Finite Group Databasebeta

← #28348055

Subgroups — #28348055

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
2#20293276C211yesmaximal, characteristic, complemented
3D4×C22⋊D107C211yesmaximal, characteristic, complemented
4#20292998C211yesmaximal, characteristic, complemented
5#20292873C211yesmaximal, characteristic, complemented
6#20292870C211yesmaximal, characteristic, complemented
7#20292830C211yesmaximal, characteristic, complemented
8#20292820C211yesmaximal, characteristic, complemented
9#20292778C211yesmaximal, characteristic, complemented
10#20292776C211yesmaximal, characteristic, complemented
11#20292772C211yesmaximal, characteristic, complemented
12#20292769C211yesmaximal, characteristic, complemented
13D8×Dic107C211yesmaximal, characteristic, complemented
14#20292703C211yesmaximal, characteristic, complemented
15C2×AGL(1,ℤ/8)×C107C211yesmaximal, characteristic, complemented
16C2×#13133817C2211yescharacteristic, complemented
17C22×C22⋊D107—21no—
18C2×D4×D107C2211yescharacteristic
19#16812133—24no—
20C2×#13133807C2211yescharacteristic, complemented
21C2×#13133806C2211yescharacteristic
22C2×#13133806C2211yescharacteristic, complemented
23C2×#13133805C2211yescharacteristic, complemented
24C2×#13133803C2211yescharacteristic, complemented
25C2×#13133801C2211yescharacteristic, complemented
26#16812107C2211yescharacteristic, complemented
27#16812105C2211yescharacteristic
28#16812092—21no—
29#16812090—21no—
30#16812088C2211yescharacteristic, complemented
31#16812085C2211yescharacteristic, complemented
32#16812083C2211yescharacteristic
33#16812081—21no—
34#16812081—21no—
35#16812080C2211yescharacteristic
36#16812078—21no—
37#16812077—21no—
38D4×Dic107C2211yescharacteristic
39D4×Dic107C2211yescharacteristic, complemented
40C4×C22⋊D107C2211yescharacteristic
41C2×Dic107⋊C4—21no—
42#16812043C2211yescharacteristic, complemented
43Dic107⋊C8C2211yescharacteristic, complemented
44#16812034C2211yescharacteristic, complemented
45#16812033C2211yescharacteristic, complemented
46#16812031C2211yescharacteristic, complemented
47#16812030C2211yescharacteristic
48#16812030C2211yescharacteristic, complemented
49#16812030C2211yescharacteristic, complemented
50C22⋊(C107⋊C8)C2211yescharacteristic, complemented
51(C2×D107)⋊C8C2211yescharacteristic, complemented
52C8×Dic107C2211yescharacteristic, complemented
53C2×#41×C107C2211yescharacteristic, complemented
54C22×D4×C107C2211yescharacteristic
55AGL(1,ℤ/8)×C107—24no—
56C2×SD16×C107C2211yescharacteristic
57C2×SD16×C107C2211yescharacteristic, complemented
58C2×D8×C107C2211yescharacteristic
59C2×D8×C107C2211yescharacteristic, complemented
60C2×M16×C107C2211yescharacteristic, complemented
61C23×D107—21no—
62#13133817—22no—
63#13133817—41no—
64C2×Dic214C2311yescharacteristic
65C2×D428C2311yescharacteristic
66C2×C22⋊D107—21no—
67C2×C22⋊D107—21no—
68C2×C22⋊D107—21no—
69C2×C22⋊D107—21no—
70C2×C22⋊D107—21no—
71C2×C22⋊D107—21no—
72C2×C22⋊D107—44no—
73C2×C22⋊D107D411yescharacteristic
74D4×D107—22no—
