Finite Group Databasebeta

← #28347948

Subgroups — #28347948

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×#16812137C211yesmaximal, characteristic, complemented
2C2×D8×D107C211yesmaximal, characteristic, complemented
3#20293274C211yesmaximal, characteristic, complemented
4#20293261C211yesmaximal, characteristic, complemented
5#20293015C211yesmaximal, characteristic, complemented
6#20292857C211yesmaximal, characteristic, complemented
7#20292841C211yesmaximal, characteristic, complemented
8#20292821C211yesmaximal, characteristic, complemented
9#20292779C211yesmaximal, characteristic, complemented
10#20292770C211yesmaximal, characteristic, complemented
11#20292762C211yesmaximal, characteristic, complemented
12#20292760C211yesmaximal, characteristic, complemented
13#20292738C211yesmaximal, characteristic, complemented
14#20292696C211yesmaximal, characteristic, complemented
15#469×C107C211yesmaximal, characteristic, complemented
16C2×#13133818C2211yescharacteristic, complemented
17C22×D428C2211yescharacteristic, complemented
18C2×D4×D107—21no—
19C2×D4×D107C2211yescharacteristic
20C2×D4×D107C2211yescharacteristic, complemented
21#16812137—24no—
22C2×#13133807C2211yescharacteristic, complemented
23C2×#13133807C2211yescharacteristic, complemented
24C2×#13133806C2211yescharacteristic, complemented
25C2×D856C2211yescharacteristic
26C2×D856C2211yescharacteristic, complemented
27C2×#13133803C2211yescharacteristic
28C2×#13133803C2211yescharacteristic, complemented
29C2×#13133803C2211yescharacteristic, complemented
30D8×D107—24no—
31C2×C8×D107C2211yescharacteristic, complemented
32#16812105C2211yescharacteristic, complemented
33#16812096C2211yescharacteristic, complemented
34#16812091—21no—
35#16812089C2211yescharacteristic, complemented
36#16812084C2211yescharacteristic
37#16812082—21no—
38#16812080—21no—
39#16812079—21no—
40#16812077—21no—
41#16812077—21no—
42#16812071C2211yescharacteristic, complemented
43#16812071C2211yescharacteristic, complemented
44C4×C22⋊D107C2211yescharacteristic
45C22⋊C4×D107—21no—
46Dic107⋊C8C2211yescharacteristic, complemented
47#16812032C2211yescharacteristic, complemented
48#16812029C2211yescharacteristic, complemented
49#16812029C2211yescharacteristic, complemented
50#16812029C2211yescharacteristic, complemented
51C22⋊(C107⋊C8)C2211yescharacteristic, complemented
52(C2×D107)⋊C8C2211yescharacteristic, complemented
53#16812019C2211yescharacteristic, complemented
54C2×D8×C107C2211yescharacteristic, complemented
55C2×M16×C107C2211yescharacteristic, complemented
56#121×C107C2211yescharacteristic, complemented
57#121×C107C2211yescharacteristic, complemented
58#106×C107C2211yescharacteristic, complemented
59#102×C107C2211yescharacteristic, complemented
60#102×C107C2211yescharacteristic, complemented
61C23×D107—21no—
62C23×D107—21no—
63C23×D107—21no—
64#13133818—22no—
65#13133818—41no—
66C2×Dic214C2311yescharacteristic
67C2×D428—22no—
68C2×D428—21no—
69C2×D428—21no—
70C2×D428—41no—
71C2×D428C2311yescharacteristic
72C2×D428C2311yescharacteristic
73C2×D428C2311yescharacteristic
74C2×C22⋊D107—21no—
75C2×C22⋊D107—21no—
76C2×C22⋊D107—21no—
77C2×C22⋊D107—21no—
78C2×C22⋊D107—21no—
79C2×C22⋊D107D411yescharacteristic
80D4×D107—22no—
81D4×D107—22no—
82D4×D107—41no—
83D4×D107—41no—
84D4×D107—44no—
85C2×C4×D107—21no—
86C2×C4×D107—21no—
87C2×C4×D107—21no—
88C2×C4×D107C2311yescharacteristic
89#13133807—22no—
90#13133807—22no—
91#13133806—41no—
92D856—22no—
93D856D414yescomplemented
94#13133803—22no—
95#13133803—22no—
96#13133803—41no—
97C2×C107⋊C8C2311yescharacteristic
98C2×C107⋊C8C2311yescharacteristic
99C8×D107—22no—
100C4⋊Dic107—21no—
101Dic107⋊C4D411yescharacteristic
102C22⋊Dic107—21no—
103C22⋊Dic107—21no—
104C22⋊Dic107D411yescharacteristic
105(C2×D107)⋊C4—21no—
