Finite Group Databasebeta

← C2×#22550481 #30652090

Subgroups — C2×#22550481 #30652090

Order 16256.

1260 conjugacy classes of subgroups; only the 485 normal subgroups are shown.

normal subgroups only

This group has 1260 conjugacy classes of subgroups — too many to draw. Only its 485 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
1#22550481C2116yesmaximal, direct product factor
2C2×#16861206C211yesmaximal, characteristic, complemented
3C2×#16861201C211yesmaximal, characteristic, complemented
4C2×#16861194C214yesmaximal, complemented
5C2×#16861183C212yesmaximal, complemented
6C2×#16861182C214yesmaximal, complemented
7C2×C4×D508C212yesmaximal, complemented
8C2×#117×C127C211yesmaximal, characteristic, complemented
9C22×D508C2212yes—
10C22×D508C2214yes—
11C22×C22⋊D127C2214yes—
12C22×C4×D127C2214yes—
13#16861206C2218yescomplemented
14#16861201C2218yescomplemented
15#16861194C22132yescomplemented
16#16861183C22116yescomplemented
17#16861182C22132yescomplemented
18C4×D508C22116yescomplemented
19C2×C4⋊Dic127C2212yes—
20C2×Dic127⋊C4C2214yes—
21C2×(C2×D127)⋊C4C2214yes—
22C2×(C2×D127)⋊C4C2214yes—
23C23×C4×C127C2211yescharacteristic
24#117×C127C2218yescomplemented
25C2×C4⋊C4×C127C2212yes—
26C2×C22⋊C4×C127C2212yes—
27C2×C42×C127C2211yescharacteristic
28C2×C42×C127C2211yescharacteristic
29C23×D127C2314yes—
30C2×D508—28no—
31C2×D508—48no—
32C2×D508C2318yes—
33C2×D508C23116yes—
34C2×C22⋊D127—48no—
35C2×C22⋊D127C23116yes—
36C22×Dic127C2314yes—
37C2×C4×D127—48no—
38C2×C4×D127C23116yes—
39C4⋊Dic127C2318yes—
40Dic127⋊C4C23116yes—
41(C2×D127)⋊C4C23116yes—
42(C2×D127)⋊C4C23116yes—
43C24×C127C2311yescharacteristic
44C22×C4×C127—24no—
45C22×C4×C127C2312yes—
46C22×C4×C127C2312yes—
47C22×C4×C127C2311yescharacteristic
48C22×C4×C127C2314yes—
49C22×C4×C127C2311yescharacteristic
50C4⋊C4×C127C2318yes—
51C22⋊C4×C127C2318yes—
52C42×C127C2314yes—
53C42×C127C2314yes—
54C22×D127—28no—
55C22×D127—28no—
56C22×D127—28no—
57C22×D127C2418yes—
58D508—216no—
59D508—416no—
60C22⋊D127—416no—
61C2×Dic127—28no—
62C2×Dic127C2418yes—
63C4×D127—416no—
64C23×C127—22no—
65C23×C127—24no—
66C23×C127C2412yes—
67C23×C127C2411yescharacteristic
68C2×C4×C127—22no—
69C2×C4×C127—24no—
70C2×C4×C127—24no—
71C2×C4×C127—28no—
72C2×C4×C127#4114yes—
73C2×C4×C127C2414yes—
74C2×C4×C127C2414yes—
75C2×C4×C127C2412yes—
76C2×C4×C127C2412yes—
77C22×C127—22no—
78C22×C127—24no—
79C22×C127—44no—
80C22×C127C2×#4112yes—
81C22×C127H(5,𝔽2)14yes—
82C22×C127C2511yescharacteristic
83C2×D127—28no—
84C2×D127—28no—
85C2×D127—28no—
86C2×D127—416no—
87C4×C127—22no—
88C4×C127—24no—
89C4×C127—24no—
90C4×C127C2×#4114yes—
91Dic127—28no—
92C2×C127—42no—
93C2×C127#54014yescomplemented
94C2×C127C22×#4111yescharacteristic
95C2×C127C2×H(5,𝔽2)12yes—
96D127—48no—
97C2×#540—1271nomaximal, Sylow
98C127C2×#54011yescharacteristic, complemented, Sylow
99#540—12716no—
100C2×#127—1271no—
101C2×#125—1271no—
102C2×#123—1274no—
103C2×#121—1272no—
