Finite Group Databasebeta

← #34109399

Subgroups — #34109399

Order 20096.

784 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19097816C211yesmaximal, characteristic, complemented
2C2×#19097795C211yesmaximal, characteristic, complemented
3C2×D8×D157C211yesmaximal, characteristic, complemented
4#23757839C211yesmaximal, characteristic, complemented
5#23757420C211yesmaximal, characteristic, complemented
6#23757419C211yesmaximal, characteristic, complemented
7#23757416C211yesmaximal, characteristic, complemented
8#23757415C211yesmaximal, characteristic, complemented
9#23757408C211yesmaximal, characteristic, complemented
10#23757407C211yesmaximal, characteristic, complemented
11#23757405C211yesmaximal, characteristic, complemented
12#23757276C211yesmaximal, characteristic, complemented
13#23757266C211yesmaximal, characteristic, complemented
14#23757238C211yesmaximal, characteristic, complemented
15#447×C157C211yesmaximal, characteristic, complemented
16#19097816—24no—
17#19097816—24no—
18C2×#14578954C2211yescharacteristic
19C2×#14578954C2211yescharacteristic, complemented
20C2×#14578953—21no—
21C2×#14578953C2211yescharacteristic
22C2×#14578953C2211yescharacteristic, complemented
23C22×C22⋊D157—21no—
24C22×C22⋊D157—21no—
25C2×D4×D157—21no—
26C2×D4×D157C2211yescharacteristic
27C2×D4×D157C2211yescharacteristic, complemented
28C2×D4×D157C2211yescharacteristic, complemented
29#19097795—24no—
30C2×#14578943C2211yescharacteristic, complemented
31C2×#14578942C2211yescharacteristic, complemented
32C2×D1256C2211yescharacteristic, complemented
33C2×#14578939C2211yescharacteristic, complemented
34C2×#14578939C2211yescharacteristic, complemented
35C2×#14578939C2211yescharacteristic, complemented
36D8×D157—24no—
37C2×#14578937C2211yescharacteristic, complemented
38C2×C8×D157C2211yescharacteristic, complemented
39#19097767—21no—
40#19097757—21no—
41#19097746C2211yescharacteristic, complemented
42#19097745C2211yescharacteristic, complemented
43#19097742C2211yescharacteristic, complemented
44#19097741—21no—
45#19097739—21no—
46#19097729C2211yescharacteristic, complemented
47#19097696C2211yescharacteristic, complemented
48#19097694C2211yescharacteristic, complemented
49#19097693C2211yescharacteristic, complemented
50#19097692C2211yescharacteristic, complemented
51#19097692C2211yescharacteristic, complemented
52#19097691C2211yescharacteristic, complemented
53C22⋊#11929434C2211yescharacteristic, complemented
54#19097683C2211yescharacteristic, complemented
55#19097683C2211yescharacteristic, complemented
56C22×D4×C157C2211yescharacteristic, complemented
57C2×D8×C157C2211yescharacteristic, complemented
58C2×D8×C157C2211yescharacteristic, complemented
59#121×C157C2211yescharacteristic, complemented
60#102×C157C2211yescharacteristic, complemented
61#102×C157C2211yescharacteristic, complemented
62C22⋊C8×C157C2211yescharacteristic, complemented
63C23×D157—21no—
64C23×D157—21no—
65C23×D157—21no—
66#14578954—22no—
67#14578954—22no—
68#14578954—22no—
69#14578954—22no—
70#14578953—28no—
71#14578953—22no—
72#14578953—22no—
73#14578953—22no—
74#14578953D414yescomplemented
75C2×Dic314D411yescharacteristic
76C2×Dic314C2311yescharacteristic
77C2×D628—21no—
78C2×D628D411yescharacteristic
79C2×D628C2311yescharacteristic
80C2×D628C2311yescharacteristic
81C2×C22⋊D157—28no—
82C2×C22⋊D157—28no—
83C2×C22⋊D157—21no—
