Finite Group Databasebeta

← C22×#15692228 #27184353

Subgroups — C22×#15692228 #27184353

Order 12928.

2528 conjugacy classes of subgroups; only the 645 normal subgroups are shown.

normal subgroups only

This group has 2528 conjugacy classes of subgroups — too many to draw. Only its 645 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⋊D101C211yesmaximal, characteristic, complemented
3C23×C22⋊D101C211yesmaximal, characteristic, complemented
4C23×C4×D101C211yesmaximal, characteristic, complemented
5C2×#15692228C2124yesmaximal, direct product factor
6C22×#13108053C211yesmaximal, characteristic, complemented
7C22×#13108051C211yesmaximal, characteristic, complemented
8C22×C22⋊C4×C101C211yesmaximal, characteristic, complemented
9C24×D101C2211yescharacteristic
10C24×D101C2211yescharacteristic
11C22×D404—28no—
12C22×D404C22112yescomplemented
13C22×C22⋊D101—28no—
14C22×C22⋊D101—28no—
15C22×C22⋊D101C22112yescomplemented
16C22×C22⋊D101C22112yescomplemented
17C23×Dic101C2211yescharacteristic
18C23×Dic101C2211yescharacteristic
19C22×C4×D101—28no—
20C22×C4×D101C22112yescomplemented
21#15692228C22164yesdirect product factor
22C2×#13108053C22112yescomplemented
23C2×#13108051C22112yescomplemented
24C25×C101C2211yescharacteristic
25C23×C4×C101C2211yescharacteristic
26C23×C4×C101C2211yescharacteristic
27C2×C22⋊C4×C101C22112yescomplemented
28C23×D101—24no—
29C23×D101—24no—
30C23×D101—24no—
31C23×D101—24no—
32C23×D101—24no—
33C23×D101D418yescomplemented
34C23×D101C2316yes—
35C23×D101C2316yes—
36C2×D404—248no—
37C2×D404C23116yescomplemented
38C2×C22⋊D101—248no—
39C2×C22⋊D101—248no—
40C2×C22⋊D101C23116yescomplemented
41C2×C22⋊D101C23116yescomplemented
42C22×Dic101—24no—
43C22×Dic101D418yescomplemented
44C22×Dic101C2316yes—
45C22×Dic101C2316yes—
46C2×C4×D101—248no—
47C2×C4×D101C23116yescomplemented
48#13108053C23116yescomplemented
49#13108051C23116yescomplemented
50C24×C101—24no—
51C24×C101—24no—
52C24×C101—24no—
53C24×C101C2316yes—
54C24×C101C2311yescharacteristic
55C22×C4×C101—24no—
56C22×C4×C101—24no—
57C22×C4×C101C2316yes—
58C22×C4×C101C2316yes—
59C22⋊C4×C101C23116yescomplemented
60C23×C101—212no—
61C23×C101—212no—
62C23×C101—212no—
63C23×C101—416no—
64C23×C101C2×D414yes—
65C23×C101C2×D414yes—
66C23×C101#4114yes—
67C23×C101C2414yes—
68C23×C101C2413yes—
69C22×D101—232no—
70C22×D101—212no—
71C22×D101—212no—
72C22×D101—212no—
73C22×D101—212no—
74C22×D101—212no—
75C22×D101—416no—
76C22×D101C2×D4124yescomplemented
77C22×D101C2414yes—
78C22×D101C2414yes—
79C2×C4×C101—212no—
80C2×C4×C101—212no—
81C2×C4×C101C2414yes—
82C2×C4×C101C2414yes—
83C22⋊D101—232no—
84C22⋊D101—232no—
85C2×Dic101—212no—
86C2×Dic101C2×D4124yescomplemented
87C2×Dic101C2414yes—
88C2×Dic101C2414yes—
89D404—232no—
90C4×D101—232no—
91C22×C101—24no—
92C22×C101—24no—
93C22×C101—24no—
94C22×C101—424no—
95C22×C101#121116yescomplemented
96C22×C101C22×D416yes—
97C22×C101C22×D416yes—
98C22×C101C2×#4116yes—
99C22×C101C2511yescharacteristic
100C2×D101—248no—
101C2×D101—24no—
102C2×D101—24no—
103C2×D101—24no—
104C2×D101—24no—
105C2×D101—24no—
106C2×D101—424no—
107C2×D101C22×D418yescomplemented
108C4×C101—24no—
109C4×C101—24no—
110Dic101—24no—
111Dic101C22×D418yescomplemented
112C2×C101—44no—
113C2×C101C2×#121112yescomplemented
114C2×C101C23×D411yescharacteristic
115C2×C101C23×D411yescharacteristic
116C2×C101C22×#4111yescharacteristic
117D101—28no—
118D101—44no—
119C22×#121—1011nomaximal, Sylow
120C101C22×#12111yescharacteristic, complemented, Sylow
121C23×D4—1011no—
122C23×D4—1011no—
123C23×D4—1011no—
124C24×C4—1011no—
125C2×#121—10124no—
126C22×C4⋊C4—1011no—
127C22×C22⋊C4—1011no—
