Finite Group Databasebeta

← #34111630

Subgroups — #34111630

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
4#23757908C211yesmaximal, characteristic, complemented
5#23757516C211yesmaximal, characteristic, complemented
6#23757506C211yesmaximal, characteristic, complemented
7#23757485C211yesmaximal, characteristic, complemented
8#23757470C211yesmaximal, characteristic, complemented
9#23757422C211yesmaximal, characteristic, complemented
10#23757415C211yesmaximal, characteristic, complemented
11#23757410C211yesmaximal, characteristic, complemented
12#23757405C211yesmaximal, characteristic, complemented
13#23757375C211yesmaximal, characteristic, complemented
14#23757346C211yesmaximal, characteristic, complemented
15#496×C157C211yesmaximal, characteristic, complemented
16C2×#14578953C2211yescharacteristic, complemented
17C2×Q8×D157C2211yescharacteristic, complemented
18C2×D4×D157—21no—
19C2×D4×D157C2211yescharacteristic
20C2×D4×D157C2211yescharacteristic
21#19097795—24no—
22C2×#14578943C2211yescharacteristic
23C2×#14578942C2211yescharacteristic, complemented
24C2×#14578942C2211yescharacteristic, complemented
25C2×#14578941C2211yescharacteristic
26SD16×D157—24no—
27C2×#14578939C2211yescharacteristic
28C2×#14578937C2211yescharacteristic, complemented
29C2×C8×D157C2211yescharacteristic, complemented
30#19097757—21no—
31#19097748—21no—
32#19097747C2211yescharacteristic, complemented
33#19097744—21no—
34#19097742—21no—
35#19097742C2211yescharacteristic
36#19097742C2211yescharacteristic
37#19097742C2211yescharacteristic, complemented
38#19097739—21no—
39#19097739—21no—
40C4×Dic314C2211yescharacteristic, complemented
41C4×D628C2211yescharacteristic
42C4×D628C2211yescharacteristic
43#19097731—21no—
44#19097705C2211yescharacteristic, complemented
45C4⋊#11929434C2211yescharacteristic, complemented
46#19097696C2211yescharacteristic, complemented
47#19097695C2211yescharacteristic, complemented
48#19097693C2211yescharacteristic
49#19097692C2211yescharacteristic, complemented
50#19097692C2211yescharacteristic, complemented
51#19097692C2211yescharacteristic, complemented
52#19097683C2211yescharacteristic, complemented
53#19097683C2211yescharacteristic, complemented
54C2×SD16×C157C2211yescharacteristic, complemented
55C2×SD16×C157C2211yescharacteristic, complemented
56C2×D8×C157C2211yescharacteristic, complemented
57C2×D8×C157C2211yescharacteristic, complemented
58#127×C157C2211yescharacteristic
59#124×C157C2211yescharacteristic, complemented
60#97×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×D628D411yescharacteristic
70C2×D628C2311yescharacteristic
71C2×C22⋊D157—21no—
72C2×C22⋊D157—21no—
73C2×C22⋊D157—21no—
74C2×C22⋊D157—21no—
75C2×C22⋊D157—21no—
76C2×C22⋊D157—21no—
77D4×D157—22no—
78D4×D157—22no—
79D4×D157—41no—
80D4×D157—41no—
81D4×D157—44no—
82C22×Dic157—21no—
83C2×C4×D157—21no—
84C2×C4×D157—21no—
85C2×C4×D157C2311yescharacteristic
86C2×C4×D157C2311yescharacteristic
87#14578943D414yescomplemented
88#14578942—22no—
89#14578942—22no—
90#14578941—22no—
91#14578939—22no—
92#14578937—22no—
93C2×#11929434C2311yescharacteristic
94C2×#11929434C2311yescharacteristic
95C8×D157—22no—
96C4⋊Dic157D411yescharacteristic
97C4⋊Dic157C2311yescharacteristic
98C4⋊Dic157C2311yescharacteristic
99#14578933—21no—
100#14578933—21no—
101C22⋊Dic157—21no—
102C22⋊Dic157—21no—
103C22⋊Dic157—21no—
104#14578931—21no—
105#14578931—21no—
106#14578931D411yescharacteristic
