Finite Group Databasebeta

← #30648086

Subgroups — #30648086

Order 16256.

762 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 762 conjugacy classes of subgroups — too many to draw. Only its 145 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
1C2×#16861254C211yesmaximal, characteristic, complemented
2#22550674C212yesmaximal, complemented
3#22550053C214yesmaximal, complemented
4#22550043C214yesmaximal, complemented
5#22549894C212yesmaximal, complemented
6C2×#16861126C211yesmaximal, characteristic, complemented
7#535×C127C211yesmaximal, characteristic, complemented
8#16861254—24no—
9#16861254—22no—
10#16861254C2214yes—
11C2×#14523248C2212yes—
12C2×#14523247—22no—
13C22×C22⋊D127—22no—
14C2×D4×D127—22no—
15#16861240C2214yescomplemented
16#16861239—22no—
17#16861235C2214yescomplemented
18#16861234—22no—
19#16861229—24no—
20C2×#14523232C2212yescomplemented
21#16861137C2218yescomplemented
22#16861126C2214yescomplemented
23C2×#41×C127C2211yescharacteristic, complemented
24C22×D4×C127C2211yescharacteristic
25#127×C127C2212yescomplemented
26#124×C127C2212yescomplemented
27#121×C127—22no—
28#120×C127—22no—
29#117×C127C2211yescharacteristic, complemented
30C23×D127—22no—
31#14523248—22no—
32#14523248—22no—
33#14523248D414yescomplemented
34#14523248C2314yes—
35#14523247—28no—
36#14523247—28no—
37C2×Dic254D412yescomplemented
38C2×D508D412yescomplemented
39C2×C22⋊D127—24no—
40C2×C22⋊D127—28no—
41C2×C22⋊D127—22no—
42C2×C22⋊D127—28no—
43D4×D127—28no—
44D4×D127—28no—
45C22×Dic127—22no—
46C2×C4×D127—22no—
47#14523238—24no—
48#14523236—24no—
49#14523235—24no—
50#14523233—24no—
51#14523232—22no—
52#14523232C2314yescomplemented
53C2×C127⋊C8—22no—
54C24×C127D412yes—
55#41×C127—22no—
56#41×C127—22no—
57C2×Q8×C127C2311yescharacteristic, complemented
58C2×D4×C127—21no—
59C2×D4×C127—24no—
60C2×D4×C127—22no—
61C2×D4×C127—22no—
62C2×D4×C127D412yes—
63C2×D4×C127C2312yes—
64C2×D4×C127C2311yescharacteristic, complemented
65C22×C4×C127—21no—
66C22×C4×C127C2311yescharacteristic
67C4⋊C4×C127—21no—
68C22⋊C4×C127—21no—
69C22⋊C4×C127—24no—
70C42×C127C2312yescomplemented
71C22×D127—22no—
72C22×D127—28no—
73C22×D127—84no—
74Dic254—22no—
75Dic254C2×D414yes—
76D508—22no—
77D508C2×D414yes—
78C22⋊D127—216no—
79C22⋊D127—24no—
80C22⋊D127—24no—
81C2×Dic127—28no—
82C2×Dic127—22no—
83C4×D127—24no—
84C4×D127—24no—
85C127⋊C8—24no—
86C23×C127—22no—
87C23×C127—22no—
88C23×C127—21no—
89C23×C127—82no—
90C23×C127C2×D414yes—
91C23×C127C2×D411yescharacteristic
92Q8×C127—22no—
93D4×C127—24no—
94D4×C127—24no—
95D4×C127—22no—
96D4×C127—41no—
97D4×C127—41no—
98D4×C127—84no—
99C2×C4×C127—22no—
100C2×C4×C127—21no—
101C2×C4×C127—42no—
102C2×C4×C127C2×D412yes—
103C2×C4×C127C2×D412yes—
104C2×C4×C127C2×D411yescharacteristic
105C2×C4×C127C2411yescharacteristic
106C22×C127—24no—
107C22×C127—41no—
108C22×C127—44no—
109C22×C127—44no—
110C22×C127—42no—
111C22×C127—41no—
112C22×C127—82no—
113C22×C127#12011yescharacteristic
114C22×C127#12011yescharacteristic
115C22×C127C22×D411yescharacteristic
116C2×D127—24no—
117C2×D127—44no—
118C2×D127—88no—
119C4×C127—42no—
120C4×C127—41no—
121C4×C127#12012yes—
122C4×C127C22×D412yes—
123Dic127—24no—
124C2×C127—21no—
125C2×C127—21no—
126C2×C127—21no—
127C2×C127—44no—
128C2×C127—81no—
129C2×C127C2×#12011yescharacteristic
130D127—44no—
131#3019—1271nomaximal, Sylow
132C127#301911yescharacteristic, complemented, Sylow
133C2×H(5,𝔽2)—1271no—
134#575—1272no—
135#535D12711yescharacteristic, complemented
136#455—1274no—
137#453—1274no—
138#421—1272no—
139C2×#100—1271no—
140H(5,𝔽2)—1274no—
141H(5,𝔽2)—2544no—
142H(5,𝔽2)—2542no—
143C2×#41—1272no—
144C2×#41—2542no—
145C2×#41C2×D12711yescharacteristic, complemented
146C22×D4—2542no—
147C22×D4—2542no—
148C22×D4C2×D12711yescharacteristic
149#137—1274no—
150#137—2542no—
151AGL(1,ℤ/8)—1274no—
152AGL(1,ℤ/8)—2542no—
153#135—2544no—
154C2×M16—1272no—
155#127C2×D12712yescomplemented
156#124C2×D12712yescomplemented
157#121—22no—
158#120—22no—
159#117C2×D12711yescharacteristic, complemented
160#104—1278no—
161#100—1274no—
162C24—2542no—
163C24C22⋊D12712yes—
164#41—22no—
165#41—22no—
166#41—1274no—
167#41—1274no—
168#41—2548no—
169#41—2548no—
170#41—2542no—
171#41—2542no—
172C2×Q8—1272no—
173C2×Q8C22×D12711yescharacteristic, complemented
174C2×D4—21no—
175C2×D4—24no—
176C2×D4—22no—
177C2×D4—22no—
178C2×D4—1272no—
179C2×D4—2548no—
180C2×D4—2548no—
181C2×D4—2544no—
182C2×D4—2548no—
183C2×D4—2542no—
184C2×D4—2548no—
185C2×D4C22⋊D12712yes—
186C2×D4C22×D12712yes—
187C2×D4C22×D12711yescharacteristic, complemented
188C22×C4—21no—
189C22×C4—2542no—
190C22×C4—2542no—
191C22×C4C22×D12711yescharacteristic
192Q16—2544no—
193SD16—2544no—
194SD16—2544no—
195D8—2544no—
196M16—1274no—
197M16—2542no—
198C2×C8—2542no—
199C4⋊C4—21no—
200C22⋊C4—21no—
201C22⋊C4—24no—
202C42C22×D12712yescomplemented
203C23—22no—
204C23—22no—
205C23—21no—
206C23—82no—
207C23—10164no—
208C23—2548no—
209C23—2542no—
210C23C2×C22⋊D12714yes—
211C23C2×C22⋊D12711yescharacteristic
212Q8—22no—
213Q8—1274no—
214Q8—2542no—
215D4—24no—
216D4—22no—
217D4—24no—
218D4—41no—
219D4—41no—
220D4—84no—
221D4—1274no—
222D4—25416no—
223D4—2544no—
224D4—2544no—
225D4—2542no—
226C2×C4—22no—
227C2×C4—21no—
228C2×C4—42no—
229C2×C4—2548no—
230C2×C4—2542no—
231C2×C4—2544no—
232C2×C4—2544no—
233C2×C4D4×D12712yes—
234C2×C4D4×D12712yes—
235C2×C4C2×C22⋊D12711yescharacteristic
236C2×C4C23×D12711yescharacteristic
237C8—2544no—
238C22—24no—
239C22—41no—
240C22—44no—
241C22—44no—
242C22—42no—
243C22—41no—
244C22—82no—
245C22—10168no—
246C22—5084no—
247C22—2544no—
248C22#1686117711yescharacteristic
249C22#1686117711yescharacteristic
250C22C22×C22⋊D12711yescharacteristic
251C4—42no—
252C4—41no—
253C4—2544no—
254C4#1686117712yes—
255C4C2×D4×D12712yes—
256C2—21no—
257C2—21no—
258C2—21no—
259C2—44no—
260C2—81no—
261C2—5084no—
262C2C2×#1686117711yescharacteristic
subgroupG/H
centerC2C2×#16861177
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#535×C127C2
socleC2×C127C2×#120
hypercenter#535D127
hypocenterC127#3019
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22×C4 ◁ #535

