Finite Group Databasebeta

← #28347983

Subgroups — #28347983

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×#16812131C211yesmaximal, characteristic, complemented
3#20293279C211yesmaximal, characteristic, complemented
4#20293274C211yesmaximal, characteristic, complemented
5#20293018C211yesmaximal, characteristic, complemented
6#20292886C211yesmaximal, characteristic, complemented
7#20292841C211yesmaximal, characteristic, complemented
8#20292831C211yesmaximal, characteristic, complemented
9#20292792C211yesmaximal, characteristic, complemented
10#20292787C211yesmaximal, characteristic, complemented
11#20292779C211yesmaximal, characteristic, complemented
12#20292770C211yesmaximal, characteristic, complemented
13#20292738C211yesmaximal, characteristic, complemented
14#20292698C211yesmaximal, characteristic, complemented
15#465×C107C211yesmaximal, characteristic, complemented
16C2×#13133819C2211yescharacteristic
17C2×#13133818C2211yescharacteristic
18C2×#13133818C2211yescharacteristic, complemented
19C2×D4×D107C2211yescharacteristic
20#16812136—24no—
21#16812131—24no—
22C2×Dic428C2211yescharacteristic, complemented
23C2×#13133808C2211yescharacteristic, complemented
24C2×#13133807C2211yescharacteristic
25C2×#13133807C2211yescharacteristic, complemented
26C2×#13133806C2211yescharacteristic
27C2×#13133805C2211yescharacteristic
28C2×D856C2211yescharacteristic
29C2×#13133803C2211yescharacteristic
30C2×#13133801C2211yescharacteristic
31#16812107C2211yescharacteristic, complemented
32#16812096C2211yescharacteristic
33#16812094—21no—
34#16812091—21no—
35#16812091—21no—
36#16812089C2211yescharacteristic
37#16812089C2211yescharacteristic
38#16812082—21no—
39#16812080—21no—
40#16812077—21no—
41#16812071C2211yescharacteristic
42#16812071C2211yescharacteristic, complemented
43C4×C22⋊D107C2211yescharacteristic
44C4⋊C4×D107—21no—
45C22⋊C4×D107—21no—
46Dic107⋊C8C2211yescharacteristic, complemented
47#16812032C2211yescharacteristic, complemented
48#16812032C2211yescharacteristic, complemented
49#16812029C2211yescharacteristic
50#16812029C2211yescharacteristic, complemented
51C22⋊(C107⋊C8)C2211yescharacteristic, complemented
52(C2×D107)⋊C8C2211yescharacteristic
53#16812019C2211yescharacteristic, complemented
54C2×SD16×C107C2211yescharacteristic
55C22×C8×C107C2211yescharacteristic, complemented
56C22⋊Q8×C107C2211yescharacteristic, complemented
57#121×C107C2211yescharacteristic
58#106×C107C2211yescharacteristic, complemented
59#103×C107C2211yescharacteristic, complemented
60#102×C107C2211yescharacteristic
61C23×D107—21no—
62#13133819—22no—
63#13133819—41no—
64#13133818—22no—
65#13133818—22no—
66#13133818—41no—
67#13133818—41no—
68C2×Dic214C2311yescharacteristic
69C2×Dic214C2311yescharacteristic
70C2×D428—21no—
71C2×D428C2311yescharacteristic
72C2×D428C2311yescharacteristic
73C2×C22⋊D107—21no—
74C2×C22⋊D107—21no—
75C2×C22⋊D107—21no—
76C2×C22⋊D107D411yescharacteristic
77D4×D107—22no—
78D4×D107—41no—
79C2×C4×D107—21no—
80C2×C4×D107—21no—
81C2×C4×D107—21no—
82C2×C4×D107—21no—
83C2×C4×D107—21no—
84C2×C4×D107C2311yescharacteristic
85Dic428—22no—
86#13133808—41no—
87#13133807—22no—
88#13133807—22no—
89#13133806—41no—
90#13133805—22no—
91D856D414yescomplemented
92#13133803—22no—
93#13133801—22no—
94C2×C107⋊C8C2311yescharacteristic
95C2×C107⋊C8C2311yescharacteristic
96C4⋊Dic107—21no—
97C4⋊Dic107—21no—
98Dic107⋊C4—21no—
99Dic107⋊C4—21no—
100Dic107⋊C4D411yescharacteristic
101C22⋊Dic107—21no—
102C22⋊Dic107—21no—
103C22⋊Dic107D411yescharacteristic
