Finite Group Databasebeta

← #28348280

Subgroups — #28348280

Order 13696.

462 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293335C211yesmaximal, characteristic, complemented
2#20293261C211yesmaximal, characteristic
3#20292972C211yesmaximal, characteristic, complemented
4#20292947C211yesmaximal, characteristic, complemented
5#20292943C211yesmaximal, characteristic, complemented
6#20292916C211yesmaximal, characteristic, complemented
7#20292859C211yesmaximal, characteristic, complemented
8#20292825C211yesmaximal, characteristic, complemented
9#20292777C211yesmaximal, characteristic, complemented
10#20292776C211yesmaximal, characteristic, complemented
11#20292712C211yesmaximal, characteristic, complemented
12C8×C22⋊D107C211yesmaximal, characteristic, complemented
13C2×#16812031C211yesmaximal, characteristic, complemented
14#102×D107C211yesmaximal, characteristic, complemented
15#465×C107C211yesmaximal, characteristic, complemented
16C22×D428C2211yescharacteristic
17C2×D4×D107—21no—
18C2×D4×D107C2211yescharacteristic
19C2×#13133805C2211yescharacteristic
20C2×#13133805C2211yescharacteristic, complemented
21C2×C8×D107C2211yescharacteristic, complemented
22#16812108C2211yescharacteristic, complemented
23#16812105C2211yescharacteristic
24#16812099—21no—
25#16812089—21no—
26#16812086—21no—
27#16812085C2211yescharacteristic, complemented
28#16812084C2211yescharacteristic
29#16812082—21no—
30#16812079—21no—
31#16812077—21no—
32Q8×Dic107C2211yescharacteristic, complemented
33#16812071C2211yescharacteristic
34C4×C22⋊D107C2211yescharacteristic
35(C22⋊D107)⋊C4—21no—
36C2×C4⋊Dic107C2211yescharacteristic, complemented
37C4⋊C4×D107C2211yescharacteristic, complemented
38#16812040C2211yescharacteristic, complemented
39#16812040C2211yescharacteristic, complemented
40Dic107⋊C8C2211yescharacteristic, complemented
41#16812031C2211yescharacteristic
42#16812031C2211yescharacteristic
43#16812031C2211yescharacteristic, complemented
44#16812031C2211yescharacteristic, complemented
45#16812030C2211yescharacteristic, complemented
46#16812030C2211yescharacteristic, complemented
47#16812029C2211yescharacteristic
48#16812029C2211yescharacteristic, complemented
49C22⋊(C107⋊C8)C2211yescharacteristic, complemented
50(C2×D107)⋊C8C2211yescharacteristic, complemented
51C8×Dic107C2211yescharacteristic, complemented
52C2×SD16×C107C2211yescharacteristic, complemented
53C22×C8×C107C2211yescharacteristic, complemented
54C22⋊Q8×C107C2211yescharacteristic, complemented
55#121×C107C2211yescharacteristic
56#106×C107C2211yescharacteristic, complemented
57#103×C107C2211yescharacteristic, complemented
58#102×C107C2211yescharacteristic, complemented
59C23×D107—21no—
60C23×D107—21no—
61C2×Dic214—21no—
62C2×D428—22no—
63C2×D428—21no—
64C2×D428—21no—
65C2×D428—41no—
