Finite Group Databasebeta

← #19143848

Subgroups — #19143848

Order 5504.

806 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#41×D43C211yesmaximal, characteristic, complemented
2C2×#13063353C211yesmaximal, characteristic, complemented
3C2×#13063350C211yesmaximal, characteristic, complemented
4C2×#13063300C211yesmaximal, characteristic, complemented
5#14583047C212yesmaximal, complemented
6#14583045C212yesmaximal, complemented
7#14583039C212yesmaximal, complemented
8#14583034C212yesmaximal, complemented
9#14582898C211yesmaximal, characteristic, complemented
10#14582862C211yesmaximal, characteristic, complemented
11C43×#417C211yesmaximal, characteristic, complemented
12C2×#10621902C2211yescharacteristic, complemented
13C2×#10621898—21no—
14C2×#10621898C2211yescharacteristic
15C2×#10621900—21no—
16C2×#10621900C2211yescharacteristic, complemented
17#41×D43—24no—
18#41×D43—24no—
19C2×Q8×D43C2211yescharacteristic, complemented
20C22×D172—21no—
21C22×D172C2211yescharacteristic, complemented
22C2×D4×D43—21no—
23C2×D4×D43C2211yescharacteristic, complemented
24C22×C4×D43—21no—
25C22×C4×D43C2211yescharacteristic
26#13063353—24no—
27#13063350—24no—
28C2×#10621873C2212yescomplemented
29C2×#10621884C2212yescomplemented
30C2×D344C2212yescomplemented
31C2×#10621882C2212yescomplemented
32C2×#10621877C2211yescharacteristic, complemented
33#13063300—21no—
34#13063300—21no—
35#13063300C2212yes—
36#13063300C2212yescomplemented
37C2×(C2×D43)⋊C4—21no—
38#13063248C2212yescomplemented
39#13063245C2212yescomplemented
40(C2×D43)⋊C8C2212yescomplemented
41(C2×D43)⋊C8C2212yescomplemented
42C2×#41×C43C2211yescharacteristic
43C2×M16×C43C2211yescharacteristic, complemented
44C2×C4⋊C4×C43C2211yescharacteristic, complemented
45#103×C43C2212yescomplemented
46#102×C43C2212yescomplemented
47C23×D43—21no—
48C23×D43—21no—
49C23×D43D411yescharacteristic
50#41×C43—22no—
51#41×C43D414yescomplemented
52C2×Q8×C43C2311yescharacteristic
53C2×D4×C43—21no—
54C2×D4×C43C2311yescharacteristic
55C22×C4×C43—21no—
56C22×C4×C43—21no—
57C22×C4×C43C2311yescharacteristic
58C2×C22⋊D43—21no—
59C2×C22⋊D43—21no—
60C2×C22⋊D43—21no—
61C22×Dic43—21no—
62C22×Dic43D411yescharacteristic
63#10621902—22no—
64#10621902—22no—
65Q8×D43—22no—
66Q8×D43—22no—
67#10621900—28no—
68#10621900—22no—
69#10621900—22no—
70D4×D43—28no—
71D4×D43—22no—
72D4×D43—22no—
73#10621898—28no—
74#10621898—22no—
75#10621898D414yescomplemented
76C2×D172—21no—
77C2×D172—21no—
78C2×D172—22no—
79C2×D172—22no—
80C2×D172—22no—
81C2×D172—22no—
82C2×D172—44no—
83C2×D172C2312yes—
84C2×D172C2311yescharacteristic
85C2×C4×D43—28no—
86C2×C4×D43—21no—
87C2×C4×D43—22no—
88C2×C4×D43—22no—
89C2×C4×D43—21no—
90C2×C4×D43—21no—
91C2×C4×D43D411yescharacteristic
92C2×C4×D43D411yescharacteristic
93C2×C4×D43C2311yescharacteristic
94C2×C4×D43C2311yescharacteristic
95C2×Dic86—21no—
96C2×Dic86C2311yescharacteristic
97M16×C43—22no—
98C2×C8×C43C2312yes—
99C4⋊C4×C43—21no—
100C4⋊C4×C43C2312yes—
101#10621884—24no—
102#10621882—24no—
103(C2×D43)⋊C4—22no—
104(C2×D43)⋊C4—22no—
105#10621877—22no—
106C2×C43⋊C8C2312yes—
107D344—24no—
108#10621873—24no—
109C23×C43—21no—
110C23×C43C2×D411yescharacteristic
111C22×D43—24no—
112C22×D43—22no—
113C22×D43—21no—
114C22×D43—22no—
115C22×D43—22no—
116C22×D43—41no—
117C22×D43—42no—
118C22×D43—42no—
119C22×D43—42no—
120C22×D43—42no—
121C22×D43—42no—
122C22×D43C2×D411yescharacteristic
123C22×D43C2×D411yescharacteristic
