Finite Group Databasebeta

← #34111749

Subgroups — #34111749

Order 20096.

556 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19097798C211yesmaximal, characteristic, complemented
2C2×#19097796C211yesmaximal, characteristic, complemented
3#23757942C211yesmaximal, characteristic, complemented
4#23757907C211yesmaximal, characteristic, complemented
5#23757648C211yesmaximal, characteristic, complemented
6#23757525C211yesmaximal, characteristic, complemented
7#23757524C211yesmaximal, characteristic, complemented
8#23757471C211yesmaximal, characteristic, complemented
9#23757467C211yesmaximal, characteristic, complemented
10#23757421C211yesmaximal, characteristic, complemented
11#23757409C211yesmaximal, characteristic, complemented
12#23757405C211yesmaximal, characteristic, complemented
13#23757358C211yesmaximal, characteristic, complemented
14#23757348C211yesmaximal, characteristic, complemented
15#449×C157C211yesmaximal, characteristic, complemented
16C2×#14578955C2211yescharacteristic, complemented
17C22×D628C2211yescharacteristic
18C2×D4×D157—21no—
19C2×D4×D157C2211yescharacteristic
20#19097798—24no—
21#19097796—24no—
22C2×#14578941C2211yescharacteristic
23C2×#14578941C2211yescharacteristic, complemented
24C2×D1256C2211yescharacteristic
25C2×D1256C2211yescharacteristic, complemented
26C2×#14578939C2211yescharacteristic
27C2×#14578939C2211yescharacteristic, complemented
28C2×#14578939C2211yescharacteristic, complemented
29C2×#14578938C2211yescharacteristic, complemented
30C2×#14578937C2211yescharacteristic, complemented
31#19097767—21no—
32#19097767C2211yescharacteristic
33#19097767C2211yescharacteristic, complemented
34#19097746C2211yescharacteristic
35#19097745—21no—
36#19097745C2211yescharacteristic, complemented
37#19097744—21no—
38#19097742—21no—
39#19097742C2211yescharacteristic, complemented
40#19097741—21no—
41#19097739—21no—
42Q8×Dic157C2211yescharacteristic, complemented
43#19097733C2211yescharacteristic
44C4×C22⋊D157—21no—
45C4×C22⋊D157C2211yescharacteristic
46#19097702C2211yescharacteristic, complemented
47#19097700C2211yescharacteristic, complemented
48#19097693C2211yescharacteristic
49#19097693C2211yescharacteristic, complemented
50#19097692C2211yescharacteristic, complemented
51#19097691C2211yescharacteristic, complemented
52#19097683C2211yescharacteristic, complemented
53C8×Dic157C2211yescharacteristic, complemented
54C2×#41×C157C2211yescharacteristic, complemented
55C2×SD16×C157C2211yescharacteristic, complemented
56C2×D8×C157C2211yescharacteristic, complemented
57#121×C157C2211yescharacteristic
58#103×C157C2211yescharacteristic, complemented
59#102×C157C2211yescharacteristic, complemented
60C22⋊C8×C157C2211yescharacteristic, complemented
61C23×D157—21no—
62C23×D157—21no—
63#14578955—22no—
64#14578955—41no—
65C2×D628—22no—
66C2×D628—21no—
67C2×D628—21no—
68C2×D628—21no—
69C2×D628—41no—
70C2×D628C2311yescharacteristic
71C2×D628C2311yescharacteristic
72C2×D628C2311yescharacteristic
73C2×C22⋊D157—21no—
74C2×C22⋊D157—21no—
