Finite Group Databasebeta

← C2×#22550618 #30652850

Subgroups — C2×#22550618 #30652850

Order 16256.

1408 conjugacy classes of subgroups; only the 469 normal subgroups are shown.

normal subgroups only

This group has 1408 conjugacy classes of subgroups — too many to draw. Only its 469 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#22550618C2116yesmaximal, direct product factor
2C2×#16861198C212yesmaximal, complemented
3C2×#16861197C212yesmaximal, complemented
4C2×#16861195C214yesmaximal, complemented
5C2×#16861179C214yesmaximal, complemented
6C2×#16861177C212yesmaximal, complemented
7C2×#124×C127C211yesmaximal, characteristic, complemented
8C24×D127C2212yescomplemented
9C22×Dic254C2212yescomplemented
10C22×D508C2212yes—
11C22×D508C2212yes—
12C22×C22⋊D127C2214yes—
13#16861198C22116yescomplemented
14#16861197C22116yescomplemented
15#16861195C22132yescomplemented
16#16861179C22132yescomplemented
17#16861177C22116yescomplemented
18C2×C22⋊Dic127C2212yescomplemented
19C2×(C2×D127)⋊C4C2214yescomplemented
20C2×(C2×D127)⋊C4C2214yescomplemented
21C2×(C2×D127)⋊C4C2214yescomplemented
22C2×C4×Dic127C2212yescomplemented
23C22×Q8×C127C2211yescharacteristic, complemented
24C22×D4×C127C2211yescharacteristic
25#124×C127C2218yescomplemented
26C2×C22⋊C4×C127C2214yescomplemented
27C2×C42×C127C2211yescharacteristic, complemented
28C23×D127—44no—
29C23×D127—48no—
30C23×D127C2314yes—
31C23×D127C2318yescomplemented
32C2×Dic254—44no—
33C2×Dic254C2318yescomplemented
34C2×D508—44no—
35C2×D508—44no—
36C2×D508C2318yes—
37C2×D508C2318yes—
38C2×C22⋊D127—48no—
39C2×C22⋊D127C23116yes—
40C22×Dic127C2314yes—
41C22⋊Dic127C2318yescomplemented
42(C2×D127)⋊C4C23116yescomplemented
43(C2×D127)⋊C4C23116yescomplemented
44(C2×D127)⋊C4C23116yescomplemented
45C4×Dic127C2318yescomplemented
46C24×C127C2312yes—
47C2×Q8×C127—42no—
48C2×Q8×C127C2314yescomplemented
49C2×D4×C127—42no—
50C2×D4×C127C2314yes—
51C22×C4×C127C2312yes—
52C22×C4×C127C2312yes—
53C22×C4×C127C2311yescharacteristic
54C22⋊C4×C127C23116yescomplemented
55C42×C127C2314yescomplemented
56C22×D127—28no—
57C22×D127—28no—
58C22×D127—28no—
59C22×D127—416no—
60C22×D127—48no—
61C22×D127—432no—
62C22×D127C2418yes—
63Dic254—48no—
64D508—48no—
65D508—48no—
66C22⋊D127—416no—
67C2×Dic127—28no—
68C2×Dic127C2418yes—
69C23×C127—24no—
70C23×C127—28no—
71C23×C127C2414yes—
72C23×C127C2411yescharacteristic
73Q8×C127—44no—
74D4×C127—44no—
75C2×C4×C127—22no—
76C2×C4×C127—24no—
77C2×C4×C127—24no—
78C2×C4×C127C2414yes—
79C2×C4×C127C2412yes—
80C2×C4×C127C2414yes—
81C22×C127—24no—
82C22×C127—28no—
83C22×C127—48no—
84C22×C127H(5,𝔽2)12yes—
85C22×C127H(5,𝔽2)14yes—
86C22×C127C2511yescharacteristic
87C2×D127—28no—
88C2×D127—28no—
89C2×D127—28no—
90C2×D127—432no—
91C2×D127—416no—
92C4×C127—22no—
93C4×C127—24no—
94C4×C127—24no—
95Dic127—28no—
96C2×C127—44no—
97C2×C127H(3,𝔽4)14yescomplemented
98C2×C127C2×H(5,𝔽2)11yescharacteristic
99C2×C127C2×H(5,𝔽2)12yes—
100D127—48no—
101C2×H(3,𝔽4)—1271nomaximal, Sylow
102C127C2×H(3,𝔽4)11yescharacteristic, complemented, Sylow
103H(3,𝔽4)—12716no—
104C2×#124—1272no—
105C2×#124—1274no—
106C2×#124—1272no—
107C2×#124D12711yescharacteristic, complemented
108C2×#120—1272no—
109C2×#120—1274no—
110C25—1272no—
111C22×Q8—1272no—
112C22×Q8C2×D12711yescharacteristic, complemented
113C22×D4—1274no—
114C22×D4—1272no—
115C22×D4—1272no—
116C22×D4C2×D12711yescharacteristic
117#124—12716no—
118#124—12732no—
