Finite Group Databasebeta

← #28347957

Subgroups — #28347957

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
2C22×#13133805C211yesmaximal, characteristic, complemented
3D4×D428C211yesmaximal, characteristic, complemented
4#20293017C211yesmaximal, characteristic, complemented
5#20292875C211yesmaximal, characteristic, complemented
6#20292861C211yesmaximal, characteristic, complemented
7#20292839C211yesmaximal, characteristic, complemented
8#20292827C211yesmaximal, characteristic, complemented
9#20292777C211yesmaximal, characteristic, complemented
10#20292772C211yesmaximal, characteristic, complemented
11#20292766C211yesmaximal, characteristic, complemented
12#20292760C211yesmaximal, characteristic, complemented
13#20292725C211yesmaximal, characteristic, complemented
14#20292699C211yesmaximal, characteristic, complemented
15#548×C107C211yesmaximal, characteristic, complemented
16C22×D428C2211yescharacteristic
17C22×D428C2211yescharacteristic
18C2×D4×D107—21no—
19#16812134—24no—
20C2×#13133805—24no—
21C2×#13133805C2211yescharacteristic
22C2×#13133805C2211yescharacteristic
23C2×#13133805C2211yescharacteristic, complemented
24C2×#13133805C2211yescharacteristic, complemented
25C2×#13133805C2211yescharacteristic, complemented
26C2×#13133805C2211yescharacteristic, complemented
27C2×#13133803C2211yescharacteristic
28C2×#13133803C2211yescharacteristic, complemented
29C2×#13133803C2211yescharacteristic, complemented
30C2×#13133802C2211yescharacteristic, complemented
31C22×C107⋊C8C2211yescharacteristic, complemented
32#16812106C2211yescharacteristic
33#16812084—21no—
34#16812083C2211yescharacteristic
35#16812083C2211yescharacteristic
36#16812079—21no—
37#16812079—21no—
38C4×D428C2211yescharacteristic
39#16812042C2211yescharacteristic, complemented
40C4⋊(C107⋊C8)C2211yescharacteristic, complemented
41#16812033C2211yescharacteristic, complemented
42#16812033C2211yescharacteristic, complemented
43#16812030C2211yescharacteristic, complemented
44#16812029C2211yescharacteristic
45#16812029C2211yescharacteristic, complemented
46#16812029C2211yescharacteristic, complemented
47C22⋊(C107⋊C8)C2211yescharacteristic, complemented
48C22⋊(C107⋊C8)C2211yescharacteristic, complemented
49#16812020C2211yescharacteristic, complemented
50C2×#41×C107C2211yescharacteristic, complemented
51C22×Q8×C107C2211yescharacteristic, complemented
52#124×C107—21no—
53#124×C107C2211yescharacteristic, complemented
54C22⋊Q8×C107—21no—
55C22⋊Q8×C107C2211yescharacteristic, complemented
56#121×C107—21no—
57#121×C107C2211yescharacteristic, complemented
58C4×Q8×C107C2211yescharacteristic, complemented
59C4×D4×C107—21no—
60C4×D4×C107C2211yescharacteristic
61C23×D107—21no—
62C23×D107—21no—
63C2×D428—22no—
64C2×D428—22no—
65C2×D428—21no—
66C2×D428—21no—
67C2×D428—21no—
68C2×D428—21no—
69C2×D428—41no—
70C2×D428—41no—
71C2×D428C2311yescharacteristic
72C2×D428C2311yescharacteristic
73C2×D428C2311yescharacteristic
74C2×C22⋊D107—21no—
75C2×C22⋊D107—21no—
76D4×D107—44no—
77C2×C4×D107—21no—
78#13133805—22no—
79#13133805—22no—
80#13133805—22no—
81#13133805—22no—
82#13133805—41no—
83#13133805D414yescomplemented
84#13133803—22no—
85#13133803—22no—
86#13133803—41no—
87#13133802—22no—
88C2×C107⋊C8—22no—
89C2×C107⋊C8C2311yescharacteristic
90C2×C107⋊C8C2311yescharacteristic
91C2×C107⋊C8C2311yescharacteristic
92C2×C107⋊C8C2311yescharacteristic
93C4⋊Dic107C2311yescharacteristic
