Finite Group Databasebeta

← #30647190

Subgroups — #30647190

Order 16256.

800 conjugacy classes of subgroups; only the 249 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 249 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
1C2×#16861254C211yesmaximal, characteristic
2#22550351C211yesmaximal, characteristic, complemented
3#22549849C214yesmaximal, complemented
4C2×#16861123C218yesmaximal, complemented
5#518×C127C211yesmaximal, characteristic, complemented
6#16861254—24no—
7#16861254—24no—
8C2×#14523248C2212yes—
9C2×#14523247C414yescomplemented
10C22×C22⋊D127C2214yes—
11C2×D4×D127C414yescomplemented
12D4×Dic127—24no—
13#16861168C2212yescomplemented
14C2×C22⋊Dic127C2214yescomplemented
15#16861123C22116yescomplemented
16C22×D4×C127C2211yescharacteristic
17C4×D4×C127—24no—
18#117×C127C2212yescomplemented
19C2×C22⋊C4×C127C2214yescomplemented
20C23×D127C2×C414yescomplemented
21#14523248—24no—
22#14523248—24no—
23#14523247—216no—
24C2×Dic254C2×C412yescomplemented
25C2×D508C2×C412yescomplemented
26C2×C22⋊D127—28no—
27C2×C22⋊D127—28no—
28C2×C22⋊D127C2×C418yescomplemented
29C2×C22⋊D127C2314yes—
30D4×D127—216no—
31C22×Dic127—24no—
32C22×Dic127C2×C414yescomplemented
33C2×C4×D127C2×C414yescomplemented
34C4⋊Dic127—22no—
35C22⋊Dic127—24no—
36C22⋊Dic127C2314yes—
37C4×Dic127—22no—
38C24×C127C2312yes—
39C2×D4×C127C2×C414yes—
40C2×D4×C127D414yes—
41C2×D4×C127D414yes—
42C22×C4×C127—24no—
43C22×C4×C127C2311yescharacteristic
44C4⋊C4×C127—22no—
45C22⋊C4×C127—24no—
46C22⋊C4×C127C2314yes—
47C42×C127—22no—
48C22×D127—28no—
49C22×D127—84no—
50C22×D127C22×C414yes—
51Dic254—24no—
52D508—24no—
53C22⋊D127—216no—
54C22⋊D127—24no—
55C22⋊D127—24no—
56C2×Dic127—28no—
57C2×Dic127—24no—
58C2×Dic127—84no—
59C2×Dic127C22×C414yes—
60C4×D127—28no—
61C23×C127—82no—
62C23×C127C22×C414yes—
63C23×C127C2×D414yes—
64C23×C127C2×D414yes—
65C23×C127C2411yescharacteristic
66D4×C127C22⋊C4116yes—
67C2×C4×C127—24no—
68C2×C4×C127—84no—
69C2×C4×C127C22×C412yes—
70C2×C4×C127C2×D412yes—
71C2×C4×C127C2×D412yes—
72C22×C127—44no—
73C22×C127—41no—
74C22×C127—44no—
75C22×C127—44no—
76C22×C127C2×C22⋊C418yes—
77C22×C127C23×C411yescharacteristic
78C22×C127C22×D411yescharacteristic
79C22×C127C22×D411yescharacteristic
80C2×D127—24no—
81C2×D127—44no—
82C2×D127—88no—
83C4×C127—44no—
84C4×C127C2×C22⋊C414yes—
85Dic127—24no—
86Dic127—44no—
87C2×C127—21no—
88C2×C127—21no—
89C2×C127—21no—
90C2×C127—28no—
91C2×C127C22×C22⋊C411yescharacteristic
92D127—44no—
93#2883—1271nomaximal, Sylow
94C127#288311yescharacteristic, complemented, Sylow
95C2×H(5,𝔽2)—1271no—
96#518—1271no—
97#518D12711yescharacteristic, complemented
98#410—1274no—
99C2×#99—1278no—
100H(5,𝔽2)—2544no—
101H(5,𝔽2)—2544no—
