Finite Group Databasebeta

← #28348293

Subgroups — #28348293

Order 13696.

408 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293315C211yesmaximal, characteristic, complemented
2#20293272C211yesmaximal, characteristic
3#20292984C211yesmaximal, characteristic, complemented
4#20292959C211yesmaximal, characteristic, complemented
5#20292954C211yesmaximal, characteristic, complemented
6#20292869C211yesmaximal, characteristic, complemented
7#20292857C211yesmaximal, characteristic, complemented
8#20292850C211yesmaximal, characteristic, complemented
9#20292790C211yesmaximal, characteristic, complemented
10#20292779C211yesmaximal, characteristic, complemented
11C4×#13133806C211yesmaximal, characteristic, complemented
12#20292692C211yesmaximal, characteristic, complemented
13#20292620C211yesmaximal, characteristic, complemented
14#102×D107C211yesmaximal, characteristic, complemented
15#478×C107C211yesmaximal, characteristic, complemented
16C2×#13133817C2211yescharacteristic
17C2×D4×D107C2211yescharacteristic
18C2×Dic428C2211yescharacteristic, complemented
19C2×#13133807C2211yescharacteristic
20C2×#13133806C2211yescharacteristic
21C2×C8×D107C2211yescharacteristic
22C2×C8×D107C2211yescharacteristic, complemented
23#16812104—21no—
24#16812100—21no—
25#16812097C2211yescharacteristic
26#16812094—21no—
27#16812093—21no—
28#16812089C2211yescharacteristic
29#16812089C2211yescharacteristic
30#16812089C2211yescharacteristic, complemented
31#16812084C2211yescharacteristic
32#16812083—21no—
33#16812079—21no—
34C4×Dic214C2211yescharacteristic, complemented
35C4×D428C2211yescharacteristic
36#16812067—21no—
37#16812067C2211yescharacteristic, complemented
38C4⋊C4×D107C2211yescharacteristic
39C2×(C2×D107)⋊C4—21no—
40#16812040C2211yescharacteristic, complemented
41#16812034C2211yescharacteristic, complemented
42#16812032C2211yescharacteristic, complemented
43#16812032C2211yescharacteristic, complemented
44#16812032C2211yescharacteristic, complemented
45#16812031C2211yescharacteristic
46#16812030C2211yescharacteristic
47#16812029C2211yescharacteristic
48#16812029C2211yescharacteristic
49(C2×D107)⋊C8C2211yescharacteristic
50(C2×D107)⋊C8C2211yescharacteristic, complemented
51C4×C107⋊C8C2211yescharacteristic, complemented
52#125×C107C2211yescharacteristic, complemented
53C4×D4×C107C2211yescharacteristic
54#107×C107C2211yescharacteristic, complemented
55#106×C107C2211yescharacteristic, complemented
56C4⋊C8×C107C2211yescharacteristic, complemented
57#102×C107C2211yescharacteristic
58#102×C107C2211yescharacteristic
59C23×D107—21no—
60#13133817—22no—
61#13133817—41no—
62C2×Dic214C2311yescharacteristic
63C2×Dic214C2311yescharacteristic
64C2×D428—21no—
65C2×D428D411yescharacteristic
66C2×D428C2311yescharacteristic
67C2×C22⋊D107—21no—
68C2×C22⋊D107—21no—
69D4×D107—22no—
70D4×D107—41no—
71C22×Dic107—21no—
72C2×C4×D107—21no—
73C2×C4×D107C2311yescharacteristic
74C2×C4×D107C2311yescharacteristic
75Dic428—41no—
76#13133807—41no—
77#13133806—22no—
78C2×C107⋊C8C2311yescharacteristic
79C2×C107⋊C8C2311yescharacteristic
80C8×D107—41no—
81C8×D107—41no—
82C4⋊Dic107—21no—
83C4⋊Dic107—21no—
84C4⋊Dic107—21no—
85C4⋊Dic107D411yescharacteristic
86C4⋊Dic107C2311yescharacteristic
87Dic107⋊C4—21no—
88Dic107⋊C4—21no—
89(C2×D107)⋊C4—21no—
90(C2×D107)⋊C4—21no—
91(C2×D107)⋊C4—21no—
92(C2×D107)⋊C4—21no—
93(C2×D107)⋊C4—21no—
94(C2×D107)⋊C4—21no—
95(C2×D107)⋊C4—21no—
96(C2×D107)⋊C4D411yescharacteristic
97(C2×D107)⋊C4D411yescharacteristic
98C4×Dic107—21no—
99C4×Dic107—21no—
100C2×D4×C107C2311yescharacteristic
101C22×C4×C107—21no—
102C2×C8×C107C2311yescharacteristic
103C2×C8×C107C2311yescharacteristic
104C4⋊C4×C107—21no—
105C4⋊C4×C107—21no—
106C4⋊C4×C107C2311yescharacteristic
107C4⋊C4×C107C2311yescharacteristic
108C4⋊C4×C107C2311yescharacteristic
109C22⋊C4×C107—21no—
110C42×C107C2311yescharacteristic
111C23×C107—21no—
112C22×D107—21no—
113C22×D107—42no—
114C22×D107—42no—
115C22×D107—81no—
116C22×D107C2×D411yescharacteristic
117C22×D107C2×D411yescharacteristic
118D4×C107—21no—
119D4×C107#4112yes—
120C2×C4×C107—21no—
121C2×C4×C107—21no—
122C2×C4×C107—21no—
123C2×C4×C107—81no—
124C2×C4×C107C2×D411yescharacteristic
125C2×C4×C107C2×D411yescharacteristic
126C2×C4×C107C2411yescharacteristic
127C22⋊D107—42no—
128C22⋊D107—42no—
129C2×Dic107—21no—
130C2×Dic107—21no—
131C2×Dic107—21no—
132C2×Dic107—42no—
133C2×Dic107C2×D411yescharacteristic
134C2×Dic107C2×D411yescharacteristic
135D428—22no—
136D428—21no—
137D428—21no—
138D428—81no—
139C4×D107—21no—
140C4×D107—21no—
141C4×D107—21no—
142C4×D107—21no—
143C4×D107—42no—
