Finite Group Databasebeta

← C2×#22550121 #30648317

Subgroups — C2×#22550121 #30648317

Order 16256.

544 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 149 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
2C22×#14523231C211yesmaximal, characteristic, complemented
3C2×#16861189C211yesmaximal, characteristic, complemented
4C2×#16861180C211yesmaximal, characteristic
5#22550121C218yesmaximal, direct product factor
6C2×#16861142C211yesmaximal, characteristic, complemented
7C2×#16861134C211yesmaximal, characteristic, complemented
8C2×#102×C127C211yesmaximal, characteristic, complemented
9C22×Dic254C2211yescharacteristic
10C22×C22⋊D127—21no—
11C22×C4×D127C2211yescharacteristic
12C2×#14523236—24no—
13C2×#14523236C2214yescomplemented
14C2×#14523231—24no—
15C2×#14523231C2214yescomplemented
16C22×C127⋊C8C2211yescharacteristic
17#16861189—22no—
18#16861189C2214yescomplemented
19#16861180—22no—
20#16861180C2214yes—
21C2×C4⋊Dic127—21no—
22C2×C4⋊Dic127C2211yescharacteristic
23C2×C22⋊Dic127—21no—
24C2×(C2×D127)⋊C4—21no—
25#16861142C2214yescomplemented
26#16861134C2214yescomplemented
27C22×D4×C127C2211yescharacteristic
28C22×C8×C127C2211yescharacteristic
29C2×C4⋊C4×C127C2211yescharacteristic
30#102×C127C2214yescomplemented
31C23×D127D411yescharacteristic
32C2×Dic254—21no—
33C2×Dic254—22no—
34C2×Dic254—22no—
35C2×Dic254C2312yes—
36C2×C22⋊D127—24no—
37C2×C22⋊D127—44no—
38C22×Dic127—21no—
39C22×Dic127—21no—
40C22×Dic127D411yescharacteristic
41C2×C4×D127—22no—
42C2×C4×D127—22no—
43C2×C4×D127D412yes—
44C2×C4×D127C2312yes—
45#14523236—28no—
46#14523231—28no—
47C2×C127⋊C8D414yescomplemented
48C2×C127⋊C8C2312yes—
49C4⋊Dic127—24no—
50C4⋊Dic127—21no—
51C4⋊Dic127C2312yes—
52C22⋊Dic127—24no—
53(C2×D127)⋊C4—24no—
54C24×C127—21no—
55C2×D4×C127—22no—
56C2×D4×C127—21no—
57C2×D4×C127—22no—
58C2×D4×C127C2312yes—
59C22×C4×C127—21no—
60C22×C4×C127C2311yescharacteristic
61C2×C8×C127—22no—
62C2×C8×C127C2312yes—
63C4⋊C4×C127—21no—
64C4⋊C4×C127C2312yes—
65C22×D127—22no—
66C22×D127—22no—
67C22×D127—22no—
68C22×D127C2×D412yes—
69Dic254—22no—
70Dic254—22no—
71Dic254—42no—
72C22⋊D127—48no—
73C2×Dic127—22no—
74C2×Dic127—22no—
75C2×Dic127—22no—
76C2×Dic127—42no—
77C2×Dic127—42no—
78C2×Dic127C2×D412yes—
79C4×D127—24no—
80C4×D127—22no—
81C4×D127—22no—
82C127⋊C8C2×D414yescomplemented
83C23×C127—22no—
84C23×C127—42no—
85C23×C127—42no—
86C23×C127—42no—
87C23×C127C2×D411yescharacteristic
88D4×C127—22no—
89D4×C127—22no—
90D4×C127—42no—
91C2×C4×C127—22no—
92C2×C4×C127—42no—
93C2×C4×C127C2×D411yescharacteristic
94C2×C4×C127C2×D412yes—
95C2×C4×C127#4112yes—
96C2×C4×C127C2411yescharacteristic
97C8×C127—22no—
98C22×C127—42no—
99C22×C127—42no—
100C22×C127—42no—
101C22×C127—84no—
102C22×C127#12112yes—
103C22×C127AGL(1,ℤ/8)12yes—
104C22×C127#13712yes—
105C22×C127C22×D411yescharacteristic
106C2×D127—22no—
107C2×D127—22no—
108C2×D127—22no—
109C2×D127—44no—
110C4×C127—42no—
111C4×C127#12112yes—
112C4×C127C22×D411yescharacteristic
113C4×C127C2×#4111yescharacteristic
114Dic127—22no—
115Dic127—42no—
116Dic127—42no—
117C2×C127—82no—
118C2×C127#46814yescomplemented
119C2×C127C2×#12111yescharacteristic
120C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
121C2×C127C2×#13711yescharacteristic
122D127—42no—
123C2×#468—1271nomaximal, Sylow
124C127C2×#46811yescharacteristic, complemented, Sylow
