Finite Group Databasebeta

← C2×#22550089 #30648152

Subgroups — C2×#22550089 #30648152

Order 16256.

564 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 157 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×#14523236C211yesmaximal, characteristic, complemented
2C2×#16861196C211yesmaximal, characteristic, complemented
3C2×D4×Dic127C211yesmaximal, characteristic, complemented
4#22550089C218yesmaximal, direct product factor
5C2×Dic127⋊C8C211yesmaximal, characteristic, complemented
6C2×#16861134C211yesmaximal, characteristic, complemented
7C2×#16861130C211yesmaximal, characteristic, complemented
8C22×D8×C127C211yesmaximal, characteristic
9C22×Dic254C2211yescharacteristic
10C23×Dic127—21no—
11C2×#14523236—24no—
12C2×#14523236C2214yescomplemented
13C22×C127⋊C8C2211yescharacteristic
14#16861196—22no—
15#16861196C2214yescomplemented
16D4×Dic127—22no—
17D4×Dic127C2214yescomplemented
18C2×C4⋊Dic127C2211yescharacteristic
19C2×C22⋊Dic127—21no—
20C2×C22⋊Dic127—21no—
21C2×C22⋊Dic127—21no—
22C2×C4×Dic127C2211yescharacteristic
23Dic127⋊C8C2214yescomplemented
24#16861134C2214yescomplemented
25#16861130C2214yescomplemented
26C22×D4×C127C2211yescharacteristic
27C22×D4×C127C2211yescharacteristic
28C2×D8×C127—24no—
29C2×D8×C127C2214yes—
30C22×C8×C127C2211yescharacteristic
31C2×Dic254—21no—
32C2×Dic254—22no—
33C2×Dic254—22no—
34C2×Dic254C2312yes—
35C22×Dic127—24no—
36C22×Dic127—21no—
37C22×Dic127—21no—
38C22×Dic127—44no—
39C22×Dic127D411yescharacteristic
40C22×Dic127D411yescharacteristic
41#14523236—28no—
42C2×C127⋊C8—22no—
43C2×C127⋊C8C2312yes—
44C4⋊Dic127—21no—
45C4⋊Dic127C2312yes—
46C22⋊Dic127—24no—
47C22⋊Dic127—24no—
48C22⋊Dic127—24no—
49C4×Dic127D412yes—
50C4×Dic127C2312yes—
51C24×C127—21no—
52C24×C127—21no—
53C2×D4×C127—21no—
54C2×D4×C127—22no—
55C2×D4×C127—21no—
56C2×D4×C127—22no—
57C2×D4×C127—22no—
58C2×D4×C127D414yes—
59C2×D4×C127C2312yes—
60C2×D4×C127C2312yes—
61C22×C4×C127C2311yescharacteristic
62D8×C127—28no—
63C2×C8×C127—22no—
64C2×C8×C127C2312yes—
65Dic254—22no—
66Dic254—22no—
67Dic254—42no—
68C2×Dic127—22no—
69C2×Dic127—22no—
70C2×Dic127—22no—
71C2×Dic127—22no—
72C2×Dic127—48no—
73C2×Dic127—42no—
74C2×Dic127—42no—
75C2×Dic127C2×D412yes—
76C2×Dic127C2×D412yes—
77C127⋊C8—22no—
78C23×C127—24no—
79C23×C127—22no—
80C23×C127—22no—
81C23×C127—42no—
82C23×C127—42no—
83C23×C127—42no—
84C23×C127—42no—
85C23×C127—42no—
86C23×C127C2×D411yescharacteristic
87D4×C127—24no—
88D4×C127—22no—
89D4×C127—22no—
90D4×C127—22no—
91D4×C127—42no—
92D4×C127C2×D414yes—
93C2×C4×C127C2×D411yescharacteristic
94C2×C4×C127C2×D412yes—
95C2×C4×C127#4112yes—
96C2×C4×C127C2411yescharacteristic
97C8×C127—22no—
98C22×C127—24no—
99C22×C127—42no—
100C22×C127—42no—
101C22×C127—48no—
102C22×C127—42no—
103C22×C127—42no—
104C22×C127—42no—
105C22×C127—84no—
106C22×C127#12112yes—
107C22×C127#13512yes—
108C22×C127AGL(1,ℤ/8)12yes—
109C22×C127C22×D411yescharacteristic
110C4×C127#12112yes—
111C4×C127C22×D411yescharacteristic
112C4×C127C2×#4111yescharacteristic
113Dic127—22no—
114Dic127—22no—
115Dic127—42no—
116Dic127—42no—
117C2×C127—44no—
118C2×C127—82no—
119C2×C127#46314yescomplemented
120C2×C127C2×#12111yescharacteristic
121C2×C127C2×#13511yescharacteristic
122C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
123C2×#463—1271nomaximal, Sylow
