Finite Group Databasebeta

← #44550818

Subgroups — #44550818

Order 32128.

598 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 598 conjugacy classes of subgroups — too many to draw. Only its 101 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×#22548684C211yesmaximal, characteristic
2C2×#22548629C212yesmaximal, complemented
3#30631543C211yesmaximal, characteristic, complemented
4#30631453C212yesmaximal, complemented
5C22×C22⋊C4×C251C211yesmaximal, characteristic, complemented
6C22×C22⋊D251C412yescomplemented
7C22×C22⋊D251C2211yescharacteristic
8#22548684—21no—
9#22548684—21no—
10#22548684—22no—
11C2×C22⋊Dic251—22no—
12C2×C22⋊Dic251C412yescomplemented
13C2×C22⋊Dic251C2212yes—
14C2×C22⋊Dic251C2211yescharacteristic
15#22548629—24no—
16C25×C251C2211yescharacteristic
17C23×C4×C251—21no—
18C2×C22⋊C4×C251—21no—
19C2×C22⋊C4×C251—24no—
20C2×C22⋊C4×C251C2212yes—
21C23×D251C2×C411yescharacteristic, complemented
22C2×C22⋊D251—22no—
23C2×C22⋊D251—22no—
24C2×C22⋊D251—22no—
25C2×C22⋊D251—21no—
26C2×C22⋊D251—21no—
27C2×C22⋊D251—22no—
28C2×C22⋊D251—44no—
29C22×Dic251—22no—
30C22×Dic251C2×C411yescharacteristic, complemented
31C22×Dic251C2×C412yescomplemented
32C22⋊Dic251—22no—
33C22⋊Dic251—22no—
34C22⋊Dic251—21no—
35C22⋊Dic251—21no—
36C22⋊Dic251—22no—
37C22⋊Dic251—22no—
38C22⋊Dic251—44no—
39C24×C251—22no—
40C24×C251—22no—
41C24×C251—44no—
42C24×C251C2×C412yes—
43C24×C251D411yescharacteristic
44C24×C251D411yescharacteristic
45C24×C251D412yes—
46C24×C251C2311yescharacteristic
47C22×C4×C251—24no—
48C22×C4×C251—22no—
49C22×C4×C251—44no—
50C22⋊C4×C251—24no—
51C22⋊C4×C251—22no—
52C22⋊C4×C251D414yescomplemented
53C22×D251—21no—
54C22×D251—22no—
55C22×D251—42no—
56C22⋊D251—44no—
57C22⋊D251—44no—
58C22⋊D251—41no—
59C22⋊D251—42no—
60C22⋊D251—41no—
61C2×Dic251—21no—
62C2×Dic251—22no—
63C2×Dic251—22no—
64C2×Dic251—42no—
65C2×Dic251—44no—
66C2×Dic251—42no—
67C23×C251—24no—
68C23×C251—21no—
69C23×C251—21no—
70C23×C251—24no—
71C23×C251—24no—
72C23×C251—24no—
73C23×C251—21no—
74C23×C251—24no—
75C23×C251—21no—
76C23×C251—42no—
77C23×C251—42no—
78C23×C251—42no—
79C23×C251—42no—
80C23×C251—48no—
81C23×C251—48no—
82C23×C251C22⋊C412yes—
83C23×C251C22⋊C412yes—
84C23×C251C22×C411yescharacteristic
85C23×C251C2×D411yescharacteristic
86C23×C251C2×D412yes—
87C23×C251C2×D411yescharacteristic
88C23×C251#4112yes—
89C2×C4×C251—24no—
90C2×C4×C251—48no—
91C2×C4×C251—44no—
92C22×C251—24no—
93C22×C251—24no—
94C22×C251—22no—
95C22×C251—22no—
96C22×C251—22no—
97C22×C251—22no—
98C22×C251—22no—
99C22×C251—22no—
100C22×C251—22no—
101C22×C251—22no—
102C22×C251—42no—
103C22×C251—42no—
104C22×C251—48no—
105C22×C251—48no—
106C22×C251—42no—
107C22×C251—42no—
108C22×C251#9914yescomplemented
109C22×C251C2×C22⋊C411yescharacteristic
110C22×C251C4×D412yes—
111C22×C251#12011yescharacteristic
112C22×C251#12112yes—
113C22×C251#12311yescharacteristic
114C2×D251—41no—
115C2×D251—42no—
116C2×D251—82no—
117C4×C251—44no—
118Dic251—41no—
119Dic251—42no—
120Dic251—82no—
121C2×C251—22no—
122C2×C251—21no—
123C2×C251—22no—
124C2×C251—24no—
125C2×C251—21no—
126C2×C251—42no—
127C2×C251#38611yescharacteristic
128C2×C251C2×#9912yes—
129D251—81no—
130C251#189811yescharacteristic, complemented, Sylow
131#1898—2511nomaximal, Sylow
132C2×#120—2511no—
