Finite Group Databasebeta

← #28345760

Subgroups — #28345760

Order 13696.

764 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812154C211yesmaximal, characteristic, complemented
2C2×#16812133C211yesmaximal, characteristic, complemented
3C2×#16812128C211yesmaximal, characteristic, complemented
4#20293206C211yesmaximal, characteristic, complemented
5#20292791C211yesmaximal, characteristic, complemented
6#20292790C211yesmaximal, characteristic, complemented
7#20292779C211yesmaximal, characteristic, complemented
8#20292778C211yesmaximal, characteristic, complemented
9#20292770C211yesmaximal, characteristic, complemented
10#20292769C211yesmaximal, characteristic, complemented
11#20292759C211yesmaximal, characteristic, complemented
12#20292625C211yesmaximal, characteristic, complemented
13#102×D107C211yesmaximal, characteristic, complemented
14#20292592C211yesmaximal, characteristic, complemented
15#447×C107C211yesmaximal, characteristic, complemented
16#16812154—24no—
17#16812154—24no—
18C2×#13133818C2211yescharacteristic
19C2×#13133818C2211yescharacteristic
20C2×#13133817—21no—
21C2×#13133817C2211yescharacteristic
22C2×#13133817C2211yescharacteristic
23C2×#13133817C2211yescharacteristic, complemented
24C22×C22⋊D107—21no—
25C22×C22⋊D107—21no—
26C2×D4×D107—21no—
27C2×D4×D107C2211yescharacteristic
28C2×D4×D107C2211yescharacteristic
29#16812133—24no—
30#16812128—24no—
31C2×Dic428C2211yescharacteristic, complemented
32C2×#13133807C2211yescharacteristic, complemented
33C2×#13133806C2211yescharacteristic, complemented
34C2×#13133806C2211yescharacteristic, complemented
35C2×#13133806C2211yescharacteristic, complemented
36C2×#13133803C2211yescharacteristic, complemented
37C2×#13133801C2211yescharacteristic, complemented
38C2×C8×D107C2211yescharacteristic, complemented
39#16812089C2211yescharacteristic, complemented
40#16812088C2211yescharacteristic, complemented
41#16812086—21no—
42#16812082—21no—
43#16812081—21no—
44#16812080—21no—
45#16812080C2211yescharacteristic, complemented
46C4⋊C4×D107C2211yescharacteristic, complemented
47#16812034C2211yescharacteristic, complemented
48#16812032C2211yescharacteristic, complemented
49#16812031C2211yescharacteristic, complemented
50#16812030C2211yescharacteristic, complemented
51#16812030C2211yescharacteristic, complemented
52#16812029C2211yescharacteristic, complemented
53C22⋊(C107⋊C8)C2211yescharacteristic, complemented
54(C2×D107)⋊C8C2211yescharacteristic, complemented
55(C2×D107)⋊C8C2211yescharacteristic, complemented
56C22×D4×C107C2211yescharacteristic
57C2×D8×C107C2211yescharacteristic, complemented
58C2×D8×C107C2211yescharacteristic, complemented
59#121×C107C2211yescharacteristic, complemented
60#102×C107C2211yescharacteristic, complemented
61#102×C107C2211yescharacteristic, complemented
62C22⋊C8×C107C2211yescharacteristic, complemented
63C23×D107—21no—
64C23×D107—21no—
65#13133818—22no—
66#13133818—22no—
67#13133818—22no—
68#13133818—22no—
69#13133817—28no—
70#13133817—22no—
71#13133817—22no—
72#13133817—22no—
73#13133817—22no—
74#13133817—22no—
75#13133817D414yescomplemented
76C2×Dic214D411yescharacteristic
77C2×Dic214C2311yescharacteristic
78C2×Dic214C2311yescharacteristic
79C2×D428D411yescharacteristic
80C2×D428C2311yescharacteristic
81C2×C22⋊D107—28no—
