Finite Group Databasebeta

← #28347952

Subgroups — #28347952

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×#16812135C211yesmaximal, characteristic, complemented
2C22×#13133806C211yesmaximal, characteristic, complemented
3#20293286C211yesmaximal, characteristic, complemented
4#20293014C211yesmaximal, characteristic, complemented
5#20292871C211yesmaximal, characteristic, complemented
6#20292861C211yesmaximal, characteristic, complemented
7#20292842C211yesmaximal, characteristic, complemented
8#20292824C211yesmaximal, characteristic, complemented
9#20292778C211yesmaximal, characteristic, complemented
10#20292770C211yesmaximal, characteristic, complemented
11#20292765C211yesmaximal, characteristic, complemented
12#20292761C211yesmaximal, characteristic, complemented
13#20292727C211yesmaximal, characteristic, complemented
14#20292699C211yesmaximal, characteristic, complemented
15D42×C107C211yesmaximal, characteristic, complemented
16C2×#13133818C2211yescharacteristic, complemented
17C22×Dic214C2211yescharacteristic, complemented
18#16812135—24no—
19C2×#13133806—24no—
20C2×#13133806C2211yescharacteristic
21C2×#13133806C2211yescharacteristic
22C2×#13133806C2211yescharacteristic, complemented
23C2×#13133806C2211yescharacteristic, complemented
24C2×#13133806C2211yescharacteristic, complemented
25C2×#13133806C2211yescharacteristic, complemented
26C2×#13133803C2211yescharacteristic
27C2×#13133803C2211yescharacteristic, complemented
28C2×#13133803C2211yescharacteristic, complemented
29C2×#13133802C2211yescharacteristic, complemented
30C22×C107⋊C8C2211yescharacteristic, complemented
31#16812097C2211yescharacteristic, complemented
32#16812095—21no—
33#16812088C2211yescharacteristic, complemented
34#16812087—21no—
35#16812083C2211yescharacteristic, complemented
36#16812081—21no—
37C4×Dic214C2211yescharacteristic, complemented
38(C22⋊D107)⋊C4—21no—
39#16812042C2211yescharacteristic, complemented
40C4⋊(C107⋊C8)C2211yescharacteristic, complemented
41#16812032C2211yescharacteristic, complemented
42#16812032C2211yescharacteristic, complemented
43#16812030C2211yescharacteristic
44#16812030C2211yescharacteristic, complemented
45#16812030C2211yescharacteristic, complemented
46#16812029C2211yescharacteristic, complemented
47C22⋊(C107⋊C8)C2211yescharacteristic, complemented
48C22⋊(C107⋊C8)C2211yescharacteristic, complemented
49#16812020C2211yescharacteristic, complemented
50C22×D4×C107—21no—
51C22×D4×C107C2211yescharacteristic
52C22×D4×C107C2211yescharacteristic
53#127×C107C2211yescharacteristic
54#121×C107—21no—
55#121×C107C2211yescharacteristic
56#121×C107C2211yescharacteristic
57#120×C107—21no—
58#120×C107—21no—
59C4×D4×C107C2211yescharacteristic
60C4×D4×C107C2211yescharacteristic
61#13133818—22no—
62#13133818—41no—
63C2×Dic214—22no—
64C2×Dic214—21no—
65C2×Dic214—41no—
66C2×Dic214C2311yescharacteristic
67C2×Dic214C2311yescharacteristic
68C2×D428C2311yescharacteristic
69C2×C22⋊D107—21no—
70C2×C22⋊D107—21no—
71C22×Dic107—21no—
72C2×C4×D107—21no—
73#13133806—22no—
74#13133806—22no—
75#13133806—22no—
76#13133806—22no—
77#13133806—41no—
78#13133806D414yescomplemented
79#13133803—22no—
80#13133803—22no—
81#13133803—41no—
82#13133802—22no—
83C2×C107⋊C8—22no—
84C2×C107⋊C8C2311yescharacteristic
85C2×C107⋊C8C2311yescharacteristic
86C2×C107⋊C8C2311yescharacteristic
87C2×C107⋊C8C2311yescharacteristic
88C4⋊Dic107C2311yescharacteristic
89Dic107⋊C4—21no—
