Finite Group Databasebeta

← C2×#22550269 #30649169

Subgroups — C2×#22550269 #30649169

Order 16256.

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

normal subgroups only

This group has 564 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×#14523238C211yesmaximal, characteristic, complemented
2C22×#14523237C211yesmaximal, characteristic
3C22×#14523235C211yesmaximal, characteristic
4C2×#16861207C211yesmaximal, characteristic, complemented
5C2×#16861198C211yesmaximal, characteristic
6#22550269C218yesmaximal, direct product factor
7C2×#16861119C211yesmaximal, characteristic, complemented
8C22×Q16×C127C211yesmaximal, characteristic, complemented
9C22×Dic254C2211yescharacteristic
10C22×D508C2211yescharacteristic
11C2×#14523238—24no—
12C2×#14523238C2214yescomplemented
13C2×#14523237—24no—
14C2×#14523237C2214yes—
15C2×#14523235—24no—
16C2×#14523235C2214yes—
17C22×C127⋊C8C2211yescharacteristic
18#16861207—22no—
19#16861207C2214yescomplemented
20#16861198—22no—
21#16861198C2214yes—
22C2×Dic127⋊C4—22no—
23C2×(C2×D127)⋊C4—22no—
24C2×C4×Dic127C2211yescharacteristic
25#16861119C2214yescomplemented
26C22×Q8×C127C2211yescharacteristic
27C22×Q8×C127C2211yescharacteristic
28C2×Q16×C127—24no—
29C2×Q16×C127C2214yescomplemented
30C22×C8×C127C2211yescharacteristic
31C23×D127—21no—
32C2×Dic254—21no—
33C2×Dic254—22no—
34C2×Dic254—22no—
35C2×Dic254C2312yes—
36C2×D508—22no—
37C2×D508—22no—
38C2×D508—21no—
39C2×D508C2312yes—
40C22×Dic127—21no—
41C22×Dic127D412yes—
42#14523238—28no—
43#14523237—28no—
44#14523235—28no—
45C2×C127⋊C8D414yes—
46C2×C127⋊C8C2312yes—
47Dic127⋊C4—28no—
48(C2×D127)⋊C4—28no—
49C4×Dic127D412yes—
50C4×Dic127C2312yes—
51C2×Q8×C127—21no—
52C2×Q8×C127—21no—
53C2×Q8×C127—22no—
54C2×Q8×C127—22no—
55C2×Q8×C127—22no—
56C2×Q8×C127—22no—
57C2×Q8×C127C2312yes—
58C2×Q8×C127C2312yes—
59C22×C4×C127—21no—
60C22×C4×C127—21no—
61C22×C4×C127C2311yescharacteristic
62Q16×C127—28no—
63C2×C8×C127D414yes—
64C2×C8×C127C2312yes—
65C22×D127—22no—
66C22×D127—42no—
67C22×D127—44no—
68Dic254—22no—
69Dic254—22no—
70Dic254—42no—
71D508—22no—
72D508—22no—
73D508—42no—
74C2×Dic127—24no—
75C2×Dic127—22no—
76C2×Dic127—42no—
77C2×Dic127C2×D414yes—
78C127⋊C8C2×D414yes—
79C23×C127C2×D411yescharacteristic
80Q8×C127—22no—
81Q8×C127—22no—
82Q8×C127—22no—
83Q8×C127—22no—
84Q8×C127—42no—
85Q8×C127—42no—
86C2×C4×C127—22no—
87C2×C4×C127—22no—
88C2×C4×C127—42no—
89C2×C4×C127—42no—
90C2×C4×C127C2×D411yescharacteristic
91C2×C4×C127C2×D412yes—
92C2×C4×C127C2×D412yes—
93C2×C4×C127C2411yescharacteristic
94C8×C127C2×D414yes—
95C22×C127#12712yes—
96C22×C127#13714yes—
97C22×C127C22×D411yescharacteristic
98C2×D127—42no—
99C2×D127—44no—
100C2×D127—84no—
101C4×C127—42no—
102C4×C127—42no—
103C4×C127#12712yes—
104C4×C127C22×D411yescharacteristic
105C4×C127C22×D411yescharacteristic
106Dic127—24no—
107Dic127—42no—
108C2×C127#49714yescomplemented
109C2×C127C2×#12711yescharacteristic
110C2×C127C2×#13712yes—
111D127—82no—
112C2×#497—1271nomaximal, Sylow
113C127C2×#49711yescharacteristic, complemented, Sylow
