Finite Group Databasebeta

← C22×#16813584 #28403744

Subgroups — C22×#16813584 #28403744

Order 14464.

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×D452C211yesmaximal, characteristic
2C23×C4×D113C212yesmaximal, complemented
3C2×#16813584C2124yesmaximal, direct product factor
4C22×#14268624C212yesmaximal, complemented
5C22×C4×Dic113C211yesmaximal, characteristic, complemented
6C22×C4⋊C4×C113C211yesmaximal, characteristic, complemented
7C24×D113C2212yes—
8C22×D452C4116yescomplemented
9C22×D452C22112yes—
10C23×Dic113C2211yescharacteristic
11C23×Dic113C2211yescharacteristic
12C22×C4×D113—216no—
13C22×C4×D113C22124yescomplemented
14#16813584C22164yesdirect product factor
15C2×#14268624C22124yescomplemented
16C2×C4×Dic113C22112yescomplemented
17C23×C4×C113C2212yes—
18C23×C4×C113C2211yescharacteristic
19C2×C4⋊C4×C113C22112yescomplemented
20C23×D113—28no—
21C23×D113—28no—
22C23×D113C2×C4116yescomplemented
23C23×D113C23112yes—
24C2×D452C2×C4196yescomplemented
25C2×D452C23116yes—
26C22×Dic113—24no—
27C22×Dic113D418yescomplemented
28C22×Dic113C2316yes—
29C22×Dic113C2316yes—
30C2×C4×D113—296no—
31C2×C4×D113C23132yescomplemented
32#14268624C23132yescomplemented
33C4×Dic113C23116yescomplemented
34C24×C113C2311yescharacteristic
35C22×C4×C113—28no—
36C22×C4×C113C2×C418yescomplemented
37C22×C4×C113C23112yes—
38C22×C4×C113C2316yes—
39C4⋊C4×C113C23116yescomplemented
40C23×C113C22×C414yes—
41C23×C113C2×D414yes—
42C23×C113#4114yes—
43C23×C113C2413yes—
44C22×D113—264no—
45C22×D113—224no—
46C22×D113—224no—
47C22×D113C22×C4148yescomplemented
48C22×D113C2418yes—
49C2×C4×C113—224no—
50C2×C4×C113C22×C4124yescomplemented
51C2×C4×C113C2418yes—
52C2×C4×C113C2414yes—
53C2×Dic113—212no—
54C2×Dic113C2×D4124yescomplemented
55C2×Dic113C2414yes—
56C2×Dic113C2414yes—
57D452C22×C4164yescomplemented
58C4×D113—264no—
59C22×C113C4×D4116yescomplemented
60C22×C113C23×C416yes—
61C22×C113C22×D416yes—
62C22×C113C2×#4116yes—
63C22×C113C2511yescharacteristic
64C2×D113—296no—
65C2×D113—28no—
66C2×D113—28no—
67C2×D113C23×C4116yescomplemented
68C4×C113—28no—
69C4×C113C23×C418yescomplemented
70Dic113—24no—
71Dic113C22×D418yescomplemented
72C2×C113C2×C4×D4112yescomplemented
73C2×C113C24×C411yescharacteristic
74C2×C113C23×D411yescharacteristic
75C2×C113C22×#4111yescharacteristic
76D113—216no—
77C22×C4×D4—1131nomaximal, Sylow
78C113C22×C4×D411yescharacteristic, complemented, Sylow
79C23×D4—1131no—
80C24×C4—1132no—
81C2×C4×D4—11324no—
82C22×C4⋊C4D11311yescharacteristic, complemented
83C22×C22⋊C4—1132no—
84C22×C42—1131no—
85C25—1132no—
86C22×D4—11316no—
87C22×D4—11312no—
88C23×C4—1131no—
89C23×C4—1131no—
90C23×C4—11324no—
91C23×C4—22616no—
92C23×C4C2×D11312yes—
93C23×C4C2×D11311yescharacteristic
94C4×D4—11364no—
95C2×C4⋊C4C2×D113112yescomplemented
96C2×C22⋊C4—11324no—
97C2×C42—11312no—