124Q8×C43—21no—
125Q8×C43C2×D412yescomplemented
126D4×C43—24no—
127D4×C43—21no—
128D4×C43C2×D412yescomplemented
129C2×C4×C43—21no—
130C2×C4×C43—24no—
131C2×C4×C43—22no—
132C2×C4×C43—42no—
133C2×C4×C43C2×D412yes—
134C2×C4×C43C2×D411yescharacteristic
135C2×C4×C43C2×D412yes—
136C2×C4×C43C2411yescharacteristic
137C22⋊D43—24no—
138C22⋊D43—24no—
139C22⋊D43—24no—
140C2×Dic43—24no—
141C2×Dic43—22no—
142C2×Dic43—21no—
143C2×Dic43C2×D411yescharacteristic
144C2×Dic43C2×D411yescharacteristic
145D172—24no—
146D172—21no—
147D172—22no—
148D172—22no—
149D172—44no—
150D172—44no—
151D172—42no—
152D172C2×D412yescomplemented
153C4×D43—24no—
154C4×D43—24no—
155C4×D43—22no—
156C4×D43—24no—
157C4×D43—22no—
158C4×D43—21no—
159C4×D43—21no—
160C4×D43—21no—
161C4×D43—21no—
162Dic86—24no—
163Dic86—21no—
164Dic86C2×D412yescomplemented
165C8×C43—22no—
166C43⋊C8—22no—
167C22×C43—21no—
168C22×C43—21no—
169C22×C43—22no—
170C22×C43—41no—
171C22×C43—41no—
172C22×C43#12012yes—
173C22×C43C22×D411yescharacteristic
174C2×D43—21no—
175C2×D43—21no—
176C2×D43—22no—
177C2×D43—21no—
178C2×D43—21no—
179C2×D43—21no—
180C2×D43—44no—
181C2×D43—48no—
182C2×D43—41no—
183C2×D43—41no—
184C2×D43—42no—
185C2×D43—42no—
186C2×D43—42no—
187C2×D43—84no—
188C2×D43AGL(1,ℤ/8)12yescomplemented
189C4×C43—22no—
190C4×C43—42no—
191C4×C43#12012yes—
192C4×C43C22×D411yescharacteristic
193C4×C43C22×D411yescharacteristic
194Dic43—21no—
195Dic43—22no—
196Dic43—21no—
197#3006—431nomaximal, Sylow
198C2×C43—22no—
199C2×C43—42no—
200C2×C43C2×#12011yescharacteristic
201C2×C43C2×AGL(1,ℤ/8)11yescharacteristic
202C2×C43#57511yescharacteristic
203D43—22no—
204D43—41no—
205D43—42no—
206D43—82no—
207C22×#41—431no—
208C2×AGL(1,ℤ/8)—431no—
209C2×AGL(1,ℤ/8)—431no—
210C2×#121—431no—
211#449—432no—
212#449—432no—
213#448—432no—
214#447—432no—
215#417—431no—
216#417D4311yescharacteristic, complemented
217#407—431no—
218C43#300611yescharacteristic, complemented, Sylow
219C2×#41—431no—
220C2×#41—431no—
221C2×#41—431no—
222C2×#41—861no—
223C2×#41—861no—
224C2×#41—864no—
225C2×#41—864no—
226C2×#41C2×D4311yescharacteristic
227C22×Q8—431no—
228C22×D4—431no—
229C22×D4—431no—
230C22×D4—861no—
231C22×D4—861no—
232C23×C4—431no—
233C23×C4—861no—
234AGL(1,ℤ/8)—864no—
235AGL(1,ℤ/8)—864no—
236C2×SD16—432no—
237C2×SD16—432no—
238C2×D8—432no—
239C2×D8—432no—
240C2×M16—431no—
241C2×M16C2×D4311yescharacteristic, complemented
242#121—432no—
243#121—432no—
244#121—861no—
245#121—861no—
246C2×C4⋊C4C2×D4311yescharacteristic, complemented
247C2×C22⋊C4—861no—
248#103—432no—
249#103C2×D4312yescomplemented
250#102—432no—
251#102C2×D4312yescomplemented
252C22⋊C8—432no—
253C22⋊C8—432no—
254C24—431no—
255C24—861no—
256C24—861no—
257#41—22no—
258#41—434no—
259#41—868no—
260#41—862no—
261#41—868no—
262#41—862no—
263#41—862no—
264#41—862no—
265#41—862no—
266#41D17214yescomplemented
267C2×Q8—431no—
268C2×Q8—861no—
269C2×Q8—862no—
270C2×Q8—862no—
271C2×Q8C22×D4311yescharacteristic
272C2×D4—21no—
273C2×D4—431no—
274C2×D4—432no—
275C2×D4—861no—
276C2×D4—862no—
277C2×D4—862no—
278C2×D4—861no—
279C2×D4—862no—
280C2×D4—868no—
281C2×D4—861no—
282C2×D4—861no—
283C2×D4—862no—
284C2×D4—862no—
285C2×D4—862no—
286C2×D4—861no—
287C2×D4—1724no—
288C2×D4C22×D4311yescharacteristic
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—431no—
292C22×C4—431no—
293C22×C4—431no—
294C22×C4—431no—
295C22×C4—431no—
296C22×C4—861no—
297C22×C4—861no—