82C2×C22⋊D107—28no—
83C2×C22⋊D107—21no—
84C2×C22⋊D107—21no—
85C2×C22⋊D107—21no—
86C2×C22⋊D107—21no—
87C2×C22⋊D107—21no—
88C2×C22⋊D107—21no—
89C2×C22⋊D107—21no—
90C2×C22⋊D107D411yescharacteristic
91C2×C22⋊D107D411yescharacteristic
92D4×D107—28no—
93D4×D107—22no—
94D4×D107—22no—
95D4×D107—22no—
96D4×D107D414yescomplemented
97C22×Dic107—21no—
98C22×Dic107—21no—
99C22×Dic107—21no—
100C2×C4×D107—21no—
101C2×C4×D107—21no—
102C2×C4×D107C2311yescharacteristic
103C2×C4×D107C2311yescharacteristic
104Dic428—22no—
105#13133807—22no—
106#13133806—22no—
107#13133806—22no—
108#13133806—22no—
109#13133803—22no—
110#13133801—22no—
111C2×C107⋊C8C2311yescharacteristic
112C2×C107⋊C8C2311yescharacteristic
113C8×D107—22no—
114C4⋊Dic107—21no—
115C4⋊Dic107C2311yescharacteristic
116Dic107⋊C4—21no—
117Dic107⋊C4—21no—
118C22⋊Dic107—21no—
119C22⋊Dic107—21no—
120(C2×D107)⋊C4—21no—
121C24×C107—21no—
122C2×D4×C107—22no—
123C2×D4×C107—22no—
124C2×D4×C107—21no—
125C2×D4×C107—21no—
126C2×D4×C107C2311yescharacteristic
127C2×D4×C107C2311yescharacteristic
128C2×D4×C107C2311yescharacteristic
129C22×C4×C107C2311yescharacteristic
130D8×C107—22no—
131D8×C107—22no—
132C2×C8×C107C2311yescharacteristic
133C2×C8×C107C2311yescharacteristic
134C4⋊C4×C107C2311yescharacteristic
135C22⋊C4×C107—21no—
136C23×C107—24no—
137C23×C107—21no—
138C23×C107—21no—
139C23×C107—21no—
140C23×C107—81no—
141C23×C107—81no—
142C23×C107C2×D411yescharacteristic
143C22×D107—21no—
144C22×D107—24no—
145C22×D107—24no—
146C22×D107—81no—
147C22×D107—81no—
148C22×D107—81no—
149C22×D107—81no—
150C22×D107C2×D411yescharacteristic
151C22×D107C2×D411yescharacteristic
152D4×C107—24no—
153D4×C107—21no—
154D4×C107—21no—
155D4×C107—21no—
156D4×C107—21no—
157D4×C107—81no—
158D4×C107C2×D412yes—
159D4×C107C2×D412yes—
160C2×C4×C107—21no—
161C2×C4×C107—21no—
162C2×C4×C107—21no—
163C2×C4×C107C2×D411yescharacteristic
164C2×C4×C107C2411yescharacteristic
165C22⋊D107—24no—
166C22⋊D107—24no—
167C22⋊D107—24no—
168C22⋊D107—24no—
169C22⋊D107—22no—
170C22⋊D107—24no—
171C22⋊D107—22no—
172C22⋊D107—42no—
173C22⋊D107—81no—
174C2×Dic107—24no—
175C2×Dic107—21no—
176C2×Dic107—24no—
177C2×Dic107—21no—
178C2×Dic107—21no—
179C2×Dic107—42no—
180C2×Dic107C2×D411yescharacteristic
181C2×Dic107C2×D411yescharacteristic
182D428—22no—
183D428—21no—
184D428C2×D412yes—
185C4×D107—24no—
186C4×D107—21no—
187C4×D107—21no—
188C4×D107—81no—
189C4×D107C2×D412yes—
190C4×D107C2×D412yes—
191Dic214—21no—
192Dic214—22no—
193Dic214—21no—
194Dic214—21no—
195Dic214C2×D412yes—
196C8×C107—21no—
197C8×C107—21no—
198C107⋊C8—21no—
199C107⋊C8—21no—
200C22×C107—22no—
201C22×C107—22no—
202C22×C107—21no—
203C22×C107—21no—
204C22×C107—21no—
205C22×C107—41no—
206C22×C107—41no—
207C22×C107—41no—
208C22×C107—41no—
209C22×C107—41no—
210C22×C107—41no—
211C22×C107—41no—
212C22×C107—84no—
213C22×C107C22×D411yescharacteristic
214C2×D107—21no—
215C2×D107—21no—
216C2×D107—21no—
217C2×D107—22no—
218C2×D107—21no—
219C2×D107—21no—
220C2×D107—41no—
221C2×D107—41no—
222C2×D107—84no—
223C2×D107—84no—
224C2×D107#12012yes—
225C4×C107—21no—
226C4×C107—41no—
227C4×C107C22×D411yescharacteristic
228C4×C107C22×D411yescharacteristic
229Dic107—21no—
230Dic107—22no—
231Dic107—41no—
232Dic107—41no—
233Dic107#12012yes—
234C2×C107—42no—
235C2×C107—42no—
