Finite Group Databasebeta

← #30647049

Subgroups — #30647049

Order 16256.

582 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 157 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
1C22×C4×Dic127C211yesmaximal, characteristic, complemented
2#22549764C214yesmaximal, complemented
3#22549757C214yesmaximal, complemented
4#22549756C212yesmaximal
5#22549748C212yesmaximal, complemented
6#22549739C211yesmaximal, characteristic, complemented
7C2×#123×C127C211yesmaximal, characteristic
8C23×Dic127C2212yes—
9C2×Dic127⋊C4C2214yes—
10C2×C22⋊Dic127C2214yes—
11C2×C22⋊Dic127C2212yes—
12C2×C4×Dic127—24no—
13C2×C4×Dic127C2214yes—
14C127⋊#95C2214yes—
15C127⋊#95C2214yes—
16C127⋊#95C2214yes—
17C22×D4×C127C2211yescharacteristic
18C23×C4×C127C2211yescharacteristic
19#123×C127—24no—
20C2×C4⋊C4×C127C2212yes—
21C2×C22⋊C4×C127C2212yes—
22C2×C22⋊C4×C127C2211yescharacteristic
23C22×Dic127—28no—
24C22×Dic127—24no—
25C22×Dic127C2314yes—
26C22×Dic127C2314yes—
27Dic127⋊C4—28no—
28C22⋊Dic127—28no—
29C22⋊Dic127—24no—
30C4×Dic127—28no—
31C24×C127C2311yescharacteristic
32C24×C127C2311yescharacteristic
33C2×D4×C127—22no—
34C2×D4×C127—24no—
35C22×C4×C127—24no—
36C22×C4×C127D414yes—
37C22×C4×C127C2312yes—
38C22×C4×C127C2311yescharacteristic
39C22×C4×C127C2312yes—
40C4⋊C4×C127—24no—
41C22⋊C4×C127—22no—
42C22⋊C4×C127—24no—
43C2×Dic127—216no—
44C2×Dic127—24no—
45C2×Dic127—24no—
46C2×Dic127—24no—
47C2×Dic127—24no—
48C2×Dic127—24no—
49C2×Dic127#4118yes—
50C23×C127—24no—
51C23×C127—21no—
52C23×C127—22no—
53C23×C127—24no—
54C23×C127C2×D412yes—
55C23×C127#4114yes—
56C23×C127C2411yescharacteristic
57D4×C127—44no—
58C2×C4×C127—28no—
59C2×C4×C127—24no—
60C2×C4×C127—22no—
61C2×C4×C127—21no—
62C2×C4×C127—22no—
63C2×C4×C127—24no—
64C2×C4×C127C2×D414yes—
65C22×C127—24no—
66C22×C127—28no—
67C22×C127—41no—
68C22×C127—42no—
69C22×C127—44no—
70C22×C127#12314yes—
71C22×C127C22×D411yescharacteristic
72C22×C127C2×#4114yes—
73C22×C127C2×#4112yes—
74C4×C127—24no—
75C4×C127—41no—
76C4×C127—42no—
77Dic127—28no—
78Dic127—44no—
79C2×C127—24no—
80C2×C127—81no—
81C2×C127C2×#12311yescharacteristic
82C2×C127#52512yes—
83C2×C127#52914yes—
84#2853—1271nomaximal, Sylow
85C127#285311yescharacteristic, complemented, Sylow
86C2×#123D12711yescharacteristic
87C22×C42—1271no—
88#388—1274no—
89#387—1274no—
90#386—1272no—
91#385—1272no—
92#381—1271no—
93C22×D4C2×D12711yescharacteristic
94C23×C4—1272no—
95C23×C4C2×D12711yescharacteristic
96#123—24no—
97C2×C4⋊C4—1274no—
98C2×C4⋊C4C2×D12712yes—
99C2×C22⋊C4—1272no—
100C2×C22⋊C4—1274no—
101C2×C22⋊C4C2×D12711yescharacteristic
102C2×C22⋊C4C2×D12712yes—
103C2×C42—1274no—
104C2×C42—2544no—
105#95—1274no—
106#95—1274no—
107#95—1274no—
108C24C22×D12711yescharacteristic
109C24C22×D12711yescharacteristic
110C2×D4—22no—
111C2×D4—24no—
112C22×C4—24no—
113C22×C4—1274no—
114C22×C4—1274no—
115C22×C4—2548no—
116C22×C4—2544no—
117C22×C4C22⋊D12714yes—
118C22×C4C22×D12711yescharacteristic
119C22×C4C22×D12712yes—
120C22×C4C22×D12712yes—
121C4⋊C4—24no—
122C4⋊C4—2548no—
123C22⋊C4—22no—
124C22⋊C4—24no—
125C22⋊C4—2548no—
126C22⋊C4—2544no—
127C42—2548no—
128C23—24no—
129C23—21no—
130C23—24no—
131C23—22no—
132C23C2×C22⋊D12712yes—
133C23#1452324714yes—
134C23C23×D12711yescharacteristic
135D4—44no—
136C2×C4—28no—
137C2×C4—24no—
138C2×C4—22no—
139C2×C4—21no—
140C2×C4—24no—
141C2×C4—22no—
142C2×C4—1278no—
143C2×C4—2544no—
144C2×C4—2544no—
145C2×C4—2544no—
146C2×C4—25416no—
147C2×C4—2544no—
148C2×C4—2544no—
149C2×C4C2×C22⋊D12714yes—
150C22—24no—
151C22—28no—
152C22—41no—
153C22—42no—
154C22—44no—
155C22#1686119014yes—
156C22C22×C22⋊D12711yescharacteristic
157C22#41×D12714yes—
158C22C2×#1452324712yes—
159C4—24no—
160C4—41no—
161C4—42no—
162C4—5084no—
163C4—2548no—
164C2—24no—
165C2—81no—
166C2C2×#1686119011yescharacteristic
167C2#2255038312yes—
168C2#2255040314yes—
subgroupG/H
center, FrattiniC23C23×D127
commutator, socle, cosocleC23×C127C24
FittingC2×#123×C127C2
prime radicalC22×D4×C127C22
hypercenterC2×#123D127
hypocenterC127#2853
LCSG ▷ C23×C127 ▷ C127
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. C22×C4×Dic127C2semidirect
2. #22549764C2semidirect
3. #22549757C2semidirect
5. #22549748C2semidirect
6. #22549739C2semidirect
85. C127#2853semidirect

Maximal subgroups

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

subgroupindex
C22×C4×Dic1272
#225497642
#225497572
#225497562
#225497482
#225497392
C2×#123×C1272
#2853127