Finite Group Databasebeta

← C2×#23757859 #34113475

Subgroups — C2×#23757859 #34113475

Order 20096.

1336 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23757859C2116yesmaximal, direct product factor
2C2×#19097766C211yesmaximal, characteristic, complemented
3C2×#19097764C211yesmaximal, characteristic, complemented
4C2×#19097757C211yesmaximal, characteristic, complemented
5C2×#19097757C211yesmaximal, characteristic, complemented
6C2×#19097756C211yesmaximal, characteristic, complemented
7C2×#19097751C211yesmaximal, characteristic, complemented
8C2×#19097745C211yesmaximal, characteristic, complemented
9C2×#19097744C211yesmaximal, characteristic, complemented
10C2×#19097742C211yesmaximal, characteristic, complemented
11C2×#19097741C211yesmaximal, characteristic, complemented
12C2×#19097739C211yesmaximal, characteristic, complemented
13C2×#19097733C211yesmaximal, characteristic, complemented
14C2×C4×C22⋊D157C211yesmaximal, characteristic, complemented
15C2×C22⋊C4×D157C211yesmaximal, characteristic, complemented
16C2×#123×C157C211yesmaximal, characteristic, complemented
17C24×D157C2211yescharacteristic
18C22×Dic314C2211yescharacteristic, complemented
19C22×D628C2211yescharacteristic
20C22×D628C2211yescharacteristic
21C22×C22⋊D157C2211yescharacteristic
22C22×C22⋊D157C2211yescharacteristic
23C22×C22⋊D157C2211yescharacteristic
24C22×C22⋊D157C2211yescharacteristic, complemented
25C22×C4×D157C2211yescharacteristic
26C22×C4×D157C2211yescharacteristic
27C22×C4×D157C2211yescharacteristic, complemented
28#19097766C2218yescomplemented
29#19097764C2218yescomplemented
30#19097757C2218yescomplemented
31#19097757C2218yescomplemented
32#19097756C2218yescomplemented
33#19097751C2218yescomplemented
34#19097745C2218yescomplemented
35#19097744C2218yescomplemented
36#19097742C2218yescomplemented
37#19097741C2218yescomplemented
38#19097739C2218yescomplemented
39#19097733C2218yescomplemented
40C4×C22⋊D157C2218yescomplemented
41C2×C4⋊Dic157C2211yescharacteristic, complemented
42C2×C4⋊Dic157C2211yescharacteristic, complemented
43C2×#14578933C2211yescharacteristic, complemented
44C2×#14578933C2211yescharacteristic, complemented
45C2×C22⋊Dic157C2211yescharacteristic
46C2×C22⋊Dic157C2211yescharacteristic
47C2×C22⋊Dic157C2211yescharacteristic, complemented
48C2×#14578931C2211yescharacteristic
49C2×#14578931C2211yescharacteristic
50C2×#14578931C2211yescharacteristic
51C2×#14578931C2211yescharacteristic
52C2×#14578931C2211yescharacteristic
53C2×#14578931C2211yescharacteristic
54C2×#14578931C2211yescharacteristic, complemented
55C2×#14578931C2211yescharacteristic, complemented
56C22⋊C4×D157C2218yescomplemented
57C2×C4×Dic157C2211yescharacteristic, complemented
58C2×C4×Dic157C2211yescharacteristic, complemented
59C22×D4×C157C2211yescharacteristic
60C23×C4×C157C2211yescharacteristic, complemented
61#123×C157C2218yescomplemented
62C2×C4⋊C4×C157C2211yescharacteristic, complemented
63C2×C4⋊C4×C157C2211yescharacteristic, complemented
64C2×C22⋊C4×C157C2211yescharacteristic
65C2×C22⋊C4×C157C2211yescharacteristic
66C2×C22⋊C4×C157C2211yescharacteristic, complemented
67C23×D157—24no—
68C23×D157—42no—
69C23×D157—42no—
70C23×D157C2311yescharacteristic
71C23×D157C2311yescharacteristic