94(C2×D107)⋊C4—21no—
95(C2×D107)⋊C4—21no—
96(C2×D107)⋊C4—21no—
97#41×C107—22no—
98#41×C107—41no—
99C2×Q8×C107—21no—
100C2×Q8×C107—22no—
101C2×Q8×C107—41no—
102C2×Q8×C107C2311yescharacteristic
103C2×Q8×C107C2311yescharacteristic
104C2×D4×C107—21no—
105C2×D4×C107—21no—
106C2×D4×C107D411yescharacteristic
107C2×D4×C107C2311yescharacteristic
108C22×C4×C107—21no—
109C22×C4×C107—21no—
110C22×C4×C107C2311yescharacteristic
111C22×C4×C107C2311yescharacteristic
112C4⋊C4×C107—21no—
113C4⋊C4×C107—21no—
114C4⋊C4×C107D411yescharacteristic
115C4⋊C4×C107C2311yescharacteristic
116C22⋊C4×C107—21no—
117C22⋊C4×C107—21no—
118C22⋊C4×C107—21no—
119C22⋊C4×C107—21no—
120C22⋊C4×C107D411yescharacteristic
121C22⋊C4×C107D411yescharacteristic
122C42×C107—21no—
123C42×C107C2311yescharacteristic
124C23×C107—21no—
125C23×C107C2×D411yescharacteristic
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—81no—
135C22×D107—81no—
136Q8×C107—21no—
137Q8×C107—21no—
138Q8×C107—21no—
139Q8×C107—42no—
140Q8×C107C2×D412yescomplemented
141D4×C107—22no—
142D4×C107—21no—
143D4×C107—21no—
144D4×C107—42no—
145D4×C107—42no—
146C2×C4×C107—21no—
147C2×C4×C107—21no—
148C2×C4×C107—21no—
149C2×C4×C107—21no—
150C2×C4×C107—21no—
151C2×C4×C107—21no—
152C2×C4×C107—21no—
153C2×C4×C107—42no—
154C2×C4×C107—42no—
155C2×C4×C107C2×D411yescharacteristic
156C2×C4×C107C2×D411yescharacteristic
157C2×C4×C107C2×D411yescharacteristic
158C2×C4×C107C2×D411yescharacteristic
159C2×C4×C107C2411yescharacteristic
160C22⋊D107—42no—
161C22⋊D107—42no—
162C2×Dic107—21no—
163D428—21no—
164D428—21no—
165D428—21no—
166D428—21no—
167D428—21no—
168D428—42no—
169D428—42no—
170D428—42no—
171D428—81no—
172D428C2×D412yes—
173C4×D107—42no—
174C107⋊C8—21no—
175C107⋊C8—21no—
176C107⋊C8—21no—
177C107⋊C8C2×D412yescomplemented
178C22×C107—21no—
179C22×C107—21no—
180C22×C107—21no—
181C22×C107—21no—
182C22×C107—21no—
183C22×C107—41no—
184C22×C107—41no—
185C22×C107—41no—
186C22×C107AGL(1,ℤ/8)12yescomplemented
187C22×C107C22×D411yescharacteristic
188C2×D107—22no—
189C2×D107—41no—
190C2×D107—41no—
191C2×D107—41no—
192C2×D107—41no—
193C2×D107—41no—
194C2×D107—41no—
195C2×D107—41no—
196C2×D107—41no—
197C2×D107—84no—
198C2×D107—84no—
199C4×C107—21no—
200C4×C107—21no—
201C4×C107—22no—
202C4×C107—21no—
203C4×C107—21no—
204C4×C107—41no—
205C4×C107—41no—
206C4×C107C22×D411yescharacteristic
207C4×C107H(5,𝔽2)11yescharacteristic
208Dic107—41no—
209C2×C107—22no—
210C2×C107—41no—
211C2×C107—81no—
212C2×C107D4211yescharacteristic
213C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
214C2×C107#57711yescharacteristic
215D107—42no—
216D107—81no—
217D107—81no—
218#3265—1071nomaximal, Sylow
219C107#326511yescharacteristic, complemented, Sylow
220C2×AGL(1,ℤ/8)—1071no—
221C22×SD16—1071no—
222#548D10711yescharacteristic, complemented
223D42—1071no—
224#496—1071no—
225#468—1071no—
226#466—1071no—
227#463—1071no—
228#460—1071no—
229#450—1071no—
230#449—1071no—
231#448—1071no—
232#447—1071no—
233#440—1071no—
234#435—1071no—
235C2×#41C2×D10711yescharacteristic, complemented
236C22×Q8C2×D10711yescharacteristic, complemented
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—1071no—
247C2×SD16—2144no—
248C2×D8—1071no—
249C2×D8—1071no—
250C2×D8—1071no—
251C2×M16—1071no—
252C22×C8—1071no—
253#127—1071no—
254#124—21no—
255#124C2×D10711yescharacteristic, complemented
256C22⋊Q8—21no—
257C22⋊Q8C2×D10711yescharacteristic, complemented
258#121—21no—
