Finite Group Databasebeta

← #34111632

Subgroups — #34111632

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×#19097796C211yesmaximal, characteristic, complemented
2C22×#14578941C211yesmaximal, characteristic, complemented
3D4×D628C211yesmaximal, characteristic, complemented
4#23757663C211yesmaximal, characteristic, complemented
5#23757521C211yesmaximal, characteristic, complemented
6#23757507C211yesmaximal, characteristic, complemented
7#23757485C211yesmaximal, characteristic, complemented
8#23757473C211yesmaximal, characteristic, complemented
9#23757423C211yesmaximal, characteristic, complemented
10#23757418C211yesmaximal, characteristic, complemented
11#23757412C211yesmaximal, characteristic, complemented
12#23757406C211yesmaximal, characteristic, complemented
13#23757371C211yesmaximal, characteristic, complemented
14#23757345C211yesmaximal, characteristic, complemented
15#548×C157C211yesmaximal, characteristic, complemented
16C22×D628C2211yescharacteristic
17C22×D628C2211yescharacteristic
18C2×D4×D157—21no—
19#19097796—24no—
20C2×#14578941—24no—
21C2×#14578941C2211yescharacteristic
22C2×#14578941C2211yescharacteristic
23C2×#14578941C2211yescharacteristic, complemented
24C2×#14578941C2211yescharacteristic, complemented
25C2×#14578941C2211yescharacteristic, complemented
26C2×#14578941C2211yescharacteristic, complemented
27C2×#14578939C2211yescharacteristic
28C2×#14578939C2211yescharacteristic, complemented
29C2×#14578939C2211yescharacteristic, complemented
30C2×#14578938C2211yescharacteristic, complemented
31C22×#11929434C2211yescharacteristic, complemented
32#19097768C2211yescharacteristic
33#19097746—21no—
34#19097745C2211yescharacteristic
35#19097745C2211yescharacteristic
36#19097741—21no—
37#19097741—21no—
38C4×D628C2211yescharacteristic
39#19097704C2211yescharacteristic, complemented
40C4⋊#11929434C2211yescharacteristic, complemented
41#19097695C2211yescharacteristic, complemented
42#19097695C2211yescharacteristic, complemented
43#19097692C2211yescharacteristic, complemented
44#19097691C2211yescharacteristic
45#19097691C2211yescharacteristic, complemented
46#19097691C2211yescharacteristic, complemented
47C22⋊#11929434C2211yescharacteristic, complemented
48C22⋊#11929434C2211yescharacteristic, complemented
49#19097682C2211yescharacteristic, complemented
50C2×#41×C157C2211yescharacteristic, complemented
51C22×Q8×C157C2211yescharacteristic, complemented
52#124×C157—21no—
53#124×C157C2211yescharacteristic, complemented
54C22⋊Q8×C157—21no—
55C22⋊Q8×C157C2211yescharacteristic, complemented
56#121×C157—21no—
57#121×C157C2211yescharacteristic, complemented
58C4×Q8×C157C2211yescharacteristic, complemented
59C4×D4×C157—21no—
60C4×D4×C157C2211yescharacteristic
61C23×D157—21no—
62C23×D157—21no—
63C2×D628—22no—
64C2×D628—22no—
65C2×D628—21no—
66C2×D628—21no—
67C2×D628—21no—
68C2×D628—21no—
69C2×D628—41no—
70C2×D628—41no—
71C2×D628C2311yescharacteristic
72C2×D628C2311yescharacteristic
73C2×D628C2311yescharacteristic
74C2×C22⋊D157—21no—
75C2×C22⋊D157—21no—
76D4×D157—44no—
77C2×C4×D157—21no—
78#14578941—22no—
79#14578941—22no—
80#14578941—22no—
81#14578941—22no—
82#14578941—41no—
83#14578941D414yescomplemented
84#14578939—22no—
85#14578939—22no—
86#14578939—41no—
87#14578938—22no—
88C2×#11929434—22no—