144Dic214—21no—
145Dic214—21no—
146Dic214—21no—
147Dic214—21no—
148C8×C107—21no—
149C8×C107—21no—
150C107⋊C8—21no—
151C107⋊C8—21no—
152C22×C107—22no—
153C22×C107—41no—
154C22×C107—41no—
155C22×C107C22×D411yescharacteristic
156C2×D107—21no—
157C2×D107—21no—
158C2×D107—21no—
159C2×D107—21no—
160C2×D107—21no—
161C2×D107—41no—
162C2×D107—41no—
163C2×D107—41no—
164C2×D107—84no—
165C2×D107#13512yes—
166C4×C107—21no—
167C4×C107—21no—
168C4×C107—41no—
169C4×C107—41no—
170C4×C107—41no—
171C4×C107C2×#4111yescharacteristic
172C4×C107H(5,𝔽2)11yescharacteristic
173Dic107—21no—
174Dic107—21no—
175Dic107—41no—
176Dic107—41no—
177Dic107—41no—
178C2×C107—42no—
179C2×C107#54611yescharacteristic
180C2×C107C2×#13511yescharacteristic
181C2×C107#57711yescharacteristic
182D107—22no—
183D107—41no—
184D107—81no—
185#3296—1071nomaximal, Sylow
186C107#329611yescharacteristic, complemented, Sylow
187#552—1071no—
188#546—1071no—
189#488—1071no—
190#483—1071no—
191#482—1071no—
192#478D10711yescharacteristic, complemented
193#467—1071no—
194#465—1071no—
195#464—1071no—
196#452—1071no—
197#450—1071no—
198C4×SD16—1071no—
199C8×D4—1071no—
200#416—1071no—
201C2×#102—1071no—
202C2×#41—1071no—
203C22×D4—1071no—
204C2×Q16—1071no—
205C2×SD16—1071no—
206C2×SD16—1071no—
207C22×C8—1071no—
208C22×C8—1071no—
209#126—2141no—
210#125—2141no—
211#125C2×D10711yescharacteristic, complemented
212#124—1071no—
213#123—2141no—
214#123—2141no—
215C22⋊Q8—1071no—
216C22⋊Q8—1071no—
217C22⋊Q8—1071no—
218#121—1071no—
219#121—2141no—
220#120—2141no—
221C4×Q8—1071no—
222C4×D4—1071no—
223C4×D4C2×D10711yescharacteristic
224#117—1071no—
225#117—2141no—
226C2×C4⋊C4—1071no—
227C2×C22⋊C4—2141no—
228#107C2×D10711yescharacteristic, complemented
229#106—1071no—
230#106C2×D10711yescharacteristic, complemented
231C4⋊C8C2×D10711yescharacteristic, complemented
232#103—1071no—
233#103—1071no—
234#103—1071no—
235#103—1071no—
236#102—1071no—
237#102—1071no—
238#102—1071no—
239#102—1071no—
240#102C2×D10711yescharacteristic
241#102C2×D10711yescharacteristic
242C22⋊C8—1071no—
243C22⋊C8—1071no—
244C4×C8—1071no—
245C24—2141no—
246#41—4281no—
247#41—2142no—
248C2×Q8—1071no—
249C2×Q8—1071no—
250C2×D4—1071no—
251C2×D4—1071no—
252C2×D4—4281no—
253C2×D4—2141no—
254C2×D4—2141no—
255C2×D4—2141no—
256C2×D4—2142no—
257C2×D4C22×D10711yescharacteristic
258C22×C4—21no—
259C22×C4—1071no—
260C22×C4—1071no—
261C22×C4—2141no—
262C22×C4—2141no—
263Q16—4281no—
264SD16—4281no—
265SD16—2142no—
266C2×C8—1071no—
267C2×C8—1071no—
268C2×C8—4281no—
269C2×C8—4281no—
270C2×C8C22×D10711yescharacteristic
271C2×C8C22×D10711yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—1071no—
275C4⋊C4—1071no—
276C4⋊C4—2141no—
277C4⋊C4—2141no—
278C4⋊C4—2141no—
279C4⋊C4—2141no—
280C4⋊C4—2141no—
281C4⋊C4C22×D10711yescharacteristic
282C4⋊C4C22×D10711yescharacteristic
283C4⋊C4C22×D10711yescharacteristic
284C22⋊C4—21no—
285C22⋊C4—1071no—
286C22⋊C4—1071no—
287C22⋊C4—2141no—
288C22⋊C4—2141no—
289C22⋊C4—2141no—
290C22⋊C4—2141no—
291C22⋊C4—2141no—
292C22⋊C4—2141no—
293C22⋊C4—2141no—
294C42—2141no—
295C42—2141no—
296C42C22×D10711yescharacteristic
297C23—21no—
298C23—8561no—
299C23—1071no—
300C23—1071no—
301C23—4282no—
302C23—4282no—
303C23—2141no—
304Q8—2141no—
305Q8—2141no—
306Q8—2141no—
307Q8—2141no—
308D4—21no—
309D4—8561no—
310D4—4282no—
311D4—4282no—
312D4—2142no—
313D4—2141no—
314D4—2141no—
315D4#1313381912yes—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—81no—
320C2×C4—1071no—
321C2×C4—1071no—
322C2×C4—4282no—
323C2×C4—4282no—
324C2×C4—2141no—
325C2×C4—2141no—
326C2×C4—2141no—
327C2×C4—2141no—
328C2×C4—2141no—
329C2×C4—2141no—
330C2×C4—2141no—
331C2×C4D4×D10711yescharacteristic
332C2×C4D4×D10711yescharacteristic
333C2×C4C23×D10711yescharacteristic
334C8—21no—
335C8—21no—
336C8—2141no—
337C8—2141no—
338C22—22no—
339C22—41no—
340C22—41no—
341C22—8564no—
342C22—1072no—
343C22—4281no—
344C22—4281no—
345C22—4281no—
346C22—2141no—
347C22—2141no—
348C22—2141no—
349C22—2141no—
350C22—2141no—
351C22C2×D4×D10711yescharacteristic
352C4—21no—
353C4—21no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—4281no—
358C4—4281no—
359C4—4281no—
360C4—2141no—
361C4—2141no—
362C4C2×#1313381911yescharacteristic
363C4#1681215511yescharacteristic
364C2—42no—
365C2—8561no—
366C2—4281no—
367C2—2142no—
368C2#2029326811yescharacteristic
369C2#135×D10711yescharacteristic
370C2#2029343711yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#478×C107C2
socleC22×C107C22×D4
prime radical#20293272C2
hypercenter#478D107
hypocenterC107#3296
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #478

