Finite Group Databasebeta

← C2×#22550206 #30648607

Subgroups — C2×#22550206 #30648607

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
1C2×#16861188C211yesmaximal, characteristic
2#22550206C218yesmaximal, direct product factor
3C2×#16861140C212yesmaximal, complemented
4C2×#16861132C212yesmaximal, complemented
5C2×C22⋊(C127⋊C8)C211yesmaximal, characteristic
6C22×C4⋊C4×C127C211yesmaximal, characteristic
7C22×Dic254C2211yescharacteristic
8C22×C127⋊C8C2212yes—
9#16861188—22no—
10#16861188C2214yes—
11C2×C4⋊Dic127C2211yescharacteristic
12C2×Dic127⋊C4—21no—
13C2×C22⋊Dic127—21no—
14#16861140C2218yescomplemented
15#16861132C2218yescomplemented
16C22⋊(C127⋊C8)C2214yes—
17C23×C4×C127—21no—
18C23×C4×C127C2211yescharacteristic
19C2×C4⋊C4×C127—22no—
20C2×C4⋊C4×C127—21no—
21C2×C4⋊C4×C127C2212yes—
22C2×C4⋊C4×C127C2214yes—
23C2×Dic254—21no—
24C2×Dic254—24no—
25C2×Dic254C2312yes—
26C22×Dic127—21no—
27C22×Dic127—21no—
28C2×C127⋊C8—24no—
29C2×C127⋊C8C2314yes—
30C4⋊Dic127—21no—
31C4⋊Dic127C2312yes—
32Dic127⋊C4—24no—
33C22⋊Dic127—24no—
34C24×C127D411yescharacteristic
35C22×C4×C127—24no—
36C22×C4×C127—24no—
37C22×C4×C127—22no—
38C22×C4×C127—44no—
39C22×C4×C127D412yes—
40C22×C4×C127D411yescharacteristic
41C22×C4×C127C2312yes—
42C22×C4×C127C2311yescharacteristic
43C4⋊C4×C127—24no—
44C4⋊C4×C127—22no—
45C4⋊C4×C127C2314yes—
46Dic254—24no—
47Dic254—42no—
48C2×Dic127—22no—
49C2×Dic127—22no—
50C2×Dic127—42no—
51C2×Dic127—42no—
52C127⋊C8—24no—
53C23×C127—22no—
54C23×C127—22no—
55C23×C127SD1614yes—
56C23×C127C2×D412yes—
57C23×C127C2×D411yescharacteristic
58C2×C4×C127—22no—
59C2×C4×C127—24no—
60C2×C4×C127—24no—
61C2×C4×C127—24no—
62C2×C4×C127—48no—
63C2×C4×C127—44no—
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—44no—
78C4×C127#12312yes—
79C4×C127C2×#4112yes—
80Dic127—42no—
81Dic127—42no—
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⋊Q8—1271no—
90C22×C4⋊C4D12711yescharacteristic
91#483—1278no—
92C2×#106—1272no—
93C2×#103—1272no—
94C2×C22⋊C8—1271no—
95C22×Q8—1271no—
96C23×C4—21no—
97C23×C4C2×D12711yescharacteristic
98C22×C8—1272no—
99C22⋊Q8—1274no—
100C22⋊Q8—2542no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—1271no—
104C2×C4⋊C4—2541no—
105C2×C4⋊C4C2×D12712yes—
106C2×C4⋊C4C2×D12714yes—
107C2×C22⋊C4—2541no—
108#106—1278no—
109#103—1278no—
110C22⋊C8—1274no—
111C24C22⋊D12711yescharacteristic
112C2×Q8—1272no—
113C2×Q8—2544no—
114C2×Q8—2541no—
115C22×C4—24no—
116C22×C4—24no—
117C22×C4—22no—
118C22×C4—44no—
119C22×C4—2541no—
120C22×C4—2541no—
121C22×C4C22⋊D12712yes—
122C22×C4C22⋊D12711yescharacteristic
123C22×C4C22×D12712yes—
124C22×C4C22×D12711yescharacteristic
125C2×C8—1274no—
126C2×C8—2544no—
127C4⋊C4—24no—
128C4⋊C4—22no—
129C4⋊C4—1272no—
130C4⋊C4—2544no—
131C4⋊C4—2541no—
132C4⋊C4C22×D12714yes—
133C22⋊C4—2544no—
134C23—22no—
135C23—22no—
136C23#1452323614yes—
137C23C2×C22⋊D12712yes—
138C23C2×C22⋊D12711yescharacteristic
139Q8—5082no—
140Q8—2544no—
141C2×C4—22no—
142C2×C4—24no—
143C2×C4—24no—
144C2×C4—24no—
145C2×C4—48no—
146C2×C4—44no—
147C2×C4—5082no—
148C2×C4—5082no—
149C2×C4—2542no—
150C2×C4—2542no—
151C2×C4C2×C22⋊D12712yes—
152C2×C4C2×C22⋊D12711yescharacteristic
153C2×C4#1452324814yes—
154C2×C4C23×D12711yescharacteristic
155C8—2544no—
156C22—22no—
157C22—22no—
158C22—28no—
159C22#1686119212yes—
160C22C2×#1452323614yes—
161C22C2×#1452323612yes—
162C22#1686124012yes—
163C22C22×C22⋊D12711yescharacteristic
164C4—22no—
165C4—44no—
166C4—5082no—
167C4—5082no—
168C4#1686119212yes—
169C4C2×#1452324812yes—
170C2—24no—
171C2#2255020614yesdirect product factor
172C2C2×#1686119211yescharacteristic
173C2C22×#1452323611yescharacteristic
174C2C2×#1686124011yescharacteristic
subgroupG/H
centerC23C2×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC22×C4⋊C4×C127C2
socleC23×C127C2×D4
prime radicalC24×C127D4
hypercenterC22×C4⋊C4D127
hypocenterC127C2×#483
LCSG ▷ C2×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
2. #22550206C2direct (N × H)
3. C2×#16861140C2semidirect
4. C2×#16861132C2semidirect
14. #16861140C22semidirect
15. #16861132C22semidirect
83. C2×C127#483semidirect
88. C127C2×#483semidirect
171. C2#22550206direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168611882
#225502062
C2×#168611402
C2×#168611322
C2×C22⋊(C127⋊C8)2
C22×C4⋊C4×C1272
C2×#483127