Finite Group Databasebeta

← #28347992

Subgroups — #28347992

Order 13696.

500 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812136C211yesmaximal, characteristic, complemented
2C2×#16812130C211yesmaximal, characteristic, complemented
3#20293280C211yesmaximal, characteristic, complemented
4#20293272C211yesmaximal, characteristic, complemented
5#20293019C211yesmaximal, characteristic, complemented
6#20292884C211yesmaximal, characteristic, complemented
7#20292859C211yesmaximal, characteristic, complemented
8#20292843C211yesmaximal, characteristic, complemented
9#20292792C211yesmaximal, characteristic, complemented
10#20292786C211yesmaximal, characteristic, complemented
11#20292776C211yesmaximal, characteristic, complemented
12#20292775C211yesmaximal, characteristic, complemented
13#20292737C211yesmaximal, characteristic, complemented
14#20292697C211yesmaximal, characteristic, complemented
15#461×C107C211yesmaximal, characteristic, complemented
16C2×#13133819C2211yescharacteristic
17C2×#13133819C2211yescharacteristic, complemented
18C2×#13133817C2211yescharacteristic
19C2×D4×D107C2211yescharacteristic
20#16812136—24no—
21#16812130—24no—
22C2×#13133808C2211yescharacteristic, complemented
23C2×#13133808C2211yescharacteristic, complemented
24C2×#13133807C2211yescharacteristic, complemented
25C2×#13133805C2211yescharacteristic
26C2×#13133805C2211yescharacteristic
27C2×D856C2211yescharacteristic
28C2×#13133803C2211yescharacteristic
29C2×#13133801C2211yescharacteristic
30C2×C8×D107C2211yescharacteristic, complemented
31#16812097C2211yescharacteristic
32#16812094—21no—
33#16812094—21no—
34#16812093—21no—
35#16812089C2211yescharacteristic
36#16812085C2211yescharacteristic
37#16812085C2211yescharacteristic, complemented
38#16812084—21no—
39#16812084C2211yescharacteristic
40#16812083—21no—
41#16812079—21no—
42C4×D428C2211yescharacteristic
43C4×D428C2211yescharacteristic, complemented
44C4⋊C4×D107—21no—
45C2×(C2×D107)⋊C4—21no—
46#16812040C2211yescharacteristic, complemented
47#16812034C2211yescharacteristic, complemented
48#16812031C2211yescharacteristic
49#16812030C2211yescharacteristic
50#16812029C2211yescharacteristic, complemented
51(C2×D107)⋊C8C2211yescharacteristic
52(C2×D107)⋊C8C2211yescharacteristic, complemented
53#16812020C2211yescharacteristic, complemented
54C2×SD16×C107C2211yescharacteristic
55C2×SD16×C107C2211yescharacteristic
56C4⋊Q8×C107C2211yescharacteristic, complemented
57C4×D4×C107C2211yescharacteristic
58C4⋊C8×C107C2211yescharacteristic, complemented
59#103×C107C2211yescharacteristic, complemented
60#103×C107C2211yescharacteristic, complemented
61C23×D107—21no—
62#13133819—22no—
63#13133819—22no—
64#13133819—41no—
65#13133819—41no—
66#13133817—22no—
67#13133817—41no—
68C2×Dic214C2311yescharacteristic
