Finite Group Databasebeta

← #28347988

Subgroups — #28347988

Order 13696.

500 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812133C211yesmaximal, characteristic, complemented
2C2×#16812128C211yesmaximal, characteristic, complemented
3#20293281C211yesmaximal, characteristic, complemented
4#20293272C211yesmaximal, characteristic, complemented
5#20293021C211yesmaximal, characteristic, complemented
6#20292886C211yesmaximal, characteristic, complemented
7#20292857C211yesmaximal, characteristic, complemented
8#20292842C211yesmaximal, characteristic, complemented
9#20292832C211yesmaximal, characteristic, complemented
10#20292790C211yesmaximal, characteristic, complemented
11#20292790C211yesmaximal, characteristic, complemented
12#20292779C211yesmaximal, characteristic, complemented
13#20292773C211yesmaximal, characteristic, complemented
14#20292700C211yesmaximal, characteristic, complemented
15#442×C107C211yesmaximal, characteristic, complemented
16C2×#13133817C2211yescharacteristic
17C2×#13133817C2211yescharacteristic
18C2×#13133817C2211yescharacteristic, complemented
19C2×D4×D107C2211yescharacteristic
20#16812133—24no—
21#16812128—24no—
22C2×Dic428C2211yescharacteristic, complemented
23C2×Dic428C2211yescharacteristic, complemented
24C2×#13133807C2211yescharacteristic
25C2×#13133807C2211yescharacteristic
26C2×#13133806C2211yescharacteristic
27C2×#13133806C2211yescharacteristic
28C2×#13133806C2211yescharacteristic, complemented
29C2×#13133803C2211yescharacteristic
30C2×#13133801C2211yescharacteristic
31C2×C8×D107C2211yescharacteristic, complemented
32#16812109C2211yescharacteristic, complemented
33#16812097C2211yescharacteristic
34#16812093—21no—
35#16812093—21no—
36#16812093—21no—
37#16812089C2211yescharacteristic
38#16812089C2211yescharacteristic
39#16812089C2211yescharacteristic, complemented
40#16812084C2211yescharacteristic
41#16812083—21no—
42#16812083—21no—
43#16812079—21no—
44C4×D428C2211yescharacteristic
45C2×C4⋊Dic107—21no—
46C2×(C2×D107)⋊C4—21no—
47C4⋊(C107⋊C8)C2211yescharacteristic, complemented
48#16812032C2211yescharacteristic, complemented
49#16812032C2211yescharacteristic, complemented
50#16812032C2211yescharacteristic, complemented
51#16812029C2211yescharacteristic
52(C2×D107)⋊C8C2211yescharacteristic
53(C2×D107)⋊C8C2211yescharacteristic, complemented
54C2×D8×C107C2211yescharacteristic
55C4×D4×C107C2211yescharacteristic
56C4×D4×C107C2211yescharacteristic, complemented
57#106×C107C2211yescharacteristic, complemented
58#102×C107C2211yescharacteristic
59#102×C107C2211yescharacteristic, complemented
60#97×C107C2211yescharacteristic, complemented
61C23×D107—21no—
62#13133817—22no—
63#13133817—22no—
64#13133817—22no—
65#13133817—41no—
66#13133817—41no—
67#13133817—41no—
68C2×Dic214C2311yescharacteristic
69C2×Dic214C2311yescharacteristic
70C2×Dic214C2311yescharacteristic
71C2×D428—21no—
72C2×D428D411yescharacteristic
73C2×D428C2311yescharacteristic
74C2×C22⋊D107—21no—
75C2×C22⋊D107—21no—
76C2×C22⋊D107—21no—
77C2×C22⋊D107—21no—
78D4×D107—22no—
79D4×D107—41no—
80C22×Dic107—21no—
81C22×Dic107—21no—
82C22×Dic107—21no—
83C2×C4×D107C2311yescharacteristic
84C2×C4×D107C2311yescharacteristic
85Dic428—22no—
86Dic428—41no—
87#13133807—22no—
88#13133807—41no—
89#13133806—22no—
90#13133806—22no—
91#13133806—22no—
92#13133803—22no—
93#13133801—22no—
