Finite Group Databasebeta

← #30648643

Subgroups — #30648643

Order 16256.

420 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 137 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#22550387C211yesmaximal, characteristic
2#22550217C212yesmaximal, complemented
3#22550211C211yesmaximal, characteristic, complemented
4#22550206C211yesmaximal, characteristic, complemented
5#22550200C212yesmaximal, complemented
6#22550194C211yesmaximal, characteristic, complemented
7#22550189C211yesmaximal, characteristic, complemented
8#22549912C212yesmaximal, complemented
9#22549870C212yesmaximal, complemented
10C2×C22⋊(C127⋊C8)C211yesmaximal, characteristic, complemented
11C2×#117×C127C211yesmaximal, characteristic, complemented
12C2×#14523248C2211yescharacteristic
13C22×C127⋊C8C2212yes—
14#16861192—21no—
15#16861188C2211yescharacteristic
16#16861188C2211yescharacteristic
17#16861183C2211yescharacteristic
18#16861183C2211yescharacteristic
19#16861178—21no—
20C4×C22⋊D127—21no—
21C4×C22⋊D127—21no—
22#16861168C2211yescharacteristic
23#16861142C2212yescomplemented
24#16861142C2212yescomplemented
25#16861140C2212yescomplemented
26#16861140C2212yescomplemented
27#16861132C2212yescomplemented
28#16861132C2212yescomplemented
29#16861129C2212yescomplemented
30#16861129C2212yescomplemented
31C22⋊(C127⋊C8)C2211yescharacteristic
32C22⋊(C127⋊C8)C2211yescharacteristic
33C22⋊(C127⋊C8)C2212yes—
34C23×C4×C127C2211yescharacteristic
35#117×C127—21no—
36#117×C127—21no—
37#117×C127C2212yes—
38#117×C127C2212yes—
39C2×C4⋊C4×C127C2211yescharacteristic
40C2×C4⋊C4×C127C2211yescharacteristic
41C2×C22⋊C4×C127—21no—
42C2×C42×C127—21no—
43#14523248—24no—
44C2×Dic254C2311yescharacteristic
45C2×D508C2311yescharacteristic
46C2×C22⋊D127—21no—
47C2×C22⋊D127—21no—
48C2×C22⋊D127—21no—
49C2×C4×D127—21no—
50C2×C127⋊C8—24no—
51C2×C127⋊C8C2312yes—
52C2×C127⋊C8C2312yes—
53C4⋊Dic127C2311yescharacteristic
54C4⋊Dic127C2311yescharacteristic
55Dic127⋊C4—21no—
56Dic127⋊C4—21no—
57C22⋊Dic127—21no—
58C22⋊Dic127—21no—
59C22⋊Dic127—21no—
60(C2×D127)⋊C4—21no—
61(C2×D127)⋊C4—21no—
62C4×Dic127—21no—
63C24×C127D411yescharacteristic
64C22×C4×C127—24no—
65C22×C4×C127—21no—
66C22×C4×C127—21no—
67C22×C4×C127D411yescharacteristic
68C22×C4×C127D411yescharacteristic
69C22×C4×C127D411yescharacteristic
70C22×C4×C127C2311yescharacteristic
71C22×C4×C127C2311yescharacteristic
72C22×C4×C127C2311yescharacteristic
73C4⋊C4×C127—21no—
74C4⋊C4×C127—21no—
75C4⋊C4×C127C2312yes—
76C4⋊C4×C127C2312yes—
77C22⋊C4×C127—22no—
78C22⋊C4×C127—22no—
79C42×C127—22no—
80C42×C127—22no—
81C22×D127—21no—
82Dic254—22no—
83D508—22no—
84C22⋊D127—42no—
85C22⋊D127—42no—
86C22⋊D127—42no—
87C2×Dic127—21no—
88C2×Dic127—21no—
89C2×Dic127—21no—
90C4×D127—42no—
91C127⋊C8—22no—
92C127⋊C8—22no—
93C23×C127—22no—
94C23×C127—22no—
95C23×C127—41no—
96C23×C127C2×D411yescharacteristic
97C23×C127C2×D411yescharacteristic
98C23×C127C2×D411yescharacteristic
99C2×C4×C127—22no—
100C2×C4×C127—22no—
101C2×C4×C127—22no—
102C2×C4×C127—22no—
103C2×C4×C127—22no—
104C2×C4×C127—22no—
105C2×C4×C127—22no—
106C2×C4×C127—42no—
107C2×C4×C127—42no—
108C2×C4×C127C2×D411yescharacteristic
109C2×C4×C127C2×D411yescharacteristic
110C2×C4×C127C2×D411yescharacteristic
111C2×C4×C127#4114yes—
112C2×C4×C127C2411yescharacteristic
113C22×C127—21no—
114C22×C127—21no—
115C22×C127—21no—
116C22×C127—21no—
117C22×C127—21no—
118C22×C127—21no—
119C22×C127—21no—
120C22×C127—44no—
121C22×C127#12312yes—
122C22×C127#13512yes—
123C22×C127C22×D411yescharacteristic
124C2×D127—41no—
125C2×D127—41no—
126C2×D127—41no—
127C4×C127—21no—
128C4×C127—21no—
129C4×C127—42no—
130C4×C127—42no—
131C4×C127#12312yes—
132C4×C127C2×#4112yes—
133Dic127—41no—
134Dic127—41no—
135Dic127—41no—
136C2×C127—22no—
137C2×C127—22no—
138C2×C127—41no—
139C2×C127C2×#12311yescharacteristic