69C2×D428—21no—
70C2×D428—21no—
71C2×D428—21no—
72C2×D428D411yescharacteristic
73C2×D428C2311yescharacteristic
74C2×D428C2311yescharacteristic
75C2×C22⋊D107—21no—
76C2×C22⋊D107—21no—
77D4×D107—22no—
78D4×D107—41no—
79C22×Dic107—21no—
80C2×C4×D107—21no—
81C2×C4×D107—21no—
82C2×C4×D107—21no—
83C2×C4×D107C2311yescharacteristic
84C2×C4×D107C2311yescharacteristic
85#13133808—22no—
86#13133808—41no—
87#13133807—22no—
88#13133805—22no—
89#13133805—41no—
90D856—22no—
91#13133803D414yescomplemented
92#13133801—22no—
93C2×C107⋊C8C2311yescharacteristic
94C2×C107⋊C8C2311yescharacteristic
95C8×D107—22no—
96C4⋊Dic107—21no—
97C4⋊Dic107D411yescharacteristic
98C4⋊Dic107C2311yescharacteristic
99Dic107⋊C4—21no—
100Dic107⋊C4—21no—
101(C2×D107)⋊C4—21no—
102(C2×D107)⋊C4—21no—
103(C2×D107)⋊C4—21no—
104(C2×D107)⋊C4—21no—
105(C2×D107)⋊C4—21no—
106(C2×D107)⋊C4—21no—
107(C2×D107)⋊C4—21no—
108(C2×D107)⋊C4D411yescharacteristic
109(C2×D107)⋊C4D411yescharacteristic
110C2×Q8×C107C2311yescharacteristic
111C2×Q8×C107C2311yescharacteristic
112C2×D4×C107C2311yescharacteristic
113C22×C4×C107—21no—
114SD16×C107—22no—
115SD16×C107—22no—
116C2×C8×C107C2311yescharacteristic
117C2×C8×C107C2311yescharacteristic
118C4⋊C4×C107—21no—
119C4⋊C4×C107—21no—
120C4⋊C4×C107C2311yescharacteristic
121C22⋊C4×C107—21no—
122C42×C107C2311yescharacteristic
123C23×C107—21no—
124C22×D107—21no—
125C22×D107—21no—
126C22×D107—42no—
127C22×D107—42no—
128C22×D107—81no—
129C22×D107C2×D411yescharacteristic
130C22×D107C2×D411yescharacteristic
131Q8×C107—21no—
132Q8×C107—21no—
133Q8×C107—21no—
134Q8×C107—21no—
135D4×C107—21no—
136D4×C107C2×D412yes—
137C2×C4×C107—21no—
138C2×C4×C107—21no—
139C2×C4×C107—21no—
140C2×C4×C107—81no—
141C2×C4×C107C2×D411yescharacteristic
142C2×C4×C107C2×D411yescharacteristic
143C2×C4×C107C2411yescharacteristic
144C22⋊D107—42no—
145C22⋊D107—42no—
146C2×Dic107—21no—
147C2×Dic107—21no—
148C2×Dic107—42no—
149C2×Dic107C2×D411yescharacteristic
150C2×Dic107C2×D411yescharacteristic
151D428—21no—
152D428—21no—
153D428—21no—
154D428—41no—
155D428—42no—
156D428—42no—
157D428—81no—
158D428C2×D412yes—
159C4×D107—21no—
160C4×D107—21no—
161C4×D107—21no—
162C4×D107—21no—
163C4×D107—42no—
164C4×D107—42no—
165C4×D107—42no—
166Dic214—21no—
167Dic214—21no—
168C8×C107—21no—
169C8×C107—21no—
170C107⋊C8—21no—
171C107⋊C8C2×D412yes—
172C22×C107—22no—
173C22×C107—41no—
174C22×C107—41no—
175C22×C107C22×D411yescharacteristic
176C2×D107—21no—
177C2×D107—21no—
178C2×D107—22no—
179C2×D107—21no—
180C2×D107—21no—
181C2×D107—21no—
182C2×D107—21no—
183C2×D107—41no—
184C2×D107—41no—
185C2×D107—41no—
186C2×D107—41no—
187C2×D107—41no—
188C2×D107—84no—
189C4×C107—21no—
190C4×C107—21no—
191C4×C107—41no—
192C4×C107—41no—
193C4×C107—41no—
194C4×C107C22×D411yescharacteristic
195C4×C107H(5,𝔽2)11yescharacteristic
