Finite Group Databasebeta

← C2×#22550204 #30648605

Subgroups — C2×#22550204 #30648605

Order 16256.

460 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 165 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×C4⋊Dic127C211yesmaximal, characteristic
2#22550204C218yesmaximal, direct product factor
3C2×#16861140C212yesmaximal, complemented
4C2×#16861133C212yesmaximal, complemented
5C2×C22⋊(C127⋊C8)C211yesmaximal, characteristic
6C2×C22⋊Q8×C127C211yesmaximal, characteristic
7C23×Dic127—21no—
8C22×C127⋊C8C2212yes—
9C2×C4⋊Dic127—21no—
10C2×C4⋊Dic127—22no—
11C2×C4⋊Dic127C2212yes—
12C2×C4⋊Dic127C2214yes—
13#16861140C2218yescomplemented
14#16861133C2218yescomplemented
15C22⋊(C127⋊C8)C2214yes—
16C22×Q8×C127C2211yescharacteristic
17C23×C4×C127C2211yescharacteristic
18C22⋊Q8×C127—22no—
19C22⋊Q8×C127C2214yes—
20C2×C4⋊C4×C127—21no—
21C2×C4⋊C4×C127C2211yescharacteristic
22C2×C22⋊C4×C127—21no—
23C22×Dic127—24no—
24C22×Dic127—22no—
25C22×Dic127—44no—
26C2×C127⋊C8—24no—
27C2×C127⋊C8C2314yes—
28C4⋊Dic127—24no—
29C4⋊Dic127—22no—
30C4⋊Dic127C2314yes—
31C24×C127D411yescharacteristic
32C2×Q8×C127—21no—
33C2×Q8×C127—24no—
34C2×Q8×C127C2312yes—
35C22×C4×C127—24no—
36C22×C4×C127—21no—
37C22×C4×C127—21no—
38C22×C4×C127D411yescharacteristic
39C22×C4×C127D412yes—
40C22×C4×C127C2312yes—
41C22×C4×C127C2311yescharacteristic
42C4⋊C4×C127—24no—
43C4⋊C4×C127—21no—
44C4⋊C4×C127C2312yes—
45C22⋊C4×C127—24no—
46C2×Dic127—24no—
47C2×Dic127—48no—
48C2×Dic127—44no—
49C127⋊C8—24no—
50C23×C127—22no—
51C23×C127—22no—
52C23×C127SD1614yes—
53C23×C127C2×D412yes—
54C23×C127C2×D411yescharacteristic
55Q8×C127—24no—
56Q8×C127—42no—
57C2×C4×C127—22no—
58C2×C4×C127—24no—
59C2×C4×C127—24no—
60C2×C4×C127—22no—
61C2×C4×C127—22no—
62C2×C4×C127—42no—
63C2×C4×C127—42no—
64C2×C4×C127C2×D412yes—
65C2×C4×C127C2×D411yescharacteristic
66C2×C4×C127#4114yes—
67C2×C4×C127C2411yescharacteristic
68C22×C127—22no—
69C22×C127—22no—
70C22×C127—28no—
71C22×C127#12312yes—
72C22×C127C2×SD1614yes—
73C22×C127C2×SD1612yes—
74C22×C127#13712yes—
75C22×C127C22×D411yescharacteristic
76C4×C127—22no—
77C4×C127—42no—
78C4×C127—42no—
79C4×C127#12312yes—
80C4×C127C2×#4112yes—
81Dic127—44no—
82C2×C127—24no—
83C2×C127#48314yescomplemented
84C2×C127C2×#12311yescharacteristic
85C2×C127C22×SD1611yescharacteristic
86C2×C127C2×#13711yescharacteristic
87C2×#483—1271nomaximal, Sylow
88C127C2×#48311yescharacteristic, complemented, Sylow
89C2×C22⋊Q8D12711yescharacteristic
90C22×C4⋊C4—1271no—
91#483—1278no—
92C2×#106—1272no—
93C2×#103—1272no—
94C2×C22⋊C8—1271no—
95C22×Q8C2×D12711yescharacteristic
96C23×C4—2541no—
97C23×C4C2×D12711yescharacteristic
98C22×C8—1272no—
99C22⋊Q8—22no—
100C22⋊Q8C2×D12714yes—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—1272no—
103C2×C4⋊C4—1274no—
104C2×C4⋊C4—2542no—
105C2×C4⋊C4—2541no—
106C2×C4⋊C4C2×D12711yescharacteristic
107C2×C22⋊C4—21no—
108#106—1278no—
109#103—1278no—
110C22⋊C8—1274no—
111C24C22⋊D12711yescharacteristic
112C2×Q8—21no—
113C2×Q8—24no—
114C2×Q8C22×D12712yes—
115C22×C4—24no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—5084no—
119C22×C4—2544no—
120C22×C4—2542no—
121C22×C4C22⋊D12712yes—
122C22×C4C22⋊D12711yescharacteristic
123C22×C4C22×D12712yes—
124C22×C4C22×D12711yescharacteristic
125C2×C8—1274no—
126C2×C8—2544no—
127C4⋊C4—24no—
128C4⋊C4—21no—
129C4⋊C4—1274no—
130C4⋊C4—2544no—
131C4⋊C4—2542no—
132C4⋊C4C22×D12712yes—
133C22⋊C4—24no—
134C23—22no—
135C23—22no—
136C23#1452323514yes—
137C23C2×C22⋊D12712yes—
138C23C2×C22⋊D12711yescharacteristic
139Q8—24no—
140Q8—42no—
141C2×C4—22no—
142C2×C4—24no—
143C2×C4—24no—
144C2×C4—22no—
145C2×C4—22no—
146C2×C4—42no—
147C2×C4—42no—
148C2×C4—5088no—
149C2×C4—5084no—
150C2×C4—2544no—
151C2×C4C2×C22⋊D12712yes—
152C2×C4C2×C22⋊D12711yescharacteristic
153C2×C4#1452324714yes—
154C2×C4C23×D12711yescharacteristic
155C8—2544no—
156C22—22no—
157C22—22no—
158C22—28no—
159C22#1686119012yes—
160C22C2×#1452323514yes—
161C22C2×#1452323512yes—
162C22#1686123912yes—
163C22C22×C22⋊D12711yescharacteristic
164C4—22no—
165C4—42no—
166C4—42no—
167C4—5084no—
168C4#1686119012yes—
169C4C2×#1452324712yes—
170C2—24no—
171C2#2255020414yesdirect product factor
172C2C2×#1686119011yescharacteristic
173C2C22×#1452323511yescharacteristic
174C2C2×#1686123911yescharacteristic
subgroupG/H
centerC23C2×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×C22⋊Q8×C127C2
socleC23×C127C2×D4
prime radicalC24×C127D4
hypercenterC2×C22⋊Q8D127
hypocenterC127C2×#483
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. #22550204C2direct (N × H)
3. C2×#16861140C2semidirect
4. C2×#16861133C2semidirect
13. #16861140C22semidirect
14. #16861133C22semidirect
83. C2×C127#483semidirect
88. C127C2×#483semidirect
171. C2#22550204direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4⋊Dic1272
#225502042
C2×#168611402
C2×#168611332
C2×C22⋊(C127⋊C8)2
C2×C22⋊Q8×C1272
C2×#483127