Complements

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

NH ≅ G/Ntype
1. C2×#16861254C2semidirect
2. #22550674C2semidirect
3. #22550053C2semidirect
4. #22550043C2semidirect
5. #22549894C2semidirect
6. C2×#16861126C2semidirect
7. #535×C127C2semidirect
15. #16861240C22semidirect
17. #16861235C22semidirect
20. C2×#14523232C22semidirect
21. #16861137C22semidirect
22. #16861126C22semidirect
23. C2×#41×C127C22semidirect
25. #127×C127C22semidirect
26. #124×C127C22semidirect
29. #117×C127C22semidirect
33. #14523248D4semidirect
37. C2×Dic254D4semidirect
38. C2×D508D4semidirect
52. #14523232C23semidirect
57. C2×Q8×C127C23semidirect
64. C2×D4×C127C23semidirect
70. C42×C127C23semidirect
132. C127#3019semidirect
135. #535D127semidirect
145. C2×#41C2×D127semidirect
155. #127C2×D127semidirect
156. #124C2×D127semidirect
159. #117C2×D127semidirect
173. C2×Q8C22×D127semidirect
187. C2×D4C22×D127semidirect
202. C42C22×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612542
#225506742
#225500532
#225500432
#225498942
C2×#168611262
#535×C1272
#3019127