94C2×C107⋊C8C2311yescharacteristic
95C2×C107⋊C8C2311yescharacteristic
96C8×D107—22no—
97C4⋊Dic107—21no—
98C4⋊Dic107—21no—
99C4⋊Dic107—21no—
100C4⋊Dic107—21no—
101C4⋊Dic107—21no—
102C4⋊Dic107D411yescharacteristic
103(C2×D107)⋊C4—21no—
104(C2×D107)⋊C4—21no—
105(C2×D107)⋊C4—21no—
106(C2×D107)⋊C4—21no—
107(C2×D107)⋊C4—21no—
108(C2×D107)⋊C4—21no—
109(C2×D107)⋊C4D411yescharacteristic
110(C2×D107)⋊C4D411yescharacteristic
111C2×D4×C107C2311yescharacteristic
112C2×D4×C107C2311yescharacteristic
113C22×C4×C107—21no—
114C22×C4×C107—21no—
115D8×C107D414yescomplemented
116C2×C8×C107C2311yescharacteristic
117C2×C8×C107C2311yescharacteristic
118C4⋊C4×C107C2311yescharacteristic
119C4⋊C4×C107C2311yescharacteristic
120C22⋊C4×C107—21no—
121C22⋊C4×C107—21no—
122C42×C107C2311yescharacteristic
123C23×C107—21no—
124C23×C107—21no—
125C22×D107—21no—
126C22×D107—42no—
127C22×D107—42no—
128C22×D107—81no—
129C22×D107C2×D411yescharacteristic
130C22×D107C2×D411yescharacteristic
131D4×C107—21no—
132D4×C107—21no—
133D4×C107C2×D412yes—
134D4×C107C2×D412yes—
135C2×C4×C107—21no—
136C2×C4×C107—21no—
137C2×C4×C107—42no—
138C2×C4×C107—81no—
139C2×C4×C107C2×D411yescharacteristic
140C2×C4×C107C2×D411yescharacteristic
141C2×C4×C107C2411yescharacteristic
142C22⋊D107—42no—
143C22⋊D107—42no—
144C22⋊D107—42no—
145C22⋊D107—42no—
146C2×Dic107—21no—
147C2×Dic107—21no—
148C2×Dic107—21no—
149C2×Dic107—42no—
150C2×Dic107—42no—
151C2×Dic107—42no—
152C2×Dic107C2×D411yescharacteristic
153C2×Dic107C2×D411yescharacteristic
154D428—21no—
155D428—21no—
156D428—41no—
157D428—81no—
158C4×D107—21no—
159C4×D107—21no—
160C4×D107—21no—
161C4×D107—21no—
162Dic214—21no—
163Dic214—21no—
164Dic214—21no—
165Dic214—21no—
166Dic214—21no—
167Dic214—21no—
168C8×C107—21no—
169C8×C107C2×D412yes—
170C107⋊C8—21no—
171C107⋊C8—21no—
172C22×C107—22no—
173C22×C107—22no—
174C22×C107—41no—
175C22×C107—41no—
176C22×C107—41no—
177C22×C107—41no—
178C22×C107C22×D411yescharacteristic
179C2×D107—21no—
180C2×D107—21no—
181C2×D107—21no—
182C2×D107—21no—
183C2×D107—21no—
184C2×D107—21no—
185C2×D107—41no—
186C2×D107—41no—
187C2×D107—41no—
188C2×D107—84no—
189C4×C107—21no—
190C4×C107—21no—
191C4×C107—41no—
192C4×C107—41no—
193C4×C107C22×D411yescharacteristic
194C4×C107H(5,𝔽2)11yescharacteristic
195Dic107—21no—
196Dic107—21no—
197Dic107—41no—
198Dic107—41no—
199Dic107—41no—
200C2×C107—42no—
201C2×C107—42no—
202C2×C107D4211yescharacteristic
203C2×C107#57511yescharacteristic
204C2×C107#57711yescharacteristic
205D107—41no—
206D107—41no—
207D107—81no—
208#3267—1071nomaximal, Sylow
209C107#326711yescharacteristic, complemented, Sylow
210C2×AGL(1,ℤ/8)—1071no—
211C2×#135—1071no—
212#547—1071no—
213#546—1071no—
214#497—1071no—
215#470—1071no—
216#465—1071no—
217#463—1071no—
218#461—1071no—
219#452—1071no—
220#452—1071no—
221#450—1071no—
222#449—1071no—
223#442D10711yescharacteristic, complemented
224#435—1071no—
225C2×#41—1071no—
226C2×#41—1071no—
227C2×#41—1071no—
228C22×D4—1071no—
229AGL(1,ℤ/8)—2144no—
230#135—2144no—
231C2×Q16—1071no—
232C2×Q16—1071no—
233C2×SD16—1071no—
234C2×SD16—1071no—
235C2×SD16—1071no—
236C2×SD16—1071no—
237C2×SD16—1071no—
238C2×D8—1071no—
239C2×D8C2×D10711yescharacteristic
240C2×M16—1071no—
241C22×C8—1071no—
242C4⋊Q8—1071no—
243#124—1071no—
244#123—2141no—
245#123—2141no—
246#123—2141no—
247C22⋊Q8—1071no—
248C22⋊Q8—1071no—
