Finite Group Databasebeta

← C2×#22550028 #30647913

Subgroups — C2×#22550028 #30647913

Order 16256.

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

normal subgroups only

This group has 996 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×D4×D127C211yesmaximal, characteristic
2C22×#14523237C211yesmaximal, characteristic, complemented
3C22×#14523236C211yesmaximal, characteristic, complemented
4C2×#16861189C211yesmaximal, characteristic, complemented
5#22550028C218yesmaximal, direct product factor
6C2×#16861129C211yesmaximal, characteristic, complemented
7C2×(C2×D127)⋊C8C211yesmaximal, characteristic, complemented
8C2×#102×C127C211yesmaximal, characteristic, complemented
9C24×D127—21no—
10C22×Dic254C2211yescharacteristic, complemented
11C22×D508C2211yescharacteristic
12C22×C22⋊D127—21no—
13C2×D4×D127—24no—
14C2×D4×D127—22no—
15C2×D4×D127—24no—
16C2×D4×D127C2214yes—
17C22×C4×D127C2211yescharacteristic
18C2×#14523237—24no—
19C2×#14523237C2214yescomplemented
20C2×#14523236—24no—
21C2×#14523236C2214yescomplemented
22C22×C127⋊C8C2211yescharacteristic, complemented
23#16861189—22no—
24#16861189C2214yescomplemented
25C2×C4⋊Dic127—21no—
26C2×(C2×D127)⋊C4—21no—
27#16861129C2214yescomplemented
28(C2×D127)⋊C8C2214yescomplemented
29C22×D4×C127C2211yescharacteristic
30C22×C8×C127C2211yescharacteristic, complemented
31C2×C4⋊C4×C127C2211yescharacteristic, complemented
32#102×C127C2214yescomplemented
33C23×D127—28no—
34C23×D127—24no—
35C23×D127—21no—
36C23×D127—44no—
37C23×D127—44no—
38C23×D127D411yescharacteristic
39C2×Dic254—21no—
40C2×Dic254—22no—
41C2×Dic254—22no—
42C2×Dic254C2312yescomplemented
43C2×D508—21no—
44C2×D508—22no—
45C2×D508D414yes—
46C2×D508C2312yes—
47C2×C22⋊D127—28no—
48C2×C22⋊D127—24no—
49D4×D127—216no—
50D4×D127—28no—
51D4×D127—28no—
52C22×Dic127—21no—
53C22×Dic127D411yescharacteristic
54C2×C4×D127—22no—
55C2×C4×D127—22no—
56C2×C4×D127D412yes—
57C2×C4×D127C2312yes—
58#14523237—28no—
59#14523236—28no—
60C2×C127⋊C8—22no—
61C2×C127⋊C8C2312yescomplemented
62C4⋊Dic127—24no—
63(C2×D127)⋊C4—24no—
64C24×C127—21no—
65C2×D4×C127—21no—
66C2×D4×C127—22no—
67C2×D4×C127D414yes—
68C2×D4×C127C2312yes—
69C22×C4×C127—21no—
70C22×C4×C127C2311yescharacteristic
71C2×C8×C127—22no—
72C2×C8×C127C2312yescomplemented
73C4⋊C4×C127—21no—
74C4⋊C4×C127C2312yescomplemented
75C22×D127—22no—
76C22×D127—22no—
77C22×D127—216no—
78C22×D127—24no—
79C22×D127—22no—
80C22×D127—48no—
81C22×D127—432no—
82C22×D127—42no—
83C22×D127—42no—
84C22×D127—48no—
85C22×D127SD1614yescomplemented
86C22×D127C2×D412yes—
87Dic254—22no—
88Dic254—22no—
89Dic254—42no—
90D508—24no—
91D508—22no—
92D508C2×D414yes—
93C22⋊D127—216no—
94C2×Dic127—22no—
95C2×Dic127—22no—
96C2×Dic127—42no—
