Finite Group Databasebeta

← #30641325

Subgroups — #30641325

Order 16256.

672 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 672 conjugacy classes of subgroups — too many to draw. Only its 189 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×(C2×D127)⋊C4C211yesmaximal, characteristic
2#22549767C214yesmaximal, complemented
3#22549765C214yesmaximal, complemented
4#22549734C212yesmaximal, complemented
5#22549731C212yesmaximal
6C4×C22⋊Dic127C211yesmaximal, characteristic, complemented
7C4×C22⋊C4×C127C211yesmaximal, characteristic, complemented
8C24×D127C2211yescharacteristic
9C23×Dic127C2211yescharacteristic
10C2×Dic127⋊C4C2214yescomplemented
11C2×C22⋊Dic127C2212yes—
12C2×(C2×D127)⋊C4C418yescomplemented
13C2×(C2×D127)⋊C4C2214yes—
14C2×(C2×D127)⋊C4C2214yes—
15C2×(C2×D127)⋊C4C2212yes—
16C2×(C2×D127)⋊C4C2212yes—
17C2×C4×Dic127C2212yescomplemented
18C127⋊#95C2214yescomplemented
19C127⋊#95C2212yescomplemented
20C23×C4×C127C2211yescharacteristic
21C2×C22⋊C4×C127C2212yes—
22C2×C42×C127C2212yescomplemented
23#95×C127C2212yescomplemented
24C23×D127—24no—
25C23×D127—42no—
26C23×D127—42no—
27C23×D127C2×C414yes—
28C23×D127C2312yes—
29C22×Dic127—42no—
30C22×Dic127C2×C414yes—
31C22×Dic127C2314yes—
32C22×Dic127C2312yes—
33Dic127⋊C4—44no—
34C22⋊Dic127—24no—
35(C2×D127)⋊C4—44no—
36(C2×D127)⋊C4—44no—
37(C2×D127)⋊C4C2×C418yes—
38(C2×D127)⋊C4C2×C418yes—
39C4×Dic127—42no—
40C24×C127C2311yescharacteristic
41C22×C4×C127—42no—
42C22×C4×C127C2×C414yes—
43C22×C4×C127C2314yes—
44C22×C4×C127C2311yescharacteristic
45C22×C4×C127C2311yescharacteristic
46C22⋊C4×C127—24no—
47C42×C127—42no—
48C22×D127—28no—
49C22×D127—24no—
50C22×D127—24no—
51C22×D127—24no—
52C22×D127—44no—
53C22×D127—44no—
54C22×D127—416no—
55C22×D127C22×C414yes—
56C2×Dic127—24no—
57C2×Dic127—24no—
58C2×Dic127—24no—
59C2×Dic127—24no—
60C2×Dic127—44no—
61C2×Dic127C22×C414yes—
62C23×C127—22no—
63C23×C127—22no—
64C23×C127C22×C412yes—
65C23×C127#4114yes—
66C23×C127C2411yescharacteristic
67C2×C4×C127—22no—
68C2×C4×C127—22no—
69C2×C4×C127—24no—
70C2×C4×C127—24no—
71C2×C4×C127—24no—
72C2×C4×C127—44no—
73C2×C4×C127C22×C412yes—
74C2×C4×C127C22×C412yes—
75C22×C127—22no—
76C22×C127—22no—
77C22×C127—28no—
78C22×C127#11714yes—
79C22×C127C23×C411yescharacteristic
80C22×C127C2×#4112yes—
81C22×C127H(5,𝔽2)12yes—
82C22×C127H(5,𝔽2)12yes—
83C2×D127—24no—
84C2×D127—24no—
85C2×D127—24no—
86C2×D127—416no—
87C2×D127—48no—
88C4×C127—22no—
89C4×C127—22no—
90C4×C127—44no—
91Dic127—24no—
92Dic127—44no—
93C2×C127—24no—
94C2×C127C2×#11711yescharacteristic
95C2×C127#51811yescharacteristic
96C2×C127#51811yescharacteristic
97C2×C127#53814yes—
98D127—44no—
99#2312—1271nomaximal, Sylow
100C127#231211yescharacteristic, complemented, Sylow
101C22×C22⋊C4—1271no—
102#388—1274no—
103#388—1274no—
104#380—1272no—
