Finite Group Databasebeta

← #449×D107 #28345658

Subgroups — #449×D107 #28345658

Order 13696.

812 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#41×D107C211yesmaximal, characteristic, complemented
2C2×SD16×D107C211yesmaximal, characteristic, complemented
3C2×D8×D107C211yesmaximal, characteristic, complemented
4#121×D107C211yesmaximal, characteristic, complemented
5#20292775C211yesmaximal, characteristic, complemented
6#20292774C211yesmaximal, characteristic, complemented
7#20292773C211yesmaximal, characteristic, complemented
8#20292772C211yesmaximal, characteristic, complemented
9#20292771C211yesmaximal, characteristic, complemented
10#20292770C211yesmaximal, characteristic, complemented
11#20292769C211yesmaximal, characteristic, complemented
12#103×D107C211yesmaximal, characteristic, complemented
13#102×D107C211yesmaximal, characteristic, complemented
14C22⋊C8×D107C211yesmaximal, characteristic, complemented
15#449×C107C211yesmaximal, characteristic, complemented
16C2×#13133819C2211yescharacteristic, complemented
17C2×#13133818—21no—
18C2×#13133818C2211yescharacteristic, complemented
19C2×#13133817—21no—
20C2×#13133817C2211yescharacteristic, complemented
21#41×D107—24no—
22#41×D107—24no—
23C2×Q8×D107C2211yescharacteristic
24C2×D4×D107—21no—
25C2×D4×D107—21no—
26C2×D4×D107C2211yescharacteristic
27C2×D4×D107C2211yescharacteristic
28C22×C4×D107—21no—
29C22×C4×D107C2211yescharacteristic
30C2×#13133807C2211yescharacteristic, complemented
31C2×#13133806C2211yescharacteristic, complemented
32C2×#13133805C2211yescharacteristic, complemented
33SD16×D107—24no—
34C2×D856C2211yescharacteristic, complemented
35C2×#13133803C2211yescharacteristic, complemented
36C2×#13133803C2211yescharacteristic, complemented
37D8×D107—24no—
38C2×C8×D107C2211yescharacteristic
39C2×C8×D107C2211yescharacteristic
40#16812084C2211yescharacteristic, complemented
41#16812083C2211yescharacteristic, complemented
42#16812082—21no—
43#16812081—21no—
44#16812080C2211yescharacteristic, complemented
45C4⋊C4×D107C2211yescharacteristic
46C22⋊C4×D107—21no—
47#16812034C2211yescharacteristic, complemented
48#16812033C2211yescharacteristic, complemented
49#16812032C2211yescharacteristic, complemented
50#16812031C2211yescharacteristic, complemented
51#16812030C2211yescharacteristic, complemented
52#16812029C2211yescharacteristic, complemented
53C22⋊(C107⋊C8)C2211yescharacteristic, complemented
54(C2×D107)⋊C8C2211yescharacteristic, complemented
55(C2×D107)⋊C8C2211yescharacteristic, complemented
56C2×#41×C107C2211yescharacteristic, complemented
57C2×SD16×C107C2211yescharacteristic, complemented
58C2×D8×C107C2211yescharacteristic, complemented
59#121×C107C2211yescharacteristic, complemented
60#103×C107C2211yescharacteristic, complemented
61#102×C107C2211yescharacteristic, complemented
62C22⋊C8×C107C2211yescharacteristic, complemented
63C23×D107—21no—
64C23×D107—21no—
65C23×D107D411yescharacteristic
66#13133819—22no—
67#13133819—22no—
68#13133818—28no—
69#13133818—22no—
70#13133818—22no—
71#13133817—28no—
72#13133817—22no—
73#13133817—22no—
74C2×Dic214—21no—
75C2×Dic214C2311yescharacteristic
76Q8×D107—22no—
77Q8×D107D414yescomplemented
78C2×D428—21no—
79C2×D428—21no—
80C2×D428C2311yescharacteristic
81C2×D428C2311yescharacteristic
82C2×C22⋊D107—21no—
83C2×C22⋊D107—21no—
84C2×C22⋊D107—21no—
85C2×C22⋊D107—21no—
86C2×C22⋊D107—21no—
87C2×C22⋊D107—21no—
