Finite Group Databasebeta

← #428×D107 #28345042

Subgroups — #428×D107 #28345042

Order 13696.

536 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×M16×D107C211yesmaximal, characteristic
2#117×D107C211yesmaximal, characteristic
3C2×C4⋊C4×D107C211yesmaximal, characteristic
4#20292670C211yesmaximal, characteristic, complemented
5#20292669C211yesmaximal, characteristic, complemented
6#20292668C212yesmaximal, complemented
7#20292667C211yesmaximal, characteristic, complemented
8#20292666C212yesmaximal, complemented
9#107×D107C212yesmaximal, complemented
10#106×D107C212yesmaximal, complemented
11#428×C107C211yesmaximal, characteristic, complemented
12C22×C4×D107—21no—
13C22×C4×D107C2211yescharacteristic
14C2×#13133802C2211yescharacteristic
15C2×#13133801C2212yes—
16M16×D107C418yescomplemented
17C2×C8×D107C2212yes—
18#16812068C2211yescharacteristic
19#16812067C2211yescharacteristic
20#16812067C2211yescharacteristic
21#16812066—21no—
22#16812065—21no—
23C2×C4⋊Dic107C2211yescharacteristic
24C2×Dic107⋊C4—21no—
25C4⋊C4×D107—21no—
26C4⋊C4×D107—21no—
27C4⋊C4×D107C2211yescharacteristic
28C4⋊C4×D107C2212yes—
29C4⋊C4×D107C2211yescharacteristic
30C4⋊C4×D107C2212yes—
31C22⋊C4×D107—21no—
32C42×D107—21no—
33#16812043C2212yescomplemented
34#16812042C2212yescomplemented
35#16812042C2212yescomplemented
36#16812041C2212yescomplemented
37#16812040C2212yescomplemented
38#16812040C2212yescomplemented
39C2×M16×C107C2211yescharacteristic
40#117×C107C2211yescharacteristic
41C2×C4⋊C4×C107C2211yescharacteristic
42#107×C107C2212yescomplemented
43#106×C107C2212yescomplemented
44C23×D107D411yescharacteristic
45C22×Dic107—21no—
46C22×Dic107D411yescharacteristic
47C2×C4×D107—21no—
48C2×C4×D107—21no—
49C2×C4×D107—22no—
50C2×C4×D107—22no—
51C2×C4×D107—44no—
52C2×C4×D107C2×C414yes—
53C2×C4×D107D411yescharacteristic
54C2×C4×D107D411yescharacteristic
55C2×C4×D107Q814yes—
56C2×C4×D107C2311yescharacteristic
57C2×C4×D107C2311yescharacteristic
58#13133802C2×C414yescomplemented
59#13133801C2×C418yescomplemented
60C2×C107⋊C8C2312yes—
61C8×D107C2×C418yescomplemented
62C4⋊Dic107—21no—
63C4⋊Dic107C2311yescharacteristic
64C4⋊Dic107C2312yes—
65C4⋊Dic107C2311yescharacteristic
66Dic107⋊C4—22no—
67Dic107⋊C4—22no—
68Dic107⋊C4—21no—
69Dic107⋊C4—21no—
70C22⋊Dic107—21no—
71(C2×D107)⋊C4—21no—
72(C2×D107)⋊C4—21no—
73C4×Dic107—21no—
74C4×Dic107—21no—
75C4×Dic107—21no—
76C22×C4×C107—21no—
77C22×C4×C107C2311yescharacteristic
78M16×C107C2×C414yescomplemented
79C2×C8×C107C2312yes—
80C4⋊C4×C107—21no—
81C4⋊C4×C107C2311yescharacteristic
82C4⋊C4×C107C2312yes—
83C4⋊C4×C107C2311yescharacteristic
84C22⋊C4×C107—21no—
85C42×C107—21no—
86C23×C107C2×D411yescharacteristic
87C22×D107—22no—
88C22×D107—22no—
89C22×D107C4⋊C414yes—
90C22×D107C2×D411yescharacteristic
91C22×D107C2×D411yescharacteristic
