Finite Group Databasebeta

← #30647052

Subgroups — #30647052

Order 16256.

582 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 165 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×#16861192C211yesmaximal, characteristic
2#22549767C214yesmaximal, complemented
3#22549760C214yesmaximal, complemented
4#22549755C212yesmaximal
5#22549749C212yesmaximal, complemented
6#22549739C211yesmaximal, characteristic, complemented
7C22×C42×C127C211yesmaximal, characteristic, complemented
8C22×C22⋊D127C2211yescharacteristic
9#16861192—24no—
10C2×Dic127⋊C4C2212yes—
11C2×Dic127⋊C4C2214yes—
12C2×C22⋊Dic127C2211yescharacteristic
13C2×C22⋊Dic127C2212yes—
14C2×(C2×D127)⋊C4C2212yes—
15C2×(C2×D127)⋊C4C2214yes—
16C127⋊#95C2214yes—
17C127⋊#95C2214yes—
18C127⋊#95C2214yes—
19C23×C4×C127C2212yes—
20C23×C4×C127C2211yescharacteristic
21C2×C42×C127—24no—
22C2×C42×C127C2214yes—
23C23×D127C2311yescharacteristic
24C2×C22⋊D127—22no—
25C2×C22⋊D127—24no—
26C22×Dic127C2311yescharacteristic
27C22×Dic127C2314yes—
28C22×Dic127C2312yes—
29Dic127⋊C4—24no—
30Dic127⋊C4—28no—
31C22⋊Dic127—22no—
32C22⋊Dic127—24no—
33(C2×D127)⋊C4—24no—
34(C2×D127)⋊C4—28no—
35C24×C127C2311yescharacteristic
36C22×C4×C127—24no—
37C22×C4×C127—24no—
38C22×C4×C127—28no—
39C22×C4×C127D414yes—
40C22×C4×C127C2312yes—
41C22×C4×C127C2314yes—
42C42×C127—28no—
43C22×D127—21no—
44C22×D127—24no—
45C22×D127—22no—
46C22⋊D127—44no—
47C2×Dic127—22no—
48C2×Dic127—21no—
49C2×Dic127—24no—
50C2×Dic127—24no—
51C2×Dic127—24no—
52C2×Dic127—24no—
53C2×Dic127—22no—
54C23×C127—24no—
55C23×C127C2×D412yes—
56C23×C127#4114yes—
57C23×C127C2411yescharacteristic
58C2×C4×C127—216no—
59C2×C4×C127—28no—
60C2×C4×C127—24no—
61C2×C4×C127—24no—
62C2×C4×C127—24no—
63C2×C4×C127C2×D414yes—
64C2×C4×C127#4118yes—
65C22×C127—24no—
66C22×C127—28no—
67C22×C127#12314yes—
68C22×C127C22×D411yescharacteristic
69C22×C127C2×#4112yes—
70C22×C127C2×#4114yes—
71C2×D127—41no—
72C2×D127—42no—
73C2×D127—44no—
74C4×C127—24no—
75C4×C127—28no—
76Dic127—41no—
77Dic127—42no—
78Dic127—44no—
79C2×C127—24no—
80C2×C127C2×#12311yescharacteristic
81C2×C127#52512yes—
82C2×C127#52914yes—
83D127—81no—
84#2853—1271nomaximal, Sylow
85C127#285311yescharacteristic, complemented, Sylow
86C2×#123—1271no—
87C22×C42D12711yescharacteristic, complemented
88#388—1274no—
89#387—1274no—
90#386—1272no—
91#385—1272no—
92#381—1271no—
93C22×D4—1271no—
94C23×C4C2×D12712yes—
95C23×C4C2×D12711yescharacteristic
96#123—2544no—
97C2×C4⋊C4—1272no—
98C2×C4⋊C4—1274no—
99C2×C22⋊C4—1271no—
100C2×C22⋊C4—1272no—
101C2×C22⋊C4—1272no—
102C2×C22⋊C4—1274no—
103C2×C42—24no—
104C2×C42C2×D12714yes—
105#95—1274no—
106#95—1274no—
107#95—1274no—
108C24—1271no—
109C24C22×D12711yescharacteristic
110C2×D4—2542no—
111C2×D4—2544no—
112C22×C4—24no—
113C22×C4—24no—
114C22×C4—28no—
115C22×C4—1274no—
116C22×C4—1271no—
117C22×C4—1272no—
118C22×C4C22⋊D12714yes—
119C22×C4C22×D12712yes—
120C22×C4C22×D12714yes—
121C4⋊C4—2544no—
122C4⋊C4—2548no—
123C22⋊C4—2542no—
124C22⋊C4—2544no—
125C22⋊C4—2544no—
126C22⋊C4—2548no—
127C42—28no—
128C23—24no—
129C23—2541no—
130C23—2544no—
131C23—2542no—
132C23C2×C22⋊D12712yes—
133C23#1452324814yes—
134C23C23×D12711yescharacteristic
135D4—5084no—
136C2×C4—216no—
137C2×C4—28no—
138C2×C4—24no—
139C2×C4—24no—
140C2×C4—24no—
141C2×C4—2542no—
142C2×C4—2541no—
143C2×C4—2544no—
144C2×C4—2544no—
145C2×C4—2544no—
146C2×C4—2544no—
147C2×C4—2542no—
148C2×C4C2×C22⋊D12714yes—
149C2×C4#1452324818yes—
150C22—24no—
151C22—28no—
152C22—5081no—
153C22—5082no—
154C22—5084no—
155C22#1686119214yes—
156C22C22×C22⋊D12711yescharacteristic
157C22C2×#1452324812yes—
158C22C2×#1452324814yes—
159C4—24no—
160C4—28no—
161C4—5081no—
162C4—5082no—
163C4—5084no—
164C2—24no—
165C2—10161no—
166C2C2×#1686119211yescharacteristic
167C2#2255038712yes—
168C2#2255041014yes—
subgroupG/H
center, FrattiniC23C23×D127
commutator, socle, cosocleC23×C127C24
FittingC22×C42×C127C2
prime radicalC22×C22⋊D127C22
hypercenterC22×C42D127
hypocenterC127#2853
LCSG ▷ C23×C127 ▷ C127
UCSC1 ◁ C23 ◁ C22×C42

Complements

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

NH ≅ G/Ntype
2. #22549767C2semidirect
3. #22549760C2semidirect
5. #22549749C2semidirect
6. #22549739C2semidirect
7. C22×C42×C127C2semidirect
85. C127#2853semidirect
87. C22×C42D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168611922
#225497672
#225497602
#225497552
#225497492
#225497392
C22×C42×C1272
#2853127