88D4×D107—28no—
89D4×D107—22no—
90D4×D107—22no—
91D4×D107—22no—
92D4×D107—44no—
93D4×D107D414yescomplemented
94C22×Dic107—21no—
95C22×Dic107D411yescharacteristic
96C2×C4×D107—28no—
97C2×C4×D107—21no—
98C2×C4×D107—21no—
99C2×C4×D107—21no—
100C2×C4×D107—22no—
101C2×C4×D107—22no—
102C2×C4×D107—21no—
103C2×C4×D107D411yescharacteristic
104C2×C4×D107D411yescharacteristic
105C2×C4×D107C2311yescharacteristic
106C2×C4×D107C2311yescharacteristic
107#13133807—22no—
108#13133806—22no—
109#13133805—22no—
110D856—22no—
111#13133803—22no—
112#13133803—22no—
113C2×C107⋊C8C2311yescharacteristic
114C2×C107⋊C8C2311yescharacteristic
115C8×D107—22no—
116C8×D107—22no—
117C4⋊Dic107C2311yescharacteristic
118Dic107⋊C4—21no—
119C22⋊Dic107—21no—
120(C2×D107)⋊C4—21no—
121(C2×D107)⋊C4—21no—
122#41×C107—22no—
123#41×C107—22no—
124C2×Q8×C107C2311yescharacteristic
125C2×D4×C107—21no—
126C2×D4×C107—21no—
127C2×D4×C107C2311yescharacteristic
128C2×D4×C107C2311yescharacteristic
129C22×C4×C107—21no—
130C22×C4×C107C2311yescharacteristic
131SD16×C107—22no—
132D8×C107—22no—
133C2×C8×C107C2311yescharacteristic
134C2×C8×C107C2311yescharacteristic
135C4⋊C4×C107C2311yescharacteristic
136C22⋊C4×C107—21no—
137C23×C107—21no—
138C23×C107—21no—
139C23×C107C2×D411yescharacteristic
140C22×D107—21no—
141C22×D107—22no—
142C22×D107—24no—
143C22×D107—22no—
144C22×D107—22no—
145C22×D107—21no—
146C22×D107—42no—
147C22×D107—42no—
148C22×D107—42no—
149C22×D107—42no—
150C22×D107—42no—
151C22×D107C2×D411yescharacteristic
152C22×D107C2×D411yescharacteristic
153Q8×C107—21no—
154Q8×C107C2×D412yescomplemented
155D4×C107—24no—
156D4×C107—21no—
157D4×C107—21no—
158D4×C107—21no—
159D4×C107—42no—
160D4×C107C2×D412yescomplemented
161C2×C4×C107—24no—
162C2×C4×C107—21no—
163C2×C4×C107—21no—
164C2×C4×C107—21no—
165C2×C4×C107—21no—
166C2×C4×C107C2×D411yescharacteristic
167C2×C4×C107C2411yescharacteristic
168C22⋊D107—24no—
169C22⋊D107—24no—
170C22⋊D107—24no—
171C22⋊D107—42no—
172C22⋊D107—42no—
173C22⋊D107—42no—
174C2×Dic107—24no—
175C2×Dic107—21no—
176C2×Dic107—22no—
177C2×Dic107—21no—
178C2×Dic107C2×D411yescharacteristic
179C2×Dic107C2×D411yescharacteristic
180D428—24no—
181D428—21no—
182D428—21no—
183D428—21no—
184D428—42no—
185D428C2×D412yescomplemented
186C4×D107—24no—
187C4×D107—24no—
188C4×D107—22no—
189C4×D107—24no—
190C4×D107—21no—
191C4×D107—21no—
192C4×D107—22no—
193C4×D107—42no—
194C4×D107C2×D412yes—
195C4×D107C2×D412yes—
196Dic214—21no—
197Dic214—24no—
198Dic214C2×D412yescomplemented
199C8×C107—21no—
200C8×C107—21no—
201C107⋊C8—21no—
202C107⋊C8—21no—
203C22×C107—21no—
204C22×C107—22no—
205C22×C107—21no—
206C22×C107—21no—
207C22×C107—41no—
208C22×C107—41no—
209C22×C107—41no—
210C22×C107—41no—
211C22×C107—41no—
212C22×C107C22×D411yescharacteristic
213C2×D107—21no—
214C2×D107—22no—
215C2×D107—21no—
216C2×D107—21no—
217C2×D107—41no—
218C2×D107—41no—
219C2×D107—44no—
220C2×D107—48no—
221C2×D107—41no—
222C2×D107—41no—
223C2×D107—41no—
224C2×D107—84no—
225C2×D107#12012yes—
226C2×D107#13512yes—
227C2×D107AGL(1,ℤ/8)12yes—
228C4×C107—22no—
229C4×C107—21no—
230C4×C107—41no—
231C4×C107C22×D411yescharacteristic
232C4×C107C22×D411yescharacteristic
233Dic107—22no—
234Dic107—21no—
235Dic107—41no—
236Dic107#12012yes—
237C2×C107—41no—
238C2×C107—42no—
239C2×C107—81no—
240C2×C107C2×#12011yescharacteristic
241C2×C107C2×#13511yescharacteristic
242C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
