Finite Group Databasebeta

← #30648229

Subgroups — #30648229

Order 16256.

526 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 141 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×#16861234C211yesmaximal, characteristic, complemented
2C22×#14523233C211yesmaximal, characteristic, complemented
3#22550350C211yesmaximal, characteristic, complemented
4#22550088C214yesmaximal, complemented
5#22550068C214yesmaximal, complemented
6#22549903C211yesmaximal, characteristic, complemented
7#22549881C211yesmaximal, characteristic, complemented
8#22549872C211yesmaximal, characteristic, complemented
9#535×C127C211yesmaximal, characteristic, complemented
10C22×D508C2211yescharacteristic
11#16861234—24no—
12C2×#14523235C2212yescomplemented
13C2×#14523233—24no—
14C2×#14523233C2214yes—
15C2×#14523233C2212yescomplemented
16C2×#14523232C2211yescharacteristic
17C22×C127⋊C8C2211yescharacteristic
18C4×D508C2214yescomplemented
19#16861171—22no—
20#16861168C2211yescharacteristic
21C22⋊C4×D127—22no—
22C4⋊(C127⋊C8)C2214yescomplemented
23#16861133C2212yescomplemented
24#16861130C2212yescomplemented
25#16861130C2214yescomplemented
26C2×#41×C127C2211yescharacteristic
27C22×D4×C127C2211yescharacteristic
28#127×C127C2212yescomplemented
29#124×C127C2212yescomplemented
30#121×C127—22no—
31#120×C127—22no—
32#117×C127C2211yescharacteristic
33C23×D127—21no—
34C2×D508—21no—
35C2×D508—22no—
36C2×D508D414yescomplemented
37C2×D508C2312yes—
38C2×C4×D127—24no—
39#14523235—24no—
40#14523233—28no—
41#14523233—24no—
42#14523232—22no—
43C2×C127⋊C8—22no—
44C2×C127⋊C8C2312yes—
45C2×C127⋊C8C2312yes—
46C4⋊Dic127C2312yes—
47C22⋊Dic127—21no—
48(C2×D127)⋊C4—24no—
49C4×Dic127—21no—
50C24×C127—21no—
51#41×C127—22no—
52#41×C127—22no—
53C2×Q8×C127C2311yescharacteristic
54C2×D4×C127—21no—
55C2×D4×C127—22no—
56C2×D4×C127—21no—
57C2×D4×C127—24no—
58C2×D4×C127—22no—
59C2×D4×C127C2312yes—
60C2×D4×C127C2311yescharacteristic
61C22×C4×C127—21no—
62C22×C4×C127C2311yescharacteristic
63C4⋊C4×C127D412yes—
64C22⋊C4×C127—24no—
65C22⋊C4×C127D412yes—
66C42×C127C2312yes—
67C22×D127—22no—
68C22×D127—24no—
69C22×D127—82no—
70D508—24no—
71D508—22no—
72D508C2×D414yes—
73C2×Dic127—22no—
74C4×D127—84no—
75C127⋊C8—22no—
76C127⋊C8—22no—
77C23×C127—22no—
78C23×C127—21no—
79C23×C127—42no—
80C23×C127—82no—
81C23×C127C2×D411yescharacteristic
82Q8×C127—21no—
83Q8×C127—21no—
84D4×C127—22no—
85D4×C127—21no—
86D4×C127—22no—
87D4×C127—21no—
88D4×C127—42no—
89D4×C127—42no—
90D4×C127—84no—
91C2×C4×C127—21no—
92C2×C4×C127—42no—
93C2×C4×C127C2×D412yes—
94C2×C4×C127C2×D414yes—
95C2×C4×C127C2×D411yescharacteristic
96C2×C4×C127#4112yes—
97C2×C4×C127C2411yescharacteristic
98C22×C127—22no—
99C22×C127—41no—
100C22×C127—42no—
101C22×C127—42no—
102C22×C127—44no—
103C22×C127—84no—
104C22×C127#12112yes—
105C22×C127C22×D411yescharacteristic
106C2×D127—24no—
107C2×D127—44no—
108C2×D127—84no—
109C4×C127—24no—
110C4×C127—41no—
111C4×C127#12112yes—
112C4×C127C22×D411yescharacteristic
113C4×C127C2×#4111yescharacteristic
114Dic127—42no—
115C2×C127—22no—
116C2×C127—81no—
117C2×C127—82no—
118C2×C127C2×#12111yescharacteristic
119C2×C127#57612yes—
120D127—44no—
121#3041—1271nomaximal, Sylow
122C127#304111yescharacteristic, complemented, Sylow
123C2×AGL(1,ℤ/8)—1271no—