98C24—11316no—
99C24—11312no—
100C24—2268no—
101C24—2268no—
102C24C22×D11311yescharacteristic
103C2×D4—11396no—
104C2×D4—11316no—
105C22×C4—28no—
106C22×C4—1138no—
107C22×C4—1136no—
108C22×C4—1136no—
109C22×C4—11332no—
110C22×C4—22696no—
111C22×C4—2264no—
112C22×C4C4×D11318yescomplemented
113C22×C4C22×D113112yes—
114C22×C4C22×D11316yes—
115C4⋊C4C22×D113116yescomplemented
116C22⋊C4—11332no—
117C42—11316no—
118C23—11348no—
119C23—1138no—
120C23—22664no—
121C23—22624no—
122C23—22624no—
123C23C2×C4×D11314yes—
124C23D4×D11314yes—
125C23#1426864814yes—
126C23C23×D11313yes—
127D4—11364no—
128C2×C4—224no—
129C2×C4—11324no—
130C2×C4—1134no—
131C2×C4—1134no—
132C2×C4—22664no—
133C2×C4—22612no—
134C2×C4C2×C4×D113124yescomplemented
135C2×C4C23×D11318yes—
136C2×C4C23×D11314yes—
137C22—11316no—
138C22—22696no—
139C22—2268no—
140C22—2268no—
141C22#16813584116yesdirect product factor
142C22C22×C4×D11316yes—
143C22C2×D4×D11316yes—
144C22C2×#1426864816yes—
145C22C24×D11311yescharacteristic
146C4—28no—
147C4—1138no—
148C4—2264no—
149C4C22×C4×D11318yescomplemented
150C2—22616no—
151C2C2×#16813584112yesdirect product factor
152C2C23×C4×D11311yescharacteristic
153C2C22×D4×D11311yescharacteristic
154C2C22×#1426864811yescharacteristic
subgroupG/H
centerC24C22×D113
commutatorC2×C113C24×C4
FrattiniC22C24×D113
FittingC22×C4⋊C4×C113C2
socleC24×C113C23
prime radicalC23×D452C2
cosocleC22×C113C25
hypercenterC22×C4⋊C4D113
hypocenterC113C22×C4×D4
LCSG ▷ C2×C113 ▷ C113
UCSC1 ◁ C24 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C23×C4×D113C2semidirect
3. C2×#16813584C2direct (N × H)
4. C22×#14268624C2semidirect
5. C22×C4×Dic113C2semidirect
6. C22×C4⋊C4×C113C2semidirect
8. C22×D452C4semidirect
13. C22×C4×D113C22semidirect
14. #16813584C22direct (N × H)
15. C2×#14268624C22semidirect
16. C2×C4×Dic113C22semidirect
19. C2×C4⋊C4×C113C22semidirect
22. C23×D113C2×C4semidirect
24. C2×D452C2×C4semidirect
27. C22×Dic113D4semidirect
31. C2×C4×D113C23semidirect
32. #14268624C23semidirect
33. C4×Dic113C23semidirect
36. C22×C4×C113C2×C4semidirect
39. C4⋊C4×C113C23semidirect
47. C22×D113C22×C4semidirect
50. C2×C4×C113C22×C4semidirect
54. C2×Dic113C2×D4semidirect
57. D452C22×C4semidirect
59. C22×C113C4×D4semidirect
67. C2×D113C23×C4semidirect
69. C4×C113C23×C4semidirect
71. Dic113C22×D4semidirect
72. C2×C113C2×C4×D4semidirect
78. C113C22×C4×D4semidirect
82. C22×C4⋊C4D113semidirect
95. C2×C4⋊C4C2×D113semidirect
112. C22×C4C4×D113semidirect
115. C4⋊C4C22×D113semidirect
134. C2×C4C2×C4×D113semidirect
141. C22#16813584direct (N × H)
149. C4C22×C4×D113semidirect
151. C2C2×#16813584direct (N × H)

Maximal subgroups

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

subgroupindex
C23×D4522
C23×C4×D1132
C2×#168135842
C22×#142686242
C22×C4×Dic1132
C22×C4⋊C4×C1132
C22×C4×D4113