Finite Group Databasebeta

← #30649103

Subgroups — #30649103

Order 16256.

484 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 121 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#22550432C211yesmaximal, characteristic, complemented
2#127×D127C211yesmaximal, characteristic
3#22550242C211yesmaximal, characteristic, complemented
4#22550241C211yesmaximal, characteristic, complemented
5#22550240C211yesmaximal, characteristic, complemented
6#22550221C214yesmaximal, complemented
7#22549879C214yesmaximal, complemented
8#22549828C211yesmaximal, characteristic, complemented
9#490×C127C211yesmaximal, characteristic, complemented
10C2×D4×D127—22no—
11C2×D4×D127C2212yes—
12C2×#14523231C2212yes—
13#16861208C2212yescomplemented
14#16861206C2211yescharacteristic
15#16861205—22no—
16#16861205C2212yes—
17#16861201C2211yescharacteristic, complemented
18#16861187—22no—
19#16861174—22no—
20#16861167C2212yes—
21C42×D127C2211yescharacteristic
22#16861131C2214yescomplemented
23#16861130C2214yescomplemented
24#16861129C2214yescomplemented
25#16861120C2211yescharacteristic, complemented
26C8×Dic127C2212yescomplemented
27#127×C127C2211yescharacteristic
28#125×C127C2211yescharacteristic, complemented
29#102×C127C2214yescomplemented
30#97×C127C2211yescharacteristic, complemented
31C23×D127—22no—
32C2×Dic254—22no—
33C2×D508—22no—
34C2×D508C2312yes—
35C2×C22⋊D127—22no—
36C2×C22⋊D127—24no—
37D4×D127—28no—
38D4×D127—48no—
39C2×C4×D127D412yes—
40C2×C4×D127C2311yescharacteristic
41#14523231—24no—
42C2×C127⋊C8C2312yes—
43C4⋊Dic127C2312yes—
44Dic127⋊C4—22no—
45Dic127⋊C4—22no—
46(C2×D127)⋊C4—22no—
47C4×Dic127—22no—
48C4×Dic127D412yes—
49C4×Dic127C2311yescharacteristic
50C2×D4×C127—22no—
51C2×D4×C127C2312yes—
52C2×C8×C127C2312yes—
53C4⋊C4×C127—22no—
54C4⋊C4×C127C2312yes—
55C42×C127C2311yescharacteristic
56C22×D127—22no—
57C22×D127—48no—
58C22×D127—44no—
59C22×D127C2×D411yescharacteristic
60Dic254—44no—
61D508—24no—
62D508—44no—
63C22⋊D127—44no—
64C22⋊D127—48no—
65C2×Dic127—22no—
66C2×Dic127C2×D411yescharacteristic
67C2×Dic127C2×D412yes—
68C4×D127—24no—
69C4×D127#4114yes—
70C127⋊C8—22no—
71C23×C127—22no—
72D4×C127—24no—
73D4×C127—44no—
74C2×C4×C127—22no—
75C2×C4×C127C2×D412yes—
76C2×C4×C127C2411yescharacteristic
77C8×C127—22no—
78C22×C127—42no—
79C22×C127—44no—
80C22×C127C22×D411yescharacteristic
81C2×D127—22no—
82C2×D127—42no—
83C2×D127—44no—
84C2×D127—88no—
85C2×D127#12412yes—
86C4×C127—22no—
87C4×C127—42no—
88C4×C127C2×#4112yes—
89Dic127—42no—
90Dic127#12412yes—
91Dic127AGL(1,ℤ/8)14yes—
92C2×C127—82no—
93C2×C127C2×#12411yescharacteristic
94C2×C127C2×AGL(1,ℤ/8)12yes—
95D127—22no—
96D127—82no—
97#3138—1271nomaximal, Sylow
98C127#313811yescharacteristic, complemented, Sylow
99#533—1271no—
100C2×#127—1271no—
101#490—1271no—
102#490—1271no—
103#490—1271no—
104#490D12711yescharacteristic, complemented
105#486—1274no—
106#418—1274no—
107C4×M16—1271no—
108C22×D4—1272no—
109C22×D4—2542no—
110C2×M16—1272no—
111C4⋊Q8—1272no—
112#127—1272no—
113#127—1271no—
114#127—2542no—
115#127C2×D12711yescharacteristic
116#125—1271no—
117#125C2×D12711yescharacteristic, complemented
118C22⋊Q8—2542no—
119C4×Q8—2542no—
120#117—1272no—
121C2×C42—1271no—
122#102—1274no—
123#102—1274no—
124#102—1274no—
125#102C2×D12714yescomplemented
126#97—1271no—
127#97C2×D12711yescharacteristic, complemented
128C4×C8—1272no—
129C24—2542no—
130C2×Q8—2542no—
131C2×D4—22no—
132C2×D4—1272no—
133C2×D4—5088no—
134C2×D4—2548no—
135C2×D4—2544no—
136C2×D4—2542no—
137C2×D4—2542no—
138C2×D4C22×D12712yes—
139C22×C4—1271no—
140C22×C4—1272no—
141M16—2544no—
142C2×C8—1272no—
143C2×C8C22×D12712yes—
144C4⋊C4—22no—
145C4⋊C4—1272no—
146C4⋊C4—2542no—
147C4⋊C4—2542no—
148C4⋊C4C22×D12712yes—
149C22⋊C4—2542no—
150C42—1272no—
151C42—1271no—
152C42—2542no—
153C42C22×D12711yescharacteristic
154C23—22no—
155C23—1271no—
156C23—5084no—
157C23—5088no—
158C23—2542no—
159Q8—5084no—
160D4—24no—
161D4—44no—
162D4—5084no—
163D4—5088no—
164D4—5084no—
165D4—2544no—
166C2×C4—22no—
167C2×C4—1271no—
168C2×C4—1272no—
169C2×C4—1274no—
170C2×C4—2544no—
171C2×C4—2542no—
172C2×C4D4×D12712yes—
173C2×C4C23×D12711yescharacteristic
174C8—22no—
175C8—2542no—
176C22—42no—
177C22—44no—
178C22—1272no—
179C22—10168no—
180C22—5082no—
181C22—5084no—
182C22—2542no—
183C22C2×D4×D12711yescharacteristic
184C4—22no—
185C4—42no—
186C4—1272no—
187C4—1274no—
188C4—5082no—
189C4#41×D12712yes—
190C2—82no—
191C2—10162no—
192C2—2542no—
193C2#124×D12711yescharacteristic
194C2AGL(1,ℤ/8)×D12712yes—
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#490×C127C2
socleC22×C127C22×D4
prime radical#127×D127C2
hypercenter#490D127
hypocenterC127#3138
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #490

Complements

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

NH ≅ G/Ntype
1. #22550432C2semidirect
3. #22550242C2semidirect
4. #22550241C2semidirect
5. #22550240C2semidirect
6. #22550221C2semidirect
7. #22549879C2semidirect
8. #22549828C2semidirect
9. #490×C127C2semidirect
13. #16861208C22semidirect
17. #16861201C22semidirect
22. #16861131C22semidirect
23. #16861130C22semidirect
24. #16861129C22semidirect
25. #16861120C22semidirect
26. C8×Dic127C22semidirect
28. #125×C127C22semidirect
29. #102×C127C22semidirect
30. #97×C127C22semidirect
98. C127#3138semidirect
104. #490D127semidirect
117. #125C2×D127semidirect
125. #102C2×D127semidirect
127. #97C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225504322
#127×D1272
#225502422
#225502412
#225502402
#225502212
#225498792
#225498282
#490×C1272
#3138127