104C2×#121—1274no—
105C2×C4×D4—1272no—
106C2×#117D12711yescharacteristic, complemented
107C22×D4—1274no—
108C22×D4—1274no—
109C22×D4—1272no—
110C23×C4—1274no—
111C23×C4C2×D12711yescharacteristic
112#127—1278no—
113#125—1278no—
114#123—12732no—
115#121—12716no—
116#121—12732no—
117C4×D4—12716no—
118#117C2×D12718yescomplemented
119C2×C4⋊C4—1272no—
120C2×C4⋊C4—1274no—
121C2×C4⋊C4C2×D12712yes—
122C2×C22⋊C4—1274no—
123C2×C22⋊C4—1274no—
124C2×C22⋊C4C2×D12712yes—
125C2×C42C2×D12711yescharacteristic
126C2×C42C2×D12711yescharacteristic
127C24—1274no—
128C24C22×D12711yescharacteristic
129C2×D4—12716no—
130C2×D4—12716no—
131C2×D4—1278no—
132C2×D4—5088no—
133C2×D4—5088no—
134C2×D4—2548no—
135C22×C4—24no—
136C22×C4—1274no—
137C22×C4—12716no—
138C22×C4—5088no—
139C22×C4C22×D12714yes—
140C22×C4C22×D12712yes—
141C22×C4C22×D12712yes—
142C22×C4C22×D12711yescharacteristic
143C22×C4C22×D12711yescharacteristic
144C4⋊C4—12716no—
145C4⋊C4—1278no—
146C4⋊C4C22×D12718yes—
147C22⋊C4—12716no—
148C22⋊C4—12716no—
149C22⋊C4C22×D12718yes—
150C42C22×D12714yes—
151C42C22×D12714yes—
152C23—22no—
153C23—24no—
154C23—1278no—
155C23—2548no—
156C23—2548no—
157C23—2548no—
158C23C23×D12712yes—
159C23C23×D12711yescharacteristic
160D4—50816no—
161D4—50816no—
162D4—25416no—
163C2×C4—22no—
164C2×C4—24no—
165C2×C4—24no—
166C2×C4—28no—
167C2×C4—1278no—
168C2×C4—50816no—
169C2×C4—2548no—
170C2×C4#1452324814yes—
171C2×C4C23×D12714yes—
172C2×C4C23×D12714yes—
173C2×C4C23×D12712yes—
174C2×C4C23×D12712yes—
175C22—22no—
176C22—24no—
177C22—44no—
178C22—50816no—
179C22—2548no—
180C22—2548no—
181C22—2548no—
182C22C2×#1452324812yes—
183C22#1686125514yes—
184C22C24×D12711yescharacteristic
185C4—22no—
186C4—24no—
187C4—24no—
188C4—2548no—
189C4C2×#1452324814yes—
190C2—42no—
191C2—5088no—
192C2#2255048114yesdirect product factor
193C2C22×#1452324811yescharacteristic
194C2C2×#1686125512yes—
subgroupG/H
centerC23C23×D127
commutator, cosocleC22×C127C25
FrattiniC22C24×D127
FittingC2×#117×C127C2
socleC23×C127C24
hypercenterC2×#117D127
hypocenterC127C2×#540
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C23 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
1. #22550481C2direct (N × H)
2. C2×#16861206C2semidirect
3. C2×#16861201C2semidirect
4. C2×#16861194C2semidirect
5. C2×#16861183C2semidirect
6. C2×#16861182C2semidirect
7. C2×C4×D508C2semidirect
8. C2×#117×C127C2semidirect
13. #16861206C22semidirect
14. #16861201C22semidirect
15. #16861194C22semidirect
16. #16861183C22semidirect
17. #16861182C22semidirect
18. C4×D508C22semidirect
24. #117×C127C22semidirect
93. C2×C127#540semidirect
98. C127C2×#540semidirect
106. C2×#117D127semidirect
118. #117C2×D127semidirect
192. C2#22550481direct (N × H)

Maximal subgroups

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

subgroupindex
#225504812
C2×#168612062
C2×#168612012
C2×#168611942
C2×#168611832
C2×#168611822
C2×C4×D5082
C2×#117×C1272
C2×#540127