Finite Group Databasebeta

← #28347943

Subgroups — #28347943

Order 13696.

568 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812133C211yesmaximal, characteristic, complemented
2C2×D8×D107C211yesmaximal, characteristic, complemented
3#20293272C211yesmaximal, characteristic, complemented
4D4×D428C211yesmaximal, characteristic, complemented
5#20293016C211yesmaximal, characteristic, complemented
6#20292881C211yesmaximal, characteristic, complemented
7#20292857C211yesmaximal, characteristic, complemented
8#20292842C211yesmaximal, characteristic, complemented
9#20292822C211yesmaximal, characteristic, complemented
10#20292779C211yesmaximal, characteristic, complemented
11#20292773C211yesmaximal, characteristic, complemented
12#20292762C211yesmaximal, characteristic, complemented
13#20292762C211yesmaximal, characteristic, complemented
14#20292700C211yesmaximal, characteristic, complemented
15#442×C107C211yesmaximal, characteristic, complemented
16C2×#13133817C2211yescharacteristic, complemented
17C22×D428—21no—
18C2×D4×D107C2211yescharacteristic
19C2×D4×D107C2211yescharacteristic, complemented
20C2×D4×D107C2211yescharacteristic, complemented
21#16812133—24no—
22C2×#13133807C2211yescharacteristic, complemented
23C2×#13133807C2211yescharacteristic, complemented
24C2×#13133806C2211yescharacteristic, complemented
25C2×D856C2211yescharacteristic, complemented
26C2×D856C2211yescharacteristic, complemented
27C2×#13133803C2211yescharacteristic
28C2×#13133803C2211yescharacteristic, complemented
29C2×#13133803C2211yescharacteristic, complemented
30D8×D107—24no—
31C2×#13133801C2211yescharacteristic, complemented
32C2×C8×D107C2211yescharacteristic, complemented
33#16812106C2211yescharacteristic, complemented
34#16812097C2211yescharacteristic, complemented
35#16812093—21no—
36#16812089C2211yescharacteristic, complemented
37#16812084C2211yescharacteristic
38#16812084C2211yescharacteristic, complemented
39#16812084C2211yescharacteristic, complemented
40#16812083—21no—
41#16812083—21no—
42#16812079—21no—
43#16812079—21no—
44#16812079—21no—
45C4×D428C2211yescharacteristic
46C2×(C2×D107)⋊C4—21no—
47C4⋊(C107⋊C8)C2211yescharacteristic, complemented
48#16812032C2211yescharacteristic, complemented
49#16812029C2211yescharacteristic, complemented
50#16812029C2211yescharacteristic, complemented
51#16812029C2211yescharacteristic, complemented
52(C2×D107)⋊C8C2211yescharacteristic, complemented
53(C2×D107)⋊C8C2211yescharacteristic, complemented
54C2×D8×C107C2211yescharacteristic
55C4×D4×C107C2211yescharacteristic, complemented
56C4×D4×C107C2211yescharacteristic, complemented
57#106×C107C2211yescharacteristic, complemented
58#102×C107C2211yescharacteristic, complemented
59#102×C107C2211yescharacteristic, complemented
60#97×C107C2211yescharacteristic, complemented
61C23×D107—21no—
62C23×D107—21no—
63C23×D107—21no—
64#13133817—22no—
65#13133817—41no—
66C2×Dic214C2311yescharacteristic
67C2×D428—21no—
68C2×D428—21no—
69C2×D428—21no—
70C2×D428—21no—
71C2×D428—21no—
72C2×D428—44no—
73C2×D428D411yescharacteristic
74C2×D428C2311yescharacteristic
75C2×D428C2311yescharacteristic
76C2×D428C2311yescharacteristic
77C2×C22⋊D107—21no—
78C2×C22⋊D107—21no—
79C2×C22⋊D107—21no—
80C2×C22⋊D107—21no—
81D4×D107—22no—
82D4×D107—22no—
83D4×D107—22no—
84D4×D107—41no—
85D4×D107—41no—
86D4×D107—41no—
87C22×Dic107—21no—
88C2×C4×D107C2311yescharacteristic
89C2×C4×D107C2311yescharacteristic
90#13133807—22no—
91#13133807—41no—
92#13133806—22no—
93D856—22no—
94D856—41no—
95#13133803—22no—
96#13133803—22no—
97#13133803—22no—
98#13133801—22no—
