Finite Group Databasebeta

← #28347959

Subgroups — #28347959

Order 13696.

546 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812134C211yesmaximal, characteristic, complemented
2C2×SD16×D107C211yesmaximal, characteristic, complemented
3#20293284C211yesmaximal, characteristic, complemented
4#20293261C211yesmaximal, characteristic, complemented
5#20293015C211yesmaximal, characteristic, complemented
6#20292871C211yesmaximal, characteristic, complemented
7#20292862C211yesmaximal, characteristic, complemented
8#20292841C211yesmaximal, characteristic, complemented
9#20292825C211yesmaximal, characteristic, complemented
10#20292780C211yesmaximal, characteristic, complemented
11#20292767C211yesmaximal, characteristic, complemented
12#20292759C211yesmaximal, characteristic, complemented
13#20292730C211yesmaximal, characteristic, complemented
14#20292701C211yesmaximal, characteristic, complemented
15#449×C107C211yesmaximal, characteristic, complemented
16C2×Q8×D107C2211yescharacteristic, complemented
17C22×D428C2211yescharacteristic
18C2×D4×D107—21no—
19C2×D4×D107C2211yescharacteristic
20#16812134—24no—
21C2×#13133807C2211yescharacteristic
22C2×#13133807C2211yescharacteristic, complemented
23C2×#13133806C2211yescharacteristic, complemented
24C2×#13133805C2211yescharacteristic
25C2×#13133805C2211yescharacteristic, complemented
26SD16×D107—24no—
27C2×#13133803C2211yescharacteristic
28C2×#13133803C2211yescharacteristic, complemented
29C2×#13133802C2211yescharacteristic, complemented
30C2×C8×D107C2211yescharacteristic, complemented
31#16812105C2211yescharacteristic
32#16812096—21no—
33#16812096C2211yescharacteristic, complemented
34#16812088—21no—
35#16812088C2211yescharacteristic, complemented
36#16812084C2211yescharacteristic
37#16812082—21no—
38#16812080—21no—
39#16812080C2211yescharacteristic, complemented
40#16812079—21no—
41#16812077—21no—
42Q8×Dic107C2211yescharacteristic, complemented
43#16812071C2211yescharacteristic
44C4×C22⋊D107—21no—
45C4×C22⋊D107C2211yescharacteristic
46#16812042C2211yescharacteristic, complemented
47Dic107⋊C8C2211yescharacteristic, complemented
48#16812032C2211yescharacteristic, complemented
49#16812031C2211yescharacteristic
50#16812031C2211yescharacteristic, complemented
51#16812030C2211yescharacteristic, complemented
52(C2×D107)⋊C8C2211yescharacteristic, complemented
53#16812019C2211yescharacteristic, complemented
54C2×#41×C107C2211yescharacteristic, complemented
55C2×SD16×C107C2211yescharacteristic, complemented
56C2×D8×C107C2211yescharacteristic, complemented
57#121×C107C2211yescharacteristic
58#103×C107C2211yescharacteristic, complemented
59#102×C107C2211yescharacteristic, complemented
60C22⋊C8×C107C2211yescharacteristic, complemented
61C23×D107—21no—
62C23×D107—21no—
63C2×Dic214—21no—
64C2×Dic214C2311yescharacteristic
65Q8×D107—22no—
66Q8×D107—41no—
67C2×D428—22no—
68C2×D428—21no—
69C2×D428—21no—
70C2×D428—41no—
