Finite Group Databasebeta

← #28348294

Subgroups — #28348294

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#20293312C211yesmaximal, characteristic, complemented
2#20293268C211yesmaximal, characteristic
3#20292983C211yesmaximal, characteristic, complemented
4#20292953C211yesmaximal, characteristic, complemented
5#20292926C211yesmaximal, characteristic, complemented
6#20292866C211yesmaximal, characteristic, complemented
7#20292857C211yesmaximal, characteristic, complemented
8#20292852C211yesmaximal, characteristic, complemented
9#20292792C211yesmaximal, characteristic, complemented
10#20292780C211yesmaximal, characteristic, complemented
11#20292714C211yesmaximal, characteristic, complemented
12(C22⋊D107)⋊C8C211yesmaximal, characteristic, complemented
13#20292622C211yesmaximal, characteristic, complemented
14C2×#16812029C211yesmaximal, characteristic, complemented
15#483×C107C211yesmaximal, characteristic, complemented
16C2×#13133819C2211yescharacteristic
17C22×D428C2211yescharacteristic
18C2×#13133808C2211yescharacteristic, complemented
19C2×#13133807C2211yescharacteristic
20C2×#13133807C2211yescharacteristic
21C22×C107⋊C8C2211yescharacteristic
22C2×C8×D107C2211yescharacteristic, complemented
23#16812104—21no—
24#16812100C2211yescharacteristic, complemented
25#16812099—21no—
26#16812098C2211yescharacteristic
27#16812094—21no—
28#16812092—21no—
29#16812089C2211yescharacteristic, complemented
30#16812088C2211yescharacteristic
31#16812084—21no—
32#16812084C2211yescharacteristic
33#16812079—21no—
34#16812074C2211yescharacteristic, complemented
35#16812071C2211yescharacteristic
36C4×C22⋊D107C2211yescharacteristic
37#16812068—21no—
38#16812067C2211yescharacteristic, complemented
39C22⋊C4×D107—21no—
40#16812042C2211yescharacteristic, complemented
41Dic107⋊C8C2211yescharacteristic, complemented
42#16812033C2211yescharacteristic, complemented
43#16812032C2211yescharacteristic, complemented
44#16812031C2211yescharacteristic
45#16812031C2211yescharacteristic
46#16812029C2211yescharacteristic
47#16812029C2211yescharacteristic
48#16812029C2211yescharacteristic
49#16812029C2211yescharacteristic
50(C2×D107)⋊C8C2211yescharacteristic, complemented
51C8×Dic107C2211yescharacteristic, complemented
52C22⋊Q8×C107C2211yescharacteristic
53C2×C4⋊C4×C107C2211yescharacteristic
54#106×C107C2211yescharacteristic, complemented
55#106×C107C2211yescharacteristic, complemented
56#103×C107C2211yescharacteristic, complemented
57#103×C107C2211yescharacteristic, complemented
58C22⋊C8×C107C2211yescharacteristic
59C23×D107—21no—
60#13133819—22no—
61#13133819—41no—
62C2×Dic214C2311yescharacteristic
63C2×D428—22no—
64C2×D428—21no—
65C2×D428—21no—
66C2×D428—21no—
67C2×D428—41no—
68C2×D428C2311yescharacteristic
69C2×D428C2311yescharacteristic
70C2×C22⋊D107D411yescharacteristic
71C2×C4×D107—21no—
72C2×C4×D107—21no—
73C2×C4×D107C2311yescharacteristic
74#13133808—41no—
75#13133807—22no—
76#13133807—41no—
77C2×C107⋊C8—41no—
78C2×C107⋊C8C2311yescharacteristic
79C2×C107⋊C8C2311yescharacteristic
80C8×D107—41no—
81C4⋊Dic107—21no—
82C4⋊Dic107—21no—
83C4⋊Dic107C2311yescharacteristic
84Dic107⋊C4—21no—
85Dic107⋊C4—21no—
86Dic107⋊C4—21no—
87Dic107⋊C4D411yescharacteristic
88C22⋊Dic107—21no—
89C22⋊Dic107D411yescharacteristic
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
97C4×Dic107—21no—
98C4×Dic107—21no—
99C4×Dic107C2311yescharacteristic
100C2×Q8×C107C2311yescharacteristic
101C22×C4×C107—21no—
102C22×C4×C107C2311yescharacteristic
103C2×C8×C107C2311yescharacteristic
104C2×C8×C107C2311yescharacteristic
105C4⋊C4×C107—21no—
106C4⋊C4×C107—21no—
107C4⋊C4×C107C2311yescharacteristic
108C4⋊C4×C107C2311yescharacteristic
109C4⋊C4×C107C2311yescharacteristic
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—
120C2×C4×C107—21no—
121C2×C4×C107—21no—
122C2×C4×C107—21no—
123C2×C4×C107—21no—
124C2×C4×C107—21no—
125C2×C4×C107—21no—
126C2×C4×C107—42no—
127C2×C4×C107C2×D411yescharacteristic
128C2×C4×C107C2411yescharacteristic
129C22⋊D107—22no—
130C2×Dic107—21no—
131C2×Dic107—21no—
132C2×Dic107C2×D411yescharacteristic
133C2×Dic107C2×D411yescharacteristic
134C2×Dic107C2×D411yescharacteristic
135D428—21no—
136D428—21no—
137D428—21no—
138D428—42no—
139D428—42no—
140D428—81no—
141D428#4112yes—
142C4×D107—21no—
143C4×D107—21no—
144C4×D107—42no—
145C4×D107—81no—
146Dic214—21no—
147Dic214—21no—
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#483D10711yescharacteristic, complemented
191#482—1071no—
192#478—1071no—
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—1071no—
211#125—2141no—
212#124—1071no—
213#123—2141no—
214#123—2141no—
215C22⋊Q8—1071no—
216C22⋊Q8—1071no—
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⋊C4C2×D10711yescharacteristic
227C2×C22⋊C4—2141no—
228#107—1071no—
229#106C2×D10711yescharacteristic, complemented
230#106C2×D10711yescharacteristic, complemented
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⋊C8C2×D10711yescharacteristic
244C4×C8—1071no—
245C24—2141no—
246#41—4281no—
247#41—2142no—
248C2×Q8—1071no—
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—21no—
259C22×C4—1071no—
260C22×C4—2141no—
261C22×C4—2141no—
262C22×C4C22×D10711yescharacteristic
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—1071no—
295C42—2141no—
296C42—2141no—
297C23—8561no—
298C23—1071no—
299C23—4282no—
300C23—4282no—
301C23—2141no—
302C23—2141no—
303C23D4×D10711yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—2141no—
307Q8—2141no—
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—42no—
323C2×C4—8561no—
324C2×C4—1071no—
325C2×C4—1071no—
326C2×C4—1071no—
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#1681213012yes—
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#1681215511yescharacteristic
364C2—22no—
365C2—8561no—
366C2—4282no—
367C2—4281no—
368C2#2029326911yescharacteristic
369C2C2×#1681213011yescharacteristic
370C2#2029343611yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#483×C107C2
socleC22×C107C22×D4
prime radical#20293268C2
hypercenter#483D107
hypocenterC107#3296
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #483

