Finite Group Databasebeta

← #34111915

Subgroups — #34111915

Order 20096.

454 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23758088C211yesmaximal, characteristic, complemented
2#23758083C211yesmaximal, characteristic, complemented
3#23758081C211yesmaximal, characteristic, complemented
4#23758077C211yesmaximal, characteristic, complemented
5#23757655C211yesmaximal, characteristic, complemented
6#23757547C211yesmaximal, characteristic, complemented
7#23757541C211yesmaximal, characteristic, complemented
8#23757539C211yesmaximal, characteristic, complemented
9#23757534C211yesmaximal, characteristic, complemented
10#23757452C211yesmaximal, characteristic, complemented
11#23757450C211yesmaximal, characteristic, complemented
12#23757447C211yesmaximal, characteristic, complemented
13#23757387C211yesmaximal, characteristic, complemented
14#23757387C211yesmaximal, characteristic, complemented
15#454×C157C211yesmaximal, characteristic, complemented
16#19097818C2211yescharacteristic, complemented
17#19097817C2211yescharacteristic
18#19097816C2211yescharacteristic
19#19097804C2211yescharacteristic
20#19097802—21no—
21#19097802—21no—
22#19097802C2211yescharacteristic
23#19097802C2211yescharacteristic, complemented
24#19097801—21no—
25#19097801C2211yescharacteristic, complemented
26#19097800—21no—
27#19097799C2211yescharacteristic
28#19097798—21no—
29#19097796—21no—
30#19097796C2211yescharacteristic, complemented
31#19097795—21no—
32#19097795—21no—
33#19097792—21no—
34#19097791—21no—
35#19097791—21no—
36#19097791C2211yescharacteristic
37#19097791C2211yescharacteristic
38#19097790—21no—
39C2×#14578944—21no—
40C2×#14578944C2211yescharacteristic, complemented
41C2×#14578942C2211yescharacteristic, complemented
42C2×#14578941—21no—
43C2×#14578941C2211yescharacteristic, complemented
44SD16×D157—21no—
45C2×#14578939C2211yescharacteristic
46D8×D157—21no—
47#19097779C2211yescharacteristic
48#19097779C2211yescharacteristic, complemented
49#19097778C2211yescharacteristic, complemented
50#19097778C2211yescharacteristic, complemented
51#19097759C2211yescharacteristic, complemented
52#19097706C2211yescharacteristic, complemented
53#19097706C2211yescharacteristic, complemented
54#19097699C2211yescharacteristic
55#19097699C2211yescharacteristic, complemented
56#19097689C2211yescharacteristic, complemented
57#19097688C2211yescharacteristic, complemented
58#19097687C2211yescharacteristic, complemented
59C4×#11929434C2211yescharacteristic, complemented
602−1+4×C157C2211yescharacteristic, complemented
61AGL(1,ℤ/8)×C157C2211yescharacteristic
62AGL(1,ℤ/8)×C157C2211yescharacteristic, complemented
63#124×C157C2211yescharacteristic, complemented
64#104×C157C2211yescharacteristic, complemented
65#104×C157C2211yescharacteristic, complemented
66#101×C157C2211yescharacteristic, complemented
67#14578955—21no—
68#14578954—21no—
69#14578954—21no—
70#14578954C2311yescharacteristic
71#14578954C2311yescharacteristic
72#14578953—21no—
73#14578953—21no—
74#14578953—21no—
75#14578953—41no—
76C2×Dic314—21no—
77C2×Dic314C2311yescharacteristic
78Q8×D157—21no—
79C2×D628—21no—
80C2×D628C2311yescharacteristic
81C2×C22⋊D157—41no—
82D4×D157—21no—
83D4×D157—21no—
84D4×D157—21no—
85D4×D157—41no—
86Dic628—21no—
87#14578944—21no—
88#14578944—21no—
89#14578944—21no—
90#14578944—21no—
91#14578944—21no—
92#14578944D411yescharacteristic
93#14578944D411yescharacteristic
94#14578943—21no—
95#14578943—21no—
96#14578942—21no—
97#14578942—21no—
98#14578942—21no—
99#14578942D411yescharacteristic
100#14578942D411yescharacteristic
