Finite Group Databasebeta

← C22×#15690782 #27154956

Subgroups — C22×#15690782 #27154956

Order 12416.

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⋊D97C211yesmaximal, characteristic, complemented
2C23×C22⋊D97C211yesmaximal, characteristic, complemented
3C24×Dic97C211yesmaximal, characteristic, complemented
4C2×#15690782C2124yesmaximal, direct product factor
5C22×#13095490C211yesmaximal, characteristic, complemented
6C22×C22⋊Dic97C211yesmaximal, characteristic, complemented
7C22×#13095488C211yesmaximal, characteristic, complemented
8C23×D4×C97C211yesmaximal, characteristic, complemented
9C24×D97C2211yescharacteristic
10C22×C22⋊D97—28no—
11C22×C22⋊D97—28no—
12C22×C22⋊D97C22112yescomplemented
13C22×C22⋊D97C22112yescomplemented
14C23×Dic97—28no—
15C23×Dic97C2211yescharacteristic
16C23×Dic97C2211yescharacteristic
17C23×Dic97C2211yescharacteristic
18C23×Dic97C22112yescomplemented
19#15690782C22164yesdirect product factor
20C2×#13095490C22112yescomplemented
21C2×C22⋊Dic97C22112yescomplemented
22C2×#13095488C22112yescomplemented
23C25×C97C2211yescharacteristic
24C25×C97C2211yescharacteristic
25C22×D4×C97—28no—
26C22×D4×C97C22112yescomplemented
27C23×C4×C97C2211yescharacteristic
28C23×D97—24no—
29C23×D97—24no—
30C23×D97—24no—
31C23×D97C2316yes—
32C2×C22⋊D97—248no—
33C2×C22⋊D97—248no—
34C2×C22⋊D97C23116yescomplemented
35C2×C22⋊D97C23116yescomplemented
36C22×Dic97—24no—
37C22×Dic97—24no—
38C22×Dic97—248no—
39C22×Dic97D418yescomplemented
40C22×Dic97C2316yes—
41C22×Dic97C2316yes—
42C22×Dic97C2316yes—
43C22×Dic97C23116yescomplemented
44#13095490C23116yescomplemented
45C22⋊Dic97C23116yescomplemented
46#13095488C23116yescomplemented
47C24×C97—24no—
48C24×C97—24no—
49C24×C97—24no—
50C24×C97—24no—
51C24×C97—24no—
52C24×C97D418yescomplemented
53C24×C97C2316yes—
54C24×C97C2316yes—
55C24×C97C2311yescharacteristic
56C2×D4×C97—248no—
57C2×D4×C97C23116yescomplemented
58C22×C4×C97—24no—
59C22×C4×C97C2316yes—
60C23×C97—232no—
61C23×C97—212no—
62C23×C97—212no—
63C23×C97—212no—
64C23×C97—212no—
65C23×C97—212no—
66C23×C97—416no—
67C23×C97C2×D414yes—
68C23×C97C2×D414yes—
69C23×C97C2×D4124yescomplemented
70C23×C97#4114yes—
71C23×C97C2414yes—
72C23×C97C2414yes—
73C23×C97C2413yes—
74C22×D97—212no—
75C22×D97—212no—
76C22×D97—212no—
77C22×D97—416no—
78C22×D97C2414yes—
79D4×C97—232no—
80C2×C4×C97—212no—
81C2×C4×C97C2414yes—
82C22⋊D97—232no—
83C22⋊D97—232no—
84C2×Dic97—212no—
85C2×Dic97—212no—
86C2×Dic97—232no—
87C2×Dic97C2×D4124yescomplemented
88C2×Dic97C2414yes—
89C2×Dic97C2414yes—
90C2×Dic97C2414yes—
91C22×C97—248no—
92C22×C97—24no—
93C22×C97—24no—
94C22×C97—24no—
95C22×C97—24no—
96C22×C97—24no—
97C22×C97—424no—
98C22×C97#121116yescomplemented
99C22×C97C22×D416yes—
100C22×C97C22×D416yes—
101C22×C97C22×D418yescomplemented
102C22×C97C2×#4116yes—
103C22×C97C2511yescharacteristic
104C2×D97—24no—
105C2×D97—24no—
106C2×D97—24no—
107C2×D97—424no—
108C4×C97—24no—
109Dic97—24no—
110Dic97—24no—
111Dic97C22×D418yescomplemented
112C2×C97—28no—
113C2×C97—44no—
114C2×C97C2×#121112yescomplemented
115C2×C97C23×D411yescharacteristic
116C2×C97C23×D411yescharacteristic
117C2×C97C22×#4111yescharacteristic
118D97—44no—
119C22×#121—971nomaximal, Sylow
120C97C22×#12111yescharacteristic, complemented, Sylow
121C23×D4—971no—
122C23×D4—971no—
123C23×D4D9711yescharacteristic, complemented
124C24×C4—971no—
125C2×#121—9724no—
126C22×C4⋊C4—971no—
127C22×C22⋊C4—971no—
128C22×C22⋊C4—971no—
129C25—971no—
130C25C2×D9711yescharacteristic
131C25C2×D9711yescharacteristic
132C22×D4—28no—
133C22×D4—9712no—
