Finite Group Databasebeta

← #44553130

Subgroups — #44553130

Order 32128.

596 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 181 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×#22548689C211yesmaximal, characteristic
2#30631587C212yesmaximal, complemented
3#30631578C212yesmaximal, complemented
4#30631570C212yesmaximal, complemented
5#30631565C212yesmaximal, complemented
6#30631563C212yesmaximal, complemented
7#30631550C211yesmaximal, characteristic, complemented
8C4×(C2×D251)⋊C4C212yesmaximal, complemented
9C4×C22⋊C4×C251C211yesmaximal, characteristic, complemented
10C22×D1004C2211yescharacteristic
11C22×C22⋊D251C2212yes—
12C22×C4×D251C2212yes—
13#22548689C418yescomplemented
14C2×C4⋊Dic251C2211yescharacteristic
15C2×Dic251⋊C4C2212yes—
16C2×C22⋊Dic251C2211yescharacteristic
17C2×C22⋊Dic251C2211yescharacteristic
18C2×(C2×D251)⋊C4C2212yes—
19C2×(C2×D251)⋊C4C2212yes—
20C2×(C2×D251)⋊C4C2212yes—
21C2×(C2×D251)⋊C4C2212yes—
22C2×C4×Dic251C2212yes—
23C251⋊#95C2212yes—
24C251⋊#95C2212yes—
25C251⋊#95C2212yes—
26C251⋊#95C2212yes—
27C23×C4×C251C2211yescharacteristic
28C2×C22⋊C4×C251C2212yes—
29C2×C42×C251C2211yescharacteristic
30C2×C42×C251C2211yescharacteristic
31#95×C251C2212yes—
32C23×D251C2312yes—
33C2×D1004—41no—
34C2×D1004—41no—
35C2×D1004C2×C414yes—
36C2×C22⋊D251—42no—
37C2×C22⋊D251—42no—
38C2×C22⋊D251C2×C418yescomplemented
39C22×Dic251C2312yes—
40C22×Dic251C2312yes—
41C22×Dic251C2312yes—
42C2×C4×D251—24no—
43C2×C4×D251—42no—
44C2×C4×D251—42no—
45C4⋊Dic251C2×C414yescomplemented
46Dic251⋊C4—42no—
47C22⋊Dic251—41no—
48C22⋊Dic251—41no—
49(C2×D251)⋊C4—24no—
50(C2×D251)⋊C4—42no—
51(C2×D251)⋊C4—42no—
52(C2×D251)⋊C4C2×C418yescomplemented
53C4×Dic251—42no—
54C24×C251C2311yescharacteristic
55C22×C4×C251—41no—
56C22×C4×C251—41no—
57C22×C4×C251C2×C414yescomplemented
58C22×C4×C251C2312yes—
59C22×C4×C251C2311yescharacteristic
60C22×C4×C251C2311yescharacteristic
61C22×C4×C251C2312yes—
62C22⋊C4×C251—24no—
63C42×C251—41no—
64C42×C251—41no—
65C22×D251—22no—
66C22×D251—22no—
67C22×D251—22no—
68C22×D251—22no—
69C22×D251—22no—
70C22×D251—22no—
71C22×D251C22×C414yes—
72D1004—44no—
73C22⋊D251—48no—
74C2×Dic251—22no—
75C2×Dic251—22no—
76C2×Dic251—22no—
77C2×Dic251—22no—
78C2×Dic251—22no—
79C2×Dic251—22no—
80C2×Dic251—22no—
81C2×Dic251—22no—
82C2×Dic251C22×C414yes—
83C4×D251—48no—
84C23×C251—22no—
85C23×C251—21no—
86C23×C251—21no—
87C23×C251—22no—
88C23×C251C22×C412yes—
89C23×C251C2411yescharacteristic
90C2×C4×C251—21no—
91C2×C4×C251—21no—
92C2×C4×C251—22no—
93C2×C4×C251—21no—
94C2×C4×C251—21no—
95C2×C4×C251—22no—
96C2×C4×C251—22no—
97C2×C4×C251—22no—
98C2×C4×C251—22no—
99C2×C4×C251—44no—
100C2×C4×C251C22×C412yes—
101C2×C4×C251C22×C412yes—
102C2×C4×C251#4114yes—
103C22×C251—22no—
104C22×C251—22no—
105C22×C251—22no—
106C22×C251—42no—
107C22×C251—42no—
108C22×C251C23×C411yescharacteristic
109C22×C251C2×#4112yes—
110C22×C251H(5,𝔽2)11yescharacteristic
111C22×C251H(5,𝔽2)11yescharacteristic
112C22×C251H(5,𝔽2)12yes—
113C2×D251—24no—
114C2×D251—24no—
115C2×D251—24no—
116C2×D251—42no—
117C2×D251—42no—
118C2×D251—42no—
119C2×D251—42no—
120C4×C251—22no—
121C4×C251—22no—
122C4×C251—24no—
123C4×C251—42no—
124Dic251—24no—
125Dic251—42no—
126Dic251—42no—
127C2×C251—42no—
128C2×C251C4×#4111yescharacteristic
129C2×C251#51811yescharacteristic
130C2×C251#51811yescharacteristic
131C2×C251#53812yes—
132C2×C251#54012yes—
133D251—44no—
134C251#233311yescharacteristic, complemented, Sylow
