Finite Group Databasebeta

← #30641550

Subgroups — #30641550

Order 16256.

1252 conjugacy classes of subgroups; only the 253 normal subgroups are shown.

normal subgroups only

This group has 1252 conjugacy classes of subgroups — too many to draw. Only its 253 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
1C23×D508C211yesmaximal, characteristic
2C2×#16861171C212yesmaximal, complemented
3C2×C22⋊C4×D127C212yesmaximal, complemented
4#22549777C212yesmaximal, complemented
5#22549754C214yesmaximal, complemented
6#22549731C212yesmaximal, complemented
7C4×C22⋊Dic127C211yesmaximal, characteristic, complemented
8#380×C127C211yesmaximal, characteristic, complemented
9C24×D127C2212yes—
10C22×D508—28no—
11C22×D508C418yescomplemented
12C22×D508C2211yescharacteristic
13C22×D508C2212yes—
14C22×D508C2211yescharacteristic
15C22×C4×D127C2214yescomplemented
16#16861171—28no—
17C2×C22⋊Dic127C2211yescharacteristic
18C2×C22⋊Dic127C2211yescharacteristic
19C2×(C2×D127)⋊C4C2214yescomplemented
20C2×(C2×D127)⋊C4C2214yescomplemented
21C2×(C2×D127)⋊C4C2214yescomplemented
22C22⋊C4×D127—28no—
23C2×C4×Dic127C2212yescomplemented
24C127⋊#95C2212yescomplemented
25C23×C4×C127C2211yescharacteristic
26C2×C4⋊C4×C127C2212yescomplemented
27C2×C22⋊C4×C127C2212yes—
28#95×C127C2212yescomplemented
29C23×D127—28no—
30C23×D127—24no—
31C23×D127—24no—
32C23×D127—44no—
33C23×D127C2×C418yescomplemented
34C23×D127C2312yes—
35C23×D127C2312yes—
36C2×D508—28no—
37C2×D508—232no—
38C2×D508—24no—
39C2×D508C2×C414yes—
40C2×D508C2×C418yescomplemented
41C2×D508C2×C414yescomplemented
42C2×D508D418yes—
43C22×Dic127C2312yes—
44C22×Dic127C2312yes—
45C2×C4×D127—28no—
46C2×C4×D127—28no—
47C2×C4×D127—44no—
48C22⋊Dic127—22no—
49C22⋊Dic127D414yescomplemented
50(C2×D127)⋊C4—28no—
51(C2×D127)⋊C4—28no—
52(C2×D127)⋊C4—44no—
53C4×Dic127—24no—
54C24×C127C2311yescharacteristic
55C22×C4×C127—24no—
56C22×C4×C127C2×C414yescomplemented
57C22×C4×C127C2314yes—
58C22×C4×C127C2311yescharacteristic
59C22×C4×C127C2311yescharacteristic
60C4⋊C4×C127—24no—
61C22⋊C4×C127—24no—
62C22×D127—22no—
63C22×D127—22no—
64C22×D127—24no—
65C22×D127—216no—
66C22×D127—22no—
67C22×D127—24no—
68C22×D127—22no—
69C22×D127—24no—
70C22×D127—216no—
71C22×D127—48no—
72C22×D127—416no—
73C22×D127—416no—
74C22×D127C22×C414yes—
75C22×D127C22×C414yes—
76C22×D127C2×D418yes—
77D508—216no—
78D508—28no—
79D508C22⋊C4116yescomplemented
80C2×Dic127—22no—
81C2×Dic127—22no—
82C2×Dic127—22no—
83C2×Dic127—22no—
84C2×Dic127—22no—
85C2×Dic127C2×D414yescomplemented
86C4×D127—416no—
87C23×C127—22no—
88C23×C127—22no—
89C23×C127C22×C412yes—
90C23×C127C2×D412yes—
91C23×C127#4112yes—
92C23×C127C2411yescharacteristic
93C2×C4×C127—22no—
94C2×C4×C127—28no—
95C2×C4×C127—24no—
96C2×C4×C127—24no—
97C2×C4×C127—24no—
98C2×C4×C127C22×C412yes—
99C2×C4×C127C22×C412yes—
100C2×C4×C127C2×D414yes—
101C22×C127—22no—
102C22×C127—22no—
103C22×C127—24no—
104C22×C127—24no—
105C22×C127C4×D414yescomplemented
106C22×C127C23×C411yescharacteristic
107C22×C127C22×D412yes—
108C22×C127C22×D411yescharacteristic
109C22×C127C2×#4111yescharacteristic
110C22×C127H(5,𝔽2)12yes—
111C2×D127—24no—
112C2×D127—24no—
113C2×D127—28no—
114C2×D127—28no—
115C2×D127—24no—
116C2×D127—24no—
117C2×D127—24no—
118C2×D127—432no—
119C2×D127—44no—
120C2×D127—44no—