102C2×#41—1274no—
103C2×#41—1272no—
104C22×D4—1274no—
105C22×D4—1274no—
106C22×D4C2×D12711yescharacteristic
107C4×D4—24no—
108C4×D4—2544no—
109#117—1272no—
110#117C2×D12712yescomplemented
111C2×C22⋊C4—1274no—
112C2×C22⋊C4C2×D12714yescomplemented
113#99—12716no—
114C24—1274no—
115C24C22×D12712yes—
116#41—25416no—
117#41—2544no—
118#41—2544no—
119C2×Q8—1272no—
120C2×D4—1278no—
121C2×D4—1274no—
122C2×D4—1272no—
123C2×D4—2548no—
124C2×D4—2548no—
125C2×D4—25416no—
126C2×D4C4×D12714yes—
127C2×D4C22⋊D12714yes—
128C2×D4D50814yes—
129C22×C4—24no—
130C22×C4—1274no—
131C22×C4—1274no—
132C22×C4—2544no—
133C22×C4C22×D12711yescharacteristic
134C4⋊C4—22no—
135C4⋊C4—2542no—
136C22⋊C4—24no—
137C22⋊C4—1274no—
138C22⋊C4—2544no—
139C22⋊C4C22×D12714yes—
140C42—22no—
141C42—2542no—
142C23—82no—
143C23—1274no—
144C23—10164no—
145C23—2548no—
146C23C2×C4×D12714yes—
147C23C2×C22⋊D12714yes—
148C23C2×D50814yes—
149C23C23×D12711yescharacteristic
150Q8—2544no—
151D4—2544no—
152D4—2544no—
153D4—25416no—
154D4—2544no—
155D4(C2×D127)⋊C4116yes—
156C2×C4—24no—
157C2×C4—84no—
158C2×C4—1274no—
159C2×C4—10164no—
160C2×C4—2544no—
161C2×C4—2548no—
162C2×C4—2548no—
163C2×C4C2×C4×D12712yes—
164C2×C4C2×C22⋊D12712yes—
165C2×C4C2×D50812yes—
166C22—44no—
167C22—44no—
168C22—41no—
169C22—44no—
170C22—10168no—
171C22—5084no—
172C22—2544no—
173C22C2×(C2×D127)⋊C418yes—
174C22C22×C4×D12711yescharacteristic
175C22C22×C22⋊D12711yescharacteristic
176C22C22×D50811yescharacteristic
177C4—44no—
178C4—5084no—
179C4—2544no—
180C4C2×(C2×D127)⋊C414yes—
181C2—21no—
182C2—21no—
183C2—21no—
184C2—28no—
185C2—5084no—
186C2C22×(C2×D127)⋊C411yescharacteristic
subgroupG/H
centerC2C22×(C2×D127)⋊C4
commutatorC22×C127C23×C4
FrattiniC23C23×D127
Fitting#518×C127C2
socleC2×C127C22×C22⋊C4
prime radicalC2×#16861254C2
cosocleC23×C127C24
hypercenter#518D127
hypocenterC127#2883
LCSG ▷ C22×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22×D4 ◁ #518

Complements

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

NH ≅ G/Ntype
2. #22550351C2semidirect
3. #22549849C2semidirect
4. C2×#16861123C2semidirect
5. #518×C127C2semidirect
9. C2×#14523247C4semidirect
11. C2×D4×D127C4semidirect
13. #16861168C22semidirect
14. C2×C22⋊Dic127C22semidirect
15. #16861123C22semidirect
18. #117×C127C22semidirect
19. C2×C22⋊C4×C127C22semidirect
20. C23×D127C2×C4semidirect
24. C2×Dic254C2×C4semidirect
25. C2×D508C2×C4semidirect
28. C2×C22⋊D127C2×C4semidirect
32. C22×Dic127C2×C4semidirect
33. C2×C4×D127C2×C4semidirect
94. C127#2883semidirect
97. #518D127semidirect
110. #117C2×D127semidirect
112. C2×C22⋊C4C2×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612542
#225503512
#225498492
C2×#168611232
#518×C1272
#2883127