66C2×D428C2311yescharacteristic
67C2×D428C2311yescharacteristic
68C2×C22⋊D107—21no—
69C2×C22⋊D107—21no—
70C2×C22⋊D107—21no—
71C2×C22⋊D107D411yescharacteristic
72D4×D107—22no—
73D4×D107—41no—
74D4×D107—44no—
75C22×Dic107—21no—
76C2×C4×D107—21no—
77C2×C4×D107—21no—
78C2×C4×D107C2311yescharacteristic
79#13133805—22no—
80#13133805—41no—
81C2×C107⋊C8C2311yescharacteristic
82C2×C107⋊C8C2311yescharacteristic
83C8×D107—41no—
84C4⋊Dic107—21no—
85C4⋊Dic107—21no—
86C4⋊Dic107C2311yescharacteristic
87C4⋊Dic107C2311yescharacteristic
88Dic107⋊C4—21no—
89Dic107⋊C4—21no—
90Dic107⋊C4—21no—
91Dic107⋊C4D411yescharacteristic
92C22⋊Dic107D411yescharacteristic
93(C2×D107)⋊C4—21no—
94(C2×D107)⋊C4—21no—
95(C2×D107)⋊C4—21no—
96(C2×D107)⋊C4D411yescharacteristic
97C4×Dic107—21no—
98C4×Dic107C2311yescharacteristic
99C2×Q8×C107C2311yescharacteristic
100C2×D4×C107—21no—
101C2×D4×C107C2311yescharacteristic
102C22×C4×C107C2311yescharacteristic
103SD16×C107—41no—
104C2×C8×C107—41no—
105C2×C8×C107C2311yescharacteristic
106C2×C8×C107C2311yescharacteristic
107C4⋊C4×C107—21no—
108C4⋊C4×C107C2311yescharacteristic
109C4⋊C4×C107C2311yescharacteristic
110C22⋊C4×C107—21no—
111C22⋊C4×C107—21no—
112C23×C107—21no—
113C23×C107C2×D411yescharacteristic
114C22×D107—21no—
115C22×D107—21no—
116C22×D107—42no—
117C22×D107—42no—
118C22×D107—42no—
119C22×D107—42no—
120C22×D107—81no—
121C22×D107—81no—
122C22×D107C2×D411yescharacteristic
123Q8×C107—21no—
124Q8×C107—21no—
125D4×C107—21no—
126D4×C107—21no—
127D4×C107—42no—
128C2×C4×C107—21no—
129C2×C4×C107—21no—
130C2×C4×C107—21no—
131C2×C4×C107—21no—
132C2×C4×C107—21no—
133C2×C4×C107C2×D411yescharacteristic
134C2×C4×C107C2411yescharacteristic
135C22⋊D107—42no—
136C22⋊D107—42no—
137C22⋊D107—81no—
138C22⋊D107SD1614yescomplemented
139C2×Dic107—21no—
140C2×Dic107—21no—
141C2×Dic107—81no—
142C2×Dic107C2×D411yescharacteristic
143C2×Dic107C2×D411yescharacteristic
144C2×Dic107C2×D411yescharacteristic
145D428—21no—
146D428—21no—
147D428—21no—
148D428—42no—
149D428—42no—
150D428#4112yes—
151C4×D107—21no—
152C4×D107—21no—
153C4×D107—42no—
154C4×D107—81no—
155Dic214—81no—
156C8×C107—21no—
157C8×C107—21no—
158C107⋊C8—21no—
159C107⋊C8—21no—
160C22×C107—21no—
161C22×C107—21no—
162C22×C107—41no—
163C22×C107—41no—
164C22×C107—41no—
165C22×C107C2×SD1612yes—
166C22×C107C22×D411yescharacteristic
167C2×D107—21no—
168C2×D107—21no—
169C2×D107—22no—
170C2×D107—41no—
171C2×D107—41no—
172C2×D107—41no—
173C2×D107—41no—
174C2×D107—41no—
175C2×D107—84no—
176C2×D107—84no—
177C2×D107C2×SD1612yes—
178C4×C107—21no—
179C4×C107—41no—
180C4×C107—41no—
181C4×C107—41no—
182C4×C107C2×#4111yescharacteristic
183C4×C107H(5,𝔽2)11yescharacteristic
184Dic107—21no—
185Dic107—21no—
186Dic107—41no—
187Dic107—41no—
188Dic107C2×SD1612yes—
189C2×C107—22no—
190C2×C107—81no—
