Finite Group Databasebeta

← C22×C4×D388 #27154908

Subgroups — C22×C4×D388 #27154908

Order 12416.

2528 conjugacy classes of subgroups; only the 981 normal subgroups are shown.

normal subgroups only

This group has 2528 conjugacy classes of subgroups — too many to draw. Only its 981 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×D388C211yesmaximal, characteristic
2C23×C4×D97C212yesmaximal, complemented
3C2×C4×D388C2124yesmaximal, direct product factor
4C22×C4⋊Dic97C211yesmaximal, characteristic, complemented
5C22×#13095488C212yesmaximal, complemented
6C22×C42×C97C211yesmaximal, characteristic, complemented
7C24×D97C2212yes—
8C22×D388C4116yesdirect product factor
9C22×D388C22112yes—
10C23×Dic97C2212yes—
11C22×C4×D97—216no—
12C22×C4×D97C22124yescomplemented
13C4×D388C22164yesdirect product factor
14C2×C4⋊Dic97C22112yescomplemented
15C2×#13095488C22124yescomplemented
16C23×C4×C97C2211yescharacteristic
17C23×C4×C97C2211yescharacteristic
18C23×C4×C97C2211yescharacteristic
19C2×C42×C97C22112yescomplemented
20C23×D97—28no—
21C23×D97—28no—
22C23×D97C2×C4116yescomplemented
23C23×D97C23112yes—
24C2×D388C2×C4196yesdirect product factor
25C2×D388C23116yes—
26C22×Dic97—28no—
27C22×Dic97C23112yes—
28C2×C4×D97—296no—
29C2×C4×D97C23132yescomplemented
30C4⋊Dic97C23116yescomplemented
31#13095488C23132yescomplemented
32C24×C97C2311yescharacteristic
33C22×C4×C97—24no—
34C22×C4×C97C2×C418yescomplemented
35C22×C4×C97D418yescomplemented
36C22×C4×C97C2316yes—
37C22×C4×C97C2316yes—
38C22×C4×C97C2316yes—
39C42×C97C23116yescomplemented
40C23×C97C22×C414yes—
41C23×C97C2×D414yes—
42C23×C97#4114yes—
43C23×C97C2413yes—
44C22×D97—264no—
45C22×D97—224no—
46C22×D97—224no—
47C22×D97C22×C4148yescomplemented
48C22×D97C2418yes—
49C2×C4×C97—212no—
50C2×C4×C97C22×C4124yescomplemented
51C2×C4×C97C2×D4124yescomplemented
52C2×C4×C97C2414yes—
53C2×C4×C97C2414yes—
54C2×C4×C97C2414yes—
55C2×Dic97—224no—
56C2×Dic97C2418yes—
57D388C22×C4164yesdirect product factor
58C4×D97—264no—
59C22×C97C4×D4116yescomplemented
60C22×C97C23×C416yes—
61C22×C97C22×D416yes—
62C22×C97C2×#4116yes—
63C22×C97C2511yescharacteristic
64C2×D97—296no—
65C2×D97—28no—
66C2×D97—28no—
67C2×D97C23×C4116yescomplemented
68C4×C97—24no—
69C4×C97C23×C418yescomplemented
70C4×C97C22×D418yescomplemented
71Dic97—28no—
72C2×C97C2×C4×D4112yescomplemented
73C2×C97C24×C411yescharacteristic
74C2×C97C23×D411yescharacteristic
75C2×C97C22×#4111yescharacteristic
76D97—216no—
77C22×C4×D4—971nomaximal, Sylow
78C97C22×C4×D411yescharacteristic, complemented, Sylow
79C23×D4—971no—
80C24×C4—972no—
81C2×C4×D4—9724no—
82C22×C4⋊C4—971no—
83C22×C22⋊C4—972no—
84C22×C42D9711yescharacteristic, complemented
85C25—972no—
86C22×D4—9716no—
87C22×D4—9712no—
88C23×C4—972no—
89C23×C4—9724no—
90C23×C4—19416no—
91C23×C4C2×D9711yescharacteristic
92C23×C4C2×D9711yescharacteristic
93C23×C4C2×D9711yescharacteristic
94C4×D4—9764no—
95C2×C4⋊C4—9712no—
96C2×C22⋊C4—9724no—
97C2×C42C2×D97112yescomplemented
98C24—9716no—
99C24—9712no—
100C24—1948no—
101C24—1948no—
102C24C22×D9711yescharacteristic
103C2×D4—9796no—
104C2×D4—9716no—
105C22×C4—24no—
106C22×C4—9712no—
107C22×C4—9732no—
108C22×C4—1948no—
109C22×C4—19496no—
110C22×C4C4×D9718yescomplemented
111C22×C4D38818yesdirect product factor
112C22×C4C22×D9716yes—
113C22×C4C22×D9716yes—
114C22×C4C22×D9716yes—
115C4⋊C4—9716no—
116C22⋊C4—9732no—
117C42C22×D97116yescomplemented
118C23—9748no—
119C23—978no—
120C23—19464no—
121C23—19424no—
122C23—19424no—
123C23C2×C4×D9714yes—
124C23C2×D38814yes—
125C23#1309551114yes—
126C23C23×D9713yes—
127D4—9764no—
128C2×C4—212no—
129C2×C4—978no—
130C2×C4—19424no—
131C2×C4—19464no—
132C2×C4C2×C4×D97124yescomplemented
133C2×C4C2×D388124yesdirect product factor
134C2×C4C23×D9714yes—
135C2×C4C23×D9714yes—
136C2×C4C23×D9714yes—
137C22—9716no—
138C22—19496no—
139C22—1948no—
140C22—1948no—
141C22C4×D388116yesdirect product factor
142C22C22×C4×D9716yes—
143C22C22×D38816yes—
144C22C2×#1309551116yes—
145C22C24×D9711yescharacteristic
146C4—24no—
147C4—1948no—
148C4C22×C4×D9718yescomplemented
149C4C22×D38818yesdirect product factor
150C2—19416no—
151C2C2×C4×D388112yesdirect product factor
152C2C23×C4×D9711yescharacteristic
153C2C23×D38811yescharacteristic
154C2C22×#1309551111yescharacteristic
subgroupG/H
centerC23×C4C2×D97
commutatorC2×C97C24×C4
FrattiniC22C24×D97
FittingC22×C42×C97C2
socleC24×C97C23
prime radicalC23×D388C2
cosocleC22×C97C25
hypercenterC22×C42D97
hypocenterC97C22×C4×D4
LCSG ▷ C2×C97 ▷ C97
UCSC1 ◁ C23×C4 ◁ C22×C42

