Finite Group Databasebeta

← #30648354

Subgroups — #30648354

Order 16256.

496 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 143 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×M16×D127C211yesmaximal, characteristic, complemented
2#22550383C211yesmaximal, characteristic
3C2×#16861185C211yesmaximal, characteristic, complemented
4#22550131C212yesmaximal, complemented
5#22550119C212yesmaximal, complemented
6#22550114C212yesmaximal, complemented
7#22550102C212yesmaximal, complemented
8#22549918C211yesmaximal, characteristic, complemented
9#22549885C211yesmaximal, characteristic, complemented
10#22549877C211yesmaximal, characteristic, complemented
11C2×#137×C127C211yesmaximal, characteristic, complemented
12C22×C4×D127C2211yescharacteristic
13C2×#14523232C2211yescharacteristic
14C2×#14523231C2212yes—
15M16×D127—24no—
16C2×C8×D127C2212yes—
17#16861190—21no—
18#16861185—21no—
19#16861185—21no—
20#16861185C2212yes—
21#16861185C2211yescharacteristic
22#16861185C2212yes—
23#16861185C2211yescharacteristic
24#16861180C2211yescharacteristic
25#16861180C2211yescharacteristic
26#16861177—21no—
27C4×C22⋊D127—21no—
28C4×C22⋊D127—21no—
29#16861168C2211yescharacteristic
30C2×C4⋊Dic127C2211yescharacteristic
31C2×Dic127⋊C4—21no—
32C2×(C2×D127)⋊C4—21no—
33#16861143C2212yescomplemented
34#16861141C2212yescomplemented
35#16861133C2212yescomplemented
36#16861133C2212yescomplemented
37#16861133C2212yescomplemented
38#16861130C2212yescomplemented
39C2×#41×C127C2211yescharacteristic
40C22×Q8×C127C2211yescharacteristic
41#137×C127—24no—
42C2×Q16×C127C2212yescomplemented
43C2×SD16×C127C2212yescomplemented
44C2×M16×C127C2211yescharacteristic
45C23×D127D411yescharacteristic
46C2×C22⋊D127—21no—
47C2×C22⋊D127—21no—
48C22×Dic127—21no—
49C22×Dic127D411yescharacteristic
50C2×C4×D127—22no—
51C2×C4×D127—22no—
52C2×C4×D127D411yescharacteristic
53C2×C4×D127D411yescharacteristic
54C2×C4×D127C2311yescharacteristic
55C2×C4×D127C2311yescharacteristic
56#14523232—22no—
57#14523231—24no—
58C2×C127⋊C8C2312yes—
59C8×D127—24no—
60C4⋊Dic127—21no—
61C4⋊Dic127C2312yes—
62C4⋊Dic127C2311yescharacteristic
63C4⋊Dic127C2311yescharacteristic
64Dic127⋊C4—22no—
65Dic127⋊C4—22no—
66Dic127⋊C4—21no—
67Dic127⋊C4—21no—
68C22⋊Dic127—21no—
69C22⋊Dic127—21no—
70C22⋊Dic127—21no—
71(C2×D127)⋊C4—22no—
72(C2×D127)⋊C4—22no—
73(C2×D127)⋊C4—21no—
74(C2×D127)⋊C4—21no—
75C4×Dic127—21no—
76#41×C127—22no—
77#41×C127—22no—
78C2×Q8×C127—21no—
79C2×Q8×C127—22no—
80C2×Q8×C127—22no—
81C2×Q8×C127C2312yes—
82C2×Q8×C127C2311yescharacteristic
83C2×D4×C127—21no—
84C2×D4×C127C2311yescharacteristic
85C22×C4×C127—21no—
86C22×C4×C127—21no—
87C22×C4×C127C2311yescharacteristic
88Q16×C127—24no—
89SD16×C127—24no—
90M16×C127D414yescomplemented
91C2×C8×C127C2312yes—
92C22×D127—22no—
93C22×D127—22no—
94C22×D127—41no—
