Finite Group Databasebeta

← C2×#22550107 #30648292

Subgroups — C2×#22550107 #30648292

Order 16256.

564 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 173 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×#14523235C211yesmaximal, characteristic, complemented
2C23×C127⋊C8C211yesmaximal, characteristic, complemented
3C2×#16861183C211yesmaximal, characteristic
4#22550107C218yesmaximal, direct product factor
5C2×#16861140C211yesmaximal, characteristic, complemented
6C2×#16861133C211yesmaximal, characteristic, complemented
7C2×#16861129C211yesmaximal, characteristic, complemented
8C2×C22⋊Q8×C127C211yesmaximal, characteristic, complemented
9C22×D508C2211yescharacteristic
10C22×C22⋊D127—21no—
11C2×#14523235—24no—
12C2×#14523235C2214yescomplemented
13C22×C127⋊C8—24no—
14C22×C127⋊C8C2211yescharacteristic
15C22×C127⋊C8C2211yescharacteristic
16C22×C127⋊C8C2214yescomplemented
17#16861183—22no—
18#16861183C2214yes—
19C2×C4⋊Dic127C2211yescharacteristic
20C2×(C2×D127)⋊C4—21no—
21#16861140C2214yescomplemented
22#16861133C2214yescomplemented
23#16861129C2214yescomplemented
24C22×Q8×C127C2211yescharacteristic
25C23×C4×C127C2211yescharacteristic
26C22⋊Q8×C127—22no—
27C22⋊Q8×C127C2214yescomplemented
28C2×C4⋊C4×C127—21no—
29C2×C4⋊C4×C127C2211yescharacteristic
30C2×C22⋊C4×C127—21no—
31C23×D127—21no—
32C2×D508—22no—
33C2×D508—21no—
34C2×D508—22no—
35C2×D508C2312yes—
36C2×C22⋊D127—24no—
37C2×C22⋊D127—44no—
38C22×Dic127—21no—
39#14523235—28no—
40C2×C127⋊C8—28no—
41C2×C127⋊C8—22no—
42C2×C127⋊C8D414yescomplemented
43C2×C127⋊C8C2312yes—
44C2×C127⋊C8C2312yes—
45C4⋊Dic127—21no—
46C4⋊Dic127C2312yes—
47(C2×D127)⋊C4—24no—
48C24×C127D411yescharacteristic
49C2×Q8×C127—21no—
50C2×Q8×C127—22no—
51C2×Q8×C127—22no—
52C2×Q8×C127C2312yes—
53C22×C4×C127—22no—
54C22×C4×C127—22no—
55C22×C4×C127—21no—
56C22×C4×C127—21no—
57C22×C4×C127D411yescharacteristic
58C22×C4×C127D412yes—
59C22×C4×C127C2312yes—
60C22×C4×C127C2311yescharacteristic
61C4⋊C4×C127—24no—
62C4⋊C4×C127—21no—
63C4⋊C4×C127C2312yes—
64C22⋊C4×C127—24no—
65C22×D127—22no—
66C22×D127—42no—
67C22×D127—42no—
68C22×D127—42no—
69D508—22no—
70D508—22no—
71D508—42no—
72C22⋊D127—48no—
73C2×Dic127—22no—
74C2×Dic127—42no—
75C127⋊C8—22no—
76C127⋊C8C2×D414yescomplemented
77C23×C127—22no—
78C23×C127—22no—
79C23×C127SD1614yescomplemented
80C23×C127C2×D412yes—
81C23×C127C2×D411yescharacteristic
82Q8×C127—22no—
83Q8×C127—22no—
84Q8×C127—42no—
85C2×C4×C127—22no—
86C2×C4×C127—24no—
87C2×C4×C127—22no—
88C2×C4×C127—22no—
89C2×C4×C127—22no—
90C2×C4×C127—22no—
91C2×C4×C127—42no—
92C2×C4×C127—42no—
93C2×C4×C127C2×D412yes—
94C2×C4×C127C2×D412yes—
95C2×C4×C127C2×D411yescharacteristic
96C2×C4×C127#4112yes—
97C2×C4×C127C2411yescharacteristic
98C22×C127—22no—
99C22×C127—22no—
100C22×C127—28no—
101C22×C127#12112yes—
102C22×C127C2×SD1612yes—
103C22×C127C2×SD1614yescomplemented
104C22×C127#13512yes—
105C22×C127C22×D411yescharacteristic
106C2×D127—42no—
107C2×D127—42no—
108C2×D127—42no—
109C2×D127—84no—
110C4×C127—22no—
111C4×C127—42no—
112C4×C127—42no—
113C4×C127#12112yes—
114C4×C127C22×D411yescharacteristic
115C4×C127C2×#4111yescharacteristic
116Dic127—42no—
117C2×C127—24no—
118C2×C127#46514yescomplemented
119C2×C127C2×#12111yescharacteristic
120C2×C127C22×SD1611yescharacteristic
121C2×C127C2×#13511yescharacteristic
122D127—82no—
123C2×#465—1271nomaximal, Sylow
124C127C2×#46511yescharacteristic, complemented, Sylow
125C22×SD16—1271no—
126C23×C8—1271no—
127C2×C22⋊Q8D12711yescharacteristic, complemented