75C2×C22⋊D157—21no—
76C2×C22⋊D157—21no—
77C2×C22⋊D157—21no—
78C2×C22⋊D157—21no—
79C2×C22⋊D157—21no—
80C2×C22⋊D157D411yescharacteristic
81D4×D157—22no—
82D4×D157—41no—
83D4×D157—44no—
84C2×C4×D157—21no—
85C2×C4×D157—21no—
86C2×C4×D157C2311yescharacteristic
87#14578941—22no—
88#14578941D414yescomplemented
89D1256—22no—
90D1256—41no—
91#14578939—22no—
92#14578939—22no—
93#14578939—22no—
94#14578938—22no—
95#14578937—22no—
96C2×#11929434C2311yescharacteristic
97C2×#11929434C2311yescharacteristic
98C4⋊Dic157—21no—
99C4⋊Dic157C2311yescharacteristic
100#14578933D411yescharacteristic
101C22⋊Dic157—21no—
102C22⋊Dic157D411yescharacteristic
103#14578931—21no—
104#14578931—21no—
105#14578931—21no—
106#14578931D411yescharacteristic
107C4×Dic157—21no—
108C4×Dic157C2311yescharacteristic
109#41×C157—22no—
110#41×C157—41no—
111C2×Q8×C157C2311yescharacteristic
112C2×D4×C157—21no—
113C2×D4×C157—21no—
114C2×D4×C157C2311yescharacteristic
115C2×D4×C157C2311yescharacteristic
116C22×C4×C157—21no—
117C22×C4×C157C2311yescharacteristic
118SD16×C157—22no—
119D8×C157—41no—
120C2×C8×C157C2311yescharacteristic
121C2×C8×C157C2311yescharacteristic
122C4⋊C4×C157C2311yescharacteristic
123C22⋊C4×C157—21no—
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—
133C22×D157C2×D411yescharacteristic
134D628—21no—
135D628—21no—
136D628—21no—
137D628—21no—
138D628—21no—
139D628—42no—
140D628—42no—
141D628—42no—
142D628C2×D412yes—
143C22⋊D157—22no—
144C22⋊D157—42no—
145C22⋊D157—42no—
146C22⋊D157—42no—
147C22⋊D157—81no—
148C22⋊D157—81no—
149C22⋊D157—81no—
150C22⋊D157—81no—
151C2×Dic157—21no—
152C2×Dic157C2×D411yescharacteristic
153C2×Dic157C2×D411yescharacteristic
154C2×Dic157C2×D411yescharacteristic
155C4×D157—21no—
156C4×D157—21no—
157C4×D157—42no—
158C4×D157—42no—
159#11929434—21no—
160#11929434C2×D412yescomplemented
161C23×C157—21no—
162C23×C157—21no—
163C23×C157C2×D411yescharacteristic
164Q8×C157—21no—
165Q8×C157C2×D412yescomplemented
166D4×C157—21no—
167D4×C157—21no—
168D4×C157—21no—
169D4×C157—21no—
170D4×C157—42no—
171D4×C157—42no—
172C2×C4×C157—21no—
173C2×C4×C157—21no—
174C2×C4×C157—21no—
175C2×C4×C157—21no—
176C2×C4×C157—42no—
177C2×C4×C157C2×D411yescharacteristic
178C2×C4×C157C2411yescharacteristic
179C8×C157—21no—
180C8×C157—21no—
181C22×C157—21no—
182C22×C157—21no—
183C22×C157—21no—
184C22×C157—41no—
185C22×C157—41no—
186C22×C157—41no—
187C22×C157—41no—
188C22×C157—41no—
189C22×C157—41no—
190C22×C157C22×D411yescharacteristic
191C2×D157—21no—
192C2×D157—22no—
193C2×D157—21no—
194C2×D157—21no—
195C2×D157—41no—
196C2×D157—41no—
197C2×D157—41no—
198C2×D157—41no—
199C2×D157—41no—
200C2×D157—41no—
201C2×D157—41no—
202C2×D157—41no—
203C2×D157—84no—
204C2×D157—84no—
205C4×C157—21no—
206C4×C157—22no—
207C4×C157—41no—
208C4×C157C22×D411yescharacteristic
209C4×C157H(5,𝔽2)11yescharacteristic
210Dic157—21no—
211Dic157—21no—
212Dic157—41no—
213Dic157AGL(1,ℤ/8)12yescomplemented
214C2×C157—41no—
215C2×C157—81no—
216C2×C157—81no—
217C2×C157D4211yescharacteristic
