Finite Group Databasebeta

← C22×#16861171 #30651900

Subgroups — C22×#16861171 #30651900

Order 16256.

2528 conjugacy classes of subgroups; only the 941 normal subgroups are shown.

normal subgroups only

This group has 2528 conjugacy classes of subgroups — too many to draw. Only its 941 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
1C23×D508C211yesmaximal, characteristic
2C23×C4×D127C212yesmaximal, complemented
3C2×#16861171C2124yesmaximal, direct product factor
4C22×(C2×D127)⋊C4C212yesmaximal, complemented
5C22×C4×Dic127C211yesmaximal, characteristic, complemented
6C22×C4⋊C4×C127C211yesmaximal, characteristic, complemented
7C24×D127C2212yes—
8C22×D508C4116yescomplemented
9C22×D508C22112yes—
10C23×Dic127C2211yescharacteristic
11C23×Dic127C2211yescharacteristic
12C22×C4×D127—216no—
13C22×C4×D127C22124yescomplemented
14#16861171C22164yesdirect product factor
15C2×(C2×D127)⋊C4C22124yescomplemented
16C2×C4×Dic127C22112yescomplemented
17C23×C4×C127C2212yes—
18C23×C4×C127C2211yescharacteristic
19C2×C4⋊C4×C127C22112yescomplemented
20C23×D127—28no—
21C23×D127—28no—
22C23×D127C2×C4116yescomplemented
23C23×D127C23112yes—
24C2×D508C2×C4196yescomplemented
25C2×D508C23116yes—
26C22×Dic127—24no—
27C22×Dic127D418yescomplemented
28C22×Dic127C2316yes—
29C22×Dic127C2316yes—
30C2×C4×D127—296no—
31C2×C4×D127C23132yescomplemented
32(C2×D127)⋊C4C23132yescomplemented
33C4×Dic127C23116yescomplemented
34C24×C127C2311yescharacteristic
35C22×C4×C127—28no—
36C22×C4×C127C2×C418yescomplemented
37C22×C4×C127C23112yes—
38C22×C4×C127C2316yes—
39C4⋊C4×C127C23116yescomplemented
40C22×D127—264no—
41C22×D127—224no—
42C22×D127—224no—
43C22×D127C22×C4148yescomplemented
44C22×D127C2418yes—
45D508C22×C4164yescomplemented
46C2×Dic127—212no—
47C2×Dic127C2×D4124yescomplemented
48C2×Dic127C2414yes—
49C2×Dic127C2414yes—
50C4×D127—264no—
51C23×C127C22×C414yes—
52C23×C127C2×D414yes—
53C23×C127#4114yes—
54C23×C127C2413yes—
55C2×C4×C127—224no—
56C2×C4×C127C22×C4124yescomplemented
57C2×C4×C127C2418yes—
58C2×C4×C127C2414yes—
59C22×C127C4×D4116yescomplemented
60C22×C127C23×C416yes—
61C22×C127C22×D416yes—
62C22×C127C2×#4116yes—
63C22×C127C2511yescharacteristic
64C2×D127—296no—
65C2×D127—28no—
66C2×D127—28no—
67C2×D127C23×C4116yescomplemented
68C4×C127—28no—
69C4×C127C23×C418yescomplemented
70Dic127—24no—
71Dic127C22×D418yescomplemented
72C2×C127C2×C4×D4112yescomplemented
73C2×C127C24×C411yescharacteristic
74C2×C127C23×D411yescharacteristic
75C2×C127C22×#4111yescharacteristic
76D127—216no—
77C22×C4×D4—1271nomaximal, Sylow
78C127C22×C4×D411yescharacteristic, complemented, Sylow
79C23×D4—1271no—
80C24×C4—1272no—
81C2×C4×D4—12724no—
82C22×C4⋊C4D12711yescharacteristic, complemented
83C22×C22⋊C4—1272no—
84C22×C42—1271no—
85C25—1272no—
86C22×D4—12716no—
87C22×D4—12712no—
88C23×C4—1271no—
89C23×C4—1271no—
90C23×C4—12724no—
91C23×C4—25416no—
92C23×C4C2×D12712yes—
93C23×C4C2×D12711yescharacteristic
