Finite Group Databasebeta

← #30641110

Subgroups — #30641110

Order 16256.

1088 conjugacy classes of subgroups; only the 245 normal subgroups are shown.

normal subgroups only

This group has 1088 conjugacy classes of subgroups — too many to draw. Only its 245 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×C4×D127C211yesmaximal, characteristic
2C2×#16861165C212yesmaximal, complemented
3#22549739C211yesmaximal, characteristic, complemented
4#22549735C214yesmaximal, complemented
5#22549731C212yesmaximal
6#22549723C214yesmaximal, complemented
7#381×C127C211yesmaximal, characteristic, complemented
8C24×D127C2211yescharacteristic
9C23×Dic127C2211yescharacteristic
10C22×C4×D127—28no—
11C22×C4×D127C418yescomplemented
12C22×C4×D127C2212yes—
13C22×C4×D127C2212yes—
14#16861165—28no—
15C2×Dic127⋊C4C2214yescomplemented
16C2×C22⋊Dic127C2212yes—
17C2×(C2×D127)⋊C4C2218yes—
18C2×C4×Dic127C2214yescomplemented
19C127⋊#95C2214yescomplemented
20C23×C4×C127C2211yescharacteristic
21C2×C22⋊C4×C127C2212yes—
22#95×C127C2214yescomplemented
23C23×D127—28no—
24C23×D127—24no—
25C23×D127C2×C414yes—
26C23×D127C2312yes—
27C22×Dic127C2×C414yes—
28C22×Dic127D418yes—
29C22×Dic127C2314yes—
30C22×Dic127C2311yescharacteristic
31C22×Dic127C2311yescharacteristic
32C2×C4×D127—232no—
33C2×C4×D127—28no—
34C2×C4×D127—28no—
35C2×C4×D127C2×C418yes—
36C2×C4×D127C2×C418yes—
37Dic127⋊C4—28no—
38C22⋊Dic127—24no—
39(C2×D127)⋊C4—216no—
40C4×Dic127—28no—
41C24×C127C2311yescharacteristic
42C22×C4×C127—24no—
43C22×C4×C127C2×C414yes—
44C22×C4×C127C2314yes—
45C22×C4×C127C2312yes—
46C22⋊C4×C127—24no—
47C22×D127—232no—
48C22×D127—28no—
49C22×D127—24no—
50C22×D127—24no—
51C22×D127—216no—
52C22×D127C22×C414yes—
53C22×D127#4118yes—
54C2×Dic127—24no—
55C2×Dic127—28no—
56C2×Dic127C22⋊C4116yescomplemented
57C2×Dic127C22×C412yes—
58C2×Dic127C22×C412yes—
59C2×Dic127C2×D414yes—
60C2×Dic127C2×D414yes—
61C4×D127—216no—
62C4×D127—216no—
63C23×C127—24no—
64C23×C127C22×C412yes—
65C23×C127C2×D414yes—
66C23×C127C2411yescharacteristic
67C2×C4×C127—28no—
68C2×C4×C127—24no—
69C2×C4×C127—24no—
70C2×C4×C127—28no—
71C2×C4×C127C22×C414yes—
72C22×C127—24no—
73C22×C127—28no—
74C22×C127C2×C22⋊C414yes—
75C22×C127C23×C411yescharacteristic
76C22×C127C22×D412yes—
77C22×C127C2×#4114yes—
78C2×D127—232no—
79C2×D127—24no—
80C2×D127—24no—
81C2×D127—216no—
82C2×D127#11718yescomplemented
83C4×C127—24no—
84C4×C127—44no—
85Dic127—44no—
86Dic127C2×C22⋊C414yes—
87Dic127C2×C22⋊C414yes—
88C2×C127—24no—
89C2×C127C22×C22⋊C411yescharacteristic
90C2×C127C2×#11712yes—
91C2×C127#52514yes—
92D127—28no—
93#2280—1271nomaximal, Sylow
