Finite Group Databasebeta

← #27145039

Subgroups — #27145039

Order 12416.

432 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 131 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
1#20260676C213yesmaximal, complemented
2#20260664C213yesmaximal
3#20260659C211yesmaximal, characteristic
4#20260586C213yesmaximal, complemented
5#20260582C213yesmaximal
6#20260580C211yesmaximal, characteristic, complemented
7#401×C97C211yesmaximal, characteristic, complemented
8C22×C4×D97C2211yescharacteristic
9C2×#13095490C2213yes—
10C2×#13095490C2213yes—
11C2×#13095488C2213yes—
12C2×#13095488C2213yes—
13C2×C4×Dic97C2213yes—
14C2×C4×Dic97C2213yes—
15#15690717C2213yes—
16#15690717C2213yes—
17#15690717C2213yes—
18#95×C97C2211yescharacteristic
19#95×C97C2213yes—
20#95×C97C2213yes—
21C23×D97C2311yescharacteristic
22C22×Dic97C2313yes—
23C22×Dic97C2313yes—
24C22×Dic97C2311yescharacteristic
25C2×C4×D97—26no—
26#13095490—26no—
27#13095490—26no—
28#13095488—26no—
29#13095488—26no—
30C4×Dic97—26no—
31C4×Dic97—26no—
32C22×C4×C97C2313yes—
33C22×C4×C97C2313yes—
34C22×C4×C97C2311yescharacteristic
35C23×C97C2411yescharacteristic
36C22×D97—23no—
37C22×D97—23no—
38C22×D97#4112yes—
39C2×C4×C97—23no—
40C2×C4×C97—23no—
41C2×C4×C97—23no—
42C2×C4×C97—23no—
43C2×C4×C97—23no—
44C2×C4×C97—23no—
45C2×C4×C97—23no—
46C2×Dic97—23no—
47C2×Dic97—23no—
48C2×Dic97—23no—
49C2×Dic97—23no—
50C2×Dic97—23no—
51C2×Dic97#4116yes—
52C2×Dic97#4116yes—
53C4×D97—44no—
54C22×C97C2×#4113yes—
55C22×C97C2×#4111yescharacteristic
56C22×C97C2×#4113yes—
57C2×D97—26no—
58C2×D97—41no—
59C2×D97—43no—
60C4×C97—41no—
61C4×C97—43no—
62C4×C97—43no—
63Dic97—26no—
64Dic97—43no—
65Dic97#12614yes—
66C2×C97C2×#12611yescharacteristic
67C2×C97#52913yes—
68C2×C97#52913yes—
69D97—42no—
70#2401—971nomaximal, Sylow
71C97#240111yescharacteristic, complemented, Sylow
72#401D9711yescharacteristic, complemented
73#400—973no—
74#397—973no—
75#396—971no—
76C4×C4⋊C4—973no—
77C4×C22⋊C4—973no—
78#376—971no—
79C23×C4—971no—
80C2×C4⋊C4—973no—
81C2×C4⋊C4—973no—
82C2×C22⋊C4—973no—
83C2×C22⋊C4—973no—
84C2×C42—973no—
85C2×C42—973no—
86#95—973no—
87#95—973no—
88#95—973no—
89#95C2×D9711yescharacteristic
90#95C2×D9713yes—
91#95C2×D9713yes—
92C24—971no—
93C22×C4—971no—
94C22×C4—973no—
95C22×C4—973no—
96C22×C4—1946no—
97C22×C4C22×D9711yescharacteristic
98C22×C4C22×D9713yes—
99C22×C4C22×D9713yes—
100C4⋊C4—1946no—
101C4⋊C4—1946no—
102C22⋊C4—1946no—
103C22⋊C4—1946no—
104C42—1946no—
105C42—1946no—
106C23—972no—
107C23—1943no—
108C23—1943no—
109C23C23×D9711yescharacteristic
110C2×C4—23no—
111C2×C4—23no—
112C2×C4—23no—
113C2×C4—23no—
114C2×C4—23no—
115C2×C4—23no—
116C2×C4—23no—
117C2×C4—976no—
118C2×C4—976no—
119C2×C4—3884no—
120C2×C4—1943no—
121C2×C4—1943no—
122C2×C4—1943no—
123C2×C4—1943no—
124C2×C4—1943no—
125C22—3881no—
126C22—3883no—
127C22—1946no—
128C22#41×D9713yes—
129C22#41×D9713yes—
130C22#41×D9711yescharacteristic
131C4—41no—
132C4—43no—
133C4—43no—
134C4—974no—
135C4—3883no—
136C4—1946no—
137C2—3882no—
138C2#126×D9711yescharacteristic
139C2#2026126813yes—
140C2#2026126813yes—
subgroupG/H
center, FrattiniC23C23×D97
commutator, socle, cosocleC23×C97C24
Fitting#401×C97C2
prime radicalC23×D97C23
hypercenter#401D97
hypocenterC97#2401
LCSG ▷ C23×C97 ▷ C97
UCSC1 ◁ C23 ◁ #401

Complements

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

NH ≅ G/Ntype
1. #20260676C2semidirect
4. #20260586C2semidirect
6. #20260580C2semidirect
7. #401×C97C2semidirect
71. C97#2401semidirect
72. #401D97semidirect

Maximal subgroups

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

subgroupindex
#202606762
#202606642
#202606592
#202605862
#202605822
#202605802
#401×C972
#240197