Finite Group Databasebeta

← #30639168

Subgroups — #30639168

Order 16256.

412 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 109 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×#16861183C211yesmaximal, characteristic
2C2×#16861131C212yesmaximal, complemented
3#22549736C211yesmaximal, characteristic, complemented
4#22549639C212yesmaximal, complemented
5C22×M16×C127C211yesmaximal, characteristic, complemented
6C22×D508C2211yescharacteristic
7C22×C22⋊D127—21no—
8#16861183—22no—
9#16861183C414yescomplemented
10C2×C4⋊Dic127C2212yes—
11C2×C4⋊Dic127C2211yescharacteristic
12C2×C22⋊Dic127—21no—
13C2×(C2×D127)⋊C4—21no—
14#16861131—24no—
15C127⋊#95—21no—
16C23×C4×C127C2211yescharacteristic
17C2×M16×C127—22no—
18C2×M16×C127—24no—
19C22×C8×C127C2212yes—
20C23×D127—21no—
21C2×D508—22no—
22C2×D508—21no—
23C2×D508—22no—
24C2×D508C2×C412yes—
25C2×C22⋊D127—24no—
26C2×C22⋊D127—44no—
27C22×Dic127—21no—
28C22×Dic127—22no—
29C4⋊Dic127—21no—
30C4⋊Dic127—22no—
31C4⋊Dic127—22no—
32C4⋊Dic127C2×C412yes—
33C22⋊Dic127—42no—
34(C2×D127)⋊C4—24no—
35C24×C127D411yescharacteristic
36C22×C4×C127—22no—
37C22×C4×C127—22no—
38C22×C4×C127C2×C412yes—
39C22×C4×C127D412yes—
40C22×C4×C127D411yescharacteristic
41C22×C4×C127C2311yescharacteristic
42M16×C127—28no—
43C2×C8×C127—22no—
44C2×C8×C127—22no—
45C2×C8×C127D414yescomplemented
46C22×D127—22no—
47C22×D127—42no—
48C22×D127—42no—
49C22×D127—41no—
50C22×D127—41no—
51D508—24no—
52D508—41no—
53D508—41no—
54C22⋊D127—48no—
55C2×Dic127—22no—
56C2×Dic127—42no—
57C2×Dic127—42no—
58C2×Dic127—42no—
59C2×Dic127—42no—
60C23×C127—22no—
61C23×C127—21no—
62C23×C127—22no—
63C23×C127—21no—
64C23×C127C22⋊C412yes—
65C23×C127C2×D411yescharacteristic
66C2×C4×C127—22no—
67C2×C4×C127—24no—
68C2×C4×C127—22no—
69C2×C4×C127—22no—
70C2×C4×C127C22⋊C412yes—
71C2×C4×C127C22×C411yescharacteristic
72C2×C4×C127C2×D412yes—
73C2×C4×C127C2×D411yescharacteristic
74C2×C4×C127#4112yes—
75C8×C127—24no—
76C22×C127—22no—
77C22×C127—22no—
78C22×C127—22no—
79C22×C127—42no—
80C22×C127—42no—
81C22×C127C2×C22⋊C411yescharacteristic
82C22×C127#12112yes—
83C22×C127AGL(1,ℤ/8)11yescharacteristic
84C22×C127AGL(1,ℤ/8)12yes—
85C22×C127#13711yescharacteristic
86C2×D127—42no—
87C2×D127—42no—
88C2×D127—42no—
89C2×D127—82no—
90C2×D127—82no—
91C4×C127—22no—
92C4×C127C4×D412yes—
93C4×C127#12011yescharacteristic
94C4×C127#12311yescharacteristic
95Dic127—42no—
96Dic127—82no—
97C2×C127—42no—
98C2×C127#38611yescharacteristic
99C2×C127#41711yescharacteristic
100C2×C127#41811yescharacteristic
101C2×C127#46812yes—
102C2×C127#46912yes—
103D127—82no—
104#1895—1271nomaximal, Sylow
105C127#189511yescharacteristic, complemented, Sylow
106C22×M16D12711yescharacteristic, complemented
107C2×#121—1271no—
