Finite Group Databasebeta

← #28346945

Subgroups — #28346945

Order 13696.

596 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812136C211yesmaximal, characteristic, complemented
2C2×#16812133C211yesmaximal, characteristic, complemented
3C2×#16812130C211yesmaximal, characteristic, complemented
4C2×D8×D107C211yesmaximal, characteristic, complemented
5#20293195C211yesmaximal, characteristic, complemented
6#20293190C211yesmaximal, characteristic, complemented
7#20293016C211yesmaximal, characteristic, complemented
8#20293015C211yesmaximal, characteristic, complemented
9#20293013C211yesmaximal, characteristic, complemented
10#20293009C211yesmaximal, characteristic, complemented
11#20293008C211yesmaximal, characteristic, complemented
12#20293003C211yesmaximal, characteristic, complemented
13#20293002C211yesmaximal, characteristic, complemented
14#20292584C211yesmaximal, characteristic, complemented
15#496×C107C211yesmaximal, characteristic, complemented
16C2×#13133819C2211yescharacteristic
17C2×#13133817C2211yescharacteristic
18C2×D4×D107C2211yescharacteristic
19C2×D4×D107C2211yescharacteristic
20#16812136—24no—
21#16812133—24no—
22#16812130—24no—
23C2×#13133808C2211yescharacteristic, complemented
24C2×#13133807C2211yescharacteristic, complemented
25C2×#13133807C2211yescharacteristic, complemented
26C2×#13133806C2211yescharacteristic, complemented
27C2×#13133805C2211yescharacteristic, complemented
28C2×D856C2211yescharacteristic
29C2×D856C2211yescharacteristic, complemented
30C2×#13133803C2211yescharacteristic
31C2×#13133803C2211yescharacteristic
32C2×#13133803C2211yescharacteristic, complemented
33C2×#13133803C2211yescharacteristic, complemented
34C2×#13133803C2211yescharacteristic, complemented
35D8×D107—24no—
36C2×#13133801C2211yescharacteristic
37C2×C8×D107C2211yescharacteristic
38#16812106C2211yescharacteristic, complemented
39#16812105C2211yescharacteristic, complemented
40#16812105C2211yescharacteristic, complemented
41#16812098C2211yescharacteristic, complemented
42#16812097C2211yescharacteristic, complemented
43#16812096C2211yescharacteristic, complemented
44#16812082—22no—
45#16812081—22no—
46#16812079—22no—
47#16812077—22no—
48#16812066C2211yescharacteristic
49#16812019C2211yescharacteristic, complemented
50C4×C107⋊C8C2211yescharacteristic, complemented
51C8×Dic107C2211yescharacteristic, complemented
52C2×SD16×C107C2211yescharacteristic, complemented
53C2×SD16×C107C2211yescharacteristic, complemented
54C2×D8×C107C2211yescharacteristic
55C2×D8×C107C2211yescharacteristic, complemented
56#127×C107C2211yescharacteristic, complemented
57#124×C107C2211yescharacteristic, complemented
58#97×C107C2211yescharacteristic, complemented
59C23×D107—21no—
60C23×D107—21no—
61#13133819—22no—
62#13133819—22no—
