Finite Group Databasebeta

← #34109441

Subgroups — #34109441

Order 20096.

764 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19097816C211yesmaximal, characteristic, complemented
2C2×#19097798C211yesmaximal, characteristic, complemented
3C2×#19097792C211yesmaximal, characteristic, complemented
4#23757853C211yesmaximal, characteristic, complemented
5#23757433C211yesmaximal, characteristic, complemented
6#23757432C211yesmaximal, characteristic, complemented
7#23757422C211yesmaximal, characteristic, complemented
8#23757420C211yesmaximal, characteristic, complemented
9#23757419C211yesmaximal, characteristic, complemented
10#23757408C211yesmaximal, characteristic, complemented
11#23757407C211yesmaximal, characteristic, complemented
12#23757271C211yesmaximal, characteristic, complemented
13#102×D157C211yesmaximal, characteristic, complemented
14#23757238C211yesmaximal, characteristic, complemented
15#450×C157C211yesmaximal, characteristic, complemented
16#19097816—24no—
17#19097816—24no—
18C2×#14578955C2211yescharacteristic, complemented
19C2×#14578954C2211yescharacteristic
20C2×#14578954C2211yescharacteristic
21C2×#14578953—21no—
22C2×#14578953C2211yescharacteristic
23C2×#14578953C2211yescharacteristic
24C22×C22⋊D157—21no—
25C22×C22⋊D157—21no—
26C2×D4×D157—21no—
27C2×D4×D157C2211yescharacteristic
28C2×D4×D157C2211yescharacteristic
29#19097798—24no—
30#19097792—24no—
31C2×#14578944C2211yescharacteristic, complemented
32C2×#14578943C2211yescharacteristic, complemented
33C2×#14578941C2211yescharacteristic, complemented
34C2×D1256C2211yescharacteristic, complemented
35C2×#14578939C2211yescharacteristic, complemented
36C2×#14578939C2211yescharacteristic, complemented
37C2×#14578937C2211yescharacteristic, complemented
38C2×C8×D157C2211yescharacteristic, complemented
39#19097749—21no—
40#19097747C2211yescharacteristic, complemented
41#19097746—21no—
42#19097746C2211yescharacteristic, complemented
43#19097745C2211yescharacteristic, complemented
44#19097744—21no—
45#19097744—21no—
46C4⋊C4×D157C2211yescharacteristic, 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
57C2×SD16×C157C2211yescharacteristic, complemented
58C2×SD16×C157C2211yescharacteristic, complemented
59C22⋊Q8×C157C2211yescharacteristic, complemented
60#102×C157C2211yescharacteristic, complemented
61#102×C157C2211yescharacteristic, complemented
62C22⋊C8×C157C2211yescharacteristic, complemented
63C23×D157—21no—
64C23×D157—21no—
65#14578955—22no—
66#14578955—22no—
67#14578954—22no—
68#14578954—22no—
69#14578954—22no—
70#14578954—22no—
71#14578953—28no—
72#14578953—22no—
73#14578953—22no—
74#14578953—22no—
75#14578953D414yescomplemented
76C2×Dic314D411yescharacteristic
77C2×Dic314C2311yescharacteristic
78C2×D628—21no—
79C2×D628—21no—
80C2×D628D411yescharacteristic
81C2×D628C2311yescharacteristic
82C2×D628C2311yescharacteristic
83C2×C22⋊D157—28no—
84C2×C22⋊D157—28no—
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×D157D414yescomplemented
98C22×Dic157—21no—
99C22×Dic157—21no—
100C2×C4×D157—21no—
101C2×C4×D157—21no—
102C2×C4×D157—21no—
103C2×C4×D157C2311yescharacteristic
