Finite Group Databasebeta

← #28346930

Subgroups — #28346930

Order 13696.

564 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812138C211yesmaximal, characteristic, complemented
2C2×#16812136C211yesmaximal, characteristic, complemented
3C2×#16812130C211yesmaximal, characteristic, complemented
4C2×SD16×D107C211yesmaximal, characteristic, complemented
5#20293202C211yesmaximal, characteristic, complemented
6#20293190C211yesmaximal, characteristic, complemented
7#20293019C211yesmaximal, characteristic, complemented
8#20293018C211yesmaximal, characteristic, complemented
9#20293014C211yesmaximal, characteristic, complemented
10#20293013C211yesmaximal, characteristic, complemented
11#20293011C211yesmaximal, characteristic, complemented
12#20293008C211yesmaximal, characteristic, complemented
13#20292998C211yesmaximal, characteristic, complemented
14#20292581C211yesmaximal, characteristic, complemented
15#492×C107C211yesmaximal, characteristic, complemented
16C2×#13133819C2211yescharacteristic
17C2×#13133817C2211yescharacteristic
18C2×Q8×D107C2211yescharacteristic
19C2×D4×D107C2211yescharacteristic
20#16812138—24no—
21#16812136—24no—
22#16812130—24no—
23C2×Dic428C2211yescharacteristic, complemented
24C2×#13133808C2211yescharacteristic, complemented
25C2×#13133808C2211yescharacteristic, complemented
26C2×#13133807C2211yescharacteristic
27C2×#13133807C2211yescharacteristic, complemented
28C2×#13133806C2211yescharacteristic
29C2×#13133806C2211yescharacteristic, complemented
30C2×#13133805C2211yescharacteristic
31C2×#13133805C2211yescharacteristic
32SD16×D107—24no—
33C2×D856C2211yescharacteristic
34C2×#13133803C2211yescharacteristic
35C2×#13133803C2211yescharacteristic, complemented
36C2×#13133801C2211yescharacteristic
37C2×C8×D107C2211yescharacteristic
38#16812107C2211yescharacteristic, complemented
39#16812105C2211yescharacteristic
40#16812098C2211yescharacteristic, complemented
41#16812097C2211yescharacteristic
42#16812097C2211yescharacteristic, complemented
43#16812096C2211yescharacteristic
44#16812094—22no—
45#16812087—22no—
46#16812081—22no—
47#16812077—22no—
48#16812066C2211yescharacteristic
49#16812020C2211yescharacteristic, complemented
50#16812019C2211yescharacteristic, complemented
51C8×Dic107C2211yescharacteristic, complemented
52C2×SD16×C107C2211yescharacteristic
53C2×SD16×C107C2211yescharacteristic
54C2×SD16×C107C2211yescharacteristic, complemented
55C2×SD16×C107C2211yescharacteristic, complemented
56C4⋊Q8×C107C2211yescharacteristic, complemented
57#127×C107C2211yescharacteristic
58C4×C8×C107C2211yescharacteristic, complemented
59C23×D107—21no—
60#13133819—22no—
61#13133819—22no—
62#13133817—22no—
63#13133817—22no—
64C2×Dic214—21no—
65C2×Dic214C2311yescharacteristic
66C2×Dic214C2311yescharacteristic
67Q8×D107—22no—
68Q8×D107—22no—
69C2×D428—21no—
70C2×D428C2311yescharacteristic
71C2×D428C2311yescharacteristic
72C2×C22⋊D107—21no—
73C2×C22⋊D107—21no—
74C2×C22⋊D107—22no—
75D4×D107—22no—
76D4×D107—22no—
77C22×Dic107—21no—
78C2×C4×D107—21no—
79C2×C4×D107—21no—
80C2×C4×D107C2311yescharacteristic
81Dic428—22no—
82#13133808—22no—
83#13133808—22no—
84#13133807—22no—
85#13133807—22no—
86#13133806—22no—
87#13133806—22no—
88#13133805—22no—
89#13133805—22no—
90D856—22no—
91#13133803—22no—
92#13133803—22no—
93#13133801D414yescomplemented
94C2×C107⋊C8C2311yescharacteristic
95C2×C107⋊C8C2311yescharacteristic
96C8×D107D414yescomplemented
97Dic107⋊C4—22no—
98Dic107⋊C4D412yes—
99C22⋊Dic107—22no—
100(C2×D107)⋊C4—22no—
101(C2×D107)⋊C4—22no—
102(C2×D107)⋊C4—22no—
103(C2×D107)⋊C4D412yes—
104C4×Dic107C2311yescharacteristic
105C2×Q8×C107C2311yescharacteristic
106C2×Q8×C107C2311yescharacteristic
107C2×D4×C107—22no—
108C2×D4×C107C2311yescharacteristic
109C2×D4×C107C2311yescharacteristic
110SD16×C107—22no—
111SD16×C107—22no—
112SD16×C107—22no—
113SD16×C107—22no—
114C2×C8×C107C2311yescharacteristic
115C2×C8×C107C2311yescharacteristic
116C4⋊C4×C107—22no—
117C42×C107C2311yescharacteristic
118C23×C107—21no—
119C23×C107—21no—
120C22×D107—21no—
121C22×D107—21no—
122C22×D107—42no—
123C22×D107—82no—
124C22×D107C2×D411yescharacteristic
125Q8×C107—21no—
126Q8×C107—21no—
127Q8×C107—21no—
128Q8×C107—21no—
129D4×C107—21no—
130D4×C107—21no—
131D4×C107—21no—
132D4×C107—21no—
133D4×C107—82no—
134C2×C4×C107—21no—
135C2×C4×C107—21no—
136C2×C4×C107C2×D412yes—
137C2×C4×C107C2411yescharacteristic
138C22⋊D107—42no—
139C22⋊D107—42no—
140C22⋊D107—82no—
141C2×Dic107—21no—
142C2×Dic107—21no—
143C2×Dic107—42no—
144C2×Dic107C2×D412yes—
145C2×Dic107C2×D411yescharacteristic
146D428—21no—
147D428—21no—
148D428—21no—
149D428—21no—
150D428—42no—
