Finite Group Databasebeta

← #30648308

Subgroups — #30648308

Order 16256.

520 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 149 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×#16861229C211yesmaximal, characteristic, complemented
2C23×C127⋊C8C211yesmaximal, characteristic, complemented
3#22550387C211yesmaximal, characteristic, complemented
4#22550115C212yesmaximal, complemented
5#22550110C212yesmaximal, complemented
6#22550107C212yesmaximal, complemented
7#22550103C212yesmaximal, complemented
8#22549912C211yesmaximal, characteristic, complemented
9#22549872C211yesmaximal, characteristic, complemented
10#22549870C211yesmaximal, characteristic, complemented
11#525×C127C211yesmaximal, characteristic, complemented
12C2×#14523248C2211yescharacteristic
13#16861229—24no—
14C2×#14523238C2211yescharacteristic, complemented
15C2×#14523236C2211yescharacteristic, complemented
16C2×#14523235C2211yescharacteristic, complemented
17C2×#14523233C2211yescharacteristic, complemented
18C22×C127⋊C8—24no—
19C22×C127⋊C8C2212yes—
20C22×C127⋊C8C2212yes—
21C22×C127⋊C8C2211yescharacteristic
22C22×C127⋊C8C2211yescharacteristic
23#16861192—21no—
24#16861188C2212yes—
25#16861183C2212yes—
26#16861178—21no—
27C4×C22⋊D127—22no—
28#16861168C2211yescharacteristic
29#16861142C2212yescomplemented
30#16861140C2212yescomplemented
31#16861133C2212yescomplemented
32#16861132C2212yescomplemented
33#16861130C2212yescomplemented
34#16861129C2212yescomplemented
35C2×#41×C127C2211yescharacteristic
36C23×C4×C127C2211yescharacteristic
37#123×C127—21no—
38C22⋊Q8×C127C2212yes—
39#121×C127C2212yes—
40#120×C127—21no—
41C4×D4×C127—22no—
42#117×C127C2211yescharacteristic
43#14523248—22no—
44#14523248—22no—
45C2×Dic254C2311yescharacteristic
46C2×D508C2311yescharacteristic
47C2×C22⋊D127—21no—
48C2×C22⋊D127—22no—
49C2×C4×D127—21no—
50#14523238—22no—
51#14523236—22no—
52#14523235—22no—
53#14523233—22no—
54C2×C127⋊C8—24no—
55C2×C127⋊C8—24no—
56C2×C127⋊C8—22no—
57C2×C127⋊C8D414yescomplemented
58C2×C127⋊C8C2312yes—
59C2×C127⋊C8C2311yescharacteristic
60C2×C127⋊C8C2311yescharacteristic
61C4⋊Dic127C2312yes—
62Dic127⋊C4—22no—
63C22⋊Dic127—22no—
64C22⋊Dic127—21no—
65(C2×D127)⋊C4—22no—
66C4×Dic127—21no—
67C24×C127D411yescharacteristic
68#41×C127—22no—
69#41×C127—22no—
70C2×Q8×C127C2311yescharacteristic
71C2×D4×C127—22no—
72C2×D4×C127—21no—
73C2×D4×C127C2311yescharacteristic
74C22×C4×C127—22no—
75C22×C4×C127—22no—
76C22×C4×C127—21no—
77C22×C4×C127D411yescharacteristic
78C22×C4×C127D412yes—
79C22×C4×C127C2312yes—
80C22×C4×C127C2311yescharacteristic
81C4⋊C4×C127—22no—
82C4⋊C4×C127C2312yes—
83C22⋊C4×C127—21no—
84C22⋊C4×C127—22no—
85C22⋊C4×C127—22no—
86C42×C127—21no—
87C22×D127—21no—
88Dic254—21no—
89Dic254—21no—
90D508—21no—
91D508—21no—
92C22⋊D127—42no—
93C22⋊D127—44no—
94C2×Dic127—22no—
95C2×Dic127—21no—
96C4×D127—42no—
97C127⋊C8—22no—
98C127⋊C8C2×D412yes—
99C127⋊C8C2×D412yes—
100C23×C127—22no—
101C23×C127—24no—
102C23×C127—21no—
103C23×C127C2×D411yescharacteristic
104C23×C127C2×D412yes—
105Q8×C127—21no—
106Q8×C127—21no—
107D4×C127—21no—
108D4×C127—21no—
109D4×C127—42no—
110D4×C127—44no—
111C2×C4×C127—22no—
112C2×C4×C127—24no—
113C2×C4×C127—22no—
114C2×C4×C127—22no—
115C2×C4×C127—22no—
116C2×C4×C127—21no—
117C2×C4×C127—42no—
118C2×C4×C127C2×D412yes—
119C2×C4×C127C2×D412yes—
120C2×C4×C127C2×D411yescharacteristic
121C2×C4×C127#4112yes—
122C2×C4×C127C2411yescharacteristic
123C22×C127—28no—
124C22×C127—22no—
125C22×C127—22no—
126C22×C127—22no—
127C22×C127—41no—
128C22×C127—42no—
129C22×C127#12112yes—
130C22×C127#13514yescomplemented
131C22×C127C22×D411yescharacteristic
132C2×D127—41no—
133C2×D127—42no—
134C4×C127—22no—
135C4×C127—41no—
136C4×C127—42no—
137C4×C127#12112yes—
138C4×C127C22×D411yescharacteristic
139C4×C127C2×#4111yescharacteristic
140Dic127—41no—
141Dic127—42no—
142C2×C127—22no—
143C2×C127—24no—
144C2×C127—81no—
145C2×C127C2×#12111yescharacteristic
146C2×C127C2×#13512yes—
147D127—81no—
