Finite Group Databasebeta

← #30648087

Subgroups — #30648087

Order 16256.

762 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 762 conjugacy classes of subgroups — too many to draw. Only its 161 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#22550674C212yesmaximal, complemented
2#22550443C211yesmaximal, characteristic, complemented
3#22550054C214yesmaximal, complemented
4#22550044C214yesmaximal, complemented
5#22549893C212yesmaximal, complemented
6C2×#16861126C211yesmaximal, characteristic, complemented
7C2×H(5,𝔽2)×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic, complemented
9#16861240—22no—
10#16861239C2214yescomplemented
11#16861235—22no—
12#16861234C2214yescomplemented
13#16861229—24no—
14C2×#14523232C2212yescomplemented
15#16861205C2212yescomplemented
16#16861196C2212yescomplemented
17#16861180—22no—
18#16861178—22no—
19#16861168C2211yescharacteristic, complemented
20#16861136C2218yescomplemented
21#16861126C2214yescomplemented
22H(5,𝔽2)×C127—24no—
23H(5,𝔽2)×C127—22no—
24H(5,𝔽2)×C127C2214yes—
25C2×#41×C127—22no—
26C2×#41×C127C2212yes—
27C22×D4×C127—22no—
28C22×D4×C127—22no—
29C22×D4×C127C2211yescharacteristic
30#14523248—22no—
31#14523248—22no—
32C2×Dic254C2311yescharacteristic, complemented
33C2×D508C2311yescharacteristic, complemented
34C2×C22⋊D127—21no—
35C2×C22⋊D127—24no—
36C2×C4×D127—21no—
37#14523238—24no—
38#14523236—24no—
39#14523235—24no—
40#14523233—24no—
41#14523232—22no—
42#14523232C2314yescomplemented
43C2×C127⋊C8—22no—
44C4⋊Dic127—21no—
45C22⋊Dic127—24no—
46C22⋊Dic127—21no—
47C4×Dic127C2312yescomplemented
48C24×C127—22no—
49C24×C127D412yes—
50#41×C127—28no—
51#41×C127—28no—
52#41×C127—22no—
53#41×C127—22no—
54#41×C127D414yescomplemented
55#41×C127C2314yes—
56C2×Q8×C127D412yescomplemented
57C2×D4×C127—24no—
58C2×D4×C127—28no—
59C2×D4×C127—28no—
60C2×D4×C127—22no—
61C2×D4×C127—28no—
62C2×D4×C127—28no—
63C2×D4×C127—22no—
64C2×D4×C127—22no—
65C2×D4×C127D412yes—
66C2×D4×C127D412yescomplemented
67C2×D4×C127C2312yes—
68C22×C4×C127—22no—
69C22×C4×C127—22no—
70C22×C4×C127C2311yescharacteristic
71C22×D127—21no—
72Dic254—22no—
73D508—22no—
74C22⋊D127—41no—
75C22⋊D127—41no—
76C22⋊D127—84no—
77C2×Dic127—22no—
78C2×Dic127—21no—
79C4×D127—42no—
80C127⋊C8—24no—
81C23×C127—28no—
82C23×C127—22no—
83C23×C127—22no—
84C23×C127—22no—
85C23×C127—82no—
86C23×C127—84no—
87C23×C127C2×D414yes—
88C23×C127C2×D411yescharacteristic
89Q8×C127—22no—
90Q8×C127C2×D414yes—
91D4×C127—216no—
92D4×C127—24no—
93D4×C127—24no—
94D4×C127—24no—
95D4×C127—22no—
96D4×C127—24no—
97D4×C127C2×D414yes—
98C2×C4×C127—28no—
99C2×C4×C127—22no—
100C2×C4×C127—24no—
101C2×C4×C127—24no—
102C2×C4×C127C2×D411yescharacteristic
103C2×C4×C127C2×D412yes—
104C2×C4×C127C2×D412yes—
105C2×C4×C127C2411yescharacteristic
106C22×C127—24no—
107C22×C127—24no—
108C22×C127—41no—
109C22×C127—44no—
110C22×C127—44no—
111C22×C127—44no—
112C22×C127—82no—
113C22×C127—88no—
114C22×C127#12011yescharacteristic
115C22×C127#12011yescharacteristic
116C22×C127C22×D411yescharacteristic
117C2×D127—42no—
118C2×D127—41no—
119C4×C127—24no—
120C4×C127#12012yes—
121C4×C127C22×D412yes—
122Dic127—42no—
123Dic127—41no—
124C2×C127—21no—
125C2×C127—21no—
126C2×C127—21no—
127C2×C127—44no—
128C2×C127—44no—
129C2×C127C2×#12011yescharacteristic
130D127—81no—
131#3019—1271nomaximal, Sylow
132C127#301911yescharacteristic, complemented, Sylow
