Finite Group Databasebeta

← #30648664

Subgroups — #30648664

Order 16256.

414 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 125 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
1#22550388C211yesmaximal, characteristic
2C2×#16861167C211yesmaximal, characteristic, complemented
3#22550220C212yesmaximal, complemented
4#22550219C212yesmaximal, complemented
5#22550203C212yesmaximal, complemented
6#22550202C212yesmaximal, complemented
7#22549919C211yesmaximal, characteristic, complemented
8#22549878C211yesmaximal, characteristic, complemented
9#22549875C211yesmaximal, characteristic, complemented
10#22549839C211yesmaximal, characteristic, complemented
11#428×C127C211yesmaximal, characteristic, complemented
12C2×#14523248C2211yescharacteristic
13C22×C4×D127C2211yescharacteristic
14C2×#14523232C2211yescharacteristic
15#16861193—21no—
16#16861189C2211yescharacteristic
17#16861189C2211yescharacteristic
18#16861184C2211yescharacteristic
19#16861184C2211yescharacteristic
20#16861179—21no—
21C4×D508—21no—
22C4×D508—21no—
23#16861167—21no—
24#16861167—21no—
25#16861167C2212yes—
26#16861167C2212yes—
27C2×C4⋊Dic127C2211yescharacteristic
28C2×(C2×D127)⋊C4—21no—
29C2×C4×Dic127—21no—
30#16861142C2212yescomplemented
31#16861140C2212yescomplemented
32#16861134C2212yescomplemented
33#16861132C2212yescomplemented
34#16861131C2212yescomplemented
35#16861129C2212yescomplemented
36(C2×D127)⋊C8C2212yes—
37(C2×D127)⋊C8C2212yes—
38C2×M16×C127C2211yescharacteristic
39#117×C127C2211yescharacteristic
40C2×C4⋊C4×C127C2211yescharacteristic
41#107×C127C2212yescomplemented
42#106×C127C2212yescomplemented
43C23×D127D411yescharacteristic
44#14523248—22no—
45#14523248—22no—
46C2×Dic254C2311yescharacteristic
47C2×D508—21no—
48C2×D508—21no—
49C2×D508C2311yescharacteristic
50C2×C22⋊D127—21no—
51C22×Dic127—21no—
52C22×Dic127D411yescharacteristic
53C2×C4×D127—21no—
54C2×C4×D127—22no—
55C2×C4×D127—22no—
56C2×C4×D127D411yescharacteristic
57C2×C4×D127D411yescharacteristic
58C2×C4×D127C2311yescharacteristic
59C2×C4×D127C2311yescharacteristic
60#14523232—22no—
61C2×C127⋊C8C2312yes—
62C4⋊Dic127—21no—
63C4⋊Dic127—21no—
64C4⋊Dic127—21no—
65C4⋊Dic127C2312yes—
66(C2×D127)⋊C4—21no—
67(C2×D127)⋊C4—21no—
68(C2×D127)⋊C4—22no—
69(C2×D127)⋊C4—22no—
70(C2×D127)⋊C4—21no—
71(C2×D127)⋊C4—21no—
72C4×Dic127—22no—
73C4×Dic127—22no—
74C22×C4×C127—21no—
75C22×C4×C127C2311yescharacteristic
76M16×C127—22no—
77C2×C8×C127C2312yes—
78C4⋊C4×C127—21no—
79C4⋊C4×C127C2311yescharacteristic
80C4⋊C4×C127C2311yescharacteristic
81C4⋊C4×C127C2312yes—
82C22⋊C4×C127—21no—
83C42×C127—21no—
84C22×D127—22no—
85C22×D127—21no—
86C22×D127—41no—
87C22×D127—41no—
88C22×D127C2×D411yescharacteristic
89C22×D127C2×D411yescharacteristic
90Dic254—21no—
91Dic254—21no—
92D508—21no—
93D508—21no—
94D508—42no—
95D508—42no—
96C22⋊D127—42no—
97C2×Dic127—22no—
98C2×Dic127—22no—
99C2×Dic127—21no—
100C2×Dic127—42no—
101C2×Dic127C2×D411yescharacteristic
102C2×Dic127C2×D411yescharacteristic
103C4×D127—22no—
104C4×D127—22no—
105C4×D127—21no—
106C4×D127—21no—
107C4×D127—21no—
108C4×D127—21no—
109C4×D127—42no—
110C127⋊C8—22no—
111C23×C127C2×D411yescharacteristic
112C2×C4×C127—21no—
113C2×C4×C127—21no—
114C2×C4×C127—22no—
115C2×C4×C127—42no—
116C2×C4×C127C2×D411yescharacteristic
117C2×C4×C127#4112yes—
118C2×C4×C127#4112yes—
119C2×C4×C127C2411yescharacteristic
120C8×C127—22no—
121C22×C127—21no—
122C22×C127—21no—
123C22×C127#12312yes—
124C22×C127C22×D411yescharacteristic
125C2×D127—21no—
126C2×D127—21no—
127C2×D127—21no—
128C2×D127—21no—
129C2×D127—21no—
130C2×D127—21no—
131C2×D127—44no—
132C2×D127—41no—
133C2×D127—41no—
134C2×D127—41no—
135C4×C127—42no—
136C4×C127—41no—
137C4×C127—41no—
138C4×C127#12312yes—
139C4×C127C2×#4111yescharacteristic
140C4×C127C2×#4111yescharacteristic
141Dic127—21no—
142Dic127—21no—
143Dic127—41no—
144Dic127—42no—
145C2×C127—22no—
146C2×C127C2×#12311yescharacteristic
147C2×C127#57511yescharacteristic
