Finite Group Databasebeta

← #480×D127 #30648598

Subgroups — #480×D127 #30648598

Order 16256.

512 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 135 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#121×D127C211yesmaximal, characteristic
2C2×C4⋊C4×D127C211yesmaximal, characteristic, complemented
3#22550191C212yesmaximal, complemented
4#22550190C211yesmaximal, characteristic, complemented
5#22550189C211yesmaximal, characteristic, complemented
6#22550188C212yesmaximal, complemented
7#22550187C211yesmaximal, characteristic, complemented
8#107×D127C212yesmaximal, complemented
9#102×D127C212yesmaximal, complemented
10C22⋊C8×D127C211yesmaximal, characteristic, complemented
11#480×C127C211yesmaximal, characteristic, complemented
12C2×D4×D127—21no—
13C2×D4×D127C2211yescharacteristic
14C22×C4×D127—21no—
15C22×C4×D127C2211yescharacteristic
16C2×C8×D127C2212yes—
17#16861184C2211yescharacteristic
18#16861183C2211yescharacteristic
19#16861182—21no—
20#16861181—21no—
21#16861180C2211yescharacteristic
22C2×C4⋊Dic127C2211yescharacteristic, complemented
23C2×Dic127⋊C4—21no—
24C4⋊C4×D127—21no—
25C4⋊C4×D127—21no—
26C4⋊C4×D127C2212yes—
27C4⋊C4×D127C2211yescharacteristic
28C4⋊C4×D127C2212yescomplemented
29C22⋊C4×D127—21no—
30#16861143C2212yescomplemented
31#16861142C2212yescomplemented
32#16861142C2212yescomplemented
33#16861131C2212yescomplemented
34#16861130C2212yescomplemented
35#16861129C2212yescomplemented
36C22⋊(C127⋊C8)C2211yescharacteristic, complemented
37(C2×D127)⋊C8C2212yescomplemented
38#121×C127C2211yescharacteristic
39C2×C4⋊C4×C127C2211yescharacteristic, complemented
40#107×C127C2212yescomplemented
41#102×C127C2212yescomplemented
42C22⋊C8×C127C2211yescharacteristic, complemented
43C23×D127—21no—
44C23×D127D411yescharacteristic
45C2×D508—21no—
46C2×D508C2311yescharacteristic
47C2×C22⋊D127—21no—
48C2×C22⋊D127—21no—
49C2×C22⋊D127—21no—
50D4×D127—24no—
51D4×D127—44no—
52C22×Dic127—21no—
53C22×Dic127D411yescharacteristic
54C2×C4×D127—24no—
55C2×C4×D127—22no—
56C2×C4×D127—22no—
57C2×C4×D127—21no—
58C2×C4×D127—44no—
59C2×C4×D127D411yescharacteristic
60C2×C4×D127D411yescharacteristic
61C2×C4×D127C2311yescharacteristic
62C2×C4×D127C2311yescharacteristic
63C2×C127⋊C8C2312yes—
64C8×D127—24no—
65C4⋊Dic127—21no—
66C4⋊Dic127C2312yes—
67C4⋊Dic127C2311yescharacteristic
68Dic127⋊C4—22no—
69Dic127⋊C4—22no—
70Dic127⋊C4—21no—
71C22⋊Dic127—21no—
72(C2×D127)⋊C4—21no—
73(C2×D127)⋊C4—21no—
74C2×D4×C127—21no—
75C2×D4×C127C2311yescharacteristic
76C22×C4×C127—21no—
77C22×C4×C127C2311yescharacteristic
78C2×C8×C127C2312yes—
79C4⋊C4×C127—21no—
80C4⋊C4×C127C2311yescharacteristic
81C4⋊C4×C127C2312yes—
82C22⋊C4×C127—21no—
83C22×D127—22no—
84C22×D127—22no—
85C22×D127—21no—
86C22×D127—42no—
87C22×D127—42no—
88C22×D127—42no—
89C22×D127D814yes—
90C22×D127C2×D411yescharacteristic