75D4×D107—41no—
76C22×Dic107—21no—
77C22×Dic107—21no—
78C22×Dic107—21no—
79C2×C4×D107C2311yescharacteristic
80#13133807—22no—
81#13133806—22no—
82#13133806—41no—
83#13133805—41no—
84#13133803—22no—
85#13133801—22no—
86C2×C107⋊C8C2311yescharacteristic
87C2×C107⋊C8C2311yescharacteristic
88C4⋊Dic107C2311yescharacteristic
89C4⋊Dic107C2311yescharacteristic
90Dic107⋊C4—21no—
91Dic107⋊C4—21no—
92Dic107⋊C4—21no—
93Dic107⋊C4—21no—
94Dic107⋊C4D411yescharacteristic
95C22⋊Dic107—21no—
96C22⋊Dic107—21no—
97C22⋊Dic107—21no—
98C22⋊Dic107—21no—
99C22⋊Dic107D411yescharacteristic
100(C2×D107)⋊C4—21no—
101(C2×D107)⋊C4—21no—
102(C2×D107)⋊C4D411yescharacteristic
103C4×Dic107C2311yescharacteristic
104C24×C107—21no—
105#41×C107—22no—
106#41×C107—41no—
107C2×Q8×C107C2311yescharacteristic
108C2×D4×C107—22no—
109C2×D4×C107—21no—
110C2×D4×C107—21no—
111C2×D4×C107—41no—
112C2×D4×C107C2311yescharacteristic
113C2×D4×C107C2311yescharacteristic
114C2×D4×C107C2311yescharacteristic
115C22×C4×C107—21no—
116C22×C4×C107C2311yescharacteristic
117SD16×C107—22no—
118SD16×C107—22no—
119D8×C107—22no—
120D8×C107D414yescomplemented
121M16×C107—22no—
122C2×C8×C107C2311yescharacteristic
123C2×C8×C107C2311yescharacteristic
124C23×C107—21no—
125C23×C107—21no—
126C23×C107—21no—
127C23×C107—42no—
128C23×C107—42no—
129C23×C107—81no—
130C23×C107C2×D411yescharacteristic
131C22×D107—21no—
132C22×D107—42no—
133C22×D107—42no—
134C22×D107—81no—
135C22×D107C2×D411yescharacteristic
136Q8×C107—21no—
137Q8×C107—21no—
138D4×C107—21no—
139D4×C107—21no—
140D4×C107—21no—
141D4×C107—21no—
142D4×C107—42no—
143D4×C107—42no—
144D4×C107C2×D412yes—
145D4×C107C2×D412yescomplemented
146C2×C4×C107—21no—
147C2×C4×C107—21no—
148C2×C4×C107—21no—
149C2×C4×C107—42no—
150C2×C4×C107C2×D411yescharacteristic
151C2×C4×C107C2411yescharacteristic
152C22⋊D107—22no—
153C22⋊D107—42no—
154C22⋊D107—42no—
155C22⋊D107—42no—
156C22⋊D107—42no—
157C22⋊D107—42no—
158C22⋊D107—81no—
159C2×Dic107—21no—
160C2×Dic107—21no—
161C2×Dic107—21no—
162C2×Dic107—42no—
163C2×Dic107—42no—
164C2×Dic107—42no—
165C2×Dic107C2×D411yescharacteristic
166C2×Dic107C2×D411yescharacteristic
167C2×Dic107C2×D411yescharacteristic
168D428—21no—
169D428—21no—
170C4×D107—21no—
171C4×D107—21no—
172Dic214—21no—
173Dic214—21no—
174C8×C107—21no—
175C8×C107C2×D412yescomplemented
176C107⋊C8—21no—
177C107⋊C8—21no—
178C22×C107—22no—
179C22×C107—22no—
180C22×C107—21no—
181C22×C107—21no—
182C22×C107—21no—
183C22×C107—41no—
184C22×C107—41no—
185C22×C107—41no—
186C22×C107—41no—
187C22×C107—41no—
188C22×C107—41no—
189C22×C107—41no—
190C22×C107—84no—
191C22×C107C22×D411yescharacteristic
192C2×D107—21no—
193C2×D107—21no—
194C2×D107—21no—
195C2×D107—41no—
196C2×D107—41no—
197C2×D107—41no—
198C2×D107—84no—
199C4×C107—21no—
200C4×C107—41no—
201C4×C107C22×D411yescharacteristic
202C4×C107H(5,𝔽2)11yescharacteristic
203Dic107—21no—
204Dic107—21no—
205Dic107—41no—
206Dic107—41no—
207Dic107—41no—
208Dic107AGL(1,ℤ/8)12yescomplemented
209C2×C107—42no—
210C2×C107—41no—
211C2×C107—42no—