135#2333—2511nomaximal, Sylow
136C2×#121—2511no—
137#390—2512no—
138#388—2512no—
139#387—2512no—
140#386—2512no—
141#386—2512no—
142#381—2511no—
143C4×C22⋊C4—2512no—
144C4×C22⋊C4D25111yescharacteristic, complemented
145C22×D4—2512no—
146C22×D4—2511no—
147C23×C4—2512no—
148C23×C4C2×D25111yescharacteristic
149#121—2518no—
150C2×C4⋊C4—2512no—
151C2×C4⋊C4—2511no—
152C2×C22⋊C4—2511no—
153C2×C22⋊C4—2511no—
154C2×C22⋊C4—2512no—
155C2×C22⋊C4—2512no—
156C2×C22⋊C4—2512no—
157C2×C22⋊C4—2512no—
158C2×C22⋊C4C2×D25112yes—
159C2×C42—2512no—
160C2×C42C2×D25111yescharacteristic
161C2×C42C2×D25111yescharacteristic
162#95—2512no—
163#95—2512no—
164#95—2512no—
165#95—2512no—
166#95C2×D25112yes—
167C24—2512no—
168C24C22×D25111yescharacteristic
169C2×D4—10041no—
170C2×D4—10041no—
171C2×D4—10042no—
172C2×D4—10042no—
173C2×D4—2518no—
174C2×D4—2514no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—10042no—
178C22×C4—10042no—
179C22×C4—5024no—
180C22×C4—2512no—
181C22×C4—2512no—
182C22×C4—2512no—
183C22×C4C4×D25114yescomplemented
184C22×C4C22×D25112yes—
185C22×C4C22×D25111yescharacteristic
186C22×C4C22×D25111yescharacteristic
187C22×C4C22×D25112yes—
188C4⋊C4—10042no—
189C4⋊C4—2514no—
190C22⋊C4—24no—
191C22⋊C4—10042no—
192C22⋊C4—10042no—
193C22⋊C4—10041no—
194C22⋊C4—10041no—
195C22⋊C4—5024no—
196C22⋊C4—2518no—
197C42—41no—
198C42—41no—
199C42—10042no—
200C23—22no—
201C23—21no—
202C23—21no—
203C23—22no—
204C23—5022no—
205C23—5022no—
206C23—5022no—
207C23—5022no—
208C23—5022no—
209C23—5022no—
210C23—2514no—
211C23C2×C4×D25112yes—
212C23C23×D25111yescharacteristic
213D4—10044no—
214D4—10048no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—44no—
225C2×C4—10048no—
226C2×C4—5022no—
227C2×C4—5022no—
228C2×C4—5022no—
229C2×C4—5022no—
230C2×C4—5022no—
231C2×C4—5022no—
232C2×C4—5022no—
233C2×C4—5022no—
234C2×C4—2514no—
235C2×C4C2×C4×D25112yes—
236C2×C4C2×C4×D25112yes—
237C2×C4#1686081014yes—
238C22—22no—
239C22—22no—
240C22—22no—
241C22—42no—
242C22—42no—
243C22—10042no—
244C22—10042no—
245C22—10042no—
246C22—10042no—
247C22—5024no—
248C22—5024no—
249C22—5024no—
250C22C22×C4×D25111yescharacteristic
251C22C2×#1686081012yes—
252C22#2254876011yescharacteristic
253C22#2254876011yescharacteristic
254C22#2254876112yes—
255C4—24no—
256C4—22no—
257C4—22no—
258C4—42no—
259C4—10042no—
260C4—10042no—
261C4—5024no—
262C2—42no—
263C2—10044no—
264C2C4×#1686081011yescharacteristic
265C2#3063216011yescharacteristic
266C2#3063216111yescharacteristic
267C2#3063227012yes—
268C2#3063228812yes—
subgroupG/H
center, FrattiniC23C23×D251
commutatorC22×C251C23×C4
FittingC4×C22⋊C4×C251C2
socle, cosocleC23×C251C24
prime radicalC2×#22548689C2
hypercenterC4×C22⋊C4D251
hypocenterC251#2333
LCSG ▷ C22×C251 ▷ C251
UCSC1 ◁ C23 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. #30631587C2semidirect
3. #30631578C2semidirect
4. #30631570C2semidirect
5. #30631565C2semidirect
6. #30631563C2semidirect
7. #30631550C2semidirect
8. C4×(C2×D251)⋊C4C2semidirect
9. C4×C22⋊C4×C251C2semidirect
13. #22548689C4semidirect
38. C2×C22⋊D251C2×C4semidirect
45. C4⋊Dic251C2×C4semidirect
52. (C2×D251)⋊C4C2×C4semidirect
57. C22×C4×C251C2×C4semidirect
134. C251#2333semidirect
144. C4×C22⋊C4D251semidirect
183. C22×C4C4×D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225486892
#306315872
#306315782
#306315702
#306315652
#306315632
#306315502
C4×(C2×D251)⋊C42
C4×C22⋊C4×C2512
#2333251