Finite Group Databasebeta

← #34111637

Subgroups — #34111637

Order 20096.

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×#19097795C211yesmaximal, characteristic, complemented
2C2×SD16×D157C211yesmaximal, characteristic, complemented
3#23757931C211yesmaximal, characteristic, complemented
4D4×D628C211yesmaximal, characteristic, complemented
5#23757663C211yesmaximal, characteristic, complemented
6#23757522C211yesmaximal, characteristic, complemented
7#23757508C211yesmaximal, characteristic, complemented
8#23757474C211yesmaximal, characteristic, complemented
9#23757422C211yesmaximal, characteristic, complemented
10#23757419C211yesmaximal, characteristic, complemented
11#23757413C211yesmaximal, characteristic, complemented
12#23757405C211yesmaximal, characteristic, complemented
13#23757373C211yesmaximal, characteristic, complemented
14#23757343C211yesmaximal, characteristic, complemented
15#463×C157C211yesmaximal, characteristic, complemented
16C2×#14578953C2211yescharacteristic, complemented
17C2×Q8×D157C2211yescharacteristic, complemented
18C22×D628—21no—
19C2×D4×D157C2211yescharacteristic
20C2×D4×D157C2211yescharacteristic
21#19097795—24no—
22C2×#14578943C2211yescharacteristic, complemented
23C2×#14578943C2211yescharacteristic, complemented
24C2×#14578942C2211yescharacteristic
25C2×#14578941C2211yescharacteristic, complemented
26C2×#14578941C2211yescharacteristic, complemented
27SD16×D157—24no—
28C2×#14578939C2211yescharacteristic, complemented
29C2×#14578939C2211yescharacteristic, complemented
30C2×#14578937C2211yescharacteristic, complemented
31C2×C8×D157C2211yescharacteristic, complemented
32#19097768C2211yescharacteristic
33#19097757—21no—
34#19097748—21no—
35#19097747C2211yescharacteristic, complemented
36#19097746C2211yescharacteristic
37#19097746C2211yescharacteristic
38#19097745—21no—
39#19097744—21no—
40#19097742C2211yescharacteristic, complemented
41#19097741—21no—
42#19097741—21no—
43C4×Dic314C2211yescharacteristic, complemented
44C4×D628C2211yescharacteristic
45#19097731—21no—
46#19097704C2211yescharacteristic, complemented
47#19097694C2211yescharacteristic, complemented
48#19097693C2211yescharacteristic, complemented
49#19097693C2211yescharacteristic, complemented
50#19097692C2211yescharacteristic
51#19097683C2211yescharacteristic, complemented
52#19097683C2211yescharacteristic, complemented
53#19097682C2211yescharacteristic, complemented
54C2×SD16×C157C2211yescharacteristic
55C2×D8×C157C2211yescharacteristic
56#124×C157C2211yescharacteristic, complemented
57C4×D4×C157C2211yescharacteristic
58C4⋊C8×C157C2211yescharacteristic, complemented
59#103×C157C2211yescharacteristic, complemented
60#102×C157C2211yescharacteristic, complemented
61C23×D157—21no—
62C23×D157—21no—
63#14578953—22no—
64#14578953—41no—
65C2×Dic314—21no—
66C2×Dic314C2311yescharacteristic
67Q8×D157—22no—
68Q8×D157—41no—
69C2×D628—21no—
70C2×D628—21no—
71C2×D628—21no—
72C2×D628—21no—
73C2×D628—44no—
74C2×D628D411yescharacteristic