104(C2×D107)⋊C4—21no—
105(C2×D107)⋊C4—21no—
106(C2×D107)⋊C4—21no—
107(C2×D107)⋊C4—21no—
108(C2×D107)⋊C4D411yescharacteristic
109C4×Dic107C2311yescharacteristic
110C2×Q8×C107C2311yescharacteristic
111C2×D4×C107—21no—
112C2×D4×C107C2311yescharacteristic
113C22×C4×C107C2311yescharacteristic
114SD16×C107—22no—
115C2×C8×C107—22no—
116C2×C8×C107C2311yescharacteristic
117C2×C8×C107C2311yescharacteristic
118C4⋊C4×C107—21no—
119C4⋊C4×C107C2311yescharacteristic
120C4⋊C4×C107C2311yescharacteristic
121C22⋊C4×C107—21no—
122C22⋊C4×C107—21no—
123C23×C107—21no—
124C23×C107C2×D411yescharacteristic
125C22×D107—21no—
126C22×D107—21no—
127C22×D107—42no—
128C22×D107—42no—
129C22×D107—81no—
130C22×D107C2×D411yescharacteristic
131Q8×C107—21no—
132Q8×C107—21no—
133D4×C107—21no—
134D4×C107—21no—
135D4×C107—81no—
136C2×C4×C107—21no—
137C2×C4×C107—21no—
138C2×C4×C107—21no—
139C2×C4×C107—21no—
140C2×C4×C107—21no—
141C2×C4×C107C2×D411yescharacteristic
142C2×C4×C107C2411yescharacteristic
143C22⋊D107—41no—
144C22⋊D107—42no—
145C22⋊D107—42no—
146C22⋊D107—42no—
147C2×Dic107—21no—
148C2×Dic107—21no—
149C2×Dic107C2×D411yescharacteristic
150C2×Dic107C2×D411yescharacteristic
151C2×Dic107C2×D411yescharacteristic
152D428—21no—
153D428—21no—
154D428—42no—
155D428C2×D412yes—
156D428C2×D412yes—
157C4×D107—21no—
158C4×D107—21no—
159C4×D107—42no—
160C4×D107—42no—
161C4×D107—42no—
162C4×D107—42no—
163C4×D107—81no—
164Dic214—21no—
165Dic214—21no—
166Dic214—21no—
167Dic214—21no—
168C8×C107—21no—
169C8×C107C2×D412yes—
170C107⋊C8—21no—
171C107⋊C8—21no—
172C22×C107—21no—
173C22×C107—21no—
174C22×C107—21no—
175C22×C107—41no—
176C22×C107—41no—
177C22×C107—41no—
178C22×C107C22×D411yescharacteristic
179C2×D107—21no—
180C2×D107—22no—
181C2×D107—22no—
182C2×D107—21no—
183C2×D107—21no—
184C2×D107—41no—
185C2×D107—41no—
186C2×D107—41no—
187C2×D107—41no—
188C2×D107—84no—
189C4×C107—21no—
190C4×C107—41no—
191C4×C107—41no—
192C4×C107—41no—
193C4×C107C22×D411yescharacteristic
194C4×C107H(5,𝔽2)11yescharacteristic
195Dic107—21no—
196Dic107—21no—
197Dic107—21no—
198Dic107—41no—
199Dic107—41no—
200C2×C107—41no—
201C2×C107—81no—
202C2×C107D4211yescharacteristic
203C2×C107#57511yescharacteristic
204C2×C107#57711yescharacteristic
205D107—41no—
206D107—42no—
207D107—42no—
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#465D10711yescharacteristic, complemented
217#463—1071no—
218#461—1071no—
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×SD16—1071no—
237C2×SD16C2×D10711yescharacteristic
238C2×D8—1071no—
239C2×D8—1071no—
240C2×M16—1071no—
241C22×C8C2×D10711yescharacteristic, complemented
242C4⋊Q8—1071no—
243#124—1071no—
244#123—2141no—
245#123—2141no—
246#123—2141no—
247C22⋊Q8—1071no—
248C22⋊Q8—1071no—
249C22⋊Q8C2×D10711yescharacteristic, complemented
250#121—2141no—
251#121—2141no—
252#121C2×D10711yescharacteristic
253#120—2141no—
254C4×D4—1071no—
255C4×D4—1071no—
256C4×D4—1071no—
257C2×C4⋊C4—2141no—
258C2×C22⋊C4—2141no—
259#106C2×D10711yescharacteristic, complemented
260C4⋊C8—1071no—
261#103—1071no—
262#103—1071no—
263#103C2×D10711yescharacteristic, complemented
264#102—1071no—
265#102—1071no—
266#102C2×D10711yescharacteristic
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×Q8—1071no—
279C2×Q8C22×D10711yescharacteristic
280C2×D4—21no—
281C2×D4—1071no—
282C2×D4—1071no—
283C2×D4—1071no—
284C2×D4—4281no—
285C2×D4—2141no—
286C2×D4—2141no—
287C2×D4—2142no—
288C2×D4—2141no—
289C2×D4—2141no—
290C2×D4C22×D10711yescharacteristic
291C22×C4—1071no—
