Finite Group Databasebeta

← C2×#22550128 #30648325

Subgroups — C2×#22550128 #30648325

Order 16256.

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

normal subgroups only

This group has 596 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
1C22×#14523233C211yesmaximal, characteristic, complemented
2C22×#14523231C211yesmaximal, characteristic, complemented
3C2×#16861184C211yesmaximal, characteristic, complemented
4C2×#16861180C211yesmaximal, characteristic, complemented
5#22550128C218yesmaximal, direct product factor
6C2×#16861140C211yesmaximal, characteristic, complemented
7C2×#16861131C211yesmaximal, characteristic, complemented
8C2×#102×C127C211yesmaximal, characteristic, complemented
9C22×D508—21no—
10C22×D508C2211yescharacteristic
11C22×C22⋊D127—21no—
12C22×C4×D127C2211yescharacteristic
13C2×#14523233—24no—
14C2×#14523233C2214yescomplemented
15C2×#14523231—24no—
16C2×#14523231C2214yescomplemented
17C22×C127⋊C8C2211yescharacteristic
18#16861184—22no—
19#16861184C2214yescomplemented
20#16861180—22no—
21#16861180C2214yescomplemented
22C2×C4⋊Dic127C2211yescharacteristic
23C2×C22⋊Dic127—21no—
24C2×(C2×D127)⋊C4—21no—
25#16861140C2214yescomplemented
26#16861131C2214yescomplemented
27C22×D4×C127C2211yescharacteristic
28C22×C8×C127C2211yescharacteristic
29C2×C4⋊C4×C127C2211yescharacteristic
30#102×C127C2214yescomplemented
31C23×D127—21no—
32C23×D127D411yescharacteristic
33C2×D508—24no—
34C2×D508—22no—
35C2×D508—21no—
36C2×D508—22no—
37C2×D508—44no—
38C2×D508C2312yes—
39C2×C22⋊D127—24no—
40C2×C22⋊D127—44no—
41C22×Dic127—21no—
42C22×Dic127D411yescharacteristic
43C2×C4×D127—22no—
44C2×C4×D127—22no—
45C2×C4×D127D412yes—
46C2×C4×D127C2312yes—
47#14523233—28no—
48#14523231—28no—
49C2×C127⋊C8D414yescomplemented
50C2×C127⋊C8C2312yes—
51C4⋊Dic127—21no—
52C4⋊Dic127C2312yes—
53C22⋊Dic127—24no—
54(C2×D127)⋊C4—24no—
55C24×C127—21no—
56C2×D4×C127—22no—
57C2×D4×C127—21no—
58C2×D4×C127—22no—
59C2×D4×C127C2312yes—
60C22×C4×C127—21no—
61C22×C4×C127C2311yescharacteristic
62C2×C8×C127—22no—
63C2×C8×C127C2312yes—
64C4⋊C4×C127—21no—
65C4⋊C4×C127C2312yes—
66C22×D127—22no—
67C22×D127—22no—
68C22×D127—22no—
69C22×D127—22no—
70C22×D127—42no—
71C22×D127—42no—
72C22×D127—42no—
73C22×D127C2×D412yes—
74D508—22no—
75D508—22no—
76D508—48no—
77D508—42no—
78C22⋊D127—48no—
79C2×Dic127—22no—
80C2×Dic127—22no—
81C2×Dic127—42no—
82C2×Dic127C2×D412yes—
83C4×D127—24no—
84C4×D127—22no—
85C4×D127—22no—
86C127⋊C8C2×D414yescomplemented
87C23×C127—22no—
88C23×C127—42no—
89C23×C127—42no—
90C23×C127—42no—
91C23×C127C2×D411yescharacteristic
92D4×C127—22no—
93D4×C127—22no—
94D4×C127—42no—
95C2×C4×C127—22no—
96C2×C4×C127—42no—
97C2×C4×C127C2×D411yescharacteristic
98C2×C4×C127C2×D412yes—
99C2×C4×C127#4112yes—
100C2×C4×C127C2411yescharacteristic
101C8×C127—22no—
102C22×C127—42no—
103C22×C127—42no—
104C22×C127—42no—
105C22×C127—84no—
106C22×C127#12112yes—
107C22×C127AGL(1,ℤ/8)12yes—
108C22×C127AGL(1,ℤ/8)12yes—
109C22×C127C22×D411yescharacteristic
110C2×D127—22no—
111C2×D127—22no—
112C2×D127—22no—
113C2×D127—44no—
114C2×D127—42no—
115C2×D127—42no—
116C2×D127—42no—
117C2×D127—84no—
118C4×C127—42no—
119C4×C127#12112yes—
120C4×C127C22×D411yescharacteristic
121C4×C127C2×#4111yescharacteristic
122Dic127—22no—
123Dic127—42no—
124C2×C127—82no—
125C2×C127#46914yescomplemented
126C2×C127C2×#12111yescharacteristic
127C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
128C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
129D127—42no—
130D127—82no—
131C2×#469—1271nomaximal, Sylow