99C2×C107⋊C8C2311yescharacteristic
100C2×C107⋊C8C2311yescharacteristic
101C8×D107—22no—
102C4⋊Dic107—21no—
103C4⋊Dic107D411yescharacteristic
104(C2×D107)⋊C4—21no—
105(C2×D107)⋊C4—21no—
106(C2×D107)⋊C4—21no—
107(C2×D107)⋊C4—21no—
108(C2×D107)⋊C4—21no—
109(C2×D107)⋊C4—21no—
110(C2×D107)⋊C4D411yescharacteristic
111(C2×D107)⋊C4D411yescharacteristic
112C2×D4×C107C2311yescharacteristic
113C2×D4×C107C2311yescharacteristic
114C22×C4×C107—21no—
115C22×C4×C107—21no—
116D8×C107D414yescomplemented
117C2×C8×C107C2311yescharacteristic
118C2×C8×C107C2311yescharacteristic
119C4⋊C4×C107C2311yescharacteristic
120C4⋊C4×C107C2311yescharacteristic
121C22⋊C4×C107—21no—
122C22⋊C4×C107—21no—
123C42×C107C2311yescharacteristic
124C23×C107—21no—
125C23×C107—21no—
126C22×D107—21no—
127C22×D107—21no—
128C22×D107—21no—
129C22×D107—42no—
130C22×D107—42no—
131C22×D107—42no—
132C22×D107—42no—
133C22×D107—42no—
134C22×D107—42no—
135C22×D107—81no—
136C22×D107—81no—
137C22×D107—81no—
138C22×D107C2×D411yescharacteristic
139C22×D107C2×D411yescharacteristic
140D4×C107—21no—
141D4×C107—21no—
142D4×C107C2×D412yes—
143D4×C107C2×D412yescomplemented
144C2×C4×C107—21no—
145C2×C4×C107—21no—
146C2×C4×C107—42no—
147C2×C4×C107—81no—
148C2×C4×C107C2×D411yescharacteristic
149C2×C4×C107C2×D411yescharacteristic
150C2×C4×C107C2411yescharacteristic
151C22⋊D107—42no—
152C22⋊D107—42no—
153C22⋊D107—42no—
154C22⋊D107—42no—
155C2×Dic107—21no—
156C2×Dic107—42no—
157C2×Dic107C2×D411yescharacteristic
158C2×Dic107C2×D411yescharacteristic
159D428—22no—
160D428—21no—
161D428—21no—
162D428—21no—
163D428—21no—
164D428—21no—
165D428—21no—
166D428—42no—
167D428—42no—
168D428—42no—
169D428—81no—
170D428—81no—
171C4×D107—21no—
172C4×D107—21no—
173C4×D107—21no—
174C4×D107—21no—
175Dic214—21no—
176Dic214—21no—
177C8×C107—21no—
178C8×C107C2×D412yescomplemented
179C107⋊C8—21no—
180C107⋊C8—21no—
181C22×C107—22no—
182C22×C107—22no—
183C22×C107—41no—
184C22×C107—41no—
185C22×C107—41no—
186C22×C107—41no—
187C22×C107C22×D411yescharacteristic
188C2×D107—21no—
189C2×D107—21no—
190C2×D107—21no—
191C2×D107—21no—
192C2×D107—21no—
193C2×D107—41no—
194C2×D107—41no—
195C2×D107—41no—
196C2×D107—41no—
197C2×D107—41no—
198C2×D107—41no—
199C2×D107—41no—
200C2×D107—41no—
201C2×D107—41no—
202C2×D107—84no—
203C2×D107—84no—
204C2×D107—84no—
205C2×D107AGL(1,ℤ/8)12yescomplemented
206C4×C107—21no—
207C4×C107—21no—
208C4×C107—41no—
209C4×C107—41no—
210C4×C107C22×D411yescharacteristic
211C4×C107H(5,𝔽2)11yescharacteristic
212Dic107—21no—
213Dic107—21no—
214Dic107—41no—
215C2×C107—42no—
216C2×C107—42no—
217C2×C107D4211yescharacteristic
218C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
219C2×C107#57611yescharacteristic
220D107—22no—
221D107—41no—
222D107—81no—
223D107—81no—
224D107—81no—
225#3264—1071nomaximal, Sylow
226C107#326411yescharacteristic, complemented, Sylow
227C2×AGL(1,ℤ/8)—1071no—
228C22×D8—1071no—
229#546—1071no—
230D42—1071no—
231#496—1071no—
232#469—1071no—
233#465—1071no—
234#463—1071no—
235#459—1071no—
236#450—1071no—
237#449—1071no—
238#447—1071no—
239#447—1071no—
240#442D10711yescharacteristic, complemented
241#435—1071no—
242C2×#41—1071no—
243C22×D4—1071no—
244C22×D4—1071no—
245C22×D4—1071no—
246C22×D4—2141no—
247AGL(1,ℤ/8)—2144no—
248C2×SD16—1071no—
249C2×SD16—1071no—
250C2×SD16—1071no—
251C2×D8—1071no—
252C2×D8—1071no—
253C2×D8—1071no—
254C2×D8—1071no—
255C2×D8—1071no—
256C2×D8—2144no—
257C2×D8C2×D10711yescharacteristic
258C2×M16—1071no—
259C22×C8—1071no—
260#127—1071no—
261#124—1071no—
262#123—2141no—
263C22⋊Q8—1071no—
264#121—1071no—
265#121—1071no—
266#121—1071no—
267#121—2141no—
