Finite Group Databasebeta

← #28349852

Subgroups — #28349852

Order 13696.

1040 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293250C212yesmaximal, complemented
2#20293249C212yesmaximal, complemented
3#20293238C211yesmaximal, characteristic, complemented
4#20293233C211yesmaximal, characteristic, complemented
5#20293232C211yesmaximal, characteristic, complemented
6#20293225C214yesmaximal, complemented
7#20293216C214yesmaximal, complemented
8#20293183C211yesmaximal, characteristic, complemented
9#20293178C211yesmaximal, characteristic, complemented
10#20293162C214yesmaximal, complemented
11#20293160C214yesmaximal, complemented
12#20293140C212yesmaximal, complemented
13C4×#13133819C212yesmaximal, complemented
14#117×D107C211yesmaximal, characteristic, complemented
15#541×C107C211yesmaximal, characteristic, complemented
16C2×#13133819C2212yes—
17C22×C4×D107C2211yescharacteristic
18#16812108C2212yescomplemented
19#16812106C2211yescharacteristic, complemented
20#16812104C2214yes—
21#16812104C2214yescomplemented
22#16812103C2214yescomplemented
23#16812101C2211yescharacteristic, complemented
24#16812099C2214yescomplemented
25#16812098C2212yescomplemented
26#16812097C2214yes—
27#16812095C2214yes—
28#16812094C2214yes—
29#16812094C2214yes—
30#16812094C2214yescomplemented
31#16812094C2214yescomplemented
32#16812091C2214yes—
33#16812089C2214yescomplemented
34#16812087C2214yes—
35#16812087C2214yescomplemented
36#16812085C2212yes—
37#16812084C2212yes—
38#16812084C2214yes—
39#16812084C2214yes—
40#16812083C2212yes—
41#16812082C2214yes—
42#16812079C2214yes—
43Q8×Dic107C2212yescomplemented
44#16812074C2214yescomplemented
45C4×D428C2214yes—
46C4×D428C2212yescomplemented
47#16812071C2214yes—
48#16812071C2214yes—
49#16812071C2214yescomplemented
50#16812071C2212yescomplemented
51(C22⋊D107)⋊C4C2214yes—
52#16812068C2211yescharacteristic
53#16812067C2212yes—
54#16812067C2212yes—
55#16812067C2214yescomplemented
56#16812066C2211yescharacteristic
57#16812066C2211yescharacteristic
58#16812065C2212yes—
59C4⋊C4×D107C2212yes—
60C22⋊C4×D107C2212yes—
61C42×D107C2211yescharacteristic
62C42×D107C2211yescharacteristic
63C42×D107C2214yescomplemented
64C4⋊Q8×C107C2211yescharacteristic, complemented
65#126×C107C2214yes—
66#125×C107C2214yescomplemented
67#125×C107C2211yescharacteristic, complemented
68C22⋊Q8×C107C2212yes—
69C4×Q8×C107C2212yescomplemented
70#117×C107C2211yescharacteristic
71C23×D107C2311yescharacteristic
72#13133819—28no—
73C2×Dic214C2314yes—
74C2×D428C2314yes—
75C2×D428C2312yes—
76C2×D428C2314yes—
77C2×D428C2314yes—
78C2×C22⋊D107C2314yes—
79C22×Dic107C2311yescharacteristic
80C2×C4×D107—24no—
81C2×C4×D107C2314yes—
82C2×C4×D107C2311yescharacteristic
83C2×C4×D107C2311yescharacteristic
84C2×C4×D107C2311yescharacteristic
85C2×C4×D107C2312yes—
86C2×C4×D107C2312yes—
87C2×C4×D107C2314yes—
88C2×C4×D107C2314yes—
89C2×C4×D107C2311yescharacteristic
90C4⋊Dic107C2312yes—
91C4⋊Dic107C2312yes—
92C4⋊Dic107C2314yes—
93Dic107⋊C4C2312yes—
94Dic107⋊C4C2312yes—
95Dic107⋊C4C2314yes—
96Dic107⋊C4C2314yes—
97Dic107⋊C4C2314yes—
98C22⋊Dic107C2312yes—
99(C2×D107)⋊C4C2312yes—
100(C2×D107)⋊C4C2312yes—
101(C2×D107)⋊C4C2314yes—
102(C2×D107)⋊C4C2314yes—
103(C2×D107)⋊C4C2314yes—
104(C2×D107)⋊C4C2314yes—
105(C2×D107)⋊C4C2314yes—
106(C2×D107)⋊C4C2314yes—
107(C2×D107)⋊C4C2314yes—
108(C2×D107)⋊C4C2314yes—
109C4×Dic107C2312yes—
110C4×Dic107C2312yes—
111C4×Dic107C2311yescharacteristic
112C4×Dic107C2311yescharacteristic
113C4×Dic107C2314yes—
114C4×Dic107C2314yes—
115C2×Q8×C107C2312yes—
116C22×C4×C107C2311yescharacteristic
117C4⋊C4×C107C2312yes—
118C4⋊C4×C107C2312yes—
119C4⋊C4×C107C2314yes—
120C4⋊C4×C107C2314yes—
121C4⋊C4×C107C2314yes—
122C4⋊C4×C107C2314yes—
123C22⋊C4×C107C2312yes—
124C22⋊C4×C107C2314yes—
125C42×C107C2311yescharacteristic
126C42×C107C2311yescharacteristic
127C42×C107C2314yes—
128C23×C107C2411yescharacteristic
129C22×D107—22no—
130C22×D107—42no—
131C22×D107C2411yescharacteristic
132C22×D107C2414yes—
133C22×D107C2411yescharacteristic
134Q8×C107—24no—
135C2×C4×C107—22no—
136C2×C4×C107C2412yes—
137C2×C4×C107C2414yes—
138C2×C4×C107C2412yes—
139C2×C4×C107C2414yes—
140C2×C4×C107C2411yescharacteristic
141C2×C4×C107C2411yescharacteristic
142C22⋊D107—44no—
143C2×Dic107—22no—
144C2×Dic107C2412yes—
145C2×Dic107C2412yes—
146C2×Dic107C2414yes—
147C2×Dic107C2411yescharacteristic
148C2×Dic107C2411yescharacteristic
149D428—28no—
150D428—24no—
151D428—44no—
152D428—44no—
153C4×D107—22no—
154C4×D107—22no—
