Finite Group Databasebeta

← C2×#22550008 #30647904

Subgroups — C2×#22550008 #30647904

Order 16256.

1048 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 1048 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, complemented
2C22×#14523233C211yesmaximal, characteristic, complemented
3C2×#16861180C211yesmaximal, characteristic, complemented
4#22550008C218yesmaximal, direct product factor
5C2×#16861131C211yesmaximal, characteristic, complemented
6C2×#16861130C211yesmaximal, characteristic, complemented
7C2×(C2×D127)⋊C8C211yesmaximal, characteristic, complemented
8C22×D8×C127C211yesmaximal, characteristic, complemented
9C24×D127—21no—
10C22×D508C2211yescharacteristic
11C22×C22⋊D127—21no—
12C22×C22⋊D127—21no—
13C2×D4×D127—24no—
14C2×D4×D127—22no—
15C2×D4×D127—24no—
16C2×D4×D127C2214yescomplemented
17C22×C4×D127C2211yescharacteristic
18C2×#14523233—24no—
19C2×#14523233C2214yescomplemented
20C22×C127⋊C8C2211yescharacteristic, complemented
21#16861180—22no—
22#16861180C2214yescomplemented
23C2×C4⋊Dic127C2211yescharacteristic, complemented
24C2×C22⋊Dic127—21no—
25#16861131C2214yescomplemented
26#16861130C2214yescomplemented
27(C2×D127)⋊C8C2214yescomplemented
28C22×D4×C127C2211yescharacteristic
29C22×D4×C127C2211yescharacteristic, complemented
30C2×D8×C127—24no—
31C2×D8×C127C2214yescomplemented
32C22×C8×C127C2211yescharacteristic, complemented
33C23×D127—28no—
34C23×D127—24no—
35C23×D127—21no—
36C23×D127—44no—
37C23×D127—44no—
38C23×D127D411yescharacteristic
39C2×D508—21no—
40C2×D508—22no—
41C2×D508D414yescomplemented
42C2×D508C2312yes—
43C2×C22⋊D127—28no—
44C2×C22⋊D127—24no—
45C2×C22⋊D127—24no—
46C2×C22⋊D127—44no—
47D4×D127—216no—
48D4×D127—28no—
49D4×D127—28no—
50C22×Dic127—21no—
51C22×Dic127D411yescharacteristic
52C2×C4×D127—22no—
53C2×C4×D127—22no—
54C2×C4×D127D412yes—
55C2×C4×D127C2312yes—
56#14523233—28no—
57C2×C127⋊C8—22no—
58C2×C127⋊C8C2312yescomplemented
59C4⋊Dic127—21no—
60C4⋊Dic127C2312yescomplemented
61C22⋊Dic127—24no—
62C24×C127—21no—
63C24×C127—21no—
64C2×D4×C127—21no—
65C2×D4×C127—22no—
66C2×D4×C127—22no—
67C2×D4×C127—22no—
68C2×D4×C127—21no—
69C2×D4×C127D414yescomplemented
70C2×D4×C127C2312yes—
71C2×D4×C127C2312yescomplemented
72C22×C4×C127C2311yescharacteristic
73D8×C127—28no—
74C2×C8×C127—22no—
75C2×C8×C127C2312yescomplemented
76C22×D127—22no—
77C22×D127—22no—
78C22×D127—216no—
79C22×D127—24no—
80C22×D127—22no—
81C22×D127—48no—
82C22×D127—432no—
83C22×D127—42no—
84C22×D127—42no—
85C22×D127—48no—
86C22×D127D814yescomplemented
87C22×D127C2×D412yes—
88D508—24no—
89D508—22no—
90D508C2×D414yescomplemented
91C22⋊D127—216no—
92C22⋊D127—48no—
93C2×Dic127—22no—
94C2×Dic127—22no—
95C2×Dic127—42no—
96C2×Dic127C2×D412yes—
97C4×D127—24no—
98C4×D127—22no—
99C4×D127—22no—
100C127⋊C8—22no—
101C23×C127—24no—
102C23×C127—22no—
103C23×C127—22no—
