Finite Group Databasebeta

← #34109356

Subgroups — #34109356

Order 20096.

764 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#41×D157C211yesmaximal, characteristic, complemented
2C2×#19097793C211yesmaximal, characteristic, complemented
3C2×#19097791C211yesmaximal, characteristic, complemented
4#23757785C211yesmaximal, characteristic, complemented
5#23757438C211yesmaximal, characteristic, complemented
6#23757436C211yesmaximal, characteristic, complemented
7#23757430C211yesmaximal, characteristic, complemented
8#23757425C211yesmaximal, characteristic, complemented
9#23757421C211yesmaximal, characteristic, complemented
10#23757419C211yesmaximal, characteristic, complemented
11#23757413C211yesmaximal, characteristic, complemented
12#23757408C211yesmaximal, characteristic, complemented
13#23757267C211yesmaximal, characteristic, complemented
14C2×#19097683C211yesmaximal, characteristic, complemented
15#416×C157C211yesmaximal, characteristic, complemented
16C2×#14578955C2211yescharacteristic
17C2×#14578954—21no—
18C2×#14578954C2211yescharacteristic
19C2×#14578954C2211yescharacteristic, complemented
20C2×#14578953—21no—
21C2×#14578953C2211yescharacteristic
22#41×D157—24no—
23#41×D157—24no—
24C2×Q8×D157C2211yescharacteristic
25C2×D4×D157—21no—
26C2×D4×D157C2211yescharacteristic
27C22×C4×D157—21no—
28C22×C4×D157C2211yescharacteristic
29#19097793—24no—
30#19097791—24no—
31C2×Dic628C2211yescharacteristic, complemented
32C2×#14578944C2211yescharacteristic, complemented
33C2×#14578943C2211yescharacteristic, complemented
34C2×#14578943C2211yescharacteristic, complemented
35C2×#14578942C2211yescharacteristic, complemented
36C2×#14578941C2211yescharacteristic, complemented
37C2×D1256C2211yescharacteristic, complemented
38C2×#14578939C2211yescharacteristic, complemented
39C22×#11929434C2211yescharacteristic, complemented
40#19097755—21no—
41#19097751C2211yescharacteristic
42#19097751C2211yescharacteristic, complemented
43#19097746C2211yescharacteristic
44#19097746C2211yescharacteristic, complemented
45#19097741—21no—
46C4×D628—21no—
47C4×D628—21no—
48#19097694C2211yescharacteristic, complemented
49#19097694C2211yescharacteristic, complemented
50#19097691C2211yescharacteristic, complemented
51#19097691C2211yescharacteristic, complemented
52#19097683C2211yescharacteristic
53#19097683C2211yescharacteristic
54#19097683C2211yescharacteristic, complemented
55#19097683C2211yescharacteristic, complemented
56C2×#41×C157C2211yescharacteristic
57C22×C8×C157C2211yescharacteristic, complemented
58#117×C157C2211yescharacteristic, complemented
59#103×C157C2211yescharacteristic, complemented
60#103×C157C2211yescharacteristic, complemented
61#102×C157C2211yescharacteristic, complemented
62#102×C157C2211yescharacteristic, complemented
63C23×D157—21no—
64C23×D157D411yescharacteristic
65#14578955—22no—
66#14578955—22no—
67#14578954—28no—
68#14578954—22no—
69#14578954—22no—
70#14578954—22no—
71#14578954D414yescomplemented
72#14578953—28no—
73#14578953—22no—
74#14578953—22no—
75C2×Dic314—21no—
76C2×Dic314C2311yescharacteristic
77C2×Dic314C2311yescharacteristic
78Q8×D157—22no—
79Q8×D157—22no—
80C2×D628—21no—
81C2×D628—21no—
82C2×D628—21no—
83C2×D628C2311yescharacteristic
