Finite Group Databasebeta

← #16835773

Subgroups — #16835773

Order 3712.

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

normal subgroups only

This group has 270 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, complemented
2#14272068C214yesmaximal, complemented
3#14271732C211yesmaximal, characteristic
4C2×#90805C2211yescharacteristic
5C22×D116C411yescharacteristic, complemented
6#11745345—22no—
7C29×#117C411yescharacteristic, complemented
8#11745257—22no—
9#11745244—22no—
10#11745237C412yescomplemented
11#11745236C2212yes—
12#11745227—22no—
13#11745215C2214yescomplemented
14C23×D29—21no—
15C22×C4×C29C2×C411yescharacteristic
16C2×C22⋊D29D412yes—
17#90805—42no—
18C2×D116—21no—
19C2×D116—24no—
20C2×D116—42no—
21C2×D116C2×C412yes—
22C2×D116C2311yescharacteristic
23C2×C4×D29D412yes—
24C2×Dic58C2×C411yescharacteristic
25C22⋊#1181—24no—
26C4⋊#1181—42no—
27C4×#1181—42no—
28C4⋊C4×C29—21no—
29C22⋊C4×C29—21no—
30C42×C29C2×C412yes—
31#90781—24no—
32C23×C29C22⋊C411yescharacteristic
33C22×D29—22no—
34C22×D29—42no—
35C22×D29—82no—
36C22×D29C2×D412yes—
37C2×#1181—44no—
38C2×C4×C29—22no—
39C2×C4×C29—22no—
40C2×C4×C29C22⋊C411yescharacteristic
41C2×C4×C29C22×C411yescharacteristic
42C22⋊D29—42no—
43C2×Dic29C22⋊C412yes—
44D116—41no—
45D116—42no—
46D116—42no—
47D116—84no—
48C4×D29—22no—
49C4×D29—22no—
50Dic58—41no—
51#2128—291nomaximal, Sylow
52C22×C29—21no—
53C22×C29—41no—
54C22×C29C2×C22⋊C411yescharacteristic
55C2×D29—22no—
56C2×D29—42no—
57C2×D29—44no—
58C2×D29—42no—
59C2×D29—84no—
60#1181—84no—
61C4×C29—21no—
62C4×C29—42no—
63C4×C29#9912yes—
64Dic29—22no—
65#535—291no—
66#410—292no—
67#353—294no—
68C2×C29—21no—
69C2×C29—41no—
70C2×C29C2×#9911yescharacteristic
71D29—42no—
72D29—82no—
73C2×#41—291no—
74C22×D4—291no—
75#127—292no—
76#124—292no—
77#121—582no—
78#120—582no—
79#117—582no—
80#117#118111yescharacteristic, complemented
81#99—294no—
82#99—582no—
83C29#212811yescharacteristic, complemented, Sylow
84C24—581no—
85#41—1162no—
86C2×Q8—291no—
87C2×D4—292no—
88C2×D4—291no—
89C2×D4—292no—
90C2×D4—584no—
91C2×D4—581no—
92C2×D4—1162no—
93C22×C4—292no—
94C22×C4C2×#118111yescharacteristic
95C4⋊C4—21no—
96C4⋊C4—1162no—
97C22⋊C4—21no—
98C22⋊C4—584no—
99C22⋊C4—584no—
100C42—1162no—
101C42C2×#118112yes—
102C23—292no—
103C23—582no—
104C23—1162no—
105C23—2322no—
106C23C22⋊#118111yescharacteristic
107Q8—1161no—
108D4—1161no—
109D4—1162no—
110D4—1162no—
111D4—1162no—
112D4—2324no—
113C2×C4—22no—
114C2×C4—22no—
115C2×C4—292no—
116C2×C4—582no—
117C2×C4—582no—
118C2×C4—1164no—
119C2×C4C22⋊#118111yescharacteristic
120C2×C4C22×#118111yescharacteristic
121C22—21no—
122C22—41no—
123C22—582no—
124C22—1162no—
125C22—1164no—
126C22—1162no—
127C22—2324no—
128C22C2×C22⋊#118111yescharacteristic
129C4—21no—
130C4—42no—
131C4—582no—
132C4—2324no—
133C4#1174521512yes—
134C2—21no—
135C2—41no—
136C2—1162no—
137C2—2322no—
138C2C2×#1174521511yescharacteristic
subgroupG/H
centerC2C2×#11745215
commutatorC2×C4×C29C22×C4
FrattiniC2×C4C22×#1181
FittingC29×#117C4
socleC2×C29C2×#99
prime radical#14271732C2
cosocleC2×D116C23
hypercenter#117#1181
hypocenterC29#2128
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
1. #14272142C2semidirect
2. #14272068C2semidirect
5. C22×D116C4semidirect
7. C29×#117C4semidirect
10. #11745237C4semidirect
13. #11745215C22semidirect
80. #117#1181semidirect
83. C29#2128semidirect

Maximal subgroups

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

subgroupindex
#142721422
#142720682
#142717322
#212829