Finite Group Databasebeta

← #31796128

Subgroups — #31796128

Order 17536.

316 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 72 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#22578424C211yesmaximal, characteristic, complemented
2C4×#14555519C211yesmaximal, characteristic, complemented
3D4⋊#11764047C214yesmaximal, complemented
4C2×D4×Dic137C211yesmaximal, characteristic
5C2×#14555519—21no—
6C4⋊#11764047C2212yescomplemented
7C22⋊#11764047—21no—
8#17984143—21no—
9#17984143—21no—
10C4×#11764047C2212yescomplemented
11C23×Dic137—21no—
12D4×Dic137—21no—
13D4×Dic137—21no—
14D4×Dic137C412yes—
15D4×Dic137C2212yes—
16C2×C4⋊Dic137C411yescharacteristic
17C2×C22⋊Dic137—21no—
18C2×C4×Dic137C2211yescharacteristic
19C22×D4×C137C411yescharacteristic
20#14555519—24no—
21C2×#11764047—22no—
22C2×#11764047—22no—
23C22×Dic137—22no—
24C22×Dic137—22no—
25C22×Dic137—21no—
26C22×Dic137—44no—
27C22×Dic137D411yescharacteristic
28C22×Dic137D411yescharacteristic
29C4⋊Dic137—21no—
30C4⋊Dic137C2×C412yes—
31C22⋊Dic137—22no—
32C22⋊Dic137—22no—
33C4×Dic137C2×C411yescharacteristic
34C4×Dic137D411yescharacteristic
35C4×Dic137D411yescharacteristic
36C4×Dic137C2311yescharacteristic
37C24×C137—21no—
38C2×D4×C137—24no—
39C2×D4×C137—21no—
40C2×D4×C137C2×C412yes—
41C22×C4×C137C2×C411yescharacteristic
42#11764047—24no—
43#11764047—42no—
44C2×Dic137—22no—
45C2×Dic137—22no—
46C2×Dic137—22no—
47C2×Dic137—44no—
48C2×Dic137—42no—
49C2×Dic137—44no—
50C2×Dic137C22⋊C411yescharacteristic
51C2×Dic137C22⋊C411yescharacteristic
52C2×Dic137C2×D411yescharacteristic
53C2×Dic137C2×D411yescharacteristic
54C23×C137—24no—
55C23×C137—22no—
56C23×C137—42no—
57C23×C137—42no—
58C23×C137C22⋊C411yescharacteristic
59D4×C137—24no—
60D4×C137—22no—
61D4×C137M1614yescomplemented
62C2×C4×C137—22no—
63C2×C4×C137C22⋊C411yescharacteristic
64C2×C4×C137C22×C411yescharacteristic
65C22×C137—21no—
66C22×C137—21no—
67C22×C137—24no—
68C22×C137—48no—
69C22×C137—42no—
70C22×C137—42no—
71C22×C137#10412yes—
72C22×C137C2×C22⋊C411yescharacteristic
73C4×C137—21no—
74C4×C137C2×M1612yes—
75Dic137—21no—
76Dic137—21no—
77Dic137—21no—
78Dic137—42no—
79Dic137AGL(1,ℤ/8)12yes—
80C2×C137—22no—
81C2×C137—44no—
82C2×C137#40711yescharacteristic
83C2×C137#41811yescharacteristic
84C2×C137C2×#10411yescharacteristic
85C137#149011yescharacteristic, complemented, Sylow
86#1490—1371nomaximal, Sylow
87C2×C4×D4—1371no—
88#432—1371no—
89C4×M16—1371no—
90D4⋊C8—1374no—
91C22×D4#1058773911yescharacteristic
92C23×C4—2741no—
93C2×M16—2741no—
94C4×D4—2741no—
95C4×D4—2741no—
96C4×D4—1372no—
97C4×D4—1372no—
98C2×C4⋊C4—1371no—
99C2×C22⋊C4—2741no—
100C2×C42—1371no—
101C4⋊C8—1372no—
102C22⋊C8—2741no—
103#97—2741no—
104#97—2741no—
105C4×C8—1372no—
106C24—21no—
107C2×D4—24no—
108C2×D4—21no—
109C2×D4C2×#1058773912yes—
110C22×C4—2741no—
111C22×C4—2742no—
112C22×C4—2742no—
113C22×C4—5484no—
114C22×C4—1371no—
115C22×C4—1371no—
116C22×C4C2×#1058773911yescharacteristic
117M16—2744no—
118C2×C8—2742no—
119C2×C8—2742no—
120C4⋊C4—2741no—
121C4⋊C4—1372no—
122C22⋊C4—2742no—
123C22⋊C4—2742no—
124C42—1371no—
125C42—1371no—
126C42—1371no—
127C42—1371no—
128C23—24no—
129C23—22no—
130C23—42no—
131C23—42no—
132C23C22⋊#1058773911yescharacteristic
133D4—24no—
134D4—22no—
135D4#1455551914yescomplemented
136C2×C4—22no—
137C2×C4—2742no—
138C2×C4—2742no—
139C2×C4—2742no—
140C2×C4—5484no—
141C2×C4—5482no—
142C2×C4—5484no—
143C2×C4—1371no—
144C2×C4—1371no—
145C2×C4—1371no—
146C2×C4—1371no—
147C2×C4C22⋊#1058773911yescharacteristic
148C2×C4C22×#1058773911yescharacteristic
149C8—2744no—
150C8—5482no—
151C22—21no—
152C22—21no—
153C22—24no—
154C22—48no—
155C22—42no—
156C22—42no—
157C22#1798415212yes—
158C22C2×C22⋊#1058773911yescharacteristic
159C4—21no—
160C4—2741no—
161C4—2741no—
162C4—2741no—
163C4—5482no—
164C4—1372no—
165C4C2×#1455551912yes—
166C2—22no—
167C2—44no—
168C2#2257837311yescharacteristic
169C2#2257839511yescharacteristic
170C2C2×#1798415211yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
Fitting, prime radicalC22×D4×C137C4
socleC22×C137C2×C22⋊C4
cosocleC4×Dic137C23
hypercenterC22×D4#10587739
hypocenterC137#1490
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. #22578424C2semidirect
2. C4×#14555519C2semidirect
3. D4⋊#11764047C2semidirect
6. C4⋊#11764047C22semidirect
10. C4×#11764047C22semidirect
61. D4×C137M16semidirect
85. C137#1490semidirect
135. D4#14555519semidirect

Maximal subgroups

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

subgroupindex
#225784242
C4×#145555192
D4⋊#117640472
C2×D4×Dic1372
#1490137