90Dic107⋊C4—21no—
91C22⋊Dic107—21no—
92(C2×D107)⋊C4—21no—
93(C2×D107)⋊C4—21no—
94(C2×D107)⋊C4—21no—
95C4×Dic107—21no—
96C24×C107—21no—
97C24×C107—21no—
98C2×D4×C107—22no—
99C2×D4×C107—22no—
100C2×D4×C107—21no—
101C2×D4×C107—21no—
102C2×D4×C107—21no—
103C2×D4×C107—21no—
104C2×D4×C107—21no—
105C2×D4×C107—21no—
106C2×D4×C107—41no—
107C2×D4×C107—41no—
108C2×D4×C107—44no—
109C2×D4×C107D411yescharacteristic
110C2×D4×C107C2311yescharacteristic
111C2×D4×C107C2311yescharacteristic
112C2×D4×C107C2311yescharacteristic
113C22×C4×C107—21no—
114C22×C4×C107C2311yescharacteristic
115C22×C4×C107C2311yescharacteristic
116C4⋊C4×C107D411yescharacteristic
117C4⋊C4×C107C2311yescharacteristic
118C22⋊C4×C107—21no—
119C22⋊C4×C107—21no—
120C22⋊C4×C107—21no—
121C22⋊C4×C107D411yescharacteristic
122C22⋊C4×C107D411yescharacteristic
123C42×C107C2311yescharacteristic
124C23×C107—21no—
125C23×C107—21no—
126C23×C107—21no—
127C23×C107—42no—
128C23×C107—42no—
129C23×C107—42no—
130C23×C107—42no—
131C23×C107—81no—
132C23×C107—81no—
133C23×C107C2×D411yescharacteristic
134C23×C107C2×D411yescharacteristic
135C22×D107—21no—
136D4×C107—22no—
137D4×C107—21no—
138D4×C107—21no—
139D4×C107—21no—
140D4×C107—21no—
141D4×C107—21no—
142D4×C107—42no—
143D4×C107—42no—
144D4×C107—42no—
145D4×C107—42no—
146D4×C107—42no—
147D4×C107—81no—
148D4×C107C2×D412yes—
149C2×C4×C107—21no—
150C2×C4×C107—21no—
151C2×C4×C107—21no—
152C2×C4×C107—21no—
153C2×C4×C107—21no—
154C2×C4×C107—42no—
155C2×C4×C107C2×D411yescharacteristic
156C2×C4×C107C2×D411yescharacteristic
157C2×C4×C107C2×D411yescharacteristic
158C2×C4×C107C2×D411yescharacteristic
159C2×C4×C107C2411yescharacteristic
160C22⋊D107—42no—
161C22⋊D107—42no—
162C2×Dic107—21no—
163C2×Dic107—21no—
164C2×Dic107—21no—
165C2×Dic107—42no—
166D428—21no—
167D428—21no—
168C4×D107—42no—
169Dic214—21no—
170Dic214—21no—
171Dic214—21no—
172Dic214—42no—
173Dic214C2×D412yescomplemented
174C107⋊C8—21no—
175C107⋊C8—21no—
176C107⋊C8—21no—
177C107⋊C8C2×D412yescomplemented
178C22×C107—21no—
179C22×C107—21no—
180C22×C107—21no—
181C22×C107—22no—
182C22×C107—21no—
183C22×C107—21no—
184C22×C107—41no—
185C22×C107—41no—
186C22×C107—41no—
187C22×C107—41no—
188C22×C107—41no—
189C22×C107—41no—
190C22×C107—41no—
191C22×C107—41no—
192C22×C107—84no—
193C22×C107—84no—
194C22×C107AGL(1,ℤ/8)12yescomplemented
195C22×C107C22×D411yescharacteristic
196C2×D107—41no—
197C2×D107—41no—
198C2×D107—41no—
199C4×C107—21no—
200C4×C107—21no—
201C4×C107—21no—
202C4×C107—21no—
203C4×C107—41no—
204C4×C107C22×D411yescharacteristic
205C4×C107H(5,𝔽2)11yescharacteristic
206Dic107—22no—
207Dic107—41no—
208Dic107—41no—
209C2×C107—22no—
210C2×C107—41no—
211C2×C107—42no—
212C2×C107—81no—
213C2×C107—81no—
214C2×C107D4211yescharacteristic
215C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
216C2×C107#57711yescharacteristic
217D107—81no—
218#3265—1071nomaximal, Sylow
219C107#326511yescharacteristic, complemented, Sylow
220C2×AGL(1,ℤ/8)—1071no—
221C22×SD16—1071no—
222#548—1071no—
223D42D10711yescharacteristic, complemented
224#496—1071no—
225#468—1071no—
226#466—1071no—
227#463—1071no—
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—21no—
238C22×D4C2×D10711yescharacteristic
239C22×D4C2×D10711yescharacteristic
240AGL(1,ℤ/8)—2144no—
241C2×SD16—1071no—
242C2×SD16—1071no—
243C2×SD16—1071no—
244C2×SD16—1071no—
245C2×SD16—1071no—
246C2×SD16—1071no—
247C2×SD16—2144no—
248C2×D8—1071no—
249C2×D8—1071no—
250C2×D8—1071no—