95C22×D127C2×D411yescharacteristic
96C22×D127C2×D411yescharacteristic
97C22⋊D127—42no—
98C22⋊D127—42no—
99C2×Dic127—22no—
100C2×Dic127—21no—
101C2×Dic127—21no—
102C2×Dic127—22no—
103C2×Dic127—42no—
104C2×Dic127C2×D411yescharacteristic
105C2×Dic127C2×D411yescharacteristic
106C4×D127—22no—
107C4×D127—22no—
108C4×D127—21no—
109C4×D127—21no—
110C4×D127—21no—
111C4×D127—21no—
112C127⋊C8—22no—
113C23×C127—21no—
114C23×C127C2×D411yescharacteristic
115Q8×C127—22no—
116Q8×C127—21no—
117Q8×C127—22no—
118Q8×C127—21no—
119Q8×C127—42no—
120D4×C127—21no—
121D4×C127—21no—
122D4×C127—42no—
123C2×C4×C127—21no—
124C2×C4×C127—22no—
125C2×C4×C127—42no—
126C2×C4×C127—42no—
127C2×C4×C127C2×D411yescharacteristic
128C2×C4×C127C2×D412yes—
129C2×C4×C127#4112yes—
130C2×C4×C127C2411yescharacteristic
131C8×C127C2×D414yes—
132C22×C127—21no—
133C22×C127—21no—
134C22×C127—41no—
135C22×C127—41no—
136C22×C127—41no—
137C22×C127#12112yes—
138C22×C127C22×D411yescharacteristic
139C2×D127—21no—
140C2×D127—21no—
141C2×D127—21no—
142C2×D127—21no—
143C2×D127—21no—
144C2×D127—44no—
145C2×D127#13712yescomplemented
146C4×C127—41no—
147C4×C127—42no—
148C4×C127#12112yes—
149C4×C127C22×D411yescharacteristic
150C4×C127C2×#4111yescharacteristic
151Dic127—21no—
152Dic127—21no—
153Dic127—41no—
154Dic127—42no—
155Dic127—41no—
156C2×C127—22no—
157C2×C127—81no—
158C2×C127C2×#12111yescharacteristic
159C2×C127C2×#13711yescharacteristic
160C2×C127#57511yescharacteristic
161D127—22no—
162D127—41no—
163#3056—1271nomaximal, Sylow
164C127#305611yescharacteristic, complemented, Sylow
165C2×#137D12711yescharacteristic, complemented
166C22×M16—1271no—
167#525—1271no—
168C2×C22⋊Q8—1271no—
169#470—1272no—
170#468—1272no—
171#467—1272no—
172#465—1272no—
173#428—1271no—
174#419—1271no—
175#417—1271no—
176C2×#41C2×D12711yescharacteristic
177C22×Q8C2×D12711yescharacteristic
178C23×C4—1271no—
179#137—24no—
180C2×Q16C2×D12712yescomplemented
181C2×SD16C2×D12712yescomplemented
182C2×M16—1271no—
183C2×M16—1272no—
184C2×M16—2544no—
185C2×M16C2×D12711yescharacteristic
186C22×C8—1272no—
187#123—2541no—
188C22⋊Q8—1272no—
189C22⋊Q8—1272no—
190C22⋊Q8—1271no—
191C22⋊Q8—1271no—
192C22⋊Q8—2541no—
193C22⋊Q8—2541no—
194#121—1271no—
195#121—1271no—
196#120—2541no—
197C4×D4—2541no—
198C4×D4—2541no—
199#117—1271no—
200C2×C4⋊C4—1271no—
201C2×C4⋊C4—2541no—
202C2×C22⋊C4—2541no—
203#107—1272no—
204#106—1272no—
205#103—1272no—
206#103—1272no—
207#103—1272no—
208#102—1272no—
209C24—1271no—
210#41—22no—
211#41—22no—
212C2×Q8—21no—
213C2×Q8—22no—
214C2×Q8—22no—
215C2×Q8C22×D12712yes—
216C2×Q8C22×D12711yescharacteristic
217C2×D4—21no—
218C2×D4—2541no—
219C2×D4—2541no—
220C2×D4C22×D12711yescharacteristic
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—1271no—
224C22×C4—1271no—
225C22×C4—1271no—
226C22×C4—1271no—
227C22×C4—1271no—
228C22×C4—2541no—
