Finite Group Databasebeta

← #30648124

Subgroups — #30648124

Order 16256.

766 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 766 conjugacy classes of subgroups — too many to draw. Only its 145 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#22550712C211yesmaximal, characteristic, complemented
2#22550388C212yesmaximal, complemented
3#22550383C211yesmaximal, characteristic, complemented
4#22550067C214yesmaximal, complemented
5#22550064C214yesmaximal, complemented
6#22549849C212yesmaximal, complemented
7#410×C127C211yesmaximal, characteristic, complemented
8#16861257—21no—
9#16861256—22no—
10#16861255—22no—
11#16861254—21no—
12#16861254C2214yes—
13C2×#14523248—22no—
14C2×#14523248C2212yes—
15#41×D127—24no—
16C22×C4×D127C2211yescharacteristic, complemented
17#16861193C2214yescomplemented
18#16861190C2212yescomplemented
19#16861189—22no—
20#16861185—21no—
21#16861184—22no—
22#16861180—21no—
23#16861179C2214yescomplemented
24#16861177C2212yescomplemented
25C4×D508—24no—
26C4×C22⋊D127—22no—
27#16861168C2211yescharacteristic, complemented
28#16861123C2218yescomplemented
29C2×#41×C127C2211yescharacteristic
30#117×C127C2212yescomplemented
31#99×C127C2214yescomplemented
32C23×D127C2311yescharacteristic
33#14523249—24no—
34#14523248—24no—
35#14523248—24no—
36#14523248—22no—
37#14523248—22no—
38#14523248—24no—
39#14523248—22no—
40#14523248—22no—
41#14523247—24no—
42#14523247—28no—
43C2×Dic254—22no—
44C2×Dic254D412yescomplemented
45Q8×D127—24no—
46C2×D508—22no—
47C2×D508—24no—
48C2×D508D412yescomplemented
49C2×C22⋊D127—24no—
50C2×C22⋊D127—22no—
51C2×C22⋊D127C2314yes—
52D4×D127—24no—
53D4×D127—28no—
54C22×Dic127C2311yescharacteristic
55C2×C4×D127—24no—
56C2×C4×D127—22no—
57C2×C4×D127—21no—
58C2×C4×D127—21no—
59C2×C4×D127—22no—
60C2×C4×D127D414yescomplemented
61C4⋊Dic127—24no—
62C4⋊Dic127—21no—
63Dic127⋊C4—22no—
64C22⋊Dic127—22no—
65C22⋊Dic127C2312yes—
66(C2×D127)⋊C4—24no—
67(C2×D127)⋊C4—24no—
68(C2×D127)⋊C4—22no—
69C4×Dic127—21no—
70#41×C127—22no—
71#41×C127—22no—
72C2×Q8×C127D411yescharacteristic, complemented
73C2×D4×C127D411yescharacteristic, complemented
74C2×D4×C127C2312yes—
75C22×C4×C127D412yescomplemented
76C22×C4×C127C2311yescharacteristic
77C4⋊C4×C127—22no—
78C22⋊C4×C127C2314yes—
79C42×C127—22no—
80C22×D127—21no—
81C22×D127—22no—
82C22×D127—24no—
83C22×D127—42no—
84C22×D127C2×D414yes—
85Dic254—24no—
86Dic254—22no—
87Dic254—22no—
88D508—24no—
89D508—22no—
90D508—22no—
91D508—48no—
92C22⋊D127—28no—
93C22⋊D127—24no—
94C22⋊D127—24no—
95C22⋊D127—44no—
96C2×Dic127—24no—
97C2×Dic127—22no—
98C2×Dic127—22no—
99C2×Dic127—21no—
100C2×Dic127C2×D414yes—
101C4×D127—28no—
102C4×D127—24no—
103C4×D127—24no—
104C4×D127—41no—
105C4×D127—42no—
106C4×D127—41no—
107C23×C127C2×D412yes—
108C23×C127C2411yescharacteristic
109Q8×C127—22no—
110D4×C127—24no—
111D4×C127—22no—
112C2×C4×C127—22no—
113C2×C4×C127—22no—
114C2×C4×C127—24no—
115C2×C4×C127C2×D412yes—
116C2×C4×C127C2×D412yes—
117C2×C4×C127C2×D412yes—
118C2×C4×C127C2×D411yescharacteristic
119C2×C4×C127C2×D411yescharacteristic
120C22×C127—22no—
121C22×C127—41no—
122C22×C127—42no—
123C22×C127C22×D412yes—
124C22×C127C22×D411yescharacteristic
125C2×D127—22no—
126C2×D127—24no—
127C2×D127—24no—
128C2×D127—41no—
129C2×D127—42no—
130C2×D127—41no—
131C2×D127—44no—
132C2×D127—84no—
133C4×C127—22no—
134C4×C127—44no—
135C4×C127#12011yescharacteristic
136C4×C127#12012yes—
137C4×C127#12011yescharacteristic
138Dic127—24no—
139Dic127—42no—
140Dic127—41no—
141C2×C127—21no—
142C2×C127—22no—
143C2×C127—42no—
144C2×C127C2×#12011yescharacteristic
145D127—42no—
146D127—44no—
147#3025—1271nomaximal, Sylow
148C127#302511yescharacteristic, complemented, Sylow
149#585—1271no—
150#525—1271no—
151#525—1272no—
152#458—1274no—
153B(4,𝔽2)—1274no—
154#410—1272no—