249C22⋊Q8—1071no—
250#121—1071no—
251#121—2141no—
252#121—2141no—
253#120—2141no—
254C4×D4—1071no—
255C4×D4C2×D10711yescharacteristic
256C4×D4C2×D10711yescharacteristic, complemented
257C2×C4⋊C4—2141no—
258C2×C22⋊C4—2141no—
259#106C2×D10711yescharacteristic, complemented
260C4⋊C8—1071no—
261#103—1071no—
262#103—1071no—
263#103—1071no—
264#102—1071no—
265#102C2×D10711yescharacteristic
266#102C2×D10711yescharacteristic, complemented
267C22⋊C8—1071no—
268C22⋊C8—1071no—
269#97C2×D10711yescharacteristic, complemented
270C24—2141no—
271#41—4281no—
272#41—4281no—
273#41—4281no—
274#41—2142no—
275#41—2142no—
276#41—2142no—
277C2×Q8—1071no—
278C2×Q8—1071no—
279C2×Q8—1071no—
280C2×D4—1071no—
281C2×D4—1071no—
282C2×D4—4281no—
283C2×D4—2141no—
284C2×D4—2141no—
285C2×D4—2141no—
286C2×D4—2141no—
287C2×D4—2141no—
288C2×D4—2142no—
289C2×D4C22×D10711yescharacteristic
290C2×D4C22×D10711yescharacteristic
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—1071no—
294C22×C4—1071no—
295C22×C4—2141no—
296C22×C4—2141no—
297C22×C4—2141no—
298Q16—4281no—
299Q16—2142no—
300SD16—4281no—
301SD16—2142no—
302SD16—2142no—
303SD16—2142no—
304SD16—2142no—
305D8—2142no—
306D8D42814yescomplemented
307M16—2142no—
308C2×C8—1071no—
309C2×C8—1071no—
310C2×C8—2142no—
311C2×C8C22×D10711yescharacteristic
312C2×C8C22×D10711yescharacteristic
313C4⋊C4—1071no—
314C4⋊C4—2141no—
315C4⋊C4—2141no—
316C4⋊C4—2141no—
317C4⋊C4—2141no—
318C4⋊C4—2141no—
319C4⋊C4C22×D10711yescharacteristic
320C4⋊C4C22×D10711yescharacteristic
321C22⋊C4—21no—
322C22⋊C4—21no—
323C22⋊C4—1071no—
324C22⋊C4—1071no—
325C22⋊C4—2141no—
326C22⋊C4—2141no—
327C22⋊C4—2141no—
328C22⋊C4—2141no—
329C22⋊C4—2141no—
330C22⋊C4—2141no—
331C42C22×D10711yescharacteristic
332C23—21no—
333C23—21no—
334C23—8561no—
335C23—1071no—
336C23—1071no—
337C23—4282no—
338C23—4282no—
339C23—2141no—
340Q8—2141no—
341Q8—2141no—
342Q8—2141no—
343Q8—2141no—
344Q8—2141no—
345Q8—2141no—
346D4—21no—
347D4—21no—
348D4—8561no—
349D4—4281no—
350D4—4282no—
351D4—4282no—
352D4—4282no—
353D4—4282no—
354D4—2141no—
355D4—2141no—
356D4C2×D42812yes—
357D4C2×D42812yes—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—42no—
361C2×C4—81no—
362C2×C4—1071no—
363C2×C4—1071no—
364C2×C4—4282no—
365C2×C4—4282no—
366C2×C4—4282no—
367C2×C4—2141no—
368C2×C4—2141no—
369C2×C4—2141no—
370C2×C4—2141no—
371C2×C4—2141no—
372C2×C4—2141no—
373C2×C4—2141no—
374C2×C4D4×D10711yescharacteristic
375C2×C4D4×D10711yescharacteristic
376C2×C4C23×D10711yescharacteristic
377C8—21no—
378C8—2141no—
379C8—2141no—
380C8C2×D42812yes—
381C22—22no—
382C22—22no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—8564no—
388C22—4281no—
389C22—4281no—
390C22—4281no—
391C22—2141no—
392C22—2141no—
393C22—2141no—
394C22—2141no—
395C22—2141no—
396C22—2141no—
397C22C2×D4×D10711yescharacteristic
398C4—21no—
399C4—21no—
400C4—41no—
401C4—41no—
402C4—4281no—
403C4—4281no—
404C4—4281no—
405C4—2141no—
406C4—2141no—
407C4C22×D42811yescharacteristic
408C4#1681215511yescharacteristic
409C2—42no—
410C2—42no—
411C2—8561no—
412C2—4281no—
413C2—4281no—
414C2D4×D42811yescharacteristic
415C2#2029342311yescharacteristic
416C2#2029343711yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#442×C107C2
socleC22×C107C22×D4
hypercenter#442D107
hypocenterC107#3267
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #442

