Finite Group Databasebeta

← #27143428

Subgroups — #27143428

Order 12416.

262 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 73 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#20260719C212yesmaximal, complemented
2#20260532C214yesmaximal, complemented
3#535×C97C211yesmaximal, characteristic
4C2×#13095495—22no—
5#15690729C2214yescomplemented
6#15690727—22no—
7C2×#41×C97C2211yescharacteristic
8C22×D4×C97C411yescharacteristic
9#127×C97C412yes—
10#124×C97C2212yes—
11#121×C97—22no—
12#120×C97—22no—
13#117×C97C411yescharacteristic
14#13095495—24no—
15#13095495—42no—
16C2×#11715895—42no—
17C24×C97—21no—
18#41×C97—42no—
19C2×Q8×C97C2311yescharacteristic
20C2×D4×C97—24no—
21C2×D4×C97—21no—
22C2×D4×C97—42no—
23C2×D4×C97C2×C412yes—
24C2×D4×C97C2×C411yescharacteristic
25C2×D4×C97D412yes—
26C22×C4×C97C2×C411yescharacteristic
27C22×C4×C97D412yes—
28C4⋊C4×C97—21no—
29C22⋊C4×C97—21no—
30C22⋊C4×C97—24no—
31C42×C97C2×C412yes—
32C23×C97—22no—
33C23×C97—42no—
34C23×C97—82no—
35C23×C97C22⋊C411yescharacteristic
36C23×C97C22⋊C412yes—
37Q8×C97—41no—
38D4×C97—41no—
39D4×C97—42no—
40D4×C97—42no—
41D4×C97—42no—
42D4×C97—84no—
43C2×C4×C97—22no—
44C2×C4×C97—22no—
45C2×C4×C97—22no—
46C2×C4×C97—22no—
47C2×C4×C97C22⋊C411yescharacteristic
48C2×C4×C97C22×C411yescharacteristic
49C2×C4×C97C2×D412yes—
50#11715895—44no—
51C22×C97—21no—
52C22×C97—22no—
53C22×C97—42no—
54C22×C97—41no—
55C22×C97—44no—
56C22×C97—42no—
57C22×C97—84no—
58C22×C97C2×C22⋊C411yescharacteristic
59C4×C97—22no—
60C4×C97—21no—
61C4×C97—42no—
62C4×C97#9912yes—
63C2×C97—21no—
64C2×C97—42no—
65C2×C97—41no—
66C2×C97—82no—
67C2×C97C2×#9911yescharacteristic
68#2134—971nomaximal, Sylow
69C97#213411yescharacteristic, complemented, Sylow
70#535D9711yescharacteristic
71#413—972no—
72#355—974no—
73C2×#41C2×D9711yescharacteristic
74C22×D4Dic9711yescharacteristic
75C2×M16—1942no—
76#127Dic9712yes—
77#124C2×D9712yes—
78#121—22no—
79#120—22no—
80#117Dic9711yescharacteristic
81#101—974no—
82#100—1942no—
83C24—21no—
84#41—42no—
85C2×Q8C22×D9711yescharacteristic
86C2×D4—24no—
87C2×D4—21no—
88C2×D4—42no—
89C2×D4C2×Dic9712yes—
90C2×D4C2×Dic9711yescharacteristic
91C2×D4C22⋊D9712yes—
92C22×C4C2×Dic9711yescharacteristic
93C22×C4C22⋊D9712yes—
94M16—3882no—
95M16—1944no—
96C2×C8—3882no—
97C4⋊C4—21no—
98C22⋊C4—21no—
99C22⋊C4—24no—
100C42C2×Dic9712yes—
101C23—22no—
102C23—42no—
103C23—82no—
104C23C22⋊Dic9711yescharacteristic
105C23C22⋊Dic9712yes—
106Q8—41no—
107D4—41no—
108D4—42no—
109D4—42no—
110D4—42no—
111D4—84no—
112C2×C4—22no—
113C2×C4—22no—
114C2×C4—22no—
115C2×C4—22no—
116C2×C4C22⋊Dic9711yescharacteristic
117C2×C4C22×Dic9711yescharacteristic
118C2×C4C2×C22⋊D9712yes—
119C8—3884no—
120C22—21no—
121C22—22no—
122C22—42no—
123C22—41no—
124C22—44no—
125C22—42no—
126C22—84no—
127C22C2×C22⋊Dic9711yescharacteristic
128C4—22no—
129C4—21no—
130C4—42no—
131C4#1569072512yes—
132C2—21no—
133C2—41no—
134C2—42no—
135C2—82no—
136C2C2×#1569072511yescharacteristic
subgroupG/H
centerC2C2×#15690725
commutatorC2×C4×C97C22×C4
FrattiniC2×Q8C22×D97
Fitting, prime radical#535×C97C2
socleC2×C97C2×#99
cosocleC2×Q8×C97C23
hypercenter#535D97
hypocenterC97#2134
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #535

Complements

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

NH ≅ G/Ntype
1. #20260719C2semidirect
2. #20260532C2semidirect
5. #15690729C22semidirect
69. C97#2134semidirect

Maximal subgroups

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

subgroupindex
#202607192
#202605322
#535×C972
#213497