Finite Group Databasebeta

← C22×#15692226 #27184351

Subgroups — C22×#15692226 #27184351

Order 12928.

2528 conjugacy classes of subgroups; only the 685 normal subgroups are shown.

normal subgroups only

This group has 2528 conjugacy classes of subgroups — too many to draw. Only its 685 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×C22⋊D101C212yesmaximal, complemented
2C23×C4×D101C211yesmaximal, characteristic, complemented
3C2×#15692226C2124yesmaximal, direct product factor
4C22×C4⋊Dic101C211yesmaximal, characteristic, complemented
5C22×C22⋊Dic101C212yesmaximal, complemented
6C23×D4×C101C211yesmaximal, characteristic, complemented
7C24×D101C2211yescharacteristic
8C22×C22⋊D101—216no—
9C22×C22⋊D101C22124yescomplemented
10C23×Dic101C2211yescharacteristic
11C23×Dic101C2212yes—
12C22×C4×D101—28no—
13C22×C4×D101C22112yescomplemented
14#15692226C22164yesdirect product factor
15C2×C4⋊Dic101C22112yescomplemented
16C2×C22⋊Dic101C22124yescomplemented
17C25×C101C2212yes—
18C22×D4×C101—28no—
19C22×D4×C101C22112yescomplemented
20C23×C4×C101C2211yescharacteristic
21C23×D101—24no—
22C23×D101—24no—
23C23×D101D418yescomplemented
24C23×D101C2316yes—
25C2×C22⋊D101—296no—
26C2×C22⋊D101C23132yescomplemented
27C22×Dic101—24no—
28C22×Dic101—28no—
29C22×Dic101C2316yes—
30C22×Dic101C23112yes—
31C2×C4×D101—248no—
32C2×C4×D101C23116yescomplemented
33C4⋊Dic101C23116yescomplemented
34C22⋊Dic101C23132yescomplemented
35C24×C101—28no—
36C24×C101—28no—
37C24×C101—28no—
38C24×C101C23112yes—
39C24×C101C2311yescharacteristic
40C2×D4×C101—248no—
41C2×D4×C101C23116yescomplemented
42C22×C4×C101D418yescomplemented
43C22×C4×C101C2316yes—
44C23×C101—224no—
45C23×C101—224no—
46C23×C101—224no—
47C23×C101—432no—
48C23×C101C2×D414yes—
49C23×C101C2×D414yes—
50C23×C101#4114yes—
51C23×C101C2418yes—
52C23×C101C2413yes—
53C22×D101—232no—
54C22×D101—212no—
55C22×D101—212no—
56C22×D101C2×D4124yescomplemented
57C22×D101C2414yes—
58D4×C101—232no—
59C2×C4×C101C2×D4124yescomplemented
60C2×C4×C101C2414yes—
61C22⋊D101—264no—
62C2×Dic101—212no—
63C2×Dic101—224no—
64C2×Dic101C2414yes—
65C2×Dic101C2418yes—
66C4×D101—232no—
67C22×C101—28no—
68C22×C101—28no—
69C22×C101—28no—
70C22×C101—448no—
71C22×C101#121116yescomplemented
72C22×C101C22×D416yes—
73C22×C101C22×D416yes—
74C22×C101C2×#4116yes—
75C22×C101C2511yescharacteristic
76C2×D101—248no—
77C2×D101—24no—
78C2×D101—24no—
79C2×D101C22×D418yescomplemented
80C4×C101C22×D418yescomplemented
81Dic101—24no—
82Dic101—28no—
83C2×C101—48no—
84C2×C101C2×#121112yescomplemented
85C2×C101C23×D411yescharacteristic
86C2×C101C23×D411yescharacteristic
87C2×C101C22×#4111yescharacteristic
88D101—28no—
89C22×#121—1011nomaximal, Sylow
90C101C22×#12111yescharacteristic, complemented, Sylow
91C23×D4—1012no—
92C23×D4D10111yescharacteristic, complemented
93C24×C4—1011no—
94C2×#121—10124no—
95C22×C4⋊C4—1011no—
96C22×C22⋊C4—1012no—
97C25—1011no—
98C25C2×D10112yes—
99C22×D4—28no—
100C22×D4—10124no—
101C22×D4—20216no—