259#121—1071no—
260#121—1071no—
261#121—2141no—
262#121C2×D10711yescharacteristic, complemented
263#120—2141no—
264#120—2141no—
265C4×Q8C2×D10711yescharacteristic, complemented
266C4×D4—21no—
267C4×D4—1071no—
268C4×D4C2×D10711yescharacteristic
269#107—1071no—
270C4⋊C8—1071no—
271#103—1071no—
272#103—1071no—
273#102—1071no—
274#102—1071no—
275#102—1071no—
276#102—1071no—
277C22⋊C8—1071no—
278C22⋊C8—1071no—
279#97—1071no—
280C24—2141no—
281C24—2141no—
282#41—22no—
283#41—41no—
284C2×Q8—21no—
285C2×Q8—22no—
286C2×Q8—41no—
287C2×Q8C22×D10711yescharacteristic
288C2×Q8C22×D10711yescharacteristic
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—1071no—
292C2×D4—1071no—
293C2×D4—1071no—
294C2×D4—4281no—
295C2×D4—4281no—
296C2×D4—4284no—
297C2×D4—2142no—
298C2×D4—2142no—
299C2×D4—2141no—
300C2×D4—2141no—
301C2×D4—2141no—
302C2×D4—2141no—
303C2×D4—2141no—
304C2×D4—2141no—
305C2×D4C22⋊D10711yescharacteristic
306C2×D4C22×D10711yescharacteristic
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—2141no—
310C22×C4C22×D10711yescharacteristic
311C22×C4C22×D10711yescharacteristic
312SD16—1074no—
313SD16—4281no—
314SD16—2142no—
315SD16—2142no—
316SD16—2142no—
317SD16—2142no—
318D8—4281no—
319D8—2142no—
320D8—2142no—
321M16—2142no—
322C2×C8—1071no—
323C2×C8—1071no—
324C2×C8—1071no—
325C2×C8—1071no—
326C2×C8—2142no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—1071no—
330C4⋊C4C22⋊D10711yescharacteristic
331C4⋊C4C22×D10711yescharacteristic
332C22⋊C4—21no—
333C22⋊C4—21no—
334C22⋊C4—21no—
335C22⋊C4—21no—
336C22⋊C4—2141no—
337C22⋊C4—2141no—
338C22⋊C4—2141no—
339C22⋊C4C22⋊D10711yescharacteristic
340C22⋊C4C22⋊D10711yescharacteristic
341C42—21no—
342C42C22×D10711yescharacteristic
343C23—21no—
344C23—8561no—
345C23—8561no—
346C23—4282no—
347C23—4282no—
348C23—4282no—
349C23—4282no—
350C23—2141no—
351C23—2141no—
352C23—2141no—
353C23C2×C22⋊D10711yescharacteristic
354C23C2×C22⋊D10711yescharacteristic
355Q8—21no—
356Q8—21no—
357Q8—21no—
358Q8—42no—
359Q8D4×D10712yescomplemented
360D4—22no—
361D4—21no—
362D4—21no—
363D4—42no—
364D4—42no—
365D4—8561no—
366D4—1072no—
367D4—4282no—
368D4—4282no—
369D4—4282no—
370D4—4282no—
371D4—4282no—
372D4—2141no—
373D4—2141no—
374D4—2141no—
375D4—2141no—
376D4—2141no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—4282no—
387C2×C4—2141no—
388C2×C4C2×C22⋊D10711yescharacteristic
389C2×C4C2×C22⋊D10711yescharacteristic
390C2×C4C2×C22⋊D10711yescharacteristic
391C2×C4C2×C22⋊D10711yescharacteristic
392C2×C4C23×D10711yescharacteristic
393C8—1072no—
394C8—2141no—
395C8—2141no—
396C8—2141no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—8564no—
406C22—8564no—
407C22—4281no—
408C22—4281no—
409C22—4281no—
410C22—4281no—
411C22—4281no—
412C22—4281no—
413C22—4281no—
414C22—4281no—
415C22—2142no—
416C22#1681213412yescomplemented
417C22C22×C22⋊D10711yescharacteristic
418C4—21no—
419C4—21no—
420C4—21no—
421C4—21no—
422C4—22no—
423C4—41no—
424C4—41no—
425C4—4281no—
426C4C2×D4×D10711yescharacteristic
427C4#1681215411yescharacteristic
428C2—22no—
429C2—41no—
430C2—81no—
431C2—8561no—
432C2—8561no—
433C2—4282no—
434C2D4×C22⋊D10711yescharacteristic
435C2C2×#1681213411yescharacteristic
436C2#2029343511yescharacteristic
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#548×C107C2
socleC22×C107C22×D4
hypercenter#548D107
hypocenterC107#3265
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C4×D4 ◁ #548

