Finite Group Databasebeta

← #417×D107 #28344969

Subgroups — #417×D107 #28344969

Order 13696.

816 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#41×D107C211yesmaximal, characteristic
2C2×M16×D107C211yesmaximal, characteristic, complemented
3C2×C4⋊C4×D107C211yesmaximal, characteristic, complemented
4#20292629C211yesmaximal, characteristic, complemented
5#20292628C211yesmaximal, characteristic, complemented
6#20292627C212yesmaximal, complemented
7#20292626C211yesmaximal, characteristic, complemented
8#20292625C212yesmaximal, complemented
9#103×D107C212yesmaximal, complemented
10#102×D107C212yesmaximal, complemented
11#417×C107C211yesmaximal, characteristic, complemented
12C2×#13133819C2211yescharacteristic
13C2×#13133818—21no—
14C2×#13133818C2211yescharacteristic
15C2×#13133817—21no—
16C2×#13133817C2211yescharacteristic
17#41×D107—24no—
18#41×D107C418yescomplemented
19C2×Q8×D107C2211yescharacteristic
20C2×D4×D107—21no—
21C2×D4×D107C2211yescharacteristic
22C22×C4×D107—21no—
23C22×C4×D107—21no—
24C22×C4×D107C2211yescharacteristic
25C2×#13133802C2211yescharacteristic, complemented
26C2×#13133801C2212yescomplemented
27M16×D107—24no—
28C2×C8×D107C2212yes—
29C2×C4⋊Dic107C2211yescharacteristic, complemented
30C2×Dic107⋊C4—21no—
31C4⋊C4×D107—21no—
32C4⋊C4×D107—21no—
33C4⋊C4×D107C2212yes—
34C4⋊C4×D107C2212yescomplemented
35#16812034C2212yescomplemented
36#16812033C2212yescomplemented
37#16812032C2212yescomplemented
38#16812031C2212yescomplemented
39#16812030C2212yescomplemented
40#16812029C2212yescomplemented
41C2×#41×C107C2211yescharacteristic
42C2×M16×C107C2211yescharacteristic, complemented
43C2×C4⋊C4×C107C2211yescharacteristic, complemented
44#103×C107C2212yescomplemented
45#102×C107C2212yescomplemented
46C23×D107—21no—
47C23×D107D411yescharacteristic
48#13133819—22no—
49#13133819C2×C414yescomplemented
50#13133818—22no—
51#13133818—28no—
52#13133818C2×C414yescomplemented
53#13133817—28no—
54#13133817—22no—
55#13133817C2×C414yescomplemented
56C2×Dic214—21no—
57C2×Dic214C2311yescharacteristic
58Q8×D107—22no—
59Q8×D107C2×C414yescomplemented
60C2×D428—21no—
61C2×D428C2311yescharacteristic
62C2×C22⋊D107—21no—
63C2×C22⋊D107—21no—
64C2×C22⋊D107—21no—
65D4×D107—28no—
66D4×D107—22no—
67D4×D107C2×C414yescomplemented
68C22×Dic107—21no—
69C22×Dic107—21no—
70C22×Dic107D411yescharacteristic
71C2×C4×D107—21no—
72C2×C4×D107—21no—
73C2×C4×D107—21no—
74C2×C4×D107—28no—
75C2×C4×D107—22no—
76C2×C4×D107—22no—
77C2×C4×D107—44no—
78C2×C4×D107C2×C414yes—
79C2×C4×D107D411yescharacteristic
80C2×C4×D107D411yescharacteristic
81C2×C4×D107D414yes—
82C2×C4×D107C2311yescharacteristic
