Finite Group Databasebeta

← #30641868

Subgroups — #30641868

Order 16256.

688 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 688 conjugacy classes of subgroups — too many to draw. Only its 173 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×#16861177C211yesmaximal, characteristic
2#22549779C211yesmaximal, characteristic, complemented
3#22549776C212yesmaximal, complemented
4#22549770C212yesmaximal, complemented
5#22549768C212yesmaximal, complemented
6#22549765C212yesmaximal, complemented
7#22549755C212yesmaximal, complemented
8#22549731C212yesmaximal, complemented
9#386×C127C211yesmaximal, characteristic, complemented
10C24×D127C2211yescharacteristic
11C22×C22⋊D127C2212yes—
12#16861177C418yescomplemented
13C2×C4⋊Dic127C2211yescharacteristic, complemented
14C2×Dic127⋊C4C2212yescomplemented
15C2×C22⋊Dic127C2212yes—
16C2×C22⋊Dic127C2211yescharacteristic
17C2×C22⋊Dic127C2212yescomplemented
18C2×(C2×D127)⋊C4C2212yes—
19C2×(C2×D127)⋊C4C2212yes—
20C2×(C2×D127)⋊C4C2212yes—
21C2×(C2×D127)⋊C4C2212yes—
22C2×(C2×D127)⋊C4C2212yes—
23C2×C4×Dic127C2211yescharacteristic, complemented
24C2×C4×Dic127C2212yescomplemented
25C127⋊#95C2212yescomplemented
26C127⋊#95C2212yescomplemented
27C22×D4×C127C2211yescharacteristic
28C23×C4×C127C2211yescharacteristic, complemented
29C2×C22⋊C4×C127C2212yes—
30C2×C22⋊C4×C127C2211yescharacteristic, complemented
31#95×C127C2212yescomplemented
32C23×D127—42no—
33C23×D127—41no—
34C23×D127—41no—
35C23×D127—42no—
36C23×D127C2×C414yescomplemented
37C23×D127C2312yes—
38C2×C22⋊D127—42no—
39C2×C22⋊D127—42no—
40C2×C22⋊D127C2×C418yes—
41C22×Dic127C2312yes—
42C22×Dic127C2312yes—
43C22×Dic127C2312yes—
44C4⋊Dic127—41no—
45Dic127⋊C4—42no—
46C22⋊Dic127—42no—
47C22⋊Dic127—42no—
48C22⋊Dic127C2×C414yescomplemented
49(C2×D127)⋊C4—42no—
50(C2×D127)⋊C4—42no—
51(C2×D127)⋊C4—42no—
52(C2×D127)⋊C4—42no—
53(C2×D127)⋊C4C2×C418yescomplemented
54C4×Dic127—41no—
55C4×Dic127—42no—
56C24×C127C2311yescharacteristic
57C24×C127C2311yescharacteristic
58C2×D4×C127—41no—
59C2×D4×C127—41no—
60C2×D4×C127C2×C414yes—
61C22×C4×C127—42no—
62C22×C4×C127—41no—
63C22×C4×C127C2312yes—
64C22×C4×C127C2312yes—
65C22×C4×C127C2311yescharacteristic
66C22⋊C4×C127—42no—
67C22⋊C4×C127—41no—
68C22×D127—22no—
69C22×D127—22no—
70C22×D127—22no—
71C22×D127—22no—
72C22×D127—22no—
73C22×D127—22no—
74C22×D127—48no—
75C22×D127—44no—
76C22×D127—48no—
77C22×D127—44no—
78C22×D127—44no—
79C22×D127C22×C414yes—
80C22⋊D127—48no—
81C2×Dic127—22no—
82C2×Dic127—22no—
83C2×Dic127—22no—
84C2×Dic127—22no—
85C2×Dic127—22no—
86C2×Dic127—22no—
87C2×Dic127—22no—
88C2×Dic127—22no—
89C2×Dic127C22×C414yes—
90C23×C127—21no—
91C23×C127—22no—
92C23×C127—21no—
93C23×C127—22no—
94C23×C127—21no—
95C23×C127—21no—
96C23×C127—22no—
97C23×C127—22no—
98C23×C127C22×C412yes—
99C23×C127C22×C412yes—
100C23×C127C2411yescharacteristic
101D4×C127—44no—
102C2×C4×C127—21no—
103C2×C4×C127—21no—
104C2×C4×C127—22no—
105C2×C4×C127—22no—
106C2×C4×C127—22no—
107C2×C4×C127—22no—
108C2×C4×C127—22no—
109C2×C4×C127—22no—
110C2×C4×C127—44no—
111C2×C4×C127C22×C412yes—
112C22×C127—22no—
113C22×C127—22no—
114C22×C127—24no—
115C22×C127—24no—
116C22×C127—42no—
117C22×C127—41no—
118C22×C127—42no—
119C22×C127—41no—
120C22×C127—41no—
121C22×C127—41no—
122C22×C127C23×C411yescharacteristic
123C22×C127H(5,𝔽2)11yescharacteristic
124C22×C127H(5,𝔽2)12yes—
125C22×C127H(5,𝔽2)11yescharacteristic
126C22×C127H(5,𝔽2)12yes—
127C2×D127—24no—
128C2×D127—24no—
129C2×D127—24no—
130C2×D127—416no—
131C2×D127—42no—
132C2×D127—42no—
133C2×D127—42no—
134C2×D127—42no—
135C4×C127—22no—
136C4×C127—42no—
137C4×C127—42no—
138Dic127—24no—
139Dic127—42no—
140Dic127—42no—
141C2×C127—42no—
142C2×C127—42no—
143C2×C127#51812yes—
144C2×C127#51811yescharacteristic
145C2×C127#56012yes—
146C2×C127#56011yescharacteristic
147C2×C127H(3,𝔽4)11yescharacteristic
148D127—44no—
149#2375—1271nomaximal, Sylow