191C2×C107#54611yescharacteristic
192C2×C107C22×SD1611yescharacteristic
193C2×C107#57611yescharacteristic
194D107—22no—
195D107—42no—
196D107—81no—
197#3295—1071nomaximal, Sylow
198C107#329511yescharacteristic, complemented, Sylow
199#554—1071no—
200D42—1071no—
201#486—1071no—
202#481—1071no—
203#481—1071no—
204#476—1071no—
205#465—1071no—
206#465D10711yescharacteristic, complemented
207#460—1071no—
208#450—1071no—
209#450—1071no—
210C4×SD16—1071no—
211C8×D4—1071no—
212C2×#102—1071no—
213C2×#102—1071no—
214C22×D4—1071no—
215C22×D4—1071no—
216C22×D4—2141no—
217C2×SD16—1071no—
218C2×SD16—1071no—
219C2×SD16C2×D10711yescharacteristic, complemented
220C22×C8—1071no—
221C22×C8C2×D10711yescharacteristic, complemented
222C4⋊Q8—1071no—
223#127—1071no—
224#125—2141no—
225C22⋊Q8—1071no—
226C22⋊Q8—2141no—
227C22⋊Q8—2141no—
228C22⋊Q8C2×D10711yescharacteristic, complemented
229#121—1071no—
230#121—2141no—
231#121C2×D10711yescharacteristic
232#120—2141no—
233#120—2141no—
234C4×Q8—1071no—
235C4×D4—1071no—
236C4×D4—1071no—
237C4×D4—2141no—
238C2×C4⋊C4—1071no—
239C2×C4⋊C4—1071no—
240#106—1071no—
241#106—1071no—
242#106C2×D10711yescharacteristic, complemented
243C4⋊C8—1071no—
244#103C2×D10711yescharacteristic, complemented
245#102—1071no—
246#102—1071no—
247#102—1071no—
248#102—1071no—
249#102—1071no—
250#102—1071no—
251#102—1071no—
252#102—1071no—
253#102C2×D10711yescharacteristic, complemented
254C22⋊C8—1071no—
255C22⋊C8—1071no—
256C4×C8—1071no—
257C24—2141no—
258C24—2141no—
259C2×Q8—2141no—
260C2×Q8C22×D10711yescharacteristic
261C2×D4—21no—
262C2×D4—1071no—
263C2×D4—1071no—
264C2×D4—1071no—
265C2×D4—4281no—
266C2×D4—4284no—
267C2×D4—4281no—
268C2×D4—2142no—
269C2×D4—2141no—
270C2×D4—2141no—
271C2×D4—2141no—
272C2×D4—2141no—
273C2×D4—2142no—
274C2×D4—2141no—
275C2×D4C22×D10711yescharacteristic
276C22×C4—1071no—
277C22×C4—2141no—
278C22×C4—2141no—
279C22×C4—2141no—
280C22×C4C22×D10711yescharacteristic
281SD16—41no—
282SD16—4281no—
283SD16—2142no—
284C2×C8—41no—
285C2×C8—1071no—
286C2×C8—1071no—
287C2×C8—4281no—
288C2×C8C22×D10711yescharacteristic
289C2×C8C22×D10711yescharacteristic
290C4⋊C4—21no—
291C4⋊C4—1071no—
292C4⋊C4—1071no—
293C4⋊C4—1071no—
294C4⋊C4—2141no—
295C4⋊C4—2141no—
296C4⋊C4—2141no—
297C4⋊C4—2141no—
298C4⋊C4—2141no—
299C4⋊C4C22×D10711yescharacteristic
300C4⋊C4C22×D10711yescharacteristic
301C22⋊C4—21no—
302C22⋊C4—21no—
303C22⋊C4—1071no—
304C22⋊C4—1071no—
305C22⋊C4—2141no—
306C22⋊C4—2141no—
307C22⋊C4—2141no—
308C42—1071no—
309C42—2141no—
310C23—21no—
311C23—8561no—
312C23—8561no—
313C23—1071no—
314C23—4282no—
315C23—4282no—
316C23—4282no—
317C23—4282no—
318C23—2141no—
319C23—2141no—
320C23D4×D10711yescharacteristic
321Q8—21no—
322Q8—21no—
323Q8—8561no—
324D4—21no—
325D4—21no—
326D4—42no—
327D4—8561no—
328D4—1074no—
329D4—1072no—