298C22×C4—868no—
299C22×C4—861no—
300C22×C4—862no—
301C22×C4—862no—
302C22×C4—861no—
303C22×C4C22×D4311yescharacteristic
304SD16—864no—
305SD16—864no—
306D8—864no—
307D8—864no—
308M16—22no—
309M16—862no—
310C2×C8—432no—
311C2×C8C22×D4312yes—
312C4⋊C4—21no—
313C4⋊C4C22×D4312yes—
314C22⋊C4—862no—
315C22⋊C4—862no—
316C23—21no—
317C23—431no—
318C23—431no—
319C23—862no—
320C23—864no—
321C23—862no—
322C23—861no—
323C23—862no—
324C23—1721no—
325C23—1722no—
326C23—1722no—
327C23—1722no—
328C23—1722no—
329C23—1722no—
330C23D4×D4311yescharacteristic
331Q8—21no—
332Q8—432no—
333Q8—864no—
334Q8—861no—
335Q8C2×D17212yescomplemented
336D4—24no—
337D4—21no—
338D4—432no—
339D4—864no—
340D4—864no—
341D4—864no—
342D4—864no—
343D4—862no—
344D4—861no—
345D4—862no—
346D4—1724no—
347D4—1724no—
348D4—1722no—
349D4C2×D17212yescomplemented
350C2×C4—21no—
351C2×C4—24no—
352C2×C4—22no—
353C2×C4—42no—
354C2×C4—431no—
355C2×C4—431no—
356C2×C4—864no—
357C2×C4—864no—
358C2×C4—864no—
359C2×C4—862no—
360C2×C4—861no—
361C2×C4—862no—
362C2×C4—862no—
363C2×C4—864no—
364C2×C4—861no—
365C2×C4—861no—
366C2×C4—861no—
367C2×C4—861no—
368C2×C4C2×D17212yes—
369C2×C4D4×D4311yescharacteristic
370C2×C4D4×D4312yes—
371C2×C4C23×D4311yescharacteristic
372C8—22no—
373C8—862no—
374C22—21no—
375C22—21no—
376C22—22no—
377C22—41no—
378C22—41no—
379C22—432no—
380C22—861no—
381C22—861no—
382C22—862no—
383C22—861no—
384C22—861no—
385C22—861no—
386C22—3444no—
387C22—1724no—
388C22—1728no—
389C22—1722no—
390C22—1722no—
391C22—1722no—
392C22—1721no—
393C22—1721no—
394C22#1306329512yes—
395C22C2×D4×D4311yescharacteristic
396C4—22no—
397C4—42no—
398C4—861no—
399C4—861no—
400C4—862no—
401C4#1306329512yes—
402C4C2×D4×D4311yescharacteristic
403C4C22×D17211yescharacteristic
404C2—22no—
405C2—42no—
406C2—862no—
407C2—3442no—
408C2—1721no—
409C2—1722no—
410C2C2×#1306329511yescharacteristic
411C2AGL(1,ℤ/8)×D4311yescharacteristic
412C2#1458369811yescharacteristic
subgroupG/H
centerC22C2×D4×D43
commutatorC2×C4×C43C24
FrattiniC2×C4C23×D43
FittingC43×#417C2
socleC22×C43C22×D4
hypercenter#417D43
hypocenterC43#3006
LCSG ▷ C2×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. C2×#41×D43C2semidirect
2. C2×#13063353C2semidirect
3. C2×#13063350C2semidirect
4. C2×#13063300C2semidirect
5. #14583047C2semidirect
6. #14583045C2semidirect
7. #14583039C2semidirect
8. #14583034C2semidirect
9. #14582898C2semidirect
10. #14582862C2semidirect
11. C43×#417C2semidirect
12. C2×#10621902C22semidirect
16. C2×#10621900C22semidirect
19. C2×Q8×D43C22semidirect
21. C22×D172C22semidirect
23. C2×D4×D43C22semidirect
28. C2×#10621873C22semidirect
29. C2×#10621884C22semidirect
30. C2×D344C22semidirect
31. C2×#10621882C22semidirect
32. C2×#10621877C22semidirect
36. #13063300C22semidirect
38. #13063248C22semidirect
39. #13063245C22semidirect
40. (C2×D43)⋊C8C22semidirect
41. (C2×D43)⋊C8C22semidirect
43. C2×M16×C43C22semidirect
44. C2×C4⋊C4×C43C22semidirect
45. #103×C43C22semidirect
46. #102×C43C22semidirect
51. #41×C43D4semidirect
75. #10621898D4semidirect
125. Q8×C43C2×D4semidirect
128. D4×C43C2×D4semidirect
152. D172C2×D4semidirect
164. Dic86C2×D4semidirect
188. C2×D43AGL(1,ℤ/8)semidirect
216. #417D43semidirect
218. C43#3006semidirect
241. C2×M16C2×D43semidirect
246. C2×C4⋊C4C2×D43semidirect
249. #103C2×D43semidirect
251. #102C2×D43semidirect
266. #41D172semidirect
335. Q8C2×D172semidirect
349. D4C2×D172semidirect