124C22×D8—1271no—
125#535D12711yescharacteristic, complemented
126#518—1271no—
127#463—1274no—
128#459—1274no—
129#424—1271no—
130#418—1271no—
131#416—1271no—
132C2×#41C2×D12711yescharacteristic
133C22×D4—1271no—
134C22×D4C2×D12711yescharacteristic
135AGL(1,ℤ/8)—2544no—
136C2×SD16—1272no—
137C2×D8—1274no—
138C2×D8—1272no—
139C2×D8—2544no—
140C2×M16—1271no—
141C22×C8—1271no—
142#127C2×D12712yescomplemented
143#124C2×D12712yescomplemented
144#121—22no—
145#120—22no—
146C4×D4—1274no—
147C4×D4—2542no—
148#117—1271no—
149#117C2×D12711yescharacteristic
150C2×C22⋊C4—2542no—
151C4⋊C8—1274no—
152#103—1272no—
153#102—1272no—
154#102—1274no—
155C24—21no—
156C24—2541no—
157#41—22no—
158#41—22no—
159C2×Q8C22×D12711yescharacteristic
160C2×D4—21no—
161C2×D4—22no—
162C2×D4—21no—
163C2×D4—24no—
164C2×D4—22no—
165C2×D4—1274no—
166C2×D4—1272no—
167C2×D4—2541no—
168C2×D4—2542no—
169C2×D4C22×D12712yes—
170C2×D4C22×D12711yescharacteristic
171C22×C4—21no—
172C22×C4—2544no—
173C22×C4C22×D12711yescharacteristic
174SD16—2544no—
175D8—2548no—
176D8—2544no—
177M16—2542no—
178C2×C8—1272no—
179C2×C8—1272no—
180C2×C8—2542no—
181C4⋊C4—1272no—
182C4⋊C4C22⋊D12712yes—
183C22⋊C4—24no—
184C22⋊C4—2541no—
185C22⋊C4—2544no—
186C22⋊C4C22⋊D12712yes—
187C42—2541no—
188C42C22×D12712yes—
189C23—22no—
190C23—21no—
191C23—42no—
192C23—82no—
193C23—10162no—
194C23—2544no—
195C23—2542no—
196C23C2×C22⋊D12711yescharacteristic
197Q8—21no—
198Q8—21no—
199D4—22no—
200D4—21no—
201D4—22no—
202D4—21no—
203D4—42no—
204D4—42no—
205D4—84no—
206D4—1274no—
207D4—2542no—
208D4—2544no—
209C2×C4—21no—
210C2×C4—42no—
211C2×C4—10164no—
212C2×C4—2542no—
213C2×C4D4×D12712yes—
214C2×C4C2×C22⋊D12714yes—
215C2×C4C2×C22⋊D12711yescharacteristic
216C2×C4#1452324712yes—
217C2×C4C23×D12711yescharacteristic
218C8—2542no—
219C8—2542no—
220C22—22no—
221C22—41no—
222C22—42no—
223C22—42no—
224C22—44no—
225C22—84no—
226C22—10164no—
227C22—5084no—
228C22—2544no—
229C22#1686118012yes—
230C22C22×C22⋊D12711yescharacteristic
231C4—24no—
232C4—41no—
233C4—5082no—
234C4#1686118012yes—
235C4C2×D4×D12711yescharacteristic
236C4C2×#1452324711yescharacteristic
237C2—22no—
238C2—82no—
239C2—81no—
240C2—5084no—
241C2C2×#1686118011yescharacteristic
242C2#2255067912yes—
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#535×C127C2
socleC22×C127C22×D4
hypercenter#535D127
hypocenterC127#3041
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ #117 ◁ #535

Complements

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

NH ≅ G/Ntype
1. C2×#16861234C2semidirect
2. C22×#14523233C2semidirect
3. #22550350C2semidirect
4. #22550088C2semidirect
5. #22550068C2semidirect
6. #22549903C2semidirect
7. #22549881C2semidirect
8. #22549872C2semidirect
9. #535×C127C2semidirect
12. C2×#14523235C22semidirect
15. C2×#14523233C22semidirect
18. C4×D508C22semidirect
22. C4⋊(C127⋊C8)C22semidirect
23. #16861133C22semidirect
24. #16861130C22semidirect
25. #16861130C22semidirect
28. #127×C127C22semidirect
29. #124×C127C22semidirect
36. C2×D508D4semidirect
122. C127#3041semidirect
125. #535D127semidirect
142. #127C2×D127semidirect
143. #124C2×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612342
C22×#145232332
#225503502
#225500882
#225500682
#225499032
#225498812
#225498722
#535×C1272
#3041127