119#124—12716no—
120#124C2×D12718yescomplemented
121#120—12716no—
122#120—12732no—
123C2×C22⋊C4—1272no—
124C2×C22⋊C4—1274no—
125C2×C22⋊C4—1274no—
126C2×C22⋊C4—1274no—
127C2×C22⋊C4C2×D12714yescomplemented
128C2×C42—1272no—
129C2×C42C2×D12711yescharacteristic, complemented
130C24—1278no—
131C24—1274no—
132C24—5084no—
133C24—5088no—
134C24C22×D12712yes—
135C2×Q8—42no—
136C2×Q8—1278no—
137C2×Q8—5084no—
138C2×Q8C22×D12714yescomplemented
139C2×D4—42no—
140C2×D4—12716no—
141C2×D4—1278no—
142C2×D4—1278no—
143C2×D4—5084no—
144C2×D4—5084no—
145C2×D4—5088no—
146C2×D4C22×D12714yes—
147C22×C4—1274no—
148C22×C4C22×D12712yes—
149C22×C4C22×D12711yescharacteristic
150C22×C4C22×D12712yes—
151C22⋊C4—1278no—
152C22⋊C4—12716no—
153C22⋊C4—12716no—
154C22⋊C4—12716no—
155C22⋊C4C22×D127116yescomplemented
156C42—1278no—
157C42C22×D12714yescomplemented
158C23—24no—
159C23—28no—
160C23—1278no—
161C23—5088no—
162C23—50816no—
163C23—50832no—
164C23—2548no—
165C23—2548no—
166C23—2548no—
167C23C23×D12714yes—
168C23C23×D12711yescharacteristic
169Q8—44no—
170Q8—5088no—
171D4—44no—
172D4—5088no—
173D4—5088no—
174D4—50816no—
175C2×C4—22no—
176C2×C4—24no—
177C2×C4—24no—
178C2×C4—1278no—
179C2×C4—2548no—
180C2×C4C23×D12714yes—
181C2×C4C23×D12712yes—
182C2×C4C23×D12714yes—
183C22—24no—
184C22—28no—
185C22—48no—
186C22—50832no—
187C22—50816no—
188C22—2548no—
189C22—2548no—
190C22—2548no—
191C22#1686125412yes—
192C22#1686125514yes—
193C22C24×D12711yescharacteristic
194C4—22no—
195C4—24no—
196C4—24no—
197C4—2548no—
198C2—44no—
199C2—5088no—
200C2#2255061814yesdirect product factor
201C2C2×#1686125411yescharacteristic
202C2C2×#1686125512yes—
subgroupG/H
centerC23C23×D127
commutator, cosocleC22×C127C25
FrattiniC22C24×D127
FittingC2×#124×C127C2
socleC23×C127C24
hypercenterC2×#124D127
hypocenterC127C2×H(3,𝔽4)
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C23 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. #22550618C2direct (N × H)
2. C2×#16861198C2semidirect
3. C2×#16861197C2semidirect
4. C2×#16861195C2semidirect
5. C2×#16861179C2semidirect
6. C2×#16861177C2semidirect
7. C2×#124×C127C2semidirect
8. C24×D127C22semidirect
9. C22×Dic254C22semidirect
13. #16861198C22semidirect
14. #16861197C22semidirect
15. #16861195C22semidirect
16. #16861179C22semidirect
17. #16861177C22semidirect
18. C2×C22⋊Dic127C22semidirect
19. C2×(C2×D127)⋊C4C22semidirect
20. C2×(C2×D127)⋊C4C22semidirect
21. C2×(C2×D127)⋊C4C22semidirect
22. C2×C4×Dic127C22semidirect
23. C22×Q8×C127C22semidirect
25. #124×C127C22semidirect
26. C2×C22⋊C4×C127C22semidirect
27. C2×C42×C127C22semidirect
31. C23×D127C23semidirect
33. C2×Dic254C23semidirect
41. C22⋊Dic127C23semidirect
42. (C2×D127)⋊C4C23semidirect
43. (C2×D127)⋊C4C23semidirect
44. (C2×D127)⋊C4C23semidirect
45. C4×Dic127C23semidirect
48. C2×Q8×C127C23semidirect
54. C22⋊C4×C127C23semidirect
55. C42×C127C23semidirect
97. C2×C127H(3,𝔽4)semidirect
102. C127C2×H(3,𝔽4)semidirect
107. C2×#124D127semidirect
112. C22×Q8C2×D127semidirect
120. #124C2×D127semidirect
127. C2×C22⋊C4C2×D127semidirect
129. C2×C42C2×D127semidirect
138. C2×Q8C22×D127semidirect
155. C22⋊C4C22×D127semidirect
157. C42C22×D127semidirect
200. C2#22550618direct (N × H)

Maximal subgroups

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

subgroupindex
#225506182
C2×#168611982
C2×#168611972
C2×#168611952
C2×#168611792
C2×#168611772
C2×#124×C1272
C2×H(3,𝔽4)127