75C2×D628C2311yescharacteristic
76C2×D628C2311yescharacteristic
77C2×C22⋊D157—21no—
78C2×C22⋊D157—21no—
79C2×C22⋊D157—21no—
80C2×C22⋊D157—21no—
81D4×D157—22no—
82D4×D157—22no—
83D4×D157—41no—
84D4×D157—41no—
85C22×Dic157—21no—
86C2×C4×D157—21no—
87C2×C4×D157C2311yescharacteristic
88C2×C4×D157C2311yescharacteristic
89#14578943—22no—
90#14578943—22no—
91#14578942D414yescomplemented
92#14578941—22no—
93#14578941—41no—
94#14578939—22no—
95#14578939—41no—
96#14578937—22no—
97C2×#11929434C2311yescharacteristic
98C2×#11929434C2311yescharacteristic
99C8×D157—22no—
100C4⋊Dic157D411yescharacteristic
101C4⋊Dic157C2311yescharacteristic
102#14578933—21no—
103#14578933—21no—
104C22⋊Dic157—21no—
105#14578931—21no—
106#14578931—21no—
107#14578931—21no—
108#14578931D411yescharacteristic
109#14578931D411yescharacteristic
110C4×Dic157—21no—
111C2×Q8×C157C2311yescharacteristic
112C2×D4×C157C2311yescharacteristic
113C2×D4×C157C2311yescharacteristic
114C22×C4×C157—21no—
115SD16×C157—22no—
116D8×C157—22no—
117C2×C8×C157C2311yescharacteristic
118C2×C8×C157C2311yescharacteristic
119C4⋊C4×C157C2311yescharacteristic
120C22⋊C4×C157—21no—
121C22⋊C4×C157—21no—
122C22⋊C4×C157—21no—
123C42×C157C2311yescharacteristic
124C22×D157—21no—
125C22×D157—21no—
126C22×D157—42no—
127C22×D157—42no—
128C22×D157—42no—
129C22×D157—42no—
130C22×D157—81no—
131C22×D157—81no—
132C22×D157C2×D411yescharacteristic
133C22×D157C2×D411yescharacteristic
134Dic314—21no—
135Dic314—42no—
136Dic314C2×D412yescomplemented
137D628—22no—
138D628—21no—
139D628—21no—
140D628—21no—
141D628—21no—
142D628—42no—
143D628—42no—
144D628—42no—
145D628—81no—
146C22⋊D157—42no—
147C22⋊D157—42no—
148C22⋊D157—42no—
149C22⋊D157—42no—
150C2×Dic157—21no—
151C2×Dic157—21no—
152C2×Dic157—42no—
153C2×Dic157C2×D411yescharacteristic
154C2×Dic157C2×D411yescharacteristic
155C4×D157—21no—
156C4×D157—21no—
157C4×D157—21no—
158C4×D157—21no—
159C4×D157—42no—
160#11929434—21no—
161#11929434C2×D412yescomplemented
162C23×C157—21no—
163C23×C157—21no—
164Q8×C157—21no—
165Q8×C157—21no—
166D4×C157—21no—
167D4×C157—21no—
168D4×C157—21no—
169D4×C157C2×D412yes—
170C2×C4×C157—21no—
171C2×C4×C157—21no—
172C2×C4×C157—42no—
173C2×C4×C157C2×D411yescharacteristic
174C2×C4×C157C2×D411yescharacteristic
175C2×C4×C157C2411yescharacteristic
176C8×C157—21no—
177C8×C157—21no—
178C22×C157—22no—
179C22×C157—41no—
180C22×C157—41no—
181C22×C157—41no—
182C22×C157—41no—
183C22×C157—41no—
184C22×C157C22×D411yescharacteristic
185C2×D157—21no—
186C2×D157—21no—
187C2×D157—21no—
188C2×D157—21no—
189C2×D157—21no—
190C2×D157—41no—
191C2×D157—41no—
192C2×D157—41no—
193C2×D157—41no—
194C2×D157—41no—
195C2×D157—41no—
196C2×D157—84no—
197C2×D157—84no—
198C2×D157AGL(1,ℤ/8)12yescomplemented
199C4×C157—21no—
200C4×C157—21no—
201C4×C157—41no—
202C4×C157—41no—
203C4×C157C22×D411yescharacteristic
204C4×C157H(5,𝔽2)11yescharacteristic
205Dic157—21no—
206Dic157—21no—
207Dic157—22no—
208Dic157—41no—
209C2×C157—42no—
210C2×C157—81no—