83C2×C4×D107C2311yescharacteristic
84#13133802—22no—
85#13133801—24no—
86C2×C107⋊C8C2312yes—
87C8×D107—24no—
88C4⋊Dic107—21no—
89C4⋊Dic107C2312yes—
90Dic107⋊C4—22no—
91Dic107⋊C4—22no—
92#41×C107—22no—
93#41×C107C2×C414yescomplemented
94C2×Q8×C107C2311yescharacteristic
95C2×D4×C107—21no—
96C2×D4×C107C2311yescharacteristic
97C22×C4×C107—21no—
98C22×C4×C107—21no—
99C22×C4×C107C2311yescharacteristic
100M16×C107—22no—
101C2×C8×C107C2312yes—
102C4⋊C4×C107—21no—
103C4⋊C4×C107C2312yes—
104C23×C107—21no—
105C23×C107C2×D411yescharacteristic
106C22×D107—22no—
107C22×D107—22no—
108C22×D107—21no—
109C22×D107—24no—
110C22×D107—42no—
111C22×D107—42no—
112C22×D107C22⋊C414yes—
113C22×D107C2×D411yescharacteristic
114C22×D107C2×D411yescharacteristic
115Q8×C107—21no—
116Q8×C107C22×C412yescomplemented
117D4×C107—24no—
118D4×C107—21no—
119D4×C107C22×C412yescomplemented
120C2×C4×C107—21no—
121C2×C4×C107—24no—
122C2×C4×C107—22no—
123C2×C4×C107—42no—
124C2×C4×C107C22×C412yes—
125C2×C4×C107C2×D411yescharacteristic
126C2×C4×C107C2×D412yes—
127C2×C4×C107C2411yescharacteristic
128C22⋊D107—24no—
129C22⋊D107—24no—
130C22⋊D107—24no—
131C2×Dic107—21no—
132C2×Dic107—24no—
133C2×Dic107—22no—
134C2×Dic107—42no—
135C2×Dic107C22⋊C414yes—
136C2×Dic107C2×D411yescharacteristic
137C2×Dic107C2×D411yescharacteristic
138D428—24no—
139D428—21no—
140D428C22×C412yescomplemented
141C4×D107—24no—
142C4×D107—24no—
143C4×D107—24no—
144C4×D107—44no—
145C4×D107—44no—
146C4×D107C22⋊C414yes—
147C4×D107C22⋊C414yes—
148C4×D107C22×C412yes—
149C4×D107C22×C412yes—
150C4×D107C2×D412yes—
151C4×D107C2×D412yes—
152Dic214—21no—
153Dic214—24no—
154Dic214C22×C412yescomplemented
155C8×C107—22no—
156C107⋊C8—22no—
157C22×C107—21no—
158C22×C107—21no—
159C22×C107—22no—
160C22×C107—41no—
161C22×C107—41no—
162C22×C107C2×C22⋊C412yes—
163C22×C107C22×D411yescharacteristic
164C2×D107—21no—
165C2×D107—21no—
166C2×D107—28no—
167C2×D107—22no—
168C2×D107—41no—
169C2×D107—41no—
170C2×D107—48no—
171C2×D107C2×C22⋊C412yes—
172C2×D107C2×C22⋊C412yes—
173C2×D107AGL(1,ℤ/8)12yes—
174C2×D107#13712yes—
175C4×C107—22no—
176C4×C107—42no—
177C4×C107C2×C22⋊C412yes—
178C4×C107C23×C411yescharacteristic
179C4×C107C22×D411yescharacteristic
180Dic107—22no—
181Dic107—42no—
182Dic107C2×C22⋊C412yes—
183Dic107C2×C22⋊C412yes—
184C2×C107—22no—
185C2×C107—42no—
186C2×C107C22×C22⋊C411yescharacteristic
187C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
188C2×C107C2×#13711yescharacteristic
189D107—22no—
190D107—42no—
191D107#41714yesdirect product factor
192C2×#417—1071nomaximal, Sylow
193C107C2×#41711yescharacteristic, complemented, Sylow
194C22×#41—1071no—
195C22×M16—1071no—