71C2×D428C2311yescharacteristic
72C2×D428C2311yescharacteristic
73C2×C22⋊D107—21no—
74C2×C22⋊D107—21no—
75C2×C22⋊D107—21no—
76C2×C22⋊D107—21no—
77C2×C22⋊D107D411yescharacteristic
78D4×D107—22no—
79D4×D107—41no—
80D4×D107—44no—
81C2×C4×D107—21no—
82C2×C4×D107—21no—
83C2×C4×D107C2311yescharacteristic
84#13133807—22no—
85#13133807—41no—
86#13133806—22no—
87#13133805—22no—
88#13133805D414yescomplemented
89#13133803—22no—
90#13133803—22no—
91#13133802—22no—
92C2×C107⋊C8C2311yescharacteristic
93C2×C107⋊C8C2311yescharacteristic
94C8×D107—22no—
95C4⋊Dic107—21no—
96C4⋊Dic107C2311yescharacteristic
97Dic107⋊C4—21no—
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)⋊C4D411yescharacteristic
107C4×Dic107—21no—
108C4×Dic107C2311yescharacteristic
109#41×C107—22no—
110#41×C107—41no—
111C2×Q8×C107C2311yescharacteristic
112C2×D4×C107—21no—
113C2×D4×C107—21no—
114C2×D4×C107C2311yescharacteristic
115C2×D4×C107C2311yescharacteristic
116C22×C4×C107—21no—
117C22×C4×C107C2311yescharacteristic
118SD16×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×C107C2×D411yescharacteristic
127C22×D107—21no—
128C22×D107—21no—
129C22×D107—42no—
130C22×D107—42no—
131C22×D107—42no—
132C22×D107—42no—
133C22×D107—81no—
134C22×D107—81no—
135C22×D107C2×D411yescharacteristic
136Q8×C107—21no—
137Q8×C107C2×D412yescomplemented
138D4×C107—21no—
139D4×C107—21no—
140D4×C107—21no—
141D4×C107—21no—
142D4×C107—42no—
143D4×C107—42no—
144C2×C4×C107—21no—
145C2×C4×C107—21no—
146C2×C4×C107—21no—
147C2×C4×C107—21no—
148C2×C4×C107—42no—
149C2×C4×C107C2×D411yescharacteristic
150C2×C4×C107C2411yescharacteristic
151C22⋊D107—22no—
152C22⋊D107—42no—
153C22⋊D107—42no—
154C22⋊D107—42no—
155C22⋊D107—81no—
156C2×Dic107—21no—
157C2×Dic107—21no—
158C2×Dic107C2×D411yescharacteristic
159C2×Dic107C2×D411yescharacteristic
160C2×Dic107C2×D411yescharacteristic
161D428—21no—
162D428—21no—
163D428—21no—
164D428—42no—
165D428—42no—
166D428C2×D412yes—
167C4×D107—21no—
168C4×D107—21no—
169C4×D107—42no—
170C4×D107—42no—
171Dic214—21no—
172Dic214—21no—
173Dic214—42no—
174C8×C107—21no—
175C8×C107—21no—
176C107⋊C8—21no—
177C107⋊C8C2×D412yescomplemented
178C22×C107—21no—
179C22×C107—21no—
180C22×C107—21no—
181C22×C107—41no—
182C22×C107—41no—
183C22×C107—41no—
184C22×C107—41no—
185C22×C107—41no—
186C22×C107—41no—
187C22×C107C22×D411yescharacteristic
188C2×D107—21no—
189C2×D107—21no—
190C2×D107—22no—
191C2×D107—41no—
192C2×D107—41no—
193C2×D107—41no—
194C2×D107—41no—
195C2×D107—41no—
196C2×D107—84no—
197C2×D107—84no—
198C2×D107AGL(1,ℤ/8)12yescomplemented
199C4×C107—22no—
200C4×C107—21no—
201C4×C107—41no—
202C4×C107C22×D411yescharacteristic
203C4×C107H(5,𝔽2)11yescharacteristic
204Dic107—21no—
205Dic107—21no—
206Dic107—21no—
207Dic107—41no—
208Dic107—41no—
209C2×C107—41no—
210C2×C107—81no—
211C2×C107—81no—
212C2×C107D4211yescharacteristic
213C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
214C2×C107#57711yescharacteristic
215D107—22no—