104C23×C127—42no—
105C23×C127—42no—
106C23×C127—42no—
107C23×C127—42no—
108C23×C127—42no—
109C23×C127C2×D411yescharacteristic
110D4×C127—24no—
111D4×C127—22no—
112D4×C127—22no—
113D4×C127—22no—
114D4×C127—42no—
115D4×C127C2×D414yescomplemented
116C2×C4×C127C2×D411yescharacteristic
117C2×C4×C127C2×D412yes—
118C2×C4×C127C2×D412yes—
119C2×C4×C127C2411yescharacteristic
120C8×C127—22no—
121C22×C127—24no—
122C22×C127—42no—
123C22×C127—42no—
124C22×C127—48no—
125C22×C127—42no—
126C22×C127—42no—
127C22×C127—42no—
128C22×C127—84no—
129C22×C127#12012yes—
130C22×C127C2×D812yes—
131C22×C127AGL(1,ℤ/8)12yes—
132C22×C127C22×D411yescharacteristic
133C2×D127—22no—
134C2×D127—22no—
135C2×D127—28no—
136C2×D127—24no—
137C2×D127—432no—
138C2×D127—42no—
139C2×D127—42no—
140C2×D127—48no—
141C2×D127C2×D814yescomplemented
142C4×C127#12012yes—
143C4×C127C22×D411yescharacteristic
144C4×C127C22×D411yescharacteristic
145Dic127—22no—
146Dic127—42no—
147C2×C127—44no—
148C2×C127—82no—
149C2×C127#44714yescomplemented
150C2×C127C2×#12011yescharacteristic
151C2×C127C22×D811yescharacteristic
152C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
153D127—24no—
154D127—44no—
155C2×#447—1271nomaximal, Sylow
156C127C2×#44711yescharacteristic, complemented, Sylow
157C23×D4—1271no—
158C22×D8—1271no—
159C22×D8D12711yescharacteristic, complemented
160C2×#121—1271no—
161#447—1278no—
162C2×#102—1271no—
163C2×#102—1271no—
164C2×C22⋊C8—1271no—
165C25—2541no—
166C22×D4—1274no—
167C22×D4—1271no—
168C22×D4—2542no—
169C22×D4—2541no—
170C22×D4—2544no—
171C22×D4—2541no—
172C22×D4—2544no—
173C22×D4C2×D12711yescharacteristic
174C22×D4C2×D12711yescharacteristic, complemented
175C23×C4—1271no—
176C2×D8—24no—
177C2×D8—1274no—
178C2×D8—2544no—
179C2×D8C2×D12714yescomplemented
180C22×C8—1271no—
181C22×C8C2×D12711yescharacteristic, complemented
182#121—1274no—
183#121—2542no—
184C2×C4⋊C4—1271no—
185C2×C22⋊C4—2541no—
186#102—1274no—
187#102—1274no—
188C22⋊C8—1274no—
189C24—21no—
190C24—21no—
191C24—1271no—
192C24—5084no—
193C24—5084no—
194C24—2544no—
195C24—2548no—
196C24—2541no—
197C2×D4—22no—
198C2×D4—21no—
199C2×D4—22no—
200C2×D4—21no—
201C2×D4—22no—
202C2×D4—1274no—
203C2×D4—1272no—
204C2×D4—5084no—
205C2×D4—2544no—
206C2×D4—2548no—
207C2×D4—25416no—
208C2×D4—2544no—
209C2×D4—2548no—
210C2×D4—2548no—
211C2×D4—2541no—
212C2×D4—2542no—
213C2×D4C22⋊D12714yescomplemented
214C2×D4C22×D12712yes—
215C2×D4C22×D12712yescomplemented
216C22×C4—1271no—
217C22×C4—1272no—
218C22×C4—1272no—
219C22×C4—2541no—
220C22×C4—2542no—
221C22×C4—2542no—
222C22×C4C22×D12711yescharacteristic
223D8—28no—
224D8—2548no—
225C2×C8—22no—
226C2×C8—1272no—
227C2×C8—2542no—
228C2×C8C22×D12712yescomplemented
229C4⋊C4—1272no—
230C4⋊C4—2541no—
231C22⋊C4—2544no—
232C23—22no—
233C23—24no—
234C23—22no—
235C23—42no—
236C23—42no—
237C23—42no—
