Finite Group Databasebeta

← #30637247

Subgroups — #30637247

Order 16256.

346 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 77 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
2C4×C22⋊Dic127C211yesmaximal, characteristic, complemented
3#22549611C214yesmaximal, complemented
4#407×C127C211yesmaximal, characteristic, complemented
5C22×D508C411yescharacteristic, complemented
6C22×C22⋊D127—21no—
7#16861183—21no—
8#16861183—21no—
9#16861183C412yescomplemented
10#16861183C2212yes—
11C2×C4⋊Dic127C411yescharacteristic, complemented
12C2×C22⋊Dic127—21no—
13C2×(C2×D127)⋊C4—21no—
14C2×C4×Dic127—21no—
15C127⋊#95C2212yescomplemented
16C23×C4×C127C2211yescharacteristic
17C2×M16×C127—21no—
18C22⋊C8×C127—21no—
19C22⋊C8×C127C2212yescomplemented
20C23×D127—21no—
21C2×D508—21no—
22C2×D508—42no—
23C2×D508C2×C412yes—
24C2×C22⋊D127—22no—
25C2×C22⋊D127—22no—
26C2×C22⋊D127—44no—
27C22×Dic127—21no—
28C22×Dic127—22no—
29C4⋊Dic127—21no—
30C4⋊Dic127C2×C412yes—
31C22⋊Dic127—24no—
32(C2×D127)⋊C4—22no—
33(C2×D127)⋊C4—22no—
34C4×Dic127—42no—
35C24×C127D411yescharacteristic
36C22×C4×C127—42no—
37C22×C4×C127C2×C411yescharacteristic
38C22×C4×C127C2×C411yescharacteristic
39C22×C4×C127D411yescharacteristic
40C22×C4×C127D411yescharacteristic
41C22×C4×C127D411yescharacteristic
42C22×C4×C127C2311yescharacteristic
43M16×C127—42no—
44C2×C8×C127—22no—
45C22×D127—22no—
46C22×D127—42no—
47C22×D127—42no—
48C22×D127—42no—
49D508—42no—
50D508—42no—
51C22⋊D127—44no—
52C22⋊D127—44no—
53C2×Dic127—24no—
54C2×Dic127—22no—
55C2×Dic127—42no—
56C2×Dic127—42no—
57C2×Dic127—42no—
58C23×C127—22no—
59C23×C127—22no—
60C23×C127—41no—
61C23×C127C22⋊C411yescharacteristic
62C23×C127C22⋊C411yescharacteristic
63C23×C127C2×D411yescharacteristic
64C2×C4×C127—22no—
65C2×C4×C127—22no—
66C2×C4×C127—22no—
67C2×C4×C127—41no—
68C2×C4×C127—41no—
69C2×C4×C127—41no—
70C2×C4×C127C22⋊C411yescharacteristic
71C2×C4×C127C22⋊C411yescharacteristic
72C2×C4×C127C22×C411yescharacteristic
73C2×C4×C127C2×D411yescharacteristic
74C8×C127—42no—
75C22×C127—21no—
76C22×C127—21no—
77C22×C127—21no—
78C22×C127—21no—
79C22×C127—21no—
80C22×C127—21no—
81C22×C127—21no—
82C22×C127—44no—
83C22×C127#10412yes—
84C22×C127C2×C22⋊C411yescharacteristic
85C22×C127AGL(1,ℤ/8)12yes—
86C2×D127—42no—
87C2×D127—42no—
88C2×D127—42no—
89C2×D127—84no—
90C4×C127—21no—
91C4×C127—21no—
92C4×C127—21no—
93C4×C127#9912yes—
94Dic127—44no—
95Dic127—42no—
96C2×C127—22no—
97C2×C127—22no—
98C2×C127—41no—
99C2×C127C2×#9911yescharacteristic
100C2×C127#41811yescharacteristic
101C2×C127C2×#10411yescharacteristic
102D127—82no—
103#1507—1271nomaximal, Sylow
104C127#150711yescharacteristic, complemented, Sylow
105C2×#121—1271no—