Complements

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

NH ≅ G/Ntype
1. #20293315C2semidirect
3. #20292984C2semidirect
4. #20292959C2semidirect
5. #20292954C2semidirect
6. #20292869C2semidirect
7. #20292857C2semidirect
8. #20292850C2semidirect
9. #20292790C2semidirect
10. #20292779C2semidirect
11. C4×#13133806C2semidirect
12. #20292692C2semidirect
13. #20292620C2semidirect
14. #102×D107C2semidirect
15. #478×C107C2semidirect
18. C2×Dic428C22semidirect
22. C2×C8×D107C22semidirect
30. #16812089C22semidirect
34. C4×Dic214C22semidirect
37. #16812067C22semidirect
40. #16812040C22semidirect
41. #16812034C22semidirect
42. #16812032C22semidirect
43. #16812032C22semidirect
44. #16812032C22semidirect
50. (C2×D107)⋊C8C22semidirect
51. C4×C107⋊C8C22semidirect
52. #125×C107C22semidirect
54. #107×C107C22semidirect
55. #106×C107C22semidirect
56. C4⋊C8×C107C22semidirect
186. C107#3296semidirect
192. #478D107semidirect
211. #125C2×D107semidirect
228. #107C2×D107semidirect
230. #106C2×D107semidirect
231. C4⋊C8C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202933152
#202932722
#202929842
#202929592
#202929542
#202928692
#202928572
#202928502
#202927902
#202927792
C4×#131338062
#202926922
#202926202
#102×D1072
#478×C1072
#3296107