104C2×C4×D157C2311yescharacteristic
105#14578944—22no—
106#14578943—22no—
107#14578941—22no—
108D1256—22no—
109#14578939—22no—
110#14578939—22no—
111#14578937—22no—
112C2×#11929434C2311yescharacteristic
113C2×#11929434C2311yescharacteristic
114C8×D157—22no—
115C4⋊Dic157C2311yescharacteristic
116#14578933—21no—
117#14578933—21no—
118#14578931—21no—
119#14578931—21no—
120#14578931—21no—
121C24×C157—21no—
122C2×Q8×C157C2311yescharacteristic
123C2×D4×C157—22no—
124C2×D4×C157—22no—
125C2×D4×C157—21no—
126C2×D4×C157C2311yescharacteristic
127C2×D4×C157C2311yescharacteristic
128C22×C4×C157C2311yescharacteristic
129SD16×C157—22no—
130SD16×C157—22no—
131C2×C8×C157C2311yescharacteristic
132C2×C8×C157C2311yescharacteristic
133C4⋊C4×C157—21no—
134C4⋊C4×C157C2311yescharacteristic
135C22⋊C4×C157—21no—
136C22×D157—21no—
137C22×D157—24no—
138C22×D157—24no—
139C22×D157—21no—
140C22×D157—81no—
141C22×D157—81no—
142C22×D157—81no—
143C22×D157—81no—
144C22×D157C2×D411yescharacteristic
145C22×D157C2×D411yescharacteristic
146Dic314—21no—
147Dic314—22no—
148Dic314C2×D412yes—
149D628—22no—
150D628—21no—
151D628—21no—
152D628—21no—
153D628—42no—
154D628—81no—
155D628C2×D412yes—
156C22⋊D157—24no—
157C22⋊D157—24no—
158C22⋊D157—24no—
159C22⋊D157—24no—
160C22⋊D157—22no—
161C22⋊D157—24no—
162C22⋊D157—22no—
163C22⋊D157—81no—
164C2×Dic157—24no—
165C2×Dic157—21no—
166C2×Dic157—24no—
167C2×Dic157—21no—
168C2×Dic157C2×D411yescharacteristic
169C2×Dic157C2×D411yescharacteristic
170C4×D157—24no—
171C4×D157—21no—
172C4×D157—21no—
173C4×D157—42no—
174C4×D157—81no—
175C4×D157C2×D412yes—
176C4×D157C2×D412yes—
177#11929434—21no—
178#11929434—21no—
179C23×C157—24no—
180C23×C157—21no—
181C23×C157—21no—
182C23×C157—81no—
183C23×C157—81no—
184C23×C157C2×D411yescharacteristic
185Q8×C157—21no—
186Q8×C157—21no—
187D4×C157—24no—
188D4×C157—21no—
189D4×C157—21no—
190D4×C157C2×D412yes—
191D4×C157C2×D412yes—
192C2×C4×C157—21no—
193C2×C4×C157—21no—
194C2×C4×C157—21no—
195C2×C4×C157—21no—
196C2×C4×C157C2×D411yescharacteristic
197C2×C4×C157C2411yescharacteristic
198C8×C157—21no—
199C8×C157—21no—
200C22×C157—21no—
201C22×C157—22no—
202C22×C157—22no—
203C22×C157—21no—
204C22×C157—21no—
205C22×C157—41no—
206C22×C157—41no—
207C22×C157—41no—
208C22×C157—41no—
209C22×C157—84no—
210C22×C157C22×D411yescharacteristic
211C2×D157—21no—
212C2×D157—21no—
213C2×D157—21no—
214C2×D157—22no—
215C2×D157—21no—
216C2×D157—21no—
217C2×D157—41no—
218C2×D157—41no—
219C2×D157—41no—
220C2×D157—41no—
221C2×D157—41no—
222C2×D157—84no—
223C2×D157—84no—
224C2×D157#12012yes—
225C4×C157—21no—
226C4×C157—41no—
227C4×C157—41no—
228C4×C157C22×D411yescharacteristic
229C4×C157C22×D411yescharacteristic
230Dic157—22no—
231Dic157—21no—
232Dic157—41no—
233Dic157#12012yes—
234C2×C157—42no—
235C2×C157—42no—
236C2×C157—41no—
237C2×C157C2×#12011yescharacteristic
238C2×C157#57611yescharacteristic
239C2×C157#57711yescharacteristic
240D157—22no—
241D157—41no—
242D157—42no—
243D157—81no—
244C157#301211yescharacteristic, complemented, Sylow
245#3012—1571nomaximal, Sylow
246C2×H(5,𝔽2)—1571no—
247C2×AGL(1,ℤ/8)—1571no—