211C2×C157D4211yescharacteristic
212C2×C157C2×AGL(1,ℤ/8)11yescharacteristic
213C2×C157#57711yescharacteristic
214D157—22no—
215D157—41no—
216D157—81no—
217D157—81no—
218C157#326511yescharacteristic, complemented, Sylow
219#3265—1571nomaximal, Sylow
220C2×AGL(1,ℤ/8)—1571no—
221C22×SD16—1571no—
222#548—1571no—
223D42—1571no—
224#496—1571no—
225#468—1571no—
226#466—1571no—
227#463D15711yescharacteristic, complemented
228#460—1571no—
229#450—1571no—
230#449—1571no—
231#448—1571no—
232#447—1571no—
233#440—1571no—
234#435—1571no—
235C2×#41—1571no—
236C22×Q8—1571no—
237C22×D4—3141no—
238C22×D4—1571no—
239C22×D4—1571no—
240AGL(1,ℤ/8)—3144no—
241C2×SD16—3144no—
242C2×SD16—1571no—
243C2×SD16—1571no—
244C2×SD16—1571no—
245C2×SD16—1571no—
246C2×SD16—1571no—
247C2×SD16C2×D15711yescharacteristic
248C2×D8—1571no—
249C2×D8—1571no—
250C2×D8C2×D15711yescharacteristic
251C2×M16—1571no—
252C22×C8—1571no—
253#127—1571no—
254#124—3141no—
255#124C2×D15711yescharacteristic, complemented
256C22⋊Q8—3141no—
257C22⋊Q8—1571no—
258#121—3141no—
259#121—3141no—
260#121—1571no—
261#121—1571no—
262#121—1571no—
263#120—3141no—
264#120—3141no—
265C4×Q8—1571no—
266C4×D4—3141no—
267C4×D4—1571no—
268C4×D4C2×D15711yescharacteristic
269#107—1571no—
270C4⋊C8C2×D15711yescharacteristic, complemented
271#103—1571no—
272#103C2×D15711yescharacteristic, complemented
273#102—1571no—
274#102—1571no—
275#102—1571no—
276#102C2×D15711yescharacteristic, complemented
277C22⋊C8—1571no—
278C22⋊C8—1571no—
279#97—1571no—
280C24—3141no—
281C24—3141no—
282#41—3142no—
283#41—6281no—
284C2×Q8—3141no—
285C2×Q8—3142no—
286C2×Q8—6281no—
287C2×Q8—1571no—
288C2×Q8C22×D15711yescharacteristic
289C2×D4—3141no—
290C2×D4—3141no—
291C2×D4—3142no—
292C2×D4—3142no—
293C2×D4—3141no—
294C2×D4—3141no—
295C2×D4—3141no—
296C2×D4—3141no—
297C2×D4—3141no—
298C2×D4—3141no—
299C2×D4—6284no—
300C2×D4—6281no—
301C2×D4—6281no—
302C2×D4—1571no—
303C2×D4—1571no—
304C2×D4—1571no—
305C2×D4C22×D15711yescharacteristic
306C2×D4C22×D15711yescharacteristic
307C22×C4—21no—
308C22×C4—3141no—
309C22×C4—3141no—
310C22×C4—1571no—
311C22×C4—1571no—
312SD16—22no—
313SD16—3142no—
314SD16—3142no—
315SD16—3142no—
316SD16—6281no—
317SD16—1574no—
318D8—22no—
319D8—3142no—
320D8—6281no—
321M16—3142no—
322C2×C8—3142no—
323C2×C8—1571no—
324C2×C8—1571no—
325C2×C8C22×D15711yescharacteristic
326C2×C8C22×D15711yescharacteristic
327C4⋊C4—3141no—
328C4⋊C4—3141no—
329C4⋊C4—1571no—
330C4⋊C4—1571no—
331C4⋊C4C22×D15711yescharacteristic
332C22⋊C4—21no—
333C22⋊C4—21no—
334C22⋊C4—21no—
335C22⋊C4—3141no—
336C22⋊C4—3141no—
337C22⋊C4—3141no—
338C22⋊C4—3141no—
339C22⋊C4—1571no—
340C22⋊C4—1571no—
341C42—3141no—
342C42C22×D15711yescharacteristic
343C23—21no—
344C23—21no—
345C23—3141no—
346C23—3141no—
347C23—6282no—
348C23—6282no—
349C23—6282no—
350C23—6282no—
351C23—1571no—
352C23—1571no—
353C23—12561no—
354C23—12561no—
355Q8—21no—
356Q8—21no—
357Q8—3141no—
358Q8—6282no—
359Q8—1572no—
360D4—21no—
361D4—21no—
362D4—21no—
363D4—3142no—