94C127#228011yescharacteristic, complemented, Sylow
95C24×C4—1271no—
96C2×#117—1272no—
97#381—1271no—
98#381D12711yescharacteristic, complemented
99#380—1274no—
100#379—1272no—
101C4×C22⋊C4—1274no—
102C25—1271no—
103C23×C4—1271no—
104C23×C4—1278no—
105C23×C4—1272no—
106C23×C4—1272no—
107C23×C4—2548no—
108C23×C4C2×D12711yescharacteristic
109#117—2548no—
110C2×C4⋊C4—1274no—
111C2×C22⋊C4—1272no—
112C2×C22⋊C4—1278no—
113C2×C22⋊C4C2×D12712yes—
114C2×C42—1274no—
115#95—1274no—
116#95C2×D12714yescomplemented
117C24—1272no—
118C24—1274no—
119C24—2548no—
120C24—2544no—
121C24C22×D12711yescharacteristic
122C22×C4—24no—
123C22×C4—1274no—
124C22×C4—1278no—
125C22×C4—1271no—
126C22×C4—1271no—
127C22×C4—1278no—
128C22×C4—1278no—
129C22×C4—1274no—
130C22×C4—2548no—
131C22×C4—2548no—
132C22×C4—25432no—
133C22×C4C4×D12714yes—
134C22×C4C22×D12714yes—
135C22×C4C22×D12712yes—
136C4⋊C4—2548no—
137C22⋊C4—24no—
138C22⋊C4—25416no—
139C22⋊C4—2544no—
140C42—2548no—
141C23—24no—
142C23—1274no—
143C23—1278no—
144C23—25416no—
145C23—25432no—
146C23—2544no—
147C23—2544no—
148C23—2548no—
149C23C2×C4×D12712yes—
150C23D4×D12714yes—
151C23C23×D12711yescharacteristic
152C2×C4—24no—
153C2×C4—28no—
154C2×C4—24no—
155C2×C4—28no—
156C2×C4—1272no—
157C2×C4—1272no—
158C2×C4—1274no—
159C2×C4—1274no—
160C2×C4—12716no—
161C2×C4—25416no—
162C2×C4—25416no—
163C2×C4—2544no—
164C2×C4—2548no—
165C2×C4C2×C4×D12714yes—
166C22—24no—
167C22—28no—
168C22—1278no—
169C22—25432no—
170C22—2544no—
171C22—2544no—
172C22—25416no—
173C22C22⋊C4×D12714yes—
174C22C22×C4×D12711yescharacteristic
175C22C2×D4×D12712yes—
176C22#41×D12714yes—
177C4—24no—
178C4—44no—
179C4—1274no—
180C4—1274no—
181C4—5084no—
182C2—24no—
183C2—2548no—
184C2C2×C22⋊C4×D12711yescharacteristic
185C2#117×D12712yes—
186C2#2255038414yes—
subgroupG/H
center, FrattiniC23C23×D127
commutatorC22×C127C23×C4
Fitting#381×C127C2
socle, cosocleC23×C127C24
prime radicalC24×D127C22
hypercenter#381D127
hypocenterC127#2280
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
2. C2×#16861165C2semidirect
3. #22549739C2semidirect
4. #22549735C2semidirect
6. #22549723C2semidirect
7. #381×C127C2semidirect
11. C22×C4×D127C4semidirect
15. C2×Dic127⋊C4C22semidirect
18. C2×C4×Dic127C22semidirect
19. C127⋊#95C22semidirect
22. #95×C127C22semidirect
56. C2×Dic127C22⋊C4semidirect
82. C2×D127#117semidirect
94. C127#2280semidirect
98. #381D127semidirect
116. #95C2×D127semidirect

Maximal subgroups

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

subgroupindex
C23×C4×D1272
C2×#168611652
#225497392
#225497352
#225497312
#225497232
#381×C1272
#2280127