134C22×D4—9712no—
135C22×D4—1948no—
136C22×D4—1948no—
137C22×D4C2×D97112yescomplemented
138C23×C4—971no—
139C23×C4—9712no—
140C23×C4—971no—
141C23×C4—971no—
142C23×C4—1948no—
143C23×C4C2×D9711yescharacteristic
144#121—9764no—
145C2×C4⋊C4—9712no—
146C2×C22⋊C4—9712no—
147C2×C22⋊C4—9712no—
148C24—24no—
149C24—24no—
150C24—24no—
151C24—24no—
152C24—24no—
153C24—976no—
154C24—1944no—
155C24—1944no—
156C24—1944no—
157C24C22⋊D9718yescomplemented
158C24C22×D9716yes—
159C24C22×D9716yes—
160C24C22×D9711yescharacteristic
161C2×D4—248no—
162C2×D4—9716no—
163C2×D4—9716no—
164C2×D4—19448no—
165C2×D4—19448no—
166C2×D4C22×D97116yescomplemented
167C22×C4—24no—
168C22×C4—978no—
169C22×C4—976no—
170C22×C4—976no—
171C22×C4—976no—
172C22×C4—9716no—
173C22×C4—19448no—
174C22×C4—1944no—
175C22×C4—1944no—
176C22×C4C22×D9716yes—
177C4⋊C4—9716no—
178C22⋊C4—9716no—
179C22⋊C4—9716no—
180C23—232no—
181C23—212no—
182C23—212no—
183C23—212no—
184C23—212no—
185C23—212no—
186C23—416no—
187C23—974no—
188C23—38816no—
189C23—19412no—
190C23—19412no—
191C23—19412no—
192C23D4×D9714yes—
193C23C2×C22⋊D9714yes—
194C23C2×C22⋊D97124yescomplemented
195C23#1309551014yes—
196C23C23×D9714yes—
197C23C23×D9714yes—
198C23C23×D9713yes—
199D4—232no—
200D4—19432no—
201D4—19432no—
202C2×C4—212no—
203C2×C4—9724no—
204C2×C4—974no—
205C2×C4—974no—
206C2×C4—974no—
207C2×C4—19432no—
208C2×C4—19412no—
209C2×C4—19412no—
210C2×C4C23×D9714yes—
211C22—248no—
212C22—24no—
213C22—24no—
214C22—24no—
215C22—24no—
216C22—24no—
217C22—424no—
218C22—38824no—
219C22—1944no—
220C22—1944no—
221C22—1944no—
222C22#15690782116yesdirect product factor
223C22C2×D4×D9716yes—
224C22C22×C22⋊D9716yes—
225C22C22×C22⋊D9718yescomplemented
226C22C2×#1309551016yes—
227C22C24×D9711yescharacteristic
228C4—24no—
229C4—978no—
230C4—1944no—
231C4—1944no—
232C2—28no—
233C2—44no—
234C2—3884no—
235C2C2×#15690782112yesdirect product factor
236C2C22×D4×D9711yescharacteristic
237C2C23×C22⋊D9711yescharacteristic
238C2C22×#1309551011yescharacteristic
subgroupG/H
centerC24C22×D97
commutator, cosocleC22×C97C25
FrattiniC22C24×D97
FittingC23×D4×C97C2
socleC24×C97C23
hypercenterC23×D4D97
hypocenterC97C22×#121
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C24 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C23×C22⋊D97C2semidirect
2. C23×C22⋊D97C2semidirect
3. C24×Dic97C2semidirect
4. C2×#15690782C2direct (N × H)
5. C22×#13095490C2semidirect
6. C22×C22⋊Dic97C2semidirect
7. C22×#13095488C2semidirect
8. C23×D4×C97C2semidirect
12. C22×C22⋊D97C22semidirect
13. C22×C22⋊D97C22semidirect
18. C23×Dic97C22semidirect
19. #15690782C22direct (N × H)
20. C2×#13095490C22semidirect
21. C2×C22⋊Dic97C22semidirect
22. C2×#13095488C22semidirect
26. C22×D4×C97C22semidirect
34. C2×C22⋊D97C23semidirect
35. C2×C22⋊D97C23semidirect
39. C22×Dic97D4semidirect
43. C22×Dic97C23semidirect
44. #13095490C23semidirect
45. C22⋊Dic97C23semidirect
46. #13095488C23semidirect
52. C24×C97D4semidirect
57. C2×D4×C97C23semidirect
69. C23×C97C2×D4semidirect
87. C2×Dic97C2×D4semidirect
98. C22×C97#121semidirect
101. C22×C97C22×D4semidirect
111. Dic97C22×D4semidirect
114. C2×C97C2×#121semidirect
120. C97C22×#121semidirect
123. C23×D4D97semidirect
137. C22×D4C2×D97semidirect
157. C24C22⋊D97semidirect
166. C2×D4C22×D97semidirect
194. C23C2×C22⋊D97semidirect
222. C22#15690782direct (N × H)
225. C22C22×C22⋊D97semidirect
235. C2C2×#15690782direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C22⋊D972
C24×Dic972
C2×#156907822
C22×#130954902
C22×C22⋊Dic972
C22×#130954882
C23×D4×C972
C22×#12197