151C4×D107—42no—
152C4×D107—42no—
153C4×D107C2×D412yes—
154C4×D107C2×D412yes—
155Dic214—21no—
156Dic214—21no—
157Dic214—21no—
158Dic214—21no—
159Dic214—42no—
160C8×C107C2×D412yes—
161C8×C107C2×D412yes—
162C107⋊C8C2×D412yes—
163C107⋊C8C2×D412yes—
164C22×C107—41no—
165C22×C107—41no—
166C22×C107—42no—
167C22×C107—42no—
168C22×C107C22×D411yescharacteristic
169C2×D107—22no—
170C2×D107—41no—
171C2×D107—41no—
172C2×D107—42no—
173C2×D107—42no—
174C2×D107—84no—
175C2×D107#12712yes—
176C4×C107—22no—
177C4×C107—41no—
178C4×C107—41no—
179C4×C107C22×D411yescharacteristic
180C4×C107C22×D411yescharacteristic
181Dic107—22no—
182Dic107—41no—
183Dic107—41no—
184Dic107#12712yes—
185C2×C107—81no—
186C2×C107—81no—
187C2×C107C2×#12711yescharacteristic
188C2×C107#57712yes—
189D107—22no—
190D107—81no—
191D107—81no—
192#3154—1071nomaximal, Sylow
193C107#315411yescharacteristic, complemented, Sylow
194C2×#137—1071no—
195C2×AGL(1,ℤ/8)—1071no—
196C2×#135—1071no—
197C22×SD16—1071no—
198#536—1071no—
199#535—1071no—
200#497—1071no—
201#497—1071no—
202#496—1071no—
203#496—1071no—
204#495—1071no—
205#495—1071no—
206#492—1071no—
207#492D10711yescharacteristic, complemented
208#405—1071no—
209C2×#41—1071no—
210C2×#41—1071no—
211C22×Q8—1071no—
212C22×D4—1071no—
213#137—2144no—
214AGL(1,ℤ/8)—2144no—
215#135—2144no—
216C2×Q16—1071no—
217C2×Q16—1071no—
218C2×Q16—1071no—
219C2×SD16—1071no—
220C2×SD16—1071no—
221C2×SD16—1071no—
222C2×SD16—1071no—
223C2×SD16—1071no—
224C2×SD16—1071no—
225C2×SD16—2144no—
226C2×SD16C2×D10711yescharacteristic
227C2×SD16C2×D10711yescharacteristic
228C2×SD16C2×D10711yescharacteristic, complemented
229C2×SD16C2×D10711yescharacteristic, complemented
230C2×D8—1071no—
231C2×D8—1071no—
232C2×D8—1071no—
233C2×M16—1071no—
234C22×C8—1071no—
235C4⋊Q8—1071no—
236C4⋊Q8C2×D10711yescharacteristic, complemented
237#127—1071no—
238#127C2×D10711yescharacteristic
239#124—1071no—
240#124—1071no—
241#124—1071no—
242#124—1071no—
243#123—2142no—
244C22⋊Q8—2142no—
245#121—2142no—
246#120—2142no—
247#117—1071no—
248#97—1071no—
249#97—1071no—
250C4×C8—1071no—
251C4×C8C2×D10711yescharacteristic, complemented
252C24—2141no—
253#41—2142no—
254#41—2142no—
255#41—2142no—
256#41—2142no—
257C2×Q8—1071no—
258C2×Q8—1071no—
259C2×Q8—2141no—
260C2×Q8—2142no—
261C2×Q8—2142no—
262C2×Q8C22×D10711yescharacteristic
263C2×Q8C22×D10711yescharacteristic
264C2×D4—22no—
265C2×D4—1071no—
266C2×D4—1071no—
267C2×D4—2141no—
268C2×D4—2142no—
269C2×D4—2142no—
270C2×D4—2142no—
271C2×D4—2141no—
272C2×D4—2141no—
273C2×D4C22×D10711yescharacteristic
274C2×D4C22×D10711yescharacteristic
275C22×C4—1071no—
276C22×C4—2141no—
277C22×C4—2141no—
278C22×C4—2141no—
279Q16—2142no—
280Q16—2142no—
281Q16—2142no—
282SD16—22no—
283SD16—22no—
284SD16—22no—
285SD16—22no—
286SD16—2142no—
287SD16—2142no—
288SD16—2142no—
289SD16—2142no—
290SD16—2142no—
291SD16—2142no—
292D8—2142no—
293D8—2142no—
294D8—2142no—
295M16—1074no—
296C2×C8—1071no—
297C2×C8—1071no—
298C2×C8—1074no—
299C2×C8C22×D10711yescharacteristic
300C2×C8C22×D10711yescharacteristic
301C4⋊C4—22no—
302C4⋊C4—1072no—
303C4⋊C4—2142no—
304C22⋊C4—1072no—
305C22⋊C4—2142no—
306C22⋊C4—2142no—
307C22⋊C4—2142no—
308C22⋊C4—2142no—
309C42—1071no—
310C42C22×D10711yescharacteristic
311C23—21no—
312C23—21no—
313C23—8562no—
314C23—1071no—
315C23—4282no—
316C23—2141no—
317C23—2141no—
318Q8—21no—
319Q8—21no—
320Q8—21no—
321Q8—21no—
322Q8—4282no—
323Q8—2141no—
324Q8—2141no—
325Q8—2141no—
326Q8—2141no—
327D4—21no—
328D4—21no—
329D4—21no—
330D4—21no—
331D4—82no—
332D4—8562no—
333D4—4282no—
334D4—4282no—
335D4—4282no—
336D4—2141no—
337D4—2141no—
338D4—2141no—
339D4—2141no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—1071no—
343C2×C4—1072no—
344C2×C4—1072no—
345C2×C4—1072no—
346C2×C4—4282no—
347C2×C4—4282no—
348C2×C4—4282no—
349C2×C4—2141no—
350C2×C4—2141no—
351C2×C4D4×D10712yes—
352C2×C4C23×D10711yescharacteristic
353C8—1072no—
354C8—1072no—
355C8D4×D10712yes—
356C8D4×D10712yes—
357C22—41no—
358C22—41no—
359C22—42no—
360C22—42no—
361C22—8564no—
362C22—1072no—
363C22—4281no—
364C22—4281no—
365C22—4282no—
366C22—4282no—
367C22—2142no—
368C22C2×D4×D10711yescharacteristic
369C4—22no—
370C4—41no—
371C4—41no—
372C4—1072no—
373C4—4281no—
374C4—4281no—
375C4—2142no—
376C4C2×D4×D10711yescharacteristic
377C4C2×D4×D10711yescharacteristic
378C2—81no—
379C2—81no—
380C2—8561no—
381C2—8561no—
382C2—2142no—
383C2#127×D10711yescharacteristic
384C2#2029343312yes—
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#492×C107C2
socleC22×C107C22×D4
hypercenter#492D107
hypocenterC107#3154
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #492

