Finite Group Databasebeta

← #28346922

Subgroups — #28346922

Order 13696.

572 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×#16812141C211yesmaximal, characteristic, complemented
2C2×#16812138C211yesmaximal, characteristic, complemented
3C2×#16812136C211yesmaximal, characteristic, complemented
4C2×#16812133C211yesmaximal, characteristic, complemented
5#20293176C211yesmaximal, characteristic, complemented
6#124×D107C211yesmaximal, characteristic, complemented
7#20293020C211yesmaximal, characteristic, complemented
8#20293018C211yesmaximal, characteristic, complemented
9#20293015C211yesmaximal, characteristic, complemented
10#20293013C211yesmaximal, characteristic, complemented
11#20293011C211yesmaximal, characteristic, complemented
12#20293009C211yesmaximal, characteristic, complemented
13#20293009C211yesmaximal, characteristic, complemented
14C4×#13133801C211yesmaximal, characteristic, complemented
15#495×C107C211yesmaximal, characteristic, complemented
16C2×#13133819C2211yescharacteristic
17C2×#13133817C2211yescharacteristic
18C2×Q8×D107C2211yescharacteristic
19C2×D4×D107C2211yescharacteristic
20#16812141—24no—
21#16812138—24no—
22#16812136—24no—
23#16812133—24no—
24C2×Dic428C2211yescharacteristic, complemented
25C2×#13133808C2211yescharacteristic, complemented
26C2×#13133808C2211yescharacteristic, complemented
27C2×#13133807C2211yescharacteristic
28C2×#13133807C2211yescharacteristic, complemented
29C2×#13133806C2211yescharacteristic
30C2×#13133806C2211yescharacteristic, complemented
31C2×#13133805C2211yescharacteristic
32C2×#13133805C2211yescharacteristic, complemented
33C2×D856C2211yescharacteristic
34C2×#13133803C2211yescharacteristic
35C2×#13133803C2211yescharacteristic
36C2×#13133801C2211yescharacteristic
37C2×#13133801C2211yescharacteristic
38#16812107C2211yescharacteristic, complemented
39#16812105C2211yescharacteristic, complemented
40#16812098C2211yescharacteristic
41#16812097C2211yescharacteristic
42#16812097C2211yescharacteristic, complemented
43#16812096C2211yescharacteristic
44#16812095—22no—
45#16812082—22no—
46(C22⋊D107)⋊C4—22no—
47C22⋊C4×D107—22no—
48C42×D107C2211yescharacteristic
49#16812019C2211yescharacteristic, complemented
50#16812019C2211yescharacteristic, complemented
51C4×C107⋊C8C2211yescharacteristic, complemented
52C2×Q16×C107C2211yescharacteristic, complemented
53C2×SD16×C107C2211yescharacteristic
54C2×SD16×C107C2211yescharacteristic, complemented
55C2×D8×C107C2211yescharacteristic
56#124×C107C2211yescharacteristic
57#124×C107C2211yescharacteristic, complemented
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—22no—
74C2×C22⋊D107—21no—
75D4×D107—22no—
76D4×D107—22no—
77C22×Dic107—21no—
78C2×C4×D107—21no—
79C2×C4×D107—21no—
80C2×C4×D107D412yes—
81C2×C4×D107C2311yescharacteristic
82Dic428—22no—
83#13133808—22no—
84#13133808—22no—
85#13133807—22no—
86#13133807—22no—
87#13133806—22no—
88#13133806—22no—
89#13133805—22no—
90#13133805—22no—
91D856—22no—
92#13133803—22no—
93#13133803—22no—
94#13133801D414yescomplemented
95#13133801D414yescomplemented
96C2×C107⋊C8C2311yescharacteristic
97C2×C107⋊C8C2311yescharacteristic
98Dic107⋊C4—22no—
99C22⋊Dic107—22no—
100(C2×D107)⋊C4—22no—
101(C2×D107)⋊C4—22no—
102(C2×D107)⋊C4—22no—
103C4×Dic107D412yes—
104C4×Dic107C2311yescharacteristic
105C2×Q8×C107C2311yescharacteristic
106C2×Q8×C107C2311yescharacteristic
107C2×D4×C107C2311yescharacteristic
108C2×D4×C107C2311yescharacteristic
109Q16×C107—22no—
110SD16×C107—22no—
111SD16×C107—22no—
112D8×C107—22no—
113C2×C8×C107C2311yescharacteristic
114C2×C8×C107C2311yescharacteristic
115C22⋊C4×C107—22no—
116C22⋊C4×C107—22no—
117C42×C107C2311yescharacteristic
118C23×C107—21no—
119C23×C107—21no—
120C22×D107—21no—
121C22×D107—21no—
122C22×D107—44no—
123C22×D107—42no—
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—
133C2×C4×C107—21no—
134C2×C4×C107—21no—
135C2×C4×C107C2×D412yes—
136C2×C4×C107C2411yescharacteristic
137C22⋊D107—44no—
138C22⋊D107—42no—
139C22⋊D107—42no—
140C2×Dic107—21no—
141C2×Dic107—21no—
142C2×Dic107—42no—
143C2×Dic107C2×D412yes—
144C2×Dic107C2×D411yescharacteristic
145D428—21no—
146D428—21no—
147D428—21no—
148D428—21no—
149D428—42no—
150C4×D107—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#57512yes—
189D107—22no—
190D107—81no—
191D107—81no—
192#3153—1071nomaximal, Sylow
193C107#315311yescharacteristic, complemented, Sylow
194C2×#137—1071no—
195C2×#137—1071no—
196C2×AGL(1,ℤ/8)—1071no—
197C2×AGL(1,ℤ/8)—1071no—
198#532—1071no—
199C2×#124—1071no—
200#497—1071no—
201#497—1071no—
202#496—1071no—
203#496—1071no—
204#495—1071no—
205#495—1071no—
206#495—1071no—
207#495D10711yescharacteristic, complemented
208C4×M16—1071no—
209C2×#41—1071no—
210C2×#41—1071no—
211C22×Q8—1071no—
212C22×D4—1071no—
213#137—2144no—
214#137—2144no—
215AGL(1,ℤ/8)—2144no—
216AGL(1,ℤ/8)—2144no—
217C2×Q16—1071no—
218C2×Q16—1071no—
219C2×Q16—1071no—
220C2×Q16C2×D10711yescharacteristic, complemented
221C2×SD16—1071no—
222C2×SD16—1071no—
223C2×SD16—1071no—
224C2×SD16—1071no—
225C2×SD16—1071no—
226C2×SD16—1071no—
227C2×SD16C2×D10711yescharacteristic
228C2×SD16C2×D10711yescharacteristic, complemented
229C2×D8—1071no—
230C2×D8—1071no—
231C2×D8—1071no—
232C2×D8C2×D10711yescharacteristic
233C2×M16—1071no—
234C2×M16—1071no—
235C4⋊Q8—1071no—
236#127—1071no—
237#124—1071no—
238#124—1071no—
239#124—1071no—
240#124—1071no—
241#124—2142no—
242#124C2×D10711yescharacteristic
243#124C2×D10711yescharacteristic, complemented
244#121—2142no—
245C4×D4—2142no—
246C2×C22⋊C4—2142no—
247C2×C42—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—1071no—
265C2×D4—1071no—
266C2×D4—2141no—
267C2×D4—2142no—
268C2×D4—2142no—
269C2×D4—2142no—
270C2×D4—2141no—
271C2×D4—2141no—
272C2×D4C22×D10711yescharacteristic
273C2×D4C22×D10711yescharacteristic
274C22×C4—1071no—
275C22×C4—1072no—
276C22×C4—2141no—
277C22×C4—2141no—
278C22×C4—2141no—
279Q16—22no—
280Q16—2142no—
281Q16—2142no—
282Q16—2142no—
283SD16—22no—
284SD16—22no—
285SD16—2142no—
286SD16—2142no—
287SD16—2142no—
288SD16—2142no—
289SD16—2142no—
290SD16—2142no—
291D8—22no—
292D8—2142no—
293D8—2142no—
294D8—2142no—
295M16—1074no—
296M16—1074no—
297C2×C8—1071no—
298C2×C8—1071no—
299C2×C8C22×D10711yescharacteristic
300C2×C8C22×D10711yescharacteristic
301C4⋊C4—2142no—
302C22⋊C4—22no—
303C22⋊C4—22no—
304C22⋊C4—2142no—
305C22⋊C4—2142no—
306C22⋊C4—2142no—
307C22⋊C4—2142no—
308C42—1072no—
309C42—1071no—
310C42C22×D10711yescharacteristic
311C23—21no—
312C23—21no—
313C23—1071no—
314C23—4282no—
315C23—4284no—
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—4282no—
332D4—4282no—
333D4—4282no—
334D4—4284no—
335D4—2141no—
336D4—2141no—
337D4—2141no—
338D4—2141no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—1071no—
342C2×C4—1072no—
343C2×C4—1072no—
344C2×C4—1072no—
345C2×C4—4282no—
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#2029342412yes—
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#495×C107C2
socleC22×C107C22×D4
hypercenter#495D107
hypocenterC107#3153
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #495