114C22×Q16—1271no—
115C22×Q16D12711yescharacteristic, complemented
116C22×SD16—1271no—
117C22×SD16—1271no—
118C2×C4⋊Q8—1271no—
119C2×#124—1271no—
120#497—1278no—
121C2×#97—1271no—
122C22×Q8—1271no—
123C22×Q8C2×D12711yescharacteristic
124C22×Q8C2×D12711yescharacteristic
125C22×D4—1271no—
126C2×Q16—24no—
127C2×Q16—1274no—
128C2×Q16—2544no—
129C2×Q16C2×D12714yescomplemented
130C2×SD16—1274no—
131C2×SD16—1274no—
132C2×SD16—2544no—
133C2×SD16—2544no—
134C22×C8—1271no—
135C22×C8C2×D12711yescharacteristic
136C4⋊Q8—1274no—
137C4⋊Q8—2542no—
138#124—1274no—
139#124—2542no—
140C2×C4⋊C4—2542no—
141C2×C22⋊C4—2542no—
142C2×C42—1271no—
143#97—1274no—
144C24—2541no—
145C2×Q8—21no—
146C2×Q8—21no—
147C2×Q8—22no—
148C2×Q8—22no—
149C2×Q8—22no—
150C2×Q8—22no—
151C2×Q8—1272no—
152C2×Q8—2542no—
153C2×Q8—2542no—
154C2×Q8—2541no—
155C2×Q8C22×D12712yes—
156C2×Q8C22×D12712yes—
157C2×D4—1272no—
158C2×D4—2541no—
159C2×D4—2542no—
160C2×D4—2542no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—1272no—
164C22×C4—2541no—
165C22×C4C22×D12711yescharacteristic
166Q16—28no—
167Q16—2548no—
168SD16—2548no—
169SD16—2548no—
170C2×C8—1274no—
171C2×C8—1272no—
172C2×C8C22⋊D12714yes—
173C2×C8C22×D12712yes—
174C4⋊C4—2548no—
175C22⋊C4—2548no—
176C42—1272no—
177C42—1272no—
178C23—5082no—
179C23—5084no—
180C23—2542no—
181C23D4×D12711yescharacteristic
182Q8—22no—
183Q8—22no—
184Q8—22no—
185Q8—22no—
186Q8—42no—
187Q8—42no—
188Q8—5082no—
189Q8—2542no—
190Q8—2542no—
191D4—5082no—
192D4—2542no—
193D4—2542no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—1274no—
199C2×C4—5082no—
200C2×C4—2542no—
201C2×C4—2544no—
202C2×C4D4×D12711yescharacteristic
203C2×C4D4×D12712yes—
204C2×C4C2×C22⋊D12712yes—
205C2×C4C23×D12711yescharacteristic
206C8—1274no—
207C8C2×C22⋊D12714yes—
208C22—10164no—
209C22—5084no—
210C22—5082no—
211C22#1686120512yes—
212C22#1686124114yes—
213C22C2×D4×D12711yescharacteristic
214C4—42no—
215C4—42no—
216C4—5082no—
217C4—2544no—
218C4#1686120512yes—
219C4C2×D4×D12711yescharacteristic
220C4C22×C22⋊D12711yescharacteristic
221C2—10162no—
222C2#2255026914yesdirect product factor
223C2C2×#1686120511yescharacteristic
224C2C2×#1686124112yes—
subgroupG/H
centerC23D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC22×Q16×C127C2
socleC23×C127C2×D4
prime radicalC22×D508C22
hypercenterC22×Q16D127
hypocenterC127C2×#497
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×Q16

Complements

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

NH ≅ G/Ntype
1. C22×#14523238C2semidirect
4. C2×#16861207C2semidirect
6. #22550269C2direct (N × H)
7. C2×#16861119C2semidirect
8. C22×Q16×C127C2semidirect
12. C2×#14523238C22semidirect
19. #16861207C22semidirect
25. #16861119C22semidirect
29. C2×Q16×C127C22semidirect
108. C2×C127#497semidirect
113. C127C2×#497semidirect
115. C22×Q16D127semidirect
129. C2×Q16C2×D127semidirect
222. C2#22550269direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145232382
C22×#145232372
C22×#145232352
C2×#168612072
C2×#168611982
#225502692
C2×#168611192
C22×Q16×C1272
C2×#497127