Complements

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

NH ≅ G/Ntype
1. C2×#16812138C2semidirect
2. C2×#16812136C2semidirect
3. C2×#16812130C2semidirect
4. C2×SD16×D107C2semidirect
5. #20293202C2semidirect
6. #20293190C2semidirect
7. #20293019C2semidirect
8. #20293018C2semidirect
9. #20293014C2semidirect
10. #20293013C2semidirect
11. #20293011C2semidirect
12. #20293008C2semidirect
13. #20292998C2semidirect
14. #20292581C2semidirect
15. #492×C107C2semidirect
23. C2×Dic428C22semidirect
24. C2×#13133808C22semidirect
25. C2×#13133808C22semidirect
27. C2×#13133807C22semidirect
29. C2×#13133806C22semidirect
35. C2×#13133803C22semidirect
38. #16812107C22semidirect
40. #16812098C22semidirect
42. #16812097C22semidirect
49. #16812020C22semidirect
50. #16812019C22semidirect
51. C8×Dic107C22semidirect
54. C2×SD16×C107C22semidirect
55. C2×SD16×C107C22semidirect
56. C4⋊Q8×C107C22semidirect
58. C4×C8×C107C22semidirect
93. #13133801D4semidirect
96. C8×D107D4semidirect
193. C107#3154semidirect
207. #492D107semidirect
228. C2×SD16C2×D107semidirect
229. C2×SD16C2×D107semidirect
236. C4⋊Q8C2×D107semidirect
251. C4×C8C2×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121382
C2×#168121362
C2×#168121302
C2×SD16×D1072
#202932022
#202931902
#202930192
#202930182
#202930142
#202930132
#202930112
#202930082
#202929982
#202925812
#492×C1072
#3154107