Finite Group Databasebeta

← #31796252

Subgroups — #31796252

Order 17536.

212 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 58 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#22578426C211yesmaximal, characteristic
2C4×#14555520C211yesmaximal, characteristic
3#22578260C212yesmaximal, complemented
4#22578260C212yesmaximal, complemented
5#125×D137C211yesmaximal, characteristic
6C2×#14555520—21no—
7#17984155—21no—
8#17984145—21no—
9#17984143C2212yes—
10#17984143C2212yes—
11C4×#11764047—21no—
12#17984080C411yescharacteristic
13#17984079C2211yescharacteristic
14#17984079C2211yescharacteristic
15#17984078—21no—
16#17984078—21no—
17C4⋊C4×D137—21no—
18C4⋊C4×D137—21no—
19C4⋊C4×D137C411yescharacteristic
20C4⋊C4×D137C411yescharacteristic
21C42×D137C2211yescharacteristic
22#125×C137C411yescharacteristic
23#14555520—24no—
24C2×#11764047—22no—
25C2×#11764047—22no—
26C2×C4×D137—21no—
27C2×C4×D137—21no—
28C2×C4×D137C2×C411yescharacteristic
29C2×C4×D137D411yescharacteristic
30C2×C4×D137D411yescharacteristic
31C4⋊Dic137—21no—
32C4⋊Dic137—21no—
33C4⋊Dic137C2×C411yescharacteristic
34C4⋊Dic137C2×C411yescharacteristic
35#14555489—21no—
36#14555489—21no—
37#14555489—21no—
38#14555489—21no—
39#14555489—21no—
40#14555489—21no—
41C4×Dic137D411yescharacteristic
42C4×Dic137D411yescharacteristic
43C4×Dic137C2311yescharacteristic
44C4⋊C4×C137—21no—
45C4⋊C4×C137—21no—
46C4⋊C4×C137C2×C411yescharacteristic
47C4⋊C4×C137C2×C411yescharacteristic
48C42×C137C2×C411yescharacteristic
49#11764047—24no—
50#11764047—42no—
51C22×D137C22⋊C411yescharacteristic
52C2×Dic137—21no—
53C2×Dic137—21no—
54C2×Dic137C22⋊C411yescharacteristic
55C2×Dic137C2×D411yescharacteristic
56C2×Dic137C2×D411yescharacteristic
57C4×D137—22no—
58C4×D137—22no—
59C4×D137—41no—
60C4×D137—42no—
61C4×D137—42no—
62C2×C4×C137—21no—
63C2×C4×C137—21no—
64C2×C4×C137C22⋊C411yescharacteristic
65C2×C4×C137C22⋊C411yescharacteristic
66C2×C4×C137C22×C411yescharacteristic
67C22×C137C2×C22⋊C411yescharacteristic
68C2×D137—21no—
69C2×D137—21no—
70C2×D137#10412yes—
71C4×C137—21no—
72C4×C137—21no—
73C4×C137—21no—
74C4×C137—41no—
75C4×C137—41no—
76Dic137—21no—
77Dic137—21no—
78Dic137—41no—
79Dic137—41no—
80Dic137#10112yes—
81C2×C137C2×#10111yescharacteristic
82C2×C137C2×#10411yescharacteristic
83C2×C137#42111yescharacteristic
84D137—22no—
85C137#152811yescharacteristic, complemented, Sylow
86#1528—1371nomaximal, Sylow
87C2×#125—1371no—
88#432—1371no—
89C4×M16—1371no—
90#330—1372no—
91#330—1372no—
92C2×M16—2741no—
93#125—2741no—
94#125—2741no—
95#125—1371no—
96#125—1371no—
97#125—1371no—
98#125#1058773911yescharacteristic
99C2×C4⋊C4—2741no—
100C2×C4⋊C4—2741no—
101C2×C4⋊C4—1371no—
102C2×C4⋊C4—1371no—
103C2×C42—1371no—
104C4⋊C8—2741no—
105C22⋊C8—2741no—
106#97—1372no—
107#97—1372no—
108C4×C8—2741no—
109C22×C4—2741no—
110C22×C4—2741no—
111C22×C4—1371no—
112C22×C4—1371no—
113C22×C4—1371no—
114M16—2744no—
115C2×C8—2742no—
116C2×C8—2742no—
117C4⋊C4—21no—
118C4⋊C4—21no—
119C4⋊C4—2741no—
120C4⋊C4—2741no—
121C4⋊C4—2741no—
122C4⋊C4—2741no—
123C4⋊C4—2741no—
124C4⋊C4—2741no—
125C4⋊C4—2741no—
126C4⋊C4—2741no—
127C4⋊C4—1371no—
128C4⋊C4—1371no—
129C4⋊C4C2×#1058773911yescharacteristic
130C4⋊C4C2×#1058773911yescharacteristic
131C42—1371no—
132C42—1371no—
133C42—1371no—
134C42C2×#1058773911yescharacteristic
135C23—1371no—
136C2×C4—21no—
137C2×C4—21no—
138C2×C4—2742no—
139C2×C4—2742no—
140C2×C4—2741no—
141C2×C4—2741no—
142C2×C4—5481no—
143C2×C4—5482no—
144C2×C4—5482no—
145C2×C4—1371no—
146C2×C4—1371no—
147C2×C4—1371no—
148C2×C4C22⋊#1058773911yescharacteristic
149C2×C4C22⋊#1058773911yescharacteristic
150C2×C4C22×#1058773911yescharacteristic
151C8—2744no—
152C8—5482no—
153C22—2741no—
154C22—2741no—
155C22—1372no—
156C22C2×C22⋊#1058773911yescharacteristic
157C4—21no—
158C4—21no—
159C4—21no—
160C4—41no—
161C4—41no—
162C4—2741no—
163C4—2741no—
164C4—5481no—
165C4—5481no—
166C4—1372no—
167C2—2742no—
168C2D137⋊#10111yescharacteristic
169C2D137⋊#10411yescharacteristic
170C2#2257840011yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
Fitting#125×C137C4
socleC22×C137C2×C22⋊C4
prime radicalC22×D137C22⋊C4
cosocleC4×Dic137C23
hypercenter#125#10587739
hypocenterC137#1528
LCSG ▷ C2×C4×C137 ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #125

Complements

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

NH ≅ G/Ntype
3. #22578260C2semidirect
4. #22578260C2semidirect
85. C137#1528semidirect

Maximal subgroups

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

subgroupindex
#225784262
C4×#145555202
#225782602
#125×D1372
#1528137