Finite Group Databasebeta

← C2×#22550056 #30648062

Subgroups — C2×#22550056 #30648062

Order 16256.

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

normal subgroups only

This group has 760 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
1C2×#16861254C211yesmaximal, characteristic
2C2×#16861242C212yesmaximal, complemented
3C2×#16861196C211yesmaximal, characteristic, complemented
4#22550056C218yesmaximal, direct product factor
5C2×#16861135C212yesmaximal, complemented
6C2×#100×C127C211yesmaximal, characteristic, complemented
7#16861254—24no—
8#16861254—22no—
9#16861254C2214yes—
10C2×#14523248C2212yes—
11C2×#14523247—22no—
12C22×Dic254C2211yescharacteristic, complemented
13C22×C22⋊D127—22no—
14C2×D4×D127—22no—
15#16861242—24no—
16#16861242C2218yescomplemented
17C2×Dic508—22no—
18C2×#14523237—22no—
19#16861196—22no—
20#16861196C2214yescomplemented
21C2×C22⋊Dic127—22no—
22C2×C4×Dic127C2211yescharacteristic, complemented
23#16861135C2218yescomplemented
24C22×D4×C127C2211yescharacteristic
25C2×M16×C127C2212yescomplemented
26#100×C127C2214yescomplemented
27C23×D127—22no—
28#14523248—22no—
29#14523248—22no—
30#14523248D414yes—
31#14523248C2314yes—
32#14523247—28no—
33#14523247—28no—
34C2×Dic254—21no—
35C2×Dic254—22no—
36C2×Dic254—22no—
37C2×Dic254D412yes—
38C2×Dic254C2312yescomplemented
39C2×D508D412yes—
40C2×C22⋊D127—24no—
41C2×C22⋊D127—28no—
42C2×C22⋊D127—22no—
43C2×C22⋊D127—28no—
44D4×D127—28no—
45D4×D127—28no—
46C22×Dic127—22no—
47C22×Dic127—21no—
48C22×Dic127—21no—
49C2×C4×D127—22no—
50Dic508—28no—
51#14523237—28no—
52C22⋊Dic127—28no—
53C4×Dic127—21no—
54C4×Dic127C2312yescomplemented
55C24×C127D412yes—
56C2×D4×C127—22no—
57C2×D4×C127—22no—
58C2×D4×C127D412yes—
59C2×D4×C127C2312yes—
60C22×C4×C127C2311yescharacteristic
61M16×C127—22no—
62M16×C127C2314yescomplemented
63C2×C8×C127—22no—
64C22×D127—22no—
65C22×D127—28no—
66C22×D127—84no—
67Dic254—22no—
68Dic254—24no—
69Dic254—42no—
70Dic254C2×D414yes—
71D508—22no—
72D508C2×D414yes—
73C22⋊D127—216no—
74C22⋊D127—24no—
75C22⋊D127—24no—
76C2×Dic127—28no—
77C2×Dic127—22no—
78C2×Dic127—22no—
79C2×Dic127—22no—
80C2×Dic127—42no—
81C2×Dic127—41no—
82C2×Dic127—41no—
83C4×D127—24no—
84C4×D127—24no—
85C23×C127—22no—
86C23×C127—22no—
87C23×C127—44no—
88C23×C127C2×D411yescharacteristic
89C23×C127C2×D414yes—
90D4×C127—24no—
91D4×C127—24no—
92C2×C4×C127C2×D412yes—
93C2×C4×C127C2×D412yes—
94C2×C4×C127C2×D411yescharacteristic
95C2×C4×C127C2411yescharacteristic
96C8×C127—24no—
97C22×C127—22no—
98C22×C127—24no—
99C22×C127—44no—
100C22×C127—44no—
101C22×C127—82no—
102C22×C127#12011yescharacteristic
103C22×C127#12011yescharacteristic
104C22×C127C22×D411yescharacteristic
105C2×D127—24no—
106C2×D127—44no—
107C2×D127—88no—
108C4×C127#12012yes—
109C4×C127C22×D412yes—
110Dic127—24no—
111Dic127—42no—
112Dic127—42no—
113C2×C127—21no—
114C2×C127—21no—
115C2×C127—44no—
116C2×C127#45512yescomplemented
117C2×C127C2×#12011yescharacteristic
118D127—44no—
119C2×#455—1271nomaximal, Sylow
120C127C2×#45511yescharacteristic, complemented, Sylow
121C2×H(5,𝔽2)—1271no—
122C2×#137—1272no—
123C2×#124—1271no—
124#455—1278no—
125C2×#104—1272no—
126C2×#100D12711yescharacteristic, complemented