292C22×C4—2141no—
293C22×C4—2141no—
294C22×C4—2141no—
295C22×C4—2141no—
296C22×C4—2141no—
297C22×C4C22×D10711yescharacteristic
298Q16—4281no—
299Q16—2142no—
300SD16—22no—
301SD16—4281no—
302SD16—2142no—
303SD16—2142no—
304SD16—2142no—
305D8—1074no—
306D8—2142no—
307M16—2142no—
308C2×C8—22no—
309C2×C8—1071no—
310C2×C8—1071no—
311C2×C8C22×D10711yescharacteristic
312C2×C8C22×D10711yescharacteristic
313C4⋊C4—21no—
314C4⋊C4—1071no—
315C4⋊C4—2141no—
316C4⋊C4—2141no—
317C4⋊C4—2141no—
318C4⋊C4—2141no—
319C4⋊C4C22×D10711yescharacteristic
320C4⋊C4C22×D10711yescharacteristic
321C22⋊C4—21no—
322C22⋊C4—21no—
323C22⋊C4—1071no—
324C22⋊C4—1071no—
325C22⋊C4—2141no—
326C22⋊C4—2141no—
327C22⋊C4—2141no—
328C22⋊C4—2141no—
329C22⋊C4—2141no—
330C22⋊C4—2141no—
331C42—1071no—
332C23—21no—
333C23—8561no—
334C23—1071no—
335C23—4282no—
336C23—4282no—
337C23—2141no—
338C23—2141no—
339C23D4×D10711yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—2141no—
343Q8—2141no—
344Q8—2141no—
345Q8—2141no—
346D4—21no—
347D4—21no—
348D4—81no—
349D4—1072no—
350D4—1072no—
351D4—4282no—
352D4—4282no—
353D4—4282no—
354D4—4281no—
355D4—4282no—
356D4—2141no—
357D4—2141no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—8561no—
364C2×C4—1071no—
365C2×C4—1071no—
366C2×C4—1071no—
367C2×C4—4282no—
368C2×C4—4282no—
369C2×C4—4282no—
370C2×C4—4282no—
371C2×C4—2141no—
372C2×C4—2141no—
373C2×C4—2141no—
374C2×C4—2141no—
375C2×C4D4×D10711yescharacteristic
376C2×C4C23×D10711yescharacteristic
377C8—21no—
378C8—2141no—
379C8—2141no—
380C8D4×D10712yes—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—8564no—
388C22—4281no—
389C22—4281no—
390C22—4281no—
391C22—4281no—
392C22—2141no—
393C22—2141no—
394C22—2142no—
395C22—2142no—
396C22—2141no—
397C22C2×D4×D10711yescharacteristic
398C4—21no—
399C4—41no—
400C4—41no—
401C4—41no—
402C4—4281no—
403C4—4281no—
404C4—2141no—
405C4—2141no—
406C4—2141no—
407C4C2×D4×D10711yescharacteristic
408C4#1681215511yescharacteristic
409C2—41no—
410C2—81no—
411C2—4281no—
412C2—4282no—
413C2—4282no—
414C2#2029326111yescharacteristic
415C2#2029342411yescharacteristic
416C2#2029343311yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#465×C107C2
socleC22×C107C22×D4
hypercenter#465D107
hypocenterC107#3267
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. C2×#16812136C2semidirect
2. C2×#16812131C2semidirect
3. #20293279C2semidirect
4. #20293274C2semidirect
5. #20293018C2semidirect
6. #20292886C2semidirect
7. #20292841C2semidirect
8. #20292831C2semidirect
9. #20292792C2semidirect
10. #20292787C2semidirect
11. #20292779C2semidirect
12. #20292770C2semidirect
13. #20292738C2semidirect
14. #20292698C2semidirect
15. #465×C107C2semidirect
18. C2×#13133818C22semidirect
22. C2×Dic428C22semidirect
23. C2×#13133808C22semidirect
25. C2×#13133807C22semidirect
31. #16812107C22semidirect
42. #16812071C22semidirect
46. Dic107⋊C8C22semidirect
47. #16812032C22semidirect
48. #16812032C22semidirect
50. #16812029C22semidirect
51. C22⋊(C107⋊C8)C22semidirect
53. #16812019C22semidirect
55. C22×C8×C107C22semidirect
56. C22⋊Q8×C107C22semidirect
58. #106×C107C22semidirect
59. #103×C107C22semidirect
91. D856D4semidirect
209. C107#3267semidirect
216. #465D107semidirect
241. C22×C8C2×D107semidirect
249. C22⋊Q8C2×D107semidirect
259. #106C2×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×#168121312
#202932792
#202932742
#202930182
#202928862
#202928412
#202928312
#202927922
#202927872
#202927792
#202927702
#202927382
#202926982
#465×C1072
#3267107