Finite Group Databasebeta

← #28348074

Subgroups — #28348074

Order 13696.

556 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812136C211yesmaximal, characteristic, complemented
2C2×#16812134C211yesmaximal, characteristic, complemented
3#20293296C211yesmaximal, characteristic, complemented
4#20293261C211yesmaximal, characteristic, complemented
5#20293002C211yesmaximal, characteristic, complemented
6#20292879C211yesmaximal, characteristic, complemented
7#20292878C211yesmaximal, characteristic, complemented
8#20292825C211yesmaximal, characteristic, complemented
9#20292821C211yesmaximal, characteristic, complemented
10#20292775C211yesmaximal, characteristic, complemented
11#20292763C211yesmaximal, characteristic, complemented
12#20292759C211yesmaximal, characteristic, complemented
13#20292712C211yesmaximal, characteristic, complemented
14#20292702C211yesmaximal, characteristic, complemented
15#449×C107C211yesmaximal, characteristic, complemented
16C2×#13133819C2211yescharacteristic, complemented
17C22×D428C2211yescharacteristic
18C2×D4×D107—21no—
19C2×D4×D107C2211yescharacteristic
20#16812136—24no—
21#16812134—24no—
22C2×#13133805C2211yescharacteristic
23C2×#13133805C2211yescharacteristic, complemented
24C2×D856C2211yescharacteristic
25C2×D856C2211yescharacteristic, complemented
26C2×#13133803C2211yescharacteristic
27C2×#13133803C2211yescharacteristic, complemented
28C2×#13133803C2211yescharacteristic, complemented
29C2×#13133802C2211yescharacteristic, complemented
30C2×#13133801C2211yescharacteristic, complemented
31#16812105—21no—
32#16812105C2211yescharacteristic
33#16812105C2211yescharacteristic, complemented
34#16812084C2211yescharacteristic
35#16812083—21no—
36#16812083C2211yescharacteristic, complemented
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#16812040C2211yescharacteristic, complemented
47Dic107⋊C8C2211yescharacteristic, complemented
48#16812031C2211yescharacteristic
49#16812031C2211yescharacteristic, complemented
50#16812030C2211yescharacteristic, complemented
51#16812029C2211yescharacteristic, complemented
52(C2×D107)⋊C8C2211yescharacteristic, complemented
53C8×Dic107C2211yescharacteristic, 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—
63#13133819—22no—
64#13133819—41no—
65C2×D428—22no—
66C2×D428—21no—
67C2×D428—21no—
68C2×D428—21no—
69C2×D428—41no—
70C2×D428C2311yescharacteristic
71C2×D428C2311yescharacteristic
72C2×D428C2311yescharacteristic
73C2×C22⋊D107—21no—
74C2×C22⋊D107—21no—
75C2×C22⋊D107—21no—
76C2×C22⋊D107—21no—
77C2×C22⋊D107—21no—
78C2×C22⋊D107—21no—
79C2×C22⋊D107—21no—
80C2×C22⋊D107D411yescharacteristic
81D4×D107—22no—
82D4×D107—41no—
83D4×D107—44no—
84C2×C4×D107—21no—
85C2×C4×D107—21no—
86C2×C4×D107C2311yescharacteristic
87#13133805—22no—
88#13133805D414yescomplemented
89D856—22no—
90D856—41no—
91#13133803—22no—
92#13133803—22no—
93#13133803—22no—
94#13133802—22no—
95#13133801—22no—
96C2×C107⋊C8C2311yescharacteristic
97C2×C107⋊C8C2311yescharacteristic
98C4⋊Dic107—21no—
99C4⋊Dic107C2311yescharacteristic
100Dic107⋊C4D411yescharacteristic
101C22⋊Dic107—21no—
102C22⋊Dic107D411yescharacteristic
103(C2×D107)⋊C4—21no—
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—21no—
130C22×D107—42no—
131C22×D107—42no—
132C22×D107—42no—
133C22×D107—42no—
134C22×D107—81no—
135C22×D107—81no—
136C22×D107C2×D411yescharacteristic
137Q8×C107—21no—
138Q8×C107C2×D412yescomplemented
139D4×C107—21no—
140D4×C107—21no—
141D4×C107—21no—
142D4×C107—21no—
143D4×C107—42no—
144D4×C107—42no—
145C2×C4×C107—21no—
146C2×C4×C107—21no—
