Finite Group Databasebeta

← C2×#22550031 #30647923

Subgroups — C2×#22550031 #30647923

Order 16256.

740 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 740 conjugacy classes of subgroups — too many to draw. Only its 181 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
1C23×Dic254C211yesmaximal, characteristic
2C22×#14523238C212yesmaximal, complemented
3#22550031C218yesmaximal, direct product factor
4C2×#16861132C212yesmaximal, complemented
5C2×C22⋊(C127⋊C8)C211yesmaximal, characteristic
6C2×C22⋊Q8×C127C211yesmaximal, characteristic
7C22×Dic254—28no—
8C22×Dic254—21no—
9C22×Dic254—22no—
10C22×Dic254C2214yes—
11C22×Dic254C2212yes—
12C23×Dic127—21no—
13C2×#14523238—28no—
14C2×#14523238C2218yescomplemented
15C22×C127⋊C8C2212yes—
16#16861132C2218yescomplemented
17C22⋊(C127⋊C8)C2214yes—
18C22×Q8×C127C2211yescharacteristic
19C23×C4×C127C2211yescharacteristic
20C22⋊Q8×C127—22no—
21C22⋊Q8×C127C2214yes—
22C2×C4⋊C4×C127—21no—
23C2×C4⋊C4×C127C2211yescharacteristic
24C2×C22⋊C4×C127—21no—
25C2×Dic254—28no—
26C2×Dic254—216no—
27C2×Dic254—216no—
28C2×Dic254—24no—
29C2×Dic254—24no—
30C2×Dic254—22no—
31C2×Dic254D418yes—
32C2×Dic254C2314yes—
33C22×Dic127—28no—
34C22×Dic127—24no—
35C22×Dic127—22no—
36#14523238—216no—
37C2×C127⋊C8—24no—
38C2×C127⋊C8C2314yes—
39C24×C127D411yescharacteristic
40C2×Q8×C127—21no—
41C2×Q8×C127—24no—
42C2×Q8×C127C2312yes—
43C22×C4×C127—24no—
44C22×C4×C127—21no—
45C22×C4×C127—21no—
46C22×C4×C127D411yescharacteristic
47C22×C4×C127D412yes—
48C22×C4×C127C2312yes—
49C22×C4×C127C2311yescharacteristic
50C4⋊C4×C127—24no—
51C4⋊C4×C127—21no—
52C4⋊C4×C127C2312yes—
53C22⋊C4×C127—24no—
54Dic254—216no—
55Dic254—28no—
56Dic254—24no—
57Dic254C2×D418yes—
58C2×Dic127—216no—
59C2×Dic127—28no—
60C2×Dic127—24no—
61C127⋊C8—24no—
62C23×C127—22no—
63C23×C127—22no—
64C23×C127Q1614yescomplemented
65C23×C127C2×D412yes—
66C23×C127C2×D411yescharacteristic
67Q8×C127—24no—
68Q8×C127—42no—
69C2×C4×C127—22no—
70C2×C4×C127—24no—
71C2×C4×C127—24no—
72C2×C4×C127—22no—
73C2×C4×C127—22no—
74C2×C4×C127—42no—
75C2×C4×C127—42no—
76C2×C4×C127C2×D414yes—
77C2×C4×C127C2×D412yes—
78C2×C4×C127C2×D411yescharacteristic
79C2×C4×C127C2411yescharacteristic
80C22×C127—22no—
81C22×C127—22no—
82C22×C127—28no—
83C22×C127#12012yes—
84C22×C127C2×Q1612yes—
85C22×C127C2×Q1614yescomplemented
86C22×C127#13712yes—
87C22×C127C22×D411yescharacteristic
88C4×C127—22no—
89C4×C127—42no—
90C4×C127—42no—
91C4×C127#12012yes—
92C4×C127C22×D412yes—
93Dic127—28no—
94C2×C127—24no—
95C2×C127#45114yescomplemented
96C2×C127C2×#12011yescharacteristic
97C2×C127C22×Q1611yescharacteristic
98C2×C127C2×#13711yescharacteristic
99C2×#451—1271nomaximal, Sylow
100C127C2×#45111yescharacteristic, complemented, Sylow
