Finite Group Databasebeta

← #27142024

Subgroups — #27142024

Order 12416.

260 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 93 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#20261502C211yesmaximal, characteristic
2#20260762C212yesmaximal, complemented
3#20260702C211yesmaximal, characteristic
4#20260505C212yesmaximal, complemented
5#410×C97C211yesmaximal, characteristic
6#15690817—24no—
7C2×#13095495—22no—
8C2×#13095495C2211yescharacteristic
9C2×#13095495C2212yes—
10C22×#11715895C2211yescharacteristic
11C4⋊#11715895—24no—
12C22⋊#11715895—22no—
13C2×#41×C97C2211yescharacteristic
14#117×C97C2212yes—
15#99×C97C414yes—
16#13095495—24no—
17#13095495—22no—
18#13095495—22no—
19#13095495—22no—
20C2×#11715895—24no—
21C2×#11715895—21no—
22C2×#11715895—22no—
23C2×#11715895—21no—
24C2×#11715895D414yes—
25#41×C97—22no—
26#41×C97—22no—
27C2×Q8×C97Q811yescharacteristic
28C2×D4×C97C2×C412yes—
29C2×D4×C97Q811yescharacteristic
30C22×C4×C97D412yes—
31C22×C4×C97C2311yescharacteristic
32C4⋊C4×C97—22no—
33C22⋊C4×C97C2×C414yes—
34C42×C97—22no—
35C23×C97C4⋊C412yes—
36C23×C97C22×C411yescharacteristic
37Q8×C97—22no—
38D4×C97—24no—
39D4×C97—22no—
40C2×C4×C97—22no—
41C2×C4×C97—22no—
42C2×C4×C97—24no—
43C2×C4×C97C4⋊C412yes—
44C2×C4×C97C2×D412yes—
45C2×C4×C97C2×D411yescharacteristic
46C2×C4×C97C2×Q811yescharacteristic
47C2×C4×C97#4112yes—
48#11715895—24no—
49#11715895—41no—
50#11715895—41no—
51C22×C97—22no—
52C22×C97—41no—
53C22×C97—42no—
54C22×C97C2×C4⋊C411yescharacteristic
55C22×C97C4×D412yes—
56C4×C97—22no—
57C4×C97—44no—
58C4×C97#12011yescharacteristic
59C4×C97C22⋊Q812yes—
60C4×C97#12311yescharacteristic
61C2×C97—22no—
62C2×C97—21no—
63C2×C97—42no—
64C2×C97#38511yescharacteristic
65#1868—971nomaximal, Sylow
66C97#186811yescharacteristic, complemented, Sylow
67#568—971no—
68#424—972no—
69#410D9711yescharacteristic
70#408—971no—
71#344—972no—
72C2×#41C2×D9711yescharacteristic
73#131—1944no—
74C2×M16—972no—
75C2×M16—971no—
76C2×M16—1942no—
77C22×C8—971no—
78#117C2×D9712yes—
79C4⋊C8—1944no—
80#99Dic9714yes—
81C22⋊C8—1942no—
82#41—22no—
83#41—22no—
84C2×Q8Dic19411yescharacteristic
85C2×D4Dic19411yescharacteristic
86C2×D4C2×Dic9712yes—
87C22×C4D38812yes—
88C22×C4C22×D9711yescharacteristic
89M16—1942no—
90M16—1944no—
91M16—1942no—
92M16—1942no—
93C2×C8—974no—
94C2×C8—1942no—
95C2×C8—1941no—
96C2×C8—1941no—
97C2×C8—1944no—
98C4⋊C4—22no—
99C22⋊C4C2×Dic9714yes—
100C42—22no—
101C23C4⋊Dic9712yes—
102C23C22×Dic9711yescharacteristic
103Q8—22no—
104D4—24no—
105D4—22no—
106C2×C4—22no—
107C2×C4—22no—
108C2×C4—24no—
109C2×C4C4⋊Dic9712yes—
110C2×C4D4×D9712yes—
111C2×C4C2×D38811yescharacteristic
112C2×C4C2×Dic19411yescharacteristic
113C2×C4#1309551012yes—
114C8—3881no—
115C8—3881no—
116C8—1944no—
117C22—22no—
118C22—41no—
119C22—42no—
120C22C2×C4⋊Dic9711yescharacteristic
121C22D4×Dic9712yes—
122C4—22no—
123C4—44no—
124C4#1569078011yescharacteristic
125C4#1569078712yes—
126C4#1569079411yescharacteristic
127C2—22no—
128C2—21no—
129C2—42no—
130C2#2026061011yescharacteristic
subgroupG/H
centerC4#15690780
commutatorC23×C97C22×C4
FrattiniC22×C4C22×D97
Fitting#410×C97C2
socleC2×C97#385
prime radicalC2×#41×C97C22
cosocleC22×C4×C97C23
hypercenter#410D97
hypocenterC97#1868
LCSG ▷ C23×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C4 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #20260762C2semidirect
4. #20260505C2semidirect
66. C97#1868semidirect

Maximal subgroups

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

subgroupindex
#202615022
#202607622
#202607022
#202605052
#410×C972
#186897