Finite Group Databasebeta

← #28347939

Subgroups — #28347939

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×#16812135C211yesmaximal, characteristic, complemented
2C2×D8×D107C211yesmaximal, characteristic, complemented
3#20293274C211yesmaximal, characteristic, complemented
4D4×C22⋊D107C211yesmaximal, characteristic, complemented
5#20293015C211yesmaximal, characteristic, complemented
6#20292878C211yesmaximal, characteristic, complemented
7#20292855C211yesmaximal, characteristic, complemented
8#20292840C211yesmaximal, characteristic, complemented
9#20292821C211yesmaximal, characteristic, complemented
10#20292779C211yesmaximal, characteristic, complemented
11#20292774C211yesmaximal, characteristic, complemented
12#20292759C211yesmaximal, characteristic, complemented
13#20292736C211yesmaximal, characteristic, complemented
14#20292701C211yesmaximal, characteristic, complemented
15#447×C107C211yesmaximal, characteristic, complemented
16C2×#13133818C2211yescharacteristic, complemented
17C22×C22⋊D107—21no—
18C2×D4×D107C2211yescharacteristic
19C2×D4×D107C2211yescharacteristic, complemented
20#16812135—24no—
21C2×#13133807C2211yescharacteristic, complemented
22C2×#13133806C2211yescharacteristic, complemented
23C2×#13133806C2211yescharacteristic, complemented
24C2×D856C2211yescharacteristic, complemented
25C2×#13133803C2211yescharacteristic
26C2×#13133803C2211yescharacteristic, complemented
27C2×#13133803C2211yescharacteristic, complemented
28D8×D107—24no—
29C2×#13133802C2211yescharacteristic, complemented
30C2×C8×D107C2211yescharacteristic, complemented
31#16812105C2211yescharacteristic, complemented
32#16812096C2211yescharacteristic, complemented
33#16812091—21no—
34#16812089C2211yescharacteristic, complemented
35#16812083C2211yescharacteristic, complemented
36#16812081—21no—
37#16812080—21no—
38#16812080C2211yescharacteristic
39#16812078—21no—
40#16812077—21no—
41#16812077—21no—
42D4×Dic107C2211yescharacteristic, complemented
43#16812071C2211yescharacteristic, complemented
44C4×C22⋊D107C2211yescharacteristic
45C22⋊C4×D107—21no—
46#16812040C2211yescharacteristic, complemented
47Dic107⋊C8C2211yescharacteristic, complemented
48#16812034C2211yescharacteristic, complemented
49#16812031C2211yescharacteristic, complemented
50#16812030C2211yescharacteristic, complemented
51#16812029C2211yescharacteristic, complemented
52(C2×D107)⋊C8C2211yescharacteristic, complemented
53#16812019C2211yescharacteristic, complemented
54C22×D4×C107C2211yescharacteristic, complemented
55C2×D8×C107C2211yescharacteristic
56C2×D8×C107C2211yescharacteristic, complemented
57#121×C107C2211yescharacteristic, complemented
58#102×C107C2211yescharacteristic, complemented
59#102×C107C2211yescharacteristic, complemented
60C22⋊C8×C107C2211yescharacteristic, complemented
61C23×D107—21no—
62C23×D107—21no—
63#13133818—22no—
64#13133818—41no—
65C2×Dic214C2311yescharacteristic
66C2×D428C2311yescharacteristic
67C2×D428C2311yescharacteristic
68C2×C22⋊D107—21no—
69C2×C22⋊D107—21no—
70C2×C22⋊D107—21no—
71C2×C22⋊D107—21no—
72C2×C22⋊D107—21no—
73C2×C22⋊D107—21no—
74C2×C22⋊D107—21no—
75C2×C22⋊D107—44no—
76C2×C22⋊D107D411yescharacteristic
77D4×D107—22no—
78D4×D107—22no—
79D4×D107—41no—
80D4×D107—41no—
81C22×Dic107—21no—
82C2×C4×D107—21no—
83C2×C4×D107—21no—
84C2×C4×D107C2311yescharacteristic
85#13133807—41no—
86#13133806—22no—
87#13133806—22no—
88D856—22no—
89#13133803—22no—
90#13133803—22no—
91#13133803D414yescomplemented
92#13133802—22no—
93C2×C107⋊C8C2311yescharacteristic
94C2×C107⋊C8C2311yescharacteristic
95C8×D107—22no—
96C4⋊Dic107—21no—
97C4⋊Dic107C2311yescharacteristic
98Dic107⋊C4D411yescharacteristic
99C22⋊Dic107—21no—
100C22⋊Dic107—21no—
101C22⋊Dic107—21no—
102C22⋊Dic107—21no—
103C22⋊Dic107D411yescharacteristic
104(C2×D107)⋊C4—21no—
105(C2×D107)⋊C4—21no—
106(C2×D107)⋊C4—21no—
107(C2×D107)⋊C4D411yescharacteristic
