Finite Group Databasebeta

← #28347962

Subgroups — #28347962

Order 13696.

546 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812133C211yesmaximal, characteristic, complemented
2C2×SD16×D107C211yesmaximal, characteristic, complemented
3#20293285C211yesmaximal, characteristic, complemented
4D4×D428C211yesmaximal, characteristic, complemented
5#20293017C211yesmaximal, characteristic, complemented
6#20292876C211yesmaximal, characteristic, complemented
7#20292862C211yesmaximal, characteristic, complemented
8#20292828C211yesmaximal, characteristic, complemented
9#20292776C211yesmaximal, characteristic, complemented
10#20292773C211yesmaximal, characteristic, complemented
11#20292767C211yesmaximal, characteristic, complemented
12#20292759C211yesmaximal, characteristic, complemented
13#20292727C211yesmaximal, characteristic, complemented
14#20292697C211yesmaximal, characteristic, complemented
15#463×C107C211yesmaximal, characteristic, complemented
16C2×#13133817C2211yescharacteristic, complemented
17C2×Q8×D107C2211yescharacteristic, complemented
18C22×D428—21no—
19C2×D4×D107C2211yescharacteristic
20C2×D4×D107C2211yescharacteristic
21#16812133—24no—
22C2×#13133807C2211yescharacteristic, complemented
23C2×#13133807C2211yescharacteristic, complemented
24C2×#13133806C2211yescharacteristic
25C2×#13133805C2211yescharacteristic, complemented
26C2×#13133805C2211yescharacteristic, complemented
27SD16×D107—24no—
28C2×#13133803C2211yescharacteristic, complemented
29C2×#13133803C2211yescharacteristic, complemented
30C2×#13133801C2211yescharacteristic, complemented
31C2×C8×D107C2211yescharacteristic, complemented
32#16812106C2211yescharacteristic
33#16812095—21no—
34#16812086—21no—
35#16812085C2211yescharacteristic, complemented
36#16812084C2211yescharacteristic
37#16812084C2211yescharacteristic
38#16812083—21no—
39#16812082—21no—
40#16812080C2211yescharacteristic, complemented
41#16812079—21no—
42#16812079—21no—
43C4×Dic214C2211yescharacteristic, complemented
44C4×D428C2211yescharacteristic
45(C22⋊D107)⋊C4—21no—
46#16812042C2211yescharacteristic, complemented
47#16812032C2211yescharacteristic, complemented
48#16812031C2211yescharacteristic, complemented
49#16812031C2211yescharacteristic, complemented
50#16812030C2211yescharacteristic
51(C2×D107)⋊C8C2211yescharacteristic, complemented
52(C2×D107)⋊C8C2211yescharacteristic, complemented
53#16812020C2211yescharacteristic, complemented
54C2×SD16×C107C2211yescharacteristic
55C2×D8×C107C2211yescharacteristic
56#124×C107C2211yescharacteristic, complemented
57C4×D4×C107C2211yescharacteristic
58C4⋊C8×C107C2211yescharacteristic, complemented
59#103×C107C2211yescharacteristic, complemented
60#102×C107C2211yescharacteristic, complemented
61C23×D107—21no—
62C23×D107—21no—
63#13133817—22no—
64#13133817—41no—
65C2×Dic214—21no—
66C2×Dic214C2311yescharacteristic
67Q8×D107—22no—
68Q8×D107—41no—
69C2×D428—21no—
70C2×D428—21no—
71C2×D428—21no—
72C2×D428—21no—
73C2×D428—44no—
