Finite Group Databasebeta

← C2×#22550054 #30648060

Subgroups — C2×#22550054 #30648060

Order 16256.

760 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 760 conjugacy classes of subgroups — too many to draw. Only its 165 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×#16861239C212yesmaximal, complemented
2C2×#16861196C211yesmaximal, characteristic, complemented
3#22550054C218yesmaximal, direct product factor
4C2×#16861136C212yesmaximal, complemented
5C2×#16861126C211yesmaximal, characteristic, complemented
6C2×H(5,𝔽2)×C127C211yesmaximal, characteristic
7C22×Dic254C2211yescharacteristic, complemented
8#16861239—24no—
9#16861239C2218yescomplemented
10C2×#14523238—22no—
11C2×#14523236—22no—
12C2×#14523232C2212yescomplemented
13#16861196—22no—
14#16861196C2214yescomplemented
15C2×C22⋊Dic127—22no—
16C2×C4×Dic127C2211yescharacteristic, complemented
17#16861136C2218yescomplemented
18#16861126C2214yescomplemented
19H(5,𝔽2)×C127—24no—
20H(5,𝔽2)×C127—22no—
21H(5,𝔽2)×C127C2214yes—
22C2×#41×C127—22no—
23C2×#41×C127C2212yes—
24C22×D4×C127—22no—
25C22×D4×C127—22no—
26C22×D4×C127C2211yescharacteristic
27C2×Dic254—21no—
28C2×Dic254—22no—
29C2×Dic254—22no—
30C2×Dic254C2312yescomplemented
31C22×Dic127—21no—
32C22×Dic127—21no—
33#14523238—28no—
34#14523236—28no—
35#14523232—22no—
36#14523232C2314yescomplemented
37C2×C127⋊C8—22no—
38C22⋊Dic127—28no—
39C4×Dic127—21no—
40C4×Dic127C2312yescomplemented
41C24×C127—22no—
42C24×C127D412yes—
43#41×C127—28no—
44#41×C127—28no—
45#41×C127—22no—
46#41×C127—22no—
47#41×C127D414yes—
48#41×C127C2314yes—
49C2×Q8×C127D412yes—
50C2×D4×C127—24no—
51C2×D4×C127—28no—
52C2×D4×C127—28no—
53C2×D4×C127—22no—
54C2×D4×C127—28no—
55C2×D4×C127—28no—
56C2×D4×C127—22no—
57C2×D4×C127—22no—
58C2×D4×C127D412yes—
59C2×D4×C127D412yes—
60C2×D4×C127C2312yes—
61C22×C4×C127—22no—
62C22×C4×C127—22no—
63C22×C4×C127C2311yescharacteristic
64Dic254—24no—
65Dic254—42no—
66C2×Dic127—22no—
67C2×Dic127—22no—
68C2×Dic127—42no—
69C2×Dic127—41no—
70C2×Dic127—41no—
71C127⋊C8—24no—
72C23×C127—28no—
73C23×C127—22no—
74C23×C127—22no—
75C23×C127—22no—
76C23×C127—44no—
77C23×C127—84no—
78C23×C127C2×D411yescharacteristic
79C23×C127C2×D414yes—
80Q8×C127—22no—
81Q8×C127C2×D414yes—
82D4×C127—216no—
83D4×C127—24no—
84D4×C127—24no—
85D4×C127—24no—
86D4×C127—22no—
87D4×C127—24no—
88D4×C127C2×D414yes—
89C2×C4×C127—28no—
90C2×C4×C127—22no—
91C2×C4×C127—24no—
92C2×C4×C127—24no—
93C2×C4×C127C2×D411yescharacteristic
94C2×C4×C127C2×D412yes—
95C2×C4×C127C2×D412yes—
96C2×C4×C127C2411yescharacteristic
97C22×C127—22no—
98C22×C127—24no—
99C22×C127—24no—
100C22×C127—44no—
101C22×C127—44no—
102C22×C127—44no—
103C22×C127—82no—
104C22×C127—88no—
105C22×C127#12011yescharacteristic
106C22×C127#12011yescharacteristic
107C22×C127C22×D411yescharacteristic
108C4×C127—24no—
109C4×C127#12012yes—
110C4×C127C22×D412yes—
111Dic127—42no—
112Dic127—42no—
113C2×C127—21no—
114C2×C127—21no—
115C2×C127—44no—
116C2×C127—44no—
117C2×C127#45512yescomplemented
118C2×C127C2×#12011yescharacteristic
119C2×#455—1271nomaximal, Sylow
120C127C2×#45511yescharacteristic, complemented, Sylow