108C4×Dic107C2311yescharacteristic
109C24×C107—21no—
110C2×D4×C107—22no—
111C2×D4×C107—21no—
112C2×D4×C107—21no—
113C2×D4×C107—41no—
114C2×D4×C107C2311yescharacteristic
115C2×D4×C107C2311yescharacteristic
116C2×D4×C107C2311yescharacteristic
117C22×C4×C107C2311yescharacteristic
118D8×C107—22no—
119D8×C107—41no—
120C2×C8×C107C2311yescharacteristic
121C2×C8×C107C2311yescharacteristic
122C4⋊C4×C107C2311yescharacteristic
123C22⋊C4×C107—21no—
124C23×C107—21no—
125C23×C107—21no—
126C23×C107—21no—
127C23×C107—42no—
128C23×C107—42no—
129C23×C107—81no—
130C23×C107C2×D411yescharacteristic
131C22×D107—21no—
132C22×D107—21no—
133C22×D107—42no—
134C22×D107—42no—
135C22×D107—42no—
136C22×D107—42no—
137C22×D107—81no—
138C22×D107—81no—
139C22×D107C2×D411yescharacteristic
140D4×C107—21no—
141D4×C107—21no—
142D4×C107—21no—
143D4×C107—21no—
144D4×C107—21no—
145D4×C107—42no—
146D4×C107—81no—
147D4×C107C2×D412yes—
148C2×C4×C107—21no—
149C2×C4×C107—21no—
150C2×C4×C107—21no—
151C2×C4×C107C2×D411yescharacteristic
152C2×C4×C107C2411yescharacteristic
153C22⋊D107—22no—
154C22⋊D107—42no—
155C22⋊D107—42no—
156C22⋊D107—42no—
157C22⋊D107—42no—
158C22⋊D107—42no—
159C22⋊D107—42no—
160C22⋊D107—81no—
161C2×Dic107—21no—
162C2×Dic107—21no—
163C2×Dic107—42no—
164C2×Dic107C2×D411yescharacteristic
165C2×Dic107C2×D411yescharacteristic
166C2×Dic107C2×D411yescharacteristic
167D428—21no—
168D428—21no—
169D428—21no—
170D428C2×D412yescomplemented
171C4×D107—21no—
172C4×D107—21no—
173C4×D107—42no—
174C4×D107—81no—
175Dic214—21no—
176Dic214—21no—
177C8×C107—21no—
178C8×C107—21no—
179C107⋊C8—21no—
180C107⋊C8C2×D412yescomplemented
181C22×C107—21no—
182C22×C107—22no—
183C22×C107—21no—
184C22×C107—21no—
185C22×C107—41no—
186C22×C107—41no—
187C22×C107—41no—
188C22×C107—41no—
189C22×C107—41no—
190C22×C107—41no—
191C22×C107—41no—
192C22×C107—41no—
193C22×C107—84no—
194C22×C107C22×D411yescharacteristic
195C2×D107—21no—
196C2×D107—21no—
197C2×D107—22no—
198C2×D107—41no—
199C2×D107—41no—
200C2×D107—41no—
201C2×D107—41no—
202C2×D107—41no—
203C2×D107—84no—
204C2×D107—84no—
205C2×D107AGL(1,ℤ/8)12yescomplemented
206C4×C107—21no—
207C4×C107—41no—
208C4×C107C22×D411yescharacteristic
209C4×C107H(5,𝔽2)11yescharacteristic
210Dic107—21no—
211Dic107—21no—
212Dic107—21no—
213Dic107—41no—
214Dic107—41no—
215C2×C107—42no—
216C2×C107—41no—
217C2×C107—81no—
218C2×C107—81no—
219C2×C107D4211yescharacteristic
220C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
221C2×C107#57611yescharacteristic
222D107—22no—
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#469—1071no—
233#465—1071no—
234#463—1071no—
235#459—1071no—
236#450—1071no—
237#449—1071no—
238#447—1071no—
239#447D10711yescharacteristic, complemented
240#442—1071no—
241#435—1071no—
242C2×#41—1071no—
243C22×D4—1071no—
244C22×D4—1071no—
245C22×D4—2141no—
246C22×D4C2×D10711yescharacteristic, complemented
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—2144no—
256C2×D8C2×D10711yescharacteristic
257C2×D8C2×D10711yescharacteristic, complemented
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—2141no—
267#121—2141no—
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#106—1071no—
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⋊C8C2×D10711yescharacteristic, complemented
286#97—1071no—
287C24—21no—
288C24—2141no—
289C24—2141no—
290#41—4281no—
291#41—2142no—
292C2×Q8—1071no—
293C2×D4—22no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—41no—
297C2×D4—1071no—
298C2×D4—1071no—
299C2×D4—1071no—
300C2×D4—4284no—
301C2×D4—4281no—
302C2×D4—4281no—
303C2×D4—2141no—
304C2×D4—2141no—
305C2×D4—2141no—
306C2×D4—2141no—
307C2×D4—2142no—
308C2×D4—2141no—
309C2×D4—2141no—
310C2×D4—2141no—
311C2×D4—2142no—
312C2×D4C22×D10711yescharacteristic
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—41no—
325D8—1074no—
326D8—2142no—
327D8—2142no—
