Finite Group Databasebeta

← #28346932

Subgroups — #28346932

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×#16812141C211yesmaximal, characteristic, complemented
2C2×#16812133C211yesmaximal, characteristic, complemented
3C2×#16812130C211yesmaximal, characteristic, complemented
4C2×SD16×D107C211yesmaximal, characteristic, complemented
5#20293197C211yesmaximal, characteristic, complemented
6#20293195C211yesmaximal, characteristic, complemented
7#20293022C211yesmaximal, characteristic, complemented
8#20293020C211yesmaximal, characteristic, complemented
9#20293017C211yesmaximal, characteristic, complemented
10#20293015C211yesmaximal, characteristic, complemented
11#20293008C211yesmaximal, characteristic, complemented
12#20293000C211yesmaximal, characteristic, complemented
13#20292998C211yesmaximal, characteristic, complemented
14#20292581C211yesmaximal, characteristic, complemented
15#495×C107C211yesmaximal, characteristic, complemented
16C2×#13133819C2211yescharacteristic
17C2×#13133817C2211yescharacteristic
18C2×Q8×D107C2211yescharacteristic
19C2×D4×D107C2211yescharacteristic
20#16812141—24no—
21#16812133—24no—
22#16812130—24no—
23C2×#13133808C2211yescharacteristic, complemented
24C2×#13133808C2211yescharacteristic, complemented
25C2×#13133807C2211yescharacteristic
26C2×#13133807C2211yescharacteristic
27C2×#13133807C2211yescharacteristic, complemented
28C2×#13133807C2211yescharacteristic, complemented
29C2×#13133806C2211yescharacteristic
30C2×#13133806C2211yescharacteristic
31C2×#13133805C2211yescharacteristic
32C2×#13133805C2211yescharacteristic, complemented
33SD16×D107—24no—
34C2×#13133803C2211yescharacteristic
35C2×#13133803C2211yescharacteristic, complemented
36C2×#13133801C2211yescharacteristic
37C2×C8×D107C2211yescharacteristic
38#16812109C2211yescharacteristic, complemented
39#16812107C2211yescharacteristic, complemented
40#16812106C2211yescharacteristic
41#16812105C2211yescharacteristic
42#16812098C2211yescharacteristic
43#16812096C2211yescharacteristic, complemented
44#16812091—22no—
45#16812086—22no—
46#16812082—22no—
47#16812079—22no—
48#16812066C2211yescharacteristic
49#16812020C2211yescharacteristic, complemented
50#16812019C2211yescharacteristic, complemented
51C8×Dic107C2211yescharacteristic, 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×D428—22no—
71C2×D428C2311yescharacteristic
72C2×D428C2311yescharacteristic
73C2×C22⋊D107—21no—
74C2×C22⋊D107—21no—
75C2×C22⋊D107—22no—
76D4×D107—22no—
77D4×D107—22no—
78C22×Dic107—21no—
79C2×C4×D107—21no—
80C2×C4×D107—21no—
81C2×C4×D107C2311yescharacteristic
82#13133808—22no—
83#13133808—22no—
84#13133807—22no—
85#13133807—22no—
86#13133807—22no—
87#13133807—22no—
88#13133806—22no—
89#13133806—22no—
90#13133805—22no—
91#13133805—22no—
92#13133803—22no—
93#13133803—22no—
94#13133801D414yescomplemented
95C2×C107⋊C8C2311yescharacteristic
96C2×C107⋊C8C2311yescharacteristic
97C8×D107D414yescomplemented
98C4⋊Dic107—22no—
99Dic107⋊C4—22no—
100Dic107⋊C4D412yes—
101C22⋊Dic107—22no—
102(C2×D107)⋊C4—22no—
103(C2×D107)⋊C4D412yes—
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—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—
133C2×C4×C107—21no—
134C2×C4×C107—21no—
135C2×C4×C107C2×D412yes—
136C2×C4×C107C2411yescharacteristic