Complements

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

NH ≅ G/Ntype
1. C2×#16812134C2semidirect
2. C22×#13133805C2semidirect
3. D4×D428C2semidirect
4. #20293017C2semidirect
5. #20292875C2semidirect
6. #20292861C2semidirect
7. #20292839C2semidirect
8. #20292827C2semidirect
9. #20292777C2semidirect
10. #20292772C2semidirect
11. #20292766C2semidirect
12. #20292760C2semidirect
13. #20292725C2semidirect
14. #20292699C2semidirect
15. #548×C107C2semidirect
23. C2×#13133805C22semidirect
24. C2×#13133805C22semidirect
25. C2×#13133805C22semidirect
26. C2×#13133805C22semidirect
28. C2×#13133803C22semidirect
29. C2×#13133803C22semidirect
30. C2×#13133802C22semidirect
31. C22×C107⋊C8C22semidirect
39. #16812042C22semidirect
40. C4⋊(C107⋊C8)C22semidirect
41. #16812033C22semidirect
42. #16812033C22semidirect
43. #16812030C22semidirect
45. #16812029C22semidirect
46. #16812029C22semidirect
47. C22⋊(C107⋊C8)C22semidirect
48. C22⋊(C107⋊C8)C22semidirect
49. #16812020C22semidirect
50. C2×#41×C107C22semidirect
51. C22×Q8×C107C22semidirect
53. #124×C107C22semidirect
55. C22⋊Q8×C107C22semidirect
57. #121×C107C22semidirect
58. C4×Q8×C107C22semidirect
83. #13133805D4semidirect
140. Q8×C107C2×D4semidirect
177. C107⋊C8C2×D4semidirect
186. C22×C107AGL(1,ℤ/8)semidirect
219. C107#3265semidirect
222. #548D107semidirect
235. C2×#41C2×D107semidirect
236. C22×Q8C2×D107semidirect
255. #124C2×D107semidirect
257. C22⋊Q8C2×D107semidirect
262. #121C2×D107semidirect
265. C4×Q8C2×D107semidirect
359. Q8D4×D107semidirect
416. C22#16812134semidirect

Maximal subgroups

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

subgroupindex
C2×#168121342
C22×#131338052
D4×D4282
#202930172
#202928752
#202928612
#202928392
#202928272
#202927772
#202927722
#202927662
#202927602
#202927252
#202926992
#548×C1072
#3265107