Finite Group Databasebeta

← C2×#22550030 #30647922

Subgroups — C2×#22550030 #30647922

Order 16256.

740 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 740 conjugacy classes of subgroups — too many to draw. Only its 173 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×Q8×D127C211yesmaximal, characteristic
2C22×#14523238C211yesmaximal, characteristic, complemented
3C2×#16861185C211yesmaximal, characteristic
4#22550030C218yesmaximal, direct product factor
5C2×#16861134C211yesmaximal, characteristic, complemented
6C2×#16861133C211yesmaximal, characteristic, complemented
7C2×(C2×D127)⋊C8C211yesmaximal, characteristic
8C22×Q16×C127C211yesmaximal, characteristic, complemented
9C22×Dic254—21no—
10C22×Dic254C2211yescharacteristic
11C2×Q8×D127—24no—
12C2×Q8×D127—22no—
13C2×Q8×D127—24no—
14C2×Q8×D127C2214yes—
15C22×C4×D127—21no—
16C22×C4×D127C2211yescharacteristic
17C2×#14523238—24no—
18C2×#14523238C2214yescomplemented
19C22×C127⋊C8C2211yescharacteristic
20#16861185—22no—
21#16861185C2214yes—
22C2×C4⋊Dic127C2211yescharacteristic
23C2×Dic127⋊C4—21no—
24C2×(C2×D127)⋊C4—21no—
25#16861134C2214yescomplemented
26#16861133C2214yescomplemented
27(C2×D127)⋊C8C2214yes—
28C22×Q8×C127C2211yescharacteristic
29C22×Q8×C127C2211yescharacteristic
30C2×Q16×C127—24no—
31C2×Q16×C127C2214yescomplemented
32C22×C8×C127C2211yescharacteristic
33C23×D127D411yescharacteristic
34C2×Dic254—28no—
35C2×Dic254—24no—
36C2×Dic254—21no—
37C2×Dic254—22no—
38C2×Dic254D414yes—
39C2×Dic254C2312yes—
40Q8×D127—216no—
41Q8×D127—28no—
42Q8×D127—28no—
43C22×Dic127—21no—
44C22×Dic127—21no—
45C22×Dic127D411yescharacteristic
46C2×C4×D127—28no—
47C2×C4×D127—24no—
48C2×C4×D127—22no—
49C2×C4×D127—22no—
50C2×C4×D127D412yes—
51C2×C4×D127C2312yes—
52#14523238—28no—
53C2×C127⋊C8—22no—
54C2×C127⋊C8C2312yes—
55C4⋊Dic127—21no—
56C4⋊Dic127C2312yes—
57Dic127⋊C4—24no—
58(C2×D127)⋊C4—24no—
59C2×Q8×C127—22no—
60C2×Q8×C127—21no—
61C2×Q8×C127—21no—
62C2×Q8×C127—22no—
63C2×Q8×C127—22no—
64C2×Q8×C127D414yes—
65C2×Q8×C127C2312yes—
66C2×Q8×C127C2312yes—
67C22×C4×C127—21no—
68C22×C4×C127—21no—
69C22×C4×C127C2311yescharacteristic
70Q16×C127—28no—
71C2×C8×C127—22no—
72C2×C8×C127C2312yes—
73C22×D127—22no—
74C22×D127—22no—
75C22×D127Q1614yescomplemented
76C22×D127C2×D412yes—
77Dic254—216no—
78Dic254—24no—
79Dic254—22no—
80Dic254C2×D414yes—
81C2×Dic127—24no—
82C2×Dic127—22no—
83C2×Dic127—22no—
84C2×Dic127—22no—
85C2×Dic127—42no—
86C2×Dic127C2×D412yes—
87C4×D127—216no—
88C4×D127—24no—
89C4×D127—22no—
90C4×D127—22no—
91C127⋊C8—22no—
92C23×C127C2×D411yescharacteristic
93Q8×C127—24no—
94Q8×C127—22no—
95Q8×C127—22no—
96Q8×C127—22no—
97Q8×C127—42no—
98Q8×C127C2×D414yes—
99C2×C4×C127—24no—
100C2×C4×C127—22no—
101C2×C4×C127—22no—
102C2×C4×C127—42no—
103C2×C4×C127C2×D411yescharacteristic
104C2×C4×C127C2×D412yes—
105C2×C4×C127C2×D412yes—
106C2×C4×C127C2411yescharacteristic
107C8×C127—22no—
108C22×C127#12012yes—
109C22×C127C2×Q1612yes—
110C22×C127#13712yes—
111C22×C127C22×D411yescharacteristic
112C2×D127—22no—
113C2×D127—22no—
114C2×D127—28no—
115C2×D127C2×Q1614yescomplemented
116C4×C127—24no—
117C4×C127—42no—
118C4×C127#12012yes—
119C4×C127C22×D411yescharacteristic
120C4×C127C22×D411yescharacteristic
121Dic127—24no—
122Dic127—22no—
123Dic127—42no—
124C2×C127#45114yescomplemented
125C2×C127C2×#12011yescharacteristic
126C2×C127C22×Q1611yescharacteristic
127C2×C127C2×#13711yescharacteristic
128D127—24no—
129C2×#451—1271nomaximal, Sylow