243D107—41no—
244D107—42no—
245D107—81no—
246D107#44914yesdirect product factor
247C2×#449—1071nomaximal, Sylow
248C107C2×#44911yescharacteristic, complemented, Sylow
249C22×#41—1071no—
250C22×SD16—1071no—
251C22×D8—1071no—
252C2×#121—1071no—
253#449—1071no—
254#449—1071no—
255#449—1071no—
256#449—1071no—
257#449—1071no—
258#449—1071no—
259#449—1071no—
260#449D10711yescharacteristic, direct product factor
261C2×#103—1071no—
262C2×#102—1071no—
263C2×C22⋊C8—1071no—
264C2×#41—1071no—
265C2×#41—1071no—
266C2×#41—1071no—
267C2×#41—2141no—
268C2×#41—2141no—
269C2×#41—2144no—
270C2×#41—2144no—
271C2×#41C2×D10711yescharacteristic, complemented
272C22×Q8—1071no—
273C22×D4—1071no—
274C22×D4—1071no—
275C22×D4—2141no—
276C22×D4—2141no—
277C23×C4—1071no—
278C23×C4—2141no—
279C2×SD16—1071no—
280C2×SD16—1071no—
281C2×SD16—1071no—
282C2×SD16—2144no—
283C2×SD16C2×D10711yescharacteristic, complemented
284C2×D8—1071no—
285C2×D8—1071no—
286C2×D8—1071no—
287C2×D8—2144no—
288C2×D8C2×D10711yescharacteristic, complemented
289C22×C8—1071no—
290C22×C8—1071no—
291#121—1071no—
292#121—1071no—
293#121—1071no—
294#121—2141no—
295#121—2141no—
296#121C2×D10711yescharacteristic, complemented
297C2×C4⋊C4—1071no—
298C2×C22⋊C4—2141no—
299#103—1071no—
300#103—1071no—
301#103—1071no—
302#103C2×D10711yescharacteristic, complemented
303#102—1071no—
304#102—1071no—
305#102—1071no—
306#102C2×D10711yescharacteristic, complemented
307C22⋊C8—1071no—
308C22⋊C8—1071no—
309C22⋊C8—1071no—
310C22⋊C8C2×D10711yescharacteristic, complemented
311C24—1071no—
312C24—2141no—
313C24—2141no—
314#41—22no—
315#41—22no—
316#41—2142no—
317#41—2142no—
318#41—2148no—
319#41—2142no—
320#41—2148no—
321#41—2142no—
322#41—2142no—
323#41—2142no—
324C2×Q8—1074no—
325C2×Q8—1071no—
326C2×Q8—2141no—
327C2×Q8—2142no—
328C2×Q8C22×D10711yescharacteristic
329C2×D4—21no—
330C2×D4—21no—
331C2×D4—1074no—
332C2×D4—1071no—
333C2×D4—1071no—
334C2×D4—4284no—
335C2×D4—2141no—
336C2×D4—2141no—
337C2×D4—2142no—
338C2×D4—2142no—
339C2×D4—2142no—
340C2×D4—2141no—
341C2×D4—2141no—
342C2×D4—2148no—
343C2×D4—2141no—
344C2×D4—2141no—
345C2×D4—2141no—
346C2×D4—2141no—
347C2×D4C22×D10711yescharacteristic
348C2×D4C22×D10711yescharacteristic
349C22×C4—21no—
350C22×C4—1071no—
351C22×C4—1071no—
352C22×C4—1071no—
353C22×C4—1071no—
354C22×C4—1071no—
355C22×C4—2142no—
356C22×C4—2142no—
357C22×C4—2141no—
358C22×C4—2141no—
359C22×C4—2141no—
360C22×C4—2148no—
361C22×C4—2141no—
362C22×C4—2141no—
363C22×C4C22×D10711yescharacteristic
364SD16—22no—
365SD16—2142no—
366SD16—2142no—
367SD16—2142no—
368D8—22no—
369D8—2142no—
370D8—2142no—
371D8—2142no—
372C2×C8—1071no—
373C2×C8—1071no—
374C2×C8—2142no—
375C2×C8—2142no—
376C2×C8C22×D10711yescharacteristic
377C2×C8C22×D10711yescharacteristic
378C4⋊C4—1071no—
379C4⋊C4—2141no—
380C4⋊C4C22×D10711yescharacteristic
381C22⋊C4—21no—
382C22⋊C4—2141no—
383C22⋊C4—2141no—
384C22⋊C4—2141no—
385C23—21no—
386C23—21no—
387C23—1071no—
388C23—1071no—
389C23—4282no—
390C23—4282no—
391C23—4282no—
392C23—4282no—
393C23—4282no—
394C23—2144no—
395C23—2142no—
396C23—2142no—
397C23—2142no—
398C23—2141no—
399C23—2141no—
400C23D4×D10711yescharacteristic
401Q8—21no—
402Q8—1072no—
403Q8—2144no—
404Q8—2141no—
405Q8D4×D10712yescomplemented
406D4—24no—
407D4—21no—
408D4—21no—
409D4—21no—
410D4—42no—
411D4—1072no—
412D4—4282no—
413D4—4282no—
414D4—4282no—
415D4—4282no—
416D4—2144no—