106(C2×D107)⋊C4—21no—
107(C2×D107)⋊C4—21no—
108(C2×D107)⋊C4—21no—
109(C2×D107)⋊C4D411yescharacteristic
110C4×Dic107C2311yescharacteristic
111C2×D4×C107—21no—
112C2×D4×C107—21no—
113C2×D4×C107C2311yescharacteristic
114C2×D4×C107C2311yescharacteristic
115C22×C4×C107C2311yescharacteristic
116D8×C107—22no—
117M16×C107—22no—
118C2×C8×C107C2311yescharacteristic
119C2×C8×C107C2311yescharacteristic
120C4⋊C4×C107C2311yescharacteristic
121C4⋊C4×C107C2311yescharacteristic
122C22⋊C4×C107—21no—
123C22⋊C4×C107—21no—
124C23×C107—21no—
125C23×C107—21no—
126C23×C107C2×D411yescharacteristic
127C22×D107—21no—
128C22×D107—21no—
129C22×D107—21no—
130C22×D107—42no—
131C22×D107—42no—
132C22×D107—42no—
133C22×D107—42no—
134C22×D107—42no—
135C22×D107—42no—
136C22×D107—81no—
137C22×D107—81no—
138C22×D107—81no—
139C22×D107C2×D411yescharacteristic
140D4×C107—21no—
141D4×C107—21no—
142D4×C107—21no—
143D4×C107—21no—
144D4×C107—42no—
145D4×C107—81no—
146C2×C4×C107—21no—
147C2×C4×C107—21no—
148C2×C4×C107—21no—
149C2×C4×C107—21no—
150C2×C4×C107C2×D411yescharacteristic
151C2×C4×C107C2411yescharacteristic
152C22⋊D107—22no—
153C22⋊D107—42no—
154C22⋊D107—42no—
155C22⋊D107—42no—
156C22⋊D107—42no—
157C22⋊D107—81no—
158C2×Dic107—21no—
159C2×Dic107C2×D411yescharacteristic
160C2×Dic107C2×D411yescharacteristic
161C2×Dic107C2×D411yescharacteristic
162D428—21no—
163D428—21no—
164D428—21no—
165D428—21no—
166D428—42no—
167D428—42no—
168D428C2×D412yes—
169D428C2×D412yescomplemented
170C4×D107—21no—
171C4×D107—21no—
172C4×D107—42no—
173C4×D107—42no—
174C4×D107—81no—
175Dic214—21no—
176Dic214—21no—
177C8×C107—21no—
178C8×C107C2×D412yescomplemented
179C107⋊C8—21no—
180C107⋊C8—21no—
181C22×C107—21no—
182C22×C107—21no—
183C22×C107—21no—
184C22×C107—41no—
185C22×C107—41no—
186C22×C107—41no—
187C22×C107—41no—
188C22×C107—41no—
189C22×C107—41no—
190C22×C107C22×D411yescharacteristic
191C2×D107—21no—
192C2×D107—21no—
193C2×D107—22no—
194C2×D107—22no—
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—41no—
208C4×C107—41no—
209C4×C107C22×D411yescharacteristic
210C4×C107H(5,𝔽2)11yescharacteristic
211Dic107—21no—
212Dic107—21no—
213Dic107—21no—
214Dic107—41no—
215C2×C107—41no—
216C2×C107—81no—
217C2×C107—81no—
218C2×C107D4211yescharacteristic
219C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
220C2×C107#57611yescharacteristic
221D107—22no—
222D107—42no—
223D107—42no—
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#469D10711yescharacteristic, complemented
233#465—1071no—
234#463—1071no—
235#459—1071no—
236#450—1071no—
237#449—1071no—
238#447—1071no—
239#447—1071no—
240#442—1071no—
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, complemented
258C2×M16C2×D10711yescharacteristic, complemented
259C22×C8—1071no—
260#127—1071no—
261#124—1071no—
262#123—2141no—
263C22⋊Q8—1071no—
264#121—1071no—
265#121—2141no—
266#121—2141no—
267#121C2×D10711yescharacteristic, complemented
268#121C2×D10711yescharacteristic, complemented
269#120—2141no—
270#120—2141no—
271#120—2141no—
272C4×D4—1071no—
273C4×D4—1071no—
274C4×D4—1071no—
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#97—1071no—
287C24—2141no—
288C24—2141no—
289C24—2141no—
290#41—4281no—
291#41—2142no—
292C2×Q8—1071no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—1071no—
296C2×D4—1071no—
297C2×D4—1071no—
298C2×D4—1071no—
299C2×D4—4281no—
300C2×D4—4284no—
301C2×D4—4281no—
302C2×D4—4281no—
303C2×D4—2142no—
304C2×D4—2141no—
305C2×D4—2141no—
306C2×D4—2141no—
307C2×D4—2141no—
308C2×D4—2141no—
309C2×D4—2142no—
310C2×D4—2142no—
311C2×D4—2141no—
312C2×D4—2141no—
313C2×D4C22×D10711yescharacteristic
314C2×D4C22×D10711yescharacteristic
315C22×C4—1071no—
316C22×C4—2141no—
317C22×C4—2141no—
318C22×C4—2141no—
319C22×C4C22×D10711yescharacteristic
320SD16—4281no—
321SD16—2142no—
