Finite Group Databasebeta

← C2×#22549916 #30647313

Subgroups — C2×#22549916 #30647313

Order 16256.

536 conjugacy classes of subgroups; only the 269 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 269 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×#14523232C211yesmaximal, characteristic
2C2×#16861168C211yesmaximal, characteristic
3#22549916C218yesmaximal, direct product factor
4C2×#16861142C212yesmaximal, complemented
5C2×#16861140C212yesmaximal, complemented
6C22×C4⋊C4×C127C211yesmaximal, characteristic
7C2×#14523232C418yescomplemented
8C2×#14523232C2214yes—
9C22×C127⋊C8C2212yes—
10#16861168—22no—
11#16861168C2214yes—
12C2×C4⋊Dic127C2211yescharacteristic
13C2×C4⋊Dic127C2211yescharacteristic
14C2×C22⋊Dic127—21no—
15C2×C4×Dic127—21no—
16#16861142C2218yescomplemented
17#16861140C2218yescomplemented
18C23×C4×C127—21no—
19C23×C4×C127C2211yescharacteristic
20C2×C4⋊C4×C127—21no—
21C2×C4⋊C4×C127—22no—
22C2×C4⋊C4×C127C2212yes—
23C2×C4⋊C4×C127C2214yes—
24C22×Dic127—21no—
25C22×Dic127—21no—
26#14523232C2×C4116yescomplemented
27C2×C127⋊C8C2×C418yescomplemented
28C2×C127⋊C8C2314yes—
29C4⋊Dic127—21no—
30C4⋊Dic127—21no—
31C4⋊Dic127C2312yes—
32C4⋊Dic127C2312yes—
33C22⋊Dic127—24no—
34C4×Dic127—24no—
35C24×C127D411yescharacteristic
36C22×C4×C127—24no—
37C22×C4×C127—22no—
38C22×C4×C127—44no—
39C22×C4×C127C2×C414yes—
40C22×C4×C127D411yescharacteristic
41C22×C4×C127D412yes—
42C22×C4×C127Q814yes—
43C22×C4×C127C2311yescharacteristic
44C22×C4×C127C2312yes—
45C4⋊C4×C127—24no—
46C4⋊C4×C127—22no—
47C4⋊C4×C127C2314yes—
48C2×Dic127—22no—
49C2×Dic127—22no—
50C2×Dic127—42no—
51C2×Dic127—42no—
52C127⋊C8C22×C418yescomplemented
53C23×C127—22no—
54C23×C127—22no—
55C23×C127C4⋊C414yes—
56C23×C127C2×D412yes—
57C23×C127C2×D411yescharacteristic
58C2×C4×C127—24no—
59C2×C4×C127—48no—
60C2×C4×C127—44no—
61C2×C4×C127C4⋊C418yes—
62C2×C4×C127C4⋊C414yes—
63C2×C4×C127C22×C412yes—
64C2×C4×C127C22×C414yes—
65C2×C4×C127C2×D412yes—
66C2×C4×C127C2×D411yescharacteristic
67C2×C4×C127C2×Q814yes—
68C2×C4×C127C2×Q812yes—
69C2×C4×C127C2411yescharacteristic
70C22×C127—22no—
71C22×C127—22no—
72C22×C127—28no—
73C22×C127C2×C4⋊C414yes—
74C22×C127C2×C4⋊C412yes—
75C22×C127AGL(1,ℤ/8)12yes—
76C22×C127#13712yes—
77C22×C127C22×D411yescharacteristic
78C4×C127—44no—
79C4×C127C2×C4⋊C412yes—
80C4×C127C2×C4⋊C414yes—
81C4×C127C23×C411yescharacteristic
82C4×C127C22×Q811yescharacteristic
83Dic127—42no—
84Dic127—42no—
85C2×C127—24no—
86C2×C127#42814yescomplemented
87C2×C127C22×C4⋊C411yescharacteristic
88C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
89C2×C127C2×#13711yescharacteristic
90C2×#428—1271nomaximal, Sylow
91C127C2×#42811yescharacteristic, complemented, Sylow