74C2×D428D411yescharacteristic
75C2×D428C2311yescharacteristic
76C2×D428C2311yescharacteristic
77C2×C22⋊D107—21no—
78C2×C22⋊D107—21no—
79C2×C22⋊D107—21no—
80C2×C22⋊D107—21no—
81D4×D107—22no—
82D4×D107—22no—
83D4×D107—41no—
84D4×D107—41no—
85C22×Dic107—21no—
86C2×C4×D107—21no—
87C2×C4×D107C2311yescharacteristic
88C2×C4×D107C2311yescharacteristic
89#13133807—22no—
90#13133807—22no—
91#13133806D414yescomplemented
92#13133805—22no—
93#13133805—41no—
94#13133803—22no—
95#13133803—41no—
96#13133801—22no—
97C2×C107⋊C8C2311yescharacteristic
98C2×C107⋊C8C2311yescharacteristic
99C8×D107—22no—
100C4⋊Dic107D411yescharacteristic
101C4⋊Dic107C2311yescharacteristic
102Dic107⋊C4—21no—
103Dic107⋊C4—21no—
104C22⋊Dic107—21no—
105(C2×D107)⋊C4—21no—
106(C2×D107)⋊C4—21no—
107(C2×D107)⋊C4—21no—
108(C2×D107)⋊C4D411yescharacteristic
109(C2×D107)⋊C4D411yescharacteristic
110C4×Dic107—21no—
111C2×Q8×C107C2311yescharacteristic
112C2×D4×C107C2311yescharacteristic
113C2×D4×C107C2311yescharacteristic
114C22×C4×C107—21no—
115SD16×C107—22no—
116D8×C107—22no—
117C2×C8×C107C2311yescharacteristic
118C2×C8×C107C2311yescharacteristic
119C4⋊C4×C107C2311yescharacteristic
120C22⋊C4×C107—21no—
121C22⋊C4×C107—21no—
122C22⋊C4×C107—21no—
123C42×C107C2311yescharacteristic
124C23×C107—21no—
125C23×C107—21no—
126C22×D107—21no—
127C22×D107—21no—
128C22×D107—42no—
129C22×D107—42no—
130C22×D107—42no—
131C22×D107—42no—
132C22×D107—81no—
133C22×D107—81no—
134C22×D107C2×D411yescharacteristic
135C22×D107C2×D411yescharacteristic
136Q8×C107—21no—
137Q8×C107—21no—
138D4×C107—21no—
139D4×C107—21no—
140D4×C107—21no—
141D4×C107C2×D412yes—
142C2×C4×C107—21no—
143C2×C4×C107—21no—
144C2×C4×C107—42no—
145C2×C4×C107C2×D411yescharacteristic
146C2×C4×C107C2×D411yescharacteristic
147C2×C4×C107C2411yescharacteristic
148C22⋊D107—42no—
149C22⋊D107—42no—
150C22⋊D107—42no—
151C22⋊D107—42no—
152C2×Dic107—21no—
153C2×Dic107—21no—
154C2×Dic107—42no—
155C2×Dic107C2×D411yescharacteristic
156C2×Dic107C2×D411yescharacteristic
157D428—22no—
158D428—21no—
159D428—21no—
160D428—21no—
161D428—21no—
162D428—42no—
163D428—42no—
164D428—42no—
165D428—81no—
166C4×D107—21no—
167C4×D107—21no—
168C4×D107—21no—
169C4×D107—21no—
170C4×D107—42no—
171Dic214—21no—
172Dic214—42no—
173Dic214C2×D412yescomplemented
174C8×C107—21no—
175C8×C107—21no—
176C107⋊C8—21no—
177C107⋊C8C2×D412yescomplemented
178C22×C107—22no—
179C22×C107—41no—
180C22×C107—41no—
181C22×C107—41no—
182C22×C107—41no—
183C22×C107—41no—
184C22×C107C22×D411yescharacteristic
185C2×D107—21no—
186C2×D107—21no—
187C2×D107—21no—
188C2×D107—21no—
189C2×D107—21no—
190C2×D107—41no—
191C2×D107—41no—
192C2×D107—41no—
193C2×D107—41no—
194C2×D107—41no—
195C2×D107—41no—
196C2×D107—84no—
197C2×D107—84no—
198C2×D107AGL(1,ℤ/8)12yescomplemented
199C4×C107—21no—
200C4×C107—21no—
201C4×C107—41no—
202C4×C107—41no—
203C4×C107C22×D411yescharacteristic
204C4×C107H(5,𝔽2)11yescharacteristic
205Dic107—22no—
206Dic107—21no—
207Dic107—21no—
208Dic107—41no—
209C2×C107—42no—
210C2×C107—81no—
