Finite Group Databasebeta

← #30648121

Subgroups — #30648121

Order 16256.

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

normal subgroups only

This group has 766 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
1#22550712C211yesmaximal, characteristic, complemented
2#22550387C212yesmaximal, complemented
3#22550065C214yesmaximal, complemented
4#22550062C214yesmaximal, complemented
5#22549850C211yesmaximal, characteristic, complemented
6#22549849C212yesmaximal, complemented
7#525×C127C211yesmaximal, characteristic, complemented
8#16861257—21no—
9#16861256—22no—
10#16861255—22no—
11#16861254—21no—
12#16861254C2214yes—
13C2×#14523248—22no—
14C2×#14523248C2212yes—
15#41×D127—24no—
16#16861192C2214yescomplemented
17#16861188—22no—
18#16861183—22no—
19#16861178C2214yescomplemented
20C4×C22⋊D127—24no—
21#16861168C2212yescomplemented
22#16861124C2214yescomplemented
23#16861123C2218yescomplemented
24C2×#41×C127C2211yescharacteristic
25C23×C4×C127C2211yescharacteristic, complemented
26#123×C127C2212yescomplemented
27C22⋊Q8×C127—21no—
28#121×C127—21no—
29#120×C127C2212yescomplemented
30C4×D4×C127—22no—
31#117×C127C2211yescharacteristic, complemented
32#14523249—24no—
33#14523248—24no—
34#14523248—24no—
35#14523248—22no—
36#14523248—22no—
37#14523248—24no—
38#14523248—22no—
39#14523248—22no—
40#14523247—24no—
41#14523247—28no—
42C2×Dic254—22no—
43C2×Dic254D412yescomplemented
44Q8×D127—24no—
45C2×D508—22no—
46C2×D508D412yescomplemented
47C2×C22⋊D127—24no—
48C2×C22⋊D127—24no—
49C2×C22⋊D127C2314yes—
50D4×D127—24no—
51D4×D127—28no—
52C2×C4×D127—24no—
53C2×C4×D127D414yescomplemented
54C4⋊Dic127—22no—
55Dic127⋊C4—24no—
56C22⋊Dic127—24no—
57C22⋊Dic127C2314yes—
58(C2×D127)⋊C4—24no—
59C4×Dic127—22no—
60C24×C127C2311yescharacteristic
61#41×C127—22no—
62#41×C127—22no—
63C2×Q8×C127D411yescharacteristic, complemented
64C2×D4×C127—22no—
65C2×D4×C127D411yescharacteristic, complemented
66C2×D4×C127C2312yes—
67C22×C4×C127—22no—
68C22×C4×C127—21no—
69C22×C4×C127—21no—
70C22×C4×C127—22no—
71C22×C4×C127D412yescomplemented
72C22×C4×C127C2311yescharacteristic
73C22×C4×C127C2311yescharacteristic
74C4⋊C4×C127—21no—
75C4⋊C4×C127—22no—
76C22⋊C4×C127—22no—
77C22⋊C4×C127—22no—
78C22⋊C4×C127C2312yes—
79C42×C127—21no—
80C22×D127—24no—
81C22×D127C2×D414yes—
82Dic254—24no—
83Dic254—22no—
84Dic254—22no—
85D508—24no—
86D508—22no—
87D508—22no—
88C22⋊D127—28no—
89C22⋊D127—24no—
90C22⋊D127—24no—
91C22⋊D127—48no—
92C2×Dic127—24no—
93C2×Dic127—24no—
94C2×Dic127C2×D414yes—
95C4×D127—28no—
96C4×D127—24no—
97C4×D127—24no—
98C23×C127—21no—
99C23×C127—22no—
100C23×C127—42no—
101C23×C127C2×D412yes—
102C23×C127C2411yescharacteristic
103Q8×C127—22no—
104D4×C127—24no—
105D4×C127—22no—
106D4×C127—44no—
107C2×C4×C127—22no—
108C2×C4×C127—22no—
109C2×C4×C127—22no—
110C2×C4×C127—21no—
111C2×C4×C127—22no—
112C2×C4×C127—41no—
113C2×C4×C127—42no—
114C2×C4×C127—41no—
115C2×C4×C127C2×D412yes—
116C2×C4×C127C2×D412yes—
117C2×C4×C127C2×D412yes—
118C2×C4×C127C2×D411yescharacteristic
119C2×C4×C127C2×D411yescharacteristic
120C22×C127—22no—
121C22×C127—24no—
122C22×C127—22no—
123C22×C127—41no—
124C22×C127—42no—
125C22×C127—42no—
126C22×C127—41no—
127C22×C127—41no—
128C22×C127C22×D412yes—
129C22×C127C22×D411yescharacteristic
130C2×D127—24no—
131C2×D127—44no—
132C2×D127—84no—
133C4×C127—22no—
134C4×C127—42no—
135C4×C127—41no—
136C4×C127#12011yescharacteristic
137C4×C127#12012yes—
138C4×C127#12011yescharacteristic
139Dic127—24no—
140Dic127—44no—
141C2×C127—22no—
142C2×C127—21no—
143C2×C127—42no—
144C2×C127—42no—
145C2×C127C2×#12011yescharacteristic
146D127—44no—
147#3025—1271nomaximal, Sylow
148C127#302511yescharacteristic, complemented, Sylow
149#585—1271no—
150#525—1272no—
151#525D12711yescharacteristic, complemented
152#458—1274no—
153B(4,𝔽2)—1274no—
154#410—1271no—