72C23×D157C2311yescharacteristic
73C23×D157C2314yes—
74C2×Dic314—42no—
75C2×Dic314C2314yescomplemented
76C2×D628—24no—
77C2×D628—42no—
78C2×D628C2314yes—
79C2×D628C2314yes—
80C2×C22⋊D157—24no—
81C2×C22⋊D157—42no—
82C2×C22⋊D157—42no—
83C2×C22⋊D157—42no—
84C2×C22⋊D157C2314yes—
85C2×C22⋊D157C2314yes—
86C2×C22⋊D157C2314yes—
87C2×C22⋊D157C2314yescomplemented
88C22×Dic157C2311yescharacteristic
89C22×Dic157C2311yescharacteristic
90C22×Dic157C2311yescharacteristic
91C22×Dic157C2311yescharacteristic
92C22×Dic157C2311yescharacteristic
93C2×C4×D157—42no—
94C2×C4×D157—42no—
95C2×C4×D157—42no—
96C2×C4×D157C2314yes—
97C2×C4×D157C2314yes—
98C2×C4×D157C2314yescomplemented
99C4⋊Dic157C2314yescomplemented
100C4⋊Dic157C2314yescomplemented
101#14578933C2314yescomplemented
102#14578933C2314yescomplemented
103C22⋊Dic157C2314yes—
104C22⋊Dic157C2314yes—
105C22⋊Dic157C2314yescomplemented
106#14578931C2314yes—
107#14578931C2314yes—
108#14578931C2314yes—
109#14578931C2314yes—
110#14578931C2314yes—
111#14578931C2314yes—
112#14578931C2314yescomplemented
113#14578931C2314yescomplemented
114C4×Dic157C2314yescomplemented
115C4×Dic157C2314yescomplemented
116C24×C157C2311yescharacteristic
117C24×C157C2311yescharacteristic
118C2×D4×C157—42no—
119C2×D4×C157C2314yes—
120C22×C4×C157—42no—
121C22×C4×C157C2311yescharacteristic
122C22×C4×C157C2311yescharacteristic
123C22×C4×C157C2311yescharacteristic
124C22×C4×C157C2311yescharacteristic
125C22×C4×C157C2311yescharacteristic
126C22×C4×C157C2314yescomplemented
127C4⋊C4×C157C2314yescomplemented
128C4⋊C4×C157C2314yescomplemented
129C22⋊C4×C157C2314yes—
130C22⋊C4×C157C2314yes—
131C22⋊C4×C157C2314yescomplemented
132C22×D157—22no—
133C22×D157—22no—
134C22×D157—22no—
135C22×D157—22no—
136C22×D157—22no—
137C22×D157—22no—
138C22×D157—22no—
139C22×D157—22no—
140C22×D157—28no—
141C22×D157—44no—
142C22×D157—44no—
143C22×D157—416no—
144C22×D157#4114yes—
145C22×D157C2412yes—
146C22×D157C2412yes—
147C22×D157C2412yes—
148Dic314—44no—
149D628—28no—
150D628—44no—
151C22⋊D157—28no—
152C22⋊D157—44no—
153C22⋊D157—44no—
154C22⋊D157—44no—
155C2×Dic157—22no—
156C2×Dic157—22no—
157C2×Dic157—22no—
158C2×Dic157—22no—
159C2×Dic157—22no—
160C2×Dic157C2412yes—
161C2×Dic157C2412yes—
162C2×Dic157C2412yes—
163C2×Dic157C2412yes—
164C2×Dic157C2412yes—
165C4×D157—44no—
166C4×D157—44no—
167C4×D157—44no—
168C23×C157—22no—
169C23×C157—22no—
170C23×C157—22no—
171C23×C157—22no—
172C23×C157—22no—
173C23×C157—22no—
174C23×C157C2412yes—
175C23×C157C2412yes—
176C23×C157C2411yescharacteristic
177D4×C157—44no—
178C2×C4×C157—22no—
179C2×C4×C157—22no—
180C2×C4×C157—22no—
181C2×C4×C157—22no—
182C2×C4×C157—22no—
183C2×C4×C157—44no—
184C2×C4×C157C2412yes—
185C2×C4×C157C2412yes—
186C2×C4×C157C2412yes—
187C2×C4×C157C2412yes—
188C2×C4×C157C2412yes—
189C22×C157—22no—
190C22×C157—22no—
191C22×C157—22no—
192C22×C157—22no—
193C22×C157—22no—
194C22×C157—22no—
195C22×C157—44no—
196C22×C157—44no—
197C22×C157C2×#4112yes—
198C22×C157H(5,𝔽2)12yes—
199C22×C157H(5,𝔽2)12yes—
200C22×C157C2511yescharacteristic
201C2×D157—28no—
202C2×D157—22no—
203C2×D157—22no—
204C2×D157—22no—
205C2×D157—22no—
206C2×D157—22no—
207C2×D157—22no—
208C2×D157—22no—
209C2×D157—22no—
210C2×D157—416no—
211C2×D157—44no—
212C2×D157—44no—
213C2×D157C2×#4114yes—