63#13133817—22no—
64#13133817—22no—
65C2×Dic214C2311yescharacteristic
66C2×D428—21no—
67C2×D428—22no—
68C2×D428C2311yescharacteristic
69C2×D428C2311yescharacteristic
70C2×D428C2311yescharacteristic
71C2×C22⋊D107—21no—
72C2×C22⋊D107—21no—
73C2×C22⋊D107—21no—
74C2×C22⋊D107—22no—
75C2×C22⋊D107—22no—
76D4×D107—22no—
77D4×D107—22no—
78D4×D107—22no—
79D4×D107—22no—
80C22×Dic107—21no—
81C2×C4×D107—21no—
82C2×C4×D107C2311yescharacteristic
83#13133808—22no—
84#13133807—22no—
85#13133807—22no—
86#13133806—22no—
87#13133805—22no—
88D856—22no—
89D856—22no—
90#13133803—22no—
91#13133803—22no—
92#13133803—22no—
93#13133803—22no—
94#13133803—22no—
95#13133801D414yescomplemented
96C2×C107⋊C8C2311yescharacteristic
97C2×C107⋊C8C2311yescharacteristic
98C8×D107D414yescomplemented
99Dic107⋊C4D412yes—
100C22⋊Dic107—22no—
101(C2×D107)⋊C4—22no—
102(C2×D107)⋊C4—22no—
103(C2×D107)⋊C4D412yes—
104C4×Dic107C2311yescharacteristic
105C2×Q8×C107C2311yescharacteristic
106C2×D4×C107—22no—
107C2×D4×C107C2311yescharacteristic
108C2×D4×C107C2311yescharacteristic
109C2×D4×C107C2311yescharacteristic
110SD16×C107—22no—
111SD16×C107—22no—
112D8×C107—22no—
113D8×C107—22no—
114C2×C8×C107C2311yescharacteristic
115C2×C8×C107C2311yescharacteristic
116C22⋊C4×C107—22no—
117C42×C107C2311yescharacteristic
118C23×C107—21no—
119C23×C107—21no—
120C23×C107—21no—
121C22×D107—21no—
122C22×D107—21no—
123C22×D107—21no—
124C22×D107—42no—
125C22×D107—42no—
126C22×D107—82no—
127C22×D107—82no—
128C22×D107C2×D411yescharacteristic
129Q8×C107—21no—
130Q8×C107—21no—
131D4×C107—21no—
132D4×C107—21no—
133D4×C107—21no—
134D4×C107—21no—
135D4×C107—21no—
136D4×C107—21no—
137D4×C107—82no—
138C2×C4×C107—21no—
139C2×C4×C107C2×D412yes—
140C2×C4×C107C2411yescharacteristic
141C22⋊D107—42no—
142C22⋊D107—42no—
143C22⋊D107—42no—
144C22⋊D107—82no—
145C22⋊D107—82no—
146C2×Dic107—21no—
147C2×Dic107—42no—
148C2×Dic107C2×D412yes—
149C2×Dic107C2×D411yescharacteristic
150D428—21no—
151D428—21no—
152D428—21no—
153D428—21no—
154D428—21no—
155D428—21no—
156D428—42no—
157D428—82no—
158C4×D107—42no—
159C4×D107C2×D412yes—
160C4×D107C2×D412yes—
161Dic214—21no—
162Dic214—21no—
163C8×C107C2×D412yes—
164C8×C107C2×D412yes—
165C107⋊C8C2×D412yes—
166C107⋊C8C2×D412yes—
167C22×C107—41no—
168C22×C107—41no—
169C22×C107—41no—
170C22×C107—42no—
171C22×C107—42no—
172C22×C107—42no—
173C22×C107C22×D411yescharacteristic
174C2×D107—22no—
175C2×D107—41no—
176C2×D107—41no—
177C2×D107—41no—
178C2×D107—42no—
179C2×D107—42no—
180C2×D107—42no—
181C2×D107—84no—
182C2×D107—84no—
183C2×D107#12712yes—
184C4×C107—22no—
185C4×C107—41no—
186C4×C107C22×D411yescharacteristic
187C4×C107C22×D411yescharacteristic
188Dic107—22no—
189Dic107—41no—
190Dic107#12712yes—
191C2×C107—81no—
192C2×C107—81no—
193C2×C107—81no—
194C2×C107C2×#12711yescharacteristic