417D4—2144no—
418D4—2144no—
419D4—2144no—
420D4—2141no—
421D4—2141no—
422D4—2141no—
423D4D4×D10712yescomplemented
424C2×C4—24no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—1071no—
430C2×C4—1072no—
431C2×C4—1072no—
432C2×C4—1071no—
433C2×C4—4282no—
434C2×C4—2142no—
435C2×C4—2142no—
436C2×C4—2144no—
437C2×C4—2144no—
438C2×C4—2144no—
439C2×C4—2142no—
440C2×C4—2141no—
441C2×C4—2144no—
442C2×C4—2141no—
443C2×C4—2141no—
444C2×C4—2141no—
445C2×C4D4×D10711yescharacteristic
446C2×C4C23×D10711yescharacteristic
447C8—21no—
448C8—21no—
449C8—2141no—
450C8—2141no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—22no—
455C22—41no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—41no—
460C22—8564no—
461C22—1072no—
462C22—1072no—
463C22—1072no—
464C22—4288no—
465C22—4284no—
466C22—4281no—
467C22—4281no—
468C22—4281no—
469C22—4281no—
470C22—4281no—
471C22—2141no—
472C22—2141no—
473C22—2142no—
474C22—2141no—
475C22C2×D4×D10711yescharacteristic
476C4—21no—
477C4—22no—
478C4—41no—
479C4—1072no—
480C4—4281no—
481C4—2141no—
482C4—2142no—
483C4C2×D4×D10711yescharacteristic
484C4C2×D4×D10711yescharacteristic
485C2—41no—
486C2—42no—
487C2—81no—
488C2—8561no—
489C2—1074no—
490C2—4282no—
491C2—4281no—
492C2#120×D10711yescharacteristic
493C2#135×D10711yescharacteristic
494C2AGL(1,ℤ/8)×D10711yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#449×C107C2
socleC22×C107C22×D4
hypercenter#449D107
hypocenterC107C2×#449
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. C2×#41×D107C2semidirect
2. C2×SD16×D107C2semidirect
3. C2×D8×D107C2semidirect
4. #121×D107C2semidirect
5. #20292775C2semidirect
6. #20292774C2semidirect
7. #20292773C2semidirect
8. #20292772C2semidirect
9. #20292771C2semidirect
10. #20292770C2semidirect
11. #20292769C2semidirect
12. #103×D107C2semidirect
13. #102×D107C2semidirect
14. C22⋊C8×D107C2semidirect
15. #449×C107C2semidirect
16. C2×#13133819C22semidirect
18. C2×#13133818C22semidirect
20. C2×#13133817C22semidirect
30. C2×#13133807C22semidirect
31. C2×#13133806C22semidirect
32. C2×#13133805C22semidirect
34. C2×D856C22semidirect
35. C2×#13133803C22semidirect
36. C2×#13133803C22semidirect
40. #16812084C22semidirect
41. #16812083C22semidirect
44. #16812080C22semidirect
47. #16812034C22semidirect
48. #16812033C22semidirect
49. #16812032C22semidirect
50. #16812031C22semidirect
51. #16812030C22semidirect
52. #16812029C22semidirect
53. C22⋊(C107⋊C8)C22semidirect
54. (C2×D107)⋊C8C22semidirect
55. (C2×D107)⋊C8C22semidirect
56. C2×#41×C107C22semidirect
57. C2×SD16×C107C22semidirect
58. C2×D8×C107C22semidirect
59. #121×C107C22semidirect
60. #103×C107C22semidirect
61. #102×C107C22semidirect
62. C22⋊C8×C107C22semidirect
77. Q8×D107D4semidirect
93. D4×D107D4semidirect
154. Q8×C107C2×D4semidirect
160. D4×C107C2×D4semidirect
185. D428C2×D4semidirect
198. Dic214C2×D4semidirect
246. D107#449direct (N × H)
248. C107C2×#449semidirect
260. #449D107direct (N × H)
271. C2×#41C2×D107semidirect
283. C2×SD16C2×D107semidirect
288. C2×D8C2×D107semidirect
296. #121C2×D107semidirect
302. #103C2×D107semidirect
306. #102C2×D107semidirect
310. C22⋊C8C2×D107semidirect
405. Q8D4×D107semidirect
423. D4D4×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×SD16×D1072
C2×D8×D1072
#121×D1072
#202927752
#202927742
#202927732
#202927722
#202927712
#202927702
#202927692
#103×D1072
#102×D1072
C22⋊C8×D1072
#449×C1072
C2×#449107