84C2×C22⋊D157—21no—
85C2×C22⋊D157—21no—
86C2×C22⋊D157—21no—
87C2×C22⋊D157—21no—
88C2×C22⋊D157—21no—
89C2×C22⋊D157—21no—
90C2×C22⋊D157—21no—
91C2×C22⋊D157D411yescharacteristic
92C2×C22⋊D157D411yescharacteristic
93D4×D157—28no—
94D4×D157—22no—
95D4×D157—22no—
96D4×D157—22no—
97D4×D157—22no—
98D4×D157—22no—
99D4×D157D414yescomplemented
100C22×Dic157—21no—
101C22×Dic157—21no—
102C2×C4×D157—21no—
103C2×C4×D157C2311yescharacteristic
104C2×C4×D157C2311yescharacteristic
105#14578943—22no—
106#14578942—22no—
107D1256—22no—
108#14578939—22no—
109#14578939—22no—
110#14578939—22no—
111#14578937—22no—
112C2×#11929434C2311yescharacteristic
113C2×#11929434C2311yescharacteristic
114C8×D157—22no—
115C4⋊Dic157C2311yescharacteristic
116C22⋊Dic157—21no—
117#14578931—21no—
118#14578931—21no—
119#14578931—21no—
120C4×Dic157—21no—
121C24×C157—21no—
122C2×D4×C157—22no—
123C2×D4×C157—22no—
124C2×D4×C157—21no—
125C2×D4×C157—21no—
126C2×D4×C157C2311yescharacteristic
127C2×D4×C157C2311yescharacteristic
128C2×D4×C157C2311yescharacteristic
129C22×C4×C157C2311yescharacteristic
130D8×C157—22no—
131D8×C157—22no—
132C2×C8×C157C2311yescharacteristic
133C2×C8×C157C2311yescharacteristic
134C4⋊C4×C157C2311yescharacteristic
135C22⋊C4×C157—21no—
136C22×D157—21no—
137C22×D157—24no—
138C22×D157—24no—
139C22×D157—21no—
140C22×D157—42no—
141C22×D157—81no—
142C22×D157—81no—
143C22×D157—81no—
144C22×D157—81no—
145C22×D157—81no—
146C22×D157—81no—
147C22×D157C2×D411yescharacteristic
148C22×D157C2×D411yescharacteristic
149Dic314—21no—
150Dic314—22no—
151Dic314C2×D412yes—
152D628—22no—
153D628—21no—
154D628—21no—
155D628—21no—
156D628—81no—
157D628C2×D412yes—
158C22⋊D157—24no—
159C22⋊D157—24no—
160C22⋊D157—24no—
161C22⋊D157—24no—
162C22⋊D157—22no—
163C22⋊D157—24no—
164C22⋊D157—22no—
165C22⋊D157—42no—
166C22⋊D157—81no—
167C22⋊D157—81no—
168C2×Dic157—24no—
169C2×Dic157—21no—
170C2×Dic157—24no—
171C2×Dic157—21no—
172C2×Dic157C2×D411yescharacteristic
173C2×Dic157C2×D411yescharacteristic
174C4×D157—24no—
175C4×D157—21no—
176C4×D157—21no—
177C4×D157C2×D412yes—
178C4×D157C2×D412yes—
179#11929434—21no—
180#11929434—21no—
181C23×C157—24no—
182C23×C157—21no—
183C23×C157—21no—
184C23×C157—21no—
185C23×C157—81no—
186C23×C157—81no—
187C23×C157C2×D411yescharacteristic
188D4×C157—24no—
189D4×C157—21no—
190D4×C157—21no—
191D4×C157—21no—
192D4×C157—21no—
193D4×C157—81no—
194D4×C157C2×D412yes—
195D4×C157C2×D412yes—
196C2×C4×C157—21no—
197C2×C4×C157—21no—
198C2×C4×C157—21no—
199C2×C4×C157C2×D411yescharacteristic
200C2×C4×C157C2411yescharacteristic
201C8×C157—21no—
202C8×C157—21no—
203C22×C157—21no—
204C22×C157—22no—
205C22×C157—22no—
206C22×C157—21no—
207C22×C157—21no—
208C22×C157—41no—
209C22×C157—41no—
210C22×C157—41no—
211C22×C157—41no—
212C22×C157—41no—
213C22×C157—41no—
214C22×C157—41no—
215C22×C157—84no—
216C22×C157C22×D411yescharacteristic
217C2×D157—21no—
218C2×D157—21no—
219C2×D157—21no—
220C2×D157—22no—
221C2×D157—21no—
222C2×D157—21no—
223C2×D157—41no—
224C2×D157—41no—
225C2×D157—41no—
226C2×D157—41no—
227C2×D157—41no—
228C2×D157—84no—
229C2×D157—84no—
230C2×D157—84no—
231C2×D157#12012yes—
232C4×C157—21no—
233C4×C157—41no—
234C4×C157C22×D411yescharacteristic
235C4×C157C22×D411yescharacteristic
236Dic157—22no—
237Dic157—21no—
238Dic157—41no—
239Dic157#12012yes—
240C2×C157—42no—
241C2×C157—42no—
242C2×C157—41no—
243C2×C157—81no—