229C22×C4—2542no—
230C22×C4—2542no—
231C22×C4C22×D12711yescharacteristic
232Q16—24no—
233SD16—24no—
234M16—2542no—
235M16—2544no—
236M16C22⋊D12714yescomplemented
237C2×C8—1272no—
238C2×C8—2544no—
239C2×C8C22×D12712yes—
240C4⋊C4—1272no—
241C4⋊C4—1271no—
242C4⋊C4—1271no—
243C4⋊C4—2542no—
244C4⋊C4—2542no—
245C4⋊C4—2541no—
246C4⋊C4—2541no—
247C4⋊C4—2541no—
248C22⋊C4—2542no—
249C22⋊C4—2542no—
250C22⋊C4—2541no—
251C22⋊C4—2541no—
252C22⋊C4—2541no—
253C22⋊C4—2541no—
254C22⋊C4—2541no—
255C42—2541no—
256C23—21no—
257C23—1271no—
258C23—1271no—
259C23—5081no—
260C23—2542no—
261C23—2542no—
262C23D4×D12711yescharacteristic
263Q8—22no—
264Q8—21no—
265Q8—22no—
266Q8—21no—
267Q8—42no—
268D4—21no—
269D4—21no—
270D4—42no—
271D4—5082no—
272D4—5082no—
273C2×C4—21no—
274C2×C4—22no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—1271no—
278C2×C4—1271no—
279C2×C4—5082no—
280C2×C4—2541no—
281C2×C4—2541no—
282C2×C4—2542no—
283C2×C4—2542no—
284C2×C4—2541no—
285C2×C4—2541no—
286C2×C4—2541no—
287C2×C4—2541no—
288C2×C4—2542no—
289C2×C4—2542no—
290C2×C4D4×D12711yescharacteristic
291C2×C4C2×C22⋊D12712yes—
292C2×C4#1452324712yes—
293C2×C4C23×D12711yescharacteristic
294C8—2542no—
295C8C2×C22⋊D12714yes—
296C22—21no—
297C22—21no—
298C22—41no—
299C22—41no—
300C22—41no—
301C22—1272no—
302C22—5084no—
303C22—2541no—
304C22—2541no—
305C22—2541no—
306C22—2541no—
307C22—2541no—
308C22#1686118012yes—
309C22C2×D4×D12711yescharacteristic
310C4—41no—
311C4—42no—
312C4—5081no—
313C4—5082no—
314C4—5081no—
315C4—2541no—
316C4—2541no—
317C4#1686118012yes—
318C4C22×C22⋊D12711yescharacteristic
319C4C2×#1452324711yescharacteristic
320C2—22no—
321C2—81no—
322C2—5081no—
323C2—2542no—
324C2C2×#1686118011yescharacteristic
325C2#137×D12711yescharacteristic
326C2#2255067511yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×#137×C127C2
socleC22×C127C22×D4
prime radical#22550383C2
hypercenterC2×#137D127
hypocenterC127#3056
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C2×M16×D127C2semidirect
3. C2×#16861185C2semidirect
4. #22550131C2semidirect
5. #22550119C2semidirect
6. #22550114C2semidirect
7. #22550102C2semidirect
8. #22549918C2semidirect
9. #22549885C2semidirect
10. #22549877C2semidirect
11. C2×#137×C127C2semidirect
33. #16861143C22semidirect
34. #16861141C22semidirect
35. #16861133C22semidirect
36. #16861133C22semidirect
37. #16861133C22semidirect
38. #16861130C22semidirect
42. C2×Q16×C127C22semidirect
43. C2×SD16×C127C22semidirect
90. M16×C127D4semidirect
145. C2×D127#137semidirect
164. C127#3056semidirect
165. C2×#137D127semidirect
180. C2×Q16C2×D127semidirect
181. C2×SD16C2×D127semidirect
236. M16C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
C2×M16×D1272
#225503832
C2×#168611852
#225501312
#225501192
#225501142
#225501022
#225499182
#225498852
#225498772
C2×#137×C1272
#3056127