211C2×C107D4211yescharacteristic
212C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
213C2×C107#57711yescharacteristic
214D107—22no—
215D107—41no—
216D107—81no—
217D107—81no—
218#3265—1071nomaximal, Sylow
219C107#326511yescharacteristic, complemented, Sylow
220C2×AGL(1,ℤ/8)—1071no—
221C22×SD16—1071no—
222#548—1071no—
223D42—1071no—
224#496—1071no—
225#468—1071no—
226#466—1071no—
227#463D10711yescharacteristic, complemented
228#460—1071no—
229#450—1071no—
230#449—1071no—
231#448—1071no—
232#447—1071no—
233#440—1071no—
234#435—1071no—
235C2×#41—1071no—
236C22×Q8—1071no—
237C22×D4—1071no—
238C22×D4—1071no—
239C22×D4—2141no—
240AGL(1,ℤ/8)—2144no—
241C2×SD16—1071no—
242C2×SD16—1071no—
243C2×SD16—1071no—
244C2×SD16—1071no—
245C2×SD16—1071no—
246C2×SD16—2144no—
247C2×SD16C2×D10711yescharacteristic
248C2×D8—1071no—
249C2×D8—1071no—
250C2×D8C2×D10711yescharacteristic
251C2×M16—1071no—
252C22×C8—1071no—
253#127—1071no—
254#124—2141no—
255#124C2×D10711yescharacteristic, complemented
256C22⋊Q8—1071no—
257C22⋊Q8—2141no—
258#121—1071no—
259#121—1071no—
260#121—1071no—
261#121—2141no—
262#121—2141no—
263#120—2141no—
264#120—2141no—
265C4×Q8—1071no—
266C4×D4—1071no—
267C4×D4—2141no—
268C4×D4C2×D10711yescharacteristic
269#107—1071no—
270C4⋊C8C2×D10711yescharacteristic, complemented
271#103—1071no—
272#103C2×D10711yescharacteristic, complemented
273#102—1071no—
274#102—1071no—
275#102—1071no—
276#102C2×D10711yescharacteristic, complemented
277C22⋊C8—1071no—
278C22⋊C8—1071no—
279#97—1071no—
280C24—2141no—
281C24—2141no—
282#41—4281no—
283#41—2142no—
284C2×Q8—1071no—
285C2×Q8—4281no—
286C2×Q8—2141no—
287C2×Q8—2142no—
288C2×Q8C22×D10711yescharacteristic
289C2×D4—1071no—
290C2×D4—1071no—
291C2×D4—1071no—
292C2×D4—4284no—
293C2×D4—4281no—
294C2×D4—4281no—
295C2×D4—2141no—
296C2×D4—2141no—
297C2×D4—2141no—
298C2×D4—2141no—
299C2×D4—2141no—
300C2×D4—2141no—
301C2×D4—2142no—
302C2×D4—2142no—
303C2×D4—2141no—
304C2×D4—2141no—
305C2×D4C22×D10711yescharacteristic
306C2×D4C22×D10711yescharacteristic
307C22×C4—21no—
308C22×C4—1071no—
309C22×C4—1071no—
310C22×C4—2141no—
311C22×C4—2141no—
312SD16—22no—
313SD16—1074no—
314SD16—4281no—
315SD16—2142no—
316SD16—2142no—
317SD16—2142no—
318D8—22no—
319D8—4281no—
320D8—2142no—
321M16—2142no—
322C2×C8—1071no—
323C2×C8—1071no—
324C2×C8—2142no—
325C2×C8C22×D10711yescharacteristic
326C2×C8C22×D10711yescharacteristic
327C4⋊C4—1071no—
328C4⋊C4—1071no—
329C4⋊C4—2141no—
330C4⋊C4—2141no—
331C4⋊C4C22×D10711yescharacteristic
332C22⋊C4—21no—
333C22⋊C4—21no—
334C22⋊C4—21no—
335C22⋊C4—1071no—
336C22⋊C4—1071no—
337C22⋊C4—2141no—
338C22⋊C4—2141no—
339C22⋊C4—2141no—
340C22⋊C4—2141no—
341C42—2141no—
342C42C22×D10711yescharacteristic
343C23—21no—
344C23—21no—
345C23—8561no—
346C23—8561no—
347C23—1071no—
348C23—1071no—
349C23—4282no—
350C23—4282no—
351C23—4282no—
352C23—4282no—
353C23—2141no—
354C23—2141no—
355Q8—21no—
356Q8—21no—
357Q8—1072no—
358Q8—4282no—
359Q8—2141no—
360D4—21no—
361D4—21no—
362D4—21no—