132C127C2×#46911yescharacteristic, complemented, Sylow
133C22×D8—1271no—
134C22×M16—1271no—
135C2×#121—1271no—
136C2×#121—1271no—
137#469—1278no—
138C2×#106—1271no—
139C2×#102—1271no—
140C2×#102D12711yescharacteristic, complemented
141C22×D4—1271no—
142C22×D4—2541no—
143C22×D4—2541no—
144C22×D4C2×D12711yescharacteristic
145C23×C4—1271no—
146C2×D8—1274no—
147C2×D8—2544no—
148C2×M16—1274no—
149C2×M16—2544no—
150C22×C8—1271no—
151C22×C8C2×D12711yescharacteristic
152#121—1274no—
153#121—1274no—
154#121—2542no—
155#121—2542no—
156C2×C4⋊C4—1271no—
157C2×C4⋊C4C2×D12711yescharacteristic
158C2×C22⋊C4—2541no—
159C2×C22⋊C4—2541no—
160#106—1274no—
161#102—1274no—
162#102C2×D12714yescomplemented
163C24—21no—
164C24—1271no—
165C24—2541no—
166C2×D4—22no—
167C2×D4—21no—
168C2×D4—22no—
169C2×D4—1272no—
170C2×D4—5084no—
171C2×D4—5084no—
172C2×D4—2544no—
173C2×D4—2544no—
174C2×D4—2541no—
175C2×D4—2542no—
176C2×D4—2542no—
177C2×D4C22×D12712yes—
178C22×C4—21no—
179C22×C4—1271no—
180C22×C4—1272no—
181C22×C4—1272no—
182C22×C4—2541no—
183C22×C4—2542no—
184C22×C4—2542no—
185C22×C4C22×D12711yescharacteristic
186D8—2548no—
187M16—2548no—
188C2×C8—22no—
189C2×C8—1274no—
190C2×C8—1272no—
191C2×C8C22×D12712yes—
192C4⋊C4—21no—
193C4⋊C4—1272no—
194C4⋊C4—2541no—
195C4⋊C4C22×D12712yes—
196C22⋊C4—2544no—
197C22⋊C4—2544no—
198C23—22no—
199C23—42no—
200C23—42no—
201C23—42no—
202C23—1272no—
203C23—5082no—
204C23—5082no—
205C23—5082no—
206C23—2542no—
207C23—2542no—
208C23—2542no—
209C23—2542no—
210C23D4×D12711yescharacteristic
211D4—22no—
212D4—22no—
213D4—42no—
214D4—5088no—
215D4—5088no—
216D4—5082no—
217D4—2542no—
218D4—2542no—
219C2×C4—22no—
220C2×C4—42no—
221C2×C4—1272no—
222C2×C4—5082no—
223C2×C4—2542no—
224C2×C4—2542no—
225C2×C4—2542no—
226C2×C4—2542no—
227C2×C4—2544no—
228C2×C4D4×D12712yes—
229C2×C4D4×D12711yescharacteristic
230C2×C4#1452324812yes—
231C2×C4C23×D12711yescharacteristic
232C8—22no—
233C8—1274no—
234C22—42no—
235C22—42no—
236C22—42no—
237C22—84no—
238C22—10164no—
239C22—5082no—
240C22—5082no—
241C22—5082no—
242C22—5084no—
243C22—2542no—
244C22—2542no—
245C22—2542no—
246C22#1686118212yes—
247C22#1686123312yes—
248C22#1686123612yes—
249C22C2×D4×D12711yescharacteristic
250C4—42no—
251C4—5082no—
252C4—2542no—
253C4#1686118212yes—
254C4C2×D4×D12711yescharacteristic
255C4C2×#1452324811yescharacteristic
256C2—82no—
257C2—10162no—
258C2—5082no—
259C2#2255012814yesdirect product factor
260C2C2×#1686118211yescharacteristic
261C2C2×#1686123311yescharacteristic
262C2C2×#1686123611yescharacteristic
subgroupG/H
centerC23D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×#102×C127C2
socleC23×C127C2×D4
hypercenterC2×#102D127
hypocenterC127C2×#469
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
1. C22×#14523233C2semidirect
2. C22×#14523231C2semidirect
3. C2×#16861184C2semidirect
4. C2×#16861180C2semidirect
5. #22550128C2direct (N × H)
6. C2×#16861140C2semidirect
7. C2×#16861131C2semidirect
8. C2×#102×C127C2semidirect
14. C2×#14523233C22semidirect
16. C2×#14523231C22semidirect
19. #16861184C22semidirect
21. #16861180C22semidirect
25. #16861140C22semidirect
26. #16861131C22semidirect
30. #102×C127C22semidirect
49. C2×C127⋊C8D4semidirect
86. C127⋊C8C2×D4semidirect
125. C2×C127#469semidirect
132. C127C2×#469semidirect
140. C2×#102D127semidirect
162. #102C2×D127semidirect
259. C2#22550128direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145232332
C22×#145232312
C2×#168611842
C2×#168611802
#225501282
C2×#168611402
C2×#168611312
C2×#102×C1272
C2×#469127