Finite Group Databasebeta

← #28346936

Subgroups — #28346936

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×#16812136C211yesmaximal, characteristic, complemented
3C2×#16812132C211yesmaximal, characteristic, complemented
4C2×SD16×D107C211yesmaximal, characteristic, complemented
5#20293202C211yesmaximal, characteristic, complemented
6#20293195C211yesmaximal, characteristic, complemented
7#20293020C211yesmaximal, characteristic, complemented
8#20293016C211yesmaximal, characteristic, complemented
9#20293013C211yesmaximal, characteristic, complemented
10#20293012C211yesmaximal, characteristic, complemented
11#20293009C211yesmaximal, characteristic, complemented
12#20293001C211yesmaximal, characteristic, complemented
13#20292998C211yesmaximal, characteristic, complemented
14#20292584C211yesmaximal, characteristic, complemented
15#497×C107C211yesmaximal, characteristic, complemented
16C2×#13133819C2211yescharacteristic
17C2×#13133819C2211yescharacteristic
18C2×Q8×D107C2211yescharacteristic
19C2×D4×D107C2211yescharacteristic
20#16812141—24no—
21#16812136—24no—
22#16812132—24no—
23C2×#13133808C2211yescharacteristic, complemented
24C2×#13133807C2211yescharacteristic
25C2×#13133807C2211yescharacteristic, complemented
26C2×#13133806C2211yescharacteristic
27C2×#13133805C2211yescharacteristic
28C2×#13133805C2211yescharacteristic
29C2×#13133805C2211yescharacteristic, complemented
30C2×#13133805C2211yescharacteristic, complemented
31C2×#13133805C2211yescharacteristic, complemented
32SD16×D107—24no—
33C2×D856C2211yescharacteristic
34C2×D856C2211yescharacteristic, complemented
35C2×#13133803C2211yescharacteristic
36C2×#13133801C2211yescharacteristic
37C2×C8×D107C2211yescharacteristic
38#16812107C2211yescharacteristic, complemented
39#16812106C2211yescharacteristic
40#16812105C2211yescharacteristic
41#16812098C2211yescharacteristic
42#16812098C2211yescharacteristic, complemented
43#16812097C2211yescharacteristic, complemented
44#16812094—22no—
45#16812087—22no—
46#16812082—22no—
47#16812079—22no—
48#16812066C2211yescharacteristic
49#16812019C2211yescharacteristic, complemented
50C4×C107⋊C8C2211yescharacteristic, complemented
51C8×Dic107C2211yescharacteristic, complemented
52C2×Q16×C107C2211yescharacteristic, complemented
53C2×Q16×C107C2211yescharacteristic, complemented
54C2×SD16×C107C2211yescharacteristic
55C2×SD16×C107C2211yescharacteristic
56C4⋊Q8×C107C2211yescharacteristic, complemented
57#124×C107C2211yescharacteristic
58#97×C107C2211yescharacteristic, complemented
59C23×D107—21no—
60#13133819—22no—
61#13133819—22no—
62#13133819—22no—
63#13133819—22no—
64C2×Dic214—21no—
65C2×Dic214C2311yescharacteristic
66Q8×D107—22no—
67Q8×D107—22no—
68C2×D428—21no—
69C2×D428—21no—
70C2×D428—22no—
71C2×D428C2311yescharacteristic
72C2×D428C2311yescharacteristic
73C2×D428C2311yescharacteristic
74C2×C22⋊D107—21no—
75C2×C22⋊D107—22no—
76D4×D107—22no—
77D4×D107—22no—
78C2×C4×D107—21no—
79C2×C4×D107—21no—
80C2×C4×D107—21no—
81C2×C4×D107C2311yescharacteristic
82#13133808—22no—
83#13133807—22no—
