Finite Group Databasebeta

← #16835776

Subgroups — #16835776

Order 3712.

238 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 54 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#14272142C212yesmaximal
2#14272070C214yesmaximal, complemented
3#14271739C211yesmaximal, characteristic
4C2×#90805C2211yescharacteristic
5C22×Dic58C411yescharacteristic, complemented
6#11745345—22no—
7C29×#117C411yescharacteristic, complemented
8#11745259—22no—
9#11745248—22no—
10#11745237C2212yes—
11#11745231C412yescomplemented
12#11745227—22no—
13#11745215C2214yes—
14C22×C4×C29C2×C411yescharacteristic
15C2×C22⋊D29D412yes—
16C22×Dic29—21no—
17#90805—42no—
18C2×D116C2311yescharacteristic
19C2×C4×D29D412yes—
20C2×Dic58—21no—
21C2×Dic58—42no—
22C2×Dic58C2×C412yes—
23C2×Dic58C2×C411yescharacteristic
24C22⋊#1181—24no—
25C4⋊#1181—42no—
26C4×#1181—42no—
27C4⋊C4×C29—21no—
28C22⋊C4×C29—21no—
29C42×C29C2×C412yes—
30#90781—24no—
31C4⋊Dic29—24no—
32C23×C29C22⋊C411yescharacteristic
33C22×D29C2×D412yes—
34C2×#1181—44no—
35C2×C4×C29—22no—
36C2×C4×C29—22no—
37C2×C4×C29C22⋊C411yescharacteristic
38C2×C4×C29C22×C411yescharacteristic
39C22⋊D29—42no—
40C2×Dic29—22no—
41C2×Dic29—42no—
42C2×Dic29C22⋊C412yes—
43D116—41no—
44C4×D29—22no—
45C4×D29—22no—
46Dic58—41no—
47Dic58—42no—
48Dic58—42no—
49#2129—291nomaximal, Sylow
50C22×C29—21no—
51C22×C29—41no—
52C22×C29C2×C22⋊C411yescharacteristic
53C2×D29—22no—
54C2×D29—42no—
55#1181—84no—
56C4×C29—21no—
57C4×C29—42no—
58C4×C29#9912yes—
59Dic29—22no—
60Dic29—42no—
61#536—291no—
62#410—292no—
63#354—294no—
64C2×C29—21no—
65C2×C29—41no—
66C2×C29C2×#9911yescharacteristic
67D29—42no—
68C2×#41—291no—
69C22×Q8—291no—
70C4⋊Q8—292no—
71#124—292no—
72#123—582no—
73C22⋊Q8—582no—
74#117—582no—
75#117#118111yescharacteristic, complemented
76#99—294no—
77#99—582no—
78C29#212911yescharacteristic, complemented, Sylow
79#41—1162no—
80C2×Q8—291no—
81C2×Q8—292no—
82C2×Q8—581no—
83C2×Q8—1162no—
84C2×D4—292no—
85C2×D4—291no—
86C22×C4—292no—
87C22×C4—581no—
88C22×C4C2×#118111yescharacteristic
89C4⋊C4—21no—
90C4⋊C4—584no—
91C4⋊C4—1162no—
92C22⋊C4—21no—
93C22⋊C4—584no—
94C22⋊C4—584no—
95C42—1162no—
96C42C2×#118112yes—
97C23—292no—
98C23C22⋊#118111yescharacteristic
99Q8—1161no—
100Q8—1162no—
101Q8—1162no—
102D4—1161no—
103D4—1162no—
104C2×C4—22no—
105C2×C4—22no—
106C2×C4—292no—
107C2×C4—582no—
108C2×C4—582no—
109C2×C4—582no—
110C2×C4—1162no—
111C2×C4—1164no—
112C2×C4C22⋊#118111yescharacteristic
113C2×C4C22×#118111yescharacteristic
114C22—21no—
115C22—41no—
116C22—582no—
117C22—1162no—
118C22C2×C22⋊#118111yescharacteristic
119C4—21no—
120C4—42no—
121C4—582no—
122C4—1162no—
123C4—2324no—
124C4#1174521512yes—
125C2—21no—
126C2—41no—
127C2—1162no—
128C2C2×#1174521511yescharacteristic
subgroupG/H
centerC2C2×#11745215
commutatorC2×C4×C29C22×C4
FrattiniC2×C4C22×#1181
FittingC29×#117C4
socleC2×C29C2×#99
prime radicalC2×#90805C22
cosocleC2×D116C23
hypercenter#117#1181
hypocenterC29#2129
LCSG ▷ C2×C4×C29 ▷ C22×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
2. #14272070C2semidirect
5. C22×Dic58C4semidirect
7. C29×#117C4semidirect
11. #11745231C4semidirect
75. #117#1181semidirect
78. C29#2129semidirect

Maximal subgroups

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

subgroupindex
#142721422
#142720702
#142717392
#212929