121C2×H(5,𝔽2)D12711yescharacteristic
122C2×#137—1272no—
123C2×#124—1271no—
124#455—1278no—
125C2×#104—1272no—
126C2×#100—1271no—
127H(5,𝔽2)—24no—
128H(5,𝔽2)—22no—
129H(5,𝔽2)C2×D12714yes—
130C2×#41—22no—
131C2×#41C2×D12712yes—
132C22×Q8—1271no—
133C22×D4—22no—
134C22×D4—22no—
135C22×D4C2×D12711yescharacteristic
136#137—1278no—
137#137—2544no—
138C2×Q16—2542no—
139C2×SD16—2542no—
140C2×M16—1272no—
141#124—1274no—
142#124—2542no—
143C2×C22⋊C4—2542no—
144C2×C42—1271no—
145#104—1278no—
146#100—1274no—
147C24—22no—
148C24C22⋊D12712yes—
149#41—28no—
150#41—28no—
151#41—22no—
152#41—22no—
153#41C22⋊D12714yes—
154#41C22×D12714yes—
155C2×Q8—1272no—
156C2×Q8—2542no—
157C2×Q8—2542no—
158C2×Q8—2541no—
159C2×Q8C22⋊D12712yes—
160C2×D4—24no—
161C2×D4—28no—
162C2×D4—28no—
163C2×D4—22no—
164C2×D4—28no—
165C2×D4—28no—
166C2×D4—22no—
167C2×D4—22no—
168C2×D4C22⋊D12712yes—
169C2×D4C22⋊D12712yes—
170C2×D4C22×D12712yes—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—2541no—
174C22×C4—2541no—
175C22×C4C22×D12711yescharacteristic
176Q16—2548no—
177SD16—2548no—
178M16—1274no—
179M16—2542no—
180C2×C8—2542no—
181C22⋊C4—2548no—
182C42—1272no—
183C42—2541no—
184C23—28no—
185C23—22no—
186C23—22no—
187C23—22no—
188C23—44no—
189C23—84no—
190C23C2×C22⋊D12711yescharacteristic
191C23C2×C22⋊D12714yes—
192Q8—22no—
193Q8—5082no—
194Q8—2544no—
195Q8C2×C22⋊D12714yes—
196D4—216no—
197D4—24no—
198D4—24no—
199D4—24no—
200D4—24no—
201D4—22no—
202D4C2×C22⋊D12714yes—
203C2×C4—28no—
204C2×C4—22no—
205C2×C4—24no—
206C2×C4—24no—
207C2×C4—5082no—
208C2×C4—5081no—
209C2×C4—5081no—
210C2×C4—2542no—
211C2×C4—2542no—
212C2×C4C2×C22⋊D12712yes—
213C2×C4C2×C22⋊D12712yes—
214C2×C4C2×C22⋊D12711yescharacteristic
215C2×C4C23×D12711yescharacteristic
216C8—2544no—
217C22—22no—
218C22—24no—
219C22—24no—
220C22—44no—
221C22—44no—
222C22—44no—
223C22—82no—
224C22—88no—
225C22#1686117811yescharacteristic
226C22#1686117811yescharacteristic
227C22C22×C22⋊D12711yescharacteristic
228C4—24no—
229C4—5082no—
230C4—5082no—
231C4#1686117812yes—
232C4C22×C22⋊D12712yes—
233C2—21no—
234C2—21no—
235C2—44no—
236C2—44no—
237C2#2255005412yesdirect product factor
238C2C2×#1686117811yescharacteristic
subgroupG/H
centerC22#16861178
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting, prime radicalC2×H(5,𝔽2)×C127C2
socleC22×C127#120
hypercenterC2×H(5,𝔽2)D127
hypocenterC127C2×#455
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. C2×#16861239C2semidirect
2. C2×#16861196C2semidirect
3. #22550054C2direct (N × H)
4. C2×#16861136C2semidirect
5. C2×#16861126C2semidirect
7. C22×Dic254C22semidirect
9. #16861239C22semidirect
12. C2×#14523232C22semidirect
14. #16861196C22semidirect
16. C2×C4×Dic127C22semidirect
17. #16861136C22semidirect
18. #16861126C22semidirect
30. C2×Dic254C23semidirect
36. #14523232C23semidirect
40. C4×Dic127C23semidirect
117. C2×C127#455semidirect
120. C127C2×#455semidirect
237. C2#22550054direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168612392
C2×#168611962
#225500542
C2×#168611362
C2×#168611262
C2×H(5,𝔽2)×C1272
C2×#455127