251C2×M16—1071no—
252C22×C8—1071no—
253#127C2×D10711yescharacteristic
254#124—1071no—
255#124—2141no—
256C22⋊Q8—1071no—
257C22⋊Q8—2141no—
258#121—21no—
259#121—1071no—
260#121—2141no—
261#121C2×D10711yescharacteristic
262#121C2×D10711yescharacteristic
263#120—21no—
264#120—21no—
265C4×Q8—1071no—
266C4×D4—2141no—
267C4×D4C2×D10711yescharacteristic
268C4×D4C2×D10711yescharacteristic
269#107—1071no—
270C4⋊C8—1071no—
271#103—1071no—
272#103—1071no—
273#102—1071no—
274#102—1071no—
275#102—1071no—
276#102—1071no—
277C22⋊C8—1071no—
278C22⋊C8—1071no—
279#97—1071no—
280C24—21no—
281C24—21no—
282#41—4281no—
283#41—2142no—
284C2×Q8—1071no—
285C2×Q8—1071no—
286C2×Q8—4281no—
287C2×Q8—2142no—
288C2×Q8—2141no—
289C2×D4—22no—
290C2×D4—22no—
291C2×D4—21no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—44no—
300C2×D4—1071no—
301C2×D4—2141no—
302C2×D4—2141no—
303C2×D4C22⋊D10711yescharacteristic
304C2×D4C22×D10711yescharacteristic
305C2×D4C22×D10711yescharacteristic
306C2×D4C22×D10711yescharacteristic
307C22×C4—21no—
308C22×C4—2141no—
309C22×C4—2141no—
310C22×C4C22×D10711yescharacteristic
311C22×C4C22×D10711yescharacteristic
312SD16—1074no—
313SD16—4281no—
314SD16—2142no—
315SD16—2142no—
316SD16—2142no—
317SD16—2142no—
318D8—4281no—
319D8—2142no—
320D8—2142no—
321M16—2142no—
322C2×C8—1071no—
323C2×C8—1071no—
324C2×C8—1071no—
325C2×C8—1071no—
326C2×C8—2142no—
327C4⋊C4—1071no—
328C4⋊C4—2141no—
329C4⋊C4—2141no—
330C4⋊C4C22⋊D10711yescharacteristic
331C4⋊C4C22×D10711yescharacteristic
332C22⋊C4—21no—
333C22⋊C4—21no—
334C22⋊C4—21no—
335C22⋊C4—2141no—
336C22⋊C4—2141no—
337C22⋊C4—2141no—
338C22⋊C4—2141no—
339C22⋊C4C22⋊D10711yescharacteristic
340C22⋊C4C22⋊D10711yescharacteristic
341C42—2141no—
342C42C22×D10711yescharacteristic
343C23—21no—
344C23—21no—
345C23—21no—
346C23—42no—
347C23—42no—
348C23—42no—
349C23—42no—
350C23—81no—
351C23—81no—
352C23—2141no—
353C23C2×C22⋊D10711yescharacteristic
354C23C2×C22⋊D10711yescharacteristic
355Q8—1072no—
356Q8—4282no—
357Q8—2141no—
358Q8—2141no—
359Q8—2141no—
360D4—22no—
361D4—21no—
362D4—21no—
363D4—21no—
364D4—21no—
365D4—21no—
366D4—42no—
367D4—42no—
368D4—42no—
369D4—42no—
370D4—42no—
371D4—81no—
372D4—4282no—
373D4—4282no—
374D4—2141no—
375D4—2141no—
376D4D4×D10712yes—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—42no—
383C2×C4—4282no—
384C2×C4—4282no—
385C2×C4—2141no—
386C2×C4—2141no—
387C2×C4—2141no—
388C2×C4C2×C22⋊D10711yescharacteristic
389C2×C4C2×C22⋊D10711yescharacteristic
390C2×C4C2×C22⋊D10711yescharacteristic
391C2×C4C2×C22⋊D10711yescharacteristic
392C2×C4C23×D10711yescharacteristic
393C8—1072no—
394C8—2141no—
395C8—2141no—
396C8—2141no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—22no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22—84no—
412C22—84no—
413C22—4281no—
414C22—4281no—
415C22—4281no—
416C22#1681213512yescomplemented
417C22C22×C22⋊D10711yescharacteristic
418C4—21no—
419C4—21no—
420C4—21no—
421C4—21no—
422C4—41no—
423C4—4281no—
424C4—4281no—
425C4—2142no—
426C4C2×D4×D10711yescharacteristic
427C4#1681215411yescharacteristic
428C2—22no—
429C2—41no—
430C2—42no—
431C2—81no—
432C2—81no—
433C2—8561no—
434C2D4×C22⋊D10711yescharacteristic
435C2C2×#1681213511yescharacteristic
436C2#2029343811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingD42×C107C2
socleC22×C107C22×D4
hypercenterD42D107
hypocenterC107#3265
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C4×D4 ◁ D42

