Finite Group Databasebeta

← #30641311

Subgroups — #30641311

Order 16256.

1104 conjugacy classes of subgroups; only the 229 normal subgroups are shown.

normal subgroups only

This group has 1104 conjugacy classes of subgroups — too many to draw. Only its 229 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
1C23×C22⋊D127C211yesmaximal, characteristic
2C2×#16861165C212yesmaximal, complemented
3#22549776C218yesmaximal, complemented
4#22549732C211yesmaximal, characteristic, complemented
5#22549731C212yesmaximal, complemented
6#379×C127C211yesmaximal, characteristic, complemented
7C24×D127C2211yescharacteristic
8C22×C22⋊D127—28no—
9C22×C22⋊D127C418yescomplemented
10C22×C22⋊D127C2214yes—
11C23×Dic127C2211yescharacteristic
12#16861165—28no—
13C2×Dic127⋊C4C2214yescomplemented
14C2×C22⋊Dic127C2212yes—
15C2×C22⋊Dic127C2214yescomplemented
16C2×(C2×D127)⋊C4C2218yescomplemented
17C2×C4×Dic127C2214yescomplemented
18C25×C127C2211yescharacteristic
19C2×C22⋊C4×C127C2212yes—
20C2×C22⋊C4×C127C2214yescomplemented
21C23×D127—24no—
22C23×D127—44no—
23C23×D127C2×C414yes—
24C23×D127C2312yes—
25C2×C22⋊D127—232no—
26C2×C22⋊D127—28no—
27C2×C22⋊D127—28no—
28C2×C22⋊D127C2×C4116yes—
29C22×Dic127C2×C414yes—
30C22×Dic127D418yes—
31C22×Dic127C2314yes—
32C22×Dic127C2312yes—
33Dic127⋊C4—28no—
34C22⋊Dic127—24no—
35C22⋊Dic127—44no—
36(C2×D127)⋊C4—48no—
37C4×Dic127—28no—
38C24×C127—24no—
39C24×C127—44no—
40C24×C127C2×C414yes—
41C24×C127C2312yes—
42C24×C127C2311yescharacteristic
43C22×C4×C127C2314yes—
44C22⋊C4×C127—24no—
45C22⋊C4×C127—44no—
46C22×D127—28no—
47C22×D127—24no—
48C22×D127—28no—
49C22×D127—48no—
50C22×D127—416no—
51C22×D127C22×C414yes—
52C22⋊D127—232no—
53C2×Dic127—24no—
54C2×Dic127—28no—
55C2×Dic127C22⋊C4116yescomplemented
56C2×Dic127C22×C414yes—
57C2×Dic127C2×D414yes—
58C2×Dic127C2×D414yes—
59C23×C127—24no—
60C23×C127—28no—
61C23×C127—24no—
62C23×C127—24no—
63C23×C127—24no—
64C23×C127—48no—
65C23×C127—416no—
66C23×C127C22×C414yes—
67C23×C127C22×C412yes—
68C23×C127C2×D414yes—
69C23×C127C2411yescharacteristic
70C2×C4×C127—24no—
71C2×C4×C127—28no—
72C22×C127—24no—
73C22×C127—28no—
74C22×C127—24no—
75C22×C127—24no—
76C22×C127—24no—
77C22×C127—416no—
78C22×C127—48no—
79C22×C127C2×C22⋊C414yes—
80C22×C127C23×C411yescharacteristic
81C22×C127C22×D412yes—
82C22×C127H(5,𝔽2)14yes—
83C2×D127—24no—
84C2×D127—28no—
85C2×D127—416no—
86C2×D127—48no—
87C4×C127—44no—
88Dic127—44no—
89Dic127C2×C22⋊C418yes—
90C2×C127—24no—
91C2×C127—44no—
92C2×C127C22×C22⋊C411yescharacteristic
93C2×C127#51812yes—
94C2×C127#53514yes—
95D127—44no—
96#2309—1271nomaximal, Sylow
97C127#230911yescharacteristic, complemented, Sylow
98C23×D4—1271no—