328D8—2142no—
329M16—2142no—
330C2×C8—1071no—
331C2×C8—1071no—
332C2×C8—2142no—
333C2×C8C22×D10711yescharacteristic
334C2×C8C22×D10711yescharacteristic
335C4⋊C4—1071no—
336C4⋊C4—1071no—
337C4⋊C4—2141no—
338C4⋊C4C22×D10711yescharacteristic
339C22⋊C4—21no—
340C22⋊C4—1071no—
341C22⋊C4—1071no—
342C22⋊C4—2141no—
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—21no—
353C23—42no—
354C23—42no—
355C23—81no—
356C23—8561no—
357C23—8561no—
358C23—1071no—
359C23—4282no—
360C23—4282no—
361C23—4282no—
362C23—4282no—
363C23—2141no—
364C23—2141no—
365C23D4×D10711yescharacteristic
366Q8—2141no—
367Q8—2141no—
368D4—21no—
369D4—21no—
370D4—21no—
371D4—21no—
372D4—21no—
373D4—42no—
374D4—81no—
375D4—8561no—
376D4—1072no—
377D4—4282no—
378D4—4282no—
379D4—4282no—
380D4—4282no—
381D4—4282no—
382D4—4282no—
383D4—2142no—
384D4—2141no—
385D4—2141no—
386D4—2141no—
387D4D4×D10712yes—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—8561no—
392C2×C4—1071no—
393C2×C4—1071no—
394C2×C4—1071no—
395C2×C4—4282no—
396C2×C4—4282no—
397C2×C4—2141no—
398C2×C4—2141no—
399C2×C4—2141no—
400C2×C4—2141no—
401C2×C4D4×D10711yescharacteristic
402C2×C4C23×D10711yescharacteristic
403C8—21no—
404C8—21no—
405C8—1072no—
406C8—2141no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—22no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22—41no—
419C22—84no—
420C22—8564no—
421C22—8564no—
422C22—1072no—
423C22—4281no—
424C22—4281no—
425C22—4281no—
426C22—4281no—
427C22—4281no—
428C22—2142no—
429C22—2141no—
430C22—2141no—
431C22C2×D4×D10711yescharacteristic
432C4—21no—
433C4—41no—
434C4—4281no—
435C4—4281no—
436C4—2141no—
437C4—2141no—
438C4—2141no—
439C4C2×D4×D10711yescharacteristic
440C4#1681215411yescharacteristic
441C2—41no—
442C2—42no—
443C2—81no—
444C2—81no—
445C2—8561no—
446C2—4282no—
447C2—2142no—
448C2#2029325911yescharacteristic
449C2AGL(1,ℤ/8)×D10711yescharacteristic
450C2#2029342811yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#447×C107C2
socleC22×C107C22×D4
hypercenter#447D107
hypocenterC107#3264
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×#16812135C2semidirect
2. C2×D8×D107C2semidirect
3. #20293274C2semidirect
4. D4×C22⋊D107C2semidirect
5. #20293015C2semidirect
6. #20292878C2semidirect
7. #20292855C2semidirect
8. #20292840C2semidirect
9. #20292821C2semidirect
10. #20292779C2semidirect
11. #20292774C2semidirect
12. #20292759C2semidirect
13. #20292736C2semidirect
14. #20292701C2semidirect
15. #447×C107C2semidirect
16. C2×#13133818C22semidirect
19. C2×D4×D107C22semidirect
21. C2×#13133807C22semidirect
22. C2×#13133806C22semidirect
23. C2×#13133806C22semidirect
24. C2×D856C22semidirect
26. C2×#13133803C22semidirect
27. C2×#13133803C22semidirect
29. C2×#13133802C22semidirect
30. C2×C8×D107C22semidirect
31. #16812105C22semidirect
32. #16812096C22semidirect
34. #16812089C22semidirect
35. #16812083C22semidirect
42. D4×Dic107C22semidirect
43. #16812071C22semidirect
46. #16812040C22semidirect
47. Dic107⋊C8C22semidirect
48. #16812034C22semidirect
49. #16812031C22semidirect
50. #16812030C22semidirect
51. #16812029C22semidirect
52. (C2×D107)⋊C8C22semidirect
53. #16812019C22semidirect
54. C22×D4×C107C22semidirect
56. C2×D8×C107C22semidirect
57. #121×C107C22semidirect
58. #102×C107C22semidirect
59. #102×C107C22semidirect
60. C22⋊C8×C107C22semidirect
91. #13133803D4semidirect
170. D428C2×D4semidirect
180. C107⋊C8C2×D4semidirect
205. C2×D107AGL(1,ℤ/8)semidirect
226. C107#3264semidirect
239. #447D107semidirect
246. C22×D4C2×D107semidirect
257. C2×D8C2×D107semidirect
268. #121C2×D107semidirect
282. #102C2×D107semidirect
283. #102C2×D107semidirect
285. C22⋊C8C2×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121352
C2×D8×D1072
#202932742
D4×C22⋊D1072
#202930152
#202928782
#202928552
#202928402
#202928212
#202927792
#202927742
#202927592
#202927362
#202927012
#447×C1072
#3264107