84C2×D628C2311yescharacteristic
85C2×C22⋊D157—21no—
86C2×C22⋊D157—21no—
87C2×C22⋊D157—21no—
88C2×C22⋊D157—21no—
89D4×D157—28no—
90D4×D157—22no—
91D4×D157—22no—
92C22×Dic157—21no—
93C22×Dic157D411yescharacteristic
94C2×C4×D157—28no—
95C2×C4×D157—21no—
96C2×C4×D157—21no—
97C2×C4×D157—21no—
98C2×C4×D157—22no—
99C2×C4×D157—22no—
100C2×C4×D157—21no—
101C2×C4×D157D411yescharacteristic
102C2×C4×D157D411yescharacteristic
103C2×C4×D157C2311yescharacteristic
104C2×C4×D157C2311yescharacteristic
105Dic628—22no—
106#14578944—22no—
107#14578943—22no—
108#14578943—22no—
109#14578942—22no—
110#14578941—22no—
111D1256—22no—
112#14578939—22no—
113C2×#11929434—22no—
114C2×#11929434C2311yescharacteristic
115C2×#11929434C2311yescharacteristic
116C4⋊Dic157—21no—
117C4⋊Dic157—21no—
118#14578931—21no—
119#14578931—21no—
120#14578931—21no—
121#14578931—21no—
122#41×C157—22no—
123#41×C157D414yescomplemented
124C2×Q8×C157C2311yescharacteristic
125C2×D4×C157—21no—
126C2×D4×C157C2311yescharacteristic
127C22×C4×C157—21no—
128C22×C4×C157C2311yescharacteristic
129C2×C8×C157—22no—
130C2×C8×C157C2311yescharacteristic
131C2×C8×C157C2311yescharacteristic
132C4⋊C4×C157C2311yescharacteristic
133C4⋊C4×C157C2311yescharacteristic
134C22⋊C4×C157—21no—
135C42×C157—21no—
136C22×D157—24no—
137C22×D157—22no—
138C22×D157—21no—
139C22×D157—22no—
140C22×D157—21no—
141C22×D157—41no—
142C22×D157—42no—
143C22×D157—42no—
144C22×D157C2×D411yescharacteristic
145C22×D157C2×D411yescharacteristic
146Dic314—24no—
147Dic314—21no—
148Dic314—21no—
149Dic314—21no—
150Dic314C2×D412yes—
151D628—24no—
152D628—21no—
153D628—21no—
154D628—21no—
155D628—42no—
156D628—42no—
157D628C2×D412yes—
158C22⋊D157—24no—
159C22⋊D157—24no—
160C22⋊D157—24no—
161C22⋊D157—42no—
162C2×Dic157—24no—
163C2×Dic157—22no—
164C2×Dic157—21no—
165C2×Dic157—21no—
166C2×Dic157C2×D411yescharacteristic
167C2×Dic157C2×D411yescharacteristic
168C4×D157—24no—
169C4×D157—24no—
170C4×D157—22no—
171C4×D157—24no—
172C4×D157—22no—
173C4×D157—21no—
174C4×D157—21no—
175C4×D157—21no—
176C4×D157—21no—
177C4×D157—42no—
178#11929434—21no—
179#11929434—21no—
180C23×C157—21no—
181C23×C157C2×D411yescharacteristic
182Q8×C157—21no—
183Q8×C157C2×D412yes—
184D4×C157—24no—
185D4×C157—21no—
186D4×C157C2×D412yes—
187C2×C4×C157—21no—
188C2×C4×C157—24no—
189C2×C4×C157—21no—
190C2×C4×C157—21no—
191C2×C4×C157C2×D412yes—
192C2×C4×C157C2×D411yescharacteristic
193C2×C4×C157C2×D412yes—
194C2×C4×C157C2411yescharacteristic
195C8×C157—21no—
196C8×C157—21no—
197C22×C157—21no—
198C22×C157—21no—
199C22×C157—22no—
200C22×C157—41no—
201C22×C157—41no—
202C22×C157#12012yes—
203C22×C157C22×D411yescharacteristic
204C2×D157—21no—
205C2×D157—21no—
206C2×D157—21no—
207C2×D157—22no—
208C2×D157—21no—
209C2×D157—21no—
210C2×D157—44no—
211C2×D157—48no—
212C2×D157—41no—
213C2×D157—41no—
214C2×D157—41no—
215C2×D157—41no—
216C2×D157—41no—
217C2×D157#13512yescomplemented
218C4×C157—22no—
219C4×C157—41no—
220C4×C157—41no—
221C4×C157#12012yes—
222C4×C157C22×D411yescharacteristic
223C4×C157C22×D411yescharacteristic
224Dic157—21no—
225Dic157—22no—
226Dic157—21no—
227Dic157—41no—
228C2×C157—22no—
229C2×C157—42no—
230C2×C157C2×#12011yescharacteristic
231C2×C157C2×#13511yescharacteristic