89C2×#11929434C2311yescharacteristic
90C2×#11929434C2311yescharacteristic
91C2×#11929434C2311yescharacteristic
92C2×#11929434C2311yescharacteristic
93C4⋊Dic157C2311yescharacteristic
94#14578931—21no—
95#14578931—21no—
96#14578931—21no—
97#41×C157—22no—
98#41×C157—41no—
99C2×Q8×C157—21no—
100C2×Q8×C157—22no—
101C2×Q8×C157—41no—
102C2×Q8×C157C2311yescharacteristic
103C2×Q8×C157C2311yescharacteristic
104C2×D4×C157—21no—
105C2×D4×C157—21no—
106C2×D4×C157D411yescharacteristic
107C2×D4×C157C2311yescharacteristic
108C22×C4×C157—21no—
109C22×C4×C157—21no—
110C22×C4×C157C2311yescharacteristic
111C22×C4×C157C2311yescharacteristic
112C4⋊C4×C157—21no—
113C4⋊C4×C157—21no—
114C4⋊C4×C157D411yescharacteristic
115C4⋊C4×C157C2311yescharacteristic
116C22⋊C4×C157—21no—
117C22⋊C4×C157—21no—
118C22⋊C4×C157—21no—
119C22⋊C4×C157—21no—
120C22⋊C4×C157D411yescharacteristic
121C22⋊C4×C157D411yescharacteristic
122C42×C157—21no—
123C42×C157C2311yescharacteristic
124C22×D157—21no—
125C22×D157—21no—
126C22×D157—21no—
127C22×D157—42no—
128C22×D157—42no—
129C22×D157—42no—
130C22×D157—42no—
131C22×D157—81no—
132C22×D157—81no—
133D628—21no—
134D628—21no—
135D628—21no—
136D628—21no—
137D628—21no—
138D628—42no—
139D628—42no—
140D628—42no—
141D628—81no—
142D628C2×D412yes—
143C22⋊D157—42no—
144C22⋊D157—42no—
145C2×Dic157—21no—
146C4×D157—42no—
147#11929434—21no—
148#11929434—21no—
149#11929434—21no—
150#11929434C2×D412yescomplemented
151C23×C157—21no—
152C23×C157C2×D411yescharacteristic
153C23×C157C2×D411yescharacteristic
154Q8×C157—21no—
155Q8×C157—21no—
156Q8×C157—21no—
157Q8×C157—42no—
158Q8×C157C2×D412yescomplemented
159D4×C157—22no—
160D4×C157—21no—
161D4×C157—21no—
162D4×C157—42no—
163D4×C157—42no—
164C2×C4×C157—21no—
165C2×C4×C157—21no—
166C2×C4×C157—21no—
167C2×C4×C157—21no—
168C2×C4×C157—21no—
169C2×C4×C157—21no—
170C2×C4×C157—21no—
171C2×C4×C157—42no—
172C2×C4×C157—42no—
173C2×C4×C157C2×D411yescharacteristic
174C2×C4×C157C2×D411yescharacteristic
175C2×C4×C157C2×D411yescharacteristic
176C2×C4×C157C2×D411yescharacteristic
177C2×C4×C157C2411yescharacteristic
178C22×C157—21no—
179C22×C157—21no—
180C22×C157—21no—
181C22×C157—21no—
182C22×C157—21no—
183C22×C157—41no—
184C22×C157—41no—
185C22×C157—41no—
186C22×C157AGL(1,ℤ/8)12yescomplemented
187C22×C157C22×D411yescharacteristic
188C2×D157—22no—
189C2×D157—41no—
190C2×D157—41no—
191C2×D157—41no—
192C2×D157—41no—
193C2×D157—41no—
194C2×D157—41no—
195C2×D157—41no—
196C2×D157—41no—
197C2×D157—84no—
198C2×D157—84no—
199C4×C157—21no—
200C4×C157—21no—
201C4×C157—22no—
202C4×C157—21no—
203C4×C157—21no—
204C4×C157—41no—
205C4×C157—41no—
206C4×C157C22×D411yescharacteristic
207C4×C157H(5,𝔽2)11yescharacteristic
208Dic157—41no—
209C2×C157—22no—
210C2×C157—41no—
211C2×C157—81no—
212C2×C157D4211yescharacteristic
213C2×C157C2×AGL(1,ℤ/8)11yescharacteristic
214C2×C157#57711yescharacteristic
215D157—42no—
216D157—81no—
217D157—81no—
218C157#326511yescharacteristic, complemented, Sylow
219#3265—1571nomaximal, Sylow
220C2×AGL(1,ℤ/8)—1571no—
221C22×SD16—1571no—
222#548D15711yescharacteristic, complemented
223D42—1571no—
224#496—1571no—
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×#41C2×D15711yescharacteristic, complemented
236C22×Q8C2×D15711yescharacteristic, complemented
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×SD16—1571no—
