Finite Group Databasebeta

← C2×#22550053 #30648059

Subgroups — C2×#22550053 #30648059

Order 16256.

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

normal subgroups only

This group has 760 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
1C2×#16861254C211yesmaximal, characteristic
2C2×#16861240C212yesmaximal, complemented
3#22550053C218yesmaximal, direct product factor
4C2×#16861137C212yesmaximal, complemented
5C2×#16861126C211yesmaximal, characteristic, complemented
6C2×#124×C127C211yesmaximal, characteristic, complemented
7#16861254—24no—
8#16861254—22no—
9#16861254C2214yes—
10C2×#14523248C2212yes—
11C2×#14523247—22no—
12C22×C22⋊D127—22no—
13C2×D4×D127—22no—
14#16861240—24no—
15#16861240C2218yescomplemented
16C2×#14523238—22no—
17C2×#14523235—22no—
18C2×#14523232C2212yescomplemented
19#16861137C2218yescomplemented
20#16861126C2214yescomplemented
21C22×Q8×C127C2211yescharacteristic, complemented
22C22×D4×C127C2211yescharacteristic
23#124×C127—22no—
24#124×C127C2214yescomplemented
25C2×C22⋊C4×C127—22no—
26C2×C42×C127C2211yescharacteristic, complemented
27C23×D127—22no—
28#14523248—22no—
29#14523248—22no—
30#14523248D414yes—
31#14523248C2314yes—
32#14523247—28no—
33#14523247—28no—
34C2×Dic254D412yes—
35C2×D508D412yes—
36C2×C22⋊D127—24no—
37C2×C22⋊D127—28no—
38C2×C22⋊D127—22no—
39C2×C22⋊D127—28no—
40D4×D127—28no—
41D4×D127—28no—
42C22×Dic127—22no—
43C2×C4×D127—22no—
44#14523238—28no—
45#14523235—28no—
46#14523232—22no—
47#14523232C2314yescomplemented
48C2×C127⋊C8—22no—
49C24×C127D412yes—
50C2×Q8×C127—21no—
51C2×Q8×C127—22no—
52C2×Q8×C127—22no—
53C2×Q8×C127C2312yescomplemented
54C2×D4×C127—22no—
55C2×D4×C127—22no—
56C2×D4×C127D412yes—
57C2×D4×C127C2312yes—
58C22×C4×C127—21no—
59C22×C4×C127—21no—
60C22×C4×C127C2311yescharacteristic
61C22⋊C4×C127—28no—
62C42×C127—21no—
63C42×C127C2312yescomplemented
64C22×D127—22no—
65C22×D127—28no—
66C22×D127—84no—
67Dic254—22no—
68Dic254C2×D414yes—
69D508—22no—
70D508C2×D414yes—
71C22⋊D127—216no—
72C22⋊D127—24no—
73C22⋊D127—24no—
74C2×Dic127—28no—
75C2×Dic127—22no—
76C4×D127—24no—
77C4×D127—24no—
78C127⋊C8—24no—
79C23×C127—22no—
80C23×C127—22no—
81C23×C127—44no—
82C23×C127C2×D411yescharacteristic
83C23×C127C2×D414yes—
84Q8×C127—24no—
85Q8×C127—42no—
86D4×C127—24no—
87D4×C127—24no—
88C2×C4×C127—22no—
89C2×C4×C127—22no—
90C2×C4×C127—42no—
91C2×C4×C127—41no—
92C2×C4×C127—41no—
93C2×C4×C127C2×D412yes—
94C2×C4×C127C2×D412yes—
95C2×C4×C127C2×D411yescharacteristic
96C2×C4×C127C2411yescharacteristic
97C22×C127—22no—
98C22×C127—24no—
99C22×C127—44no—
100C22×C127—44no—
101C22×C127—82no—
102C22×C127#12011yescharacteristic
103C22×C127#12011yescharacteristic
104C22×C127C22×D411yescharacteristic
105C2×D127—24no—
106C2×D127—44no—
107C2×D127—88no—
108C4×C127—42no—
109C4×C127—42no—
110C4×C127#12012yes—
111C4×C127C22×D412yes—
112Dic127—24no—
113C2×C127—21no—
114C2×C127—21no—
115C2×C127—44no—
116C2×C127#45512yescomplemented
117C2×C127C2×#12011yescharacteristic
118D127—44no—
119C2×#455—1271nomaximal, Sylow
120C127C2×#45511yescharacteristic, complemented, Sylow
121C2×H(5,𝔽2)—1271no—
122C2×#137—1272no—
123C2×#124D12711yescharacteristic, complemented
124#455—1278no—
125C2×#104—1272no—
126C2×#100—1271no—