155#410—1272no—
1562−1+4—2542no—
1572−1+4—2541no—
158H(5,𝔽2)—1274no—
159H(5,𝔽2)—2542no—
160H(5,𝔽2)—2541no—
161C2×#41—1272no—
162C2×#41—2544no—
163C2×#41—2542no—
164C2×#41C2×D12711yescharacteristic
165C23×C4C2×D12711yescharacteristic, complemented
166#123—1274no—
167#123C2×D12712yescomplemented
168C22⋊Q8—21no—
169C22⋊Q8—2542no—
170#121—21no—
171#121—2542no—
172#120—1274no—
173#120C2×D12712yescomplemented
174C4×D4—22no—
175C4×D4—2544no—
176#117—1272no—
177#117C2×D12711yescharacteristic, complemented
178#99—1274no—
179#99—1278no—
180C24C22×D12711yescharacteristic
181#41—22no—
182#41—22no—
183#41—2544no—
184#41—2544no—
185#41—2548no—
186#41—2544no—
187#41—2544no—
188#41—2542no—
189#41—2542no—
190#41—2542no—
191#41—2542no—
192#41—2544no—
193C2×Q8—1272no—
194C2×Q8—2542no—
195C2×Q8—2544no—
196C2×Q8C22⋊D12711yescharacteristic, complemented
197C2×D4—22no—
198C2×D4—1274no—
199C2×D4—1272no—
200C2×D4—2542no—
201C2×D4—2548no—
202C2×D4—2544no—
203C2×D4—2544no—
204C2×D4—2544no—
205C2×D4C22⋊D12711yescharacteristic, complemented
206C2×D4C22×D12712yes—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—1274no—
212C22×C4—2544no—
213C22×C4C22⋊D12712yescomplemented
214C22×C4C22×D12711yescharacteristic
215C22×C4C22×D12711yescharacteristic
216C4⋊C4—21no—
217C4⋊C4—22no—
218C4⋊C4—2544no—
219C4⋊C4—2542no—
220C22⋊C4—22no—
221C22⋊C4—22no—
222C22⋊C4—1274no—
223C22⋊C4—2544no—
224C22⋊C4—2544no—
225C22⋊C4C22×D12712yes—
226C42—21no—
227C42—2542no—
228C23—22no—
229C23—21no—
230C23—42no—
231C23—1274no—
232C23—2544no—
233C23C2×C22⋊D12712yes—
234C23C23×D12711yescharacteristic
235Q8—22no—
236Q8—2544no—
237Q8—2542no—
238Q8—2542no—
239D4—24no—
240D4—22no—
241D4—44no—
242D4—5088no—
243D4—2544no—
244D4—2548no—
245D4—2544no—
246D4—2544no—
247D4—2542no—
248D4—2542no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—41no—
255C2×C4—42no—
256C2×C4—41no—
257C2×C4—1274no—
258C2×C4—2548no—
259C2×C4—2544no—
260C2×C4—2544no—
261C2×C4—2544no—
262C2×C4—2544no—
263C2×C4D4×D12712yes—
264C2×C4D4×D12712yes—
265C2×C4C2×C22⋊D12712yes—
266C2×C4C2×C22⋊D12711yescharacteristic
267C2×C4C2×C22⋊D12711yescharacteristic
268C22—22no—
269C22—24no—
270C22—22no—
271C22—41no—
272C22—42no—
273C22—41no—
274C22—42no—
275C22—41no—
276C22—10164no—
277C22—5084no—
278C22—2544no—
279C22C2×D4×D12712yes—
280C22C22×C22⋊D12711yescharacteristic
281C4—22no—
282C4—42no—
283C4—41no—
284C4—5084no—
285C4—2544no—
286C4#1686117711yescharacteristic
287C4#1686117712yes—
288C4#1686117711yescharacteristic
289C2—21no—
290C2—22no—
291C2—42no—
292C2—42no—
293C2—5084no—
294C2C2×#1686117711yescharacteristic
subgroupG/H
centerC4#16861177
commutator, cosocleC23×C127C24
FrattiniC23C23×D127
Fitting#525×C127C2
socleC2×C127C2×#120
hypercenter#525D127
hypocenterC127#3025
LCSG ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C4 ◁ C22×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. #22550712C2semidirect
2. #22550387C2semidirect
3. #22550065C2semidirect
4. #22550062C2semidirect
5. #22549850C2semidirect
6. #22549849C2semidirect
7. #525×C127C2semidirect
16. #16861192C22semidirect
19. #16861178C22semidirect
21. #16861168C22semidirect
22. #16861124C22semidirect
23. #16861123C22semidirect
25. C23×C4×C127C22semidirect
26. #123×C127C22semidirect
29. #120×C127C22semidirect
31. #117×C127C22semidirect
43. C2×Dic254D4semidirect
46. C2×D508D4semidirect
53. C2×C4×D127D4semidirect
63. C2×Q8×C127D4semidirect
65. C2×D4×C127D4semidirect
71. C22×C4×C127D4semidirect
148. C127#3025semidirect
151. #525D127semidirect
165. C23×C4C2×D127semidirect
167. #123C2×D127semidirect
173. #120C2×D127semidirect
177. #117C2×D127semidirect
196. C2×Q8C22⋊D127semidirect
205. C2×D4C22⋊D127semidirect
213. C22×C4C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
#225507122
#225503872
#225500652
#225500622
#225498502
#225498492
#525×C1272
#3025127