128C22×C22⋊C4D10111yescharacteristic, complemented
129C25—1011no—
130C25—1011no—
131C25C2×D10111yescharacteristic
132C22×D4—10112no—
133C22×D4—10112no—
134C22×D4—10112no—
135C22×D4—2028no—
136C22×D4—2028no—
137C22×D4—2028no—
138C23×C4—1011no—
139C23×C4—1011no—
140C23×C4—10112no—
141C23×C4—2028no—
142C23×C4C2×D10111yescharacteristic
143C23×C4C2×D10111yescharacteristic
144#121—10164no—
145C2×C4⋊C4—10112no—
146C2×C22⋊C4—10112no—
147C2×C22⋊C4C2×D101112yescomplemented
148C24—24no—
149C24—24no—
150C24—24no—
151C24—1018no—
152C24—1016no—
153C24—1016no—
154C24—2024no—
155C24—2024no—
156C24—2024no—
157C24—2024no—
158C24—2024no—
159C24C22×D10116yes—
160C24C22×D10111yescharacteristic
161C2×D4—10116no—
162C2×D4—10116no—
163C2×D4—10116no—
164C2×D4—20248no—
165C2×D4—20248no—
166C2×D4—20248no—
167C22×C4—24no—
168C22×C4—24no—
169C22×C4—1018no—
170C22×C4—1016no—
171C22×C4—1016no—
172C22×C4—10116no—
173C22×C4—20248no—
174C22×C4—2024no—
175C22×C4C22×D10116yes—
176C22×C4C22×D10116yes—
177C4⋊C4—10116no—
178C22⋊C4—10116no—
179C22⋊C4C22×D101116yescomplemented
180C23—212no—
181C23—212no—
182C23—212no—
183C23—416no—
184C23—10124no—
185C23—1014no—
186C23—1014no—
187C23—40416no—
188C23—20212no—
189C23—20212no—
190C23—20212no—
191C23—20212no—
192C23—20212no—
193C23—20232no—
194C23D4×D10114yes—
195C23D4×D10114yes—
196C23#1310807414yes—
197C23C23×D10114yes—
198C23C23×D10113yes—
199D4—20232no—
200D4—20232no—
201D4—20232no—
202C2×C4—212no—
203C2×C4—212no—
204C2×C4—10124no—
205C2×C4—1014no—
206C2×C4—1014no—
207C2×C4—20232no—
208C2×C4—20212no—
209C2×C4C23×D10114yes—
210C2×C4C23×D10114yes—
211C22—24no—
212C22—24no—
213C22—24no—
214C22—424no—
215C22—1018no—
216C22—40424no—
217C22—2024no—
218C22—2024no—
219C22—2024no—
220C22—2024no—
221C22—2024no—
222C22—20248no—
223C22#15692228116yesdirect product factor
224C22C2×D4×D10116yes—
225C22C2×D4×D10116yes—
226C22C2×#1310807416yes—
227C22C24×D10111yescharacteristic
228C4—24no—
229C4—24no—
230C4—1018no—
231C4—2024no—
232C2—44no—
233C2—4044no—
234C2—2028no—
235C2C2×#15692228112yesdirect product factor
236C2C22×D4×D10111yescharacteristic
237C2C22×D4×D10111yescharacteristic
238C2C22×#1310807411yescharacteristic
subgroupG/H
centerC24C22×D101
commutator, cosocleC22×C101C25
FrattiniC22C24×D101
FittingC22×C22⋊C4×C101C2
socleC24×C101C23
hypercenterC22×C22⋊C4D101
hypocenterC101C22×#121
LCSG ▷ C22×C101 ▷ C101
UCSC1 ◁ C24 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C23×D404C2semidirect
2. C23×C22⋊D101C2semidirect
3. C23×C22⋊D101C2semidirect
4. C23×C4×D101C2semidirect
5. C2×#15692228C2direct (N × H)
6. C22×#13108053C2semidirect
7. C22×#13108051C2semidirect
8. C22×C22⋊C4×C101C2semidirect
12. C22×D404C22semidirect
15. C22×C22⋊D101C22semidirect
16. C22×C22⋊D101C22semidirect
20. C22×C4×D101C22semidirect
21. #15692228C22direct (N × H)
22. C2×#13108053C22semidirect
23. C2×#13108051C22semidirect
27. C2×C22⋊C4×C101C22semidirect
33. C23×D101D4semidirect
37. C2×D404C23semidirect
40. C2×C22⋊D101C23semidirect
41. C2×C22⋊D101C23semidirect
43. C22×Dic101D4semidirect
47. C2×C4×D101C23semidirect
48. #13108053C23semidirect
49. #13108051C23semidirect
59. C22⋊C4×C101C23semidirect
76. C22×D101C2×D4semidirect
86. C2×Dic101C2×D4semidirect
95. C22×C101#121semidirect
107. C2×D101C22×D4semidirect
111. Dic101C22×D4semidirect
113. C2×C101C2×#121semidirect
120. C101C22×#121semidirect
128. C22×C22⋊C4D101semidirect
147. C2×C22⋊C4C2×D101semidirect
179. C22⋊C4C22×D101semidirect
223. C22#15692228direct (N × H)
235. C2C2×#15692228direct (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
C23×C4×D1012
C2×#156922282
C22×#131080532
C22×#131080512
C22×C22⋊C4×C1012
C22×#121101