92C2×C4×C107—21no—
93C2×C4×C107—21no—
94C2×C4×C107—22no—
95C2×C4×C107—42no—
96C2×C4×C107C22×C412yes—
97C2×C4×C107C2×D411yescharacteristic
98C2×C4×C107C2×Q812yes—
99C2×C4×C107C2411yescharacteristic
100C2×Dic107—21no—
101C2×Dic107—21no—
102C2×Dic107—22no—
103C2×Dic107—42no—
104C2×Dic107C4⋊C414yes—
105C2×Dic107C2×D411yescharacteristic
106C2×Dic107C2×D411yescharacteristic
107C4×D107—44no—
108C4×D107—42no—
109C4×D107—44no—
110C4×D107—42no—
111C4×D107C4⋊C414yes—
112C4×D107C4⋊C414yes—
113C4×D107C22×C412yes—
114C4×D107C22×C412yes—
115C4×D107C2×Q812yes—
116C4×D107C2×Q812yes—
117C8×C107C22×C414yescomplemented
118C107⋊C8C22×C414yescomplemented
119C22×C107—21no—
120C22×C107—21no—
121C22×C107C2×C4⋊C412yes—
122C22×C107C22×D411yescharacteristic
123C2×D107—21no—
124C2×D107—21no—
125C2×D107—28no—
126C2×D107C2×C4⋊C412yes—
127C2×D107C2×C4⋊C412yes—
128C2×D107AGL(1,ℤ/8)12yes—
129C2×D107#13712yes—
130C4×C107—42no—
131C4×C107—41no—
132C4×C107—41no—
133C4×C107C2×C4⋊C412yes—
134C4×C107C23×C411yescharacteristic
135C4×C107C22×Q811yescharacteristic
136Dic107—42no—
137Dic107—41no—
138Dic107—41no—
139Dic107C2×C4⋊C412yes—
140Dic107C2×C4⋊C412yes—
141C2×C107—22no—
142C2×C107C22×C4⋊C411yescharacteristic
143C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
144C2×C107C2×#13711yescharacteristic
145D107—22no—
146D107#42814yesdirect product factor
147C2×#428—1071nomaximal, Sylow
148C107C2×#42811yescharacteristic, complemented, Sylow
149C22×M16—1071no—
150C2×#117—1071no—
151C22×C4⋊C4—1071no—
152#428—1071no—
153#428—1071no—
154#428—1071no—
155#428—1072no—
156#428—1072no—
157#428D10711yescharacteristic, direct product factor
158C2×#107—1072no—
159C2×#106—1072no—
160C23×C4—1071no—
161C23×C4—2141no—
162C2×M16—1071no—
163C2×M16—1078no—
164C2×M16—1072no—
165C2×M16C2×D10711yescharacteristic
166C22×C8—1072no—
167#117—1071no—
168#117—1071no—
169#117—1071no—
170#117—2141no—
171#117—2141no—
172#117C2×D10711yescharacteristic
173C2×C4⋊C4—1071no—
174C2×C4⋊C4—1072no—
175C2×C4⋊C4—1072no—
176C2×C4⋊C4—1071no—
177C2×C4⋊C4—1071no—
178C2×C4⋊C4—2141no—
179C2×C4⋊C4—2141no—
180C2×C4⋊C4—2141no—
181C2×C4⋊C4C2×D10711yescharacteristic
182C2×C22⋊C4—2141no—
183C2×C42—2141no—
184#107—1072no—
185#107—1072no—
186#107—1072no—
187#107C2×D10712yescomplemented
188#106—1072no—
189#106—1072no—
190#106—1072no—
191#106C2×D10712yescomplemented
192C24—1071no—
193C22×C4—21no—
194C22×C4—1071no—
195C22×C4—1074no—
196C22×C4—1071no—
197C22×C4—1071no—
198C22×C4—1074no—
199C22×C4—1071no—
200C22×C4—1071no—
201C22×C4—4284no—
202C22×C4—2141no—
203C22×C4—2141no—
204C22×C4—2142no—
205C22×C4—2142no—
206C22×C4—2141no—
207C22×C4C22×D10711yescharacteristic
208M16—1074no—
209M16—1078no—
210M16C4×D10714yescomplemented
211C2×C8—1072no—
212C2×C8—1078no—
213C2×C8C22×D10712yes—
214C4⋊C4—21no—
215C4⋊C4—1071no—
216C4⋊C4—1072no—