248C2×#135—1571no—
249#537—1571no—
250#452—1571no—
251#452—1571no—
252#450—1571no—
253#450D15711yescharacteristic, complemented
254#449—1571no—
255#449—1571no—
256#447—1571no—
257#447—1571no—
258#417—1571no—
259C2×#102—1571no—
260#408—1571no—
261H(5,𝔽2)—3144no—
262H(5,𝔽2)—3144no—
263C2×#41—3141no—
264C2×#41—1571no—
265C2×#41—1571no—
266C2×#41—1571no—
267C2×#41—1571no—
268C2×#41—1571no—
269C22×D4—3141no—
270C22×D4—3141no—
271C22×D4—3141no—
272C22×D4—1571no—
273C22×D4—1571no—
274C22×D4C2×D15711yescharacteristic
275AGL(1,ℤ/8)—3144no—
276#135—3144no—
277C2×Q16—1571no—
278C2×SD16—1571no—
279C2×SD16—1571no—
280C2×SD16C2×D15711yescharacteristic, complemented
281C2×SD16C2×D15711yescharacteristic, complemented
282C2×D8—1571no—
283C2×D8—1571no—
284C2×D8—1571no—
285C2×M16—1571no—
286C22×C8—1571no—
287C22⋊Q8—3141no—
288C22⋊Q8—1571no—
289C22⋊Q8C2×D15711yescharacteristic, complemented
290#121—3141no—
291#121—3141no—
292#121—3141no—
293#121—1571no—
294#121—1571no—
295C2×C4⋊C4—1571no—
296#103—1571no—
297#103—1571no—
298#102—1571no—
299#102—1571no—
300#102—1571no—
301#102—1571no—
302#102C2×D15711yescharacteristic, complemented
303#102C2×D15711yescharacteristic, complemented
304C22⋊C8—1571no—
305C22⋊C8—1571no—
306C22⋊C8—1571no—
307C22⋊C8C2×D15711yescharacteristic, complemented
308C24—21no—
309C24—3141no—
310C24—3141no—
311#41—3142no—
312#41—3142no—
313#41—3142no—
314#41—3142no—
315#41—3148no—
316#41—3142no—
317#41—3142no—
318#41—3142no—
319#41—3142no—
320#41—3142no—
321#41—1574no—
322C2×Q8—1571no—
323C2×Q8—1571no—
324C2×Q8C22×D15711yescharacteristic
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—21no—
328C2×D4—3141no—
329C2×D4—3141no—
330C2×D4—3148no—
331C2×D4—3142no—
332C2×D4—3148no—
333C2×D4—3148no—
334C2×D4—3141no—
335C2×D4—3141no—
336C2×D4—3141no—
337C2×D4—3142no—
338C2×D4—3141no—
339C2×D4—3142no—
340C2×D4—3141no—
341C2×D4—3141no—
342C2×D4—1571no—
343C2×D4—1571no—
344C2×D4—1574no—
345C2×D4—1571no—
346C2×D4—1571no—
347C2×D4—1571no—
348C2×D4C22×D15711yescharacteristic
349C2×D4C22×D15711yescharacteristic
350C22×C4—3141no—
351C22×C4—3141no—
352C22×C4—3141no—
353C22×C4—3141no—
354C22×C4—3141no—
355C22×C4—1571no—
356C22×C4—1571no—
357C22×C4C22×D15711yescharacteristic
358Q16—3142no—
359SD16—22no—
360SD16—22no—
361SD16—3142no—
362SD16—3142no—
363D8—3142no—
364D8—3142no—
365D8—3142no—
366M16—3142no—
367C2×C8—3142no—
368C2×C8—1571no—
369C2×C8—1571no—
370C2×C8C22×D15711yescharacteristic
371C2×C8C22×D15711yescharacteristic
372C4⋊C4—21no—
373C4⋊C4—3141no—
374C4⋊C4—3141no—
375C4⋊C4—1571no—
376C4⋊C4C22×D15711yescharacteristic
377C22⋊C4—21no—
378C22⋊C4—3141no—
379C22⋊C4—3141no—
380C22⋊C4—3141no—
381C23—24no—
382C23—21no—
383C23—21no—
384C23—81no—
385C23—81no—
386C23—3144no—
387C23—3144no—
388C23—3141no—
389C23—3141no—
390C23—1571no—
391C23—1571no—
392C23—12561no—
393C23—12561no—
394C23—12561no—
395C23—12561no—
396C23D4×D15711yescharacteristic
397Q8—21no—
398Q8—21no—
399Q8—3142no—
400Q8—3141no—
401Q8—1572no—
402D4—24no—
403D4—21no—
404D4—21no—
405D4—3142no—
406D4—3144no—
407D4—3144no—
408D4—3144no—
409D4—3144no—
410D4—3142no—
411D4—3142no—
412D4—3144no—
