Finite Group Databasebeta

← #27150003

Subgroups — #27150003

Order 12416.

348 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 133 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#20260677C213yesmaximal
2#20260669C213yesmaximal
3#20260667C213yesmaximal
4#20260656C213yesmaximal
5#20260605C211yesmaximal, characteristic
6#20260603C211yesmaximal, characteristic
7#383×C97C211yesmaximal, characteristic
8C22×Dic194C2211yescharacteristic
9C2×C4⋊Dic97C2213yes—
10C2×C4⋊Dic97C2213yes—
11C2×C4⋊Dic97C2213yes—
12C2×#13095490C2213yes—
13C2×#13095490C2213yes—
14#15690717C2213yes—
15#15690717C2213yes—
16#15690717C2213yes—
17#15690717C2213yes—
18C2×C42×C97C2211yescharacteristic
19#95×C97C2213yes—
20#95×C97C2213yes—
21C2×Dic194—43no—
22C22×Dic97C2313yes—
23C22×Dic97C2313yes—
24C22×Dic97C2311yescharacteristic
25C22×Dic97C2311yescharacteristic
26C4⋊Dic97—43no—
27C4⋊Dic97—43no—
28C4⋊Dic97—43no—
29#13095490—43no—
30#13095490—43no—
31C22×C4×C97C2311yescharacteristic
32C22×C4×C97C2313yes—
33C22×C4×C97C2313yes—
34C42×C97Q814yescomplemented
35C23×C97C2411yescharacteristic
36C2×C4×C97—23no—
37C2×C4×C97—23no—
38C2×C4×C97—23no—
39C2×C4×C97—23no—
40C2×C4×C97—26no—
41C2×C4×C97C2×Q816yes—
42C2×Dic97—23no—
43C2×Dic97—23no—
44C2×Dic97—23no—
45C2×Dic97—23no—
46C2×Dic97—23no—
47C2×Dic97—23no—
48C2×Dic97—26no—
49Dic194—44no—
50C22×C97C22×Q811yescharacteristic
51C22×C97H(5,𝔽2)13yes—
52C22×C972−1+413yes—
53C4×C97—26no—
54C4×C97—41no—
55C4×C97—43no—
56Dic97—41no—
57Dic97—41no—
58Dic97—43no—
59Dic97—43no—
60C2×C97#54413yes—
61C2×C97#56011yescharacteristic
62C2×C97#56313yes—
63#2843—971nomaximal, Sylow
64C97#284311yescharacteristic, complemented, Sylow
65#400—973no—
66#398—973no—
67#398—973no—
68#395—973no—
69#384—971no—
70#383—971no—
71#383D9711yescharacteristic
72C22×Q8—971no—
73C2×C4⋊C4—973no—
74C2×C4⋊C4—973no—
75C2×C4⋊C4—973no—
76C2×C4⋊C4—973no—
77C2×C4⋊C4—973no—
78C2×C42C2×D9711yescharacteristic
79#95—973no—
80#95—973no—
81#95—973no—
82#95—973no—
83#95C2×D9713yes—
84#95C2×D9713yes—
85C2×Q8—3883no—
86C22×C4—973no—
87C22×C4—971no—
88C22×C4—971no—
89C22×C4—973no—
90C22×C4C22×D9713yes—
91C22×C4C22×D9711yescharacteristic
92C22×C4C22×D9713yes—
93C4⋊C4—3883no—
94C4⋊C4—3883no—
95C4⋊C4—3883no—
96C4⋊C4—3883no—
97C4⋊C4—3883no—
98C42Dic19414yescomplemented
99C23C23×D9711yescharacteristic
100Q8—3884no—
101C2×C4—23no—
102C2×C4—23no—
103C2×C4—23no—
104C2×C4—23no—
105C2×C4—26no—
106C2×C4—1943no—
107C2×C4—1943no—
108C2×C4—1943no—
109C2×C4—1943no—
110C2×C4—1943no—
111C2×C4—1943no—
112C2×C4—1946no—
113C2×C4C2×Dic19416yes—
114C22C22×Dic19411yescharacteristic
115C22#1569085613yes—
116C22#1569085713yes—
117C4—26no—
118C4—41no—
119C4—43no—
120C4—3881no—
121C4—3881no—
122C4—3883no—
123C4—3883no—
124C2#2026136313yes—
125C2#2026147611yescharacteristic
126C2#2026148713yes—
subgroupG/H
center, FrattiniC23C23×D97
commutator, socle, prime radical, cosocleC23×C97C24
Fitting#383×C97C2
hypercenter#383D97
hypocenterC97#2843
LCSG ▷ C23×C97 ▷ C97
UCSC1 ◁ C23 ◁ #383

Complements

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

NH ≅ G/Ntype
34. C42×C97Q8semidirect
64. C97#2843semidirect
98. C42Dic194semidirect

Maximal subgroups

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

subgroupindex
#202606772
#202606692
#202606672
#202606562
#202606052
#202606032
#383×C972
#284397