105#379—1272no—
106C4×C22⋊C4—1271no—
107C4×C22⋊C4D12711yescharacteristic, complemented
108C25—1271no—
109C23×C4—1271no—
110C23×C4C2×D12711yescharacteristic
111C2×C4⋊C4—1274no—
112C2×C22⋊C4—1278no—
113C2×C22⋊C4—1272no—
114C2×C22⋊C4—1272no—
115C2×C22⋊C4—1274no—
116C2×C22⋊C4—1274no—
117C2×C22⋊C4—1272no—
118C2×C22⋊C4C2×D12712yes—
119C2×C42—1272no—
120C2×C42C2×D12712yescomplemented
121#95—1272no—
122#95—1274no—
123#95C2×D12712yescomplemented
124C24—1274no—
125C24—1272no—
126C24—5082no—
127C24—5082no—
128C24—2544no—
129C24C22×D12711yescharacteristic
130C22×C4—42no—
131C22×C4—1274no—
132C22×C4—1272no—
133C22×C4—1274no—
134C22×C4—5082no—
135C22×C4C4×D12714yes—
136C22×C4C22×D12714yes—
137C22×C4C22×D12711yescharacteristic
138C22×C4C22×D12711yescharacteristic
139C4⋊C4—5084no—
140C22⋊C4—24no—
141C22⋊C4—1278no—
142C22⋊C4—1278no—
143C22⋊C4—5084no—
144C22⋊C4—5084no—
145C22⋊C4—2544no—
146C42—42no—
147C42—5082no—
148C23—22no—
149C23—22no—
150C23—1274no—
151C23—50816no—
152C23—5084no—
153C23—5084no—
154C23—2544no—
155C23—2544no—
156C23—2544no—
157C23—2548no—
158C23C2×C4×D12712yes—
159C23#1452324814yes—
160C23C23×D12711yescharacteristic
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—24no—
164C2×C4—24no—
165C2×C4—24no—
166C2×C4—44no—
167C2×C4—1274no—
168C2×C4—5084no—
169C2×C4—2544no—
170C2×C4—2544no—
171C2×C4—2544no—
172C2×C4—2544no—
173C2×C4C2×C4×D12712yes—
174C2×C4C2×C4×D12712yes—
175C22—22no—
176C22—22no—
177C22—28no—
178C22—50816no—
179C22—5088no—
180C22—2544no—
181C22—2544no—
182C22—2544no—
183C22#1686116614yes—
184C22C22×C4×D12711yescharacteristic
185C22C2×#1452324812yes—
186C22#1686125412yes—
187C22#1686125512yes—
188C4—22no—
189C4—22no—
190C4—44no—
191C4—5084no—
192C4—2544no—
193C2—24no—
194C2—5084no—
195C2C2×#1686116611yescharacteristic
196C2#2255034911yescharacteristic
197C2#2255035011yescharacteristic
198C2#2255045914yes—
subgroupG/H
center, FrattiniC23C23×D127
commutatorC22×C127C23×C4
FittingC4×C22⋊C4×C127C2
socle, cosocleC23×C127C24
prime radicalC24×D127C22
hypercenterC4×C22⋊C4D127
hypocenterC127#2312
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C23 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. #22549767C2semidirect
3. #22549765C2semidirect
4. #22549734C2semidirect
6. C4×C22⋊Dic127C2semidirect
7. C4×C22⋊C4×C127C2semidirect
10. C2×Dic127⋊C4C22semidirect
12. C2×(C2×D127)⋊C4C4semidirect
17. C2×C4×Dic127C22semidirect
18. C127⋊#95C22semidirect
19. C127⋊#95C22semidirect
22. C2×C42×C127C22semidirect
23. #95×C127C22semidirect
100. C127#2312semidirect
107. C4×C22⋊C4D127semidirect
120. C2×C42C2×D127semidirect
123. #95C2×D127semidirect

Maximal subgroups

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

subgroupindex
C22×(C2×D127)⋊C42
#225497672
#225497652
#225497342
#225497312
C4×C22⋊Dic1272
C4×C22⋊C4×C1272
#2312127