238C23—42no—
239C23—42no—
240C23—1274no—
241C23—1272no—
242C23—5088no—
243C23—5088no—
244C23—50832no—
245C23—5082no—
246C23—5082no—
247C23—25416no—
248C23—2542no—
249C23—2544no—
250C23—2542no—
251C23—2542no—
252C23D4×D12711yescharacteristic
253D4—22no—
254D4—22no—
255D4—22no—
256D4—24no—
257D4—42no—
258D4—1274no—
259D4—5088no—
260D4—25416no—
261D4—2542no—
262D4—2544no—
263D4C2×C22⋊D12714yescomplemented
264C2×C4—1272no—
265C2×C4—5082no—
266C2×C4—2542no—
267C2×C4—2542no—
268C2×C4—2542no—
269C2×C4—2542no—
270C2×C4—2544no—
271C2×C4D4×D12712yes—
272C2×C4D4×D12711yescharacteristic
273C2×C4C2×C22⋊D12712yes—
274C2×C4C23×D12711yescharacteristic
275C8—22no—
276C8—2542no—
277C22—24no—
278C22—42no—
279C22—42no—
280C22—48no—
281C22—42no—
282C22—42no—
283C22—42no—
284C22—84no—
285C22—1274no—
286C22—50832no—
287C22—5082no—
288C22—5082no—
289C22—5088no—
290C22—2544no—
291C22—2542no—
292C22—2542no—
293C22—2548no—
294C22#1686117712yes—
295C22D8×D12712yes—
296C22#1686123312yes—
297C22C2×D4×D12711yescharacteristic
298C4—5082no—
299C4—2542no—
300C4#1686117712yes—
301C4C2×D4×D12711yescharacteristic
302C4C22×C22⋊D12711yescharacteristic
303C2—44no—
304C2—82no—
305C2—5084no—
306C2—2544no—
307C2#2255000814yesdirect product factor
308C2C2×#1686117711yescharacteristic
309C2C2×D8×D12711yescharacteristic
310C2C2×#1686123311yescharacteristic
subgroupG/H
centerC23D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC22×D8×C127C2
socleC23×C127C2×D4
hypercenterC22×D8D127
hypocenterC127C2×#447
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×D4×D127C2semidirect
2. C22×#14523233C2semidirect
3. C2×#16861180C2semidirect
4. #22550008C2direct (N × H)
5. C2×#16861131C2semidirect
6. C2×#16861130C2semidirect
7. C2×(C2×D127)⋊C8C2semidirect
8. C22×D8×C127C2semidirect
16. C2×D4×D127C22semidirect
19. C2×#14523233C22semidirect
20. C22×C127⋊C8C22semidirect
22. #16861180C22semidirect
23. C2×C4⋊Dic127C22semidirect
25. #16861131C22semidirect
26. #16861130C22semidirect
27. (C2×D127)⋊C8C22semidirect
29. C22×D4×C127C22semidirect
31. C2×D8×C127C22semidirect
32. C22×C8×C127C22semidirect
41. C2×D508D4semidirect
58. C2×C127⋊C8C23semidirect
60. C4⋊Dic127C23semidirect
69. C2×D4×C127D4semidirect
71. C2×D4×C127C23semidirect
75. C2×C8×C127C23semidirect
86. C22×D127D8semidirect
90. D508C2×D4semidirect
115. D4×C127C2×D4semidirect
141. C2×D127C2×D8semidirect
149. C2×C127#447semidirect
156. C127C2×#447semidirect
159. C22×D8D127semidirect
174. C22×D4C2×D127semidirect
179. C2×D8C2×D127semidirect
181. C22×C8C2×D127semidirect
213. C2×D4C22⋊D127semidirect
215. C2×D4C22×D127semidirect
228. C2×C8C22×D127semidirect
263. D4C2×C22⋊D127semidirect
307. C2#22550008direct (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×#145232332
C2×#168611802
#225500082
C2×#168611312
C2×#168611302
C2×(C2×D127)⋊C82
C22×D8×C1272
C2×#447127