Finite Group Databasebeta

← #28348302

Subgroups — #28348302

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#20293319C211yesmaximal, characteristic, complemented
2#20293268C211yesmaximal, characteristic
3#20292983C211yesmaximal, characteristic, complemented
4#20292955C211yesmaximal, characteristic, complemented
5#20292954C211yesmaximal, characteristic, complemented
6#20292930C211yesmaximal, characteristic, complemented
7#20292859C211yesmaximal, characteristic, complemented
8#20292851C211yesmaximal, characteristic, complemented
9#20292788C211yesmaximal, characteristic, complemented
10#20292777C211yesmaximal, characteristic, complemented
11#20292712C211yesmaximal, characteristic, complemented
12C8×C22⋊D107C211yesmaximal, characteristic, complemented
13#20292622C211yesmaximal, characteristic, complemented
14C2×#16812031C211yesmaximal, characteristic, complemented
15#467×C107C211yesmaximal, characteristic, complemented
16C2×#13133819C2211yescharacteristic
17C22×D428C2211yescharacteristic
18C2×#13133805C2211yescharacteristic
19C2×#13133805C2211yescharacteristic
20C2×C8×D107C2211yescharacteristic, complemented
21#16812104—21no—
22#16812100—21no—
23#16812100C2211yescharacteristic, complemented
24#16812098C2211yescharacteristic
25#16812094—21no—
26#16812093—21no—
27#16812085C2211yescharacteristic, complemented
28#16812084—21no—
29#16812084C2211yescharacteristic
30#16812079—21no—
31Q8×Dic107C2211yescharacteristic, complemented
32#16812071C2211yescharacteristic
33C4×C22⋊D107C2211yescharacteristic
34#16812067C2211yescharacteristic, complemented
35#16812065—21no—
36C2×C4⋊Dic107C2211yescharacteristic
37C22⋊C4×D107—21no—
38#16812040C2211yescharacteristic, complemented
39#16812040C2211yescharacteristic, complemented
40Dic107⋊C8C2211yescharacteristic, complemented
41#16812033C2211yescharacteristic, complemented
42#16812033C2211yescharacteristic, complemented
43#16812031C2211yescharacteristic
44#16812031C2211yescharacteristic
45#16812031C2211yescharacteristic
46#16812031C2211yescharacteristic
47#16812029C2211yescharacteristic
48#16812029C2211yescharacteristic
49C22⋊(C107⋊C8)C2211yescharacteristic
50(C2×D107)⋊C8C2211yescharacteristic, complemented
51C8×Dic107C2211yescharacteristic, complemented
52C2×Q16×C107C2211yescharacteristic, complemented
53C22×C8×C107C2211yescharacteristic
54C22⋊Q8×C107C2211yescharacteristic
55C22⋊Q8×C107C2211yescharacteristic
56#107×C107C2211yescharacteristic, complemented
57#103×C107C2211yescharacteristic, complemented
58#103×C107C2211yescharacteristic, complemented
59C23×D107—21no—
60#13133819—22no—
61#13133819—41no—
62C2×D428—22no—
63C2×D428—21no—
64C2×D428—21no—
65C2×D428—21no—
66C2×D428—41no—
67C2×D428C2311yescharacteristic
68C2×D428C2311yescharacteristic
69C2×C22⋊D107D411yescharacteristic
70C22×Dic107—21no—
71C2×C4×D107—21no—
72C2×C4×D107—21no—
73C2×C4×D107C2311yescharacteristic
74#13133805—22no—
75#13133805—41no—
76C2×C107⋊C8C2311yescharacteristic
77C2×C107⋊C8C2311yescharacteristic
78C8×D107—41no—
79C4⋊Dic107—21no—
80C4⋊Dic107—21no—
81C4⋊Dic107—21no—
82C4⋊Dic107C2311yescharacteristic
83C4⋊Dic107C2311yescharacteristic
84Dic107⋊C4—21no—
85Dic107⋊C4—21no—
86Dic107⋊C4D411yescharacteristic
87C22⋊Dic107D411yescharacteristic
88(C2×D107)⋊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)⋊C4D411yescharacteristic