155C4×D107—22no—
156C4×D107—28no—
157C4×D107—42no—
158C4×D107—44no—
159C4×D107—44no—
160C4×D107—42no—
161C4×D107#4114yes—
162Dic214—44no—
163C22×C107—21no—
164C22×C107—22no—
165C22×C107C2511yescharacteristic
166C2×D107—22no—
167C2×D107—21no—
168C2×D107—24no—
169C2×D107—24no—
170C2×D107—24no—
171C2×D107—22no—
172C2×D107—44no—
173C2×D107C2×#4112yes—
174C4×C107—21no—
175C4×C107—24no—
176C4×C107—24no—
177C4×C107—22no—
178C4×C107—22no—
179C4×C107C2×#4112yes—
180Dic107—21no—
181Dic107—22no—
182Dic107—24no—
183Dic107—22no—
184Dic107C2×#4112yes—
185C2×C107—41no—
186C2×C107C22×#4111yescharacteristic
187C2×C107#58512yes—
188D107—22no—
189D107—44no—
190D107—41no—
191#3455—1071nomaximal, Sylow
192C107#345511yescharacteristic, complemented, Sylow
193#542—1072no—
194#542—1072no—
195#541—1071no—
196#541D10711yescharacteristic, complemented
197#540—1071no—
198#540—1071no—
199#539—1074no—
200#538—1074no—
201#533—1071no—
202#532—1071no—
203#529—1074no—
204#529—1074no—
205#525—1072no—
206C4×#41—1072no—
207C2×#117—1071no—
208C2×#41—1072no—
209C23×C4—1071no—
210C4⋊Q8—1072no—
211C4⋊Q8C2×D10711yescharacteristic, complemented
212#127—1071no—
213#126—1074no—
214#126—1074no—
215#126—1074no—
216#126C2×D10714yes—
217#125—1074no—
218#125—1071no—
219#125C2×D10714yescomplemented
220#125C2×D10711yescharacteristic, complemented
221#124—1072no—
222#124—1074no—
223#124—1074no—
224#123—1074no—
225#123—1074no—
226#123—1074no—
227#123—1074no—
228#123—1074no—
229C22⋊Q8—1072no—
230C22⋊Q8—1074no—
231C22⋊Q8—1074no—
232C22⋊Q8—1074no—
233C22⋊Q8C2×D10712yes—
234#121—1072no—
235#121—1072no—
236#121—1074no—
237#121—1074no—
238#121—1074no—
239#120—1074no—
240C4×Q8—1072no—
241C4×Q8—1074no—
242C4×Q8C2×D10712yescomplemented
243C4×D4—1074no—
244C4×D4—1072no—
245C4×D4—1074no—
246C4×D4—1074no—
247C4×D4—1074no—
248C4×D4—1074no—
249C4×D4—1072no—
250#117—1071no—
251#117—1072no—
252#117—1072no—
253#117—1074no—
254#117—1072no—
255#117—1071no—
256#117—1071no—
257#117C2×D10711yescharacteristic
258C2×C4⋊C4—1072no—
259C2×C22⋊C4—1072no—
260C2×C42—1071no—
261C2×C42—1071no—
262C2×C42—1074no—
263C24—1071no—
264#41—2148no—
265C2×Q8—1074no—
266C2×Q8C22×D10712yes—
267C2×D4—1074no—
268C2×D4—1074no—
269C2×D4—1074no—
270C2×D4—1074no—
271C2×D4—1072no—
272C22×C4—1071no—
273C22×C4—1071no—
274C22×C4—1072no—
275C22×C4—1072no—
276C22×C4—1074no—
277C22×C4—1074no—
278C22×C4—1074no—
279C22×C4—1071no—
280C22×C4—1071no—
281C22×C4—1071no—
282C22×C4—2144no—
283C22×C4C22×D10711yescharacteristic
284C4⋊C4—1072no—
285C4⋊C4—1072no—
286C4⋊C4—1072no—
287C4⋊C4—1072no—
288C4⋊C4—1074no—
289C4⋊C4—1074no—
290C4⋊C4—1074no—
291C4⋊C4—1074no—
292C4⋊C4C22×D10712yes—
293C4⋊C4C22×D10712yes—
294C4⋊C4C22×D10714yes—
295C4⋊C4C22×D10714yes—
296C4⋊C4C22×D10714yes—
297C4⋊C4C22×D10714yes—
298C22⋊C4—1072no—
299C22⋊C4—1072no—
300C22⋊C4—1072no—
301C22⋊C4—1074no—
302C22⋊C4—1074no—
303C22⋊C4—1074no—
304C22⋊C4—1074no—
305C22⋊C4—1074no—
306C22⋊C4—1074no—
307C22⋊C4—1074no—
308C22⋊C4—1074no—
309C22⋊C4C22×D10712yes—
310C22⋊C4C22×D10714yes—
311C42—1072no—
312C42—1072no—
313C42—1074no—
314C42—1071no—
315C42—1071no—
316C42—1074no—
317C42C22×D10711yescharacteristic
318C42C22×D10711yescharacteristic
319C42C22×D10714yes—
320C23—1071no—
321C23—1071no—
322C23—1074no—
323C23—4282no—
324C23—2142no—
325C23C23×D10711yescharacteristic
326Q8—24no—
327Q8—4284no—
328D4—4284no—
329D4—4284no—
330D4—4284no—
331D4—2148no—
332D4—2144no—
333C2×C4—22no—
334C2×C4—1071no—
335C2×C4—1074no—
336C2×C4—1072no—
337C2×C4—1072no—
338C2×C4—1074no—
339C2×C4—1071no—
340C2×C4—4282no—
341C2×C4—4284no—
342C2×C4—4282no—
343C2×C4—4284no—
344C2×C4—2142no—
345C2×C4—2148no—
346C2×C4—2142no—
347C2×C4—2142no—
348C2×C4—2142no—
349C2×C4C23×D10712yes—
350C2×C4C23×D10714yes—
351C2×C4C23×D10714yes—
352C2×C4C23×D10712yes—
353C2×C4C23×D10711yescharacteristic
354C2×C4C23×D10711yescharacteristic
355C22—21no—
356C22—22no—
357C22—1072no—
358C22—4284no—
359C22—2141no—
360C22—2144no—
361C22—2144no—
362C22—2144no—
363C22—2142no—
364C22—2142no—
365C22C24×D10711yescharacteristic
366C4—21no—
367C4—24no—
368C4—22no—
369C4—22no—
370C4—24no—
371C4—1072no—
372C4—2141no—
373C4—2142no—
374C4—2144no—
375C4—2142no—
376C4#41×D10712yes—
377C2—41no—
378C2—4281no—
379C2—4284no—
380C2—2142no—
381C2C2×#41×D10711yescharacteristic
382C2#2029346412yes—
subgroupG/H
center, FrattiniC22C24×D107
commutator, socle, cosocleC22×C107C25
Fitting#541×C107C2
hypercenter#541D107
hypocenterC107#3455
LCSG ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ #541