268#121—2141no—
269#120—2141no—
270#120—2141no—
271#120—2141no—
272C4×D4—1071no—
273C4×D4C2×D10711yescharacteristic, complemented
274C4×D4C2×D10711yescharacteristic, complemented
275C2×C22⋊C4—2141no—
276#106C2×D10711yescharacteristic, complemented
277C4⋊C8—1071no—
278#103—1071no—
279#102—1071no—
280#102—1071no—
281#102—1071no—
282#102C2×D10711yescharacteristic, complemented
283#102C2×D10711yescharacteristic, complemented
284C22⋊C8—1071no—
285C22⋊C8—1071no—
286#97C2×D10711yescharacteristic, complemented
287C24—2141no—
288C24—2141no—
289C24—2141no—
290#41—4281no—
291#41—2142no—
292C2×Q8—1071no—
293C2×D4—1071no—
294C2×D4—1071no—
295C2×D4—1071no—
296C2×D4—1071no—
297C2×D4—4284no—
298C2×D4—4281no—
299C2×D4—4281no—
300C2×D4—4281no—
301C2×D4—2141no—
302C2×D4—2141no—
303C2×D4—2141no—
304C2×D4—2141no—
305C2×D4—2141no—
306C2×D4—2141no—
307C2×D4—2141no—
308C2×D4—2142no—
309C2×D4—2142no—
310C2×D4—2142no—
311C2×D4—2141no—
312C2×D4—2141no—
313C2×D4C22×D10711yescharacteristic
314C2×D4C22×D10711yescharacteristic
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—1071no—
318C22×C4—1071no—
319C22×C4—2141no—
320SD16—4281no—
321SD16—2142no—
322SD16—2142no—
323D8—4281no—
324D8—2142no—
325D8—2142no—
326D8—2142no—
327D8—2142no—
328D8D42814yescomplemented
329M16—2142no—
330C2×C8—1071no—
331C2×C8—1071no—
332C2×C8—2142no—
333C2×C8C22×D10711yescharacteristic
334C2×C8C22×D10711yescharacteristic
335C4⋊C4—1071no—
336C4⋊C4—2141no—
337C4⋊C4C22×D10711yescharacteristic
338C4⋊C4C22×D10711yescharacteristic
339C22⋊C4—21no—
340C22⋊C4—21no—
341C22⋊C4—1071no—
342C22⋊C4—1071no—
343C22⋊C4—2141no—
344C22⋊C4—2141no—
345C22⋊C4—2141no—
346C22⋊C4—2141no—
347C22⋊C4—2141no—
348C22⋊C4—2141no—
349C42C22×D10711yescharacteristic
350C23—21no—
351C23—21no—
352C23—8561no—
353C23—8561no—
354C23—8561no—
355C23—1071no—
356C23—1071no—
357C23—4282no—
358C23—4282no—
359C23—4282no—
360C23—4282no—
361C23—4282no—
362C23—4282no—
363C23—2141no—
364C23—2141no—
365C23—2141no—
366Q8—2141no—
367Q8—2141no—
368D4—21no—
369D4—21no—
370D4—8561no—
371D4—8561no—
372D4—4282no—
373D4—4282no—
374D4—4282no—
375D4—4282no—
376D4—4282no—
377D4—4282no—
378D4—4282no—
379D4—2142no—
380D4—2141no—
381D4—2141no—
382D4—2141no—
383D4—2141no—
384D4—2141no—
385D4—2141no—
386D4C2×D42812yes—
387D4C2×D42812yescomplemented
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—42no—
391C2×C4—81no—
392C2×C4—1071no—
393C2×C4—1071no—
394C2×C4—4282no—
395C2×C4—2141no—
396C2×C4—2141no—
397C2×C4—2141no—
398C2×C4—2141no—
399C2×C4—2141no—
400C2×C4D4×D10711yescharacteristic
401C2×C4D4×D10711yescharacteristic
402C2×C4C23×D10711yescharacteristic
403C8—21no—
404C8—2141no—
405C8—2141no—
406C8C2×D42812yescomplemented
407C22—22no—
408C22—22no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—8564no—
414C22—8564no—
415C22—8564no—
416C22—1072no—
417C22—4281no—
418C22—4281no—
419C22—4281no—
420C22—4281no—
421C22—4281no—
422C22—4281no—
423C22—4281no—
424C22—4281no—
425C22—4281no—
426C22—2141no—
427C22—2141no—
428C22—2141no—
429C22—2141no—
430C22—2141no—
431C22C2×D4×D10711yescharacteristic
432C4—21no—
433C4—21no—
434C4—41no—
435C4—41no—
436C4—4281no—
437C4—2141no—
438C4—2141no—
439C4C22×D42811yescharacteristic
440C4#1681215511yescharacteristic
441C2—42no—
442C2—42no—
443C2—8561no—
444C2—8561no—
445C2—8561no—
446C2—4281no—
447C2—2142no—
448C2D4×D42811yescharacteristic
449C2AGL(1,ℤ/8)×D10711yescharacteristic
450C2#2029343111yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#442×C107C2
socleC22×C107C22×D4
hypercenter#442D107
hypocenterC107#3264
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #442

