Finite Group Databasebeta

← C22×#15690778 #27154951

Subgroups — C22×#15690778 #27154951

Order 12416.

4544 conjugacy classes of subgroups; only the 725 normal subgroups are shown.

normal subgroups only

This group has 4544 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
1C25×D97C211yesmaximal, characteristic, complemented
2C23×C22⋊D97C212yesmaximal, complemented
3C2×#15690778C2124yesmaximal, direct product factor
4C22×C22⋊Dic97C211yesmaximal, characteristic, complemented
5C22×#13095488C212yesmaximal, complemented
6C23×D4×C97C211yesmaximal, characteristic, complemented
7C24×D97—216no—
8C24×D97—28no—
9C24×D97C2212yes—
10C24×D97C22112yescomplemented
11C22×C22⋊D97—216no—
12C22×C22⋊D97C22124yescomplemented
13C23×Dic97C2212yescomplemented
14#15690778C22164yesdirect product factor
15C2×C22⋊Dic97C22112yescomplemented
16C2×#13095488C22124yescomplemented
17C25×C97C2211yescharacteristic
18C25×C97C2211yescharacteristic, complemented
19C22×D4×C97—28no—
20C22×D4×C97C22112yescomplemented
21C23×C4×C97C2211yescharacteristic, complemented
22C23×D97—2128no—
23C23×D97—28no—
24C23×D97—28no—
25C23×D97—248no—
26C23×D97—296no—
27C23×D97D4116yescomplemented
28C23×D97C23112yes—
29C23×D97C23116yescomplemented
30C2×C22⋊D97—296no—
31C2×C22⋊D97C23132yescomplemented
32C22×Dic97—28no—
33C22×Dic97C23112yescomplemented
34C22⋊Dic97C23116yescomplemented
35#13095488C23132yescomplemented
36C24×C97—28no—
37C24×C97—28no—
38C24×C97—24no—
39C24×C97D418yescomplemented
40C24×C97C2316yes—
41C24×C97C2311yescharacteristic
42C24×C97C2316yescomplemented
43C2×D4×C97—248no—
44C2×D4×C97C23116yescomplemented
45C22×C4×C97—24no—
46C22×C4×C97C2316yescomplemented
47C23×C97—232no—
48C23×C97—224no—
49C23×C97—212no—
50C23×C97—224no—
51C23×C97—416no—
52C23×C97C2×D418yes—
53C23×C97C2×D414yes—
54C23×C97C2×D4124yescomplemented
55C23×C97C2414yes—
56C23×C97C2413yes—
57C23×C97C2414yescomplemented
58C22×D97—2384no—
59C22×D97—264no—
60C22×D97—224no—
61C22×D97—224no—
62C22×D97—232no—
63C22×D97—264no—
64C22×D97C2×D4148yescomplemented
65C22×D97C2418yes—
66D4×C97—232no—
67C2×C4×C97—212no—
68C2×C4×C97C2414yescomplemented
69C22⋊D97—264no—
70C2×Dic97—224no—
71C2×Dic97C2418yescomplemented
72C22×C97—248no—
73C22×C97—28no—
74C22×C97—28no—
75C22×C97—24no—
76C22×C97—424no—
77C22×C97#120116yescomplemented
78C22×C97C22×D4112yes—
79C22×C97C22×D416yes—
80C22×C97C22×D418yescomplemented
81C22×C97C2511yescharacteristic
82C2×D97—2128no—
83C2×D97—296no—
84C2×D97—28no—
85C2×D97—28no—
86C2×D97C22×D4116yescomplemented
87C4×C97—24no—
88Dic97—28no—
89C2×C97—28no—
90C2×C97—44no—
91C2×C97C2×#120112yescomplemented
92C2×C97C23×D412yes—
93C2×C97C23×D411yescharacteristic
94D97—216no—
95C22×#120—971nomaximal, Sylow
96C97C22×#12011yescharacteristic, complemented, Sylow
97C26—971no—
98C23×D4—972no—
99C23×D4D9711yescharacteristic, complemented
100C2×#120—9724no—
101C22×C22⋊C4—971no—
102C22×C22⋊C4—972no—
103C25—972no—
104C25—9712no—
105C25—19416no—
106C25—1948no—
107C25C2×D9711yescharacteristic
108C25C2×D9711yescharacteristic, complemented
109C22×D4—28no—
110C22×D4—9724no—
111C22×D4—19416no—
112C22×D4C2×D97112yescomplemented
113C23×C4—972no—
114C23×C4C2×D9711yescharacteristic, complemented
115#120—9764no—
116C2×C22⋊C4—9712no—
117C2×C22⋊C4—9724no—