413D4—3141no—
414D4—3141no—
415D4—3141no—
416D4—6282no—
417D4—1572no—
418D4—12561no—
419D4—12561no—
420D4D4×D15712yes—
421D4D4×D15712yes—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—3144no—
427C2×C4—3144no—
428C2×C4—3141no—
429C2×C4—3144no—
430C2×C4—3141no—
431C2×C4—3141no—
432C2×C4—3141no—
433C2×C4—6282no—
434C2×C4—1571no—
435C2×C4—1571no—
436C2×C4—1572no—
437C2×C4—1572no—
438C2×C4—12561no—
439C2×C4D4×D15711yescharacteristic
440C2×C4C23×D15711yescharacteristic
441C8—21no—
442C8—21no—
443C8—3141no—
444C8—3141no—
445C22—21no—
446C22—22no—
447C22—22no—
448C22—21no—
449C22—21no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—41no—
454C22—84no—
455C22—3141no—
456C22—3141no—
457C22—3141no—
458C22—3142no—
459C22—3141no—
460C22—3141no—
461C22—6281no—
462C22—6281no—
463C22—6281no—
464C22—6281no—
465C22—6281no—
466C22—1572no—
467C22—12564no—
468C22—12564no—
469C22C2×D4×D15711yescharacteristic
470C4—21no—
471C4—41no—
472C4—41no—
473C4—3142no—
474C4—3141no—
475C4—6281no—
476C4—1572no—
477C4C2×D4×D15711yescharacteristic
478C4C2×D4×D15711yescharacteristic
479C2—41no—
480C2—42no—
481C2—42no—
482C2—3142no—
483C2—6281no—
484C2—6282no—
485C2—12561no—
486C2#120×D15711yescharacteristic
487C2#2375807711yescharacteristic
488C2#2375807911yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#450×C157C2
socleC22×C157C22×D4
hypercenter#450D157
hypocenterC157#3012
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. C2×#19097816C2semidirect
2. C2×#19097798C2semidirect
3. C2×#19097792C2semidirect
4. #23757853C2semidirect
5. #23757433C2semidirect
6. #23757432C2semidirect
7. #23757422C2semidirect
8. #23757420C2semidirect
9. #23757419C2semidirect
10. #23757408C2semidirect
11. #23757407C2semidirect
12. #23757271C2semidirect
13. #102×D157C2semidirect
14. #23757238C2semidirect
15. #450×C157C2semidirect
18. C2×#14578955C22semidirect
31. C2×#14578944C22semidirect
32. C2×#14578943C22semidirect
33. C2×#14578941C22semidirect
34. C2×D1256C22semidirect
35. C2×#14578939C22semidirect
36. C2×#14578939C22semidirect
37. C2×#14578937C22semidirect
38. C2×C8×D157C22semidirect
40. #19097747C22semidirect
42. #19097746C22semidirect
43. #19097745C22semidirect
46. C4⋊C4×D157C22semidirect
47. #19097696C22semidirect
48. #19097694C22semidirect
49. #19097693C22semidirect
50. #19097692C22semidirect
51. #19097692C22semidirect
52. #19097691C22semidirect
53. C22⋊#11929434C22semidirect
54. #19097683C22semidirect
55. #19097683C22semidirect
57. C2×SD16×C157C22semidirect
58. C2×SD16×C157C22semidirect
59. C22⋊Q8×C157C22semidirect
60. #102×C157C22semidirect
61. #102×C157C22semidirect
62. C22⋊C8×C157C22semidirect
75. #14578953D4semidirect
97. D4×D157D4semidirect
244. C157#3012semidirect
253. #450D157semidirect
280. C2×SD16C2×D157semidirect
281. C2×SD16C2×D157semidirect
289. C22⋊Q8C2×D157semidirect
302. #102C2×D157semidirect
303. #102C2×D157semidirect
307. 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×#190977982
C2×#190977922
#237578532
#237574332
#237574322
#237574222
#237574202
#237574192
#237574082
#237574072
#237572712
#102×D1572
#237572382
#450×C1572
#3012157