212C2×C107—81no—
213C2×C107D4211yescharacteristic
214C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
215C2×C107#57711yescharacteristic
216D107—41no—
217D107—81no—
218#3272—1071nomaximal, Sylow
219C107#327211yescharacteristic, complemented, Sylow
220C2×AGL(1,ℤ/8)—1071no—
221C2×AGL(1,ℤ/8)D10711yescharacteristic, complemented
222#547—1071no—
223D42—1071no—
224#492—1071no—
225#468—1071no—
226#468—1071no—
227#461—1071no—
228#459—1071no—
229#450—1071no—
230#450—1071no—
231#449—1071no—
232#449—1071no—
233C4×D8—1071no—
234#436—1071no—
235C2×#41—1071no—
236C2×#41C2×D10711yescharacteristic, complemented
237C22×D4—1071no—
238C22×D4—2141no—
239C22×D4C2×D10711yescharacteristic
240AGL(1,ℤ/8)—24no—
241AGL(1,ℤ/8)—2144no—
242C2×SD16—1071no—
243C2×SD16—1071no—
244C2×SD16—1071no—
245C2×SD16—1071no—
246C2×SD16C2×D10711yescharacteristic
247C2×SD16C2×D10711yescharacteristic, complemented
248C2×D8—1071no—
249C2×D8C2×D10711yescharacteristic
250C2×D8C2×D10711yescharacteristic, complemented
251C2×M16—1071no—
252C2×M16C2×D10711yescharacteristic, complemented
253C4⋊Q8—1071no—
254#127—1071no—
255#123—2141no—
256#123—2141no—
257C22⋊Q8—1071no—
258C22⋊Q8—1071no—
259#121—1071no—
260#121—1071no—
261#121—2141no—
262#121—2141no—
263#120—2141no—
264#120—2141no—
265C4×D4—1071no—
266C4×D4—1071no—
267C4×D4—1071no—
268C2×C4⋊C4—2141no—
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—
279C4×C8—1071no—
280C24—21no—
281C24—2141no—
282#41—22no—
283#41—41no—
284#41—4281no—
285#41—2142no—
286C2×Q8—1071no—
287C2×Q8C22×D10711yescharacteristic
288C2×D4—22no—
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—41no—
292C2×D4—1071no—
293C2×D4—1071no—
294C2×D4—4284no—
295C2×D4—4281no—
296C2×D4—2141no—
297C2×D4—2141no—
298C2×D4—2141no—
299C2×D4—2141no—
300C2×D4—2141no—
301C2×D4—2141no—
302C2×D4—2142no—
303C2×D4C22×D10711yescharacteristic
304C2×D4C22×D10711yescharacteristic
305C2×D4C22×D10711yescharacteristic
306C22×C4—21no—
307C22×C4—1071no—
308C22×C4—2141no—
309C22×C4—2141no—
310C22×C4—2141no—
311C22×C4C22×D10711yescharacteristic
312SD16—22no—
313SD16—22no—
314SD16—4281no—
315SD16—4281no—
316SD16—2142no—
317SD16—2142no—
318D8—22no—
319D8—2142no—
320D8C22⋊D10714yescomplemented
321M16—22no—
322M16—2142no—
323C2×C8—1071no—
324C2×C8—1071no—
325C2×C8C22×D10711yescharacteristic
326C2×C8C22×D10711yescharacteristic
327C4⋊C4—1071no—
328C4⋊C4—1071no—
329C4⋊C4—1071no—
330C4⋊C4—2141no—
331C4⋊C4—2141no—
332C4⋊C4—2141no—
333C4⋊C4—2141no—
334C22⋊C4—1071no—
335C22⋊C4—1071no—
336C22⋊C4—2141no—
337C22⋊C4—2141no—
338C22⋊C4—2141no—
339C22⋊C4—2141no—
340C22⋊C4—2141no—
341C22⋊C4—2141no—
342C42—1071no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—42no—
347C23—42no—
348C23—81no—
349C23—8561no—
350C23—1071no—
351C23—4282no—
352C23—4282no—
353C23—2141no—
354C23D4×D10711yescharacteristic
355Q8—21no—
356Q8—21no—
357Q8—2141no—
358Q8—2141no—
359D4—21no—
360D4—21no—
361D4—21no—
362D4—21no—
363D4—42no—
364D4—42no—
365D4—8561no—
366D4—4282no—
367D4—4282no—
368D4—4282no—
369D4—4282no—