150C127#237511yescharacteristic, complemented, Sylow
151C2×#120—1271no—
152#390—1271no—
153#390—1272no—
154#388—1272no—
155#388—1272no—
156#388—1272no—
157#386—1272no—
158#386D12711yescharacteristic, complemented
159#379—1272no—
160C25—1271no—
161C22×D4—1272no—
162C22×D4C2×D12711yescharacteristic
163C23×C4C2×D12711yescharacteristic, complemented
164#120—1278no—
165C2×C4⋊C4—1272no—
166C2×C4⋊C4—1271no—
167C2×C22⋊C4—1271no—
168C2×C22⋊C4—1272no—
169C2×C22⋊C4—1272no—
170C2×C22⋊C4—1272no—
171C2×C22⋊C4—1272no—
172C2×C22⋊C4—1272no—
173C2×C22⋊C4—1272no—
174C2×C22⋊C4—1272no—
175C2×C22⋊C4C2×D12712yes—
176C2×C22⋊C4C2×D12711yescharacteristic, complemented
177C2×C42—1272no—
178C2×C42—1271no—
179#95—1272no—
180#95—1272no—
181#95C2×D12712yescomplemented
182C24—1274no—
183C24—1272no—
184C24—5082no—
185C24—5081no—
186C24—5081no—
187C24—5082no—
188C24C22×D12711yescharacteristic
189C24C22×D12711yescharacteristic
190C2×D4—41no—
191C2×D4—41no—
192C2×D4—1278no—
193C2×D4—5082no—
194C2×D4—5082no—
195C2×D4C4×D12714yes—
196C22×C4—42no—
197C22×C4—41no—
198C22×C4—1272no—
199C22×C4—1272no—
200C22×C4—1272no—
201C22×C4C22×D12712yes—
202C22×C4C22×D12712yes—
203C22×C4C22×D12711yescharacteristic
204C4⋊C4—5082no—
205C4⋊C4—5081no—
206C22⋊C4—42no—
207C22⋊C4—41no—
208C22⋊C4—1274no—
209C22⋊C4—1278no—
210C22⋊C4—5082no—
211C22⋊C4—5082no—
212C22⋊C4—5082no—
213C22⋊C4—5082no—
214C22⋊C4—5082no—
215C22⋊C4—5082no—
216C42—5081no—
217C42—5082no—
218C23—21no—
219C23—22no—
220C23—21no—
221C23—22no—
222C23—21no—
223C23—21no—
224C23—22no—
225C23—22no—
226C23—1274no—
227C23—5088no—
228C23—5084no—
229C23—5088no—
230C23—5084no—
231C23—5084no—
232C23—2542no—
233C23—2542no—
234C23—2542no—
235C23—2542no—
236C23—2542no—
237C23—2542no—
238C23C2×C4×D12712yes—
239C23C2×C4×D12712yes—
240C23C23×D12711yescharacteristic
241D4—44no—
242D4—5088no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—44no—
252C2×C4—1274no—
253C2×C4—2542no—
254C2×C4—2542no—
255C2×C4—2542no—
256C2×C4—2542no—
257C2×C4—2542no—
258C2×C4—2542no—
259C2×C4—2542no—
260C2×C4—2542no—
261C2×C4C2×C4×D12712yes—
262C22—24no—
263C22—24no—
264C22—22no—
265C22—22no—
266C22—42no—
267C22—42no—
268C22—41no—
269C22—41no—
270C22—41no—
271C22—41no—
272C22—50816no—
273C22—5082no—
274C22—5082no—
275C22—5082no—
276C22—5082no—
277C22—2544no—
278C22—2544no—
279C22—2544no—
280C22C22×C4×D12711yescharacteristic
281C22#1686125412yes—
282C22#1686125411yescharacteristic
283C22#1686125411yescharacteristic
284C22#1686125512yes—
285C4—22no—
286C4—42no—
287C4—42no—
288C4—5082no—
289C4—5082no—
290C4—2544no—
291C2—42no—
292C2—42no—
293C2—5084no—
294C2#2255034812yes—
295C2#2255034911yescharacteristic
296C2#2255061412yes—
297C2#2255061611yescharacteristic
298C2#2255061911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D127
commutatorC22×C127C23×C4
Fitting#386×C127C2
socle, cosocleC23×C127C24
prime radicalC2×#16861177C2
hypercenter#386D127
hypocenterC127#2375
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. #22549779C2semidirect
3. #22549776C2semidirect
4. #22549770C2semidirect
5. #22549768C2semidirect
6. #22549765C2semidirect
7. #22549755C2semidirect
8. #22549731C2semidirect
9. #386×C127C2semidirect
12. #16861177C4semidirect
13. C2×C4⋊Dic127C22semidirect
14. C2×Dic127⋊C4C22semidirect
17. C2×C22⋊Dic127C22semidirect
23. C2×C4×Dic127C22semidirect
24. C2×C4×Dic127C22semidirect
25. C127⋊#95C22semidirect
26. C127⋊#95C22semidirect
28. C23×C4×C127C22semidirect
30. C2×C22⋊C4×C127C22semidirect
31. #95×C127C22semidirect
36. C23×D127C2×C4semidirect
48. C22⋊Dic127C2×C4semidirect
53. (C2×D127)⋊C4C2×C4semidirect
150. C127#2375semidirect
158. #386D127semidirect
163. C23×C4C2×D127semidirect
176. C2×C22⋊C4C2×D127semidirect
181. #95C2×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168611772
#225497792
#225497762
#225497702
#225497682
#225497652
#225497552
#225497312
#386×C1272
#2375127