133C2×H(5,𝔽2)D12711yescharacteristic, complemented
134#575—1272no—
135#535—1271no—
136#455—1274no—
137#453—1274no—
138#421—1272no—
139C2×#100—1271no—
140H(5,𝔽2)—24no—
141H(5,𝔽2)—22no—
142H(5,𝔽2)C2×D12714yes—
143C2×#41—22no—
144C2×#41—1271no—
145C2×#41C2×D12712yes—
146C22×D4—22no—
147C22×D4—22no—
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#127—1272no—
156#124—1272no—
157#121—2542no—
158#120—2542no—
159#117—1271no—
160#104—1278no—
161#100—1274no—
162C24—22no—
163C24C22⋊D12712yes—
164#41—28no—
165#41—28no—
166#41—22no—
167#41—22no—
168#41—2542no—
169#41—2542no—
170#41C22⋊D12714yescomplemented
171#41C22×D12714yes—
172C2×Q8—1271no—
173C2×Q8C22⋊D12712yescomplemented
174C2×D4—24no—
175C2×D4—28no—
176C2×D4—28no—
177C2×D4—22no—
178C2×D4—28no—
179C2×D4—28no—
180C2×D4—22no—
181C2×D4—22no—
182C2×D4—1271no—
183C2×D4—2544no—
184C2×D4—2541no—
185C2×D4C22⋊D12712yes—
186C2×D4C22⋊D12712yescomplemented
187C2×D4C22×D12712yes—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—2541no—
191C22×C4C22×D12711yescharacteristic
192Q16—2544no—
193SD16—2544no—
194SD16—2544no—
195D8—2544no—
196M16—1274no—
197M16—2542no—
198C2×C8—2542no—
199C4⋊C4—2541no—
200C22⋊C4—2541no—
201C22⋊C4—2544no—
202C42—1272no—
203C23—28no—
204C23—22no—
205C23—22no—
206C23—22no—
207C23—82no—
208C23—84no—
209C23—2541no—
210C23C2×C22⋊D12714yes—
211C23C2×C22⋊D12711yescharacteristic
212Q8—22no—
213Q8—2542no—
214Q8C2×C22⋊D12714yes—
215D4—216no—
216D4—24no—
217D4—24no—
218D4—24no—
219D4—24no—
220D4—22no—
221D4—10164no—
222D4—5081no—
223D4—5081no—
224D4—2542no—
225D4C2×C22⋊D12714yes—
226C2×C4—28no—
227C2×C4—22no—
228C2×C4—24no—
229C2×C4—24no—
230C2×C4—5082no—
231C2×C4—2542no—
232C2×C4—2541no—
233C2×C4C2×C22⋊D12712yes—
234C2×C4C2×C22⋊D12712yes—
235C2×C4C2×C22⋊D12711yescharacteristic
236C2×C4C23×D12711yescharacteristic
237C8—2544no—
238C22—24no—
239C22—24no—
240C22—41no—
241C22—44no—
242C22—44no—
243C22—44no—
244C22—82no—
245C22—88no—
246C22—5082no—
247C22—5081no—
248C22#1686117811yescharacteristic
249C22#1686117811yescharacteristic
250C22C22×C22⋊D12711yescharacteristic
251C4—24no—
252C4—5082no—
253C4—5081no—
254C4#1686117812yes—
255C4C22×C22⋊D12712yes—
256C2—21no—
257C2—21no—
258C2—21no—
259C2—44no—
260C2—44no—
261C2—10161no—
262C2C2×#1686117811yescharacteristic
subgroupG/H
centerC2C2×#16861178
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×H(5,𝔽2)×C127C2
socleC2×C127C2×#120
hypercenterC2×H(5,𝔽2)D127
hypocenterC127#3019
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #22550674C2semidirect
2. #22550443C2semidirect
3. #22550054C2semidirect
4. #22550044C2semidirect
5. #22549893C2semidirect
6. C2×#16861126C2semidirect
7. C2×H(5,𝔽2)×C127C2semidirect
8. C2×#14523248C22semidirect
10. #16861239C22semidirect
12. #16861234C22semidirect
14. C2×#14523232C22semidirect
15. #16861205C22semidirect
16. #16861196C22semidirect
19. #16861168C22semidirect
20. #16861136C22semidirect
21. #16861126C22semidirect
32. C2×Dic254C23semidirect
33. C2×D508C23semidirect
42. #14523232C23semidirect
47. C4×Dic127C23semidirect
54. #41×C127D4semidirect
56. C2×Q8×C127D4semidirect
66. C2×D4×C127D4semidirect
132. C127#3019semidirect
133. C2×H(5,𝔽2)D127semidirect
170. #41C22⋊D127semidirect
173. C2×Q8C22⋊D127semidirect
186. C2×D4C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
#225506742
#225504432
#225500542
#225500442
#225498932
C2×#168611262
C2×H(5,𝔽2)×C1272
#3019127