216D107—42no—
217D107—81no—
218#3265—1071nomaximal, Sylow
219C107#326511yescharacteristic, complemented, Sylow
220C2×AGL(1,ℤ/8)—1071no—
221C22×SD16—1071no—
222#548—1071no—
223D42—1071no—
224#496—1071no—
225#468—1071no—
226#466—1071no—
227#463—1071no—
228#460—1071no—
229#450—1071no—
230#449D10711yescharacteristic, complemented
231#448—1071no—
232#447—1071no—
233#440—1071no—
234#435—1071no—
235C2×#41C2×D10711yescharacteristic, complemented
236C22×Q8—1071no—
237C22×D4—1071no—
238C22×D4—1071no—
239C22×D4—2141no—
240AGL(1,ℤ/8)—2144no—
241C2×SD16—1071no—
242C2×SD16—1071no—
243C2×SD16—1071no—
244C2×SD16—1071no—
245C2×SD16—1071no—
246C2×SD16—2144no—
247C2×SD16C2×D10711yescharacteristic, complemented
248C2×D8—1071no—
249C2×D8—1071no—
250C2×D8C2×D10711yescharacteristic, complemented
251C2×M16—1071no—
252C22×C8—1071no—
253#127—1071no—
254#124—1071no—
255#124—2141no—
256C22⋊Q8—1071no—
257C22⋊Q8—2141no—
258#121—1071no—
259#121—1071no—
260#121—2141no—
261#121—2141no—
262#121C2×D10711yescharacteristic
263#120—2141no—
264#120—2141no—
265C4×Q8—1071no—
266C4×D4—1071no—
267C4×D4—1071no—
268C4×D4—2141no—
269#107—1071no—
270C4⋊C8—1071no—
271#103—1071no—
272#103C2×D10711yescharacteristic, complemented
273#102—1071no—
274#102—1071no—
275#102—1071no—
276#102C2×D10711yescharacteristic, complemented
277C22⋊C8—1071no—
278C22⋊C8C2×D10711yescharacteristic, complemented
279#97—1071no—
280C24—2141no—
281C24—2141no—
282#41—22no—
283#41—41no—
284C2×Q8—1071no—
285C2×Q8—4281no—
286C2×Q8—2141no—
287C2×Q8—2142no—
288C2×Q8C22×D10711yescharacteristic
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—1071no—
292C2×D4—1071no—
293C2×D4—1071no—
294C2×D4—4281no—
295C2×D4—4284no—
296C2×D4—4281no—
297C2×D4—2142no—
298C2×D4—2141no—
299C2×D4—2141no—
300C2×D4—2141no—
301C2×D4—2141no—
302C2×D4—2142no—
303C2×D4—2141no—
304C2×D4—2141no—
305C2×D4C22×D10711yescharacteristic
306C2×D4C22×D10711yescharacteristic
307C22×C4—21no—
308C22×C4—1071no—
309C22×C4—2141no—
310C22×C4—2141no—
311C22×C4C22×D10711yescharacteristic
312SD16—22no—
313SD16—1074no—
314SD16—4281no—
315SD16—2142no—
316SD16—2142no—
317SD16—2142no—
318D8—41no—
319D8—2142no—
320D8—2142no—
321M16—2142no—
322C2×C8—1071no—
323C2×C8—1071no—
324C2×C8—2142no—
325C2×C8C22×D10711yescharacteristic
326C2×C8C22×D10711yescharacteristic
327C4⋊C4—1071no—
328C4⋊C4—1071no—
329C4⋊C4—2141no—
330C4⋊C4—2141no—
331C4⋊C4C22×D10711yescharacteristic
332C22⋊C4—21no—
333C22⋊C4—1071no—
334C22⋊C4—1071no—
335C22⋊C4—2141no—
336C22⋊C4—2141no—
337C22⋊C4—2141no—
338C22⋊C4—2141no—
339C22⋊C4—2141no—
340C22⋊C4—2141no—
341C42—1071no—
342C42—2141no—
343C23—21no—
344C23—21no—
345C23—8561no—
346C23—8561no—
347C23—1071no—
348C23—4282no—
349C23—4282no—
350C23—4282no—
351C23—4282no—
352C23—2141no—
353C23—2141no—
354C23D4×D10711yescharacteristic
355Q8—21no—
356Q8—4282no—
357Q8—2141no—
358Q8—2141no—
359Q8D4×D10712yescomplemented
360D4—21no—
361D4—21no—
362D4—21no—
363D4—21no—
