Finite Group Databasebeta

← #31796061

Subgroups — #31796061

Order 17536.

204 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 60 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#22578423C211yesmaximal, characteristic
2#22578419C211yesmaximal, characteristic
3#22578249C212yesmaximal
4#22578248C212yesmaximal
5C2×#17984078C211yesmaximal, characteristic
6#17984155—21no—
7C4⋊#11764047—21no—
8C22⋊#11764047C2212yes—
9C22⋊#11764047C2212yes—
10#17984143—21no—
11C4×#11764047—21no—
12#17984078—21no—
13#17984078—21no—
14#17984078—21no—
15#17984078—21no—
16C2×C4⋊Dic137C2211yescharacteristic
17C2×#14555489C411yescharacteristic
18C2×#14555489C411yescharacteristic
19C2×#14555489C411yescharacteristic
20C2×#14555489C2211yescharacteristic
21C2×C4×Dic137C2211yescharacteristic
22C2×C4⋊C4×C137C411yescharacteristic
23C2×#11764047—22no—
24C2×#11764047—22no—
25C22×Dic137C2×C411yescharacteristic
26C22×Dic137C2×C411yescharacteristic
27C22×Dic137C2×C411yescharacteristic
28C22×Dic137C2311yescharacteristic
29C4⋊Dic137—21no—
30C4⋊Dic137—21no—
31#14555489—21no—
32#14555489—21no—
33#14555489—21no—
34#14555489—21no—
35#14555489—21no—
36#14555489—21no—
37#14555489—21no—
38#14555489—21no—
39C4×Dic137D411yescharacteristic
40C4×Dic137D411yescharacteristic
41C4×Dic137D411yescharacteristic
42C4×Dic137D411yescharacteristic
43C22×C4×C137C2×C411yescharacteristic
44C22×C4×C137C2×C411yescharacteristic
45C22×C4×C137C2×C411yescharacteristic
46C4⋊C4×C137—21no—
47C4⋊C4×C137—21no—
48#11764047—42no—
49#11764047—42no—
50C2×Dic137—22no—
51C2×Dic137—21no—
52C2×Dic137—21no—
53C2×Dic137—22no—
54C2×Dic137—41no—
55C2×Dic137—41no—
56C2×Dic137C22⋊C411yescharacteristic
57C2×Dic137C22⋊C411yescharacteristic
58C2×Dic137C2×D411yescharacteristic
59C2×Dic137C2×D411yescharacteristic
60C23×C137C22×C411yescharacteristic
61C2×C4×C137—22no—
62C2×C4×C137—21no—
63C2×C4×C137—21no—
64C2×C4×C137—22no—
65C2×C4×C137—41no—
66C2×C4×C137C22⋊C411yescharacteristic
67C2×C4×C137C22⋊C411yescharacteristic
68C22×C137—21no—
69C22×C137—21no—
70C22×C137C2×C22⋊C411yescharacteristic
71C22×C137#13112yes—
72C4×C137—21no—
73C4×C137—21no—
74C4×C137—41no—
75C4×C137—41no—
76Dic137—21no—
77Dic137—21no—
78Dic137—21no—
79Dic137—41no—
80Dic137—41no—
81Dic137#10112yes—
82C2×C137—22no—
83C2×C137#40811yescharacteristic
84C2×C137#41011yescharacteristic
85C2×C137C2×#10111yescharacteristic
86C137#147011yescharacteristic, complemented, Sylow
87#1470—1371nomaximal, Sylow
88C2×#125—1371no—
89#432—1371no—
90#431—1371no—
91#324—1372no—
92#324—1372no—
93#125—2741no—
94#125—2741no—
95#125—2741no—
96#125—2741no—
97C2×C4⋊C4—1371no—
98C2×C4⋊C4—1371no—
99C2×C4⋊C4—1371no—
100C2×C4⋊C4—1371no—
101C2×C4⋊C4—1371no—
102C2×C4⋊C4#1058773911yescharacteristic
103C2×C42—1371no—
104C4⋊C8—2741no—
105C4⋊C8—2741no—
106C22⋊C8—1372no—
107C22⋊C8—1372no—
108#97—2741no—
109C4×C8—2741no—
110C22×C4—1371no—
111C22×C4—1371no—
112C22×C4—1371no—
113C22×C4—1371no—
114C22×C4C2×#1058773911yescharacteristic
115C22×C4C2×#1058773911yescharacteristic
116C22×C4C2×#1058773911yescharacteristic
117C2×C8—2742no—
118C2×C8—2742no—
119C4⋊C4—21no—
120C4⋊C4—21no—
121C4⋊C4—2741no—
122C4⋊C4—2741no—
123C4⋊C4—2741no—
124C4⋊C4—2741no—
125C4⋊C4—2741no—
126C4⋊C4—2741no—
127C4⋊C4—2741no—
128C4⋊C4—2741no—
129C4⋊C4—2741no—
130C4⋊C4—2741no—
131C42—1371no—
132C42—1371no—
133C42—1371no—
134C42—1371no—
135C23C22×#1058773911yescharacteristic
136C2×C4—22no—
137C2×C4—21no—
138C2×C4—21no—
139C2×C4—22no—
140C2×C4—41no—
141C2×C4—2742no—
142C2×C4—2741no—
143C2×C4—2741no—
144C2×C4—2742no—
145C2×C4—5481no—
146C2×C4—5481no—
147C2×C4—1371no—
148C2×C4—1371no—
149C2×C4—1371no—
150C2×C4—1371no—
151C2×C4C22⋊#1058773911yescharacteristic
152C2×C4C22⋊#1058773911yescharacteristic
153C8—5482no—
154C8—5482no—
155C22—21no—
156C22—21no—
157C22C2×C22⋊#1058773911yescharacteristic
158C22#1798417512yes—
159C4—21no—
160C4—21no—
161C4—41no—
162C4—41no—
163C4—2741no—
164C4—2741no—
165C4—2741no—
166C4—5481no—
167C4—5481no—
168C4—1372no—
169C2—22no—
170C2#2257837611yescharacteristic
171C2#2257838211yescharacteristic
172C2D137⋊#10111yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10587739
commutator, prime radicalC23×C137C22×C4
FrattiniC23C22×#10587739
FittingC2×C4⋊C4×C137C4
socleC22×C137C2×C22⋊C4
cosocleC22×Dic137C23
hypercenterC2×C4⋊C4#10587739
hypocenterC137#1470
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
86. C137#1470semidirect

Maximal subgroups

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

subgroupindex
#225784232
#225784192
#225782492
#225782482
C2×#179840782
#1470137