Finite Group Databasebeta

← #27144312

Subgroups — #27144312

Order 12416.

1160 conjugacy classes of subgroups; only the 229 normal subgroups are shown.

normal subgroups only

This group has 1160 conjugacy classes of subgroups — too many to draw. Only its 229 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
2C2×C22⋊C4×D97C212yesmaximal, complemented
3#20260611C218yesmaximal, complemented
4#20260596C212yesmaximal, complemented
5#20260591C211yesmaximal, characteristic, complemented
6#379×C97C211yesmaximal, characteristic, complemented
7C24×D97C2211yescharacteristic
8C22×C22⋊D97—28no—
9C22×C22⋊D97C418yescomplemented
10C22×C22⋊D97C2214yes—
11C23×Dic97C2211yescharacteristic
12C22×C4×D97C2214yescomplemented
13C2×C22⋊Dic97C2212yes—
14C2×C22⋊Dic97C2214yescomplemented
15C2×#13095488C2214yescomplemented
16C22⋊C4×D97—28no—
17#15690717C2218yescomplemented
18C25×C97C2211yescharacteristic
19C2×C22⋊C4×C97C2212yes—
20C2×C22⋊C4×C97C2214yescomplemented
21C23×D97—44no—
22C23×D97C2×C414yes—
23C23×D97D418yes—
24C23×D97C2312yes—
25C2×C22⋊D97—232no—
26C2×C22⋊D97—28no—
27C2×C22⋊D97—28no—
28C2×C22⋊D97C2×C4116yescomplemented
29C22×Dic97—24no—
30C22×Dic97C2×C414yescomplemented
31C22×Dic97C2314yes—
32C22×Dic97C2312yes—
33C2×C4×D97—28no—
34C2×C4×D97—48no—
35C22⋊Dic97—24no—
36C22⋊Dic97—44no—
37#13095488—28no—
38C24×C97—24no—
39C24×C97—44no—
40C24×C97C2×C414yescomplemented
41C24×C97C2312yes—
42C24×C97C2311yescharacteristic
43C22×C4×C97C2314yes—
44C22⋊C4×C97—24no—
45C22⋊C4×C97—44no—
46C23×C97—24no—
47C23×C97—28no—
48C23×C97—24no—
49C23×C97—24no—
50C23×C97—24no—
51C23×C97—48no—
52C23×C97—416no—
53C23×C97C22×C414yes—
54C23×C97C22×C412yes—
55C23×C97C2×D414yes—
56C23×C97C2411yescharacteristic
57C22×D97—28no—
58C22×D97—48no—
59C22×D97—416no—
60C22×D97C22⋊C4116yescomplemented
61C22×D97C22×C414yes—
62C22×D97C2×D414yes—
63C22×D97C2×D414yes—
64C2×C4×C97—24no—
65C2×C4×C97—28no—
66C22⋊D97—232no—
67C2×Dic97—28no—
68C2×Dic97—24no—
69C2×Dic97—24no—
70C2×Dic97—28no—
71C2×Dic97C22×C414yes—
72C4×D97—416no—
73C22×C97—24no—
74C22×C97—28no—
75C22×C97—24no—
76C22×C97—24no—
77C22×C97—24no—
78C22×C97—416no—
79C22×C97—48no—
80C22×C97C2×C22⋊C414yes—
81C22×C97C23×C411yescharacteristic
82C22×C97C22×D412yes—
83C22×C97H(5,𝔽2)14yes—
84C2×D97—28no—
85C2×D97—28no—
86C2×D97—28no—
87C2×D97—416no—
88C2×D97C2×C22⋊C418yes—
89C4×C97—44no—
90Dic97—24no—
91Dic97—44no—
92C2×C97—24no—
93C2×C97—44no—
94C2×C97C22×C22⋊C411yescharacteristic
95C2×C97#51812yes—
96C2×C97#53414yes—
97D97—28no—
98#2308—971nomaximal, Sylow
99C97#230811yescharacteristic, complemented, Sylow
100C23×D4—971no—
101C22×C22⋊C4—972no—
102#386—978no—
103#381—972no—
104#379—971no—
