Finite Group Databasebeta

← #31796689

Subgroups — #31796689

Order 17536.

316 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 70 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#22578375C212yesmaximal
2#22578283C214yesmaximal, complemented
3C2×C4⋊C4×D137C211yesmaximal, characteristic
4C2×#14555519C2214yes—
5#17984145—22no—
6#17984145—22no—
7C22×C4×D137C412yes—
8C22×C4×D137C2211yescharacteristic
9C2×C4⋊Dic137C411yescharacteristic
10C2×#14555489C2212yes—
11C4⋊C4×D137—22no—
12C4⋊C4×D137—22no—
13C2×C4⋊C4×C137C411yescharacteristic
14#14555519—28no—
15C2×#11764047—24no—
16C23×D137C2×C411yescharacteristic
17C22×Dic137C2×C412yes—
18C22×Dic137C2311yescharacteristic
19C2×C4×D137—24no—
20C2×C4×D137—22no—
21C2×C4×D137—22no—
22C2×C4×D137—44no—
23C2×C4×D137D412yes—
24C2×C4×D137D412yes—
25C4⋊Dic137—22no—
26#14555489—24no—
27C22×C4×C137C2×C412yes—
28C22×C4×C137C2×C411yescharacteristic
29C4⋊C4×C137—22no—
30#11764047—44no—
31C22×D137—22no—
32C22×D137—22no—
33C22×D137—22no—
34C22×D137C22⋊C411yescharacteristic
35C22×D137C22⋊C411yescharacteristic
36C2×Dic137—22no—
37C2×Dic137—42no—
38C2×Dic137C2×D412yes—
39C2×Dic137#4114yes—
40C4×D137—24no—
41C4×D137—24no—
42C4×D137—44no—
43C4×D137—44no—
44C23×C137C22×C411yescharacteristic
45C2×C4×C137—22no—
46C2×C4×C137—22no—
47C2×C4×C137—42no—
48C2×C4×C137C22⋊C412yes—
49C22×C137—21no—
50C22×C137—21no—
51C22×C137C2×C22⋊C411yescharacteristic
52C22×C137#11712yes—
53C2×D137—28no—
54C2×D137—21no—
55C2×D137—21no—
56C2×D137—21no—
57C2×D137—21no—
58C2×D137#10012yes—
59C2×D137#10112yes—
60C4×C137—22no—
61C4×C137—42no—
62Dic137—42no—
63Dic137#12314yes—
64C2×C137—22no—
65C2×C137#38111yescharacteristic
66C2×C137C2×#10011yescharacteristic
67C2×C137C2×#10111yescharacteristic
68D137—22no—
69D137—22no—
70C137#182111yescharacteristic, complemented, Sylow
71#1821—1371nomaximal, Sylow
72C22×C4⋊C4—1371no—
73#407—1372no—
74#343—1374no—
75C23×C4—1371no—
76C23×C4—1372no—
77C2×M16—1374no—
78C2×C4⋊C4—2742no—
79C2×C4⋊C4—2742no—
80C2×C4⋊C4—1372no—
81C2×C4⋊C4—1371no—
82C2×C4⋊C4#1058773911yescharacteristic
83C22⋊C8—2742no—
84C22⋊C8—2742no—
85C24—1371no—
86C22×C4—2744no—
87C22×C4—2742no—
88C22×C4—2742no—
89C22×C4—5484no—
90C22×C4—1372no—
91C22×C4—1372no—
92C22×C4—1372no—
93C22×C4—1371no—
94C22×C4C2×#1058773912yes—
95C22×C4C2×#1058773911yescharacteristic
96M16—2748no—
97C2×C8—2744no—
98C4⋊C4—22no—
99C4⋊C4—2744no—
100C4⋊C4—2742no—
101C23—2742no—
102C23—2742no—
103C23—2742no—
104C23—1371no—
105C23—1371no—
106C23C22×#1058773911yescharacteristic
107C2×C4—22no—
108C2×C4—22no—
109C2×C4—42no—
110C2×C4—2744no—
111C2×C4—2744no—
112C2×C4—2742no—
113C2×C4—5484no—
114C2×C4—5484no—
115C2×C4—5482no—
116C2×C4—1372no—
117C2×C4—1374no—
118C2×C4C22⋊#1058773912yes—
119C8—5484no—
120C22—21no—
121C22—21no—
122C22—2748no—
123C22—2741no—
124C22—2741no—
125C22—2741no—
126C22—2741no—
127C22—1372no—
128C22—1372no—
129C22C2×C22⋊#1058773911yescharacteristic
130C22#1798416712yes—
131C4—22no—
132C4—42no—
133C4—5482no—
134C4—1374no—
135C2—22no—
136C2—2742no—
137C2—2742no—
138C2#2257834611yescharacteristic
139C2D137⋊#10011yescharacteristic
140C2D137⋊#10111yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC23×C137C22×C4
FrattiniC23C22×#10587739
FittingC2×C4⋊C4×C137C4
socleC22×C137C2×C22⋊C4
prime radicalC23×D137C2×C4
cosocleC22×Dic137C23
hypercenterC2×C4⋊C4#10587739
hypocenterC137#1821
LCSG ▷ C23×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #22578283C2semidirect
70. C137#1821semidirect

Maximal subgroups

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

subgroupindex
#225783752
#225782832
C2×C4⋊C4×D1372
#1821137