217C4⋊C4—1071no—
218C4⋊C4—2142no—
219C4⋊C4—2142no—
220C4⋊C4—2141no—
221C4⋊C4—2141no—
222C4⋊C4—2141no—
223C4⋊C4C22×D10711yescharacteristic
224C4⋊C4C22×D10712yes—
225C4⋊C4C22×D10711yescharacteristic
226C22⋊C4—21no—
227C22⋊C4—2141no—
228C22⋊C4—2141no—
229C22⋊C4—2141no—
230C42—21no—
231C42—2141no—
232C42—2141no—
233C42—2141no—
234C23—1071no—
235C23—1074no—
236C23—1071no—
237C23—2142no—
238C23—2142no—
239C23D4×D10711yescharacteristic
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—42no—
244C2×C4—1074no—
245C2×C4—1071no—
246C2×C4—1071no—
247C2×C4—1074no—
248C2×C4—1074no—
249C2×C4—1072no—
250C2×C4—1072no—
251C2×C4—1072no—
252C2×C4—1072no—
253C2×C4—4284no—
254C2×C4—4282no—
255C2×C4—4282no—
256C2×C4—4284no—
257C2×C4—4282no—
258C2×C4—2141no—
259C2×C4—2141no—
260C2×C4—2142no—
261C2×C4C2×C4×D10712yes—
262C2×C4D4×D10711yescharacteristic
263C2×C4Q8×D10712yes—
264C2×C4C23×D10711yescharacteristic
265C8—1074no—
266C8C2×C4×D10714yescomplemented
267C22—21no—
268C22—21no—
269C22—1072no—
270C22—1072no—
271C22—1072no—
272C22—1072no—
273C22—2148no—
274C22—2141no—
275C22—2141no—
276C22C4⋊C4×D10712yes—
277C22C2×D4×D10711yescharacteristic
278C4—42no—
279C4—41no—
280C4—41no—
281C4—1072no—
282C4—1072no—
283C4—4282no—
284C4—4281no—
285C4—4281no—
286C4C4⋊C4×D10712yes—
287C4C22×C4×D10711yescharacteristic
288C4C2×Q8×D10711yescharacteristic
289C2—22no—
290C2—1074no—
291C2—2142no—
292C2C2×C4⋊C4×D10711yescharacteristic
293C2AGL(1,ℤ/8)×D10711yescharacteristic
294C2#137×D10711yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutatorC4×C107C23×C4
FrattiniC2×C4C23×D107
Fitting#428×C107C2
socleC22×C107C22×D4
prime radicalC23×D107D4
cosocleC2×C4×C107C24
hypercenter#428D107
hypocenterC107C2×#428
LCSG ▷ C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
4. #20292670C2semidirect
5. #20292669C2semidirect
6. #20292668C2semidirect
7. #20292667C2semidirect
8. #20292666C2semidirect
9. #107×D107C2semidirect
10. #106×D107C2semidirect
11. #428×C107C2semidirect
16. M16×D107C4semidirect
33. #16812043C22semidirect
34. #16812042C22semidirect
35. #16812042C22semidirect
36. #16812041C22semidirect
37. #16812040C22semidirect
38. #16812040C22semidirect
42. #107×C107C22semidirect
43. #106×C107C22semidirect
58. #13133802C2×C4semidirect
59. #13133801C2×C4semidirect
61. C8×D107C2×C4semidirect
78. M16×C107C2×C4semidirect
117. C8×C107C22×C4semidirect
118. C107⋊C8C22×C4semidirect
146. D107#428direct (N × H)
148. C107C2×#428semidirect
157. #428D107direct (N × H)
187. #107C2×D107semidirect
191. #106C2×D107semidirect
210. M16C4×D107semidirect
266. C8C2×C4×D107semidirect

Maximal subgroups

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

subgroupindex
C2×M16×D1072
#117×D1072
C2×C4⋊C4×D1072
#202926702
#202926692
#202926682
#202926672
#202926662
#107×D1072
#106×D1072
#428×C1072
C2×#428107