106#407D12711yescharacteristic, complemented
107C4×C22⋊C4—1271no—
108#327—1274no—
109C22×D4—1271no—
110C22×D4—2541no—
111C23×C4C2×D12711yescharacteristic
112C2×M16—21no—
113#121—1272no—
114#121—1272no—
115#121—2541no—
116#121—2541no—
117C2×C4⋊C4—1271no—
118C2×C22⋊C4—2541no—
119C2×C22⋊C4—2541no—
120C2×C42—2541no—
121C22⋊C8—21no—
122C22⋊C8C2×D12712yescomplemented
123#95—1272no—
124C24—2541no—
125C24D50811yescharacteristic
126C2×D4—1272no—
127C2×D4—5082no—
128C2×D4—5084no—
129C2×D4—2542no—
130C2×D4—2542no—
131C2×D4—2541no—
132C22×C4—42no—
133C22×C4—2541no—
134C22×C4—2542no—
135C22×C4C4×D12711yescharacteristic
136C22×C4C4×D12711yescharacteristic
137C22×C4C22⋊D12711yescharacteristic
138C22×C4C22⋊D12711yescharacteristic
139C22×C4D50811yescharacteristic
140C22×C4C22×D12711yescharacteristic
141M16—42no—
142C2×C8—22no—
143C4⋊C4—1272no—
144C4⋊C4—2541no—
145C22⋊C4—2544no—
146C22⋊C4—2542no—
147C22⋊C4—2542no—
148C42—5082no—
149C23—22no—
150C23—22no—
151C23—41no—
152C23—5082no—
153C23—5082no—
154C23—5082no—
155C23—2542no—
156C23(C2×D127)⋊C411yescharacteristic
157C23(C2×D127)⋊C411yescharacteristic
158C23C2×D50811yescharacteristic
159D4—5082no—
160D4—5084no—
161D4—5084no—
162D4—5082no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—5082no—
170C2×C4—5082no—
171C2×C4—5082no—
172C2×C4—2544no—
173C2×C4—2542no—
174C2×C4(C2×D127)⋊C411yescharacteristic
175C2×C4(C2×D127)⋊C411yescharacteristic
176C2×C4C2×C4×D12711yescharacteristic
177C2×C4C2×C22⋊D12711yescharacteristic
178C8—42no—
179C22—21no—
180C22—21no—
181C22—21no—
182C22—21no—
183C22—21no—
184C22—21no—
185C22—21no—
186C22—44no—
187C22—10164no—
188C22—5082no—
189C22—5082no—
190C22—5082no—
191C22#1686113512yes—
192C22C2×(C2×D127)⋊C411yescharacteristic
193C22#1686123712yes—
194C4—21no—
195C4—21no—
196C4—21no—
197C4—5084no—
198C4—5082no—
199C4#1686112312yes—
200C2—22no—
201C2—22no—
202C2—41no—
203C2—10162no—
204C2C2×#1686112311yescharacteristic
205C2#2254988211yescharacteristic
206C2C2×#1686113511yescharacteristic
subgroupG/H
centerC22C2×(C2×D127)⋊C4
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#407×C127C2
socleC22×C127C2×C22⋊C4
prime radicalC2×#16861183C2
cosocleC22×C4×C127C23
hypercenter#407D127
hypocenterC127#1507
LCSG ▷ C2×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
2. C4×C22⋊Dic127C2semidirect
3. #22549611C2semidirect
4. #407×C127C2semidirect
5. C22×D508C4semidirect
9. #16861183C4semidirect
11. C2×C4⋊Dic127C4semidirect
15. C127⋊#95C22semidirect
19. C22⋊C8×C127C22semidirect
104. C127#1507semidirect
106. #407D127semidirect
122. C22⋊C8C2×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168611832
C4×C22⋊Dic1272
#225496112
#407×C1272
#1507127