130C127C2×#45111yescharacteristic, complemented, Sylow
131C23×Q8—1271no—
132C22×Q16—1271no—
133C22×Q16D12711yescharacteristic, complemented
134C2×C22⋊Q8—1271no—
135#451—1278no—
136C2×#103—1271no—
137C2×#103—1271no—
138C2×C22⋊C8—1271no—
139C22×Q8—1274no—
140C22×Q8—1271no—
141C22×Q8—2541no—
142C22×Q8—2542no—
143C22×Q8—2544no—
144C22×Q8—2544no—
145C22×Q8C2×D12711yescharacteristic
146C22×Q8C2×D12711yescharacteristic
147C23×C4—1271no—
148C23×C4—2541no—
149C2×Q16—24no—
150C2×Q16—1274no—
151C2×Q16—2544no—
152C2×Q16C2×D12714yescomplemented
153C22×C8—1271no—
154C22×C8C2×D12711yescharacteristic
155C22⋊Q8—1274no—
156C22⋊Q8—2542no—
157C2×C4⋊C4—1271no—
158C2×C4⋊C4—2541no—
159C2×C22⋊C4—2541no—
160#103—1274no—
161#103—1274no—
162C22⋊C8—1274no—
163C24—1271no—
164C2×Q8—21no—
165C2×Q8—22no—
166C2×Q8—21no—
167C2×Q8—22no—
168C2×Q8—22no—
169C2×Q8—1274no—
170C2×Q8—1272no—
171C2×Q8—2544no—
172C2×Q8—2548no—
173C2×Q8—2548no—
174C2×Q8—25416no—
175C2×Q8—2548no—
176C2×Q8—2541no—
177C2×Q8—2542no—
178C2×Q8C22⋊D12714yes—
179C2×Q8C22×D12712yes—
180C2×Q8C22×D12712yes—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—1271no—
184C22×C4—1272no—
185C22×C4—1272no—
186C22×C4—2544no—
187C22×C4—2548no—
188C22×C4—2541no—
189C22×C4—2541no—
190C22×C4—2542no—
191C22×C4—2542no—
192C22×C4C22×D12711yescharacteristic
193Q16—28no—
194Q16—2548no—
195C2×C8—22no—
196C2×C8—1272no—
197C2×C8—2542no—
198C2×C8C22×D12712yes—
199C4⋊C4—1272no—
200C4⋊C4—2544no—
201C4⋊C4—2541no—
202C22⋊C4—2544no—
203C23—1274no—
204C23—1272no—
205C23—2542no—
206C23—2542no—
207C23D4×D12711yescharacteristic
208Q8—22no—
209Q8—24no—
210Q8—22no—
211Q8—22no—
212Q8—42no—
213Q8—1274no—
214Q8—25416no—
215Q8—2542no—
216Q8—2544no—
217Q8C2×C22⋊D12714yes—
218C2×C4—22no—
219C2×C4—24no—
220C2×C4—22no—
221C2×C4—42no—
222C2×C4—1272no—
223C2×C4—5082no—
224C2×C4—25416no—
225C2×C4—2542no—
226C2×C4—2542no—
227C2×C4—2542no—
228C2×C4—2544no—
229C2×C4—2542no—
230C2×C4—2542no—
231C2×C4—2544no—
232C2×C4D4×D12712yes—
233C2×C4D4×D12711yescharacteristic
234C2×C4C2×C22⋊D12712yes—
235C2×C4C23×D12711yescharacteristic
236C8—22no—
237C8—2542no—
238C22—1274no—
239C22—2548no—
240C22—2542no—
241C22—2542no—
242C22#1686117712yes—
243C22Q16×D12712yes—
244C22#1686124112yes—
245C22C2×D4×D12711yescharacteristic
246C4—24no—
247C4—42no—
248C4—5082no—
249C4—2542no—
250C4—2544no—
251C4#1686117712yes—
252C4C2×D4×D12711yescharacteristic
253C4C22×C22⋊D12711yescharacteristic
254C2—2544no—
255C2#2255003014yesdirect product factor
256C2C2×#1686117711yescharacteristic
257C2C2×Q16×D12711yescharacteristic
258C2C2×#1686124111yescharacteristic
subgroupG/H
centerC23D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC22×Q16×C127C2
socleC23×C127C2×D4
prime radicalC23×D127D4
hypercenterC22×Q16D127
hypocenterC127C2×#451
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
2. C22×#14523238C2semidirect
4. #22550030C2direct (N × H)
5. C2×#16861134C2semidirect
6. C2×#16861133C2semidirect
8. C22×Q16×C127C2semidirect
18. C2×#14523238C22semidirect
25. #16861134C22semidirect
26. #16861133C22semidirect
31. C2×Q16×C127C22semidirect
75. C22×D127Q16semidirect
115. C2×D127C2×Q16semidirect
124. C2×C127#451semidirect
130. C127C2×#451semidirect
133. C22×Q16D127semidirect
152. C2×Q16C2×D127semidirect
255. C2#22550030direct (N × H)

Maximal subgroups

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

subgroupindex
C22×Q8×D1272
C22×#145232382
C2×#168611852
#225500302
C2×#168611342
C2×#168611332
C2×(C2×D127)⋊C82
C22×Q16×C1272
C2×#451127