99C2×#117—1272no—
100#390—1278no—
101#379—1271no—
102#379—1272no—
103#379D12711yescharacteristic, complemented
104C25—1271no—
105C25C2×D12711yescharacteristic
106C22×D4—1278no—
107C22×D4—1274no—
108C22×D4—2548no—
109C23×C4—1271no—
110#117—2548no—
111C2×C4⋊C4—1274no—
112C2×C22⋊C4—1274no—
113C2×C22⋊C4—1278no—
114C2×C22⋊C4—1272no—
115C2×C22⋊C4C2×D12712yes—
116C2×C22⋊C4C2×D12714yescomplemented
117C2×C42—1274no—
118C24—24no—
119C24—44no—
120C24—1274no—
121C24—1272no—
122C24—5084no—
123C24—2544no—
124C24C4×D12714yes—
125C24C22×D12712yes—
126C24C22×D12711yescharacteristic
127C2×D4—12716no—
128C2×D4—2548no—
129C2×D4—2548no—
130C2×D4—25432no—
131C22×C4—1274no—
132C22×C4—1278no—
133C22×C4—1272no—
134C22×C4—1274no—
135C22×C4C22×D12714yes—
136C4⋊C4—2548no—
137C22⋊C4—24no—
138C22⋊C4—44no—
139C22⋊C4—5088no—
140C22⋊C4—5084no—
141C22⋊C4—2544no—
142C42—2548no—
143C23—24no—
144C23—24no—
145C23—24no—
146C23—24no—
147C23—28no—
148C23—48no—
149C23—416no—
150C23—1274no—
151C23—50816no—
152C23—5088no—
153C23—2544no—
154C23—2548no—
155C23—2548no—
156C23C2×C4×D12714yes—
157C23C2×C4×D12712yes—
158C23D4×D12714yes—
159C23C23×D12711yescharacteristic
160D4—25432no—
161C2×C4—24no—
162C2×C4—28no—
163C2×C4—1274no—
164C2×C4—1274no—
165C2×C4—1274no—
166C2×C4—12716no—
167C2×C4—2544no—
168C2×C4—2548no—
169C22—24no—
170C22—28no—
171C22—24no—
172C22—24no—
173C22—24no—
174C22—416no—
175C22—48no—
176C22—50816no—
177C22—5088no—
178C22—2544no—
179C22—2548no—
180C22C22⋊C4×D12714yes—
181C22C22×C4×D12711yescharacteristic
182C22C2×D4×D12712yes—
183C22#1686125414yes—
184C4—44no—
185C4—1278no—
186C4—5084no—
187C2—24no—
188C2—44no—
189C2—5084no—
190C2C2×C22⋊C4×D12711yescharacteristic
191C2#2255034912yes—
192C2#2255044114yes—
subgroupG/H
center, FrattiniC23C23×D127
commutatorC22×C127C23×C4
Fitting#379×C127C2
socle, cosocleC23×C127C24
prime radicalC23×C22⋊D127C2
hypercenter#379D127
hypocenterC127#2309
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C23 ◁ #379

Complements

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

NH ≅ G/Ntype
2. C2×#16861165C2semidirect
3. #22549776C2semidirect
4. #22549732C2semidirect
5. #22549731C2semidirect
6. #379×C127C2semidirect
9. C22×C22⋊D127C4semidirect
13. C2×Dic127⋊C4C22semidirect
15. C2×C22⋊Dic127C22semidirect
16. C2×(C2×D127)⋊C4C22semidirect
17. C2×C4×Dic127C22semidirect
20. C2×C22⋊C4×C127C22semidirect
55. C2×Dic127C22⋊C4semidirect
97. C127#2309semidirect
103. #379D127semidirect
116. C2×C22⋊C4C2×D127semidirect

Maximal subgroups

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

subgroupindex
C23×C22⋊D1272
C2×#168611652
#225497762
#225497322
#225497312
#379×C1272
#2309127