95C4×Dic107—21no—
96C4×Dic107—21no—
97C4×Dic107C2311yescharacteristic
98C2×Q8×C107C2311yescharacteristic
99C2×Q8×C107C2311yescharacteristic
100C22×C4×C107C2311yescharacteristic
101Q16×C107—41no—
102C2×C8×C107—41no—
103C2×C8×C107C2311yescharacteristic
104C2×C8×C107C2311yescharacteristic
105C4⋊C4×C107—21no—
106C4⋊C4×C107—21no—
107C4⋊C4×C107C2311yescharacteristic
108C4⋊C4×C107C2311yescharacteristic
109C22⋊C4×C107—21no—
110C22⋊C4×C107—21no—
111C23×C107C2×D411yescharacteristic
112C22×D107—21no—
113C22×D107—21no—
114C22×D107—42no—
115C22×D107—42no—
116C22×D107—81no—
117C22×D107C2×D411yescharacteristic
118Q8×C107—21no—
119Q8×C107—21no—
120Q8×C107—21no—
121Q8×C107—21no—
122C2×C4×C107—21no—
123C2×C4×C107—21no—
124C2×C4×C107—21no—
125C2×C4×C107—21no—
126C2×C4×C107—21no—
127C2×C4×C107—21no—
128C2×C4×C107C2×D411yescharacteristic
129C2×C4×C107C2411yescharacteristic
130C22⋊D107—22no—
131C2×Dic107—21no—
132C2×Dic107—21no—
133C2×Dic107—42no—
134C2×Dic107C2×D411yescharacteristic
135C2×Dic107C2×D411yescharacteristic
136C2×Dic107C2×D411yescharacteristic
137D428—21no—
138D428—21no—
139D428—21no—
140D428—42no—
141D428—42no—
142D428—81no—
143D428#4112yes—
144C4×D107—21no—
145C4×D107—21no—
146C4×D107—42no—
147C4×D107—81no—
148C8×C107—21no—
149C8×C107—21no—
150C107⋊C8—21no—
151C107⋊C8—21no—
152C22×C107—21no—
153C22×C107—21no—
154C22×C107#13512yes—
155C22×C107C22×D411yescharacteristic
156C2×D107—21no—
157C2×D107—22no—
158C2×D107—21no—
159C2×D107—21no—
160C2×D107—41no—
161C2×D107—41no—
162C2×D107—41no—
163C2×D107—41no—
164C2×D107—41no—
165C2×D107—84no—
166C4×C107—21no—
167C4×C107—41no—
168C4×C107—41no—
169C4×C107—41no—
170C4×C107—41no—
171C4×C107C2×#4111yescharacteristic
172C4×C107H(5,𝔽2)11yescharacteristic
173Dic107—21no—
174Dic107—21no—
175Dic107—21no—
176Dic107—41no—
177Dic107—41no—
178C2×C107—22no—
179C2×C107#54611yescharacteristic
180C2×C107C2×#13511yescharacteristic
181C2×C107#57711yescharacteristic
182D107—41no—
183D107—42no—
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#478—1071no—
193#467D10711yescharacteristic, complemented
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×Q16C2×D10711yescharacteristic, complemented
205C2×SD16—1071no—
206C2×SD16—1071no—
207C22×C8—1071no—
208C22×C8C2×D10711yescharacteristic
209#126—2141no—
210#125—1071no—
211#125—2141no—
212#124—1071no—
213#123—2141no—
214#123—2141no—
215C22⋊Q8—1071no—
216C22⋊Q8C2×D10711yescharacteristic
217C22⋊Q8C2×D10711yescharacteristic
218#121—1071no—
219#121—2141no—
220#120—2141no—
221C4×Q8—1071no—
222C4×D4—1071no—
223C4×D4—1071no—
224#117—1071no—
225#117—2141no—
226C2×C4⋊C4—1071no—
227C2×C22⋊C4—2141no—
228#107C2×D10711yescharacteristic, complemented
229#106—1071no—
230#106—1071no—
231C4⋊C8—1071no—
232#103—1071no—
233#103—1071no—
234#103C2×D10711yescharacteristic, complemented
235#103C2×D10711yescharacteristic, complemented
236#102—1071no—
237#102—1071no—
238#102—1071no—
239#102—1071no—
240#102—1071no—
241#102—1071no—
242C22⋊C8—1071no—
243C22⋊C8—1071no—
244C4×C8—1071no—
245C24—2141no—
246#41—4281no—
247#41—2142no—
248C2×Q8C22×D10711yescharacteristic
249C2×Q8C22×D10711yescharacteristic
250C2×D4—1071no—
251C2×D4—1071no—
252C2×D4—1071no—
253C2×D4—4281no—
254C2×D4—2141no—
255C2×D4—2142no—
256C2×D4—2141no—
257C2×D4—2141no—
258C22×C4—1071no—
259C22×C4—2141no—
260C22×C4—2141no—