133C22×C22⋊C4D25111yescharacteristic, complemented
134C2×#99—2512no—
135#379—2511no—
136#342—2512no—
137C25C2×D25111yescharacteristic
138C22×D4—2512no—
139C22×D4—2511no—
140C23×C4—21no—
141#120—5021no—
142#120—5021no—
143#120—5022no—
144C2×C22⋊C4—21no—
145C2×C22⋊C4—24no—
146C2×C22⋊C4—5022no—
147C2×C22⋊C4—2512no—
148C2×C22⋊C4—2512no—
149C2×C22⋊C4—2511no—
150C2×C22⋊C4C2×D25112yes—
151#99—5024no—
152C24—22no—
153C24—22no—
154C24—44no—
155C24—2511no—
156C24C4×D25112yes—
157C24C22⋊D25111yescharacteristic
158C24C22⋊D25111yescharacteristic
159C24D100412yes—
160C24C22×D25111yescharacteristic
161C2×D4—10044no—
162C2×D4—5022no—
163C2×D4—5022no—
164C2×D4—5022no—
165C2×D4—5021no—
166C2×D4—5021no—
167C2×D4—5022no—
168C22×C4—24no—
169C22×C4—22no—
170C22×C4—44no—
171C22×C4—5022no—
172C22×C4—2512no—
173C22×C4—2511no—
174C22⋊C4—24no—
175C22⋊C4—22no—
176C22⋊C4—10044no—
177C22⋊C4—5022no—
178C22⋊C4—5022no—
179C22⋊C4—5022no—
180C22⋊C4—5022no—
181C22⋊C4—5021no—
182C22⋊C4—5021no—
183C22⋊C4C22⋊D25114yescomplemented
184C23—24no—
185C23—21no—
186C23—21no—
187C23—24no—
188C23—24no—
189C23—24no—
190C23—21no—
191C23—24no—
192C23—21no—
193C23—42no—
194C23—42no—
195C23—42no—
196C23—42no—
197C23—48no—
198C23—48no—
199C23—10042no—
200C23—5022no—
201C23—5021no—
202C23(C2×D251)⋊C412yes—
203C23(C2×D251)⋊C412yes—
204C23C2×C4×D25111yescharacteristic
205C23C2×C22⋊D25111yescharacteristic
206C23C2×C22⋊D25112yes—
207C23C2×D100411yescharacteristic
208C23#1686081012yes—
209D4—10044no—
210D4—10044no—
211D4—10041no—
212D4—10042no—
213D4—10041no—
214C2×C4—24no—
215C2×C4—48no—
216C2×C4—44no—
217C2×C4—10044no—
218C2×C4—10042no—
219C2×C4—10042no—
220C2×C4—5021no—
221C2×C4—5022no—
222C2×C4—5022no—
223C22—24no—
224C22—24no—
225C22—22no—
226C22—22no—
227C22—22no—
228C22—22no—
229C22—22no—
230C22—22no—
231C22—22no—
232C22—22no—
233C22—42no—
234C22—42no—
235C22—48no—
236C22—48no—
237C22—42no—
238C22—42no—
239C22—20082no—
240C22—10041no—
241C22—10042no—
242C22#2254862914yescomplemented
243C22C2×(C2×D251)⋊C411yescharacteristic
244C22C4×C22⋊D25112yes—
245C22#2254868411yescharacteristic
246C22#2254868912yes—
247C22#2254869811yescharacteristic
248C4—44no—
249C4—20082no—
250C4—10041no—
251C4—10042no—
252C2—21no—
253C2—22no—
254C2—24no—
255C2—21no—
256C2—22no—
257C2—42no—
258C2—20081no—
259C2#3063156611yescharacteristic
260C2C2×#2254862912yes—
subgroupG/H
centerC22C2×(C2×D251)⋊C4
commutatorC23×C251C22×C4
FrattiniC24C22×D251
FittingC22×C22⋊C4×C251C2
socleC22×C251C2×C22⋊C4
prime radicalC2×#22548684C2
cosocleC24×C251C23
hypercenterC22×C22⋊C4D251
hypocenterC251#1898
LCSG ▷ C23×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C25 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#22548629C2semidirect
3. #30631543C2semidirect
4. #30631453C2semidirect
5. C22×C22⋊C4×C251C2semidirect
6. C22×C22⋊D251C4semidirect
12. C2×C22⋊Dic251C4semidirect
21. C23×D251C2×C4semidirect
30. C22×Dic251C2×C4semidirect
31. C22×Dic251C2×C4semidirect
52. C22⋊C4×C251D4semidirect
108. C22×C251#99semidirect
130. C251#1898semidirect
133. C22×C22⋊C4D251semidirect
183. C22⋊C4C22⋊D251semidirect
242. C22#22548629semidirect

Maximal subgroups

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

subgroupindex
C2×#225486842
C2×#225486292
#306315432
#306314532
C22×C22⋊C4×C2512
#1898251