248C2×D8—1571no—
249C2×D8—1571no—
250C2×D8—1571no—
251C2×M16—1571no—
252C22×C8—1571no—
253#127—1571no—
254#124—21no—
255#124C2×D15711yescharacteristic, complemented
256C22⋊Q8—21no—
257C22⋊Q8C2×D15711yescharacteristic, complemented
258#121—21no—
259#121—3141no—
260#121—1571no—
261#121—1571no—
262#121C2×D15711yescharacteristic, complemented
263#120—3141no—
264#120—3141no—
265C4×Q8C2×D15711yescharacteristic, complemented
266C4×D4—21no—
267C4×D4—1571no—
268C4×D4C2×D15711yescharacteristic
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#97—1571no—
280C24—3141no—
281C24—3141no—
282#41—22no—
283#41—41no—
284C2×Q8—21no—
285C2×Q8—22no—
286C2×Q8—41no—
287C2×Q8C22×D15711yescharacteristic
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×D4—1571no—
305C2×D4C22⋊D15711yescharacteristic
306C2×D4C22×D15711yescharacteristic
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—3141no—
310C22×C4C22×D15711yescharacteristic
311C22×C4C22×D15711yescharacteristic
312SD16—3142no—
313SD16—3142no—
314SD16—3142no—
315SD16—3142no—
316SD16—6281no—
317SD16—1574no—
318D8—3142no—
319D8—3142no—
320D8—6281no—
321M16—3142no—
322C2×C8—3142no—
323C2×C8—1571no—
324C2×C8—1571no—
325C2×C8—1571no—
326C2×C8—1571no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—1571no—
330C4⋊C4C22⋊D15711yescharacteristic
331C4⋊C4C22×D15711yescharacteristic
332C22⋊C4—21no—
333C22⋊C4—21no—
334C22⋊C4—21no—
335C22⋊C4—21no—
336C22⋊C4—3141no—
337C22⋊C4—3141no—
338C22⋊C4—3141no—
339C22⋊C4C22⋊D15711yescharacteristic
340C22⋊C4C22⋊D15711yescharacteristic
341C42—21no—
342C42C22×D15711yescharacteristic
343C23—21no—
344C23—3141no—
345C23—3141no—
346C23—3141no—
347C23—6282no—
348C23—6282no—
349C23—6282no—
350C23—6282no—
351C23—12561no—
352C23—12561no—
353C23C2×C22⋊D15711yescharacteristic
354C23C2×C22⋊D15711yescharacteristic
355Q8—21no—
356Q8—21no—
357Q8—21no—
358Q8—42no—
359Q8D4×D15712yescomplemented
360D4—22no—
361D4—21no—
362D4—21no—
363D4—42no—
364D4—42no—
365D4—3141no—
366D4—3141no—
367D4—3141no—
368D4—3141no—
369D4—3141no—
370D4—6282no—
371D4—6282no—
372D4—6282no—
373D4—6282no—
374D4—6282no—
375D4—1572no—
376D4—12561no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—3141no—
387C2×C4—6282no—
388C2×C4C2×C22⋊D15711yescharacteristic
389C2×C4C2×C22⋊D15711yescharacteristic
390C2×C4C2×C22⋊D15711yescharacteristic
391C2×C4C2×C22⋊D15711yescharacteristic
392C2×C4C23×D15711yescharacteristic
393C8—3141no—
394C8—3141no—
395C8—3141no—
396C8—1572no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—3142no—
406C22—6281no—
407C22—6281no—
408C22—6281no—
409C22—6281no—
410C22—6281no—
411C22—6281no—
412C22—6281no—
413C22—6281no—
414C22—12564no—
415C22—12564no—
416C22#1909779612yescomplemented
417C22C22×C22⋊D15711yescharacteristic
418C4—21no—
419C4—21no—
420C4—21no—
421C4—22no—
422C4—21no—
423C4—41no—
424C4—41no—
425C4—6281no—
426C4C2×D4×D15711yescharacteristic
427C4#1909781611yescharacteristic
428C2—22no—
429C2—41no—
430C2—81no—
431C2—6282no—
432C2—12561no—
433C2—12561no—
434C2D4×C22⋊D15711yescharacteristic
435C2C2×#1909779611yescharacteristic
436C2#2375808111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#548×C157C2
socleC22×C157C22×D4
hypercenter#548D157
hypocenterC157#3265
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C4×D4 ◁ #548