148#3050—1271nomaximal, Sylow
149C127#305011yescharacteristic, complemented, Sylow
150C2×#135—1271no—
151C23×C8—1271no—
152#525—1271no—
153#525D12711yescharacteristic, complemented
154#467—1272no—
155#466—1272no—
156#465—1272no—
157#465—1272no—
158#427—1271no—
159#416—1271no—
160#416—1271no—
161C2×#41—1271no—
162C2×#41C2×D12711yescharacteristic
163C23×C4C2×D12711yescharacteristic
164#135—2544no—
165C2×Q16—1271no—
166C2×SD16—1271no—
167C2×SD16—1271no—
168C2×D8—1271no—
169C22×C8—1271no—
170C22×C8—1271no—
171C22×C8—1272no—
172C22×C8—1272no—
173C22×C8—2544no—
174#123—21no—
175#123—2541no—
176C22⋊Q8—1272no—
177C22⋊Q8C2×D12712yes—
178#121—1272no—
179#121C2×D12712yes—
180#120—21no—
181#120—2541no—
182C4×D4—22no—
183C4×D4—2542no—
184#117—1271no—
185#117C2×D12711yescharacteristic
186#107—1272no—
187#106—1272no—
188#103—1272no—
189#103—1272no—
190#102—1272no—
191#102—1272no—
192C24C22⋊D12711yescharacteristic
193#41—22no—
194#41—22no—
195#41—2542no—
196#41—2542no—
197C2×Q8—1271no—
198C2×Q8C22×D12711yescharacteristic
199C2×D4—22no—
200C2×D4—21no—
201C2×D4—1271no—
202C2×D4—2542no—
203C2×D4—2541no—
204C2×D4C22×D12711yescharacteristic
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—21no—
208C22×C4—2541no—
209C22×C4C22⋊D12711yescharacteristic
210C22×C4C22⋊D12712yes—
211C22×C4C22×D12712yes—
212C22×C4C22×D12711yescharacteristic
213Q16—2542no—
214SD16—2542no—
215SD16—2542no—
216D8—2542no—
217C2×C8—1274no—
218C2×C8—1271no—
219C2×C8—1271no—
220C2×C8—1272no—
221C2×C8—2544no—
222C2×C8—2544no—
223C2×C8—2542no—
224C4⋊C4—22no—
225C4⋊C4—1272no—
226C4⋊C4—2542no—
227C4⋊C4C22×D12712yes—
228C22⋊C4—21no—
229C22⋊C4—22no—
230C22⋊C4—22no—
231C22⋊C4—2541no—
232C22⋊C4—2542no—
233C22⋊C4—2542no—
234C42—21no—
235C42—2541no—
236C23—24no—
237C23—22no—
238C23—21no—
239C23—2541no—
240C23C2×C22⋊D12712yes—
241C23C2×C22⋊D12711yescharacteristic
242Q8—21no—
243Q8—21no—
244Q8—2541no—
245Q8—2541no—
246D4—21no—
247D4—21no—
248D4—44no—
249D4—42no—
250D4—5084no—
251D4—5082no—
252D4—2541no—
253D4—2541no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—24no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—42no—
261C2×C4—5082no—
262C2×C4—2542no—
263C2×C4—2541no—
264C2×C4D4×D12712yes—
265C2×C4C2×C22⋊D12712yes—
266C2×C4C2×C22⋊D12711yescharacteristic
267C2×C4#1452324712yes—
268C2×C4C23×D12711yescharacteristic
269C8—1272no—
270C8—1272no—
271C8—2542no—
272C22—28no—
273C22—22no—
274C22—22no—
275C22—22no—
276C22—41no—
277C22—42no—
278C22—5081no—
279C22—5082no—
280C22#1686118112yes—
281C22#1686122914yescomplemented
282C22C22×C22⋊D12711yescharacteristic
283C4—22no—
284C4—41no—
285C4—42no—
286C4—5081no—
287C4—5082no—
288C4#1686118112yes—
289C4C2×D4×D12711yescharacteristic
290C4C2×#1452324711yescharacteristic
291C2—24no—
292C2—22no—
293C2—81no—
294C2—10161no—
295C2C2×#1686118111yescharacteristic
296C2C2×#1686122912yes—
subgroupG/H
centerC2×C4C2×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#525×C127C2
socleC22×C127C22×D4
hypercenter#525D127
hypocenterC127#3050
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C2×#16861229C2semidirect
2. C23×C127⋊C8C2semidirect
3. #22550387C2semidirect
4. #22550115C2semidirect
5. #22550110C2semidirect
6. #22550107C2semidirect
7. #22550103C2semidirect
8. #22549912C2semidirect
9. #22549872C2semidirect
10. #22549870C2semidirect
11. #525×C127C2semidirect
14. C2×#14523238C22semidirect
15. C2×#14523236C22semidirect
16. C2×#14523235C22semidirect
17. C2×#14523233C22semidirect
29. #16861142C22semidirect
30. #16861140C22semidirect
31. #16861133C22semidirect
32. #16861132C22semidirect
33. #16861130C22semidirect
34. #16861129C22semidirect
57. C2×C127⋊C8D4semidirect
130. C22×C127#135semidirect
149. C127#3050semidirect
153. #525D127semidirect
281. C22#16861229semidirect

Maximal subgroups

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

subgroupindex
C2×#168612292
C23×C127⋊C82
#225503872
#225501152
#225501102
#225501072
#225501032
#225499122
#225498722
#225498702
#525×C1272
#3050127