322SD16—2142no—
323D8—22no—
324D8—1074no—
325D8—4281no—
326D8—2142no—
327D8—2142no—
328D8—2142no—
329M16—22no—
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—
349C42—1071no—
350C23—21no—
351C23—21no—
352C23—8561no—
353C23—8561no—
354C23—8561no—
355C23—1071no—
356C23—4282no—
357C23—4282no—
358C23—4282no—
359C23—4282no—
360C23—4282no—
361C23—4282no—
362C23—2141no—
363C23—2141no—
364C23—2141no—
365C23D4×D10711yescharacteristic
366Q8—2141no—
367Q8—2141no—
368D4—21no—
369D4—21no—
370D4—21no—
371D4—21no—
372D4—42no—
373D4—81no—
374D4—8561no—
375D4—1072no—
376D4—1072no—
377D4—4282no—
378D4—4282no—
379D4—4282no—
380D4—4282no—
381D4—4282no—
382D4—4282no—
383D4—2142no—
384D4—2141no—
385D4—2141no—
386D4—2141no—
387D4—2141no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—8561no—
393C2×C4—1071no—
394C2×C4—1071no—
395C2×C4—1071no—
396C2×C4—4282no—
397C2×C4—4282no—
398C2×C4—2141no—
399C2×C4—2141no—
400C2×C4—2141no—
401C2×C4D4×D10711yescharacteristic
402C2×C4C23×D10711yescharacteristic
403C8—21no—
404C8—2141no—
405C8—2141no—
406C8D4×D10712yescomplemented
407C22—21no—
408C22—21no—
409C22—21no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—8564no—
417C22—8564no—
418C22—8564no—
419C22—1072no—
420C22—4281no—
421C22—4281no—
422C22—4281no—
423C22—4281no—
424C22—4281no—
425C22—4281no—
426C22—4281no—
427C22—2142no—
428C22—2142no—
429C22—2141no—
430C22—2141no—
431C22C2×D4×D10711yescharacteristic
432C4—21no—
433C4—41no—
434C4—41no—
435C4—4281no—
436C4—2141no—
437C4—2141no—
438C4—2141no—
439C4C2×D4×D10711yescharacteristic
440C4#1681215511yescharacteristic
441C2—41no—
442C2—81no—
443C2—81no—
444C2—8561no—
445C2—4282no—
446C2—4282no—
447C2—2142no—
448C2#2029326111yescharacteristic
449C2AGL(1,ℤ/8)×D10711yescharacteristic
450C2#2029343111yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#469×C107C2
socleC22×C107C22×D4
hypercenter#469D107
hypocenterC107#3264
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. C2×#16812137C2semidirect
2. C2×D8×D107C2semidirect
3. #20293274C2semidirect
4. #20293261C2semidirect
5. #20293015C2semidirect
6. #20292857C2semidirect
7. #20292841C2semidirect
8. #20292821C2semidirect
9. #20292779C2semidirect
10. #20292770C2semidirect
11. #20292762C2semidirect
12. #20292760C2semidirect
13. #20292738C2semidirect
14. #20292696C2semidirect
15. #469×C107C2semidirect
16. C2×#13133818C22semidirect
17. C22×D428C22semidirect
20. C2×D4×D107C22semidirect
22. C2×#13133807C22semidirect
23. C2×#13133807C22semidirect
24. C2×#13133806C22semidirect
26. C2×D856C22semidirect
28. C2×#13133803C22semidirect
29. C2×#13133803C22semidirect
31. C2×C8×D107C22semidirect
32. #16812105C22semidirect
33. #16812096C22semidirect
35. #16812089C22semidirect
42. #16812071C22semidirect
43. #16812071C22semidirect
46. Dic107⋊C8C22semidirect
47. #16812032C22semidirect
48. #16812029C22semidirect
49. #16812029C22semidirect
50. #16812029C22semidirect
51. C22⋊(C107⋊C8)C22semidirect
52. (C2×D107)⋊C8C22semidirect
53. #16812019C22semidirect
54. C2×D8×C107C22semidirect
55. C2×M16×C107C22semidirect
56. #121×C107C22semidirect
57. #121×C107C22semidirect
58. #106×C107C22semidirect
59. #102×C107C22semidirect
60. #102×C107C22semidirect
93. D856D4semidirect
169. D428C2×D4semidirect
178. C8×C107C2×D4semidirect
205. C2×D107AGL(1,ℤ/8)semidirect
226. C107#3264semidirect
232. #469D107semidirect
257. C2×D8C2×D107semidirect
258. C2×M16C2×D107semidirect
267. #121C2×D107semidirect
268. #121C2×D107semidirect
276. #106C2×D107semidirect
282. #102C2×D107semidirect
283. #102C2×D107semidirect
406. C8D4×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121372
C2×D8×D1072
#202932742
#202932612
#202930152
#202928572
#202928412
#202928212
#202927792
#202927702
#202927622
#202927602
#202927382
#202926962
#469×C1072
#3264107