196C22×C4⋊C4—1071no—
197#417—1071no—
198#417—1071no—
199#417—1071no—
200#417—1072no—
201#417—1072no—
202#417D10711yescharacteristic, direct product factor
203C2×#103—1072no—
204C2×#102—1072no—
205C2×#41—1078no—
206C2×#41—1071no—
207C2×#41—1071no—
208C2×#41—1071no—
209C2×#41—2141no—
210C2×#41—2141no—
211C2×#41—2144no—
212C2×#41C2×D10711yescharacteristic
213C22×Q8—1071no—
214C22×D4—1071no—
215C22×D4—2141no—
216C23×C4—1071no—
217C23×C4—2141no—
218C23×C4—2141no—
219C2×M16—1071no—
220C2×M16—1072no—
221C2×M16—2144no—
222C2×M16C2×D10711yescharacteristic, complemented
223C22×C8—1072no—
224C2×C4⋊C4—1071no—
225C2×C4⋊C4—1072no—
226C2×C4⋊C4—1072no—
227C2×C4⋊C4—2141no—
228C2×C4⋊C4—2141no—
229C2×C4⋊C4—2141no—
230C2×C4⋊C4C2×D10711yescharacteristic, complemented
231#103—1072no—
232#103—1072no—
233#103—1072no—
234#103C2×D10712yescomplemented
235#102—1072no—
236#102—1072no—
237#102—1072no—
238#102C2×D10712yescomplemented
239C24—1071no—
240C24—2141no—
241#41—22no—
242#41—1074no—
243#41—1074no—
244#41—1074no—
245#41—2148no—
246#41—2148no—
247#41—2142no—
248#41—2142no—
249#41—2142no—
250#41C4×D10714yescomplemented
251C2×Q8—1074no—
252C2×Q8—1071no—
253C2×Q8—2141no—
254C2×Q8—2142no—
255C2×Q8C22×D10711yescharacteristic
256C2×D4—21no—
257C2×D4—1074no—
258C2×D4—1071no—
259C2×D4—2141no—
260C2×D4—2142no—
261C2×D4—2141no—
262C2×D4—2148no—
263C2×D4—2141no—
264C2×D4—2141no—
265C2×D4C22×D10711yescharacteristic
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—1071no—
269C22×C4—1074no—
270C22×C4—1071no—
271C22×C4—1071no—
272C22×C4—1074no—
273C22×C4—1071no—
274C22×C4—1071no—
275C22×C4—4284no—
276C22×C4—2141no—
277C22×C4—2148no—
278C22×C4—2142no—
279C22×C4—2142no—
280C22×C4—2141no—
281C22×C4—2141no—
282C22×C4—2141no—
283C22×C4—2141no—
284C22×C4C22×D10711yescharacteristic
285M16—22no—
286M16—2144no—
287M16—2142no—
288C2×C8—1072no—
289C2×C8—2144no—
290C2×C8C22×D10712yes—
291C4⋊C4—21no—
292C4⋊C4—1072no—
293C4⋊C4—2142no—
294C4⋊C4—2142no—
295C4⋊C4—2141no—
296C4⋊C4C22×D10712yes—
297C23—21no—
298C23—1071no—
299C23—1074no—
300C23—1071no—
301C23—4282no—
302C23—4282no—
303C23—2144no—
304C23—2142no—
305C23—2142no—
306C23—2141no—
307C23D4×D10711yescharacteristic
308Q8—21no—
309Q8—1072no—
310Q8—2144no—
311Q8—2141no—
312Q8C2×C4×D10712yescomplemented
313D4—24no—
314D4—21no—
315D4—1072no—
316D4—2144no—
317D4—2144no—
318D4—2144no—
319D4—2144no—
320D4—2141no—
321D4C2×C4×D10712yescomplemented
322C2×C4—21no—
323C2×C4—24no—
324C2×C4—22no—
325C2×C4—42no—
326C2×C4—1074no—
327C2×C4—1071no—
328C2×C4—1071no—
329C2×C4—1074no—
330C2×C4—1074no—
331C2×C4—1072no—
332C2×C4—1072no—
333C2×C4—1072no—
334C2×C4—1072no—
335C2×C4—4284no—