147C2×C4×C107—21no—
148C2×C4×C107—21no—
149C2×C4×C107—42no—
150C2×C4×C107C2×D411yescharacteristic
151C2×C4×C107C2411yescharacteristic
152C22⋊D107—22no—
153C22⋊D107—42no—
154C22⋊D107—42no—
155C22⋊D107—42no—
156C22⋊D107—81no—
157C22⋊D107—81no—
158C22⋊D107—81no—
159C22⋊D107—81no—
160C2×Dic107—21no—
161C2×Dic107C2×D411yescharacteristic
162C2×Dic107C2×D411yescharacteristic
163C2×Dic107C2×D411yescharacteristic
164D428—21no—
165D428—21no—
166D428—21no—
167D428—21no—
168D428—21no—
169D428—42no—
170D428—42no—
171D428—42no—
172D428C2×D412yes—
173C4×D107—21no—
174C4×D107—21no—
175C4×D107—42no—
176C4×D107—42no—
177C8×C107—21no—
178C8×C107—21no—
179C107⋊C8—21no—
180C107⋊C8C2×D412yescomplemented
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—22no—
193C2×D107—21no—
194C2×D107—21no—
195C2×D107—41no—
196C2×D107—41no—
197C2×D107—41no—
198C2×D107—41no—
199C2×D107—41no—
200C2×D107—41no—
201C2×D107—41no—
202C2×D107—41no—
203C2×D107—84no—
204C2×D107—84no—
205C4×C107—22no—
206C4×C107—21no—
207C4×C107—41no—
208C4×C107C22×D411yescharacteristic
209C4×C107H(5,𝔽2)11yescharacteristic
210Dic107—21no—
211Dic107—21no—
212Dic107—41no—
213Dic107AGL(1,ℤ/8)12yescomplemented
214C2×C107—41no—
215C2×C107—81no—
216C2×C107—81no—
217C2×C107D4211yescharacteristic
218C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
219C2×C107#57611yescharacteristic
220D107—41no—
221D107—42no—
222D107—81no—
223D107—81no—
224#3274—1071nomaximal, Sylow
225C107#327411yescharacteristic, complemented, Sylow
226C2×AGL(1,ℤ/8)—1071no—
227C2×AGL(1,ℤ/8)—1071no—
228#550—1071no—
229D42—1071no—
230#493—1071no—
231#469—1071no—
232#469—1071no—
233#460—1071no—
234#459—1071no—
235#449—1071no—
236#449D10711yescharacteristic, complemented
237#447—1071no—
238#447—1071no—
239C4×SD16—1071no—
240#436—1071no—
241C2×#41—1071no—
242C2×#41C2×D10711yescharacteristic, complemented
243C22×D4—1071no—
244C22×D4—1071no—
245C22×D4—2141no—
246AGL(1,ℤ/8)—2144no—
247AGL(1,ℤ/8)—2144no—
248C2×SD16—1071no—
249C2×SD16—1071no—
250C2×SD16C2×D10711yescharacteristic, complemented
251C2×D8—1071no—
252C2×D8—1071no—
253C2×D8—1071no—
254C2×D8—1071no—
255C2×D8—1071no—
256C2×D8C2×D10711yescharacteristic, complemented
257C2×M16—1071no—
258C2×M16—1071no—
259#127—1071no—
260#127—1071no—
261#127—2141no—
262#121—1071no—
263#121—1071no—
264#121—1071no—
265#121—2141no—
266#121—2141no—
267#121—2141no—
268#121C2×D10711yescharacteristic
269#120—2141no—
270#120—2141no—
271C4×Q8—1071no—
272C4×D4—1071no—
273C4×D4—1071no—
274C4×D4—2141no—
275#106—1071no—
276C4⋊C8—1071no—
277#103C2×D10711yescharacteristic, complemented
278#102—1071no—
279#102—1071no—
280#102—1071no—
281#102—1071no—
282#102C2×D10711yescharacteristic, complemented
283C22⋊C8—1071no—
284C22⋊C8C2×D10711yescharacteristic, complemented
285C4×C8—1071no—
286C24—2141no—
287C24—2141no—
288#41—22no—
289#41—41no—
290#41—4281no—
291#41—2142no—
292C2×Q8C22×D10711yescharacteristic
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—2142no—
303C2×D4—2141no—
304C2×D4—2141no—
305C2×D4—2141no—
306C2×D4—2141no—
307C2×D4—2141no—
308C2×D4—2141no—
309C2×D4—2141no—
310C2×D4—2142no—
311C2×D4—2141no—
312C2×D4—2141no—
313C2×D4—2141no—
314C2×D4C22×D10711yescharacteristic
315C2×D4C22×D10711yescharacteristic
316C22×C4—21no—
317C22×C4—1071no—
318C22×C4—2141no—
319C22×C4—2141no—
320C22×C4C22×D10711yescharacteristic
321SD16—22no—
322SD16—1074no—
323SD16—2142no—
324D8—41no—
325D8—4281no—
326D8—2142no—
327D8—2142no—
328D8—2142no—
329D8—2142no—
330M16—2142no—
331M16—2142no—
332C2×C8—1071no—
333C2×C8—1071no—
334C2×C8C22×D10711yescharacteristic