330D4—4282no—
331D4—4282no—
332D4—4282no—
333D4—4282no—
334D4—2141no—
335D4—2141no—
336D4—2141no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—8561no—
343C2×C4—8561no—
344C2×C4—1071no—
345C2×C4—1071no—
346C2×C4—1071no—
347C2×C4—4282no—
348C2×C4—2141no—
349C2×C4—2141no—
350C2×C4—2141no—
351C2×C4—2141no—
352C2×C4D4×D10711yescharacteristic
353C2×C4C23×D10711yescharacteristic
354C8—21no—
355C8—21no—
356C8—2141no—
357C8—2141no—
358C22—21no—
359C22—21no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—8564no—
364C22—8564no—
365C22—1072no—
366C22—4281no—
367C22—4281no—
368C22—4281no—
369C22—4281no—
370C22—4281no—
371C22—2142no—
372C22—2141no—
373C22—2141no—
374C22SD16×D10712yes—
375C22C2×D4×D10711yescharacteristic
376C4—21no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—1072no—
381C4—4281no—
382C4—4281no—
383C4—2141no—
384C4—2141no—
385C4#41×D10711yescharacteristic
386C4#1681215411yescharacteristic
387C2—22no—
388C2—81no—
389C2—8561no—
390C2—4282no—
391C2—2142no—
392C2#2029326611yescharacteristic
393C2C2×SD16×D10711yescharacteristic
394C2#2029342911yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#465×C107C2
socleC22×C107C22×D4
prime radical#20293261C2
hypercenter#465D107
hypocenterC107#3295
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. #20293335C2semidirect
3. #20292972C2semidirect
4. #20292947C2semidirect
5. #20292943C2semidirect
6. #20292916C2semidirect
7. #20292859C2semidirect
8. #20292825C2semidirect
9. #20292777C2semidirect
10. #20292776C2semidirect
11. #20292712C2semidirect
12. C8×C22⋊D107C2semidirect
13. C2×#16812031C2semidirect
14. #102×D107C2semidirect
15. #465×C107C2semidirect
20. C2×#13133805C22semidirect
21. C2×C8×D107C22semidirect
22. #16812108C22semidirect
27. #16812085C22semidirect
32. Q8×Dic107C22semidirect
36. C2×C4⋊Dic107C22semidirect
37. C4⋊C4×D107C22semidirect
38. #16812040C22semidirect
39. #16812040C22semidirect
40. Dic107⋊C8C22semidirect
43. #16812031C22semidirect
44. #16812031C22semidirect
45. #16812030C22semidirect
46. #16812030C22semidirect
48. #16812029C22semidirect
49. C22⋊(C107⋊C8)C22semidirect
50. (C2×D107)⋊C8C22semidirect
51. C8×Dic107C22semidirect
52. C2×SD16×C107C22semidirect
53. C22×C8×C107C22semidirect
54. C22⋊Q8×C107C22semidirect
56. #106×C107C22semidirect
57. #103×C107C22semidirect
58. #102×C107C22semidirect
138. C22⋊D107SD16semidirect
198. C107#3295semidirect
206. #465D107semidirect
219. C2×SD16C2×D107semidirect
221. C22×C8C2×D107semidirect
228. C22⋊Q8C2×D107semidirect
242. #106C2×D107semidirect
244. #103C2×D107semidirect
253. #102C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202933352
#202932612
#202929722
#202929472
#202929432
#202929162
#202928592
#202928252
#202927772
#202927762
#202927122
C8×C22⋊D1072
C2×#168120312
#102×D1072
#465×C1072
#3295107