127H(5,𝔽2)—1274no—
128H(5,𝔽2)—2544no—
129H(5,𝔽2)—2542no—
130C2×#41—1272no—
131C2×#41—2542no—
132C22×Q8—1271no—
133C22×D4—2542no—
134C22×D4—2542no—
135C22×D4C2×D12711yescharacteristic
136#137—1278no—
137#137—2544no—
138C2×Q16—2542no—
139C2×SD16—2542no—
140C2×M16C2×D12712yescomplemented
141#124—1274no—
142#124—2542no—
143C2×C22⋊C4—2542no—
144C2×C42—1271no—
145#104—1278no—
146#100C2×D12714yescomplemented
147C24—2542no—
148C24D50812yes—
149#41—1274no—
150#41—1274no—
151#41—2548no—
152#41—2548no—
153#41—2542no—
154#41—2542no—
155C2×Q8—1272no—
156C2×Q8—1272no—
157C2×Q8—2542no—
158C2×Q8—2542no—
159C2×Q8—2541no—
160C2×D4—22no—
161C2×D4—22no—
162C2×D4—1272no—
163C2×D4—2548no—
164C2×D4—2548no—
165C2×D4—2544no—
166C2×D4—2548no—
167C2×D4—2542no—
168C2×D4—2548no—
169C2×D4D50812yes—
170C2×D4C22×D12712yes—
171C22×C4—2541no—
172C22×C4—2541no—
173C22×C4—2542no—
174C22×C4—2542no—
175C22×C4C22×D12711yescharacteristic
176Q16—2548no—
177SD16—2548no—
178M16—22no—
179M16C22×D12714yescomplemented
180C2×C8—22no—
181C22⋊C4—2548no—
182C42—1272no—
183C42—2541no—
184C23—22no—
185C23—22no—
186C23—44no—
187C23—10164no—
188C23—2548no—
189C23—2542no—
190C23C2×D50811yescharacteristic
191C23C2×D50814yes—
192Q8—1274no—
193Q8—5082no—
194Q8—2544no—
195Q8—2542no—
196D4—24no—
197D4—24no—
198D4—1274no—
199D4—25416no—
200D4—2544no—
201D4—2544no—
202D4—2542no—
203C2×C4—5082no—
204C2×C4—5081no—
205C2×C4—5081no—
206C2×C4—2542no—
207C2×C4—2542no—
208C2×C4—2548no—
209C2×C4—2542no—
210C2×C4—2544no—
211C2×C4—2544no—
212C2×C4D4×D12712yes—
213C2×C4D4×D12712yes—
214C2×C4C2×D50811yescharacteristic
215C2×C4C23×D12711yescharacteristic
216C8—24no—
217C22—22no—
218C22—24no—
219C22—44no—
220C22—44no—
221C22—82no—
222C22—10168no—
223C22—5084no—
224C22—2544no—
225C22#1686117911yescharacteristic
226C22#1686117911yescharacteristic
227C22C22×D50811yescharacteristic
228C4—5082no—
229C4—5082no—
230C4—2544no—
231C4#1686117912yes—
232C4C2×D4×D12712yes—
233C2—21no—
234C2—21no—
235C2—44no—
236C2—5084no—
237C2#2255005612yesdirect product factor
238C2C2×#1686117911yescharacteristic
subgroupG/H
centerC22#16861179
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×#100×C127C2
socleC22×C127#120
prime radicalC2×#16861254C2
hypercenterC2×#100D127
hypocenterC127C2×#455
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#100

Complements

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

NH ≅ G/Ntype
2. C2×#16861242C2semidirect
3. C2×#16861196C2semidirect
4. #22550056C2direct (N × H)
5. C2×#16861135C2semidirect
6. C2×#100×C127C2semidirect
12. C22×Dic254C22semidirect
16. #16861242C22semidirect
20. #16861196C22semidirect
22. C2×C4×Dic127C22semidirect
23. #16861135C22semidirect
25. C2×M16×C127C22semidirect
26. #100×C127C22semidirect
38. C2×Dic254C23semidirect
54. C4×Dic127C23semidirect
62. M16×C127C23semidirect
116. C2×C127#455semidirect
120. C127C2×#455semidirect
126. C2×#100D127semidirect
140. C2×M16C2×D127semidirect
146. #100C2×D127semidirect
179. M16C22×D127semidirect
237. C2#22550056direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168612542
C2×#168612422
C2×#168611962
#225500562
C2×#168611352
C2×#100×C1272
C2×#455127