127H(5,𝔽2)—1274no—
128H(5,𝔽2)—2544no—
129H(5,𝔽2)—2542no—
130C2×#41—1272no—
131C2×#41—2542no—
132C22×Q8C2×D12711yescharacteristic, complemented
133C22×D4—2542no—
134C22×D4—2542no—
135C22×D4C2×D12711yescharacteristic
136#137—1278no—
137#137—2544no—
138C2×Q16—2542no—
139C2×SD16—2542no—
140C2×M16—1272no—
141#124—22no—
142#124C2×D12714yescomplemented
143C2×C22⋊C4—22no—
144C2×C42C2×D12711yescharacteristic, complemented
145#104—1278no—
146#100—1274no—
147C24—2542no—
148C24C22⋊D12712yes—
149#41—1274no—
150#41—1274no—
151#41—2548no—
152#41—2548no—
153#41—2542no—
154#41—2542no—
155C2×Q8—21no—
156C2×Q8—22no—
157C2×Q8—22no—
158C2×Q8—1272no—
159C2×Q8C22×D12712yescomplemented
160C2×D4—22no—
161C2×D4—22no—
162C2×D4—1272no—
163C2×D4—2548no—
164C2×D4—2548no—
165C2×D4—2544no—
166C2×D4—2548no—
167C2×D4—2542no—
168C2×D4—2548no—
169C2×D4C22⋊D12712yes—
170C2×D4C22×D12712yes—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—2542no—
174C22×C4—2542no—
175C22×C4C22×D12711yescharacteristic
176Q16—2548no—
177SD16—2548no—
178M16—1274no—
179M16—2542no—
180C2×C8—2542no—
181C22⋊C4—28no—
182C42—21no—
183C42C22×D12712yescomplemented
184C23—22no—
185C23—22no—
186C23—44no—
187C23—10164no—
188C23—2548no—
189C23—2542no—
190C23C2×C22⋊D12711yescharacteristic
191C23C2×C22⋊D12714yes—
192Q8—24no—
193Q8—42no—
194Q8—1274no—
195Q8—2542no—
196D4—24no—
197D4—24no—
198D4—1274no—
199D4—25416no—
200D4—2544no—
201D4—2544no—
202D4—2542no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—42no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—2548no—
209C2×C4—2542no—
210C2×C4—2544no—
211C2×C4—2544no—
212C2×C4D4×D12712yes—
213C2×C4D4×D12712yes—
214C2×C4C2×C22⋊D12711yescharacteristic
215C2×C4C23×D12711yescharacteristic
216C8—2544no—
217C22—22no—
218C22—24no—
219C22—44no—
220C22—44no—
221C22—82no—
222C22—10168no—
223C22—5084no—
224C22—2544no—
225C22#1686117711yescharacteristic
226C22#1686117711yescharacteristic
227C22C22×C22⋊D12711yescharacteristic
228C4—42no—
229C4—42no—
230C4—2544no—
231C4#1686117712yes—
232C4C2×D4×D12712yes—
233C2—21no—
234C2—21no—
235C2—44no—
236C2—5084no—
237C2#2255005312yesdirect product factor
238C2C2×#1686117711yescharacteristic
subgroupG/H
centerC22#16861177
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×#124×C127C2
socleC22×C127#120
prime radicalC2×#16861254C2
hypercenterC2×#124D127
hypocenterC127C2×#455
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
2. C2×#16861240C2semidirect
3. #22550053C2direct (N × H)
4. C2×#16861137C2semidirect
5. C2×#16861126C2semidirect
6. C2×#124×C127C2semidirect
15. #16861240C22semidirect
18. C2×#14523232C22semidirect
19. #16861137C22semidirect
20. #16861126C22semidirect
21. C22×Q8×C127C22semidirect
24. #124×C127C22semidirect
26. C2×C42×C127C22semidirect
47. #14523232C23semidirect
53. C2×Q8×C127C23semidirect
63. C42×C127C23semidirect
116. C2×C127#455semidirect
120. C127C2×#455semidirect
123. C2×#124D127semidirect
132. C22×Q8C2×D127semidirect
142. #124C2×D127semidirect
144. C2×C42C2×D127semidirect
159. C2×Q8C22×D127semidirect
183. C42C22×D127semidirect
237. C2#22550053direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168612542
C2×#168612402
#225500532
C2×#168611372
C2×#168611262
C2×#124×C1272
C2×#455127