Finite Group Databasebeta

← C2×#22550619 #30652851

Subgroups — C2×#22550619 #30652851

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#22550619C2116yesmaximal, direct product factor
2C2×#16861196C213yesmaximal, complemented
3C2×#16861195C216yesmaximal, complemented
4C2×#16861178C212yesmaximal, complemented
5C2×#16861177C213yesmaximal, complemented
6C2×#120×C127C211yesmaximal, characteristic, complemented
7C24×D127C2211yescharacteristic, complemented
8C22×Dic254C2213yescomplemented
9C22×C22⋊D127C2216yes—
10#16861196C22124yescomplemented
11#16861195C22148yescomplemented
12#16861178C22116yescomplemented
13#16861177C22124yescomplemented
14C2×C22⋊Dic127C2213yescomplemented
15C2×C22⋊Dic127C2216yescomplemented
16C2×(C2×D127)⋊C4C2216yescomplemented
17C2×C4×Dic127C2213yescomplemented
18C25×C127C2211yescharacteristic, complemented
19C22×D4×C127C2213yes—
20#120×C127C2218yescomplemented
21C2×C22⋊C4×C127C2213yescomplemented
22C23×D127—46no—
23C23×D127C2312yes—
24C23×D127C2314yescomplemented
25C2×Dic254—46no—
26C2×Dic254C23112yescomplemented
27C2×C22⋊D127—412no—
28C2×C22⋊D127C23124yes—
29C22×Dic127C2316yes—
30C22⋊Dic127C23124yescomplemented
31C22⋊Dic127C23112yescomplemented
32(C2×D127)⋊C4C23124yescomplemented
33C4×Dic127C23112yescomplemented
34C24×C127—46no—
35C24×C127C2313yes—
36C24×C127C2311yescharacteristic
37C24×C127C2314yescomplemented
38C2×D4×C127—46no—
39C2×D4×C127C23112yes—
40C22×C4×C127C2313yes—
41C22⋊C4×C127C23112yescomplemented
42C22×D127—212no—
43C22×D127—412no—
44C22×D127—416no—
45C22×D127C2414yes—
46Dic254—412no—
47C22⋊D127—424no—
48C2×Dic127—212no—
49C2×Dic127C24112yes—
50C23×C127—26no—
51C23×C127—26no—
52C23×C127—212no—
53C23×C127—412no—
54C23×C127—416no—
55C23×C127C2416yes—
56C23×C127C2412yes—
57C23×C127C2411yescharacteristic
58D4×C127—412no—
59C2×C4×C127—26no—
60C2×C4×C127C2416yes—
61C22×C127—26no—
62C22×C127—26no—
63C22×C127—212no—
64C22×C127—416no—
65C22×C127—412no—
66C22×C127—44no—
67C22×C127H(5,𝔽2)16yes—
68C22×C127C2511yescharacteristic
69C2×D127—212no—
70C2×D127—416no—
71C2×D127—48no—
72C4×C127—26no—
73Dic127—212no—
74C2×C127—42no—
75C2×C127—46no—
76C2×C127H(3,𝔽4)14yescomplemented
77C2×C127C2×H(5,𝔽2)13yes—
78D127—44no—
79C2×H(3,𝔽4)—1271nomaximal, Sylow
80C127C2×H(3,𝔽4)11yescharacteristic, complemented, Sylow
81H(3,𝔽4)—12716no—
82C2×#124—1273no—
83C2×#124—1276no—
84C2×#120—1272no—
85C2×#120—1273no—
86C2×#120D12711yescharacteristic, complemented
87C25—1271no—
88C25C2×D12711yescharacteristic, complemented
89C22×Q8—1273no—
90C22×D4—1276no—
91C22×D4C2×D12713yes—
92#124—12724no—
93#124—12748no—
94#120—12716no—
95#120—12724no—
96#120C2×D12718yescomplemented
97C2×C22⋊C4—1273no—
98C2×C22⋊C4—1276no—
99C2×C22⋊C4—1276no—
100C2×C22⋊C4C2×D12713yescomplemented
101C2×C42—1273no—
102C24—46no—
103C24—1274no—
104C24—1272no—
105C24—5086no—
106C24C22×D12711yescharacteristic
107C24C22×D12713yes—
108C24C22×D12714yescomplemented
109C2×Q8—12712no—
110C2×Q8—5086no—
111C2×D4—46no—
112C2×D4—12724no—
113C2×D4—50812no—
114C2×D4C22×D127112yes—
115C22×C4—1276no—
116C22×C4C22×D12713yes—
117C22⋊C4—12712no—
118C22⋊C4—12724no—
119C22⋊C4—12724no—
120C22⋊C4C22×D127112yescomplemented
121C42—12712no—
122C23—26no—
123C23—26no—
124C23—212no—
125C23—412no—
126C23—416no—
127C23—1274no—
128C23—50812no—
129C23—50816no—
130C23—25412no—
131C23C23×D12712yes—
132C23C23×D12716yes—
133C23C23×D12711yescharacteristic
134Q8—50812no—
135D4—412no—
136D4—50824no—
137C2×C4—26no—
138C2×C4—12712no—
139C2×C4—25412no—
140C2×C4C23×D12716yes—
141C22—26no—
142C22—26no—
143C22—212no—
144C22—416no—
145C22—412no—
146C22—44no—
147C22—50816no—
148C22—5088no—
149C22—25412no—
150C22#1686125416yes—
151C22C24×D12711yescharacteristic
152C4—26no—
153C4—25412no—
154C2—46no—
155C2—42no—
156C2—5084no—
157C2#2255061914yesdirect product factor
158C2C2×#1686125413yes—
subgroupG/H
centerC23C23×D127
commutator, cosocleC22×C127C25
FrattiniC22C24×D127
FittingC2×#120×C127C2
socleC23×C127C24
hypercenterC2×#120D127
hypocenterC127C2×H(3,𝔽4)
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C23 ◁ C2×#120