105#379D9711yescharacteristic, complemented
106C25—971no—
107C25C2×D9711yescharacteristic
108C22×D4—978no—
109C22×D4—974no—
110C22×D4—1948no—
111C23×C4—971no—
112C23×C4—974no—
113C2×C22⋊C4—974no—
114C2×C22⋊C4—972no—
115C2×C22⋊C4—974no—
116C2×C22⋊C4—1948no—
117C2×C22⋊C4C2×D9712yes—
118C2×C22⋊C4C2×D9714yescomplemented
119#95—978no—
120C24—24no—
121C24—44no—
122C24—974no—
123C24—978no—
124C24—972no—
125C24—3884no—
126C24C4×D9714yescomplemented
127C24C22×D9712yes—
128C24C22×D9711yescharacteristic
129C2×D4—9716no—
130C2×D4—19432no—
131C2×D4—1948no—
132C2×D4—1948no—
133C22×C4—974no—
134C22×C4—972no—
135C22×C4—974no—
136C22×C4—3888no—
137C22×C4—1948no—
138C22×C4—1944no—
139C22×C4C22×D9714yes—
140C22⋊C4—24no—
141C22⋊C4—44no—
142C22⋊C4—3884no—
143C22⋊C4—1948no—
144C22⋊C4—1944no—
145C23—24no—
146C23—28no—
147C23—24no—
148C23—24no—
149C23—24no—
150C23—48no—
151C23—416no—
152C23—9716no—
153C23—974no—
154C23—974no—
155C23—974no—
156C23—3888no—
157C23—38816no—
158C23—1948no—
159C23C2×C4×D9714yes—
160C23C2×C4×D9712yes—
161C23D4×D9714yes—
162C23C23×D9711yescharacteristic
163D4—19432no—
164C2×C4—24no—
165C2×C4—28no—
166C2×C4—974no—
167C2×C4—38816no—
168C2×C4—1944no—
169C2×C4—1948no—
170C2×C4—1948no—
171C2×C4—1944no—
172C22—24no—
173C22—28no—
174C22—24no—
175C22—24no—
176C22—24no—
177C22—416no—
178C22—48no—
179C22—978no—
180C22—38816no—
181C22—1948no—
182C22—1948no—
183C22—1948no—
184C22C22⋊C4×D9714yes—
185C22C22×C4×D9711yescharacteristic
186C22C2×D4×D9712yes—
187C22#1569085514yes—
188C4—44no—
189C4—3884no—
190C4—1944no—
191C2—24no—
192C2—44no—
193C2—1948no—
194C2C2×C22⋊C4×D9711yescharacteristic
195C2#2026120812yes—
196C2#2026129514yes—
subgroupG/H
center, FrattiniC23C23×D97
commutatorC22×C97C23×C4
Fitting#379×C97C2
socle, cosocleC23×C97C24
prime radicalC23×C22⋊D97C2
hypercenter#379D97
hypocenterC97#2308
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ #379

Complements

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

NH ≅ G/Ntype
2. C2×C22⋊C4×D97C2semidirect
3. #20260611C2semidirect
4. #20260596C2semidirect
5. #20260591C2semidirect
6. #379×C97C2semidirect
9. C22×C22⋊D97C4semidirect
12. C22×C4×D97C22semidirect
14. C2×C22⋊Dic97C22semidirect
15. C2×#13095488C22semidirect
17. #15690717C22semidirect
20. C2×C22⋊C4×C97C22semidirect
28. C2×C22⋊D97C2×C4semidirect
30. C22×Dic97C2×C4semidirect
40. C24×C97C2×C4semidirect
60. C22×D97C22⋊C4semidirect
99. C97#2308semidirect
105. #379D97semidirect
118. C2×C22⋊C4C2×D97semidirect
126. C24C4×D97semidirect

Maximal subgroups

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

subgroupindex
C23×C22⋊D972
C2×C22⋊C4×D972
#202606112
#202605962
#202605912
#379×C972
#230897