124C127C2×#46311yescharacteristic, complemented, Sylow
125C22×SD16—1271no—
126C22×D8D12711yescharacteristic
127C2×#124—1271no—
128C2×C4×D4—1271no—
129#463—1278no—
130C2×C4⋊C8—1271no—
131C2×#103—1271no—
132C2×#102—1271no—
133C22×Q8—1271no—
134C22×D4C2×D12711yescharacteristic
135C22×D4C2×D12711yescharacteristic
136C23×C4—2541no—
137C2×SD16—1274no—
138C2×SD16—2544no—
139C2×D8—24no—
140C2×D8C2×D12714yes—
141C22×C8—1271no—
142C22×C8C2×D12711yescharacteristic
143#124—1274no—
144#124—2542no—
145C4×D4—1274no—
146C4×D4—2542no—
147C2×C4⋊C4—1271no—
148C2×C22⋊C4—2541no—
149C2×C22⋊C4—2541no—
150C2×C22⋊C4—2541no—
151C2×C42—1271no—
152C4⋊C8—1274no—
153#103—1274no—
154#102—1274no—
155C24—21no—
156C24—21no—
157C2×Q8—1272no—
158C2×Q8—2542no—
159C2×Q8—2542no—
160C2×Q8—2541no—
161C2×D4—21no—
162C2×D4—21no—
163C2×D4—22no—
164C2×D4—22no—
165C2×D4—22no—
166C2×D4C22⋊D12714yes—
167C2×D4C22×D12712yes—
168C2×D4C22×D12712yes—
169C22×C4—1271no—
170C22×C4—1271no—
171C22×C4—5084no—
172C22×C4—2544no—
173C22×C4—2541no—
174C22×C4—2541no—
175C22×C4C22×D12711yescharacteristic
176SD16—2548no—
177D8—28no—
178C2×C8—22no—
179C2×C8—1272no—
180C2×C8—2542no—
181C2×C8C22×D12712yes—
182C4⋊C4—1272no—
183C4⋊C4—2541no—
184C22⋊C4—2544no—
185C22⋊C4—2544no—
186C22⋊C4—2544no—
187C42—1272no—
188C42—1272no—
189C23—22no—
190C23—24no—
191C23—22no—
192C23—42no—
193C23—42no—
194C23—42no—
195C23—42no—
196C23—42no—
197C23D4×D12711yescharacteristic
198Q8—5082no—
199Q8—2542no—
200Q8—2542no—
201D4—22no—
202D4—22no—
203D4—22no—
204D4—24no—
205D4—42no—
206D4C2×C22⋊D12714yes—
207C2×C4—1272no—
208C2×C4—1272no—
209C2×C4—5082no—
210C2×C4—5082no—
211C2×C4—5088no—
212C2×C4—2542no—
213C2×C4—2542no—
214C2×C4—2542no—
215C2×C4—2542no—
216C2×C4D4×D12711yescharacteristic
217C2×C4C2×C22⋊D12712yes—
218C2×C4#1452324712yes—
219C2×C4C23×D12711yescharacteristic
220C8—22no—
221C8—2542no—
222C22—24no—
223C22—48no—
224C22—42no—
225C22—42no—
226C22—42no—
227C22—42no—
228C22—42no—
229C22—84no—
230C22#1686118112yes—
231C22#1686122812yes—
232C22#1686123312yes—
233C22C2×D4×D12711yescharacteristic
234C4—5082no—
235C4—5082no—
236C4—2542no—
237C4—2542no—
238C4#1686118112yes—
239C4C22×C22⋊D12711yescharacteristic
240C4C2×#1452324711yescharacteristic
241C2—44no—
242C2—82no—
243C2#2255008914yesdirect product factor
244C2C2×#1686118111yescharacteristic
245C2C2×#1686122811yescharacteristic
246C2C2×#1686123311yescharacteristic
subgroupG/H
centerC23D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting, prime radicalC22×D8×C127C2
socleC23×C127C2×D4
hypercenterC22×D8D127
hypocenterC127C2×#463
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×D8

Complements

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

NH ≅ G/Ntype
1. C22×#14523236C2semidirect
2. C2×#16861196C2semidirect
3. C2×D4×Dic127C2semidirect
4. #22550089C2direct (N × H)
5. C2×Dic127⋊C8C2semidirect
6. C2×#16861134C2semidirect
7. C2×#16861130C2semidirect
12. C2×#14523236C22semidirect
15. #16861196C22semidirect
17. D4×Dic127C22semidirect
23. Dic127⋊C8C22semidirect
24. #16861134C22semidirect
25. #16861130C22semidirect
119. C2×C127#463semidirect
124. C127C2×#463semidirect
243. C2#22550089direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145232362
C2×#168611962
C2×D4×Dic1272
#225500892
C2×Dic127⋊C82
C2×#168611342
C2×#168611302
C22×D8×C1272
C2×#463127