Complements

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

NH ≅ G/Ntype
1. C2×#16812135C2semidirect
2. C22×#13133806C2semidirect
3. #20293286C2semidirect
4. #20293014C2semidirect
5. #20292871C2semidirect
6. #20292861C2semidirect
7. #20292842C2semidirect
8. #20292824C2semidirect
9. #20292778C2semidirect
10. #20292770C2semidirect
11. #20292765C2semidirect
12. #20292761C2semidirect
13. #20292727C2semidirect
14. #20292699C2semidirect
15. D42×C107C2semidirect
16. C2×#13133818C22semidirect
17. C22×Dic214C22semidirect
22. C2×#13133806C22semidirect
23. C2×#13133806C22semidirect
24. C2×#13133806C22semidirect
25. C2×#13133806C22semidirect
27. C2×#13133803C22semidirect
28. C2×#13133803C22semidirect
29. C2×#13133802C22semidirect
30. C22×C107⋊C8C22semidirect
31. #16812097C22semidirect
33. #16812088C22semidirect
35. #16812083C22semidirect
37. C4×Dic214C22semidirect
39. #16812042C22semidirect
40. C4⋊(C107⋊C8)C22semidirect
41. #16812032C22semidirect
42. #16812032C22semidirect
44. #16812030C22semidirect
45. #16812030C22semidirect
46. #16812029C22semidirect
47. C22⋊(C107⋊C8)C22semidirect
48. C22⋊(C107⋊C8)C22semidirect
49. #16812020C22semidirect
78. #13133806D4semidirect
173. Dic214C2×D4semidirect
177. C107⋊C8C2×D4semidirect
194. C22×C107AGL(1,ℤ/8)semidirect
219. C107#3265semidirect
223. D42D107semidirect
416. C22#16812135semidirect

Maximal subgroups

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

subgroupindex
C2×#168121352
C22×#131338062
#202932862
#202930142
#202928712
#202928612
#202928422
#202928242
#202927782
#202927702
#202927652
#202927612
#202927272
#202926992
D42×C1072
#3265107