121C2×D127C2×C22⋊C418yescomplemented
122C4×C127—22no—
123C4×C127—44no—
124C4×C127C2×C22⋊C414yescomplemented
125Dic127—24no—
126Dic127—42no—
127C2×C127—24no—
128C2×C127C22×C22⋊C411yescharacteristic
129C2×C127C2×C4×D411yescharacteristic
130C2×C127#51811yescharacteristic
131C2×C127D4212yes—
132C2×C127#54612yes—
133D127—28no—
134D127—44no—
135D4×C22⋊C4—1271nomaximal, Sylow
136C127D4×C22⋊C411yescharacteristic, complemented, Sylow
137C23×D4—1271no—
138C2×C4×D4—1272no—
139C22×C22⋊C4—1272no—
140#390—1272no—
141#386—1274no—
142#380D12711yescharacteristic, complemented
143#379—1272no—
144C4×C22⋊C4—1271no—
145C25—1272no—
146C22×D4—1278no—
147C22×D4—1271no—
148C22×D4—1272no—
149C22×D4—1271no—
150C22×D4—2548no—
151C23×C4—1274no—
152C23×C4C2×D12711yescharacteristic
153C4×D4—2548no—
154C2×C4⋊C4C2×D12712yescomplemented
155C2×C22⋊C4—1271no—
156C2×C22⋊C4—1274no—
157C2×C22⋊C4—1271no—
158C2×C22⋊C4—1274no—
159C2×C22⋊C4—1274no—
160C2×C22⋊C4—2548no—
161C2×C22⋊C4C2×D12712yes—
162C2×C42—1272no—
163#95—1272no—
164#95C2×D12712yescomplemented
165C24—1272no—
166C24—1278no—
167C24—1272no—
168C24—5084no—
169C24—2548no—
170C24—2544no—
171C24—2544no—
172C24C22×D12711yescharacteristic
173C2×D4—1274no—
174C2×D4—1278no—
175C2×D4—1274no—
176C2×D4—1278no—
177C2×D4—25432no—
178C2×D4—2548no—
179C2×D4—2544no—
180C22×C4—24no—
181C22×C4—1272no—
182C22×C4—1272no—
183C22×C4—5084no—
184C22×C4—2548no—
185C22×C4—2548no—
186C22×C4C4×D12714yescomplemented
187C22×C4C22×D12714yes—
188C22×C4C22×D12711yescharacteristic
189C22×C4C22×D12711yescharacteristic
190C4⋊C4—24no—
191C22⋊C4—24no—
192C22⋊C4—1274no—
193C22⋊C4—5084no—
194C22⋊C4—2548no—
195C22⋊C4—2548no—
196C22⋊C4—2542no—
197C42—2544no—
198C23—22no—
199C23—22no—
200C23—1274no—
201C23—1278no—
202C23—1274no—
203C23—50816no—
204C23—50816no—
205C23—5088no—
206C23—25416no—
207C23—2542no—
208C23—2544no—
209C23—2542no—
210C23—2544no—
211C23—25416no—
212C23—2542no—
213C23—2542no—
214C23—2544no—
215C23C2×C4×D12712yes—
216C23D4×D12712yes—
217C23#1452324912yes—
218C23C23×D12711yescharacteristic
219D4—12716no—
220D4—25416no—
221D4—2548no—
222C2×C4—22no—
223C2×C4—28no—
224C2×C4—24no—
225C2×C4—24no—
226C2×C4—24no—
227C2×C4—1274no—
228C2×C4—50816no—
229C2×C4—2542no—
230C2×C4—2542no—
231C2×C4—2542no—
232C2×C4—2542no—
233C2×C4—2542no—
234C2×C4C2×C4×D12712yes—
235C2×C4C2×C4×D12712yes—
236C2×C4D4×D12714yes—
237C22—22no—
238C22—22no—
239C22—24no—
240C22—24no—
241C22—1278no—
242C22—50832no—
243C22—5084no—
244C22—5084no—
245C22—2544no—
246C22—2544no—
247C22—2544no—
248C22—2544no—
249C22—2544no—
250C22—2548no—
251C22—2548no—
252C22#1686117114yescomplemented
253C22C22×C4×D12711yescharacteristic
254C22C2×D4×D12712yes—
255C22C2×D4×D12711yescharacteristic
256C22C2×#1452324911yescharacteristic
257C22#1686125512yes—
258C4—22no—
259C4—44no—
260C4—5082no—
261C4—2544no—
262C4C22⋊C4×D12714yescomplemented
263C2—24no—
264C2—5084no—
265C2—2548no—
266C2C2×C22⋊C4×D12711yescharacteristic
267C2C2×#1686117111yescharacteristic
268C2#2255035011yescharacteristic
269C2#2255051012yes—
270C2#2255051712yes—
subgroupG/H
center, FrattiniC23C23×D127
commutatorC22×C127C23×C4
Fitting#380×C127C2
socle, cosocleC23×C127C24
prime radicalC23×D508C2
hypercenter#380D127
hypocenterC127D4×C22⋊C4
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C23 ◁ #380