107#14578931D411yescharacteristic
108C4×Dic157—21no—
109C2×Q8×C157C2311yescharacteristic
110C2×D4×C157—21no—
111C2×D4×C157—21no—
112C2×D4×C157C2311yescharacteristic
113C2×D4×C157C2311yescharacteristic
114C2×D4×C157C2311yescharacteristic
115SD16×C157—22no—
116SD16×C157—41no—
117D8×C157—22no—
118D8×C157—41no—
119C2×C8×C157C2311yescharacteristic
120C2×C8×C157C2311yescharacteristic
121C22⋊C4×C157—21no—
122C22⋊C4×C157—21no—
123C42×C157C2311yescharacteristic
124C22×D157—21no—
125C22×D157—42no—
126C22×D157—42no—
127C22×D157—42no—
128C22×D157—42no—
129C22×D157—81no—
130C22×D157—81no—
131C22×D157C2×D411yescharacteristic
132C22×D157C2×D411yescharacteristic
133Dic314—21no—
134Dic314—42no—
135Dic314C2×D412yescomplemented
136D628—22no—
137D628—21no—
138D628C2×D412yes—
139C22⋊D157—42no—
140C22⋊D157—42no—
141C22⋊D157—42no—
142C22⋊D157—42no—
143C22⋊D157—42no—
144C22⋊D157—42no—
145C2×Dic157—21no—
146C2×Dic157—21no—
147C2×Dic157—21no—
148C2×Dic157—42no—
149C2×Dic157C2×D411yescharacteristic
150C2×Dic157C2×D411yescharacteristic
151C4×D157—21no—
152C4×D157—21no—
153C4×D157—21no—
154C4×D157—21no—
155C4×D157—42no—
156C4×D157—42no—
157#11929434—21no—
158#11929434—21no—
159C23×C157—21no—
160C23×C157—21no—
161C23×C157—21no—
162Q8×C157—21no—
163Q8×C157—21no—
164D4×C157—21no—
165D4×C157—21no—
166D4×C157—21no—
167D4×C157—21no—
168D4×C157—21no—
169D4×C157—21no—
170D4×C157—42no—
171D4×C157—81no—
172C2×C4×C157—21no—
173C2×C4×C157C2×D411yescharacteristic
174C2×C4×C157C2×D411yescharacteristic
175C2×C4×C157C2411yescharacteristic
176C8×C157—21no—
177C8×C157C2×D412yescomplemented
178C22×C157—41no—
179C22×C157—41no—
180C22×C157—41no—
181C22×C157—41no—
182C22×C157—41no—
183C22×C157—41no—
184C22×C157—41no—
185C22×C157—41no—
186C22×C157—41no—
187C22×C157C22×D411yescharacteristic
188C2×D157—21no—
189C2×D157—21no—
190C2×D157—21no—
191C2×D157—22no—
192C2×D157—21no—
193C2×D157—21no—
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×C157C22×D411yescharacteristic
203C4×C157H(5,𝔽2)11yescharacteristic
204Dic157—21no—
205Dic157—21no—
206Dic157—22no—
207Dic157—41no—
208Dic157—41no—
209C2×C157—81no—
210C2×C157—81no—
211C2×C157—81no—
212C2×C157D4211yescharacteristic
213C2×C157C2×AGL(1,ℤ/8)11yescharacteristic
214C2×C157#57711yescharacteristic
215D157—22no—
216D157—41no—
217D157—42no—
218C157#326511yescharacteristic, complemented, Sylow
219#3265—1571nomaximal, Sylow
220C2×AGL(1,ℤ/8)—1571no—
221C22×SD16—1571no—
222#548—1571no—
223D42—1571no—
224#496D15711yescharacteristic, complemented
225#468—1571no—
226#466—1571no—
227#463—1571no—
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×SD16C2×D15711yescharacteristic, complemented
247C2×SD16C2×D15711yescharacteristic, complemented
248C2×D8—1571no—
249C2×D8C2×D15711yescharacteristic, complemented
250C2×D8C2×D15711yescharacteristic, complemented
251C2×M16—1571no—
252C22×C8—1571no—
253#127C2×D15711yescharacteristic
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×D4—1571no—
269#107—1571no—
270C4⋊C8—1571no—
271#103—1571no—
272#103—1571no—
273#102—1571no—
274#102—1571no—
275#102—1571no—
276#102—1571no—
277C22⋊C8—1571no—
278C22⋊C8—1571no—
279#97C2×D15711yescharacteristic, complemented
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—21no—
290C2×D4—21no—
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—6281no—
300C2×D4—6281no—
301C2×D4—6284no—
302C2×D4—1571no—
303C2×D4—1571no—
304C2×D4C22×D15711yescharacteristic
305C2×D4C22×D15711yescharacteristic