108C2×#102—1272no—
109#380—1271no—
110#340—1272no—
111C22×D4—1271no—
112C22×D4—2541no—
113C23×C4C2×D12711yescharacteristic
114C2×M16—22no—
115C2×M16—24no—
116C22×C8C2×D12712yes—
117#121—1274no—
118#121—2542no—
119C2×C4⋊C4—1271no—
120C2×C4⋊C4—1272no—
121C2×C22⋊C4—2541no—
122C2×C22⋊C4—2541no—
123#102—2544no—
124#95—2541no—
125C24—2541no—
126C24D50811yescharacteristic
127C2×D4—1272no—
128C2×D4—5084no—
129C2×D4—2544no—
130C2×D4—2541no—
131C2×D4—2542no—
132C2×D4—2542no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—2542no—
136C22×C4—2541no—
137C22×C4C4×D12712yes—
138C22×C4C22⋊D12712yes—
139C22×C4D50811yescharacteristic
140C22×C4C22×D12711yescharacteristic
141M16—28no—
142C2×C8—22no—
143C2×C8—22no—
144C2×C8C22⋊D12714yescomplemented
145C4⋊C4—1272no—
146C4⋊C4—2542no—
147C4⋊C4—2542no—
148C4⋊C4—2541no—
149C22⋊C4—5082no—
150C22⋊C4—2544no—
151C23—21no—
152C23—22no—
153C23—21no—
154C23—22no—
155C23—5082no—
156C23—5081no—
157C23—5081no—
158C23—5082no—
159C23—2542no—
160C23(C2×D127)⋊C412yes—
161C23C2×D50811yescharacteristic
162D4—5088no—
163D4—5081no—
164D4—5081no—
165D4—2544no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—24no—
169C2×C4—22no—
170C2×C4—5082no—
171C2×C4—5082no—
172C2×C4—5082no—
173C2×C4—5082no—
174C2×C4—2542no—
175C2×C4(C2×D127)⋊C412yes—
176C2×C4C2×C4×D12711yescharacteristic
177C2×C4C2×C22⋊D12711yescharacteristic
178C2×C4C2×C22⋊D12712yes—
179C2×C4#1452324812yes—
180C8—24no—
181C22—22no—
182C22—22no—
183C22—22no—
184C22—42no—
185C22—42no—
186C22—10162no—
187C22—10162no—
188C22—5082no—
189C22—5082no—
190C22—5082no—
191C22C2×(C2×D127)⋊C411yescharacteristic
192C22#1686118312yes—
193C22#1686123712yes—
194C22#1686123711yescharacteristic
195C22#1686124211yescharacteristic
196C4—22no—
197C4—10162no—
198C4—5082no—
199C4C4×C22⋊D12712yes—
200C4#1686117811yescharacteristic
201C4#1686119211yescharacteristic
202C2—42no—
203C2—10162no—
204C2#2254975511yescharacteristic
205C2#2254987811yescharacteristic
206C2#2254988211yescharacteristic
207C2#2255012212yes—
208C2#2255012912yes—
subgroupG/H
centerC23C2×D508
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
FittingC22×M16×C127C2
socleC23×C127C2×D4
prime radicalC2×#16861183C2
cosocleC22×C4×C127C23
hypercenterC22×M16D127
hypocenterC127#1895
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×M16

Complements

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

NH ≅ G/Ntype
2. C2×#16861131C2semidirect
3. #22549736C2semidirect
4. #22549639C2semidirect
5. C22×M16×C127C2semidirect
9. #16861183C4semidirect
45. C2×C8×C127D4semidirect
105. C127#1895semidirect
106. C22×M16D127semidirect
144. C2×C8C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168611832
C2×#168611312
#225497362
#225496392
C22×M16×C1272
#1895127