Finite Group Databasebeta

← C22×#15692251 #27184403

Subgroups — C22×#15692251 #27184403

Order 12928.

3000 conjugacy classes of subgroups; only the 725 normal subgroups are shown.

normal subgroups only

This group has 3000 conjugacy classes of subgroups — too many to draw. Only its 725 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×D404C211yesmaximal, characteristic, complemented
2C23×C22⋊D101C214yesmaximal, complemented
3C2×#15692251C2124yesmaximal, direct product factor
4C22×C4×Dic101C211yesmaximal, characteristic, complemented
5C23×D4×C101C211yesmaximal, characteristic, complemented
6C24×D101C2212yes—
7C22×D404—28no—
8C22×D404C22112yescomplemented
9C22×C22⋊D101—232no—
10C22×C22⋊D101C22148yescomplemented
11C23×Dic101C2212yes—
12#15692251C22164yesdirect product factor
13C2×C4×Dic101C22112yescomplemented
14C25×C101C2212yes—
15C22×D4×C101—28no—
16C22×D4×C101C22112yescomplemented
17C23×C4×C101C2211yescharacteristic
18C23×D101—216no—
19C23×D101—28no—
20C23×D101C23112yes—
21C2×D404—248no—
22C2×D404C23116yescomplemented
23C2×C22⋊D101—2192no—
24C2×C22⋊D101C23164yescomplemented
25C22×Dic101D4116yescomplemented
26C22×Dic101C23112yes—
27C4×Dic101C23116yescomplemented
28C24×C101—216no—
29C24×C101—28no—
30C24×C101C23112yes—
31C24×C101C2311yescharacteristic
32C2×D4×C101—248no—
33C2×D4×C101C23116yescomplemented
34C22×C4×C101D418yescomplemented
35C22×C4×C101C2316yes—
36C23×C101—224no—
37C23×C101—248no—
38C23×C101—432no—
39C23×C101C2×D418yes—
40C23×C101C2×D414yes—
41C23×C101C2418yes—
42C23×C101C2413yes—
43C22×D101—248no—
44C22×D101—224no—
45C22×D101—432no—
46C22×D101C2418yes—
47D4×C101—232no—
48C2×C4×C101C2×D4124yescomplemented
49C2×C4×C101C2414yes—
50C22⋊D101—2128no—
51C2×Dic101C2×D4148yescomplemented
52C2×Dic101C2418yes—
53D404—232no—
54C22×C101—216no—
55C22×C101—28no—
56C22×C101—448no—
57C22×C101#127116yescomplemented
58C22×C101C22×D4112yes—
59C22×C101C22×D416yes—
60C22×C101C2511yescharacteristic
61C2×D101—28no—
62C2×D101—216no—
63C2×D101—448no—
64C4×C101C22×D418yescomplemented
65Dic101C22×D4116yescomplemented
66C2×C101—48no—
67C2×C101C2×#127112yescomplemented
68C2×C101C23×D412yes—
69C2×C101C23×D411yescharacteristic
70D101—48no—
71C22×#127—1011nomaximal, Sylow
72C101C22×#12711yescharacteristic, complemented, Sylow
73C23×D4—1014no—
74C23×D4—1011no—
75C23×D4D10111yescharacteristic, complemented
76C2×#127—10124no—
77C22×C42—1011no—
78C25—1012no—
79C25C2×D10112yes—
80C22×D4—28no—
81C22×D4—10148no—
82C22×D4—10112no—
83C22×D4—20232no—
84C22×D4—2028no—
85C22×D4C2×D101112yescomplemented
86C23×C4—1012no—
87C23×C4C2×D10111yescharacteristic
88#127—10164no—
89C2×C42—10112no—
90C24—216no—
91C24—28no—
92C24—10112no—
93C24—20216no—
94C24—2028no—
95C24C22×D101112yes—
96C24C22×D10111yescharacteristic
97C2×D4—248no—
98C2×D4—10164no—
99C2×D4—10116no—
100C2×D4—202192no—
101C2×D4—20248no—
102C2×D4C22×D101116yescomplemented
103C22×C4—10116no—
104C22×C4—10112no—
105C22×C4C22⋊D10118yescomplemented
106C22×C4C22×D10116yes—
107C42—10116no—
108C23—248no—
109C23—224no—
110C23—432no—
111C23—1018no—
112C23—40432no—
113C23—20248no—
114C23—20224no—
115C23D4×D10118yes—
116C23C2×C22⋊D10114yes—
117C23C23×D10118yes—
118C23C23×D10113yes—
119D4—232no—
120D4—202128no—
121D4—20232no—
122C2×C4—10148no—
123C2×C4—1018no—
124C2×C4C2×C22⋊D101124yescomplemented
125C2×C4C23×D10114yes—
126C22—216no—
127C22—28no—
128C22—448no—
129C22—40448no—
130C22—20216no—
131C22—2028no—
132C22#15692251116yesdirect product factor
133C22C2×D4×D101112yes—
134C22C22×C22⋊D10116yes—
135C22C24×D10111yescharacteristic
136C4—10116no—
137C4C22×C22⋊D10118yescomplemented
138C2—48no—
139C2—4048no—
140C2C2×#15692251112yesdirect product factor
141C2C22×D4×D10112yes—
142C2C23×C22⋊D10111yescharacteristic
subgroupG/H
centerC24C22×D101
commutator, cosocleC22×C101C25
FrattiniC22C24×D101
FittingC23×D4×C101C2
socleC24×C101C23
hypercenterC23×D4D101
hypocenterC101C22×#127
LCSG ▷ C22×C101 ▷ C101
UCSC1 ◁ C24 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C23×D404C2semidirect
2. C23×C22⋊D101C2semidirect
3. C2×#15692251C2direct (N × H)
4. C22×C4×Dic101C2semidirect
5. C23×D4×C101C2semidirect
8. C22×D404C22semidirect
10. C22×C22⋊D101C22semidirect
12. #15692251C22direct (N × H)
13. C2×C4×Dic101C22semidirect
16. C22×D4×C101C22semidirect
22. C2×D404C23semidirect
24. C2×C22⋊D101C23semidirect
25. C22×Dic101D4semidirect
27. C4×Dic101C23semidirect
33. C2×D4×C101C23semidirect
34. C22×C4×C101D4semidirect
48. C2×C4×C101C2×D4semidirect
51. C2×Dic101C2×D4semidirect
57. C22×C101#127semidirect
64. C4×C101C22×D4semidirect
65. Dic101C22×D4semidirect
67. C2×C101C2×#127semidirect
72. C101C22×#127semidirect
75. C23×D4D101semidirect
85. C22×D4C2×D101semidirect
102. C2×D4C22×D101semidirect
105. C22×C4C22⋊D101semidirect
124. C2×C4C2×C22⋊D101semidirect
132. C22#15692251direct (N × H)
137. C4C22×C22⋊D101semidirect
140. C2C2×#15692251direct (N × H)

Maximal subgroups

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

subgroupindex
C23×D4042
C23×C22⋊D1012
C2×#156922512
C22×C4×Dic1012
C23×D4×C1012
C22×#127101