92C22×M16—1271no—
93C2×#117—1271no—
94C22×C4⋊C4D12711yescharacteristic
95#428—1278no—
96C2×#107—1272no—
97C2×#106—1272no—
98C23×C4—21no—
99C23×C4C2×D12711yescharacteristic
100C2×M16—1278no—
101C2×M16—1274no—
102C22×C8—1272no—
103#117—1274no—
104#117—2542no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—1271no—
108C2×C4⋊C4—1271no—
109C2×C4⋊C4C2×D12712yes—
110C2×C4⋊C4C2×D12714yes—
111C2×C22⋊C4—2541no—
112C2×C42—2541no—
113#107—1278no—
114#106—1278no—
115C24C22⋊D12711yescharacteristic
116C22×C4—24no—
117C22×C4—22no—
118C22×C4—44no—
119C22×C4—2541no—
120C22×C4—2541no—
121C22×C4C4×D12714yes—
122C22×C4C22⋊D12712yes—
123C22×C4C22⋊D12711yescharacteristic
124C22×C4Dic25414yes—
125C22×C4C22×D12712yes—
126C22×C4C22×D12711yescharacteristic
127M16—12716no—
128C2×C8—1278no—
129C2×C8—1274no—
130C4⋊C4—24no—
131C4⋊C4—22no—
132C4⋊C4—1272no—
133C4⋊C4—1272no—
134C4⋊C4—2541no—
135C4⋊C4—2541no—
136C4⋊C4C22×D12714yes—
137C22⋊C4—2544no—
138C42—2544no—
139C23—22no—
140C23—22no—
141C23Dic127⋊C414yes—
142C23C2×C22⋊D12712yes—
143C23C2×C22⋊D12711yescharacteristic
144C2×C4—24no—
145C2×C4—48no—
146C2×C4—44no—
147C2×C4—5082no—
148C2×C4—5082no—
149C2×C4—2542no—
150C2×C4—2542no—
151C2×C4Dic127⋊C418yes—
152C2×C4Dic127⋊C414yes—
153C2×C4C2×C4×D12712yes—
154C2×C4C2×C4×D12714yes—
155C2×C4C2×C22⋊D12712yes—
156C2×C4C2×C22⋊D12711yescharacteristic
157C2×C4C2×Dic25414yes—
158C2×C4C2×Dic25412yes—
159C2×C4C23×D12711yescharacteristic
160C8—1278no—
161C22—22no—
162C22—22no—
163C22—28no—
164C22C2×Dic127⋊C414yes—
165C22C2×Dic127⋊C412yes—
166C22#1686123512yes—
167C22#1686124012yes—
168C22C22×C22⋊D12711yescharacteristic
169C4—44no—
170C4—5082no—
171C4—5082no—
172C4C2×Dic127⋊C412yes—
173C4C2×Dic127⋊C414yes—
174C4C22×C4×D12711yescharacteristic
175C4C22×Dic25411yescharacteristic
176C2—24no—
177C2#2254991614yesdirect product factor
178C2C22×Dic127⋊C411yescharacteristic
179C2C2×#1686123511yescharacteristic
180C2C2×#1686124011yescharacteristic
subgroupG/H
centerC23C2×C22⋊D127
commutatorC4×C127C23×C4
FrattiniC2×C4C23×D127
FittingC22×C4⋊C4×C127C2
socleC23×C127C2×D4
prime radicalC24×C127D4
cosocleC2×C4×C127C24
hypercenterC22×C4⋊C4D127
hypocenterC127C2×#428
LCSG ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #22549916C2direct (N × H)
4. C2×#16861142C2semidirect
5. C2×#16861140C2semidirect
7. C2×#14523232C4semidirect
16. #16861142C22semidirect
17. #16861140C22semidirect
26. #14523232C2×C4semidirect
27. C2×C127⋊C8C2×C4semidirect
52. C127⋊C8C22×C4semidirect
86. C2×C127#428semidirect
91. C127C2×#428semidirect
177. C2#22549916direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145232322
C2×#168611682
#225499162
C2×#168611422
C2×#168611402
C22×C4⋊C4×C1272
C2×#428127