Finite Group Databasebeta

← #27150052

Subgroups — #27150052

Order 12416.

582 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C2×#15690804C211yesmaximal, characteristic, complemented
2C2×C42×D97C211yesmaximal, characteristic, complemented
3#20260627C213yesmaximal, complemented
4#20260620C213yesmaximal, complemented
5#20260613C213yesmaximal
6#20260609C213yesmaximal, complemented
7#383×C97C211yesmaximal, characteristic, complemented
8C22×D388C2211yescharacteristic
9C22×C4×D97C2213yes—
10#15690804—24no—
11C2×C4⋊Dic97C2213yes—
12C2×#13095490C2213yes—
13C2×#13095488C2213yes—
14C2×#13095488C2213yes—
15C2×#13095488C2213yes—
16C2×C4×Dic97C2213yes—
17C42×D97—24no—
18#15690717C2213yes—
19#15690717C2213yes—
20C2×C42×C97C2211yescharacteristic
21#95×C97C2213yes—
22#95×C97C2213yes—
23C23×D97C2311yescharacteristic
24C23×D97C2311yescharacteristic
25C2×D388—26no—
26C22×Dic97C2313yes—
27C22×Dic97C2313yes—
28C2×C4×D97—26no—
29C2×C4×D97—26no—
30C2×C4×D97—26no—
31C4⋊Dic97—26no—
32#13095490—26no—
33#13095488—26no—
34#13095488—26no—
35#13095488—26no—
36C4×Dic97—26no—
37C22×C4×C97C2313yes—
38C22×C4×C97C2311yescharacteristic
39C22×C4×C97C2313yes—
40C42×C97D414yescomplemented
41C23×C97C2411yescharacteristic
42C22×D97—21no—
43C22×D97—23no—
44C22×D97—21no—
45C22×D97—23no—
46C22×D97—23no—
47C22×D97#4116yes—
48C2×C4×C97—23no—
49C2×C4×C97—23no—
50C2×C4×C97—23no—
51C2×C4×C97—23no—
52C2×C4×C97—23no—
53C2×C4×C97—23no—
54C2×C4×C97C2×D416yes—
55C2×Dic97—23no—
56C2×Dic97—23no—
57C2×Dic97—23no—
58C2×Dic97—23no—
59C2×Dic97—23no—
60C2×Dic97#4116yes—
61D388—44no—
62C4×D97—224no—
63C22×C97C22×D411yescharacteristic
64C22×C97C2×#4113yes—
65C22×C97C2×#4113yes—
66C2×D97—26no—
67C2×D97—26no—
68C2×D97—43no—
69C2×D97—41no—
70C2×D97—43no—
71C2×D97#12614yescomplemented
72C4×C97—26no—
73C4×C97—41no—
74C4×C97—43no—
75Dic97—26no—
76Dic97—43no—
77C2×C97#52513yes—
78C2×C97C2×#12611yescharacteristic
79C2×C97#52913yes—
80D97—24no—
81D97—81no—
82#2852—971nomaximal, Sylow
83C97#285211yescharacteristic, complemented, Sylow
84C2×#126—971no—
85C22×C42—971no—
86#388—973no—
87#387—973no—
88#386—973no—
89#385—973no—
90#383D9711yescharacteristic, complemented
91C22×D4—971no—
92C23×C4—973no—
93#126—1944no—
94C2×C4⋊C4—973no—
95C2×C4⋊C4—973no—
96C2×C22⋊C4—973no—
97C2×C22⋊C4—973no—
98C2×C22⋊C4—973no—
99C2×C42—973no—
100C2×C42—1944no—
101C2×C42C2×D9711yescharacteristic
102#95—973no—
103#95—973no—
104#95C2×D9713yes—
105#95C2×D9713yes—
106C24—971no—
107C24—971no—
108C2×D4—1946no—
109C22×C4—973no—
110C22×C4—973no—
111C22×C4—1946no—
112C22×C4—1946no—
113C22×C4—1946no—
114C22×C4C22×D9711yescharacteristic
115C22×C4C22×D9713yes—
116C22×C4C22×D9713yes—
117C4⋊C4—1946no—
118C4⋊C4—1946no—
119C22⋊C4—1946no—
120C22⋊C4—1946no—
121C22⋊C4—1946no—
122C42—1946no—
123C42D38814yescomplemented
124C23—976no—
125C23—1941no—
126C23—1943no—
127C23—1943no—
128C23—1941no—
129C23—1943no—
130C23C23×D9711yescharacteristic
131D4—3884no—
132C2×C4—23no—
133C2×C4—23no—
134C2×C4—23no—
135C2×C4—23no—
136C2×C4—23no—
137C2×C4—23no—
138C2×C4—976no—
139C2×C4—19424no—
140C2×C4—1943no—
141C2×C4—1943no—
142C2×C4—1943no—
143C2×C4—1943no—
144C2×C4—1943no—
145C2×C4C2×D38816yes—
146C22—974no—
147C22—3883no—
148C22—3881no—
149C22—3883no—
150C22—1946no—
151C22—1946no—
152C22C22×D38811yescharacteristic
153C22#41×D9713yes—
154C22#41×D9713yes—
155C4—26no—
156C4—41no—
157C4—43no—
158C4—3883no—
159C4—1946no—
160C2—7761no—
161C2—1944no—
162C2#2026124713yes—
163C2#126×D9711yescharacteristic
164C2#2026126813yes—
subgroupG/H
center, FrattiniC23C23×D97
commutator, socle, cosocleC23×C97C24
Fitting#383×C97C2
prime radicalC22×D388C22
hypercenter#383D97
hypocenterC97#2852
LCSG ▷ C23×C97 ▷ C97
UCSC1 ◁ C23 ◁ #383

Complements

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

NH ≅ G/Ntype
1. C2×#15690804C2semidirect
2. C2×C42×D97C2semidirect
3. #20260627C2semidirect
4. #20260620C2semidirect
6. #20260609C2semidirect
7. #383×C97C2semidirect
40. C42×C97D4semidirect
71. C2×D97#126semidirect
83. C97#2852semidirect
90. #383D97semidirect
123. C42D388semidirect

Maximal subgroups

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

subgroupindex
C2×#156908042
C2×C42×D972
#202606272
#202606202
#202606132
#202606092
#383×C972
#285297