102C22×D4C2×D101112yescomplemented
103C23×C4—1011no—
104C23×C4—1012no—
105C23×C4—10112no—
106C23×C4—2028no—
107C23×C4C2×D10111yescharacteristic
108#121—10164no—
109C2×C4⋊C4—10112no—
110C2×C22⋊C4—10124no—
111C24—28no—
112C24—28no—
113C24—28no—
114C24—1018no—
115C24—1016no—
116C24—2024no—
117C24—2024no—
118C24C22×D101112yes—
119C24C22×D10111yescharacteristic
120C2×D4—248no—
121C2×D4—10132no—
122C2×D4—20296no—
123C2×D4C22×D101116yescomplemented
124C22×C4—1016no—
125C22×C4—10112no—
126C22×C4—10116no—
127C22×C4—2028no—
128C22×C4—2024no—
129C22×C4—20248no—
130C22×C4C22⋊D10118yescomplemented
131C22×C4C22×D10116yes—
132C4⋊C4—10116no—
133C22⋊C4—10132no—
134C23—224no—
135C23—224no—
136C23—224no—
137C23—432no—
138C23—10124no—
139C23—1014no—
140C23—20212no—
141C23—20212no—
142C23—20232no—
143C23D4×D10114yes—
144C23C2×C22⋊D10114yes—
145C23#1310807314yes—
146C23C23×D10118yes—
147C23C23×D10113yes—
148D4—232no—
149D4—20264no—
150C2×C4—1014no—
151C2×C4—1018no—
152C2×C4—20224no—
153C2×C4—20212no—
154C2×C4—20232no—
155C2×C4C2×C22⋊D101124yescomplemented
156C2×C4C23×D10114yes—
157C22—28no—
158C22—28no—
159C22—28no—
160C22—448no—
161C22—1018no—
162C22—2024no—
163C22—2024no—
164C22—20248no—
165C22#15692226116yesdirect product factor
166C22C2×D4×D10116yes—
167C22C22×C22⋊D10116yes—
168C22C2×#1310807316yes—
169C22C24×D10111yescharacteristic
170C4—2028no—
171C4—2024no—
172C4C22×C22⋊D10118yescomplemented
173C2—48no—
174C2—2028no—
175C2C2×#15692226112yesdirect product factor
176C2C22×D4×D10111yescharacteristic
177C2C23×C22⋊D10111yescharacteristic
178C2C22×#1310807311yescharacteristic
subgroupG/H
centerC24C22×D101
commutator, cosocleC22×C101C25
FrattiniC22C24×D101
FittingC23×D4×C101C2
socleC24×C101C23
hypercenterC23×D4D101
hypocenterC101C22×#121
LCSG ▷ C22×C101 ▷ C101
UCSC1 ◁ C24 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C23×C22⋊D101C2semidirect
2. C23×C4×D101C2semidirect
3. C2×#15692226C2direct (N × H)
4. C22×C4⋊Dic101C2semidirect
5. C22×C22⋊Dic101C2semidirect
6. C23×D4×C101C2semidirect
9. C22×C22⋊D101C22semidirect
13. C22×C4×D101C22semidirect
14. #15692226C22direct (N × H)
15. C2×C4⋊Dic101C22semidirect
16. C2×C22⋊Dic101C22semidirect
19. C22×D4×C101C22semidirect
23. C23×D101D4semidirect
26. C2×C22⋊D101C23semidirect
32. C2×C4×D101C23semidirect
33. C4⋊Dic101C23semidirect
34. C22⋊Dic101C23semidirect
41. C2×D4×C101C23semidirect
42. C22×C4×C101D4semidirect
56. C22×D101C2×D4semidirect
59. C2×C4×C101C2×D4semidirect
71. C22×C101#121semidirect
79. C2×D101C22×D4semidirect
80. C4×C101C22×D4semidirect
84. C2×C101C2×#121semidirect
90. C101C22×#121semidirect
92. C23×D4D101semidirect
102. C22×D4C2×D101semidirect
123. C2×D4C22×D101semidirect
130. C22×C4C22⋊D101semidirect
155. C2×C4C2×C22⋊D101semidirect
165. C22#15692226direct (N × H)
172. C4C22×C22⋊D101semidirect
175. C2C2×#15692226direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C22⋊D1012
C23×C4×D1012
C2×#156922262
C22×C4⋊Dic1012
C22×C22⋊Dic1012
C23×D4×C1012
C22×#121101