97C2×Dic127C2×D412yes—
98C4×D127—24no—
99C4×D127—22no—
100C4×D127—22no—
101C127⋊C8—22no—
102C23×C127—24no—
103C23×C127—22no—
104C23×C127—42no—
105C23×C127—42no—
106C23×C127C2×D411yescharacteristic
107D4×C127—24no—
108D4×C127—22no—
109D4×C127C2×D414yes—
110C2×C4×C127—22no—
111C2×C4×C127—42no—
112C2×C4×C127C2×D411yescharacteristic
113C2×C4×C127C2×D412yes—
114C2×C4×C127C2×D412yes—
115C2×C4×C127C2411yescharacteristic
116C8×C127—22no—
117C22×C127—24no—
118C22×C127—42no—
119C22×C127—42no—
120C22×C127—48no—
121C22×C127#12012yes—
122C22×C127C2×SD1612yes—
123C22×C127AGL(1,ℤ/8)12yes—
124C22×C127C22×D411yescharacteristic
125C2×D127—22no—
126C2×D127—22no—
127C2×D127—28no—
128C2×D127—24no—
129C2×D127—432no—
130C2×D127—42no—
131C2×D127—42no—
132C2×D127—48no—
133C2×D127C2×SD1614yescomplemented
134C4×C127—42no—
135C4×C127#12012yes—
136C4×C127C22×D411yescharacteristic
137C4×C127C22×D411yescharacteristic
138Dic127—22no—
139Dic127—42no—
140C2×C127—44no—
141C2×C127#45014yescomplemented
142C2×C127C2×#12011yescharacteristic
143C2×C127C22×SD1611yescharacteristic
144C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
145D127—24no—
146D127—44no—
147C2×#450—1271nomaximal, Sylow
148C127C2×#45011yescharacteristic, complemented, Sylow
149C23×D4—1271no—
150C22×SD16—1271no—
151C22×SD16—1271no—
152C2×C22⋊Q8—1271no—
153#450—1278no—
154C2×#102—1271no—
155C2×#102D12711yescharacteristic, complemented
156C2×C22⋊C8—1271no—
157C25—2541no—
158C22×Q8—1271no—
159C22×D4—1274no—
160C22×D4—1271no—
161C22×D4—2542no—
162C22×D4—2541no—
163C22×D4—2544no—
164C22×D4—2544no—
165C22×D4C2×D12711yescharacteristic
166C23×C4—1271no—
167C2×SD16—1274no—
168C2×SD16—1274no—
169C2×SD16—2544no—
170C2×SD16—2544no—
171C22×C8—1271no—
172C22×C8C2×D12711yescharacteristic, complemented
173C22⋊Q8—1274no—
174C22⋊Q8—2542no—
175C2×C4⋊C4—2541no—
176C2×C4⋊C4C2×D12711yescharacteristic, complemented
177C2×C22⋊C4—2541no—
178#102—1274no—
179#102C2×D12714yescomplemented
180C22⋊C8—1274no—
181C24—21no—
182C24—1271no—
183C24—5084no—
184C24—5084no—
185C24—2544no—
186C24—2548no—
187C24—2541no—
188C2×Q8—1272no—
189C2×Q8—2542no—
190C2×Q8—2542no—
191C2×Q8—2541no—
192C2×D4—22no—
193C2×D4—21no—
194C2×D4—1274no—
195C2×D4—1272no—
196C2×D4—2544no—
197C2×D4—2548no—
198C2×D4—25416no—
199C2×D4—2548no—
200C2×D4—2548no—
201C2×D4—2541no—
202C2×D4—2542no—
203C2×D4D50814yes—
204C2×D4C22×D12712yes—
205C22×C4—21no—
206C22×C4—1271no—
207C22×C4—1272no—
208C22×C4—1272no—
209C22×C4—2541no—
210C22×C4—2542no—
211C22×C4—2542no—
212C22×C4C22×D12711yescharacteristic
213SD16—2548no—
214SD16—2548no—
215C2×C8—22no—
216C2×C8—1272no—
217C2×C8—2542no—
218C2×C8C22×D12712yescomplemented
219C4⋊C4—21no—
220C4⋊C4—2544no—
221C4⋊C4C22×D12712yescomplemented