306C2×D4C22×D15711yescharacteristic
307C22×C4—3141no—
308C22×C4—3141no—
309C22×C4—3141no—
310C22×C4—1571no—
311C22×C4—1571no—
312SD16—22no—
313SD16—41no—
314SD16—3142no—
315SD16—3142no—
316SD16—3142no—
317SD16—1574no—
318D8—22no—
319D8—41no—
320D8—3142no—
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⋊C4—1571no—
332C22⋊C4—21no—
333C22⋊C4—21no—
334C22⋊C4—3141no—
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—21no—
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—21no—
364D4—21no—
365D4—21no—
366D4—42no—
367D4—81no—
368D4—3142no—
369D4—3141no—
370D4—6282no—
371D4—6282no—
372D4—6282no—
373D4—6282no—
374D4—6282no—
375D4—6282no—
376D4—1572no—
377C2×C4—21no—
378C2×C4—3141no—
379C2×C4—3141no—
380C2×C4—3141no—
381C2×C4—3141no—
382C2×C4—3141no—
383C2×C4—3141no—
384C2×C4—3141no—
385C2×C4—6282no—
386C2×C4—6282no—
387C2×C4—6282no—
388C2×C4—1571no—
389C2×C4—1571no—
390C2×C4D4×D15711yescharacteristic
391C2×C4D4×D15711yescharacteristic
392C2×C4C23×D15711yescharacteristic
393C8—21no—
394C8—3141no—
395C8—3141no—
396C8D4×D15712yescomplemented
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—3141no—
407C22—3141no—
408C22—3141no—
409C22—3141no—
410C22—3142no—
411C22—3141no—
412C22—6281no—
413C22—6281no—
414C22—1572no—
415C22—12564no—
416C22—12564no—
417C22C2×D4×D15711yescharacteristic
418C4—21no—
419C4—21no—
420C4—41no—
421C4—3141no—
422C4—3141no—
423C4—3142no—
424C4—6281no—
425C4—6281no—
426C4C2×D4×D15711yescharacteristic
427C4#1909781611yescharacteristic
428C2—81no—
429C2—81no—
430C2—81no—
431C2—3142no—
432C2—6281no—
433C2—6282no—
434C2#2375790811yescharacteristic
435C2AGL(1,ℤ/8)×D15711yescharacteristic
436C2#2375808311yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#496×C157C2
socleC22×C157C22×D4
hypercenter#496D157
hypocenterC157#3265
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×#19097795C2semidirect
2. C2×SD16×D157C2semidirect
3. #23757931C2semidirect
4. #23757908C2semidirect
5. #23757516C2semidirect
6. #23757506C2semidirect
7. #23757485C2semidirect
8. #23757470C2semidirect
9. #23757422C2semidirect
10. #23757415C2semidirect
11. #23757410C2semidirect
12. #23757405C2semidirect
13. #23757375C2semidirect
14. #23757346C2semidirect
15. #496×C157C2semidirect
16. C2×#14578953C22semidirect
17. C2×Q8×D157C22semidirect
23. C2×#14578942C22semidirect
24. C2×#14578942C22semidirect
28. C2×#14578937C22semidirect
29. C2×C8×D157C22semidirect
32. #19097747C22semidirect
37. #19097742C22semidirect
40. C4×Dic314C22semidirect
44. #19097705C22semidirect
45. C4⋊#11929434C22semidirect
46. #19097696C22semidirect
47. #19097695C22semidirect
49. #19097692C22semidirect
50. #19097692C22semidirect
51. #19097692C22semidirect
52. #19097683C22semidirect
53. #19097683C22semidirect
54. C2×SD16×C157C22semidirect
55. C2×SD16×C157C22semidirect
56. C2×D8×C157C22semidirect
57. C2×D8×C157C22semidirect
59. #124×C157C22semidirect
60. #97×C157C22semidirect
87. #14578943D4semidirect
135. Dic314C2×D4semidirect
177. C8×C157C2×D4semidirect
198. C2×D157AGL(1,ℤ/8)semidirect
218. C157#3265semidirect
224. #496D157semidirect
246. C2×SD16C2×D157semidirect
247. C2×SD16C2×D157semidirect
249. C2×D8C2×D157semidirect
250. C2×D8C2×D157semidirect
255. #124C2×D157semidirect
279. #97C2×D157semidirect
396. C8D4×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
#237579082
#237575162
#237575062
#237574852
#237574702
#237574222
#237574152
#237574102
#237574052
#237573752
#237573462
#496×C1572
#3265157