336C2×C4—4284no—
337C2×C4—4282no—
338C2×C4—2144no—
339C2×C4—2144no—
340C2×C4—2144no—
341C2×C4—2142no—
342C2×C4—2141no—
343C2×C4—2144no—
344C2×C4C2×C4×D10712yes—
345C2×C4D4×D10711yescharacteristic
346C2×C4D4×D10712yes—
347C2×C4C23×D10711yescharacteristic
348C8—22no—
349C8—2142no—
350C22—21no—
351C22—21no—
352C22—22no—
353C22—41no—
354C22—41no—
355C22—1072no—
356C22—1072no—
357C22—1072no—
358C22—1072no—
359C22—4288no—
360C22—4281no—
361C22—4281no—
362C22—2148no—
363C22—2142no—
364C22—2141no—
365C22—2141no—
366C22C22⋊C4×D10712yes—
367C22C2×D4×D10711yescharacteristic
368C4—22no—
369C4—42no—
370C4—1072no—
371C4—1072no—
372C4—4282no—
373C4—2142no—
374C4C22⋊C4×D10712yes—
375C4C22×C4×D10711yescharacteristic
376C4C2×D4×D10711yescharacteristic
377C2—22no—
378C2—42no—
379C2—1074no—
380C2—4282no—
381C2—2142no—
382C2C2×C22⋊C4×D10711yescharacteristic
383C2AGL(1,ℤ/8)×D10711yescharacteristic
384C2#137×D10711yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutatorC4×C107C23×C4
FrattiniC2×C4C23×D107
Fitting#417×C107C2
socleC22×C107C22×D4
prime radicalC2×#41×D107C2
cosocleC2×C4×C107C24
hypercenter#417D107
hypocenterC107C2×#417
LCSG ▷ C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
2. C2×M16×D107C2semidirect
3. C2×C4⋊C4×D107C2semidirect
4. #20292629C2semidirect
5. #20292628C2semidirect
6. #20292627C2semidirect
7. #20292626C2semidirect
8. #20292625C2semidirect
9. #103×D107C2semidirect
10. #102×D107C2semidirect
11. #417×C107C2semidirect
18. #41×D107C4semidirect
25. C2×#13133802C22semidirect
26. C2×#13133801C22semidirect
29. C2×C4⋊Dic107C22semidirect
34. C4⋊C4×D107C22semidirect
35. #16812034C22semidirect
36. #16812033C22semidirect
37. #16812032C22semidirect
38. #16812031C22semidirect
39. #16812030C22semidirect
40. #16812029C22semidirect
42. C2×M16×C107C22semidirect
43. C2×C4⋊C4×C107C22semidirect
44. #103×C107C22semidirect
45. #102×C107C22semidirect
49. #13133819C2×C4semidirect
52. #13133818C2×C4semidirect
55. #13133817C2×C4semidirect
59. Q8×D107C2×C4semidirect
67. D4×D107C2×C4semidirect
93. #41×C107C2×C4semidirect
116. Q8×C107C22×C4semidirect
119. D4×C107C22×C4semidirect
140. D428C22×C4semidirect
154. Dic214C22×C4semidirect
191. D107#417direct (N × H)
193. C107C2×#417semidirect
202. #417D107direct (N × H)
222. C2×M16C2×D107semidirect
230. C2×C4⋊C4C2×D107semidirect
234. #103C2×D107semidirect
238. #102C2×D107semidirect
250. #41C4×D107semidirect
312. Q8C2×C4×D107semidirect
321. D4C2×C4×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#41×D1072
C2×M16×D1072
C2×C4⋊C4×D1072
#202926292
#202926282
#202926272
#202926262
#202926252
#103×D1072
#102×D1072
#417×C1072
C2×#417107