Complements

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

NH ≅ G/Ntype
1. #20293250C2semidirect
2. #20293249C2semidirect
3. #20293238C2semidirect
4. #20293233C2semidirect
5. #20293232C2semidirect
6. #20293225C2semidirect
7. #20293216C2semidirect
8. #20293183C2semidirect
9. #20293178C2semidirect
10. #20293162C2semidirect
11. #20293160C2semidirect
12. #20293140C2semidirect
13. C4×#13133819C2semidirect
14. #117×D107C2semidirect
15. #541×C107C2semidirect
18. #16812108C22semidirect
19. #16812106C22semidirect
21. #16812104C22semidirect
22. #16812103C22semidirect
23. #16812101C22semidirect
24. #16812099C22semidirect
25. #16812098C22semidirect
30. #16812094C22semidirect
31. #16812094C22semidirect
33. #16812089C22semidirect
35. #16812087C22semidirect
43. Q8×Dic107C22semidirect
44. #16812074C22semidirect
46. C4×D428C22semidirect
49. #16812071C22semidirect
50. #16812071C22semidirect
55. #16812067C22semidirect
63. C42×D107C22semidirect
64. C4⋊Q8×C107C22semidirect
66. #125×C107C22semidirect
67. #125×C107C22semidirect
69. C4×Q8×C107C22semidirect
192. C107#3455semidirect
196. #541D107semidirect
211. C4⋊Q8C2×D107semidirect
219. #125C2×D107semidirect
220. #125C2×D107semidirect
242. C4×Q8C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202932502
#202932492
#202932382
#202932332
#202932322
#202932252
#202932162
#202931832
#202931782
#202931622
#202931602
#202931402
C4×#131338192
#117×D1072
#541×C1072
#3455107