261C22×C4—2141no—
262C22×C4C22×D10711yescharacteristic
263Q16—41no—
264SD16—4281no—
265SD16—2142no—
266C2×C8—41no—
267C2×C8—1071no—
268C2×C8—1071no—
269C2×C8—4281no—
270C2×C8C22×D10711yescharacteristic
271C2×C8C22×D10711yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—1071no—
275C4⋊C4—1071no—
276C4⋊C4—1071no—
277C4⋊C4—2141no—
278C4⋊C4—2141no—
279C4⋊C4—2141no—
280C4⋊C4—2141no—
281C4⋊C4—2141no—
282C4⋊C4C22×D10711yescharacteristic
283C4⋊C4C22×D10711yescharacteristic
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—1071no—
287C22⋊C4—1071no—
288C22⋊C4—2141no—
289C22⋊C4—2141no—
290C22⋊C4—2141no—
291C22⋊C4—2141no—
292C22⋊C4—2141no—
293C22⋊C4—2141no—
294C42—1071no—
295C42—2141no—
296C42—2141no—
297C23—8561no—
298C23—1071no—
299C23—4282no—
300C23—4282no—
301C23—2141no—
302C23—2141no—
303C23D4×D10711yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—21no—
307Q8—21no—
308D4—8561no—
309D4—1072no—
310D4—4282no—
311D4—4282no—
312D4—2142no—
313D4—2141no—
314D4—2141no—
315D4—2141no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—8561no—
323C2×C4—1071no—
324C2×C4—1071no—
325C2×C4—1071no—
326C2×C4—4282no—
327C2×C4—4282no—
328C2×C4—2141no—
329C2×C4—2141no—
330C2×C4—2141no—
331C2×C4—2141no—
332C2×C4D4×D10711yescharacteristic
333C2×C4C23×D10711yescharacteristic
334C8—21no—
335C8—21no—
336C8—2141no—
337C8—2141no—
338C22—21no—
339C22—21no—
340C22—8564no—
341C22—4281no—
342C22—4281no—
343C22—4281no—
344C22—4281no—
345C22—4281no—
346C22—2141no—
347C22—2141no—
348C22—2142no—
349C22—2141no—
350C22#1681213212yes—
351C22C2×D4×D10711yescharacteristic
352C4—21no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—4281no—
358C4—4281no—
359C4—2141no—
360C4—2141no—
361C4—2141no—
362C4#41×D10711yescharacteristic
363C4#1681215411yescharacteristic
364C2—22no—
365C2—8561no—
366C2—4282no—
367C2—4281no—
368C2#2029326611yescharacteristic
369C2C2×#1681213211yescharacteristic
370C2#2029343411yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#467×C107C2
socleC22×C107C22×D4
prime radical#20293268C2
hypercenter#467D107
hypocenterC107#3296
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
1. #20293319C2semidirect
3. #20292983C2semidirect
4. #20292955C2semidirect
5. #20292954C2semidirect
6. #20292930C2semidirect
7. #20292859C2semidirect
8. #20292851C2semidirect
9. #20292788C2semidirect
10. #20292777C2semidirect
11. #20292712C2semidirect
12. C8×C22⋊D107C2semidirect
13. #20292622C2semidirect
14. C2×#16812031C2semidirect
15. #467×C107C2semidirect
20. C2×C8×D107C22semidirect
23. #16812100C22semidirect
27. #16812085C22semidirect
31. Q8×Dic107C22semidirect
34. #16812067C22semidirect
38. #16812040C22semidirect
39. #16812040C22semidirect
40. Dic107⋊C8C22semidirect
41. #16812033C22semidirect
42. #16812033C22semidirect
50. (C2×D107)⋊C8C22semidirect
51. C8×Dic107C22semidirect
52. C2×Q16×C107C22semidirect
56. #107×C107C22semidirect
57. #103×C107C22semidirect
58. #103×C107C22semidirect
186. C107#3296semidirect
193. #467D107semidirect
204. C2×Q16C2×D107semidirect
228. #107C2×D107semidirect
234. #103C2×D107semidirect
235. #103C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202933192
#202932682
#202929832
#202929552
#202929542
#202929302
#202928592
#202928512
#202927882
#202927772
#202927122
C8×C22⋊D1072
#202926222
C2×#168120312
#467×C1072
#3296107