195C2×C107#57612yes—
196D107—22no—
197D107—81no—
198D107—81no—
199D107—81no—
200#3155—1071nomaximal, Sylow
201C107#315511yescharacteristic, complemented, Sylow
202C2×AGL(1,ℤ/8)—1071no—
203C2×AGL(1,ℤ/8)—1071no—
204C2×#135—1071no—
205C22×D8—1071no—
206#535—1071no—
207#535—1071no—
208#496—1071no—
209#496—1071no—
210#496—1071no—
211#496D10711yescharacteristic, complemented
212#495—1071no—
213#495—1071no—
214#493—1071no—
215#493—1071no—
216#405—1071no—
217C2×#41—1071no—
218C2×#41—1071no—
219C22×D4—1071no—
220C22×D4—1071no—
221AGL(1,ℤ/8)—2144no—
222AGL(1,ℤ/8)—2144no—
223#135—2144no—
224C2×Q16—1071no—
225C2×SD16—1071no—
226C2×SD16—1071no—
227C2×SD16—1071no—
228C2×SD16—1071no—
229C2×SD16C2×D10711yescharacteristic, complemented
230C2×SD16C2×D10711yescharacteristic, complemented
231C2×D8—1071no—
232C2×D8—1071no—
233C2×D8—1071no—
234C2×D8—1071no—
235C2×D8—1071no—
236C2×D8—1071no—
237C2×D8—1071no—
238C2×D8—2144no—
239C2×D8C2×D10711yescharacteristic
240C2×D8C2×D10711yescharacteristic, complemented
241C2×M16—1071no—
242C22×C8—1071no—
243#127—1071no—
244#127—1071no—
245#127—1071no—
246#127C2×D10711yescharacteristic, complemented
247#124—1071no—
248#124—1071no—
249#124—1071no—
250#124C2×D10711yescharacteristic, complemented
251#121—2142no—
252#121—2142no—
253#120—2142no—
254#120—2142no—
255#117—1071no—
256#97—1071no—
257#97C2×D10711yescharacteristic, complemented
258C4×C8—1071no—
259C4×C8—1071no—
260C24—2141no—
261C24—2141no—
262#41—2142no—
263#41—2142no—
264#41—2142no—
265#41—2142no—
266C2×Q8—1071no—
267C2×Q8C22×D10711yescharacteristic
268C2×D4—22no—
269C2×D4—1071no—
270C2×D4—1071no—
271C2×D4—1071no—
272C2×D4—2141no—
273C2×D4—2142no—
274C2×D4—2142no—
275C2×D4—2142no—
276C2×D4—2142no—
277C2×D4—2142no—
278C2×D4—2142no—
279C2×D4—2141no—
280C2×D4—2141no—
281C2×D4—2141no—
282C2×D4—2142no—
283C2×D4C22×D10711yescharacteristic
284C2×D4C22×D10711yescharacteristic
285C2×D4C22×D10711yescharacteristic
286C22×C4—1071no—
287C22×C4—2141no—
288C22×C4—2141no—
289Q16—2142no—
290SD16—22no—
291SD16—22no—
292SD16—2142no—
293SD16—2142no—
294SD16—2142no—
295SD16—2142no—
296D8—22no—
297D8—22no—
298D8—2142no—
299D8—2142no—
300D8—2142no—
301D8—2142no—
302D8—2142no—
303D8—2142no—
304D8—2142no—
305M16—1074no—
306C2×C8—1071no—
307C2×C8—1071no—
308C2×C8—1074no—
309C2×C8C22×D10711yescharacteristic
310C2×C8C22×D10711yescharacteristic
311C4⋊C4—1072no—
312C22⋊C4—22no—
313C22⋊C4—1072no—
314C22⋊C4—2142no—
315C22⋊C4—2142no—
316C22⋊C4—2142no—
317C42—1071no—
318C42C22×D10711yescharacteristic
319C23—21no—
320C23—21no—
321C23—21no—
322C23—8562no—
323C23—8562no—
324C23—1071no—
325C23—4282no—
326C23—4282no—
327C23—2141no—
328C23—2141no—
329C23—2141no—
330Q8—21no—
331Q8—21no—
332Q8—2141no—
333Q8—2141no—
334D4—21no—
335D4—21no—
336D4—21no—
337D4—21no—