236C2×C107—41no—
237C2×C107—81no—
238C2×C107C2×#12011yescharacteristic
239C2×C107#57611yescharacteristic
240C2×C107#57711yescharacteristic
241D107—22no—
242D107—41no—
243D107—42no—
244#3012—1071nomaximal, Sylow
245C107#301211yescharacteristic, complemented, Sylow
246C2×H(5,𝔽2)—1071no—
247C2×AGL(1,ℤ/8)—1071no—
248C2×#135—1071no—
249#537—1071no—
250#452—1071no—
251#452—1071no—
252#450—1071no—
253#450—1071no—
254#449—1071no—
255#449—1071no—
256#447—1071no—
257#447D10711yescharacteristic, complemented
258#417—1071no—
259C2×#102—1071no—
260#408—1071no—
261H(5,𝔽2)—2144no—
262H(5,𝔽2)—2144no—
263C2×#41—1071no—
264C2×#41—1071no—
265C2×#41—1071no—
266C2×#41—1071no—
267C2×#41—1071no—
268C2×#41—2141no—
269C22×D4—1071no—
270C22×D4—1071no—
271C22×D4—2141no—
272C22×D4—2141no—
273C22×D4—2141no—
274C22×D4C2×D10711yescharacteristic
275AGL(1,ℤ/8)—2144no—
276#135—2144no—
277C2×Q16—1071no—
278C2×SD16—1071no—
279C2×SD16—1071no—
280C2×SD16—1071no—
281C2×SD16—1071no—
282C2×D8—1071no—
283C2×D8C2×D10711yescharacteristic, complemented
284C2×D8C2×D10711yescharacteristic, complemented
285C2×M16—1071no—
286C22×C8—1071no—
287C22⋊Q8—1071no—
288C22⋊Q8—1071no—
289C22⋊Q8—2141no—
290#121—1071no—
291#121—2141no—
292#121—2141no—
293#121—2141no—
294#121C2×D10711yescharacteristic, complemented
295C2×C4⋊C4—1071no—
296#103—1071no—
297#103—1071no—
298#102—1071no—
299#102—1071no—
300#102—1071no—
301#102—1071no—
302#102C2×D10711yescharacteristic, complemented
303#102C2×D10711yescharacteristic, complemented
304C22⋊C8—1071no—
305C22⋊C8—1071no—
306C22⋊C8—1071no—
307C22⋊C8C2×D10711yescharacteristic, complemented
308C24—21no—
309C24—2141no—
310C24—2141no—
311#41—1074no—
312#41—2142no—
313#41—2142no—
314#41—2142no—
315#41—2142no—
316#41—2142no—
317#41—2142no—
318#41—2142no—
319#41—2142no—
320#41—2148no—
321#41—2142no—
322C2×Q8—1071no—
323C2×Q8—1071no—
324C2×Q8—1071no—
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—21no—
328C2×D4—21no—
329C2×D4—1074no—
330C2×D4—1071no—
331C2×D4—1071no—
332C2×D4—1071no—
333C2×D4—1071no—
334C2×D4—2142no—
335C2×D4—2141no—
336C2×D4—2141no—
337C2×D4—2142no—
338C2×D4—2148no—
339C2×D4—2141no—
340C2×D4—2141no—
341C2×D4—2141no—
342C2×D4—2141no—
343C2×D4—2148no—
344C2×D4—2148no—
345C2×D4—2141no—
346C2×D4—2142no—
347C2×D4C22×D10711yescharacteristic
348C2×D4C22×D10711yescharacteristic
349C2×D4C22×D10711yescharacteristic
350C22×C4—1071no—
351C22×C4—1071no—
352C22×C4—2141no—
353C22×C4—2141no—
354C22×C4—2141no—
355C22×C4—2141no—
356C22×C4—2141no—
357C22×C4C22×D10711yescharacteristic
358Q16—2142no—
359SD16—2142no—
360SD16—2142no—
361SD16—2142no—
362SD16—2142no—
363D8—22no—
364D8—22no—
365D8—2142no—
366M16—2142no—
367C2×C8—1071no—
368C2×C8—1071no—
369C2×C8—2142no—
370C2×C8C22×D10711yescharacteristic
371C2×C8C22×D10711yescharacteristic
372C4⋊C4—1071no—
373C4⋊C4—2141no—
374C4⋊C4—2141no—
375C4⋊C4—2141no—
376C4⋊C4C22×D10711yescharacteristic
377C22⋊C4—21no—
378C22⋊C4—2141no—
379C22⋊C4—2141no—
380C22⋊C4—2141no—
381C23—24no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—81no—
386C23—81no—
387C23—8561no—
388C23—8561no—
389C23—8561no—
390C23—8561no—
391C23—1071no—
392C23—1071no—
393C23—2144no—
394C23—2144no—
395C23—2141no—
396C23D4×D10711yescharacteristic
397Q8—1072no—
398Q8—2142no—
399Q8—2141no—
400Q8—2141no—
401Q8—2141no—