364D4—42no—
365D4—42no—
366D4—8561no—
367D4—1072no—
368D4—4282no—
369D4—4282no—
370D4—4282no—
371D4—4282no—
372D4—4282no—
373D4—2142no—
374D4—2141no—
375D4—2141no—
376D4—2141no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—42no—
382C2×C4—1071no—
383C2×C4—1071no—
384C2×C4—1071no—
385C2×C4—4282no—
386C2×C4—4282no—
387C2×C4—2141no—
388C2×C4—2141no—
389C2×C4—2141no—
390C2×C4—2141no—
391C2×C4D4×D10711yescharacteristic
392C2×C4C23×D10711yescharacteristic
393C8—21no—
394C8—21no—
395C8—1072no—
396C8—2141no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—8564no—
407C22—8564no—
408C22—1072no—
409C22—4281no—
410C22—4281no—
411C22—4281no—
412C22—4281no—
413C22—4281no—
414C22—2142no—
415C22—2141no—
416C22—2141no—
417C22C2×D4×D10711yescharacteristic
418C4—21no—
419C4—22no—
420C4—41no—
421C4—4281no—
422C4—4281no—
423C4—2141no—
424C4—2141no—
425C4—2141no—
426C4C2×D4×D10711yescharacteristic
427C4#1681215411yescharacteristic
428C2—41no—
429C2—81no—
430C2—81no—
431C2—8561no—
432C2—4282no—
433C2—2142no—
434C2#2029325911yescharacteristic
435C2AGL(1,ℤ/8)×D10711yescharacteristic
436C2#2029343411yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#449×C107C2
socleC22×C107C22×D4
hypercenter#449D107
hypocenterC107#3265
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×#16812134C2semidirect
2. C2×SD16×D107C2semidirect
3. #20293284C2semidirect
4. #20293261C2semidirect
5. #20293015C2semidirect
6. #20292871C2semidirect
7. #20292862C2semidirect
8. #20292841C2semidirect
9. #20292825C2semidirect
10. #20292780C2semidirect
11. #20292767C2semidirect
12. #20292759C2semidirect
13. #20292730C2semidirect
14. #20292701C2semidirect
15. #449×C107C2semidirect
16. C2×Q8×D107C22semidirect
22. C2×#13133807C22semidirect
23. C2×#13133806C22semidirect
25. C2×#13133805C22semidirect
28. C2×#13133803C22semidirect
29. C2×#13133802C22semidirect
30. C2×C8×D107C22semidirect
33. #16812096C22semidirect
35. #16812088C22semidirect
39. #16812080C22semidirect
42. Q8×Dic107C22semidirect
46. #16812042C22semidirect
47. Dic107⋊C8C22semidirect
48. #16812032C22semidirect
50. #16812031C22semidirect
51. #16812030C22semidirect
52. (C2×D107)⋊C8C22semidirect
53. #16812019C22semidirect
54. C2×#41×C107C22semidirect
55. C2×SD16×C107C22semidirect
56. C2×D8×C107C22semidirect
58. #103×C107C22semidirect
59. #102×C107C22semidirect
60. C22⋊C8×C107C22semidirect
88. #13133805D4semidirect
137. Q8×C107C2×D4semidirect
177. C107⋊C8C2×D4semidirect
198. C2×D107AGL(1,ℤ/8)semidirect
219. C107#3265semidirect
230. #449D107semidirect
235. C2×#41C2×D107semidirect
247. C2×SD16C2×D107semidirect
250. C2×D8C2×D107semidirect
272. #103C2×D107semidirect
276. #102C2×D107semidirect
278. C22⋊C8C2×D107semidirect
359. Q8D4×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121342
C2×SD16×D1072
#202932842
#202932612
#202930152
#202928712
#202928622
#202928412
#202928252
#202927802
#202927672
#202927592
#202927302
#202927012
#449×C1072
#3265107