363D4—8561no—
364D4—4282no—
365D4—4282no—
366D4—4282no—
367D4—4282no—
368D4—4282no—
369D4—4282no—
370D4—4282no—
371D4—2142no—
372D4—2141no—
373D4—2141no—
374D4—2141no—
375D4—2141no—
376D4D4×D10712yes—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—42no—
380C2×C4—1071no—
381C2×C4—1071no—
382C2×C4—4282no—
383C2×C4—4282no—
384C2×C4—2141no—
385C2×C4—2141no—
386C2×C4—2141no—
387C2×C4—2141no—
388C2×C4—2141no—
389C2×C4—2141no—
390C2×C4D4×D10711yescharacteristic
391C2×C4D4×D10711yescharacteristic
392C2×C4C23×D10711yescharacteristic
393C8—21no—
394C8—21no—
395C8—1072no—
396C8—2141no—
397C22—22no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—8564no—
404C22—8564no—
405C22—1072no—
406C22—4281no—
407C22—4281no—
408C22—4281no—
409C22—4281no—
410C22—4281no—
411C22—4281no—
412C22—2141no—
413C22—2141no—
414C22—2141no—
415C22—2141no—
416C22—2141no—
417C22C2×D4×D10711yescharacteristic
418C4—21no—
419C4—21no—
420C4—41no—
421C4—41no—
422C4—4281no—
423C4—2141no—
424C4—2141no—
425C4—2142no—
426C4C2×D4×D10711yescharacteristic
427C4#1681215511yescharacteristic
428C2—42no—
429C2—81no—
430C2—8561no—
431C2—8561no—
432C2—4281no—
433C2—2142no—
434C2#2029326111yescharacteristic
435C2AGL(1,ℤ/8)×D10711yescharacteristic
436C2#2029343411yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#463×C107C2
socleC22×C107C22×D4
hypercenter#463D107
hypocenterC107#3265
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. C2×#16812133C2semidirect
2. C2×SD16×D107C2semidirect
3. #20293285C2semidirect
4. D4×D428C2semidirect
5. #20293017C2semidirect
6. #20292876C2semidirect
7. #20292862C2semidirect
8. #20292828C2semidirect
9. #20292776C2semidirect
10. #20292773C2semidirect
11. #20292767C2semidirect
12. #20292759C2semidirect
13. #20292727C2semidirect
14. #20292697C2semidirect
15. #463×C107C2semidirect
16. C2×#13133817C22semidirect
17. C2×Q8×D107C22semidirect
22. C2×#13133807C22semidirect
23. C2×#13133807C22semidirect
25. C2×#13133805C22semidirect
26. C2×#13133805C22semidirect
28. C2×#13133803C22semidirect
29. C2×#13133803C22semidirect
30. C2×#13133801C22semidirect
31. C2×C8×D107C22semidirect
35. #16812085C22semidirect
40. #16812080C22semidirect
43. C4×Dic214C22semidirect
46. #16812042C22semidirect
47. #16812032C22semidirect
48. #16812031C22semidirect
49. #16812031C22semidirect
51. (C2×D107)⋊C8C22semidirect
52. (C2×D107)⋊C8C22semidirect
53. #16812020C22semidirect
56. #124×C107C22semidirect
58. C4⋊C8×C107C22semidirect
59. #103×C107C22semidirect
60. #102×C107C22semidirect
91. #13133806D4semidirect
173. Dic214C2×D4semidirect
177. C107⋊C8C2×D4semidirect
198. C2×D107AGL(1,ℤ/8)semidirect
219. C107#3265semidirect
227. #463D107semidirect
255. #124C2×D107semidirect
270. C4⋊C8C2×D107semidirect
272. #103C2×D107semidirect
276. #102C2×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121332
C2×SD16×D1072
#202932852
D4×D4282
#202930172
#202928762
#202928622
#202928282
#202927762
#202927732
#202927672
#202927592
#202927272
#202926972
#463×C1072
#3265107