335C2×C8C22×D10711yescharacteristic
336C4⋊C4—1071no—
337C4⋊C4—1071no—
338C4⋊C4—2141no—
339C4⋊C4C22×D10711yescharacteristic
340C22⋊C4—21no—
341C22⋊C4—1071no—
342C22⋊C4—1071no—
343C22⋊C4—2141no—
344C22⋊C4—2141no—
345C22⋊C4—2141no—
346C22⋊C4—2141no—
347C42—1071no—
348C42—2141no—
349C23—21no—
350C23—21no—
351C23—8561no—
352C23—8561no—
353C23—1071no—
354C23—4282no—
355C23—4282no—
356C23—4282no—
357C23—4282no—
358C23—2141no—
359C23—2141no—
360C23—2141no—
361C23D4×D10711yescharacteristic
362Q8—21no—
363Q8D4×D10712yescomplemented
364D4—21no—
365D4—21no—
366D4—21no—
367D4—21no—
368D4—42no—
369D4—42no—
370D4—8561no—
371D4—8561no—
372D4—8561no—
373D4—8561no—
374D4—1072no—
375D4—4282no—
376D4—4282no—
377D4—4282no—
378D4—4282no—
379D4—4282no—
380D4—4282no—
381D4—2142no—
382D4—2141no—
383D4—2141no—
384D4—2141no—
385D4—2141no—
386D4—2141no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—42no—
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×C4D4×D10711yescharacteristic
401C2×C4C23×D10711yescharacteristic
402C8—21no—
403C8—21no—
404C8—1072no—
405C8—2141no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—8564no—
416C22—8564no—
417C22—4281no—
418C22—4281no—
419C22—4281no—
420C22—4281no—
421C22—4281no—
422C22—4281no—
423C22—4281no—
424C22—4281no—
425C22—2141no—
426C22—2141no—
427C22—2141no—
428C22—2142no—
429C22C2×D4×D10711yescharacteristic
430C4—21no—
431C4—22no—
432C4—41no—
433C4—1072no—
434C4—4281no—
435C4—2141no—
436C4—2141no—
437C4C2×D4×D10711yescharacteristic
438C4#1681215411yescharacteristic
439C2—41no—
440C2—81no—
441C2—81no—
442C2—8561no—
443C2—8561no—
444C2—4282no—
445C2—4281no—
446C2#2029325911yescharacteristic
447C2AGL(1,ℤ/8)×D10711yescharacteristic
448C2#2029342911yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#449×C107C2
socleC22×C107C22×D4
hypercenter#449D107
hypocenterC107#3274
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×#16812136C2semidirect
2. C2×#16812134C2semidirect
3. #20293296C2semidirect
4. #20293261C2semidirect
5. #20293002C2semidirect
6. #20292879C2semidirect
7. #20292878C2semidirect
8. #20292825C2semidirect
9. #20292821C2semidirect
10. #20292775C2semidirect
11. #20292763C2semidirect
12. #20292759C2semidirect
13. #20292712C2semidirect
14. #20292702C2semidirect
15. #449×C107C2semidirect
16. C2×#13133819C22semidirect
23. C2×#13133805C22semidirect
25. C2×D856C22semidirect
27. C2×#13133803C22semidirect
28. C2×#13133803C22semidirect
29. C2×#13133802C22semidirect
30. C2×#13133801C22semidirect
33. #16812105C22semidirect
36. #16812083C22semidirect
39. #16812080C22semidirect
42. Q8×Dic107C22semidirect
46. #16812040C22semidirect
47. Dic107⋊C8C22semidirect
49. #16812031C22semidirect
50. #16812030C22semidirect
51. #16812029C22semidirect
52. (C2×D107)⋊C8C22semidirect
53. C8×Dic107C22semidirect
54. C2×#41×C107C22semidirect
55. C2×SD16×C107C22semidirect
56. C2×D8×C107C22semidirect
58. #103×C107C22semidirect
59. #102×C107C22semidirect
60. C22⋊C8×C107C22semidirect
88. #13133805D4semidirect
138. Q8×C107C2×D4semidirect
180. C107⋊C8C2×D4semidirect
213. Dic107AGL(1,ℤ/8)semidirect
225. C107#3274semidirect
236. #449D107semidirect
242. C2×#41C2×D107semidirect
250. C2×SD16C2×D107semidirect
256. C2×D8C2×D107semidirect
277. #103C2×D107semidirect
282. #102C2×D107semidirect
284. C22⋊C8C2×D107semidirect
363. Q8D4×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121362
C2×#168121342
#202932962
#202932612
#202930022
#202928792
#202928782
#202928252
#202928212
#202927752
#202927632
#202927592
#202927122
#202927022
#449×C1072
#3274107