84#13133807—22no—
85#13133806—22no—
86#13133805—22no—
87#13133805—22no—
88#13133805—22no—
89#13133805—22no—
90#13133805—22no—
91D856—22no—
92D856—22no—
93#13133803—22no—
94#13133801D414yescomplemented
95C2×C107⋊C8C2311yescharacteristic
96C2×C107⋊C8C2311yescharacteristic
97C8×D107D414yescomplemented
98Dic107⋊C4—22no—
99Dic107⋊C4D412yes—
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×Q8×C107C2311yescharacteristic
108C2×D4×C107C2311yescharacteristic
109Q16×C107—22no—
110Q16×C107—22no—
111SD16×C107—22no—
112SD16×C107—22no—
113C2×C8×C107C2311yescharacteristic
114C2×C8×C107C2311yescharacteristic
115C4⋊C4×C107—22no—
116C22⋊C4×C107—22no—
117C42×C107C2311yescharacteristic
118C23×C107—21no—
119C22×D107—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—
129Q8×C107—21no—
130Q8×C107—21no—
131D4×C107—21no—
132D4×C107—21no—
133C2×C4×C107—21no—
134C2×C4×C107—21no—
135C2×C4×C107—21no—
136C2×C4×C107C2×D412yes—
137C2×C4×C107C2411yescharacteristic
138C22⋊D107—42no—
139C22⋊D107—82no—
140C2×Dic107—21no—
141C2×Dic107C2×D412yes—
142C2×Dic107C2×D411yescharacteristic
143D428—21no—
144D428—21no—
145D428—21no—
146D428—21no—
147D428—21no—
148D428—21no—
149D428—42no—
150D428—42no—
151D428—82no—
152C4×D107—42no—
153C4×D107—42no—
154C4×D107—42no—
155C4×D107C2×D412yes—
156C4×D107C2×D412yes—
157Dic214—21no—
158Dic214—21no—
159Dic214—42no—
160C8×C107C2×D412yes—
161C8×C107C2×D412yes—
162C107⋊C8C2×D412yes—
163C107⋊C8C2×D412yes—
164C22×C107—41no—
165C22×C107—42no—
166C22×C107C22×D411yescharacteristic
167C2×D107—22no—
168C2×D107—41no—
169C2×D107—41no—
170C2×D107—41no—
171C2×D107—42no—
172C2×D107—42no—
173C2×D107—42no—
174C2×D107—84no—
175C2×D107#12712yes—
176C4×C107—22no—
177C4×C107—41no—
178C4×C107—41no—
179C4×C107—41no—
180C4×C107C22×D411yescharacteristic
181C4×C107C22×D411yescharacteristic
182Dic107—22no—
183Dic107—41no—
184Dic107#12712yes—
185C2×C107—81no—
186C2×C107C2×#12711yescharacteristic
187C2×C107#57712yes—
188D107—22no—
189D107—81no—
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#497D10711yescharacteristic, complemented
202#496—1071no—
203#496—1071no—
204#495—1071no—
205#495—1071no—
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×Q16C2×D10711yescharacteristic, complemented
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
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#127—1071no—
239#124—1071no—
240#124—1071no—
241#124—1071no—
242#124C2×D10711yescharacteristic
243#123—2142no—
244C22⋊Q8—2142no—
245#121—2142no—
246#120—2142no—
247#117—1071no—
248#97—1071no—
249#97C2×D10711yescharacteristic, complemented
250C4×C8—1071no—
251C4×C8—1071no—
252C24—2141no—
253#41—2142no—
254#41—2142no—
255#41—2142no—
256#41—2142no—
257C2×Q8—1071no—
258C2×Q8—2141no—
259C2×Q8—2142no—
260C2×Q8—2142no—
261C2×Q8C22×D10711yescharacteristic
262C2×Q8C22×D10711yescharacteristic
263C2×Q8C22×D10711yescharacteristic
264C2×D4—1071no—
265C2×D4—1071no—
266C2×D4—1071no—
267C2×D4—2141no—
268C2×D4—2141no—
269C2×D4—2142no—
270C2×D4—2142no—
271C2×D4—2142no—
272C2×D4—2141no—
273C2×D4—2142no—
