊C8—22no—
98C251⋊C8C2×D412yes—
99C251⋊C8C2×D412yes—
100C23×C251—22no—
101C23×C251—24no—
102C23×C251—21no—
103C23×C251C2×D411yescharacteristic
104C23×C251C2×D412yes—
105Q8×C251—21no—
106Q8×C251—21no—
107D4×C251—21no—
108D4×C251—21no—
109D4×C251—42no—
110D4×C251—44no—
111C2×C4×C251—22no—
112C2×C4×C251—24no—
113C2×C4×C251—22no—
114C2×C4×C251—22no—
115C2×C4×C251—22no—
116C2×C4×C251—21no—
117C2×C4×C251—42no—
118C2×C4×C251C2×D412yes—
119C2×C4×C251C2×D412yes—
120C2×C4×C251C2×D411yescharacteristic
121C2×C4×C251#4112yes—
122C2×C4×C251C2411yescharacteristic
123C22×C251—28no—
124C22×C251—22no—
125C22×C251—22no—
126C22×C251—22no—
127C22×C251—41no—
128C22×C251—42no—
129C22×C251#12112yes—
130C22×C251#13514yescomplemented
131C22×C251C22×D411yescharacteristic
132C2×D251—41no—
133C2×D251—42no—
134C4×C251—22no—
135C4×C251—41no—
136C4×C251—42no—
137C4×C251#12112yes—
138C4×C251C22×D411yescharacteristic
139C4×C251C2×#4111yescharacteristic
140Dic251—41no—
141Dic251—42no—
142C2×C251—22no—
143C2×C251—24no—
144C2×C251—81no—
145C2×C251C2×#12111yescharacteristic
146C2×C251C2×#13512yes—
147D251—81no—
148C251#305011yescharacteristic, complemented, Sylow
149#3050—2511nomaximal, Sylow
150C2×#135—2511no—
151C23×C8—2511no—
152#525—2511no—
153#525D25111yescharacteristic, complemented
154#467—2512no—
155#466—2512no—
156#465—2512no—
157#465—2512no—
158#427—2511no—
159#416—2511no—
160#416—2511no—
161C2×#41—2511no—
162C2×#41C2×D25111yescharacteristic
163C23×C4C2×D25111yescharacteristic
164#135—5024no—
165C2×Q16—2511no—
166C2×SD16—2511no—
167C2×SD16—2511no—
168C2×D8—2511no—
169C22×C8—5024no—
170C22×C8—2511no—
171C22×C8—2512no—
172C22×C8—2512no—
173C22×C8—2511no—
174#123—21no—
175#123—5021no—
176C22⋊Q8—2512no—
177C22⋊Q8C2×D25112yes—
178#121—2512no—
179#121C2×D25112yes—
180#120—21no—
181#120—5021no—
182C4×D4—22no—
183C4×D4—5022no—
184#117—2511no—
185#117C2×D25111yescharacteristic
186#107—2512no—
187#106—2512no—
188#103—2512no—
189#103—2512no—
190#102—2512no—
191#102—2512no—
192C24C22⋊D25111yescharacteristic
193#41—22no—
194#41—22no—
195#41—5022no—
196#41—5022no—
197C2×Q8—2511no—
198C2×Q8C22×D25111yescharacteristic
199C2×D4—22no—
200C2×D4—21no—
201C2×D4—5022no—
202C2×D4—5021no—
203C2×D4—2511no—
204C2×D4C22×D25111yescharacteristic
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—21no—
208C22×C4—5021no—
209C22×C4C22⋊D25111yescharacteristic
210C22×C4C22⋊D25112yes—
211C22×C4C22×D25112yes—
212C22×C4C22×D25111yescharacteristic
213Q16—5022no—
214SD16—5022no—
215SD16—5022no—
216D8—5022no—
217C2×C8—5022no—
218C2×C8—5024no—
219C2×C8—5024no—
220C2×C8—2512no—
221C2×C8—2514no—
222C2×C8—2511no—
223C2×C8—2511no—
224C4⋊C4—22no—
225C4⋊C4—5022no—
226C4⋊C4—2512no—
227C4⋊C4C22×D25112yes—
228C22⋊C4—21no—
229C22⋊C4—22no—
230C22⋊C4—22no—
231C22⋊C4—5021no—
232C22⋊C4—5022no—
233C22⋊C4—5022no—
234C42—21no—
235C42—5021no—
236C23—22no—
237C23—24no—
238C23—21no—
239C23—5021no—
240C23C2×C22⋊D25111yescharacteristic
241C23C2×C22⋊D25112yes—
242Q8—21no—
243Q8—21no—
244Q8—5021no—
245Q8—5021no—
246D4—21no—
247D4—21no—
248D4—42no—
249D4—44no—
250D4—10042no—
251D4—10044no—
252D4—5021no—
253D4—5021no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—24no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—42no—
261C2×C4—10042no—
262C2×C4—5022no—
263C2×C4—5021no—
264C2×C4D4×D25112yes—
265C2×C4C2×C22⋊D25112yes—
266C2×C4C2×C22⋊D25111yescharacteristic
267C2×C4#1686080912yes—
268C2×C4C23×D25111yescharacteristic
269C8—5022no—
270C8—2512no—
271C8—2512no—
272C22—28no—
273C22—22no—
274C22—22no—
275C22—22no—
276C22—42no—
277C22—41no—
278C22—10042no—
279C22—10041no—
280C22#2254868712yes—
281C22#2254873514yescomplemented
282C22C22×C22⋊D25111yescharacteristic
283C4—22no—
284C4—42no—
285C4—41no—
286C4—10042no—
287C4—10041no—
288C4#2254868712yes—
289C4C2×D4×D25111yescharacteristic
290C4C2×#1686080911yescharacteristic
291C2—22no—
292C2—24no—
293C2—81no—
294C2—20081no—
295C2C2×#2254868711yescharacteristic
296C2C2×#2254873512yes—
subgroupG/H
centerC2×C4C2×C22⋊D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#525×C251C2
socleC22×C251C22×D4
hypercenter#525D251
hypocenterC251#3050
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C2×#22548735C2semidirect
2. C23×C251⋊C8C2semidirect
3. #30632198C2semidirect
4. #30631926C2semidirect
5. #30631921C2semidirect
6. #30631918C2semidirect
7. #30631914C2semidirect
8. #30631723C2semidirect
9. #30631683C2semidirect
10. #30631681C2semidirect
11. #525×C251C2semidirect
14. C2×#16860800C22semidirect
15. C2×#16860798C22semidirect
16. C2×#16860797C22semidirect
17. C2×#16860795C22semidirect
29. #22548648C22semidirect
30. #22548646C22semidirect
31. #22548639C22semidirect
32. #22548638C22semidirect
33. #22548636C22semidirect
34. #22548635C22semidirect
57. C2×C251⋊C8D4semidirect
130. C22×C251#135semidirect
148. C251#3050semidirect
153. #525D251semidirect
281. C22#22548735semidirect

Maximal subgroups

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

subgroupindex
C2×#225487352
C23×C251⋊C82
#306321982
#306319262
#306319212
#306319182
#306319142
#306317232
#306316832
#306316812
#525×C2512
#3050251