101#14578941—21no—
102#14578941—21no—
103#14578941—21no—
104#14578941—21no—
105#14578941—21no—
106#14578941D411yescharacteristic
107#14578941D411yescharacteristic
108D1256—21no—
109#14578939—21no—
110#14578939—21no—
111#14578939—21no—
112#14578939D411yescharacteristic
113#14578939D411yescharacteristic
114#14578938—21no—
115#14578938—21no—
116#14578938C2311yescharacteristic
117#14578938C2311yescharacteristic
118#14578937—21no—
119#14578937—21no—
120C2×#11929434—21no—
121C2×#11929434—21no—
122C2×#11929434C2311yescharacteristic
123C2×#11929434C2311yescharacteristic
124C8×D157—21no—
125C8×D157—21no—
126#14578931—41no—
127#41×C157—21no—
128#41×C157—21no—
129#41×C157—41no—
130#41×C157C2311yescharacteristic
131#41×C157C2311yescharacteristic
132C2×Q8×C157—21no—
133C2×Q8×C157—21no—
134C2×Q8×C157C2311yescharacteristic
135C2×D4×C157C2311yescharacteristic
136SD16×C157—21no—
137SD16×C157—21no—
138D8×C157—21no—
139D8×C157—21no—
140M16×C157C2311yescharacteristic
141M16×C157C2311yescharacteristic
142C22⋊C4×C157—41no—
143C42×C157C2311yescharacteristic
144C22×D157—21no—
145C22×D157—41no—
146C22×D157—41no—
147Dic314—21no—
148Dic314—21no—
149Dic314—41no—
150Dic314C2×D411yescharacteristic
151Dic314C2×D411yescharacteristic
152D628—21no—
153D628—21no—
154D628—41no—
155D628C2×D411yescharacteristic
156D628C2×D411yescharacteristic
157C22⋊D157—21no—
158C22⋊D157—21no—
159C22⋊D157—41no—
160C22⋊D157—41no—
161C22⋊D157—41no—
162C22⋊D157—41no—
163C2×Dic157—21no—
164C2×Dic157—41no—
165C2×Dic157—41no—
166C4×D157—21no—
167C4×D157—21no—
168C4×D157—41no—
169C4×D157—41no—
170#11929434—21no—
171#11929434—21no—
172#11929434C2×D411yescharacteristic
173#11929434C2×D411yescharacteristic
174#11929434C2×D411yescharacteristic
175#11929434C2×D411yescharacteristic
176C23×C157—21no—
177Q8×C157—21no—
178Q8×C157—21no—
179Q8×C157—41no—
180Q8×C157C2×D411yescharacteristic
181Q8×C157C2×D411yescharacteristic
182D4×C157—21no—
183D4×C157—21no—
184D4×C157—21no—
185D4×C157—21no—
186D4×C157—41no—
187D4×C157C2×D411yescharacteristic
188D4×C157C2×D411yescharacteristic
189C2×C4×C157—21no—
190C2×C4×C157—21no—
191C2×C4×C157—21no—
192C2×C4×C157—21no—
193C2×C4×C157—41no—
194C2×C4×C157—41no—
195C2×C4×C157C2411yescharacteristic
196C8×C157—21no—
197C8×C157—21no—
198C22×C157—21no—
199C22×C157—21no—
200C22×C157—41no—
201C22×C157—81no—
202C22×C157H(5,𝔽2)11yescharacteristic
203C2×D157—21no—
204C2×D157—21no—
205C2×D157—41no—
206C2×D157—81no—
207C2×D157—161no—
208C2×D157—161no—
209C4×C157—21no—
210C4×C157—21no—
211C4×C157—41no—
212C4×C157—41no—
213C4×C157C22×D411yescharacteristic
214C4×C157C22×D411yescharacteristic
215Dic157—21no—
216Dic157—21no—
217Dic157—41no—
218C2×C157—21no—
219C2×C157—41no—
220C2×C157—41no—
221C2×C157—81no—
222C2×C157D4211yescharacteristic
223D157—41no—
224D157—41no—
225D157—81no—
226C157#329011yescharacteristic, complemented, Sylow
227#3290—1571nomaximal, Sylow
228#578—1571no—
229#577—1571no—
230#577—1571no—
231#576—1571no—
232#495—1571no—
233#473—1571no—
234#472—1571no—
235#471—1571no—
236#471—1571no—
237#455—1571no—
238#455—1571no—
239#454—1571no—
240#454D15711yescharacteristic, complemented
241#443—1571no—
242#443—1571no—
2432−1+4—1571no—
2442−1+4C2×D15711yescharacteristic, complemented
245H(5,𝔽2)—1571no—
246H(5,𝔽2)—1571no—
247#137—3141no—
248#137—3141no—
249#137—3141no—
250#137—3141no—
251#137—1571no—
252#137—1571no—
253#137—1571no—
254#137—1571no—
255AGL(1,ℤ/8)—3141no—
256AGL(1,ℤ/8)—3141no—
257AGL(1,ℤ/8)—3141no—
258AGL(1,ℤ/8)—3141no—
259AGL(1,ℤ/8)—1571no—
260AGL(1,ℤ/8)—1571no—
261AGL(1,ℤ/8)C2×D15711yescharacteristic