338D4—21no—
339D4—21no—
340D4—82no—
341D4—8562no—
342D4—8562no—
343D4—8562no—
344D4—4282no—
345D4—4282no—
346D4—4282no—
347D4—4282no—
348D4—2141no—
349D4—2141no—
350D4—2141no—
351D4—2141no—
352D4—2141no—
353D4—2141no—
354C2×C4—21no—
355C2×C4—1071no—
356C2×C4—1072no—
357C2×C4—1072no—
358C2×C4—1072no—
359C2×C4—4282no—
360C2×C4—4282no—
361C2×C4—2141no—
362C2×C4D4×D10712yes—
363C2×C4C23×D10711yescharacteristic
364C8—1072no—
365C8—1072no—
366C8D4×D10712yes—
367C8D4×D10712yes—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—42no—
372C22—42no—
373C22—42no—
374C22—8564no—
375C22—8564no—
376C22—1072no—
377C22—4281no—
378C22—4281no—
379C22—4281no—
380C22—4282no—
381C22—4282no—
382C22—4282no—
383C22—2142no—
384C22C2×D4×D10711yescharacteristic
385C4—22no—
386C4—41no—
387C4—1072no—
388C4—4281no—
389C4—2142no—
390C4C2×D4×D10711yescharacteristic
391C4C2×D4×D10711yescharacteristic
392C2—81no—
393C2—81no—
394C2—81no—
395C2—8561no—
396C2—8561no—
397C2—8561no—
398C2—2142no—
399C2#127×D10711yescharacteristic
400C2#2029343112yes—
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#496×C107C2
socleC22×C107C22×D4
hypercenter#496D107
hypocenterC107#3155
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×#16812136C2semidirect
2. C2×#16812133C2semidirect
3. C2×#16812130C2semidirect
4. C2×D8×D107C2semidirect
5. #20293195C2semidirect
6. #20293190C2semidirect
7. #20293016C2semidirect
8. #20293015C2semidirect
9. #20293013C2semidirect
10. #20293009C2semidirect
11. #20293008C2semidirect
12. #20293003C2semidirect
13. #20293002C2semidirect
14. #20292584C2semidirect
15. #496×C107C2semidirect
23. C2×#13133808C22semidirect
24. C2×#13133807C22semidirect
25. C2×#13133807C22semidirect
26. C2×#13133806C22semidirect
27. C2×#13133805C22semidirect
29. C2×D856C22semidirect
32. C2×#13133803C22semidirect
33. C2×#13133803C22semidirect
34. C2×#13133803C22semidirect
38. #16812106C22semidirect
39. #16812105C22semidirect
40. #16812105C22semidirect
41. #16812098C22semidirect
42. #16812097C22semidirect
43. #16812096C22semidirect
49. #16812019C22semidirect
50. C4×C107⋊C8C22semidirect
51. C8×Dic107C22semidirect
52. C2×SD16×C107C22semidirect
53. C2×SD16×C107C22semidirect
55. C2×D8×C107C22semidirect
56. #127×C107C22semidirect
57. #124×C107C22semidirect
58. #97×C107C22semidirect
95. #13133801D4semidirect
98. C8×D107D4semidirect
201. C107#3155semidirect
211. #496D107semidirect
229. C2×SD16C2×D107semidirect
230. C2×SD16C2×D107semidirect
240. C2×D8C2×D107semidirect
246. #127C2×D107semidirect
250. #124C2×D107semidirect
257. #97C2×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121362
C2×#168121332
C2×#168121302
C2×D8×D1072
#202931952
#202931902
#202930162
#202930152
#202930132
#202930092
#202930082
#202930032
#202930022
#202925842
#496×C1072
#3155107