Complements

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

NH ≅ G/Ntype
1. #20293312C2semidirect
3. #20292983C2semidirect
4. #20292953C2semidirect
5. #20292926C2semidirect
6. #20292866C2semidirect
7. #20292857C2semidirect
8. #20292852C2semidirect
9. #20292792C2semidirect
10. #20292780C2semidirect
11. #20292714C2semidirect
12. (C22⋊D107)⋊C8C2semidirect
13. #20292622C2semidirect
14. C2×#16812029C2semidirect
15. #483×C107C2semidirect
18. C2×#13133808C22semidirect
22. C2×C8×D107C22semidirect
24. #16812100C22semidirect
29. #16812089C22semidirect
34. #16812074C22semidirect
38. #16812067C22semidirect
40. #16812042C22semidirect
41. Dic107⋊C8C22semidirect
42. #16812033C22semidirect
43. #16812032C22semidirect
50. (C2×D107)⋊C8C22semidirect
51. C8×Dic107C22semidirect
54. #106×C107C22semidirect
55. #106×C107C22semidirect
56. #103×C107C22semidirect
57. #103×C107C22semidirect
186. C107#3296semidirect
190. #483D107semidirect
229. #106C2×D107semidirect
230. #106C2×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
#202933122
#202932682
#202929832
#202929532
#202929262
#202928662
#202928572
#202928522
#202927922
#202927802
#202927142
(C22⋊D107)⋊C82
#202926222
C2×#168120292
#483×C1072
#3296107