Complements

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

NH ≅ G/Ntype
1. C2×#16812133C2semidirect
2. C2×D8×D107C2semidirect
3. #20293272C2semidirect
4. D4×D428C2semidirect
5. #20293016C2semidirect
6. #20292881C2semidirect
7. #20292857C2semidirect
8. #20292842C2semidirect
9. #20292822C2semidirect
10. #20292779C2semidirect
11. #20292773C2semidirect
12. #20292762C2semidirect
13. #20292762C2semidirect
14. #20292700C2semidirect
15. #442×C107C2semidirect
16. C2×#13133817C22semidirect
19. C2×D4×D107C22semidirect
20. C2×D4×D107C22semidirect
22. C2×#13133807C22semidirect
23. C2×#13133807C22semidirect
24. C2×#13133806C22semidirect
25. C2×D856C22semidirect
26. C2×D856C22semidirect
28. C2×#13133803C22semidirect
29. C2×#13133803C22semidirect
31. C2×#13133801C22semidirect
32. C2×C8×D107C22semidirect
33. #16812106C22semidirect
34. #16812097C22semidirect
36. #16812089C22semidirect
38. #16812084C22semidirect
39. #16812084C22semidirect
47. C4⋊(C107⋊C8)C22semidirect
48. #16812032C22semidirect
49. #16812029C22semidirect
50. #16812029C22semidirect
51. #16812029C22semidirect
52. (C2×D107)⋊C8C22semidirect
53. (C2×D107)⋊C8C22semidirect
55. C4×D4×C107C22semidirect
56. C4×D4×C107C22semidirect
57. #106×C107C22semidirect
58. #102×C107C22semidirect
59. #102×C107C22semidirect
60. #97×C107C22semidirect
116. D8×C107D4semidirect
143. D4×C107C2×D4semidirect
178. C8×C107C2×D4semidirect
205. C2×D107AGL(1,ℤ/8)semidirect
226. C107#3264semidirect
240. #442D107semidirect
273. C4×D4C2×D107semidirect
274. C4×D4C2×D107semidirect
276. #106C2×D107semidirect
282. #102C2×D107semidirect
283. #102C2×D107semidirect
286. #97C2×D107semidirect
328. D8D428semidirect
387. D4C2×D428semidirect
406. C8C2×D428semidirect

Maximal subgroups

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

subgroupindex
C2×#168121332
C2×D8×D1072
#202932722
D4×D4282
#202930162
#202928812
#202928572
#202928422
#202928222
#202927792
#202927732
#202927622
#202927002
#442×C1072
#3264107