Finite Group Databasebeta

← #30649087

Subgroups — #30649087

Order 16256.

404 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 125 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#22550507C211yesmaximal, characteristic, complemented
2#22550440C211yesmaximal, characteristic
3#22550239C212yesmaximal, complemented
4#22550237C212yesmaximal, complemented
5#22550232C212yesmaximal, complemented
6#22550221C212yesmaximal, complemented
7#22549882C211yesmaximal, characteristic, complemented
8#22549872C211yesmaximal, characteristic, complemented
9C2×#16861129C211yesmaximal, characteristic, complemented
10#22549834C211yesmaximal, characteristic, complemented
11#417×C127C211yesmaximal, characteristic, complemented
12C22×D508C2211yescharacteristic
13C22×C127⋊C8C2211yescharacteristic
14#16861208—21no—
15#16861208C2212yescomplemented
16#16861205C2212yes—
17#16861200—21no—
18#16861200C2212yescomplemented
19#16861198C2212yes—
20#16861188—22no—
21#16861183—21no—
22#16861183—21no—
23#16861177—22no—
24#16861168C2211yescharacteristic
25#16861168C2211yescharacteristic
26#16861133C2211yescharacteristic, complemented
27#16861133C2211yescharacteristic, complemented
28#16861131C2212yes—
29#16861131C2212yes—
30#16861130C2211yescharacteristic, complemented
31#16861130C2211yescharacteristic, complemented
32#16861129C2212yes—
33#16861129C2212yes—
34#16861119C2212yescomplemented
35C8×Dic127C2212yescomplemented
36C2×#41×C127C2211yescharacteristic
37C2×M16×C127C2211yescharacteristic
38C2×C4⋊C4×C127C2211yescharacteristic
39#103×C127C2212yescomplemented
40#102×C127C2212yescomplemented
41C23×D127—21no—
42C2×Dic254—22no—
43C2×D508—21no—
44C2×D508—22no—
45C2×D508—22no—
46C2×D508C2312yes—
47C2×C22⋊D127—22no—
48C2×C22⋊D127—22no—
49C2×C127⋊C8—22no—
50C2×C127⋊C8C2311yescharacteristic
51C2×C127⋊C8C2311yescharacteristic
52C4⋊Dic127—22no—
53C4⋊Dic127D411yescharacteristic
54C4⋊Dic127D411yescharacteristic
55C4⋊Dic127C2311yescharacteristic
56C4⋊Dic127C2311yescharacteristic
57Dic127⋊C4—22no—
58Dic127⋊C4—22no—
59C22⋊Dic127—21no—
60C22⋊Dic127D412yes—
61(C2×D127)⋊C4—22no—
62(C2×D127)⋊C4—22no—
63C4×Dic127—21no—
64C4×Dic127C2312yes—
65#41×C127—22no—
66#41×C127—22no—
67C2×Q8×C127C2311yescharacteristic
68C2×D4×C127—21no—
69C2×D4×C127C2311yescharacteristic
70C22×C4×C127—21no—
71C22×C4×C127—21no—
72C22×C4×C127C2311yescharacteristic
73M16×C127—22no—
74C2×C8×C127C2312yes—
75C4⋊C4×C127—21no—
76C4⋊C4×C127C2312yes—
77C22×D127—22no—
78C22×D127—42no—
79C22×D127—81no—
80C22×D127—81no—
81Dic254—82no—
82D508—22no—
83D508—22no—
84D508—42no—
85C22⋊D127—82no—
86C22⋊D127—82no—
87C2×Dic127—21no—
88C2×Dic127—21no—
89C2×Dic127C2×D412yes—
90C2×Dic127C2×D412yes—
91C127⋊C8—21no—
92C127⋊C8—21no—
93C23×C127—21no—
94C23×C127C2×D411yescharacteristic
95Q8×C127—21no—
96Q8×C127—21no—
97D4×C127—21no—
98D4×C127—21no—
99D4×C127—42no—
100C2×C4×C127—22no—
101C2×C4×C127—21no—
102C2×C4×C127—42no—
103C2×C4×C127—81no—
104C2×C4×C127C2×D411yescharacteristic
105C2×C4×C127#4112yes—
106C2×C4×C127#4112yes—
107C2×C4×C127C2411yescharacteristic
108C8×C127—22no—
109C22×C127—21no—
110C22×C127—21no—
111C22×C127—41no—
112C22×C127—41no—
113C22×C127—41no—
114C22×C127#12412yes—
115C22×C127C22×D411yescharacteristic
116C2×D127—42no—
117C2×D127—42no—
118C2×D127—42no—
119C2×D127—84no—
120C4×C127—41no—
121C4×C127—42no—
122C4×C127#12412yes—
123C4×C127C2×#4111yescharacteristic
124C4×C127C2×#4111yescharacteristic
125Dic127—22no—
126Dic127—22no—
127Dic127—41no—
128Dic127—41no—
129C2×C127—22no—
130C2×C127—81no—
131C2×C127C2×#12411yescharacteristic
132C2×C127#57611yescharacteristic
133C2×C127#57711yescharacteristic