232C2×C157#57511yescharacteristic
233D157—22no—
234D157—41no—
235D157—42no—
236D157—81no—
237C157#300211yescharacteristic, complemented, Sylow
238#3002—1571nomaximal, Sylow
239C22×#41—1571no—
240C2×#135—1571no—
241C2×#135—1571no—
242#525—1571no—
243#452—1571no—
244#452—1571no—
245#451—1571no—
246#450—1571no—
247#449—1571no—
248#449—1571no—
249#448—1571no—
250#447—1571no—
251#416—1571no—
252#416D15711yescharacteristic, complemented
253C2×C22⋊C8—1571no—
254C2×#41—3141no—
255C2×#41—3144no—
256C2×#41—3144no—
257C2×#41—3141no—
258C2×#41—1571no—
259C2×#41—1571no—
260C2×#41—1571no—
261C2×#41—1571no—
262C2×#41C2×D15711yescharacteristic
263C22×Q8—1571no—
264C22×D4—3141no—
265C22×D4—1571no—
266C23×C4—3141no—
267C23×C4—1571no—
268#135—3144no—
269#135—3144no—
270C2×Q16—1571no—
271C2×Q16—1571no—
272C2×SD16—1571no—
273C2×SD16—1571no—
274C2×SD16—1571no—
275C2×SD16—1571no—
276C2×D8—1571no—
277C2×D8—1571no—
278C22×C8—1571no—
279C22×C8C2×D15711yescharacteristic, complemented
280#123—3141no—
281C22⋊Q8—1571no—
282C22⋊Q8—1571no—
283#121—1571no—
284#121—1571no—
285#120—3141no—
286C4×D4—3141no—
287C4×D4—3141no—
288#117C2×D15711yescharacteristic, complemented
289#103—1571no—
290#103—1571no—
291#103C2×D15711yescharacteristic, complemented
292#103C2×D15711yescharacteristic, complemented
293#102—1571no—
294#102—1571no—
295#102C2×D15711yescharacteristic, complemented
296#102C2×D15711yescharacteristic, complemented
297C22⋊C8—1571no—
298C22⋊C8—1571no—
299C22⋊C8—1571no—
300C22⋊C8—1571no—
301C24—3141no—
302C24—1571no—
303#41—22no—
304#41—3148no—
305#41—3148no—
306#41—3142no—
307#41—3142no—
308#41—3142no—
309#41—3142no—
310#41—3142no—
311#41—3142no—
312#41—3142no—
313#41—1574no—
314#41D62814yescomplemented
315C2×Q8—3141no—
316C2×Q8—3142no—
317C2×Q8—3142no—
318C2×Q8—1571no—
319C2×Q8—1571no—
320C2×Q8C22×D15711yescharacteristic
321C2×D4—21no—
322C2×D4—3141no—
323C2×D4—3141no—
324C2×D4—3141no—
325C2×D4—3148no—
326C2×D4—3142no—
327C2×D4—3141no—
328C2×D4—3141no—
329C2×D4—3142no—
330C2×D4—3141no—
331C2×D4—3141no—
332C2×D4—1571no—
333C2×D4—1571no—
334C2×D4C22×D15711yescharacteristic
335C22×C4—21no—
336C22×C4—3148no—
337C22×C4—3141no—
338C22×C4—3141no—
339C22×C4—3141no—
340C22×C4—3142no—
341C22×C4—3141no—
342C22×C4—3142no—
343C22×C4—3141no—
344C22×C4—1571no—
345C22×C4—1571no—
346C22×C4—1571no—
347C22×C4—1571no—
348C22×C4—1571no—
349C22×C4C22×D15711yescharacteristic
350Q16—3142no—
351Q16—3142no—
352SD16—3142no—
353SD16—3142no—
354SD16—3142no—
355SD16—3142no—
356D8—3142no—
357D8—3142no—
358C2×C8—22no—
359C2×C8—3142no—
360C2×C8—1571no—
361C2×C8—1571no—
362C2×C8C22×D15711yescharacteristic
363C2×C8C22×D15711yescharacteristic
364C4⋊C4—3141no—
365C4⋊C4—3141no—
366C4⋊C4C22×D15711yescharacteristic
367C4⋊C4C22×D15711yescharacteristic
368C22⋊C4—21no—
369C22⋊C4—3141no—
370C22⋊C4—3141no—
371C22⋊C4—3141no—
372C22⋊C4—3141no—
373C42—21no—
374C23—21no—
375C23—3144no—
376C23—3142no—
377C23—3142no—
378C23—3141no—
379C23—3141no—
380C23—6281no—
381C23—6282no—
382C23—6282no—
383C23—1571no—
384C23—1571no—
385C23D4×D15711yescharacteristic
386Q8—21no—
387Q8—3144no—
388Q8—3141no—
389Q8—3141no—
390Q8—3141no—
391Q8—1572no—
392Q8C2×D62812yes—
393D4—24no—
394D4—21no—
395D4—3144no—
396D4—3144no—
397D4—3144no—
