Finite Group Databasebeta

← C2×#22550189 #30648601

Subgroups — C2×#22550189 #30648601

Order 16256.

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

normal subgroups only

This group has 512 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×#16861183C211yesmaximal, characteristic
2#22550189C218yesmaximal, direct product factor
3C2×#16861142C212yesmaximal, complemented
4C2×#16861129C212yesmaximal, complemented
5C2×C22⋊(C127⋊C8)C211yesmaximal, characteristic, complemented
6C22×C4⋊C4×C127C211yesmaximal, characteristic, complemented
7C22×D508C2211yescharacteristic
8C22×C22⋊D127—21no—
9C22×C127⋊C8C2212yes—
10#16861183—22no—
11#16861183C2214yes—
12C2×C4⋊Dic127C2211yescharacteristic
13C2×(C2×D127)⋊C4—21no—
14#16861142C2218yescomplemented
15#16861129C2218yescomplemented
16C22⋊(C127⋊C8)C2214yescomplemented
17C23×C4×C127—21no—
18C23×C4×C127C2211yescharacteristic
19C2×C4⋊C4×C127—22no—
20C2×C4⋊C4×C127—21no—
21C2×C4⋊C4×C127C2212yes—
22C2×C4⋊C4×C127C2214yescomplemented
23C23×D127—21no—
24C2×D508—21no—
25C2×D508—24no—
26C2×D508C2312yes—
27C2×C22⋊D127—24no—
28C2×C22⋊D127—44no—
29C22×Dic127—21no—
30C2×C127⋊C8—24no—
31C2×C127⋊C8C2314yes—
32C4⋊Dic127—21no—
33C4⋊Dic127C2312yes—
34(C2×D127)⋊C4—24no—
35C24×C127D411yescharacteristic
36C22×C4×C127—24no—
37C22×C4×C127—22no—
38C22×C4×C127—24no—
39C22×C4×C127—44no—
40C22×C4×C127D411yescharacteristic
41C22×C4×C127D412yes—
42C22×C4×C127C2312yes—
43C22×C4×C127C2311yescharacteristic
44C4⋊C4×C127—22no—
45C4⋊C4×C127—24no—
46C4⋊C4×C127C2314yes—
47C22×D127—22no—
48C22×D127—42no—
49C22×D127—42no—
50C22×D127—42no—
51D508—24no—
52D508—42no—
53C22⋊D127—48no—
54C2×Dic127—22no—
55C2×Dic127—42no—
56C127⋊C8—24no—
57C23×C127—22no—
58C23×C127—22no—
59C23×C127D814yes—
60C23×C127C2×D412yes—
61C23×C127C2×D411yescharacteristic
62C2×C4×C127—22no—
63C2×C4×C127—24no—
64C2×C4×C127—24no—
65C2×C4×C127—24no—
66C2×C4×C127—48no—
67C2×C4×C127—44no—
68C2×C4×C127C2×D412yes—
69C2×C4×C127C2×D411yescharacteristic
70C2×C4×C127#4114yes—
71C2×C4×C127C2411yescharacteristic
72C22×C127—22no—
73C22×C127—22no—
74C22×C127—28no—
75C22×C127#12312yes—
76C22×C127C2×D814yes—
77C22×C127C2×D812yes—
78C22×C127#13712yes—
79C22×C127C22×D411yescharacteristic
80C2×D127—42no—
81C2×D127—42no—
82C2×D127—42no—
83C2×D127—84no—
84C4×C127—22no—
85C4×C127—44no—
86C4×C127#12312yes—
87C4×C127C2×#4112yes—
88Dic127—42no—
89C2×C127—24no—
90C2×C127#48014yescomplemented
91C2×C127C2×#12311yescharacteristic
92C2×C127C22×D811yescharacteristic
93C2×C127C2×#13711yescharacteristic
94D127—82no—
95C2×#480—1271nomaximal, Sylow
96C127C2×#48011yescharacteristic, complemented, Sylow
97C2×#121—1271no—
98C22×C4⋊C4D12711yescharacteristic, complemented
