Finite Group Databasebeta

← C2×#22550014 #30647917

Subgroups — C2×#22550014 #30647917

Order 16256.

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

normal subgroups only

This group has 792 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, complemented
2C22×#14523236C212yesmaximal, complemented
3#22550014C218yesmaximal, direct product factor
4C2×#16861132C212yesmaximal, complemented
5C2×C22⋊(C127⋊C8)C211yesmaximal, characteristic, complemented
6C2×#121×C127C211yesmaximal, characteristic
7C22×Dic254—28no—
8C22×Dic254—21no—
9C22×Dic254—22no—
10C22×Dic254C2212yes—
11C22×Dic254C2214yescomplemented
12C23×Dic127—21no—
13C2×#14523236—28no—
14C2×#14523236C2218yescomplemented
15C22×C127⋊C8C2212yes—
16#16861132C2218yescomplemented
17C22⋊(C127⋊C8)C2214yescomplemented
18C22×D4×C127—21no—
19C22×D4×C127C2211yescharacteristic
20C23×C4×C127C2211yescharacteristic
21#121×C127—22no—
22#121×C127C2214yes—
23C2×C4⋊C4×C127C2211yescharacteristic
24C2×C22⋊C4×C127—21no—
25C2×Dic254—28no—
26C2×Dic254—216no—
27C2×Dic254—216no—
28C2×Dic254—24no—
29C2×Dic254—22no—
30C2×Dic254—24no—
31C2×Dic254D418yescomplemented
32C2×Dic254C2314yes—
33C22×Dic127—28no—
34C22×Dic127—22no—
35C22×Dic127—24no—
36#14523236—216no—
37C2×C127⋊C8—24no—
38C2×C127⋊C8C2314yes—
39C24×C127—21no—
40C24×C127D411yescharacteristic
41C2×D4×C127—24no—
42C2×D4×C127—24no—
43C2×D4×C127—21no—
44C2×D4×C127—44no—
45C2×D4×C127C2312yes—
46C22×C4×C127—24no—
47C22×C4×C127—21no—
48C22×C4×C127D411yescharacteristic
49C22×C4×C127D412yes—
50C22×C4×C127C2312yes—
51C22×C4×C127C2311yescharacteristic
52C4⋊C4×C127—21no—
53C4⋊C4×C127C2312yes—
54C22⋊C4×C127—24no—
55Dic254—216no—
56Dic254—28no—
57Dic254—24no—
58Dic254C2×D418yescomplemented
59C2×Dic127—216no—
60C2×Dic127—28no—
61C2×Dic127—24no—
62C127⋊C8—24no—
63C23×C127—22no—
64C23×C127—22no—
65C23×C127—22no—
66C23×C127—42no—
67C23×C127—42no—
68C23×C127—42no—
69C23×C127SD1614yescomplemented
70C23×C127C2×D412yes—
71C23×C127C2×D411yescharacteristic
72D4×C127—24no—
73D4×C127—48no—
74D4×C127—42no—
75C2×C4×C127—22no—
76C2×C4×C127—24no—
77C2×C4×C127—24no—
78C2×C4×C127—22no—
79C2×C4×C127—42no—
80C2×C4×C127C2×D414yes—
81C2×C4×C127C2×D412yes—
82C2×C4×C127C2×D411yescharacteristic
83C2×C4×C127C2411yescharacteristic
84C22×C127—22no—
85C22×C127—22no—
86C22×C127—28no—
87C22×C127—42no—
88C22×C127—42no—
89C22×C127—42no—
90C22×C127—84no—
91C22×C127#12012yes—
92C22×C127C2×SD1612yes—
93C22×C127C2×SD1614yescomplemented
94C22×C127#13712yes—
95C22×C127C22×D411yescharacteristic
96C4×C127—22no—
97C4×C127—42no—
98C4×C127#12012yes—
99C4×C127C22×D412yes—
100Dic127—28no—
101C2×C127—24no—
102C2×C127—82no—
103C2×C127#44814yescomplemented
104C2×C127C2×#12011yescharacteristic
105C2×C127C22×SD1611yescharacteristic
106C2×C127C2×#13711yescharacteristic
107C2×#448—1271nomaximal, Sylow
108C127C2×#44811yescharacteristic, complemented, Sylow
109C23×Q8—1271no—
110C22×SD16—1272no—
111C2×#121D12711yescharacteristic