155#410D12711yescharacteristic, complemented
1562−1+4—2542no—
1572−1+4—2541no—
158H(5,𝔽2)—1274no—
159H(5,𝔽2)—2542no—
160H(5,𝔽2)—2541no—
161C2×#41—1272no—
162C2×#41—2544no—
163C2×#41—2542no—
164C2×#41C2×D12711yescharacteristic
165C23×C4—1271no—
166#123—1272no—
167#123—1274no—
168C22⋊Q8—2542no—
169C22⋊Q8—2541no—
170#121—2542no—
171#121—2541no—
172#120—1272no—
173#120—1274no—
174C4×D4—2544no—
175C4×D4—2542no—
176#117—1271no—
177#117C2×D12712yescomplemented
178#99—1278no—
179#99C2×D12714yescomplemented
180C24—1271no—
181#41—22no—
182#41—22no—
183#41—2544no—
184#41—2544no—
185#41—2548no—
186#41—2544no—
187#41—2544no—
188#41—2542no—
189#41—2542no—
190#41—2542no—
191#41—2542no—
192#41—2544no—
193C2×Q8—1272no—
194C2×Q8—2542no—
195C2×Q8—2544no—
196C2×Q8D50811yescharacteristic, complemented
197C2×D4—1274no—
198C2×D4—1272no—
199C2×D4—2544no—
200C2×D4—2542no—
201C2×D4—2548no—
202C2×D4—2544no—
203C2×D4—2542no—
204C2×D4—2544no—
205C2×D4D50811yescharacteristic, complemented
206C2×D4C22×D12712yes—
207C22×C4—1274no—
208C22×C4—1271no—
209C22×C4—2544no—
210C22×C4—2542no—
211C22×C4—2542no—
212C22×C4—2541no—
213C22×C4—2541no—
214C22×C4D50812yescomplemented
215C22×C4C22×D12711yescharacteristic
216C4⋊C4—22no—
217C4⋊C4—2544no—
218C4⋊C4—2542no—
219C4⋊C4—2541no—
220C22⋊C4—1272no—
221C22⋊C4—2544no—
222C22⋊C4—2544no—
223C22⋊C4—2542no—
224C22⋊C4—2542no—
225C22⋊C4C22×D12714yes—
226C42—22no—
227C42—2541no—
228C23—1274no—
229C23—5082no—
230C23—2544no—
231C23—2542no—
232C23—2541no—
233C23C2×D50812yes—
234C23C23×D12711yescharacteristic
235Q8—22no—
236Q8—2544no—
237Q8—2542no—
238Q8—2542no—
239D4—24no—
240D4—22no—
241D4—5088no—
242D4—5084no—
243D4—2544no—
244D4—2548no—
245D4—2544no—
246D4—2544no—
247D4—2542no—
248D4—2542no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—24no—
252C2×C4—1274no—
253C2×C4—5081no—
254C2×C4—5082no—
255C2×C4—5081no—
256C2×C4—2548no—
257C2×C4—2542no—
258C2×C4—2542no—
259C2×C4—2541no—
260C2×C4—2544no—
261C2×C4—2544no—
262C2×C4—2544no—
263C2×C4D4×D12712yes—
264C2×C4D4×D12712yes—
265C2×C4C2×D50812yes—
266C2×C4C2×D50811yescharacteristic
267C2×C4C2×D50811yescharacteristic
268C22—22no—
269C22—41no—
270C22—42no—
271C22—10164no—
272C22—5084no—
273C22—5082no—
274C22—5081no—
275C22—5081no—
276C22—2544no—
277C22—2542no—
278C22—2544no—
279C22C2×D4×D12712yes—
280C22C22×D50811yescharacteristic
281C4—22no—
282C4—44no—
283C4—5082no—
284C4—5081no—
285C4—2544no—
286C4#1686117911yescharacteristic
287C4#1686117912yes—
288C4#1686117911yescharacteristic
289C2—22no—
290C2—21no—
291C2—42no—
292C2—5084no—
293C2—5082no—
294C2C2×#1686117911yescharacteristic
subgroupG/H
centerC4#16861179
commutator, cosocleC23×C127C24
FrattiniC23C23×D127
Fitting#410×C127C2
socleC2×C127C2×#120
hypercenter#410D127
hypocenterC127#3025
LCSG ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C4 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
1. #22550712C2semidirect
2. #22550388C2semidirect
3. #22550383C2semidirect
4. #22550067C2semidirect
5. #22550064C2semidirect
6. #22549849C2semidirect
7. #410×C127C2semidirect
16. C22×C4×D127C22semidirect
17. #16861193C22semidirect
18. #16861190C22semidirect
23. #16861179C22semidirect
24. #16861177C22semidirect
27. #16861168C22semidirect
28. #16861123C22semidirect
30. #117×C127C22semidirect
31. #99×C127C22semidirect
44. C2×Dic254D4semidirect
48. C2×D508D4semidirect
60. C2×C4×D127D4semidirect
72. C2×Q8×C127D4semidirect
73. C2×D4×C127D4semidirect
75. C22×C4×C127D4semidirect
148. C127#3025semidirect
155. #410D127semidirect
177. #117C2×D127semidirect
179. #99C2×D127semidirect
196. C2×Q8D508semidirect
205. C2×D4D508semidirect
214. C22×C4D508semidirect

Maximal subgroups

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

subgroupindex
#225507122
#225503882
#225503832
#225500672
#225500642
#225498492
#410×C1272
#3025127