91C22×D127C2×D411yescharacteristic
92D508—22no—
93D508—42no—
94C22⋊D127—42no—
95C22⋊D127—42no—
96C22⋊D127—42no—
97C2×Dic127—22no—
98C2×Dic127—21no—
99C2×Dic127—22no—
100C2×Dic127—42no—
101C2×Dic127C2×D411yescharacteristic
102C2×Dic127C2×D411yescharacteristic
103C4×D127—22no—
104C4×D127—22no—
105C4×D127—22no—
106C4×D127—44no—
107C4×D127—44no—
108C4×D127—42no—
109C4×D127#4114yes—
110C127⋊C8—22no—
111C23×C127—21no—
112C23×C127C2×D411yescharacteristic
113D4×C127—22no—
114D4×C127—42no—
115C2×C4×C127—22no—
116C2×C4×C127—21no—
117C2×C4×C127—22no—
118C2×C4×C127—42no—
119C2×C4×C127C2×D411yescharacteristic
120C2×C4×C127C2411yescharacteristic
121C8×C127—22no—
122C22×C127—21no—
123C22×C127—21no—
124C22×C127—41no—
125C22×C127—41no—
126C22×C127—41no—
127C22×C127C2×D812yes—
128C22×C127C22×D411yescharacteristic
129C2×D127—21no—
130C2×D127—21no—
131C2×D127—28no—
132C2×D127—41no—
133C2×D127—41no—
134C2×D127—41no—
135C2×D127—84no—
136C2×D127#12312yes—
137C2×D127C2×D812yes—
138C2×D127C2×D812yes—
139C2×D127#13712yes—
140C4×C127—21no—
141C4×C127—42no—
142C4×C127—41no—
143C4×C127C2×#4112yes—
144Dic127—21no—
145Dic127—42no—
146Dic127—41no—
147Dic127#12312yes—
148C2×C127—22no—
149C2×C127—81no—
150C2×C127C2×#12311yescharacteristic
151C2×C127C22×D811yescharacteristic
152C2×C127C2×#13711yescharacteristic
153D127—22no—
154D127—81no—
155D127#48014yesdirect product factor
156C2×#480—1271nomaximal, Sylow
157C127C2×#48011yescharacteristic, complemented, Sylow
158C2×#121—1271no—
159C22×C4⋊C4—1271no—
160#480—1271no—
161#480—1271no—
162#480—1272no—
163#480—1272no—
164#480—1271no—
165#480D12711yescharacteristic, direct product factor
166C2×#107—1272no—
167C2×#102—1272no—
168C2×C22⋊C8—1271no—
169C22×D4—1271no—
170C22×D4—2541no—
171C23×C4—1271no—
172C23×C4—2541no—
173C22×C8—1272no—
174#121—1271no—
175#121—1271no—
176#121—1271no—
177#121—2541no—
178#121—2541no—
179#121C2×D12711yescharacteristic
180C2×C4⋊C4—1271no—
181C2×C4⋊C4—1272no—
182C2×C4⋊C4—1272no—
183C2×C4⋊C4—1271no—
184C2×C4⋊C4—2541no—
185C2×C4⋊C4—2541no—
186C2×C4⋊C4—2541no—
187C2×C4⋊C4C2×D12711yescharacteristic, complemented
188C2×C22⋊C4—2541no—
189#107—1272no—
190#107—1272no—
191#107—1272no—
192#107C2×D12712yescomplemented
193#102—1272no—
194#102—1272no—
195#102—1272no—
196#102C2×D12712yescomplemented
197C22⋊C8—1271no—
198C22⋊C8—1272no—
199C22⋊C8C2×D12711yescharacteristic, complemented
200C24—1271no—
201C24—2541no—
202C2×D4—21no—
203C2×D4—1271no—
204C2×D4—5084no—
205C2×D4—2541no—
206C2×D4—2544no—
207C2×D4—2541no—
208C2×D4—2541no—
209C2×D4—2541no—
210C2×D4C22×D12711yescharacteristic
211C22×C4—21no—
212C22×C4—1271no—
213C22×C4—1271no—
214C22×C4—1271no—
215C22×C4—1271no—
216C22×C4—1271no—
217C22×C4—5084no—
218C22×C4—2544no—
219C22×C4—2541no—
220C22×C4—2542no—
221C22×C4—2542no—
222C22×C4—2541no—
223C22×C4C22×D12711yescharacteristic
