Finite Group Databasebeta

← #27150001

Subgroups — #27150001

Order 12416.

548 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 125 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#20260671C213yesmaximal, complemented
2#20260659C212yesmaximal, complemented
3#20260657C216yesmaximal, complemented
4#20260590C213yesmaximal
5#399×C97C211yesmaximal, characteristic, complemented
6C24×D97C2211yescharacteristic
7C2×C4⋊Dic97C2213yescomplemented
8C2×#13095490C2216yescomplemented
9C2×C22⋊Dic97C2213yes—
10C2×#13095488C2216yes—
11C2×#13095488C2216yes—
12#15690717C2213yescomplemented
13C2×C4⋊C4×C97C2213yescomplemented
14C2×C22⋊C4×C97C2213yes—
15#95×C97C2211yescharacteristic, complemented
16C23×D97—43no—
17C23×D97—43no—
18C23×D97Q814yes—
19C23×D97C2312yes—
20C22×Dic97C2316yes—
21C4⋊Dic97—43no—
22#13095490—46no—
23C22⋊Dic97—43no—
24#13095488—46no—
25#13095488—46no—
26C24×C97C2311yescharacteristic
27C22×C4×C97C2313yes—
28C22×C4×C97C2313yes—
29C4⋊C4×C97—43no—
30C22⋊C4×C97—43no—
31C23×C97—23no—
32C23×C97—23no—
33C23×C97C2×Q812yes—
34C23×C97C2411yescharacteristic
35C22×D97—26no—
36C22×D97—26no—
37C22×D97—412no—
38C22×D97—44no—
39C22×D97—412no—
40C22×D97C2×Q814yes—
41C2×C4×C97—23no—
42C2×C4×C97—23no—
43C2×C4×C97—26no—
44C2×C4×C97—26no—
45C2×Dic97—26no—
46C2×Dic97—26no—
47C2×Dic97—26no—
48C22×C97—26no—
49C22×C97—41no—
50C22×C97—43no—
51C22×C97C22×Q811yescharacteristic
52C22×C97H(5,𝔽2)13yes—
53C22×C97H(5,𝔽2)13yes—
54C2×D97—212no—
55C2×D97—416no—
56C2×D97—42no—
57C2×D97—46no—
58C4×C97—43no—
59C4×C97—43no—
60Dic97—46no—
61C2×C97—42no—
62C2×C97#54313yes—
63C2×C97#56011yescharacteristic
64C2×C97#56013yes—
65D97—44no—
66#2842—971nomaximal, Sylow
67C97#284211yescharacteristic, complemented, Sylow
68#399—973no—
69#399D9711yescharacteristic, complemented
70#396—976no—
71#396—972no—
72#379—973no—
73C25—971no—
74C2×C4⋊C4—976no—
75C2×C4⋊C4—973no—
76C2×C4⋊C4C2×D9713yescomplemented
77C2×C22⋊C4—973no—
78C2×C22⋊C4—976no—
79C2×C22⋊C4—976no—
80C2×C22⋊C4C2×D9713yes—
81#95—973no—
82#95C2×D9711yescharacteristic, complemented
83C24—974no—
84C24—972no—
85C24—3883no—
86C24—3883no—
87C24C22×D9711yescharacteristic
88C22×C4—976no—
89C22×C4C22×D9713yes—
90C22×C4C22×D9713yes—
91C4⋊C4—43no—
92C4⋊C4—3886no—
93C4⋊C4—3883no—
94C22⋊C4—43no—
95C22⋊C4—3886no—
96C22⋊C4—3886no—
97C22⋊C4—3883no—
98C23—23no—
99C23—23no—
100C23—974no—
101C23—38812no—
102C23—3884no—
103C23—38812no—
104C23—1946no—
105C23—1946no—
106C23Q8×D9712yes—
107C23C23×D9711yescharacteristic
108C2×C4—23no—
109C2×C4—23no—
110C2×C4—26no—
111C2×C4—26no—
112C2×C4—1946no—
113C2×C4—1946no—
114C2×C4—1946no—
115C22—26no—
116C22—41no—
117C22—43no—
118C22—38816no—
119C22—3882no—
120C22—3886no—
121C22—19412no—
122C22C2×Q8×D9711yescharacteristic
123C22#1569085513yes—
124C22#1569085613yes—
125C4—43no—
126C4—43no—
127C4—3886no—
128C2—42no—
129C2—3884no—
130C2#2026136013yes—
131C2#2026147211yescharacteristic
132C2#2026147513yes—
subgroupG/H
center, FrattiniC23C23×D97
commutator, socle, cosocleC23×C97C24
Fitting#399×C97C2
prime radicalC24×D97C22
hypercenter#399D97
hypocenterC97#2842
LCSG ▷ C23×C97 ▷ C97
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. #20260671C2semidirect
2. #20260659C2semidirect
3. #20260657C2semidirect
5. #399×C97C2semidirect
7. C2×C4⋊Dic97C22semidirect
8. C2×#13095490C22semidirect
12. #15690717C22semidirect
13. C2×C4⋊C4×C97C22semidirect
15. #95×C97C22semidirect
67. C97#2842semidirect
69. #399D97semidirect
76. C2×C4⋊C4C2×D97semidirect
82. #95C2×D97semidirect

Maximal subgroups

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

subgroupindex
#202606712
#202606592
#202606572
#202605902
#399×C972
#284297