Finite Group Databasebeta

← #30649116

Subgroups — #30649116

Order 16256.

400 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 133 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#22550433C211yesmaximal, characteristic, complemented
2#22550235C214yesmaximal, complemented
3#22550230C214yesmaximal, complemented
4#22549885C212yesmaximal, complemented
5#22549881C212yesmaximal
6C4×#14523232C211yesmaximal, characteristic, complemented
7C2×#124×C127C211yesmaximal, characteristic
8C2×#14523232C2212yes—
9#16861209C2212yescomplemented
10#16861201C2212yescomplemented
11#16861188—22no—
12C4×Dic254—22no—
13#16861168C2212yes—
14#16861133C2218yescomplemented
15#16861130C2218yes—
16#16861120C2212yescomplemented
17C4×C127⋊C8C2212yescomplemented
18C22×Q8×C127C2211yescharacteristic
19C22×D4×C127C2211yescharacteristic
20#124×C127—22no—
21#124×C127C2212yes—
22#124×C127C2212yes—
23C2×C22⋊C4×C127—22no—
24C2×C42×C127C2211yescharacteristic
25C2×Dic254—22no—
26#14523232—24no—
27C2×C127⋊C8C2314yes—
28C4⋊Dic127—22no—
29C4⋊Dic127C2314yes—
30Dic127⋊C4—24no—
31C22⋊Dic127—22no—
32C4×Dic127—22no—
33C24×C127—21no—
34C2×Q8×C127—21no—
35C2×Q8×C127—24no—
36C2×Q8×C127C2312yes—
37C2×D4×C127—21no—
38C2×D4×C127—24no—
39C2×D4×C127C2312yes—
40C22×C4×C127—21no—
41C22×C4×C127D412yes—
42C22×C4×C127C2311yescharacteristic
43C22⋊C4×C127—24no—
44C22⋊C4×C127—24no—
45C42×C127D412yes—
46C42×C127C2311yescharacteristic
47C42×C127C2311yescharacteristic
48Dic254—44no—
49C2×Dic127—24no—
50C127⋊C8—24no—
51C23×C127—22no—
52C23×C127—44no—
53C23×C127—42no—
54C23×C127C2×D411yescharacteristic
55Q8×C127—24no—
56Q8×C127—42no—
57D4×C127—24no—
58D4×C127—42no—
59C2×C4×C127—22no—
60C2×C4×C127—42no—
61C2×C4×C127—42no—
62C2×C4×C127C2×D412yes—
63C2×C4×C127C2×D412yes—
64C2×C4×C127C2×D411yescharacteristic
65C2×C4×C127#4114yes—
66C2×C4×C127C2411yescharacteristic
67C22×C127—22no—
68C22×C127—42no—
69C22×C127—44no—
70C22×C127—84no—
71C22×C127#12412yes—
72C22×C127C22×D411yescharacteristic
73C4×C127—22no—
74C4×C127—22no—
75C4×C127—42no—
76C4×C127#12412yes—
77C4×C127C2×#4112yes—
78Dic127—44no—
79C2×C127—22no—
80C2×C127—82no—
81C2×C127C2×#12411yescharacteristic
82C2×C127#57512yes—
83#3140—1271nomaximal, Sylow
84C127#314011yescharacteristic, complemented, Sylow
85#533—1271no—
86C2×#124D12711yescharacteristic
87#489—1274no—
88#488—1274no—
89#419—1272no—
90#418—1272no—
91C4×M16—1271no—
92C22×Q8C2×D12711yescharacteristic
93C22×D4C2×D12711yescharacteristic
94C2×M16—1272no—
95C4⋊Q8—1272no—
96#125—1272no—
97#124—22no—
98#124C2×D12712yes—
99#124C2×D12712yes—
100C22⋊Q8—2542no—
101C4×Q8—2542no—
102#117—1272no—
103C2×C22⋊C4—22no—
104C2×C42C2×D12711yescharacteristic
105#103—1278no—
106#102—1278no—
107#97—1272no—
108C4×C8—1272no—
109C24—21no—
110C2×Q8—21no—
111C2×Q8—24no—
112C2×Q8—2542no—
113C2×Q8C22×D12712yes—
114C2×D4—21no—
115C2×D4—24no—
116C2×D4C22×D12712yes—
117C22×C4—21no—
118C22×C4C22⋊D12712yes—
119C22×C4C22×D12711yescharacteristic
120M16—2544no—
121C2×C8—1274no—
122C4⋊C4—1274no—
123C4⋊C4—2544no—
124C4⋊C4—2542no—
125C22⋊C4—24no—
126C22⋊C4—24no—
127C22⋊C4—2542no—
128C42—2542no—
129C42C22⋊D12712yes—
130C42C22×D12711yescharacteristic
131C42C22×D12711yescharacteristic
132C23—22no—
133C23—44no—
134C23—42no—
135C23C2×C22⋊D12711yescharacteristic
136Q8—24no—
137Q8—42no—
138Q8—5084no—
139D4—24no—
140D4—42no—
141C2×C4—22no—
142C2×C4—42no—
143C2×C4—42no—
144C2×C4—2544no—
145C2×C4C2×C22⋊D12712yes—
146C2×C4C2×C22⋊D12712yes—
147C2×C4C2×C22⋊D12711yescharacteristic
148C2×C4#1452324714yes—
149C2×C4C23×D12711yescharacteristic
150C8—2544no—
151C22—22no—
152C22—42no—
153C22—44no—
154C22—84no—
155C22#1686119612yes—
156C22C22×C22⋊D12711yescharacteristic
157C4—22no—
158C4—22no—
159C4—42no—
160C4—5084no—
161C4#1686119612yes—
162C4C2×#1452324712yes—
163C2—22no—
164C2—82no—
165C2C2×#1686119611yescharacteristic
166C2#2255067412yes—
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×#124×C127C2
socleC22×C127C22×D4
prime radicalC22×D4×C127C22
hypercenterC2×#124D127
hypocenterC127#3140
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. #22550433C2semidirect
2. #22550235C2semidirect
3. #22550230C2semidirect
4. #22549885C2semidirect
6. C4×#14523232C2semidirect
9. #16861209C22semidirect
10. #16861201C22semidirect
14. #16861133C22semidirect
16. #16861120C22semidirect
17. C4×C127⋊C8C22semidirect
84. C127#3140semidirect

Maximal subgroups

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

subgroupindex
#225504332
#225502352
#225502302
#225498852
#225498812
C4×#145232322
C2×#124×C1272
#3140127