128C2×#121—1271no—
129#465—1278no—
130C2×#106—1271no—
131C2×#103—1271no—
132C2×#102—1271no—
133C22×Q8C2×D12711yescharacteristic
134C22×D4—1271no—
135C22×D4—2541no—
136C23×C4C2×D12711yescharacteristic
137C2×SD16—1274no—
138C2×SD16—2544no—
139C22×C8—1271no—
140C22×C8—1271no—
141C22×C8—1274no—
142C22×C8—2544no—
143C22⋊Q8—22no—
144C22⋊Q8C2×D12714yescomplemented
145#121—1274no—
146#121—2542no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—1271no—
149C2×C4⋊C4C2×D12711yescharacteristic
150C2×C22⋊C4—21no—
151C2×C22⋊C4—2541no—
152#106—1274no—
153#103—1274no—
154#102—1274no—
155C24—2541no—
156C24C22⋊D12711yescharacteristic
157C2×Q8—21no—
158C2×Q8—22no—
159C2×Q8—22no—
160C2×Q8C22×D12712yes—
161C2×D4—1272no—
162C2×D4—5084no—
163C2×D4—2544no—
164C2×D4—2542no—
165C2×D4—2541no—
166C2×D4—2542no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—2541no—
172C22×C4C22⋊D12712yes—
173C22×C4C22⋊D12711yescharacteristic
174C22×C4C22×D12712yes—
175C22×C4C22×D12711yescharacteristic
176SD16—2548no—
177C2×C8—1274no—
178C2×C8—1272no—
179C2×C8—1272no—
180C2×C8—2548no—
181C2×C8—2542no—
182C4⋊C4—24no—
183C4⋊C4—21no—
184C4⋊C4—1272no—
185C4⋊C4—2541no—
186C4⋊C4C22×D12712yes—
187C22⋊C4—24no—
188C22⋊C4—2544no—
189C23—22no—
190C23—22no—
191C23—5082no—
192C23—5082no—
193C23—5082no—
194C23—2542no—
195C23#1452323514yescomplemented
196C23C2×C22⋊D12712yes—
197C23C2×C22⋊D12711yescharacteristic
198Q8—22no—
199Q8—22no—
200Q8—42no—
201D4—5088no—
202D4—5082no—
203D4—2542no—
204D4—2542no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—24no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—5082no—
214C2×C4—2542no—
215C2×C4D4×D12712yes—
216C2×C4C2×C22⋊D12712yes—
217C2×C4C2×C22⋊D12711yescharacteristic
218C2×C4#1452324712yes—
219C2×C4C23×D12711yescharacteristic
220C8—1274no—
221C8—2542no—
222C22—28no—
223C22—22no—
224C22—22no—
225C22—10164no—
226C22—5082no—
227C22—5082no—
228C22—5082no—
229C22#1686118112yes—
230C22C2×#1452323512yes—
231C22C2×#1452323514yescomplemented
232C22#1686122912yes—
233C22C22×C22⋊D12711yescharacteristic
234C4—22no—
235C4—42no—
236C4—42no—
237C4—5082no—
238C4#1686118112yes—
239C4C2×D4×D12711yescharacteristic
240C4C2×#1452324711yescharacteristic
241C2—24no—
242C2—10162no—
243C2#2255010714yesdirect product factor
244C2C2×#1686118111yescharacteristic
245C2C22×#1452323511yescharacteristic
246C2C2×#1686122911yescharacteristic
subgroupG/H
centerC23C2×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×C22⋊Q8×C127C2
socleC23×C127C2×D4
prime radicalC2×#16861183C2
hypercenterC2×C22⋊Q8D127
hypocenterC127C2×#465
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C22×#14523235C2semidirect
2. C23×C127⋊C8C2semidirect
4. #22550107C2direct (N × H)
5. C2×#16861140C2semidirect
6. C2×#16861133C2semidirect
7. C2×#16861129C2semidirect
8. C2×C22⋊Q8×C127C2semidirect
12. C2×#14523235C22semidirect
16. C22×C127⋊C8C22semidirect
21. #16861140C22semidirect
22. #16861133C22semidirect
23. #16861129C22semidirect
27. C22⋊Q8×C127C22semidirect
42. C2×C127⋊C8D4semidirect
76. C127⋊C8C2×D4semidirect
79. C23×C127SD16semidirect
103. C22×C127C2×SD16semidirect
118. C2×C127#465semidirect
124. C127C2×#465semidirect
127. C2×C22⋊Q8D127semidirect
144. C22⋊Q8C2×D127semidirect
195. C23#14523235semidirect
231. C22C2×#14523235semidirect
243. C2#22550107direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145232352
C23×C127⋊C82
C2×#168611832
#225501072
C2×#168611402
C2×#168611332
C2×#168611292
C2×C22⋊Q8×C1272
C2×#465127