Finite Group Databasebeta

← #30648660

Subgroups — #30648660

Order 16256.

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

normal subgroups only

This group has 414 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#22550383C211yesmaximal, characteristic
2C2×#16861167C211yesmaximal, characteristic, complemented
3#22550216C212yesmaximal, complemented
4#22550215C212yesmaximal, complemented
5#22550199C212yesmaximal, complemented
6#22550198C212yesmaximal, complemented
7#22549918C211yesmaximal, characteristic, complemented
8#22549916C211yesmaximal, characteristic, complemented
9#22549877C211yesmaximal, characteristic, complemented
10#22549839C211yesmaximal, characteristic, complemented
11#417×C127C211yesmaximal, characteristic, complemented
12C22×C4×D127C2211yescharacteristic
13C2×#14523232C2211yescharacteristic
14#16861190—21no—
15#16861185C2211yescharacteristic
16#16861185C2211yescharacteristic
17#16861180C2211yescharacteristic
18#16861180C2211yescharacteristic
19#16861177—21no—
20C4×C22⋊D127—21no—
21C4×C22⋊D127—21no—
22#16861168C2211yescharacteristic
23#16861167—21no—
24#16861167—21no—
25#16861167C2212yes—
26#16861167C2212yes—
27C2×C4⋊Dic127C2211yescharacteristic
28C2×(C2×D127)⋊C4—21no—
29C2×C4×Dic127—21no—
30#16861143C2212yescomplemented
31#16861142C2212yescomplemented
32#16861141C2212yescomplemented
33#16861140C2212yescomplemented
34#16861133C2212yescomplemented
35#16861130C2212yescomplemented
36(C2×D127)⋊C8C2212yes—
37(C2×D127)⋊C8C2212yes—
38C2×#41×C127C2211yescharacteristic
39C2×M16×C127C2211yescharacteristic
40C2×C4⋊C4×C127C2211yescharacteristic
41#103×C127C2212yescomplemented
42#102×C127C2212yescomplemented
43C23×D127D411yescharacteristic
44C2×C22⋊D127—21no—
45C2×C22⋊D127—21no—
46C22×Dic127—21no—
47C22×Dic127D411yescharacteristic
48C2×C4×D127—22no—
49C2×C4×D127—22no—
50C2×C4×D127D411yescharacteristic
51C2×C4×D127D411yescharacteristic
52C2×C4×D127C2311yescharacteristic
53C2×C4×D127C2311yescharacteristic
54#14523232—22no—
55C2×C127⋊C8C2312yes—
56C4⋊Dic127—21no—
57C4⋊Dic127C2311yescharacteristic
58C4⋊Dic127C2311yescharacteristic
59C4⋊Dic127C2312yes—
60Dic127⋊C4—21no—
61Dic127⋊C4—21no—
62C22⋊Dic127—21no—
63C22⋊Dic127—21no—
64C22⋊Dic127—21no—
65(C2×D127)⋊C4—21no—
66(C2×D127)⋊C4—21no—
67(C2×D127)⋊C4—22no—
68(C2×D127)⋊C4—22no—
69C4×Dic127—22no—
70C4×Dic127—22no—
71C4×Dic127—21no—
72#41×C127—22no—
73#41×C127—22no—
74C2×Q8×C127C2311yescharacteristic
75C2×D4×C127—21no—
76C2×D4×C127C2311yescharacteristic
77C22×C4×C127—21no—
78C22×C4×C127—21no—
79C22×C4×C127C2311yescharacteristic
80M16×C127—22no—
81C2×C8×C127C2312yes—
82C4⋊C4×C127—21no—
83C4⋊C4×C127C2312yes—
84C22×D127—22no—
85C22×D127—41no—
86C22×D127—41no—
87C22×D127C2×D411yescharacteristic
88C22×D127C2×D411yescharacteristic
89C22⋊D127—42no—
90C22⋊D127—42no—
91C2×Dic127—22no—
92C2×Dic127—22no—
93C2×Dic127—21no—
94C2×Dic127—21no—
95C2×Dic127—42no—
96C2×Dic127C2×D411yescharacteristic
97C2×Dic127C2×D411yescharacteristic
98C4×D127—22no—
99C4×D127—22no—
100C4×D127—21no—
101C4×D127—21no—
102C4×D127—21no—
103C4×D127—21no—
104C127⋊C8—22no—
105C23×C127—21no—
106C23×C127C2×D411yescharacteristic
107Q8×C127—21no—
108Q8×C127—21no—
109D4×C127—21no—
110D4×C127—21no—
111D4×C127—42no—
112C2×C4×C127—22no—
113C2×C4×C127—21no—
114C2×C4×C127—42no—
115C2×C4×C127—42no—
116C2×C4×C127C2×D411yescharacteristic
117C2×C4×C127#4112yes—
118C2×C4×C127#4112yes—
119C2×C4×C127C2411yescharacteristic
120C8×C127—22no—
121C22×C127—21no—
122C22×C127—21no—
123C22×C127—41no—
124C22×C127—41no—
125C22×C127—41no—
126C22×C127#12312yes—
127C22×C127C22×D411yescharacteristic
128C2×D127—21no—
129C2×D127—21no—
130C2×D127—21no—
131C2×D127—21no—
132C2×D127—21no—
133C2×D127—21no—
134C2×D127—44no—
135C4×C127—41no—
136C4×C127—42no—
137C4×C127#12312yes—
138C4×C127C2×#4111yescharacteristic
139C4×C127C2×#4111yescharacteristic
140Dic127—21no—
141Dic127—21no—
142Dic127—42no—
143Dic127—41no—
144Dic127—41no—
145C2×C127—22no—
146C2×C127—81no—
147C2×C127C2×#12311yescharacteristic