Complements

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

NH ≅ G/Ntype
1. C2×#16812141C2semidirect
2. C2×#16812138C2semidirect
3. C2×#16812136C2semidirect
4. C2×#16812133C2semidirect
5. #20293176C2semidirect
6. #124×D107C2semidirect
7. #20293020C2semidirect
8. #20293018C2semidirect
9. #20293015C2semidirect
10. #20293013C2semidirect
11. #20293011C2semidirect
12. #20293009C2semidirect
13. #20293009C2semidirect
14. C4×#13133801C2semidirect
15. #495×C107C2semidirect
24. C2×Dic428C22semidirect
25. C2×#13133808C22semidirect
26. C2×#13133808C22semidirect
28. C2×#13133807C22semidirect
30. C2×#13133806C22semidirect
32. C2×#13133805C22semidirect
38. #16812107C22semidirect
39. #16812105C22semidirect
42. #16812097C22semidirect
49. #16812019C22semidirect
50. #16812019C22semidirect
51. C4×C107⋊C8C22semidirect
52. C2×Q16×C107C22semidirect
54. C2×SD16×C107C22semidirect
57. #124×C107C22semidirect
58. C4×C8×C107C22semidirect
94. #13133801D4semidirect
95. #13133801D4semidirect
193. C107#3153semidirect
207. #495D107semidirect
220. C2×Q16C2×D107semidirect
228. C2×SD16C2×D107semidirect
243. #124C2×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×#168121412
C2×#168121382
C2×#168121362
C2×#168121332
#202931762
#124×D1072
#202930202
#202930182
#202930152
#202930132
#202930112
#202930092
C4×#131338012
#495×C1072
#3153107