224C2×C8—1272no—
225C2×C8—2544no—
226C2×C8C22×D12712yes—
227C4⋊C4—21no—
228C4⋊C4—1272no—
229C4⋊C4—1271no—
230C4⋊C4—2542no—
231C4⋊C4—2542no—
232C4⋊C4—2541no—
233C4⋊C4—2541no—
234C4⋊C4C22×D12712yes—
235C4⋊C4C22×D12711yescharacteristic
236C22⋊C4—21no—
237C22⋊C4—2541no—
238C22⋊C4—2541no—
239C22⋊C4—2541no—
240C23—21no—
241C23—1271no—
242C23—1274no—
243C23—1271no—
244C23—5082no—
245C23—5082no—
246C23—5082no—
247C23—2542no—
248C23—2542no—
249C23—2541no—
250C23D4×D12711yescharacteristic
251D4—22no—
252D4—42no—
253D4—5082no—
254D4—5082no—
255D4—5082no—
256D4—5082no—
257D4—2542no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—42no—
262C2×C4—1271no—
263C2×C4—1274no—
264C2×C4—1271no—
265C2×C4—5084no—
266C2×C4—5084no—
267C2×C4—5082no—
268C2×C4—5082no—
269C2×C4—2542no—
270C2×C4—2542no—
271C2×C4—2542no—
272C2×C4—2541no—
273C2×C4—2542no—
274C2×C4—2542no—
275C2×C4D4×D12711yescharacteristic
276C2×C4C23×D12711yescharacteristic
277C8—22no—
278C8—2542no—
279C22—21no—
280C22—21no—
281C22—41no—
282C22—41no—
283C22—41no—
284C22—1272no—
285C22—1272no—
286C22—1272no—
287C22—1272no—
288C22—10164no—
289C22—5081no—
290C22—5081no—
291C22—5081no—
292C22—2548no—
293C22—2541no—
294C22—2541no—
295C22D8×D12712yes—
296C22C2×D4×D12711yescharacteristic
297C4—21no—
298C4—42no—
299C4—41no—
300C4—1272no—
301C4—5082no—
302C4—5081no—
303C4—2541no—
304C4#41×D12712yes—
305C2—22no—
306C2—81no—
307C2—1274no—
308C2—10161no—
309C2—2542no—
310C2#123×D12711yescharacteristic
311C2C2×D8×D12711yescharacteristic
312C2#137×D12711yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#480×C127C2
socleC22×C127C22×D4
prime radical#121×D127C2
hypercenter#480D127
hypocenterC127C2×#480
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×D127C2semidirect
3. #22550191C2semidirect
4. #22550190C2semidirect
5. #22550189C2semidirect
6. #22550188C2semidirect
7. #22550187C2semidirect
8. #107×D127C2semidirect
9. #102×D127C2semidirect
10. C22⋊C8×D127C2semidirect
11. #480×C127C2semidirect
22. C2×C4⋊Dic127C22semidirect
28. C4⋊C4×D127C22semidirect
30. #16861143C22semidirect
31. #16861142C22semidirect
32. #16861142C22semidirect
33. #16861131C22semidirect
34. #16861130C22semidirect
35. #16861129C22semidirect
36. C22⋊(C127⋊C8)C22semidirect
37. (C2×D127)⋊C8C22semidirect
39. C2×C4⋊C4×C127C22semidirect
40. #107×C127C22semidirect
41. #102×C127C22semidirect
42. C22⋊C8×C127C22semidirect
155. D127#480direct (N × H)
157. C127C2×#480semidirect
165. #480D127direct (N × H)
187. C2×C4⋊C4C2×D127semidirect
192. #107C2×D127semidirect
196. #102C2×D127semidirect
199. C22⋊C8C2×D127semidirect

Maximal subgroups

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

subgroupindex
#121×D1272
C2×C4⋊C4×D1272
#225501912
#225501902
#225501892
#225501882
#225501872
#107×D1272
#102×D1272
C22⋊C8×D1272
#480×C1272
C2×#480127