125C22×SD16—1271no—
126C22×M16—1271no—
127C2×C22⋊Q8—1271no—
128C2×#121—1271no—
129#468—1278no—
130C2×#107—1271no—
131C2×#103—1271no—
132C2×#102D12711yescharacteristic, complemented
133C22×Q8—1271no—
134C22×D4—2541no—
135C22×D4C2×D12711yescharacteristic
136C23×C4—1271no—
137C2×SD16—1274no—
138C2×SD16—2544no—
139C2×M16—1274no—
140C2×M16—2544no—
141C22×C8—1271no—
142C22×C8C2×D12711yescharacteristic
143C22⋊Q8—1274no—
144C22⋊Q8—2542no—
145#121—1274no—
146#121—2542no—
147C2×C4⋊C4—1271no—
148C2×C4⋊C4—2541no—
149C2×C4⋊C4C2×D12711yescharacteristic
150C2×C22⋊C4—2541no—
151C2×C22⋊C4—2541no—
152#107—1274no—
153#103—1274no—
154#102C2×D12714yescomplemented
155C24—21no—
156C24—1271no—
157C2×Q8—1272no—
158C2×Q8—2542no—
159C2×Q8—2542no—
160C2×Q8—2541no—
161C2×D4—22no—
162C2×D4—21no—
163C2×D4—22no—
164C2×D4—5084no—
165C2×D4—2544no—
166C2×D4C22×D12712yes—
167C22×C4—21no—
168C22×C4—1271no—
169C22×C4—1272no—
170C22×C4—1272no—
171C22×C4—2541no—
172C22×C4—2541no—
173C22×C4—2542no—
174C22×C4—2542no—
175C22×C4C22×D12711yescharacteristic
176SD16—2548no—
177M16—2548no—
178C2×C8—22no—
179C2×C8—1274no—
180C2×C8—1272no—
181C2×C8C22×D12712yes—
182C4⋊C4—21no—
183C4⋊C4—1272no—
184C4⋊C4—2544no—
185C4⋊C4—2541no—
186C4⋊C4C22×D12712yes—
187C22⋊C4—2544no—
188C22⋊C4—2544no—
189C23—22no—
190C23—42no—
191C23—42no—
192C23—42no—
193C23—1272no—
194C23—2542no—
195C23—2542no—
196C23—2542no—
197C23D4×D12711yescharacteristic
198Q8—5082no—
199Q8—2542no—
200Q8—2542no—
201D4—22no—
202D4—22no—
203D4—42no—
204D4—5088no—
205C2×C4—22no—
206C2×C4—42no—
207C2×C4—1272no—
208C2×C4—5082no—
209C2×C4—5082no—
210C2×C4—2542no—
211C2×C4—2542no—
212C2×C4—2542no—
213C2×C4—2542no—
214C2×C4—2542no—
215C2×C4—2544no—
216C2×C4D4×D12712yes—
217C2×C4D4×D12711yescharacteristic
218C2×C4#1452324812yes—
219C2×C4C23×D12711yescharacteristic
220C8—22no—
221C8—1274no—
222C22—42no—
223C22—42no—
224C22—42no—
225C22—84no—
226C22—5084no—
227C22—2542no—
228C22—2542no—
229C22—2542no—
230C22#1686118212yes—
231C22#1686123312yes—
232C22#1686123812yes—
233C22C2×D4×D12711yescharacteristic
234C4—42no—
235C4—5082no—
236C4—5082no—
237C4—2542no—
238C4#1686118212yes—
239C4C2×D4×D12711yescharacteristic
240C4C2×#1452324811yescharacteristic
241C2—82no—
242C2—5082no—
243C2#2255012114yesdirect product factor
244C2C2×#1686118211yescharacteristic
245C2C2×#1686123311yescharacteristic
246C2C2×#1686123811yescharacteristic
subgroupG/H
centerC23D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×#102×C127C2
socleC23×C127C2×D4
prime radicalC2×#16861180C2
hypercenterC2×#102D127
hypocenterC127C2×#468
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
1. C22×#14523236C2semidirect
2. C22×#14523231C2semidirect
3. C2×#16861189C2semidirect
5. #22550121C2direct (N × H)
6. C2×#16861142C2semidirect
7. C2×#16861134C2semidirect
8. C2×#102×C127C2semidirect
13. C2×#14523236C22semidirect
15. C2×#14523231C22semidirect
18. #16861189C22semidirect
25. #16861142C22semidirect
26. #16861134C22semidirect
30. #102×C127C22semidirect
47. C2×C127⋊C8D4semidirect
82. C127⋊C8C2×D4semidirect
118. C2×C127#468semidirect
124. C127C2×#468semidirect
132. C2×#102D127semidirect
154. #102C2×D127semidirect
243. C2#22550121direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145232362
C22×#145232312
C2×#168611892
C2×#168611802
#225501212
C2×#168611422
C2×#168611342
C2×#102×C1272
C2×#468127