274C2×D4C22×D10711yescharacteristic
275C22×C4—1071no—
276C22×C4—2141no—
277C22×C4—2141no—
278C22×C4—2141no—
279Q16—22no—
280Q16—22no—
281Q16—2142no—
282SD16—22no—
283SD16—22no—
284SD16—2142no—
285SD16—2142no—
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—22no—
305C22⋊C4—1072no—
306C22⋊C4—2142no—
307C22⋊C4—2142no—
308C22⋊C4—2142no—
309C42—1071no—
310C42C22×D10711yescharacteristic
311C23—21no—
312C23—8562no—
313C23—1071no—
314C23—4282no—
315C23—2141no—
316C23—2141no—
317C23—2141no—
318Q8—21no—
319Q8—21no—
320Q8—21no—
321Q8—21no—
322Q8—21no—
323Q8—21no—
324Q8—4282no—
325Q8—2141no—
326Q8—2141no—
327D4—21no—
328D4—21no—
329D4—8562no—
330D4—8562no—
331D4—4282no—
332D4—4282no—
333D4—4282no—
334D4—2141no—
335D4—2141no—
336D4—2141no—
337D4—2141no—
338D4—2141no—
339D4—2141no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—1071no—
344C2×C4—1072no—
345C2×C4—1072no—
346C2×C4—1072no—
347C2×C4—4282no—
348C2×C4—4282no—
349C2×C4—4282no—
350C2×C4—2141no—
351C2×C4D4×D10712yes—
352C2×C4C23×D10711yescharacteristic
353C8—1072no—
354C8—1072no—
355C8D4×D10712yes—
356C8D4×D10712yes—
357C22—41no—
358C22—42no—
359C22—8564no—
360C22—1072no—
361C22—4281no—
362C22—4281no—
363C22—4281no—
364C22—4282no—
365C22—4282no—
366C22—4282no—
367C22—2142no—
368C22C2×D4×D10711yescharacteristic
369C4—22no—
370C4—41no—
371C4—41no—
372C4—41no—
373C4—1072no—
374C4—4281no—
375C4—2142no—
376C4C2×D4×D10711yescharacteristic
377C4C2×D4×D10711yescharacteristic
378C2—81no—
379C2—8561no—
380C2—8561no—
381C2—8561no—
382C2—2142no—
383C2#127×D10711yescharacteristic
384C2#2029343612yes—
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#497×C107C2
socleC22×C107C22×D4
hypercenter#497D107
hypocenterC107#3154
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. C2×#16812141C2semidirect
2. C2×#16812136C2semidirect
3. C2×#16812132C2semidirect
4. C2×SD16×D107C2semidirect
5. #20293202C2semidirect
6. #20293195C2semidirect
7. #20293020C2semidirect
8. #20293016C2semidirect
9. #20293013C2semidirect
10. #20293012C2semidirect
11. #20293009C2semidirect
12. #20293001C2semidirect
13. #20292998C2semidirect
14. #20292584C2semidirect
15. #497×C107C2semidirect
23. C2×#13133808C22semidirect
25. C2×#13133807C22semidirect
29. C2×#13133805C22semidirect
30. C2×#13133805C22semidirect
31. C2×#13133805C22semidirect
34. C2×D856C22semidirect
38. #16812107C22semidirect
42. #16812098C22semidirect
43. #16812097C22semidirect
49. #16812019C22semidirect
50. C4×C107⋊C8C22semidirect
51. C8×Dic107C22semidirect
52. C2×Q16×C107C22semidirect
53. C2×Q16×C107C22semidirect
56. C4⋊Q8×C107C22semidirect
58. #97×C107C22semidirect
94. #13133801D4semidirect
97. C8×D107D4semidirect
193. C107#3154semidirect
201. #497D107semidirect
217. C2×Q16C2×D107semidirect
218. C2×Q16C2×D107semidirect
236. C4⋊Q8C2×D107semidirect
249. #97C2×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121412
C2×#168121362
C2×#168121322
C2×SD16×D1072
#202932022
#202931952
#202930202
#202930162
#202930132
#202930122
#202930092
#202930012
#202929982
#202925842
#497×C1072
#3154107