Complements

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

NH ≅ G/Ntype
1. #22550619C2direct (N × H)
2. C2×#16861196C2semidirect
3. C2×#16861195C2semidirect
4. C2×#16861178C2semidirect
5. C2×#16861177C2semidirect
6. C2×#120×C127C2semidirect
7. C24×D127C22semidirect
8. C22×Dic254C22semidirect
10. #16861196C22semidirect
11. #16861195C22semidirect
12. #16861178C22semidirect
13. #16861177C22semidirect
14. C2×C22⋊Dic127C22semidirect
15. C2×C22⋊Dic127C22semidirect
16. C2×(C2×D127)⋊C4C22semidirect
17. C2×C4×Dic127C22semidirect
18. C25×C127C22semidirect
20. #120×C127C22semidirect
21. C2×C22⋊C4×C127C22semidirect
24. C23×D127C23semidirect
26. C2×Dic254C23semidirect
30. C22⋊Dic127C23semidirect
31. C22⋊Dic127C23semidirect
32. (C2×D127)⋊C4C23semidirect
33. C4×Dic127C23semidirect
37. C24×C127C23semidirect
41. C22⋊C4×C127C23semidirect
76. C2×C127H(3,𝔽4)semidirect
80. C127C2×H(3,𝔽4)semidirect
86. C2×#120D127semidirect
88. C25C2×D127semidirect
96. #120C2×D127semidirect
100. C2×C22⋊C4C2×D127semidirect
108. C24C22×D127semidirect
120. C22⋊C4C22×D127semidirect
157. C2#22550619direct (N × H)

Maximal subgroups

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

subgroupindex
#225506192
C2×#168611962
C2×#168611952
C2×#168611782
C2×#168611772
C2×#120×C1272
C2×H(3,𝔽4)127