101C23×Q8—1271no—
102C22×Q16—1272no—
103C2×C22⋊Q8D12711yescharacteristic
104#451—1278no—
105C2×#103—1272no—
106C2×C22⋊C8—1271no—
107C22×Q8—1272no—
108C22×Q8—1274no—
109C22×Q8—2548no—
110C22×Q8—2542no—
111C22×Q8—2541no—
112C22×Q8C2×D12711yescharacteristic
113C23×C4—2541no—
114C23×C4C2×D12711yescharacteristic
115C2×Q16—1278no—
116C2×Q16—2548no—
117C22×C8—1272no—
118C22⋊Q8—22no—
119C22⋊Q8C2×D12714yes—
120C2×C4⋊C4—21no—
121C2×C4⋊C4C2×D12711yescharacteristic
122C2×C22⋊C4—21no—
123#103—1278no—
124C22⋊C8—1274no—
125C24C22⋊D12711yescharacteristic
126C2×Q8—21no—
127C2×Q8—24no—
128C2×Q8—1278no—
129C2×Q8—1274no—
130C2×Q8—25416no—
131C2×Q8—2548no—
132C2×Q8—25416no—
133C2×Q8—2544no—
134C2×Q8—2544no—
135C2×Q8—2542no—
136C2×Q8C22×D12712yes—
137C22×C4—24no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—2548no—
141C22×C4—2544no—
142C22×C4—2542no—
143C22×C4C22⋊D12712yes—
144C22×C4C22⋊D12711yescharacteristic
145C22×C4C22×D12712yes—
146C22×C4C22×D12711yescharacteristic
147Q16—25416no—
148C2×C8—1274no—
149C2×C8—2544no—
150C4⋊C4—24no—
151C4⋊C4—21no—
152C4⋊C4C22×D12712yes—
153C22⋊C4—24no—
154C23—22no—
155C23—22no—
156C23#1452323814yescomplemented
157C23C2×C22⋊D12712yes—
158C23C2×C22⋊D12711yescharacteristic
159Q8—24no—
160Q8—42no—
161Q8—1278no—
162Q8—25416no—
163Q8—2544no—
164Q8—2548no—
165C2×C4—22no—
166C2×C4—24no—
167C2×C4—24no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—42no—
171C2×C4—42no—
172C2×C4—25416no—
173C2×C4—2544no—
174C2×C4—2548no—
175C2×C4D4×D12714yes—
176C2×C4C2×C22⋊D12712yes—
177C2×C4C2×C22⋊D12711yescharacteristic
178C2×C4C23×D12711yescharacteristic
179C8—2544no—
180C22—28no—
181C22—22no—
182C22—22no—
183C22#1686117712yes—
184C22C2×#1452323812yes—
185C22C2×#1452323814yescomplemented
186C22#1686123912yes—
187C22C22×C22⋊D12711yescharacteristic
188C4—22no—
189C4—42no—
190C4—42no—
191C4—2548no—
192C4#1686117712yes—
193C4C2×D4×D12712yes—
194C2—24no—
195C2#2255003114yesdirect product factor
196C2C2×#1686117711yescharacteristic
197C2C22×#1452323811yescharacteristic
198C2C2×#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×#451
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. C22×#14523238C2semidirect
3. #22550031C2direct (N × H)
4. C2×#16861132C2semidirect
14. C2×#14523238C22semidirect
16. #16861132C22semidirect
64. C23×C127Q16semidirect
85. C22×C127C2×Q16semidirect
95. C2×C127#451semidirect
100. C127C2×#451semidirect
156. C23#14523238semidirect
185. C22C2×#14523238semidirect
195. C2#22550031direct (N × H)

Maximal subgroups

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

subgroupindex
C23×Dic2542
C22×#145232382
#225500312
C2×#168611322
C2×C22⋊(C127⋊C8)2
C2×C22⋊Q8×C1272
C2×#451127