99#480—1278no—
100C2×#107—1272no—
101C2×#102—1272no—
102C2×C22⋊C8—1271no—
103C22×D4—1271no—
104C22×D4—2541no—
105C23×C4—21no—
106C23×C4C2×D12711yescharacteristic
107C22×C8—1272no—
108#121—1274no—
109#121—2542no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—1271no—
113C2×C4⋊C4C2×D12712yes—
114C2×C4⋊C4C2×D12714yescomplemented
115C2×C22⋊C4—2541no—
116#107—1278no—
117#102—1278no—
118C22⋊C8—1274no—
119C24—2541no—
120C24C22⋊D12711yescharacteristic
121C2×D4—1272no—
122C2×D4—5084no—
123C2×D4—2544no—
124C2×D4—2541no—
125C2×D4—2544no—
126C22×C4—24no—
127C22×C4—24no—
128C22×C4—22no—
129C22×C4—44no—
130C22×C4—2541no—
131C22×C4C22⋊D12712yes—
132C22×C4C22⋊D12711yescharacteristic
133C22×C4C22×D12712yes—
134C22×C4C22×D12711yescharacteristic
135C2×C8—1274no—
136C2×C8—2544no—
137C4⋊C4—22no—
138C4⋊C4—24no—
139C4⋊C4—1272no—
140C4⋊C4—2541no—
141C4⋊C4C22×D12714yes—
142C22⋊C4—2544no—
143C23—22no—
144C23—22no—
145C23—5082no—
146C23—5082no—
147C23—5082no—
148C23—2542no—
149C23#1452323314yes—
150C23C2×C22⋊D12712yes—
151C23C2×C22⋊D12711yescharacteristic
152D4—5088no—
153D4—5082no—
154D4—2544no—
155C2×C4—22no—
156C2×C4—24no—
157C2×C4—24no—
158C2×C4—24no—
159C2×C4—48no—
160C2×C4—44no—
161C2×C4—5082no—
162C2×C4—2542no—
163C2×C4C2×C22⋊D12712yes—
164C2×C4C2×C22⋊D12711yescharacteristic
165C2×C4#1452324814yes—
166C2×C4C23×D12711yescharacteristic
167C8—2544no—
168C22—22no—
169C22—22no—
170C22—28no—
171C22—10164no—
172C22—5082no—
173C22—5082no—
174C22—5082no—
175C22#1686119212yes—
176C22C2×#1452323314yes—
177C22C2×#1452323312yes—
178C22#1686124012yes—
179C22C22×C22⋊D12711yescharacteristic
180C4—22no—
181C4—44no—
182C4—5082no—
183C4#1686119212yes—
184C4C2×#1452324812yes—
185C2—24no—
186C2—10162no—
187C2#2255018914yesdirect product factor
188C2C2×#1686119211yescharacteristic
189C2C22×#1452323311yescharacteristic
190C2C2×#1686124011yescharacteristic
subgroupG/H
centerC23C2×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC22×C4⋊C4×C127C2
socleC23×C127C2×D4
prime radicalC2×#16861183C2
hypercenterC22×C4⋊C4D127
hypocenterC127C2×#480
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. #22550189C2direct (N × H)
3. C2×#16861142C2semidirect
4. C2×#16861129C2semidirect
5. C2×C22⋊(C127⋊C8)C2semidirect
6. C22×C4⋊C4×C127C2semidirect
14. #16861142C22semidirect
15. #16861129C22semidirect
16. C22⋊(C127⋊C8)C22semidirect
22. C2×C4⋊C4×C127C22semidirect
90. C2×C127#480semidirect
96. C127C2×#480semidirect
98. C22×C4⋊C4D127semidirect
114. C2×C4⋊C4C2×D127semidirect
187. C2#22550189direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168611832
#225501892
C2×#168611422
C2×#168611292
C2×C22⋊(C127⋊C8)2
C22×C4⋊C4×C1272
C2×#480127