Finite Group Databasebeta

← #30650526

Subgroups — #30650526

Order 16256.

476 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 117 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#22550680C212yesmaximal, complemented
2#22550679C212yesmaximal, complemented
3#22550252C211yesmaximal, characteristic, complemented
4#22550145C212yesmaximal, complemented
5#22550144C212yesmaximal, complemented
6#22550045C212yesmaximal, complemented
7#22550043C211yesmaximal, characteristic, complemented
8#22549990C212yesmaximal, complemented
9#453×C127C211yesmaximal, characteristic, complemented
10#16861255C2212yescomplemented
11#16861254C2211yescharacteristic, complemented
12#16861237C2212yescomplemented
13#16861236—22no—
14#16861235—22no—
15#16861235C2212yescomplemented
16#16861234—22no—
17#16861234C2212yescomplemented
18#16861233—22no—
19#16861230—22no—
20#16861229—22no—
21#16861229C2212yes—
22C2×#14523233—22no—
23C2×#14523233C2212yescomplemented
24C2×#14523233C2212yescomplemented
25D8×D127—22no—
26#16861217C2212yescomplemented
27#16861216C2212yescomplemented
28#16861206C2211yescharacteristic, complemented
29#16861144C2212yescomplemented
30#16861137C2212yescomplemented
31#16861126C2211yescharacteristic, complemented
32#16861125C2212yescomplemented
33C4×C127⋊C8C2211yescharacteristic, complemented
34H(5,𝔽2)×C127C2211yescharacteristic, complemented
35AGL(1,ℤ/8)×C127C2212yescomplemented
36#127×C127C2211yescharacteristic, complemented
37#104×C127C2212yescomplemented
38#100×C127C2211yescharacteristic, complemented
39#14523249—22no—
40#14523248—22no—
41#14523248C2312yes—
42#14523247—22no—
43C2×D508—22no—
44C2×D508—41no—
45C2×D508C2312yes—
46C2×C22⋊D127—41no—
47D4×D127—22no—
48D4×D127—22no—
49D4×D127—42no—
50#14523238D412yescomplemented
51#14523237—22no—
52#14523236—22no—
53#14523236D412yescomplemented
54#14523235—22no—
55#14523235D412yescomplemented
56D1016—22no—
57#14523233—22no—
58#14523233—22no—
59#14523233—22no—
60#14523233—22no—
61#14523233—22no—
62#14523233—22no—
63#14523233D412yescomplemented
64#14523232—22no—
65#14523232C2312yes—
66#14523231—22no—
67C2×C127⋊C8—22no—
68C2×C127⋊C8C2312yes—
69C8×D127—22no—
70#41×C127—22no—
71#41×C127C2312yes—
72C2×D4×C127—22no—
73C2×D4×C127—41no—
74C2×D4×C127—41no—
75C2×D4×C127C2311yescharacteristic
76C2×D4×C127C2311yescharacteristic
77SD16×C127—22no—
78D8×C127—22no—
79M16×C127C2312yes—
80C42×C127C2311yescharacteristic
81C22×D127—22no—
82C22×D127—41no—
83C22×D127—42no—
84Dic254C2×D412yes—
85D508—22no—
86D508—22no—
87D508—42no—
88D508—82no—
89D508C2×D412yes—
90C22⋊D127—22no—
91C22⋊D127—42no—
92C22⋊D127—42no—
93C2×Dic127—41no—
94C4×D127—22no—
95C4×D127—42no—
96C127⋊C8—22no—
97C127⋊C8C2×D412yes—
98C127⋊C8C2×D412yes—
99C23×C127—21no—
100C23×C127—21no—
101C23×C127—41no—
102Q8×C127C2×D412yes—
103D4×C127—22no—
104D4×C127—22no—
105D4×C127—22no—
106D4×C127—42no—
107D4×C127—82no—
108D4×C127C2×D412yes—
109C2×C4×C127—21no—
110C2×C4×C127—22no—
111C2×C4×C127—41no—
112C2×C4×C127C2411yescharacteristic
113C8×C127—22no—
114C22×C127—22no—
115C22×C127—41no—
116C22×C127—41no—
117C22×C127—81no—
118C22×C127—81no—
119C22×C127—161no—
120C22×C127H(5,𝔽2)11yescharacteristic
121C2×D127—22no—
122C2×D127—42no—
123C2×D127—82no—
124C2×D127—161no—
125C2×D127—162no—
126C4×C127—22no—
127C4×C127—22no—
128C4×C127C22×D412yes—
129Dic127—22no—
130C2×C127—21no—
131C2×C127—42no—
132C2×C127—81no—
133C2×C127—81no—
134C2×C127D4211yescharacteristic
135D127—42no—
136D127—82no—
137#3292—1271nomaximal, Sylow
138C127#329211yescharacteristic, complemented, Sylow
139#576—1272no—
140#576—1272no—
141#493—1271no—
142#472—1272no—
143#472—1272no—
144#453—1271no—
145#453—1272no—
146#453D12711yescharacteristic, complemented
147#443—1272no—
148H(5,𝔽2)—1272no—
149H(5,𝔽2)—1271no—
150H(5,𝔽2)C2×D12711yescharacteristic, complemented
151AGL(1,ℤ/8)—1272no—
152AGL(1,ℤ/8)—1272no—
153AGL(1,ℤ/8)—1272no—
154AGL(1,ℤ/8)—2542no—
155AGL(1,ℤ/8)—2542no—
156AGL(1,ℤ/8)—2542no—
157AGL(1,ℤ/8)—2542no—
158AGL(1,ℤ/8)C2×D12712yescomplemented
159#135—1272no—
160#135—2542no—
161#135—2542no—
162C2×D8—1272no—
163C2×D8—1272no—
164C2×D8—2542no—
165C2×D8—2542no—
166#131—1272no—
167#131—1272no—
168#127—1271no—
169#127C2×D12711yescharacteristic, complemented
170#108—1272no—
171#104—1272no—
172#104C2×D12712yescomplemented
173#100—1271no—
174#100—1272no—
175#100C2×D12711yescharacteristic, complemented
176C4×C8—1271no—
177#41—22no—
178#41—1272no—
179#41—2542no—
180#41—2542no—
181#41—2542no—
182#41C22×D12712yes—
183C2×D4—22no—
184C2×D4—41no—
185C2×D4—41no—
186C2×D4—1272no—
187C2×D4—5081no—
188C2×D4—5082no—
189C2×D4—5081no—
190C2×D4—2542no—
191C2×D4—2542no—
192C2×D4—2542no—
193C2×D4C22×D12711yescharacteristic
194C2×D4C22×D12711yescharacteristic
195Q16—1272no—
196SD16—22no—
197SD16—1272no—
198SD16—1272no—
199SD16—2542no—
200SD16—2542no—
201SD16—2542no—
202D8—22no—
203D8—1272no—
204D8—2542no—
205D8—2542no—
206D8—2542no—
207D8—2542no—
208D8—2542no—
209D8—2542no—
210D8—2542no—
211M16—1272no—
212M16—2542no—
213M16—2542no—
214M16C22×D12712yes—
215C2×C8—1272no—
216C2×C8—2542no—
217C2×C8—2542no—
218C42C22×D12711yescharacteristic
219C23—21no—
220C23—21no—
221C23—41no—
222C23—5082no—
223C23—5081no—
224C23—2542no—
225Q8—1272no—
226Q8D4×D12712yes—
227D4—22no—
228D4—22no—
229D4—22no—
230D4—42no—
231D4—82no—
232D4—1272no—
233D4—10162no—
234D4—5082no—
235D4—5082no—
236D4—5082no—
237D4—2542no—
238D4—2542no—
239D4—2542no—
240D4D4×D12712yes—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—41no—
244C2×C4—5082no—
245C2×C4—5081no—
246C2×C4—2542no—
247C2×C4C23×D12711yescharacteristic
248C8—22no—
249C8—1272no—
250C8—1272no—
251C8—2542no—
252C22—22no—
253C22—41no—
254C22—41no—
255C22—81no—
256C22—81no—
257C22—161no—
258C22—20322no—
259C22—20321no—
260C22—10162no—
261C22—5082no—
262C22—2542no—
263C22#1686125411yescharacteristic
264C4—22no—
265C4—22no—
266C4—2542no—
267C4C2×D4×D12712yes—
268C2—21no—
269C2—42no—
270C2—81no—
271C2—81no—
272C2—10162no—
273C2—5082no—
274C2#2255050811yescharacteristic
subgroupG/H
centerC2#22550508
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#453×C127C2
socleC2×C127D42
hypercenter#453D127
hypocenterC127#3292
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ #453