Complements

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

NH ≅ G/Ntype
1. C2×#19097796C2semidirect
2. C22×#14578941C2semidirect
3. D4×D628C2semidirect
4. #23757663C2semidirect
5. #23757521C2semidirect
6. #23757507C2semidirect
7. #23757485C2semidirect
8. #23757473C2semidirect
9. #23757423C2semidirect
10. #23757418C2semidirect
11. #23757412C2semidirect
12. #23757406C2semidirect
13. #23757371C2semidirect
14. #23757345C2semidirect
15. #548×C157C2semidirect
23. C2×#14578941C22semidirect
24. C2×#14578941C22semidirect
25. C2×#14578941C22semidirect
26. C2×#14578941C22semidirect
28. C2×#14578939C22semidirect
29. C2×#14578939C22semidirect
30. C2×#14578938C22semidirect
31. C22×#11929434C22semidirect
39. #19097704C22semidirect
40. C4⋊#11929434C22semidirect
41. #19097695C22semidirect
42. #19097695C22semidirect
43. #19097692C22semidirect
45. #19097691C22semidirect
46. #19097691C22semidirect
47. C22⋊#11929434C22semidirect
48. C22⋊#11929434C22semidirect
49. #19097682C22semidirect
50. C2×#41×C157C22semidirect
51. C22×Q8×C157C22semidirect
53. #124×C157C22semidirect
55. C22⋊Q8×C157C22semidirect
57. #121×C157C22semidirect
58. C4×Q8×C157C22semidirect
83. #14578941D4semidirect
150. #11929434C2×D4semidirect
158. Q8×C157C2×D4semidirect
186. C22×C157AGL(1,ℤ/8)semidirect
218. C157#3265semidirect
222. #548D157semidirect
235. C2×#41C2×D157semidirect
236. C22×Q8C2×D157semidirect
255. #124C2×D157semidirect
257. C22⋊Q8C2×D157semidirect
262. #121C2×D157semidirect
265. C4×Q8C2×D157semidirect
359. Q8D4×D157semidirect
416. C22#19097796semidirect

Maximal subgroups

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

subgroupindex
C2×#190977962
C22×#145789412
D4×D6282
#237576632
#237575212
#237575072
#237574852
#237574732
#237574232
#237574182
#237574122
#237574062
#237573712
#237573452
#548×C1572
#3265157