398D4—3144no—
399D4—3141no—
400D4—3141no—
401D4—3141no—
402D4—6282no—
403D4—6282no—
404D4—6282no—
405D4—1572no—
406D4C2×D62812yes—
407C2×C4—21no—
408C2×C4—24no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—3144no—
412C2×C4—3144no—
413C2×C4—3144no—
414C2×C4—3142no—
415C2×C4—3141no—
416C2×C4—3142no—
417C2×C4—3144no—
418C2×C4—3142no—
419C2×C4—3141no—
420C2×C4—3141no—
421C2×C4—3141no—
422C2×C4—3141no—
423C2×C4—3141no—
424C2×C4—6282no—
425C2×C4—1571no—
426C2×C4—1571no—
427C2×C4D4×D15712yes—
428C2×C4D4×D15711yescharacteristic
429C2×C4C2×D62812yes—
430C2×C4C23×D15711yescharacteristic
431C8—21no—
432C8—21no—
433C8—3141no—
434C8—3141no—
435C22—21no—
436C22—21no—
437C22—22no—
438C22—41no—
439C22—41no—
440C22—3141no—
441C22—3141no—
442C22—3141no—
443C22—3142no—
444C22—3141no—
445C22—3141no—
446C22—6284no—
447C22—6288no—
448C22—6281no—
449C22—6281no—
450C22—6281no—
451C22—6281no—
452C22—6281no—
453C22—1572no—
454C22#1909774112yes—
455C22C2×D4×D15711yescharacteristic
456C4—22no—
457C4—41no—
458C4—41no—
459C4—3141no—
460C4—3142no—
461C4—3141no—
462C4—6281no—
463C4#1909774112yes—
464C4C2×D4×D15711yescharacteristic
465C4C22×D62811yescharacteristic
466C2—22no—
467C2—42no—
468C2—3142no—
469C2—6281no—
470C2—6282no—
471C2—12561no—
472C2C2×#1909774111yescharacteristic
473C2#135×D15711yescharacteristic
474C2#2375807011yescharacteristic
subgroupG/H
centerC2×C4D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#416×C157C2
socleC22×C157C22×D4
hypercenter#416D157
hypocenterC157#3002
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. C2×#41×D157C2semidirect
2. C2×#19097793C2semidirect
3. C2×#19097791C2semidirect
4. #23757785C2semidirect
5. #23757438C2semidirect
6. #23757436C2semidirect
7. #23757430C2semidirect
8. #23757425C2semidirect
9. #23757421C2semidirect
10. #23757419C2semidirect
11. #23757413C2semidirect
12. #23757408C2semidirect
13. #23757267C2semidirect
14. C2×#19097683C2semidirect
15. #416×C157C2semidirect
19. C2×#14578954C22semidirect
31. C2×Dic628C22semidirect
32. C2×#14578944C22semidirect
33. C2×#14578943C22semidirect
34. C2×#14578943C22semidirect
35. C2×#14578942C22semidirect
36. C2×#14578941C22semidirect
37. C2×D1256C22semidirect
38. C2×#14578939C22semidirect
39. C22×#11929434C22semidirect
42. #19097751C22semidirect
44. #19097746C22semidirect
48. #19097694C22semidirect
49. #19097694C22semidirect
50. #19097691C22semidirect
51. #19097691C22semidirect
54. #19097683C22semidirect
55. #19097683C22semidirect
57. C22×C8×C157C22semidirect
58. #117×C157C22semidirect
59. #103×C157C22semidirect
60. #103×C157C22semidirect
61. #102×C157C22semidirect
62. #102×C157C22semidirect
71. #14578954D4semidirect
123. #41×C157D4semidirect
217. C2×D157#135semidirect
237. C157#3002semidirect
252. #416D157semidirect
279. C22×C8C2×D157semidirect
288. #117C2×D157semidirect
291. #103C2×D157semidirect
292. #103C2×D157semidirect
295. #102C2×D157semidirect
296. #102C2×D157semidirect
314. #41D628semidirect

Maximal subgroups

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

subgroupindex
C2×#41×D1572
C2×#190977932
C2×#190977912
#237577852
#237574382
#237574362
#237574302
#237574252
#237574212
#237574192
#237574132
#237574082
#237572672
C2×#190976832
#416×C1572
#3002157