Complements

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

NH ≅ G/Ntype
2. C23×C4×D97C2semidirect
3. C2×C4×D388C2direct (N × H)
4. C22×C4⋊Dic97C2semidirect
5. C22×#13095488C2semidirect
6. C22×C42×C97C2semidirect
8. C22×D388C4direct (N × H)
12. C22×C4×D97C22semidirect
13. C4×D388C22direct (N × H)
14. C2×C4⋊Dic97C22semidirect
15. C2×#13095488C22semidirect
19. C2×C42×C97C22semidirect
22. C23×D97C2×C4semidirect
24. C2×D388C2×C4direct (N × H)
29. C2×C4×D97C23semidirect
30. C4⋊Dic97C23semidirect
31. #13095488C23semidirect
34. C22×C4×C97C2×C4semidirect
35. C22×C4×C97D4semidirect
39. C42×C97C23semidirect
47. C22×D97C22×C4semidirect
50. C2×C4×C97C22×C4semidirect
51. C2×C4×C97C2×D4semidirect
57. D388C22×C4direct (N × H)
59. C22×C97C4×D4semidirect
67. C2×D97C23×C4semidirect
69. C4×C97C23×C4semidirect
70. C4×C97C22×D4semidirect
72. C2×C97C2×C4×D4semidirect
78. C97C22×C4×D4semidirect
84. C22×C42D97semidirect
97. C2×C42C2×D97semidirect
110. C22×C4C4×D97semidirect
111. C22×C4D388direct (N × H)
117. C42C22×D97semidirect
132. C2×C4C2×C4×D97semidirect
133. C2×C4C2×D388direct (N × H)
141. C22C4×D388direct (N × H)
148. C4C22×C4×D97semidirect
149. C4C22×D388direct (N × H)
151. C2C2×C4×D388direct (N × H)

Maximal subgroups

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

subgroupindex
C23×D3882
C23×C4×D972
C2×C4×D3882
C22×C4⋊Dic972
C22×#130954882
C22×C42×C972
C22×C4×D497