244C2×C157C2×#12011yescharacteristic
245C2×C157#57611yescharacteristic
246C2×C157#57611yescharacteristic
247D157—22no—
248D157—41no—
249D157—42no—
250D157—81no—
251C157#300811yescharacteristic, complemented, Sylow
252#3008—1571nomaximal, Sylow
253C2×H(5,𝔽2)—1571no—
254C2×AGL(1,ℤ/8)—1571no—
255C22×D8—1571no—
256#535—1571no—
257#449—1571no—
258#449—1571no—
259#449—1571no—
260#449—1571no—
261#447—1571no—
262#447—1571no—
263#447—1571no—
264#447D15711yescharacteristic, complemented
265#418—1571no—
266#416—1571no—
267#408—1571no—
268H(5,𝔽2)—3144no—
269H(5,𝔽2)—3144no—
270C2×#41—3141no—
271C2×#41—1571no—
272C2×#41—1571no—
273C2×#41—1571no—
274C2×#41—1571no—
275C22×D4—3141no—
276C22×D4—3141no—
277C22×D4—3141no—
278C22×D4—1571no—
279C22×D4—1571no—
280C22×D4—1571no—
281C22×D4C2×D15711yescharacteristic, complemented
282AGL(1,ℤ/8)—3144no—
283C2×SD16—1571no—
284C2×SD16—1571no—
285C2×D8—3144no—
286C2×D8—1571no—
287C2×D8—1571no—
288C2×D8—1571no—
289C2×D8—1571no—
290C2×D8C2×D15711yescharacteristic, complemented
291C2×D8C2×D15711yescharacteristic, complemented
292C2×M16—1571no—
293C22×C8—1571no—
294#127—3141no—
295#124—3141no—
296#121—1571no—
297#121—1571no—
298#121—1571no—
299#121C2×D15711yescharacteristic, complemented
300#120—3141no—
301#120—3141no—
302#117—1571no—
303#103—1571no—
304#103—1571no—
305#102—1571no—
306#102—1571no—
307#102—1571no—
308#102—1571no—
309#102C2×D15711yescharacteristic, complemented
310#102C2×D15711yescharacteristic, complemented
311C22⋊C8—1571no—
312C22⋊C8—1571no—
313C22⋊C8—1571no—
314C22⋊C8C2×D15711yescharacteristic, complemented
315C24—21no—
316C24—3141no—
317C24—3141no—
318C24—3141no—
319#41—3142no—
320#41—3142no—
321#41—3142no—
322#41—3142no—
323#41—3148no—
324#41—3142no—
325#41—3142no—
326#41—3142no—
327#41—1574no—
328C2×Q8—1571no—
329C2×Q8—1571no—
330C2×D4—22no—
331C2×D4—22no—
332C2×D4—21no—
333C2×D4—21no—
334C2×D4—3141no—
335C2×D4—3148no—
336C2×D4—3142no—
337C2×D4—3148no—
338C2×D4—3148no—
339C2×D4—3141no—
340C2×D4—3141no—
341C2×D4—3141no—
342C2×D4—3142no—
343C2×D4—3141no—
344C2×D4—3141no—
345C2×D4—3142no—
346C2×D4—3142no—
347C2×D4—3142no—
348C2×D4—3141no—
349C2×D4—3141no—
350C2×D4—3141no—
351C2×D4—1571no—
352C2×D4—1571no—
353C2×D4—1574no—
354C2×D4—1571no—
355C2×D4—1571no—
356C2×D4—1571no—
357C2×D4C22×D15711yescharacteristic
358C2×D4C22×D15711yescharacteristic
359C2×D4C22×D15711yescharacteristic
360C22×C4—3141no—
361C22×C4—3141no—
362C22×C4—3141no—
363C22×C4—1571no—
364C22×C4—1571no—
365C22×C4C22×D15711yescharacteristic
366SD16—3142no—
367SD16—3142no—
368D8—22no—
369D8—22no—
370D8—3142no—
371D8—3142no—
372D8—3142no—
373D8—3142no—
374M16—3142no—
375C2×C8—3142no—
376C2×C8—1571no—
377C2×C8—1571no—
378C2×C8C22×D15711yescharacteristic
379C2×C8C22×D15711yescharacteristic
380C4⋊C4—1571no—
381C4⋊C4C22×D15711yescharacteristic
382C22⋊C4—21no—
383C22⋊C4—3141no—
384C22⋊C4—3141no—
385C22⋊C4—3141no—
386C22⋊C4—3141no—
387C42—3141no—
388C23—24no—
389C23—21no—
390C23—21no—
391C23—21no—
392C23—81no—
393C23—81no—
394C23—3144no—
395C23—3144no—
396C23—3141no—
397C23—3141no—
398C23—6282no—
399C23—1571no—
400C23—1571no—
401C23—12561no—
402C23—12561no—
403C23—12561no—
404C23—12561no—
405C23—12561no—
406C23—12561no—
407C23D4×D15711yescharacteristic
408Q8—3142no—
409Q8—3141no—
410Q8—1572no—
411D4—24no—
412D4—21no—
413D4—21no—
414D4—21no—
415D4—21no—
416D4—81no—
417D4—3142no—
418D4—3144no—