Complements

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

NH ≅ G/Ntype
2. C2×#16861171C2semidirect
3. C2×C22⋊C4×D127C2semidirect
4. #22549777C2semidirect
5. #22549754C2semidirect
6. #22549731C2semidirect
7. C4×C22⋊Dic127C2semidirect
8. #380×C127C2semidirect
11. C22×D508C4semidirect
15. C22×C4×D127C22semidirect
19. C2×(C2×D127)⋊C4C22semidirect
20. C2×(C2×D127)⋊C4C22semidirect
21. C2×(C2×D127)⋊C4C22semidirect
23. C2×C4×Dic127C22semidirect
24. C127⋊#95C22semidirect
26. C2×C4⋊C4×C127C22semidirect
28. #95×C127C22semidirect
33. C23×D127C2×C4semidirect
40. C2×D508C2×C4semidirect
41. C2×D508C2×C4semidirect
49. C22⋊Dic127D4semidirect
56. C22×C4×C127C2×C4semidirect
79. D508C22⋊C4semidirect
85. C2×Dic127C2×D4semidirect
105. C22×C127C4×D4semidirect
121. C2×D127C2×C22⋊C4semidirect
124. C4×C127C2×C22⋊C4semidirect
136. C127D4×C22⋊C4semidirect
142. #380D127semidirect
154. C2×C4⋊C4C2×D127semidirect
164. #95C2×D127semidirect
186. C22×C4C4×D127semidirect
252. C22#16861171semidirect
262. C4C22⋊C4×D127semidirect

Maximal subgroups

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

subgroupindex
C23×D5082
C2×#168611712
C2×C22⋊C4×D1272
#225497772
#225497542
#225497312
C4×C22⋊Dic1272
#380×C1272
D4×C22⋊C4127