148C2×C127#57511yescharacteristic
149D127—41no—
150D127—41no—
151D127—81no—
152#3093—1271nomaximal, Sylow
153C127#309311yescharacteristic, complemented, Sylow
154#525—1271no—
155C2×#117—1271no—
156#485—1272no—
157#485—1272no—
158#482—1272no—
159#482—1272no—
160#428—1271no—
161#428D12711yescharacteristic, complemented
162#417—1271no—
163#417—1271no—
164#407—1271no—
165C2×#41—1271no—
166C23×C4—1271no—
167C2×M16—1271no—
168C2×M16C2×D12711yescharacteristic
169#123—2541no—
170C22⋊Q8—1271no—
171C22⋊Q8—1271no—
172#121—1271no—
173#121—1271no—
174#120—2541no—
175C4×D4—2541no—
176C4×D4—2541no—
177#117—1272no—
178#117—1272no—
179#117—2541no—
180#117—2541no—
181#117C2×D12711yescharacteristic
182C2×C4⋊C4—1271no—
183C2×C4⋊C4C2×D12711yescharacteristic
184C2×C22⋊C4—2541no—
185C2×C42—2541no—
186#107—1272no—
187#107C2×D12712yescomplemented
188#106—1272no—
189#106C2×D12712yescomplemented
190#103—1272no—
191#103—1272no—
192#102—1272no—
193#102—1272no—
194C22⋊C8—1272no—
195C22⋊C8—1272no—
196C24—1271no—
197#41—2542no—
198#41—2542no—
199C2×Q8—1271no—
200C2×D4—1271no—
201C2×D4—2541no—
202C2×D4—2541no—
203C2×D4—2541no—
204C22×C4—21no—
205C22×C4—1271no—
206C22×C4—1271no—
207C22×C4—1271no—
208C22×C4—1271no—
209C22×C4—1271no—
210C22×C4—2541no—
211C22×C4—2541no—
212C22×C4—2542no—
213C22×C4—2542no—
214C22×C4C22×D12711yescharacteristic
215M16—22no—
216M16—2542no—
217C2×C8—1272no—
218C2×C8C22×D12712yes—
219C4⋊C4—21no—
220C4⋊C4—1272no—
221C4⋊C4—2541no—
222C4⋊C4—2541no—
223C4⋊C4—2541no—
224C4⋊C4C22×D12712yes—
225C4⋊C4C22×D12711yescharacteristic
226C4⋊C4C22×D12711yescharacteristic
227C22⋊C4—21no—
228C22⋊C4—2541no—
229C22⋊C4—2541no—
230C22⋊C4—2542no—
231C22⋊C4—2542no—
232C22⋊C4—2541no—
233C22⋊C4—2541no—
234C42—21no—
235C42—2542no—
236C42—2542no—
237C23—1271no—
238C23—1271no—
239C23—5081no—
240C23—5081no—
241C23—2541no—
242C23—2542no—
243C23D4×D12711yescharacteristic
244Q8—2541no—
245Q8—2541no—
246D4—5082no—
247D4—5082no—
248D4—5082no—
249D4—2541no—
250D4—2541no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—42no—
255C2×C4—1271no—
256C2×C4—1271no—
257C2×C4—5082no—
258C2×C4—5082no—
259C2×C4—2542no—
260C2×C4—2541no—
261C2×C4—2542no—
262C2×C4—2542no—
263C2×C4—2542no—
264C2×C4—2541no—
265C2×C4—2541no—
266C2×C4—2541no—
267C2×C4—2541no—
268C2×C4D4×D12711yescharacteristic
269C2×C4#1452324812yes—
270C2×C4#1452324912yes—
271C2×C4C23×D12711yescharacteristic
272C8—22no—
273C8—2542no—
274C22—21no—
275C22—21no—
276C22—5084no—
277C22—5081no—
278C22—5081no—
279C22—5081no—
280C22—2541no—
281C22—2541no—
282C22—2541no—
283C22—2541no—
284C22—2541no—
285C22—2541no—
286C22#1686119412yes—
287C22C2×D4×D12711yescharacteristic
288C4—42no—
289C4—41no—
290C4—41no—
291C4—5081no—
292C4—5082no—
293C4—2541no—
294C4—2541no—
295C4#1686119412yes—
296C4C2×#1452324811yescharacteristic
297C4C2×#1452324911yescharacteristic
298C2—22no—
299C2—10161no—
300C2—5081no—
301C2—5081no—
302C2C2×#1686119411yescharacteristic
303C2#2255067211yescharacteristic
304C2#2255067511yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#428×C127C2
socleC22×C127C22×D4
prime radical#22550388C2
hypercenter#428D127
hypocenterC127#3093
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
2. C2×#16861167C2semidirect
3. #22550220C2semidirect
4. #22550219C2semidirect
5. #22550203C2semidirect
6. #22550202C2semidirect
7. #22549919C2semidirect
8. #22549878C2semidirect
9. #22549875C2semidirect
10. #22549839C2semidirect
11. #428×C127C2semidirect
30. #16861142C22semidirect
31. #16861140C22semidirect
32. #16861134C22semidirect
33. #16861132C22semidirect
34. #16861131C22semidirect
35. #16861129C22semidirect
41. #107×C127C22semidirect
42. #106×C127C22semidirect
153. C127#3093semidirect
161. #428D127semidirect
187. #107C2×D127semidirect
189. #106C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225503882
C2×#168611672
#225502202
#225502192
#225502032
#225502022
#225499192
#225498782
#225498752
#225498392
#428×C1272
#3093127