419D4—3144no—
420D4—3144no—
421D4—3144no—
422D4—3142no—
423D4—3142no—
424D4—3144no—
425D4—3141no—
426D4—3141no—
427D4—3141no—
428D4—6282no—
429D4—1572no—
430D4—12561no—
431D4—12561no—
432D4—12561no—
433D4D4×D15712yes—
434D4D4×D15712yes—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—3144no—
439C2×C4—3144no—
440C2×C4—3141no—
441C2×C4—3144no—
442C2×C4—3141no—
443C2×C4—3141no—
444C2×C4—3141no—
445C2×C4—1571no—
446C2×C4—1571no—
447C2×C4—1572no—
448C2×C4—1572no—
449C2×C4D4×D15711yescharacteristic
450C2×C4C23×D15711yescharacteristic
451C8—21no—
452C8—21no—
453C8—3141no—
454C8—3141no—
455C22—21no—
456C22—22no—
457C22—22no—
458C22—21no—
459C22—21no—
460C22—41no—
461C22—41no—
462C22—41no—
463C22—41no—
464C22—41no—
465C22—41no—
466C22—41no—
467C22—84no—
468C22—3141no—
469C22—3141no—
470C22—3141no—
471C22—3142no—
472C22—3141no—
473C22—3141no—
474C22—6281no—
475C22—6281no—
476C22—6281no—
477C22—6281no—
478C22—6281no—
479C22—1572no—
480C22—12564no—
481C22—12564no—
482C22—12564no—
483C22C2×D4×D15711yescharacteristic
484C4—21no—
485C4—41no—
486C4—3142no—
487C4—3141no—
488C4—6281no—
489C4—1572no—
490C4C2×D4×D15711yescharacteristic
491C4C2×D4×D15711yescharacteristic
492C2—41no—
493C2—42no—
494C2—42no—
495C2—81no—
496C2—3142no—
497C2—6281no—
498C2—6282no—
499C2—12561no—
500C2#120×D15711yescharacteristic
501C2#2375807411yescharacteristic
502C2#2375807711yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#447×C157C2
socleC22×C157C22×D4
hypercenter#447D157
hypocenterC157#3008
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. C2×#19097816C2semidirect
2. C2×#19097795C2semidirect
3. C2×D8×D157C2semidirect
4. #23757839C2semidirect
5. #23757420C2semidirect
6. #23757419C2semidirect
7. #23757416C2semidirect
8. #23757415C2semidirect
9. #23757408C2semidirect
10. #23757407C2semidirect
11. #23757405C2semidirect
12. #23757276C2semidirect
13. #23757266C2semidirect
14. #23757238C2semidirect
15. #447×C157C2semidirect
19. C2×#14578954C22semidirect
22. C2×#14578953C22semidirect
27. C2×D4×D157C22semidirect
28. C2×D4×D157C22semidirect
30. C2×#14578943C22semidirect
31. C2×#14578942C22semidirect
32. C2×D1256C22semidirect
33. C2×#14578939C22semidirect
34. C2×#14578939C22semidirect
35. C2×#14578939C22semidirect
37. C2×#14578937C22semidirect
38. C2×C8×D157C22semidirect
41. #19097746C22semidirect
42. #19097745C22semidirect
43. #19097742C22semidirect
46. #19097729C22semidirect
47. #19097696C22semidirect
48. #19097694C22semidirect
49. #19097693C22semidirect
50. #19097692C22semidirect
51. #19097692C22semidirect
52. #19097691C22semidirect
53. C22⋊#11929434C22semidirect
54. #19097683C22semidirect
55. #19097683C22semidirect
56. C22×D4×C157C22semidirect
57. C2×D8×C157C22semidirect
58. C2×D8×C157C22semidirect
59. #121×C157C22semidirect
60. #102×C157C22semidirect
61. #102×C157C22semidirect
62. C22⋊C8×C157C22semidirect
74. #14578953D4semidirect
99. D4×D157D4semidirect
251. C157#3008semidirect
264. #447D157semidirect
281. C22×D4C2×D157semidirect
290. C2×D8C2×D157semidirect
291. C2×D8C2×D157semidirect
299. #121C2×D157semidirect
309. #102C2×D157semidirect
310. #102C2×D157semidirect
314. C22⋊C8C2×D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190978162
C2×#190977952
C2×D8×D1572
#237578392
#237574202
#237574192
#237574162
#237574152
#237574082
#237574072
#237574052
#237572762
#237572662
#237572382
#447×C1572
#3008157