222C22⋊C4—2544no—
223C23—22no—
224C23—24no—
225C23—42no—
226C23—42no—
227C23—1274no—
228C23—1272no—
229C23—5088no—
230C23—50832no—
231C23—5088no—
232C23—5082no—
233C23—5082no—
234C23—25416no—
235C23—2542no—
236C23—2544no—
237C23—2542no—
238C23—2542no—
239C23D4×D12711yescharacteristic
240Q8—5082no—
241Q8—2542no—
242Q8—2542no—
243D4—22no—
244D4—24no—
245D4—1274no—
246D4—25416no—
247D4—2542no—
248D4—2544no—
249D4C2×D50814yes—
250C2×C4—22no—
251C2×C4—42no—
252C2×C4—1272no—
253C2×C4—5082no—
254C2×C4—2542no—
255C2×C4—2542no—
256C2×C4—2542no—
257C2×C4—2542no—
258C2×C4—2544no—
259C2×C4D4×D12712yes—
260C2×C4D4×D12711yescharacteristic
261C2×C4C2×D50812yes—
262C2×C4C23×D12711yescharacteristic
263C8—22no—
264C8—2542no—
265C22—24no—
266C22—48no—
267C22—42no—
268C22—42no—
269C22—1274no—
270C22—50832no—
271C22—5088no—
272C22—5082no—
273C22—5082no—
274C22—2544no—
275C22—2548no—
276C22—2542no—
277C22—2542no—
278C22#1686117912yes—
279C22SD16×D12712yes—
280C22#1686123312yes—
281C22C2×D4×D12711yescharacteristic
282C4—42no—
283C4—5082no—
284C4—2542no—
285C4#1686117912yes—
286C4C2×D4×D12711yescharacteristic
287C4C22×D50811yescharacteristic
288C2—44no—
289C2—5084no—
290C2—2544no—
291C2#2255002814yesdirect product factor
292C2C2×#1686117911yescharacteristic
293C2C2×SD16×D12711yescharacteristic
294C2C2×#1686123311yescharacteristic
subgroupG/H
centerC23D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×#102×C127C2
socleC23×C127C2×D4
prime radicalC22×D4×D127C2
hypercenterC2×#102D127
hypocenterC127C2×#450
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
2. C22×#14523237C2semidirect
3. C22×#14523236C2semidirect
4. C2×#16861189C2semidirect
5. #22550028C2direct (N × H)
6. C2×#16861129C2semidirect
7. C2×(C2×D127)⋊C8C2semidirect
8. C2×#102×C127C2semidirect
10. C22×Dic254C22semidirect
19. C2×#14523237C22semidirect
21. C2×#14523236C22semidirect
22. C22×C127⋊C8C22semidirect
24. #16861189C22semidirect
27. #16861129C22semidirect
28. (C2×D127)⋊C8C22semidirect
30. C22×C8×C127C22semidirect
31. C2×C4⋊C4×C127C22semidirect
32. #102×C127C22semidirect
42. C2×Dic254C23semidirect
61. C2×C127⋊C8C23semidirect
72. C2×C8×C127C23semidirect
74. C4⋊C4×C127C23semidirect
85. C22×D127SD16semidirect
133. C2×D127C2×SD16semidirect
141. C2×C127#450semidirect
148. C127C2×#450semidirect
155. C2×#102D127semidirect
172. C22×C8C2×D127semidirect
176. C2×C4⋊C4C2×D127semidirect
179. #102C2×D127semidirect
218. C2×C8C22×D127semidirect
221. C4⋊C4C22×D127semidirect
291. C2#22550028direct (N × H)

Maximal subgroups

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

subgroupindex
C22×D4×D1272
C22×#145232372
C22×#145232362
C2×#168611892
#225500282
C2×#168611292
C2×(C2×D127)⋊C82
C2×#102×C1272
C2×#450127