218C2×C157C2×AGL(1,ℤ/8)11yescharacteristic
219C2×C157#57611yescharacteristic
220D157—41no—
221D157—42no—
222D157—81no—
223D157—81no—
224C157#327411yescharacteristic, complemented, Sylow
225#3274—1571nomaximal, Sylow
226C2×AGL(1,ℤ/8)—1571no—
227C2×AGL(1,ℤ/8)—1571no—
228#550—1571no—
229D42—1571no—
230#493—1571no—
231#469—1571no—
232#469—1571no—
233#460—1571no—
234#459—1571no—
235#449—1571no—
236#449D15711yescharacteristic, complemented
237#447—1571no—
238#447—1571no—
239C4×SD16—1571no—
240#436—1571no—
241C2×#41—1571no—
242C2×#41C2×D15711yescharacteristic, complemented
243C22×D4—3141no—
244C22×D4—1571no—
245C22×D4—1571no—
246AGL(1,ℤ/8)—3144no—
247AGL(1,ℤ/8)—3144no—
248C2×SD16—1571no—
249C2×SD16—1571no—
250C2×SD16C2×D15711yescharacteristic, complemented
251C2×D8—1571no—
252C2×D8—1571no—
253C2×D8—1571no—
254C2×D8—1571no—
255C2×D8—1571no—
256C2×D8C2×D15711yescharacteristic, complemented
257C2×M16—1571no—
258C2×M16—1571no—
259#127—3141no—
260#127—1571no—
261#127—1571no—
262#121—3141no—
263#121—3141no—
264#121—3141no—
265#121—1571no—
266#121—1571no—
267#121—1571no—
268#121C2×D15711yescharacteristic
269#120—3141no—
270#120—3141no—
271C4×Q8—1571no—
272C4×D4—3141no—
273C4×D4—1571no—
274C4×D4—1571no—
275#106—1571no—
276C4⋊C8—1571no—
277#103C2×D15711yescharacteristic, complemented
278#102—1571no—
279#102—1571no—
280#102—1571no—
281#102—1571no—
282#102C2×D15711yescharacteristic, complemented
283C22⋊C8—1571no—
284C22⋊C8C2×D15711yescharacteristic, complemented
285C4×C8—1571no—
286C24—3141no—
287C24—3141no—
288#41—22no—
289#41—41no—
290#41—3142no—
291#41—6281no—
292C2×Q8C22×D15711yescharacteristic
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—3142no—
296C2×D4—3141no—
297C2×D4—3141no—
298C2×D4—3142no—
299C2×D4—3141no—
300C2×D4—3141no—
301C2×D4—3141no—
302C2×D4—3141no—
303C2×D4—3141no—
304C2×D4—3141no—
305C2×D4—3141no—
306C2×D4—3141no—
307C2×D4—6281no—
308C2×D4—6281no—
309C2×D4—6284no—
310C2×D4—1571no—
311C2×D4—1571no—
312C2×D4—1571no—
313C2×D4—1571no—
314C2×D4C22×D15711yescharacteristic
315C2×D4C22×D15711yescharacteristic
316C22×C4—21no—
317C22×C4—3141no—
318C22×C4—3141no—
319C22×C4—1571no—
320C22×C4C22×D15711yescharacteristic
321SD16—22no—
322SD16—3142no—
323SD16—1574no—
324D8—41no—
325D8—3142no—
326D8—3142no—
327D8—3142no—
328D8—3142no—
329D8—6281no—
330M16—3142no—
331M16—3142no—
332C2×C8—1571no—
333C2×C8—1571no—
334C2×C8C22×D15711yescharacteristic
335C2×C8C22×D15711yescharacteristic
336C4⋊C4—3141no—
337C4⋊C4—1571no—
338C4⋊C4—1571no—
339C4⋊C4C22×D15711yescharacteristic
340C22⋊C4—21no—
341C22⋊C4—3141no—
342C22⋊C4—3141no—
343C22⋊C4—3141no—
344C22⋊C4—3141no—
345C22⋊C4—1571no—
346C22⋊C4—1571no—
347C42—3141no—
348C42—1571no—
349C23—21no—
350C23—21no—
351C23—3141no—
352C23—3141no—
353C23—3141no—
354C23—6282no—
355C23—6282no—
356C23—6282no—
357C23—6282no—
358C23—1571no—
359C23—12561no—
360C23—12561no—
361C23D4×D15711yescharacteristic
362Q8—21no—
363Q8D4×D15712yescomplemented
364D4—21no—
365D4—21no—
366D4—21no—
367D4—21no—
368D4—42no—
369D4—42no—
370D4—3142no—