Complements

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

NH ≅ G/Ntype
1. #22550680C2semidirect
2. #22550679C2semidirect
3. #22550252C2semidirect
4. #22550145C2semidirect
5. #22550144C2semidirect
6. #22550045C2semidirect
7. #22550043C2semidirect
8. #22549990C2semidirect
9. #453×C127C2semidirect
10. #16861255C22semidirect
11. #16861254C22semidirect
12. #16861237C22semidirect
15. #16861235C22semidirect
17. #16861234C22semidirect
23. C2×#14523233C22semidirect
24. C2×#14523233C22semidirect
26. #16861217C22semidirect
27. #16861216C22semidirect
28. #16861206C22semidirect
29. #16861144C22semidirect
30. #16861137C22semidirect
31. #16861126C22semidirect
32. #16861125C22semidirect
33. C4×C127⋊C8C22semidirect
34. H(5,𝔽2)×C127C22semidirect
35. AGL(1,ℤ/8)×C127C22semidirect
36. #127×C127C22semidirect
37. #104×C127C22semidirect
38. #100×C127C22semidirect
50. #14523238D4semidirect
53. #14523236D4semidirect
55. #14523235D4semidirect
63. #14523233D4semidirect
138. C127#3292semidirect
146. #453D127semidirect
150. H(5,𝔽2)C2×D127semidirect
158. AGL(1,ℤ/8)C2×D127semidirect
169. #127C2×D127semidirect
172. #104C2×D127semidirect
175. #100C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225506802
#225506792
#225502522
#225501452
#225501442
#225500452
#225500432
#225499902
#453×C1272
#3292127