Finite Group Databasebeta

← #30647640

Subgroups — #30647640

Order 16256.

458 conjugacy classes of subgroups; only the 217 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 217 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#22550643C212yesmaximal, complemented
2#22549998C214yesmaximal, complemented
3#22549993C214yesmaximal, complemented
4#22549925C211yesmaximal, characteristic, complemented
5C2×#16861136C212yesmaximal, complemented
6M16×Dic127C211yesmaximal, characteristic, complemented
7#575×C127C211yesmaximal, characteristic
8#16861216—24no—
9#16861216C2218yescomplemented
10C2×#14523232—24no—
11C2×#14523232C2212yes—
12C2×#14523232C2211yescharacteristic
13C2×C4×Dic127C2211yescharacteristic
14#16861145C2214yescomplemented
15#16861136C2214yescomplemented
16#16861136C2214yescomplemented
17#16861119C2212yescomplemented
18C8×Dic127C2212yescomplemented
19C2×#41×C127C2212yes—
20#137×C127C414yescomplemented
21AGL(1,ℤ/8)×C127C414yescomplemented
22#135×C127C2214yes—
23C2×M16×C127C2211yescharacteristic
24C22×Dic127—21no—
25#14523232—28no—
26#14523232—22no—
27#14523232D414yes—
28#14523232C2314yes—
29C2×C127⋊C8—28no—
30C2×C127⋊C8—22no—
31C2×C127⋊C8C2312yes—
32C4×Dic127—21no—
33C4×Dic127C2311yescharacteristic
34C4×Dic127C2311yescharacteristic
35#41×C127—22no—
36#41×C127—22no—
37#41×C127C2×C414yes—
38#41×C127C2314yes—
39C2×Q8×C127C2×C412yes—
40C2×D4×C127—22no—
41C2×D4×C127C2×C412yes—
42C22×C4×C127—22no—
43C22×C4×C127C2311yescharacteristic
44Q16×C127C2×C414yescomplemented
45SD16×C127C2×C418yescomplemented
46D8×C127C2×C414yescomplemented
47M16×C127C2×C414yes—
48C2×C8×C127C2312yes—
49C2×Dic127—21no—
50C2×Dic127—21no—
51C2×Dic127—42no—
52C127⋊C8—24no—
53C127⋊C8C2×D414yes—
54C23×C127—22no—
55C23×C127#4111yescharacteristic
56Q8×C127—22no—
57Q8×C127C22×C414yes—
58D4×C127—24no—
59D4×C127—24no—
60D4×C127—22no—
61D4×C127C22×C414yes—
62C2×C4×C127—22no—
63C2×C4×C127—24no—
64C2×C4×C127—24no—
65C2×C4×C127C22×C412yes—
66C2×C4×C127C2×D412yes—
67C2×C4×C127#4111yescharacteristic
68C2×C4×C127C2411yescharacteristic
69C8×C127C22×C414yes—
70C22×C127—24no—
71C22×C127—41no—
72C22×C127—82no—
73C22×C127C4×D412yes—
74C22×C127C2×#4111yescharacteristic
75C4×C127—24no—
76C4×C127C4×D412yes—
77C4×C127C23×C411yescharacteristic
78C4×C127C22×D411yescharacteristic
79Dic127—22no—
80Dic127—41no—
81C2×C127—21no—
82C2×C127—22no—
83C2×C127—44no—
84C2×C127C2×C4×D411yescharacteristic
85#2956—1271nomaximal, Sylow
86C127#295611yescharacteristic, complemented, Sylow
87#575D12711yescharacteristic
88#568—1272no—
89#444—1274no—
90#443—1274no—
91#430—1271no—
92C2×#104—1272no—
93C4×M16—1271no—
94C2×#41C2×D12712yes—
95#137Dic12714yescomplemented
96AGL(1,ℤ/8)Dic12714yescomplemented
97#135C2×D12714yes—
98#131—1278no—
99#131—2544no—
100C2×M16—1271no—
101C2×M16—1272no—
102C2×M16—2544no—
103C2×M16C2×D12711yescharacteristic
104C2×C42—1271no—
105#108—1274no—
106#104—1274no—
107#104—1274no—
108#97—1272no—
109C4×C8—1272no—
110#41—22no—
111#41—22no—
112#41C2×Dic12714yes—
113#41C22×D12714yes—
114C2×Q8C2×Dic12712yes—
115C2×D4—22no—
116C2×D4C2×Dic12712yes—
117C22×C4—22no—
118C22×C4—2541no—
119C22×C4C22×D12711yescharacteristic
120Q16C2×Dic12714yescomplemented
121SD16C2×Dic12718yescomplemented
122D8C2×Dic12714yescomplemented
123M16—1274no—
124M16—1274no—
125M16—2548no—
126M16—2542no—
127M16C2×Dic12714yes—
128C2×C8—1272no—
129C2×C8—2548no—
130C2×C8—2542no—
131C2×C8C22×D12712yes—
132C42—1271no—
133C42—1271no—
134C42—2541no—
135C23—22no—
136C23#1452324711yescharacteristic
137Q8—22no—
138Q8C22×Dic12714yes—
139D4—24no—
140D4—24no—
141D4—22no—
142D4C22×Dic12714yes—
143C2×C4—22no—
144C2×C4—24no—
145C2×C4—24no—
146C2×C4—5082no—
147C2×C4—2541no—
148C2×C4—2541no—
149C2×C4C22×Dic12712yes—
150C2×C4D4×D12712yes—
151C2×C4#1452324711yescharacteristic
152C2×C4C23×D12711yescharacteristic
153C8—1274no—
154C8—2544no—
155C8C22×Dic12714yes—
156C22—24no—
157C22—41no—
158C22—82no—
159C22D4×Dic12712yes—
160C22C2×#1452324711yescharacteristic
161C4—24no—
162C4—5081no—
163C4—2542no—
164C4D4×Dic12712yes—
165C4C23×Dic12711yescharacteristic
166C4C2×D4×D12711yescharacteristic
167C2—21no—
168C2—22no—
169C2—44no—
170C2C2×D4×Dic12711yescharacteristic
subgroupG/H
centerC4C2×D4×D127
commutatorC4×C127C23×C4
FrattiniC2×C4C23×D127
Fitting, prime radical#575×C127C2
socleC2×C127C2×C4×D4
cosocleC2×C4×C127C24
hypercenter#575D127
hypocenterC127#2956
LCSG ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C4 ◁ C22×C4 ◁ #575

Complements

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

NH ≅ G/Ntype
1. #22550643C2semidirect
2. #22549998C2semidirect
3. #22549993C2semidirect
4. #22549925C2semidirect
5. C2×#16861136C2semidirect
6. M16×Dic127C2semidirect
9. #16861216C22semidirect
14. #16861145C22semidirect
15. #16861136C22semidirect
16. #16861136C22semidirect
17. #16861119C22semidirect
18. C8×Dic127C22semidirect
20. #137×C127C4semidirect
21. AGL(1,ℤ/8)×C127C4semidirect
44. Q16×C127C2×C4semidirect
45. SD16×C127C2×C4semidirect
46. D8×C127C2×C4semidirect
86. C127#2956semidirect
95. #137Dic127semidirect
96. AGL(1,ℤ/8)Dic127semidirect
120. Q16C2×Dic127semidirect
121. SD16C2×Dic127semidirect
122. D8C2×Dic127semidirect

Maximal subgroups

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

subgroupindex
#225506432
#225499982
#225499932
#225499252
C2×#168611362
M16×Dic1272
#575×C1272
#2956127