Finite Group Databasebeta

← C2×#22550235 #30649058

Subgroups — C2×#22550235 #30649058

Order 16256.

436 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 149 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×#16861209C211yesmaximal, characteristic, complemented
2#22550235C218yesmaximal, direct product factor
3C2×#16861133C212yesmaximal, complemented
4C2×#16861130C212yesmaximal
5C2×#16861120C211yesmaximal, characteristic, complemented
6C2×#124×C127C211yesmaximal, characteristic
7C22×Dic254—21no—
8C22×C127⋊C8C2212yes—
9#16861209—22no—
10#16861209C2214yescomplemented
11C2×C4⋊Dic127—21no—
12C2×C4⋊Dic127C2212yes—
13#16861133C2218yescomplemented
14#16861130C2218yes—
15#16861120C2214yescomplemented
16C22×Q8×C127C2211yescharacteristic
17C22×D4×C127C2211yescharacteristic
18#124×C127—22no—
19#124×C127C2214yes—
20C2×C22⋊C4×C127—21no—
21C2×C22⋊C4×C127—21no—
22C2×C42×C127C2211yescharacteristic
23C2×Dic254—24no—
24C2×Dic254—44no—
25C22×Dic127—22no—
26C2×C127⋊C8—24no—
27C2×C127⋊C8C2314yes—
28C4⋊Dic127—22no—
29C4⋊Dic127—24no—
30C4⋊Dic127C2314yes—
31C24×C127—21no—
32C2×Q8×C127—21no—
33C2×Q8×C127—24no—
34C2×Q8×C127C2312yes—
35C2×D4×C127—21no—
36C2×D4×C127—24no—
37C2×D4×C127C2312yes—
38C22×C4×C127—21no—
39C22×C4×C127D411yescharacteristic
40C22×C4×C127D411yescharacteristic
41C22×C4×C127C2311yescharacteristic
42C22⋊C4×C127—24no—
43C22⋊C4×C127—24no—
44C42×C127D412yes—
45C42×C127C2312yes—
46Dic254—48no—
47C2×Dic127—24no—
48C2×Dic127—44no—
49C127⋊C8—24no—
50C23×C127—22no—
51C23×C127—42no—
52C23×C127—42no—
53C23×C127—42no—
54C23×C127C2×D411yescharacteristic
55Q8×C127—24no—
56Q8×C127—42no—
57D4×C127—24no—
58D4×C127—42no—
59C2×C4×C127—22no—
60C2×C4×C127—22no—
61C2×C4×C127—22no—
62C2×C4×C127—42no—
63C2×C4×C127C2×D412yes—
64C2×C4×C127C2×D412yes—
65C2×C4×C127C2×D411yescharacteristic
66C2×C4×C127#4114yes—
67C2×C4×C127C2411yescharacteristic
68C22×C127—42no—
69C22×C127—42no—
70C22×C127—42no—
71C22×C127—84no—
72C22×C127#12412yes—
73C22×C127AGL(1,ℤ/8)12yes—
74C22×C127#13712yes—
75C22×C127C22×D411yescharacteristic
76C4×C127—22no—
77C4×C127—22no—
78C4×C127—42no—
79C4×C127#12412yes—
80C4×C127C2×#4112yes—
81Dic127—44no—
82C2×C127—82no—
83C2×C127#48914yescomplemented
84C2×C127C2×#12411yescharacteristic
85C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
86C2×C127C2×#13711yescharacteristic
87C2×#489—1271nomaximal, Sylow
88C127C2×#48911yescharacteristic, complemented, Sylow
89C2×C4⋊Q8—1271no—
90C2×#124D12711yescharacteristic
91#489—1278no—
92C2×#103—1272no—
93C2×#102—1272no—
94C2×#97—1271no—
95C22×Q8—2541no—
96C22×Q8C2×D12711yescharacteristic
97C22×D4C2×D12711yescharacteristic
98C22×C8—1272no—
99C4⋊Q8—1274no—
100C4⋊Q8—2542no—
101#124—22no—
102#124C2×D12714yes—
103C2×C4⋊C4—1272no—
104C2×C4⋊C4—2541no—
105C2×C22⋊C4—21no—
106C2×C22⋊C4—21no—
107C2×C42C2×D12711yescharacteristic
108#103—1278no—
109#102—1278no—
110#97—1274no—
111C24—21no—
112C2×Q8—21no—
113C2×Q8—24no—
114C2×Q8—5084no—
115C2×Q8—2544no—
116C2×Q8C22×D12712yes—
117C2×D4—21no—
118C2×D4—24no—
119C2×D4C22×D12712yes—
120C22×C4—21no—
121C22×C4—2542no—
122C22×C4C22⋊D12711yescharacteristic
123C22×C4C22⋊D12711yescharacteristic
124C22×C4C22×D12711yescharacteristic
125C2×C8—1274no—
126C2×C8—2544no—
127C4⋊C4—1274no—
128C4⋊C4—2542no—
129C4⋊C4—2544no—
130C22⋊C4—24no—
131C22⋊C4—24no—
132C42C22⋊D12712yes—
133C42C22×D12712yes—
134C23—22no—
135C23—42no—
136C23—42no—
137C23—42no—
138C23C2×C22⋊D12711yescharacteristic
139Q8—24no—
140Q8—42no—
141Q8—5088no—
142D4—24no—
143D4—42no—
144C2×C4—22no—
145C2×C4—22no—
146C2×C4—22no—
147C2×C4—42no—
148C2×C4—5084no—
149C2×C4—2544no—
150C2×C4C2×C22⋊D12712yes—
151C2×C4C2×C22⋊D12712yes—
152C2×C4C2×C22⋊D12711yescharacteristic
153C2×C4#1452324714yes—
154C2×C4C23×D12711yescharacteristic
155C8—2544no—
156C22—42no—
157C22—42no—
158C22—42no—
159C22—84no—
160C22#1686119612yes—
161C22#1686123412yes—
162C22#1686123912yes—
163C22C22×C22⋊D12711yescharacteristic
164C4—22no—
165C4—22no—
166C4—42no—
167C4—5084no—
168C4#1686119612yes—
169C4C2×#1452324712yes—
170C2—82no—
171C2#2255023514yesdirect product factor
172C2C2×#1686119611yescharacteristic
173C2C2×#1686123411yescharacteristic
174C2C2×#1686123911yescharacteristic
subgroupG/H
centerC23C2×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×#124×C127C2
socleC23×C127C2×D4
prime radicalC22×D4×C127C22
hypercenterC2×#124D127
hypocenterC127C2×#489
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. C2×#16861209C2semidirect
2. #22550235C2direct (N × H)
3. C2×#16861133C2semidirect
5. C2×#16861120C2semidirect
10. #16861209C22semidirect
13. #16861133C22semidirect
15. #16861120C22semidirect
83. C2×C127#489semidirect
88. C127C2×#489semidirect
171. C2#22550235direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168612092
#225502352
C2×#168611332
C2×#168611302
C2×#168611202
C2×#124×C1272
C2×#489127