196Dic107—21no—
197Dic107—21no—
198Dic107—41no—
199Dic107—41no—
200C2×C107—42no—
201C2×C107D4211yescharacteristic
202C2×C107#57511yescharacteristic
203C2×C107#57711yescharacteristic
204D107—41no—
205D107—41no—
206D107—42no—
207D107—81no—
208#3267—1071nomaximal, Sylow
209C107#326711yescharacteristic, complemented, Sylow
210C2×AGL(1,ℤ/8)—1071no—
211C2×#135—1071no—
212#547—1071no—
213#546—1071no—
214#497—1071no—
215#470—1071no—
216#465—1071no—
217#463—1071no—
218#461D10711yescharacteristic, complemented
219#452—1071no—
220#452—1071no—
221#450—1071no—
222#449—1071no—
223#442—1071no—
224#435—1071no—
225C2×#41—1071no—
226C2×#41—1071no—
227C2×#41—1071no—
228C22×D4—1071no—
229AGL(1,ℤ/8)—2144no—
230#135—2144no—
231C2×Q16—1071no—
232C2×Q16—1071no—
233C2×SD16—1071no—
234C2×SD16—1071no—
235C2×SD16—1071no—
236C2×SD16C2×D10711yescharacteristic
237C2×SD16C2×D10711yescharacteristic
238C2×D8—1071no—
239C2×D8—1071no—
240C2×M16—1071no—
241C22×C8—1071no—
242C4⋊Q8C2×D10711yescharacteristic, complemented
243#124—1071no—
244#123—2141no—
245#123—2141no—
246#123—2141no—
247C22⋊Q8—1071no—
248C22⋊Q8—1071no—
249C22⋊Q8—1071no—
250#121—1071no—
251#121—2141no—
252#121—2141no—
253#120—2141no—
254C4×D4—1071no—
255C4×D4—1071no—
256C4×D4C2×D10711yescharacteristic
257C2×C4⋊C4—2141no—
258C2×C22⋊C4—2141no—
259#106—1071no—
260C4⋊C8C2×D10711yescharacteristic, complemented
261#103—1071no—
262#103C2×D10711yescharacteristic, complemented
263#103C2×D10711yescharacteristic, complemented
264#102—1071no—
265#102—1071no—
266#102—1071no—
267C22⋊C8—1071no—
268C22⋊C8—1071no—
269#97—1071no—
270C24—2141no—
271#41—4281no—
272#41—4281no—
273#41—4281no—
274#41—2142no—
275#41—2142no—
276#41—2142no—
277C2×Q8—1071no—
278C2×Q8C22×D10711yescharacteristic
279C2×Q8C22×D10711yescharacteristic
280C2×D4—1071no—
281C2×D4—1071no—
282C2×D4—1071no—
283C2×D4—4281no—
284C2×D4—2141no—
285C2×D4—2141no—
286C2×D4—2141no—
287C2×D4—2141no—
288C2×D4—2141no—
289C2×D4—2142no—
290C2×D4C22×D10711yescharacteristic
291C22×C4—21no—
292C22×C4—1071no—
293C22×C4—1071no—
294C22×C4—2141no—
295C22×C4—2141no—
296C22×C4—2141no—
297C22×C4—2141no—
298Q16—4281no—
299Q16—2142no—
300SD16—22no—
301SD16—22no—
302SD16—4281no—
303SD16—2142no—
304SD16—2142no—
305D8—1074no—
306D8—2142no—
307M16—2142no—
308C2×C8—1071no—
309C2×C8—1071no—
310C2×C8—2142no—
311C2×C8C22×D10711yescharacteristic
312C2×C8C22×D10711yescharacteristic
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—1071no—
316C4⋊C4—1071no—
317C4⋊C4—2141no—
318C4⋊C4—2141no—
319C4⋊C4—2141no—
320C4⋊C4C22×D10711yescharacteristic
321C22⋊C4—21no—
322C22⋊C4—1071no—
323C22⋊C4—1071no—
324C22⋊C4—2141no—
325C22⋊C4—2141no—
326C22⋊C4—2141no—
327C22⋊C4—2141no—
328C22⋊C4—2141no—
329C22⋊C4—2141no—
330C22⋊C4—2141no—
331C42C22×D10711yescharacteristic
332C23—21no—
333C23—8561no—