148C2×C127#57511yescharacteristic
149C2×C127#57511yescharacteristic
150D127—41no—
151D127—41no—
152#3093—1271nomaximal, Sylow
153C127#309311yescharacteristic, complemented, Sylow
154#525—1271no—
155C2×#117—1271no—
156#485—1272no—
157#485—1272no—
158#482—1272no—
159#482—1272no—
160#428—1271no—
161#428—1271no—
162#417—1271no—
163#417D12711yescharacteristic, complemented
164#407—1271no—
165C2×#41C2×D12711yescharacteristic
166C23×C4—1271no—
167C2×M16—1271no—
168C2×M16C2×D12711yescharacteristic
169#123—2541no—
170C22⋊Q8—1271no—
171C22⋊Q8—1271no—
172#121—1271no—
173#121—1271no—
174#120—2541no—
175C4×D4—2541no—
176C4×D4—2541no—
177#117—1271no—
178#117—1272no—
179#117—1272no—
180#117—2541no—
181#117—2541no—
182C2×C4⋊C4—1271no—
183C2×C4⋊C4C2×D12711yescharacteristic
184C2×C22⋊C4—2541no—
185C2×C42—2541no—
186#107—1272no—
187#107—1272no—
188#106—1272no—
189#106—1272no—
190#103—1272no—
191#103C2×D12712yescomplemented
192#102—1272no—
193#102C2×D12712yescomplemented
194C22⋊C8—1272no—
195C22⋊C8—1272no—
196C24—1271no—
197#41—22no—
198#41—22no—
199C2×Q8C22×D12711yescharacteristic
200C2×D4—21no—
201C2×D4—2541no—
202C2×D4—2541no—
203C2×D4C22×D12711yescharacteristic
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—1271no—
207C22×C4—1271no—
208C22×C4—1271no—
209C22×C4—1271no—
210C22×C4—1271no—
211C22×C4—2541no—
212C22×C4—2542no—
213C22×C4—2542no—
214C22×C4C22×D12711yescharacteristic
215M16—22no—
216M16—2542no—
217C2×C8—1272no—
218C2×C8C22×D12712yes—
219C4⋊C4—21no—
220C4⋊C4—1272no—
221C4⋊C4—1271no—
222C4⋊C4—1271no—
223C4⋊C4—2541no—
224C4⋊C4—2541no—
225C4⋊C4—2541no—
226C4⋊C4C22×D12712yes—
227C22⋊C4—2542no—
228C22⋊C4—2542no—
229C22⋊C4—2541no—
230C22⋊C4—2541no—
231C22⋊C4—2541no—
232C22⋊C4—2541no—
233C22⋊C4—2541no—
234C42—2542no—
235C42—2542no—
236C42—2541no—
237C23—21no—
238C23—1271no—
239C23—1271no—
240C23—5081no—
241C23—5081no—
242C23—2542no—
243C23D4×D12711yescharacteristic
244Q8—21no—
245Q8—21no—
246D4—21no—
247D4—21no—
248D4—42no—
249D4—5082no—
250D4—5082no—
251C2×C4—22no—
252C2×C4—21no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—1271no—
256C2×C4—1271no—
257C2×C4—5082no—
258C2×C4—2542no—
259C2×C4—2541no—
260C2×C4—2541no—
261C2×C4—2542no—
262C2×C4—2542no—
263C2×C4—2542no—
264C2×C4—2541no—
265C2×C4—2541no—
266C2×C4—2541no—
267C2×C4—2541no—
268C2×C4D4×D12711yescharacteristic
269C2×C4#1452324712yes—
270C2×C4#1452324812yes—
271C2×C4C23×D12711yescharacteristic
272C8—22no—
273C8—2542no—
274C22—21no—
275C22—21no—
276C22—41no—
277C22—41no—
278C22—41no—
279C22—5084no—
280C22—2541no—
281C22—2541no—
282C22—2541no—
283C22—2541no—
284C22—2541no—
285C22—2541no—
286C22#1686119112yes—
287C22C2×D4×D12711yescharacteristic
288C4—41no—
289C4—42no—
290C4—5081no—
291C4—5082no—
292C4—5081no—
293C4—2541no—
294C4—2541no—
295C4#1686119112yes—
296C4C2×#1452324711yescharacteristic
297C4C2×#1452324811yescharacteristic
298C2—22no—
299C2—81no—
300C2—5081no—
301C2—5081no—
302C2C2×#1686119111yescharacteristic
303C2#2255067211yescharacteristic
304C2#2255067511yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#417×C127C2
socleC22×C127C22×D4
prime radical#22550383C2
hypercenter#417D127
hypocenterC127#3093
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
2. C2×#16861167C2semidirect
3. #22550216C2semidirect
4. #22550215C2semidirect
5. #22550199C2semidirect
6. #22550198C2semidirect
7. #22549918C2semidirect
8. #22549916C2semidirect
9. #22549877C2semidirect
10. #22549839C2semidirect
11. #417×C127C2semidirect
30. #16861143C22semidirect
31. #16861142C22semidirect
32. #16861141C22semidirect
33. #16861140C22semidirect
34. #16861133C22semidirect
35. #16861130C22semidirect
41. #103×C127C22semidirect
42. #102×C127C22semidirect
153. C127#3093semidirect
163. #417D127semidirect
191. #103C2×D127semidirect
193. #102C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225503832
C2×#168611672
#225502162
#225502152
#225501992
#225501982
#225499182
#225499162
#225498772
#225498392
#417×C1272
#3093127