112#448—1278no—
113C2×#103—1272no—
114C2×C22⋊C8—1271no—
115C22×Q8—1272no—
116C22×Q8—1274no—
117C22×Q8—2548no—
118C22×Q8—2542no—
119C22×Q8—2541no—
120C22×D4—21no—
121C22×D4C2×D12711yescharacteristic
122C23×C4—2541no—
123C23×C4C2×D12711yescharacteristic
124C2×SD16—1278no—
125C2×SD16—2548no—
126C22×C8—1272no—
127#121—22no—
128#121C2×D12714yes—
129C2×C4⋊C4C2×D12711yescharacteristic
130C2×C22⋊C4—21no—
131#103—1278no—
132C22⋊C8—1274no—
133C24—21no—
134C24C22⋊D12711yescharacteristic
135C2×Q8—1278no—
136C2×Q8—1274no—
137C2×Q8—25416no—
138C2×Q8—2548no—
139C2×Q8—25416no—
140C2×Q8—2544no—
141C2×Q8—2544no—
142C2×Q8—2542no—
143C2×D4—24no—
144C2×D4—24no—
145C2×D4—21no—
146C2×D4—44no—
147C2×D4C22×D12712yes—
148C22×C4—24no—
149C22×C4—21no—
150C22×C4—2548no—
151C22×C4—2544no—
152C22×C4—2542no—
153C22×C4C22⋊D12712yes—
154C22×C4C22⋊D12711yescharacteristic
155C22×C4C22×D12712yes—
156C22×C4C22×D12711yescharacteristic
157SD16—25416no—
158C2×C8—1274no—
159C2×C8—2544no—
160C4⋊C4—21no—
161C4⋊C4C22×D12712yes—
162C22⋊C4—24no—
163C23—22no—
164C23—22no—
165C23—22no—
166C23—42no—
167C23—42no—
168C23—42no—
169C23#1452323614yescomplemented
170C23C2×C22⋊D12712yes—
171C23C2×C22⋊D12711yescharacteristic
172Q8—1278no—
173Q8—25416no—
174Q8—2544no—
175Q8—2548no—
176D4—24no—
177D4—48no—
178D4—42no—
179C2×C4—22no—
180C2×C4—24no—
181C2×C4—24no—
182C2×C4—22no—
183C2×C4—42no—
184C2×C4—25416no—
185C2×C4—2544no—
186C2×C4—2548no—
187C2×C4D4×D12714yes—
188C2×C4C2×C22⋊D12712yes—
189C2×C4C2×C22⋊D12711yescharacteristic
190C2×C4C23×D12711yescharacteristic
191C8—2544no—
192C22—22no—
193C22—22no—
194C22—28no—
195C22—42no—
196C22—42no—
197C22—42no—
198C22—84no—
199C22#1686117712yes—
200C22C2×#1452323612yes—
201C22C2×#1452323614yescomplemented
202C22#1686123912yes—
203C22C22×C22⋊D12711yescharacteristic
204C4—22no—
205C4—42no—
206C4—2548no—
207C4#1686117712yes—
208C4C2×D4×D12712yes—
209C2—24no—
210C2—82no—
211C2#2255001414yesdirect product factor
212C2C2×#1686117711yescharacteristic
213C2C22×#1452323611yescharacteristic
214C2C2×#1686123911yescharacteristic
subgroupG/H
centerC23C2×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting, prime radicalC2×#121×C127C2
socleC23×C127C2×D4
hypercenterC2×#121D127
hypocenterC127C2×#448
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C23×Dic254C2semidirect
2. C22×#14523236C2semidirect
3. #22550014C2direct (N × H)
4. C2×#16861132C2semidirect
5. C2×C22⋊(C127⋊C8)C2semidirect
11. C22×Dic254C22semidirect
14. C2×#14523236C22semidirect
16. #16861132C22semidirect
17. C22⋊(C127⋊C8)C22semidirect
31. C2×Dic254D4semidirect
58. Dic254C2×D4semidirect
69. C23×C127SD16semidirect
93. C22×C127C2×SD16semidirect
103. C2×C127#448semidirect
108. C127C2×#448semidirect
169. C23#14523236semidirect
201. C22C2×#14523236semidirect
211. C2#22550014direct (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×#145232362
#225500142
C2×#168611322
C2×C22⋊(C127⋊C8)2
C2×#121×C1272
C2×#448127