370D4—4282no—
371D4—2142no—
372D4—2141no—
373D4—2141no—
374D4C2×C22⋊D10712yes—
375D4C2×C22⋊D10712yescomplemented
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—42no—
380C2×C4—1071no—
381C2×C4—1071no—
382C2×C4—1071no—
383C2×C4—4282no—
384C2×C4—4282no—
385C2×C4—4282no—
386C2×C4—2141no—
387C2×C4—2141no—
388C2×C4—2141no—
389C2×C4—2141no—
390C2×C4—2141no—
391C2×C4D4×D10711yescharacteristic
392C2×C4C23×D10711yescharacteristic
393C8—21no—
394C8—2141no—
395C8—2141no—
396C8C2×C22⋊D10712yescomplemented
397C22—21no—
398C22—21no—
399C22—21no—
400C22—22no—
401C22—22no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22—41no—
409C22—84no—
410C22—8564no—
411C22—4281no—
412C22—4281no—
413C22—4281no—
414C22—2141no—
415C22—2141no—
416C22—2141no—
417C22C2×D4×D10711yescharacteristic
418C4—21no—
419C4—41no—
420C4—1072no—
421C4—4281no—
422C4—4281no—
423C4—4281no—
424C4—2141no—
425C4—2141no—
426C4C22×C22⋊D10711yescharacteristic
427C4#1681215411yescharacteristic
428C2—41no—
429C2—42no—
430C2—42no—
431C2—81no—
432C2—8561no—
433C2—4281no—
434C2D4×C22⋊D10711yescharacteristic
435C2AGL(1,ℤ/8)×D10711yescharacteristic
436C2#2029343711yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC2×AGL(1,ℤ/8)×C107C2
socleC22×C107C22×D4
hypercenterC2×AGL(1,ℤ/8)D107
hypocenterC107#3272
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#16812133C2semidirect
2. #20293276C2semidirect
3. D4×C22⋊D107C2semidirect
4. #20292998C2semidirect
5. #20292873C2semidirect
6. #20292870C2semidirect
7. #20292830C2semidirect
8. #20292820C2semidirect
9. #20292778C2semidirect
10. #20292776C2semidirect
11. #20292772C2semidirect
12. #20292769C2semidirect
13. D8×Dic107C2semidirect
14. #20292703C2semidirect
15. C2×AGL(1,ℤ/8)×C107C2semidirect
16. C2×#13133817C22semidirect
20. C2×#13133807C22semidirect
22. C2×#13133806C22semidirect
23. C2×#13133805C22semidirect
24. C2×#13133803C22semidirect
25. C2×#13133801C22semidirect
26. #16812107C22semidirect
30. #16812088C22semidirect
31. #16812085C22semidirect
39. D4×Dic107C22semidirect
42. #16812043C22semidirect
43. Dic107⋊C8C22semidirect
44. #16812034C22semidirect
45. #16812033C22semidirect
46. #16812031C22semidirect
48. #16812030C22semidirect
49. #16812030C22semidirect
50. C22⋊(C107⋊C8)C22semidirect
51. (C2×D107)⋊C8C22semidirect
52. C8×Dic107C22semidirect
53. C2×#41×C107C22semidirect
57. C2×SD16×C107C22semidirect
59. C2×D8×C107C22semidirect
60. C2×M16×C107C22semidirect
120. D8×C107D4semidirect
145. D4×C107C2×D4semidirect
175. C8×C107C2×D4semidirect
208. Dic107AGL(1,ℤ/8)semidirect
219. C107#3272semidirect
221. C2×AGL(1,ℤ/8)D107semidirect
236. C2×#41C2×D107semidirect
247. C2×SD16C2×D107semidirect
250. C2×D8C2×D107semidirect
252. C2×M16C2×D107semidirect
320. D8C22⋊D107semidirect
375. D4C2×C22⋊D107semidirect
396. C8C2×C22⋊D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121332
#202932762
D4×C22⋊D1072
#202929982
#202928732
#202928702
#202928302
#202928202
#202927782
#202927762
#202927722
#202927692
D8×Dic1072
#202927032
C2×AGL(1,ℤ/8)×C1072
#3272107