94C4×D4—12764no—
95C2×C4⋊C4C2×D127112yescomplemented
96C2×C22⋊C4—12724no—
97C2×C42—12712no—
98C24—12716no—
99C24—12712no—
100C24—2548no—
101C24—2548no—
102C24C22×D12711yescharacteristic
103C2×D4—12796no—
104C2×D4—12716no—
105C22×C4—28no—
106C22×C4—1278no—
107C22×C4—1276no—
108C22×C4—1276no—
109C22×C4—12732no—
110C22×C4—25496no—
111C22×C4—2544no—
112C22×C4C4×D12718yescomplemented
113C22×C4C22×D127112yes—
114C22×C4C22×D12716yes—
115C4⋊C4C22×D127116yescomplemented
116C22⋊C4—12732no—
117C42—12716no—
118C23—12748no—
119C23—1278no—
120C23—25424no—
121C23—25424no—
122C23—25464no—
123C23C2×C4×D12714yes—
124C23D4×D12714yes—
125C23#1452324914yes—
126C23C23×D12713yes—
127D4—12764no—
128C2×C4—224no—
129C2×C4—12724no—
130C2×C4—1274no—
131C2×C4—1274no—
132C2×C4—25464no—
133C2×C4—25412no—
134C2×C4C2×C4×D127124yescomplemented
135C2×C4C23×D12718yes—
136C2×C4C23×D12714yes—
137C22—12716no—
138C22—25496no—
139C22—2548no—
140C22—2548no—
141C22#16861171116yesdirect product factor
142C22C22×C4×D12716yes—
143C22C2×D4×D12716yes—
144C22C2×#1452324916yes—
145C22C24×D12711yescharacteristic
146C4—28no—
147C4—1278no—
148C4—2544no—
149C4C22×C4×D12718yescomplemented
150C2—25416no—
151C2C2×#16861171112yesdirect product factor
152C2C23×C4×D12711yescharacteristic
153C2C22×D4×D12711yescharacteristic
154C2C22×#1452324911yescharacteristic
subgroupG/H
centerC24C22×D127
commutatorC2×C127C24×C4
FrattiniC22C24×D127
FittingC22×C4⋊C4×C127C2
socleC24×C127C23
prime radicalC23×D508C2
cosocleC22×C127C25
hypercenterC22×C4⋊C4D127
hypocenterC127C22×C4×D4
LCSG ▷ C2×C127 ▷ C127
UCSC1 ◁ C24 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C23×C4×D127C2semidirect
3. C2×#16861171C2direct (N × H)
4. C22×(C2×D127)⋊C4C2semidirect
5. C22×C4×Dic127C2semidirect
6. C22×C4⋊C4×C127C2semidirect
8. C22×D508C4semidirect
13. C22×C4×D127C22semidirect
14. #16861171C22direct (N × H)
15. C2×(C2×D127)⋊C4C22semidirect
16. C2×C4×Dic127C22semidirect
19. C2×C4⋊C4×C127C22semidirect
22. C23×D127C2×C4semidirect
24. C2×D508C2×C4semidirect
27. C22×Dic127D4semidirect
31. C2×C4×D127C23semidirect
32. (C2×D127)⋊C4C23semidirect
33. C4×Dic127C23semidirect
36. C22×C4×C127C2×C4semidirect
39. C4⋊C4×C127C23semidirect
43. C22×D127C22×C4semidirect
45. D508C22×C4semidirect
47. C2×Dic127C2×D4semidirect
56. C2×C4×C127C22×C4semidirect
59. C22×C127C4×D4semidirect
67. C2×D127C23×C4semidirect
69. C4×C127C23×C4semidirect
71. Dic127C22×D4semidirect
72. C2×C127C2×C4×D4semidirect
78. C127C22×C4×D4semidirect
82. C22×C4⋊C4D127semidirect
95. C2×C4⋊C4C2×D127semidirect
112. C22×C4C4×D127semidirect
115. C4⋊C4C22×D127semidirect
134. C2×C4C2×C4×D127semidirect
141. C22#16861171direct (N × H)
149. C4C22×C4×D127semidirect
151. C2C2×#16861171direct (N × H)

Maximal subgroups

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

subgroupindex
C23×D5082
C23×C4×D1272
C2×#168611712
C22×(C2×D127)⋊C42
C22×C4×Dic1272
C22×C4⋊C4×C1272
C22×C4×D4127