140C2×C127C2×#13511yescharacteristic
141C2×C127#57511yescharacteristic
142D127—81no—
143#3091—1271nomaximal, Sylow
144C127#309111yescharacteristic, complemented, Sylow
145#525—1271no—
146C2×#117D12711yescharacteristic, complemented
147#485—1272no—
148#484—1271no—
149#483—1271no—
150#482—1272no—
151#481—1271no—
152#480—1271no—
153#427—1272no—
154#416—1272no—
155C2×C22⋊C8—1271no—
156C2×#41—1271no—
157C23×C4C2×D12711yescharacteristic
158C22×C8—1272no—
159#123—2541no—
160C22⋊Q8—1271no—
161C22⋊Q8—1271no—
162#121—1271no—
163#121—1271no—
164#120—2541no—
165C4×D4—2541no—
166C4×D4—2541no—
167#117—21no—
168#117—21no—
169#117—1271no—
170#117C2×D12712yes—
171#117C2×D12712yes—
172C2×C4⋊C4C2×D12711yescharacteristic
173C2×C4⋊C4C2×D12711yescharacteristic
174C2×C22⋊C4—21no—
175C2×C42—21no—
176#107—1272no—
177#107—1272no—
178#106—1272no—
179#106—1272no—
180#103—1272no—
181#103—1272no—
182#102—1272no—
183#102—1272no—
184C22⋊C8—1271no—
185C22⋊C8—1271no—
186C22⋊C8—1272no—
187C24C22⋊D12711yescharacteristic
188#41—2544no—
189C2×Q8—1271no—
190C2×D4—1271no—
191C2×D4—2541no—
192C2×D4—2541no—
193C2×D4—2541no—
194C22×C4—24no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—2541no—
198C22×C4C22⋊D12711yescharacteristic
199C22×C4C22⋊D12711yescharacteristic
200C22×C4C22⋊D12711yescharacteristic
201C22×C4C22×D12711yescharacteristic
202C22×C4C22×D12711yescharacteristic
203C22×C4C22×D12711yescharacteristic
204C2×C8—1272no—
205C2×C8—1272no—
206C2×C8—2544no—
207C4⋊C4—21no—
208C4⋊C4—21no—
209C4⋊C4—1271no—
210C4⋊C4—1271no—
211C4⋊C4—2541no—
212C4⋊C4—2541no—
213C4⋊C4C22×D12712yes—
214C4⋊C4C22×D12712yes—
215C22⋊C4—22no—
216C22⋊C4—22no—
217C22⋊C4—2541no—
218C22⋊C4—2541no—
219C22⋊C4—2541no—
220C22⋊C4—2541no—
221C22⋊C4—2541no—
222C42—22no—
223C42—22no—
224C42—2541no—
225C23—22no—
226C23—22no—
227C23—41no—
228C23—2541no—
229C23C2×C22⋊D12711yescharacteristic
230C23C2×C22⋊D12711yescharacteristic
231C23C2×C22⋊D12711yescharacteristic
232Q8—2542no—
233D4—5082no—
234D4—5082no—
235D4—5082no—
236D4—2542no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—5082no—
247C2×C4—2541no—
248C2×C4—2541no—
249C2×C4—2541no—
250C2×C4C2×C22⋊D12711yescharacteristic
251C2×C4C2×C22⋊D12711yescharacteristic
252C2×C4C2×C22⋊D12711yescharacteristic
253C2×C4#1452324814yes—
254C2×C4C23×D12711yescharacteristic
255C8—2542no—
256C8—2542no—
257C22—21no—
258C22—21no—
259C22—21no—
260C22—21no—
261C22—21no—
262C22—21no—
263C22—21no—
264C22—44no—
265C22—5081no—
266C22—5081no—
267C22—5081no—
268C22#1686119212yes—
269C22#1686122912yes—
270C22C22×C22⋊D12711yescharacteristic
271C4—21no—
272C4—21no—
273C4—42no—
274C4—42no—
275C4—5081no—
276C4—5081no—
277C4—5081no—
278C4#1686119212yes—
279C4C2×#1452324812yes—
280C2—22no—
281C2—22no—
282C2—41no—
283C2—10161no—
284C2C2×#1686119211yescharacteristic
285C2C2×#1686122911yescharacteristic
286C2#2255067411yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×#117×C127C2
socleC22×C127C22×D4
prime radical#22550387C2
hypercenterC2×#117D127
hypocenterC127#3091
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. #22550217C2semidirect
3. #22550211C2semidirect
4. #22550206C2semidirect
5. #22550200C2semidirect
6. #22550194C2semidirect
7. #22550189C2semidirect
8. #22549912C2semidirect
9. #22549870C2semidirect
10. C2×C22⋊(C127⋊C8)C2semidirect
11. C2×#117×C127C2semidirect
23. #16861142C22semidirect
24. #16861142C22semidirect
25. #16861140C22semidirect
26. #16861140C22semidirect
27. #16861132C22semidirect
28. #16861132C22semidirect
29. #16861129C22semidirect
30. #16861129C22semidirect
144. C127#3091semidirect
146. C2×#117D127semidirect

Maximal subgroups

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

subgroupindex
#225503872
#225502172
#225502112
#225502062
#225502002
#225501942
#225501892
#225499122
#225498702
C2×C22⋊(C127⋊C8)2
C2×#117×C1272
#3091127