Finite Group Databasebeta

← #27144116

Subgroups — #27144116

Order 12416.

1088 conjugacy classes of subgroups; only the 245 normal subgroups are shown.

normal subgroups only

This group has 1088 conjugacy classes of subgroups — too many to draw. Only its 245 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×C4×D97C211yesmaximal, characteristic
2C2×#15690766C212yesmaximal, complemented
3#20260598C211yesmaximal, characteristic, complemented
4#20260594C214yesmaximal, complemented
5#20260590C212yesmaximal
6#20260582C214yesmaximal, complemented
7#381×C97C211yesmaximal, characteristic, complemented
8C24×D97C2211yescharacteristic
9C23×Dic97C2211yescharacteristic
10C22×C4×D97—28no—
11C22×C4×D97C418yescomplemented
12C22×C4×D97C2212yes—
13C22×C4×D97C2212yes—
14#15690766—28no—
15C2×#13095490C2214yescomplemented
16C2×C22⋊Dic97C2212yes—
17C2×#13095488C2218yes—
18C2×C4×Dic97C2214yescomplemented
19#15690717C2214yescomplemented
20C23×C4×C97C2211yescharacteristic
21C2×C22⋊C4×C97C2212yes—
22#95×C97C2214yescomplemented
23C23×D97—24no—
24C23×D97—28no—
25C23×D97C2×C414yes—
26C23×D97C2312yes—
27C22×Dic97C2×C414yes—
28C22×Dic97D418yes—
29C22×Dic97C2314yes—
30C22×Dic97C2311yescharacteristic
31C22×Dic97C2311yescharacteristic
32C2×C4×D97—232no—
33C2×C4×D97—28no—
34C2×C4×D97—28no—
35C2×C4×D97C2×C418yes—
36C2×C4×D97C2×C418yes—
37#13095490—28no—
38C22⋊Dic97—24no—
39#13095488—216no—
40C4×Dic97—28no—
41C24×C97C2311yescharacteristic
42C22×C4×C97—24no—
43C22×C4×C97C2×C414yes—
44C22×C4×C97C2314yes—
45C22×C4×C97C2312yes—
46C22⋊C4×C97—24no—
47C23×C97—24no—
48C23×C97C22×C412yes—
49C23×C97C2×D414yes—
50C23×C97C2411yescharacteristic
51C22×D97—232no—
52C22×D97—28no—
53C22×D97—24no—
54C22×D97—24no—
55C22×D97—216no—
56C22×D97C22×C414yes—
57C22×D97#4118yes—
58C2×C4×C97—28no—
59C2×C4×C97—24no—
60C2×C4×C97—24no—
61C2×C4×C97—28no—
62C2×C4×C97C22×C414yes—
63C2×Dic97—24no—
64C2×Dic97—28no—
65C2×Dic97C22⋊C4116yescomplemented
66C2×Dic97C22×C412yes—
67C2×Dic97C22×C412yes—
68C2×Dic97C2×D414yes—
69C2×Dic97C2×D414yes—
70C4×D97—216no—
71C4×D97—216no—
72C22×C97—24no—
73C22×C97—28no—
74C22×C97C2×C22⋊C414yes—
75C22×C97C23×C411yescharacteristic
76C22×C97C22×D412yes—
77C22×C97C2×#4114yes—
78C2×D97—232no—
79C2×D97—24no—
80C2×D97—24no—
81C2×D97—216no—
82C2×D97#11718yescomplemented
83C4×C97—24no—
84C4×C97—44no—
85Dic97—44no—
86Dic97C2×C22⋊C414yes—
87Dic97C2×C22⋊C414yes—
88C2×C97—24no—
89C2×C97C22×C22⋊C411yescharacteristic
90C2×C97C2×#11712yes—
91C2×C97#52514yes—
92D97—28no—
93#2280—971nomaximal, Sylow
94C97#228011yescharacteristic, complemented, Sylow
95C24×C4—971no—
96C2×#117—972no—
97#381—971no—
98#381D9711yescharacteristic, complemented
99#380—974no—
100#379—972no—
101C4×C22⋊C4—974no—
102C25—971no—
103C23×C4—971no—
104C23×C4—978no—
105C23×C4—972no—
106C23×C4—972no—
107C23×C4—1948no—
108C23×C4C2×D9711yescharacteristic
109#117—1948no—
110C2×C4⋊C4—974no—
111C2×C22⋊C4—972no—
112C2×C22⋊C4—978no—
113C2×C22⋊C4C2×D9712yes—
114C2×C42—974no—
115#95—974no—
116#95C2×D9714yescomplemented
117C24—972no—
118C24—974no—
119C24—1948no—
120C24—1944no—
121C24C22×D9711yescharacteristic
122C22×C4—24no—
123C22×C4—974no—
124C22×C4—978no—
125C22×C4—971no—
126C22×C4—971no—
127C22×C4—978no—
128C22×C4—978no—
129C22×C4—974no—
130C22×C4—19432no—
131C22×C4—1948no—
132C22×C4—1948no—
133C22×C4C4×D9714yes—
134C22×C4C22×D9714yes—
135C22×C4C22×D9712yes—
136C4⋊C4—1948no—
137C22⋊C4—24no—
138C22⋊C4—19416no—
139C22⋊C4—1944no—
140C42—1948no—
141C23—24no—
142C23—974no—
143C23—978no—
144C23—19416no—
145C23—19432no—
146C23—1948no—
147C23—1944no—
148C23—1944no—
149C23C2×C4×D9712yes—
150C23D4×D9714yes—
151C23C23×D9711yescharacteristic
152C2×C4—28no—
153C2×C4—24no—
154C2×C4—24no—
155C2×C4—28no—
156C2×C4—9716no—
157C2×C4—972no—
158C2×C4—972no—
159C2×C4—974no—
160C2×C4—974no—
161C2×C4—19416no—
162C2×C4—19416no—
163C2×C4—1944no—
164C2×C4—1948no—
165C2×C4C2×C4×D9714yes—
166C22—24no—
167C22—28no—
168C22—978no—
169C22—19432no—
170C22—1944no—
171C22—1944no—
172C22—19416no—
173C22C22⋊C4×D9714yes—
174C22C22×C4×D9711yescharacteristic
175C22C2×D4×D9712yes—
176C22#41×D9714yes—
177C4—24no—
178C4—44no—
179C4—974no—
180C4—974no—
181C4—3884no—
182C2—24no—
183C2—1948no—
184C2C2×C22⋊C4×D9711yescharacteristic
185C2#117×D9712yes—
186C2#2026124314yes—
subgroupG/H
center, FrattiniC23C23×D97
commutatorC22×C97C23×C4
Fitting#381×C97C2
socle, cosocleC23×C97C24
prime radicalC24×D97C22
hypercenter#381D97
hypocenterC97#2280
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
2. C2×#15690766C2semidirect
3. #20260598C2semidirect
4. #20260594C2semidirect
6. #20260582C2semidirect
7. #381×C97C2semidirect
11. C22×C4×D97C4semidirect
15. C2×#13095490C22semidirect
18. C2×C4×Dic97C22semidirect
19. #15690717C22semidirect
22. #95×C97C22semidirect
65. C2×Dic97C22⋊C4semidirect
82. C2×D97#117semidirect
94. C97#2280semidirect
98. #381D97semidirect
116. #95C2×D97semidirect

Maximal subgroups

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

subgroupindex
C23×C4×D972
C2×#156907662
#202605982
#202605942
#202605902
#202605822
#381×C972
#228097