402D4—24no—
403D4—21no—
404D4—21no—
405D4—21no—
406D4—21no—
407D4—81no—
408D4—8561no—
409D4—1072no—
410D4—4282no—
411D4—2142no—
412D4—2142no—
413D4—2144no—
414D4—2142no—
415D4—2144no—
416D4—2144no—
417D4—2144no—
418D4—2144no—
419D4—2141no—
420D4D4×D10712yes—
421D4D4×D10712yes—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—8561no—
426C2×C4—1071no—
427C2×C4—1072no—
428C2×C4—1072no—
429C2×C4—1071no—
430C2×C4—4282no—
431C2×C4—2144no—
432C2×C4—2144no—
433C2×C4—2141no—
434C2×C4—2141no—
435C2×C4—2144no—
436C2×C4—2141no—
437C2×C4—2141no—
438C2×C4—2141no—
439C2×C4D4×D10711yescharacteristic
440C2×C4C23×D10711yescharacteristic
441C8—21no—
442C8—21no—
443C8—2141no—
444C8—2141no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22—22no—
449C22—22no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—41no—
454C22—41no—
455C22—41no—
456C22—41no—
457C22—84no—
458C22—8564no—
459C22—8564no—
460C22—1072no—
461C22—4281no—
462C22—4281no—
463C22—2141no—
464C22—2141no—
465C22—2142no—
466C22—2141no—
467C22—2141no—
468C22—2141no—
469C22C2×D4×D10711yescharacteristic
470C4—21no—
471C4—41no—
472C4—1072no—
473C4—4281no—
474C4—4281no—
475C4—2141no—
476C4—2142no—
477C4C2×D4×D10711yescharacteristic
478C4C2×D4×D10711yescharacteristic
479C2—41no—
480C2—42no—
481C2—42no—
482C2—81no—
483C2—4282no—
484C2—4281no—
485C2—2142no—
486C2#120×D10711yescharacteristic
487C2#2029342811yescharacteristic
488C2#2029343711yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#447×C107C2
socleC22×C107C22×D4
hypercenter#447D107
hypocenterC107#3012
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. C2×#16812154C2semidirect
2. C2×#16812133C2semidirect
3. C2×#16812128C2semidirect
4. #20293206C2semidirect
5. #20292791C2semidirect
6. #20292790C2semidirect
7. #20292779C2semidirect
8. #20292778C2semidirect
9. #20292770C2semidirect
10. #20292769C2semidirect
11. #20292759C2semidirect
12. #20292625C2semidirect
13. #102×D107C2semidirect
14. #20292592C2semidirect
15. #447×C107C2semidirect
23. C2×#13133817C22semidirect
31. C2×Dic428C22semidirect
32. C2×#13133807C22semidirect
33. C2×#13133806C22semidirect
34. C2×#13133806C22semidirect
35. C2×#13133806C22semidirect
36. C2×#13133803C22semidirect
37. C2×#13133801C22semidirect
38. C2×C8×D107C22semidirect
39. #16812089C22semidirect
40. #16812088C22semidirect
45. #16812080C22semidirect
46. C4⋊C4×D107C22semidirect
47. #16812034C22semidirect
48. #16812032C22semidirect
49. #16812031C22semidirect
50. #16812030C22semidirect
51. #16812030C22semidirect
52. #16812029C22semidirect
53. C22⋊(C107⋊C8)C22semidirect
54. (C2×D107)⋊C8C22semidirect
55. (C2×D107)⋊C8C22semidirect
57. C2×D8×C107C22semidirect
58. C2×D8×C107C22semidirect
59. #121×C107C22semidirect
60. #102×C107C22semidirect
61. #102×C107C22semidirect
62. C22⋊C8×C107C22semidirect
75. #13133817D4semidirect
96. D4×D107D4semidirect
245. C107#3012semidirect
257. #447D107semidirect
283. C2×D8C2×D107semidirect
284. C2×D8C2×D107semidirect
294. #121C2×D107semidirect
302. #102C2×D107semidirect
303. #102C2×D107semidirect
307. C22⋊C8C2×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121542
C2×#168121332
C2×#168121282
#202932062
#202927912
#202927902
#202927792
#202927782
#202927702
#202927692
#202927592
#202926252
#102×D1072
#202925922
#447×C1072
#3012107