371D4—3141no—
372D4—3141no—
373D4—3141no—
374D4—3141no—
375D4—3141no—
376D4—6282no—
377D4—6282no—
378D4—6282no—
379D4—6282no—
380D4—6282no—
381D4—6282no—
382D4—1572no—
383D4—12561no—
384D4—12561no—
385D4—12561no—
386D4—12561no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—42no—
392C2×C4—3141no—
393C2×C4—3141no—
394C2×C4—3141no—
395C2×C4—6282no—
396C2×C4—6282no—
397C2×C4—1571no—
398C2×C4—1571no—
399C2×C4—1571no—
400C2×C4D4×D15711yescharacteristic
401C2×C4C23×D15711yescharacteristic
402C8—21no—
403C8—21no—
404C8—3141no—
405C8—1572no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—3141no—
416C22—3142no—
417C22—3141no—
418C22—3141no—
419C22—6281no—
420C22—6281no—
421C22—6281no—
422C22—6281no—
423C22—6281no—
424C22—6281no—
425C22—6281no—
426C22—6281no—
427C22—12564no—
428C22—12564no—
429C22C2×D4×D15711yescharacteristic
430C4—21no—
431C4—22no—
432C4—41no—
433C4—3141no—
434C4—3141no—
435C4—6281no—
436C4—1572no—
437C4C2×D4×D15711yescharacteristic
438C4#1909781611yescharacteristic
439C2—41no—
440C2—81no—
441C2—81no—
442C2—6281no—
443C2—6282no—
444C2—12561no—
445C2—12561no—
446C2#2375790511yescharacteristic
447C2AGL(1,ℤ/8)×D15711yescharacteristic
448C2#2375807511yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#449×C157C2
socleC22×C157C22×D4
hypercenter#449D157
hypocenterC157#3274
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. C2×#19097798C2semidirect
2. C2×#19097796C2semidirect
3. #23757942C2semidirect
4. #23757907C2semidirect
5. #23757648C2semidirect
6. #23757525C2semidirect
7. #23757524C2semidirect
8. #23757471C2semidirect
9. #23757467C2semidirect
10. #23757421C2semidirect
11. #23757409C2semidirect
12. #23757405C2semidirect
13. #23757358C2semidirect
14. #23757348C2semidirect
15. #449×C157C2semidirect
16. C2×#14578955C22semidirect
23. C2×#14578941C22semidirect
25. C2×D1256C22semidirect
27. C2×#14578939C22semidirect
28. C2×#14578939C22semidirect
29. C2×#14578938C22semidirect
30. C2×#14578937C22semidirect
33. #19097767C22semidirect
36. #19097745C22semidirect
39. #19097742C22semidirect
42. Q8×Dic157C22semidirect
46. #19097702C22semidirect
47. #19097700C22semidirect
49. #19097693C22semidirect
50. #19097692C22semidirect
51. #19097691C22semidirect
52. #19097683C22semidirect
53. C8×Dic157C22semidirect
54. C2×#41×C157C22semidirect
55. C2×SD16×C157C22semidirect
56. C2×D8×C157C22semidirect
58. #103×C157C22semidirect
59. #102×C157C22semidirect
60. C22⋊C8×C157C22semidirect
88. #14578941D4semidirect
160. #11929434C2×D4semidirect
165. Q8×C157C2×D4semidirect
213. Dic157AGL(1,ℤ/8)semidirect
224. C157#3274semidirect
236. #449D157semidirect
242. C2×#41C2×D157semidirect
250. C2×SD16C2×D157semidirect
256. C2×D8C2×D157semidirect
277. #103C2×D157semidirect
282. #102C2×D157semidirect
284. C22⋊C8C2×D157semidirect
363. Q8D4×D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190977982
C2×#190977962
#237579422
#237579072
#237576482
#237575252
#237575242
#237574712
#237574672
#237574212
#237574092
#237574052
#237573582
#237573482
#449×C1572
#3274157