334C23—1071no—
335C23—1071no—
336C23—4282no—
337C23—4282no—
338C23—2141no—
339C23—2141no—
340Q8—21no—
341Q8—21no—
342Q8—21no—
343Q8—21no—
344Q8—2141no—
345Q8—2141no—
346D4—21no—
347D4—8561no—
348D4—1072no—
349D4—4281no—
350D4—4282no—
351D4—4282no—
352D4—4282no—
353D4—4282no—
354D4—2141no—
355D4—2141no—
356D4—2141no—
357D4D4×D10712yes—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—81no—
362C2×C4—1071no—
363C2×C4—1071no—
364C2×C4—4282no—
365C2×C4—4282no—
366C2×C4—4282no—
367C2×C4—4282no—
368C2×C4—2141no—
369C2×C4—2141no—
370C2×C4—2141no—
371C2×C4—2141no—
372C2×C4—2141no—
373C2×C4—2141no—
374C2×C4D4×D10711yescharacteristic
375C2×C4D4×D10711yescharacteristic
376C2×C4C23×D10711yescharacteristic
377C8—21no—
378C8—21no—
379C8—1072no—
380C8—2141no—
381C22—22no—
382C22—41no—
383C22—41no—
384C22—8564no—
385C22—4281no—
386C22—4281no—
387C22—4281no—
388C22—4281no—
389C22—4281no—
390C22—2141no—
391C22—2141no—
392C22—2141no—
393C22—2141no—
394C22—2142no—
395C22—2141no—
396C22—2141no—
397C22C2×D4×D10711yescharacteristic
398C4—21no—
399C4—21no—
400C4—41no—
401C4—41no—
402C4—41no—
403C4—4281no—
404C4—4281no—
405C4—2141no—
406C4—2141no—
407C4C2×D4×D10711yescharacteristic
408C4#1681215511yescharacteristic
409C2—42no—
410C2—8561no—
411C2—4281no—
412C2—4282no—
413C2—4281no—
414C2#2029326111yescharacteristic
415C2#2029342611yescharacteristic
416C2#2029343311yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#461×C107C2
socleC22×C107C22×D4
hypercenter#461D107
hypocenterC107#3267
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. C2×#16812136C2semidirect
2. C2×#16812130C2semidirect
3. #20293280C2semidirect
4. #20293272C2semidirect
5. #20293019C2semidirect
6. #20292884C2semidirect
7. #20292859C2semidirect
8. #20292843C2semidirect
9. #20292792C2semidirect
10. #20292786C2semidirect
11. #20292776C2semidirect
12. #20292775C2semidirect
13. #20292737C2semidirect
14. #20292697C2semidirect
15. #461×C107C2semidirect
17. C2×#13133819C22semidirect
22. C2×#13133808C22semidirect
23. C2×#13133808C22semidirect
24. C2×#13133807C22semidirect
30. C2×C8×D107C22semidirect
37. #16812085C22semidirect
43. C4×D428C22semidirect
46. #16812040C22semidirect
47. #16812034C22semidirect
50. #16812029C22semidirect
52. (C2×D107)⋊C8C22semidirect
53. #16812020C22semidirect
56. C4⋊Q8×C107C22semidirect
58. C4⋊C8×C107C22semidirect
59. #103×C107C22semidirect
60. #103×C107C22semidirect
91. #13133803D4semidirect
209. C107#3267semidirect
218. #461D107semidirect
242. C4⋊Q8C2×D107semidirect
260. C4⋊C8C2×D107semidirect
262. #103C2×D107semidirect
263. #103C2×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121362
C2×#168121302
#202932802
#202932722
#202930192
#202928842
#202928592
#202928432
#202927922
#202927862
#202927762
#202927752
#202927372
#202926972
#461×C1072
#3267107