134D127—82no—
135#3136—1271nomaximal, Sylow
136C127#313611yescharacteristic, complemented, Sylow
137#544—1271no—
138#535—1271no—
139#490—1272no—
140#489—1272no—
141#488—1272no—
142#486—1272no—
143#418—1271no—
144#417D12711yescharacteristic, complemented
145#416—1271no—
146C2×#102—1271no—
147#405—1271no—
148C2×#41C2×D12711yescharacteristic
149C22×D4—1271no—
150C2×M16C2×D12711yescharacteristic
151C22×C8—1271no—
152C4⋊Q8—1272no—
153C4⋊Q8—2541no—
154#127—1272no—
155#125—1272no—
156#125—2541no—
157#124—1272no—
158C22⋊Q8—2542no—
159#121—2541no—
160#121—2541no—
161#120—2542no—
162#117—1271no—
163#117—1271no—
164C2×C4⋊C4C2×D12711yescharacteristic
165#103—1271no—
166#103—1271no—
167#103C2×D12712yescomplemented
168#102—1271no—
169#102—1271no—
170#102—1272no—
171#102—1272no—
172#102—1272no—
173#102—1272no—
174#102C2×D12712yescomplemented
175#97—1272no—
176C4×C8—1272no—
177C24—2541no—
178#41—22no—
179#41—22no—
180C2×Q8—2542no—
181C2×Q8C22×D12711yescharacteristic
182C2×D4—21no—
183C2×D4—1272no—
184C2×D4—2542no—
185C2×D4—2542no—
186C2×D4—2541no—
187C2×D4—2542no—
188C2×D4—2542no—
189C2×D4C22×D12711yescharacteristic
190C22×C4—21no—
191C22×C4—21no—
192C22×C4C22×D12711yescharacteristic
193M16—22no—
194C2×C8—1271no—
195C2×C8—1271no—
196C2×C8—2542no—
197C2×C8C22×D12712yes—
198C4⋊C4—21no—
199C4⋊C4—1271no—
200C4⋊C4—1271no—
201C4⋊C4—1271no—
202C4⋊C4—1271no—
203C4⋊C4—2542no—
204C4⋊C4—2542no—
205C4⋊C4—2542no—
206C4⋊C4C22×D12712yes—
207C22⋊C4—1272no—
208C22⋊C4—2542no—
209C22⋊C4—2542no—
210C22⋊C4—2541no—
211C42—1272no—
212C42—2541no—
213C23—21no—
214C23—10161no—
215C23—10161no—
216C23—5082no—
217C23—2542no—
218C23D4×D12711yescharacteristic
219Q8—21no—
220Q8—21no—
221Q8—10162no—
222D4—21no—
223D4—21no—
224D4—42no—
225D4—10162no—
226D4—10162no—
227D4—5082no—
228D4—2542no—
229D4—2542no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—42no—
233C2×C4—81no—
234C2×C4—1272no—
235C2×C4—1272no—
236C2×C4—2541no—
237C2×C4—2541no—
238C2×C4D4×D12711yescharacteristic
239C2×C4#1452324712yes—
240C2×C4#1452324812yes—
241C2×C4C23×D12711yescharacteristic
242C8—22no—
243C8—2541no—
244C8—2541no—
245C22—21no—
246C22—21no—
247C22—41no—
248C22—41no—
249C22—41no—
250C22—10164no—
251C22—5082no—
252C22—5082no—
253C22—5082no—
254C22#1686119512yes—
255C22C2×D4×D12711yescharacteristic
256C4—41no—
257C4—42no—
258C4—5081no—
259C4—5081no—
260C4—2542no—
261C4—2542no—
262C4#1686119512yes—
263C4C2×#1452324711yescharacteristic
264C4C2×#1452324811yescharacteristic
265C2—22no—
266C2—81no—
267C2—10162no—
268C2C2×#1686119511yescharacteristic
269C2#2255068011yescharacteristic
270C2#2255068511yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#417×C127C2
socleC22×C127C22×D4
prime radical#22550440C2
hypercenter#417D127
hypocenterC127#3136
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. #22550507C2semidirect
3. #22550239C2semidirect
4. #22550237C2semidirect
5. #22550232C2semidirect
6. #22550221C2semidirect
7. #22549882C2semidirect
8. #22549872C2semidirect
9. C2×#16861129C2semidirect
10. #22549834C2semidirect
11. #417×C127C2semidirect
15. #16861208C22semidirect
18. #16861200C22semidirect
26. #16861133C22semidirect
27. #16861133C22semidirect
30. #16861130C22semidirect
31. #16861130C22semidirect
34. #16861119C22semidirect
35. C8×Dic127C22semidirect
39. #103×C127C22semidirect
40. #102×C127C22semidirect
136. C127#3136semidirect
144. #417D127semidirect
167. #103C2×D127semidirect
174. #102C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225505072
#225504402
#225502392
#225502372
#225502322
#225502212
#225498822
#225498722
C2×#168611292
#225498342
#417×C1272
#3136127