262AGL(1,ℤ/8)C2×D15711yescharacteristic, complemented
263#135—3141no—
264#135—3141no—
265#135—3141no—
266#135—3141no—
267#135—1571no—
268#135—1571no—
269C2×Q16—3141no—
270C2×Q16—1571no—
271C2×SD16—3141no—
272C2×SD16—3141no—
273C2×SD16—1571no—
274C2×SD16—1571no—
275C2×D8—3141no—
276C2×D8—1571no—
277#131—1571no—
278#131—1571no—
279#131—1571no—
280#131—1571no—
281#124—1571no—
282#124C2×D15711yescharacteristic, complemented
283#108—1571no—
284#108—1571no—
285#104—1571no—
286#104—1571no—
287#104C2×D15711yescharacteristic, complemented
288#104C2×D15711yescharacteristic, complemented
289#101—1571no—
290#101C2×D15711yescharacteristic, complemented
291#100—1571no—
292#100—1571no—
293C4×C8—1571no—
294#41—21no—
295#41—21no—
296#41—41no—
297#41—3141no—
298#41—3141no—
299#41—3141no—
300#41—3141no—
301#41—3141no—
302#41—3141no—
303#41—6281no—
304#41—1571no—
305#41—1571no—
306#41C22×D15711yescharacteristic
307#41C22×D15711yescharacteristic
308C2×Q8—21no—
309C2×Q8—21no—
310C2×Q8—3141no—
311C2×Q8—3141no—
312C2×Q8—1571no—
313C2×Q8C22×D15711yescharacteristic
314C2×D4—3141no—
315C2×D4—3141no—
316C2×D4—3141no—
317C2×D4—3141no—
318C2×D4—6281no—
319C2×D4—6281no—
320C2×D4—1571no—
321C2×D4C22×D15711yescharacteristic
322Q16—3141no—
323Q16—3141no—
324Q16—3141no—
325Q16—3141no—
326Q16—3141no—
327Q16—3141no—
328Q16—1571no—
329Q16—1571no—
330SD16—21no—
331SD16—21no—
332SD16—3141no—
333SD16—3141no—
334SD16—3141no—
335SD16—3141no—
336SD16—3141no—
337SD16—3141no—
338SD16—3141no—
339SD16—3141no—
340SD16—3141no—
341SD16—3141no—
342SD16—1571no—
343SD16—1571no—
344SD16—1571no—
345SD16—1571no—
346D8—21no—
347D8—21no—
348D8—3141no—
349D8—3141no—
350D8—3141no—
351D8—3141no—
352D8—1571no—
353D8—1571no—
354M16—3141no—
355M16—3141no—
356M16—3141no—
357M16—3141no—
358M16—1571no—
359M16—1571no—
360M16C22×D15711yescharacteristic
361M16C22×D15711yescharacteristic
362C2×C8—3141no—
363C2×C8—3141no—
364C2×C8—3141no—
365C2×C8—3141no—
366C2×C8—1571no—
367C2×C8—1571no—
368C22⋊C4—41no—
369C22⋊C4—6281no—
370C42C22×D15711yescharacteristic
371C23—21no—
372C23—3141no—
373C23—6281no—
374C23—6281no—
375Q8—21no—
376Q8—21no—
377Q8—41no—
378Q8—3141no—
379Q8—3141no—
380Q8—6281no—
381Q8—1571no—
382Q8—1571no—
383Q8D4×D15711yescharacteristic
384Q8D4×D15711yescharacteristic
385D4—21no—
386D4—21no—
387D4—21no—
388D4—21no—
389D4—41no—
390D4—3141no—
391D4—3141no—
392D4—3141no—
393D4—3141no—
394D4—6281no—
395D4—6281no—
396D4—6281no—
397D4—6281no—
398D4—6281no—
399D4—1571no—
400D4—1571no—
401D4D4×D15711yescharacteristic
402D4D4×D15711yescharacteristic
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—3141no—
410C2×C4—3141no—
411C2×C4—3141no—
412C2×C4—6281no—
413C2×C4—6281no—
414C2×C4—6281no—
415C2×C4—6281no—
416C2×C4C23×D15711yescharacteristic
417C8—21no—
418C8—21no—
419C8—3141no—
420C8—3141no—
421C8—1571no—
422C8—1571no—
423C8—1571no—
424C8—1571no—
425C22—21no—
426C22—21no—
427C22—41no—
428C22—81no—
429C22—3141no—
430C22—3141no—
431C22—6281no—
432C22—25121no—
433C22—25121no—
434C22—12561no—
435C22#1909781611yescharacteristic
436C4—21no—
437C4—21no—
438C4—41no—
439C4—41no—
440C4—3141no—
441C4—3141no—
442C4—6281no—
443C4C2×D4×D15711yescharacteristic
444C4C2×D4×D15711yescharacteristic
445C2—21no—
446C2—41no—
447C2—41no—
448C2—81no—
449C2—6281no—
450C2—6281no—
451C2—12561no—
452C2#2375790511yescharacteristic
subgroupG/H
centerC2#23757905
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#454×C157C2
socleC2×C157D42
hypercenter#454D157
hypocenterC157#3290
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C2×C4 ◁ #454