364D4—3141no—
365D4—3141no—
366D4—3141no—
367D4—3141no—
368D4—6282no—
369D4—6282no—
370D4—6282no—
371D4—6282no—
372D4—6282no—
373D4—6282no—
374D4—6282no—
375D4—12561no—
376D4D4×D15712yes—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—42no—
380C2×C4—3141no—
381C2×C4—3141no—
382C2×C4—3141no—
383C2×C4—3141no—
384C2×C4—3141no—
385C2×C4—3141no—
386C2×C4—6282no—
387C2×C4—6282no—
388C2×C4—1571no—
389C2×C4—1571no—
390C2×C4D4×D15711yescharacteristic
391C2×C4D4×D15711yescharacteristic
392C2×C4C23×D15711yescharacteristic
393C8—21no—
394C8—21no—
395C8—3141no—
396C8—1572no—
397C22—22no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—3141no—
404C22—3141no—
405C22—3141no—
406C22—3141no—
407C22—3141no—
408C22—6281no—
409C22—6281no—
410C22—6281no—
411C22—6281no—
412C22—6281no—
413C22—6281no—
414C22—1572no—
415C22—12564no—
416C22—12564no—
417C22C2×D4×D15711yescharacteristic
418C4—21no—
419C4—21no—
420C4—41no—
421C4—41no—
422C4—3141no—
423C4—3141no—
424C4—3142no—
425C4—6281no—
426C4C2×D4×D15711yescharacteristic
427C4#1909781711yescharacteristic
428C2—42no—
429C2—81no—
430C2—3142no—
431C2—6281no—
432C2—12561no—
433C2—12561no—
434C2#2375790711yescharacteristic
435C2AGL(1,ℤ/8)×D15711yescharacteristic
436C2#2375808011yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#463×C157C2
socleC22×C157C22×D4
hypercenter#463D157
hypocenterC157#3265
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. C2×#19097795C2semidirect
2. C2×SD16×D157C2semidirect
3. #23757931C2semidirect
4. D4×D628C2semidirect
5. #23757663C2semidirect
6. #23757522C2semidirect
7. #23757508C2semidirect
8. #23757474C2semidirect
9. #23757422C2semidirect
10. #23757419C2semidirect
11. #23757413C2semidirect
12. #23757405C2semidirect
13. #23757373C2semidirect
14. #23757343C2semidirect
15. #463×C157C2semidirect
16. C2×#14578953C22semidirect
17. C2×Q8×D157C22semidirect
22. C2×#14578943C22semidirect
23. C2×#14578943C22semidirect
25. C2×#14578941C22semidirect
26. C2×#14578941C22semidirect
28. C2×#14578939C22semidirect
29. C2×#14578939C22semidirect
30. C2×#14578937C22semidirect
31. C2×C8×D157C22semidirect
35. #19097747C22semidirect
40. #19097742C22semidirect
43. C4×Dic314C22semidirect
46. #19097704C22semidirect
47. #19097694C22semidirect
48. #19097693C22semidirect
49. #19097693C22semidirect
51. #19097683C22semidirect
52. #19097683C22semidirect
53. #19097682C22semidirect
56. #124×C157C22semidirect
58. C4⋊C8×C157C22semidirect
59. #103×C157C22semidirect
60. #102×C157C22semidirect
91. #14578942D4semidirect
136. Dic314C2×D4semidirect
161. #11929434C2×D4semidirect
198. C2×D157AGL(1,ℤ/8)semidirect
218. C157#3265semidirect
227. #463D157semidirect
255. #124C2×D157semidirect
270. C4⋊C8C2×D157semidirect
272. #103C2×D157semidirect
276. #102C2×D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190977952
C2×SD16×D1572
#237579312
D4×D6282
#237576632
#237575222
#237575082
#237574742
#237574222
#237574192
#237574132
#237574052
#237573732
#237573432
#463×C1572
#3265157