118C24—28no—
119C24—28no—
120C24—24no—
121C24—9716no—
122C24—9712no—
123C24—9716no—
124C24—19496no—
125C24—194128no—
126C24—1948no—
127C24—1948no—
128C24—19448no—
129C24C22⋊D9718yescomplemented
130C24C22×D9716yes—
131C24C22×D9711yescharacteristic
132C24C22×D9716yescomplemented
133C2×D4—248no—
134C2×D4—9732no—
135C2×D4—19496no—
136C2×D4C22×D97116yescomplemented
137C22×C4—24no—
138C22×C4—9712no—
139C22×C4—1948no—
140C22×C4C22×D9716yescomplemented
141C22⋊C4—9716no—
142C22⋊C4—9732no—
143C23—232no—
144C23—224no—
145C23—224no—
146C23—212no—
147C23—416no—
148C23—9748no—
149C23—978no—
150C23—19464no—
151C23—194384no—
152C23—19464no—
153C23—19424no—
154C23—19424no—
155C23—19432no—
156C23D4×D9718yes—
157C23C2×C22⋊D9714yes—
158C23C2×C22⋊D97124yescomplemented
159C23C23×D9714yes—
160C23C23×D9713yes—
161C23C23×D9714yescomplemented
162D4—232no—
163D4—19464no—
164C2×C4—212no—
165C2×C4—978no—
166C2×C4—19424no—
167C2×C4C23×D9714yescomplemented
168C22—248no—
169C22—28no—
170C22—28no—
171C22—24no—
172C22—424no—
173C22—9716no—
174C22—194128no—
175C22—19496no—
176C22—1948no—
177C22—1948no—
178C22#15690778116yesdirect product factor
179C22C2×D4×D97112yes—
180C22C22×C22⋊D9716yes—
181C22C22×C22⋊D9718yescomplemented
182C22C24×D9711yescharacteristic
183C4—24no—
184C4—1948no—
185C2—28no—
186C2—44no—
187C2—19416no—
188C2C2×#15690778112yesdirect product factor
189C2C22×D4×D9712yes—
190C2C23×C22⋊D9711yescharacteristic
subgroupG/H
centerC24C22×D97
commutator, cosocleC22×C97C25
FrattiniC22C24×D97
FittingC23×D4×C97C2
socleC24×C97C23
hypercenterC23×D4D97
hypocenterC97C22×#120
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C24 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C25×D97C2semidirect
2. C23×C22⋊D97C2semidirect
3. C2×#15690778C2direct (N × H)
4. C22×C22⋊Dic97C2semidirect
5. C22×#13095488C2semidirect
6. C23×D4×C97C2semidirect
10. C24×D97C22semidirect
12. C22×C22⋊D97C22semidirect
13. C23×Dic97C22semidirect
14. #15690778C22direct (N × H)
15. C2×C22⋊Dic97C22semidirect
16. C2×#13095488C22semidirect
18. C25×C97C22semidirect
20. C22×D4×C97C22semidirect
21. C23×C4×C97C22semidirect
27. C23×D97D4semidirect
29. C23×D97C23semidirect
31. C2×C22⋊D97C23semidirect
33. C22×Dic97C23semidirect
34. C22⋊Dic97C23semidirect
35. #13095488C23semidirect
39. C24×C97D4semidirect
42. C24×C97C23semidirect
44. C2×D4×C97C23semidirect
46. C22×C4×C97C23semidirect
54. C23×C97C2×D4semidirect
57. C23×C97C24semidirect
64. C22×D97C2×D4semidirect
68. C2×C4×C97C24semidirect
71. C2×Dic97C24semidirect
77. C22×C97#120semidirect
80. C22×C97C22×D4semidirect
86. C2×D97C22×D4semidirect
91. C2×C97C2×#120semidirect
96. C97C22×#120semidirect
99. C23×D4D97semidirect
108. C25C2×D97semidirect
112. C22×D4C2×D97semidirect
114. C23×C4C2×D97semidirect
129. C24C22⋊D97semidirect
132. C24C22×D97semidirect
136. C2×D4C22×D97semidirect
140. C22×C4C22×D97semidirect
158. C23C2×C22⋊D97semidirect
161. C23C23×D97semidirect
167. C2×C4C23×D97semidirect
178. C22#15690778direct (N × H)
181. C22C22×C22⋊D97semidirect
188. C2C2×#15690778direct (N × H)

Maximal subgroups

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

subgroupindex
C25×D972
C23×C22⋊D972
C2×#156907782
C22×C22⋊Dic972
C22×#130954882
C23×D4×C972
C22×#12097