Complements

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

NH ≅ G/Ntype
1. #23758088C2semidirect
2. #23758083C2semidirect
3. #23758081C2semidirect
4. #23758077C2semidirect
5. #23757655C2semidirect
6. #23757547C2semidirect
7. #23757541C2semidirect
8. #23757539C2semidirect
9. #23757534C2semidirect
10. #23757452C2semidirect
11. #23757450C2semidirect
12. #23757447C2semidirect
13. #23757387C2semidirect
14. #23757387C2semidirect
15. #454×C157C2semidirect
16. #19097818C22semidirect
23. #19097802C22semidirect
25. #19097801C22semidirect
30. #19097796C22semidirect
40. C2×#14578944C22semidirect
41. C2×#14578942C22semidirect
43. C2×#14578941C22semidirect
48. #19097779C22semidirect
49. #19097778C22semidirect
50. #19097778C22semidirect
51. #19097759C22semidirect
52. #19097706C22semidirect
53. #19097706C22semidirect
55. #19097699C22semidirect
56. #19097689C22semidirect
57. #19097688C22semidirect
58. #19097687C22semidirect
59. C4×#11929434C22semidirect
60. 2−1+4×C157C22semidirect
62. AGL(1,ℤ/8)×C157C22semidirect
63. #124×C157C22semidirect
64. #104×C157C22semidirect
65. #104×C157C22semidirect
66. #101×C157C22semidirect
226. C157#3290semidirect
240. #454D157semidirect
244. 2−1+4C2×D157semidirect
262. AGL(1,ℤ/8)C2×D157semidirect
282. #124C2×D157semidirect
287. #104C2×D157semidirect
288. #104C2×D157semidirect
290. #101C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237580882
#237580832
#237580812
#237580772
#237576552
#237575472
#237575412
#237575392
#237575342
#237574522
#237574502
#237574472
#237573872
#454×C1572
#3290157