Complements

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

NH ≅ G/Ntype
1. C2×#16812133C2semidirect
2. C2×#16812128C2semidirect
3. #20293281C2semidirect
4. #20293272C2semidirect
5. #20293021C2semidirect
6. #20292886C2semidirect
7. #20292857C2semidirect
8. #20292842C2semidirect
9. #20292832C2semidirect
10. #20292790C2semidirect
11. #20292790C2semidirect
12. #20292779C2semidirect
13. #20292773C2semidirect
14. #20292700C2semidirect
15. #442×C107C2semidirect
18. C2×#13133817C22semidirect
22. C2×Dic428C22semidirect
23. C2×Dic428C22semidirect
28. C2×#13133806C22semidirect
31. C2×C8×D107C22semidirect
32. #16812109C22semidirect
39. #16812089C22semidirect
47. C4⋊(C107⋊C8)C22semidirect
48. #16812032C22semidirect
49. #16812032C22semidirect
50. #16812032C22semidirect
53. (C2×D107)⋊C8C22semidirect
56. C4×D4×C107C22semidirect
57. #106×C107C22semidirect
59. #102×C107C22semidirect
60. #97×C107C22semidirect
115. D8×C107D4semidirect
209. C107#3267semidirect
223. #442D107semidirect
256. C4×D4C2×D107semidirect
259. #106C2×D107semidirect
266. #102C2×D107semidirect
269. #97C2×D107semidirect
306. D8D428semidirect

Maximal subgroups

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

subgroupindex
C2×#168121332
C2×#168121282
#202932812
#202932722
#202930212
#202928862
#202928572
#202928422
#202928322
#202927902
#202927792
#202927732
#202927002
#442×C1072
#3267107