137C22⋊D107—42no—
138C22⋊D107—42no—
139C22⋊D107—82no—
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—
150D428—82no—
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#495D10711yescharacteristic, complemented
206#492—1071no—
207#492—1071no—
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×Q16C2×D10711yescharacteristic, complemented
219C2×SD16—1071no—
220C2×SD16—1071no—
221C2×SD16—1071no—
222C2×SD16—1071no—
223C2×SD16—1071no—
224C2×SD16—1071no—
225C2×SD16—1071no—
226C2×SD16—1071no—
227C2×SD16—2144no—
228C2×SD16C2×D10711yescharacteristic
229C2×SD16C2×D10711yescharacteristic, complemented
230C2×D8—1071no—
231C2×D8—1071no—
232C2×D8C2×D10711yescharacteristic
233C2×M16—1071no—
234C22×C8—1071no—
235C4⋊Q8—1071no—
236C4⋊Q8—1071no—
237#127—1071no—
238#127—1071no—
239#124—1071no—
240#124—1071no—
241#124C2×D10711yescharacteristic
242#124C2×D10711yescharacteristic, complemented
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—1071no—
265C2×D4—1071no—
266C2×D4—2141no—
267C2×D4—2142no—
268C2×D4—2142no—
269C2×D4—2142no—
270C2×D4—2141no—
271C2×D4—2141no—
272C2×D4—2142no—
273C2×D4C22×D10711yescharacteristic
274C2×D4C22×D10711yescharacteristic
275C22×C4—1071no—
276C22×C4—2141no—
277C22×C4—2141no—
278C22×C4—2141no—
279Q16—22no—
280Q16—2142no—
281Q16—2142no—
282SD16—22no—
283SD16—22no—
284SD16—2142no—
285SD16—2142no—
286SD16—2142no—
287SD16—2142no—
288SD16—2142no—
289SD16—2142no—
290SD16—2142no—
291SD16—2142no—
292D8—22no—
293D8—2142no—
294D8—2142no—
295M16—1074no—
296C2×C8—1071no—
297C2×C8—1071no—
298C2×C8—1074no—
299C2×C8C22×D10711yescharacteristic
300C2×C8C22×D10711yescharacteristic
301C4⋊C4—1072no—
302C4⋊C4—2142no—
303C4⋊C4—2142no—
304C22⋊C4—22no—
305C22⋊C4—22no—
306C22⋊C4—1072no—
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—8562no—
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#2029343412yes—
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#495×C107C2
socleC22×C107C22×D4
hypercenter#495D107
hypocenterC107#3154
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×#16812133C2semidirect
3. C2×#16812130C2semidirect
4. C2×SD16×D107C2semidirect
5. #20293197C2semidirect
6. #20293195C2semidirect
7. #20293022C2semidirect
8. #20293020C2semidirect
9. #20293017C2semidirect
10. #20293015C2semidirect
11. #20293008C2semidirect
12. #20293000C2semidirect
13. #20292998C2semidirect
14. #20292581C2semidirect
15. #495×C107C2semidirect
23. C2×#13133808C22semidirect
24. C2×#13133808C22semidirect
27. C2×#13133807C22semidirect
28. C2×#13133807C22semidirect
32. C2×#13133805C22semidirect
35. C2×#13133803C22semidirect
38. #16812109C22semidirect
39. #16812107C22semidirect
43. #16812096C22semidirect
49. #16812020C22semidirect
50. #16812019C22semidirect
51. C8×Dic107C22semidirect
52. C2×Q16×C107C22semidirect
54. C2×SD16×C107C22semidirect
57. #124×C107C22semidirect
58. C4×C8×C107C22semidirect
94. #13133801D4semidirect
97. C8×D107D4semidirect
193. C107#3154semidirect
205. #495D107semidirect
218. C2×Q16C2×D107semidirect
229. C2×SD16C2×D107semidirect
242. #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×#168121332
C2×#168121302
C2×SD16×D1072
#202931972
#202931952
#202930222
#202930202
#202930172
#202930152
#202930082
#202930002
#202929982
#202925812
#495×C1072
#3154107