214C4×C157—22no—
215C4×C157—22no—
216C4×C157—22no—
217C4×C157—22no—
218C4×C157—22no—
219Dic157—22no—
220Dic157—22no—
221Dic157—22no—
222Dic157—22no—
223Dic157—22no—
224C2×C157—42no—
225C2×C157—42no—
226C2×C157#53814yescomplemented
227C2×C157C22×#4111yescharacteristic
228C2×C157C2×H(5,𝔽2)11yescharacteristic
229C2×C157C2×H(5,𝔽2)11yescharacteristic
230D157—24no—
231D157—42no—
232D157—42no—
233C157C2×#53811yescharacteristic, complemented, Sylow
234C2×#538—1571nomaximal, Sylow
235#538—15716no—
236C2×#126—1571no—
237C2×#126—1571no—
238C2×#124—1571no—
239C2×#124—1571no—
240C2×#123—1571no—
241C2×#123D15711yescharacteristic, complemented
242C2×C22⋊Q8—1571no—
243C2×#121—1571no—
244C2×#121—1571no—
245C2×#121—1571no—
246C2×#120—1571no—
247C2×#120—1571no—
248C2×C4×D4—1571no—
249C2×C4×D4—1571no—
250C22×C22⋊C4—1571no—
251C25—1571no—
252C22×Q8—1571no—
253C22×D4—1571no—
254C22×D4—1571no—
255C22×D4—1571no—
256C22×D4—1571no—
257C22×D4—1571no—
258C22×D4—1571no—
259C22×D4C2×D15711yescharacteristic
260C23×C4—1571no—
261C23×C4—1571no—
262C23×C4—1571no—
263C23×C4C2×D15711yescharacteristic, complemented
264#126—1578no—
265#126—1578no—
266#124—1578no—
267#124—1578no—
268#123—1578no—
269#123C2×D15718yescomplemented
270C22⋊Q8—1578no—
271#121—1578no—
272#121—1578no—
273#121—1578no—
274#120—1578no—
275#120—1578no—
276C4×D4—1578no—
277C4×D4—1578no—
278C2×C4⋊C4—1571no—
279C2×C4⋊C4—1571no—
280C2×C4⋊C4—1571no—
281C2×C4⋊C4—1571no—
282C2×C4⋊C4C2×D15711yescharacteristic, complemented
283C2×C4⋊C4C2×D15711yescharacteristic, complemented
284C2×C22⋊C4—1571no—
285C2×C22⋊C4—1571no—
286C2×C22⋊C4—1571no—
287C2×C22⋊C4—1571no—
288C2×C22⋊C4—1571no—
289C2×C22⋊C4—1571no—
290C2×C22⋊C4—1571no—
291C2×C22⋊C4—1571no—
292C2×C22⋊C4—1571no—
293C2×C22⋊C4—1571no—
294C2×C22⋊C4—1571no—
295C2×C22⋊C4—1578no—
296C2×C22⋊C4C2×D15711yescharacteristic
297C2×C22⋊C4C2×D15711yescharacteristic
298C2×C22⋊C4C2×D15711yescharacteristic, complemented
299C2×C42—1571no—
300C2×C42—1571no—
301C24—3144no—
302C24—6282no—
303C24—6282no—
304C24—1571no—
305C24—1571no—
306C24—1571no—
307C24—1574no—
308C24C22×D15711yescharacteristic
309C24C22×D15711yescharacteristic
310C2×Q8—6282no—
311C2×Q8—1574no—
312C2×D4—42no—
313C2×D4—3144no—
314C2×D4—3144no—
315C2×D4—6282no—
316C2×D4—6282no—
317C2×D4—6282no—
318C2×D4—6282no—
319C2×D4—1574no—
320C2×D4—1574no—
321C2×D4—1574no—
322C2×D4—1574no—
323C2×D4—1574no—
324C2×D4—1574no—
325C2×D4C22×D15714yes—
326C22×C4—42no—
327C22×C4—6282no—
328C22×C4—6282no—
329C22×C4—6282no—
330C22×C4—1571no—
331C22×C4—1571no—
332C22×C4—1571no—
333C22×C4—1571no—
334C22×C4—1571no—
335C22×C4—1574no—
336C22×C4—1574no—
337C22×C4—1574no—
338C22×C4C22×D15711yescharacteristic
339C22×C4C22×D15711yescharacteristic
340C22×C4C22×D15711yescharacteristic
341C22×C4C22×D15711yescharacteristic
342C22×C4C22×D15711yescharacteristic
343C22×C4C22×D15714yescomplemented
344C4⋊C4—1574no—
345C4⋊C4—1574no—
346C4⋊C4—1574no—
347C4⋊C4—1574no—
348C4⋊C4C22×D15714yescomplemented
349C4⋊C4C22×D15714yescomplemented
350C22⋊C4—1574no—
351C22⋊C4—1574no—
352C22⋊C4—1574no—
353C22⋊C4—1574no—
354C22⋊C4—1574no—
355C22⋊C4—1574no—
356C22⋊C4—1574no—
357C22⋊C4—1574no—
358C22⋊C4—1574no—
359C22⋊C4—1574no—
360C22⋊C4—1574no—
361C22⋊C4C22×D15714yes—
362C22⋊C4C22×D15714yes—
363C22⋊C4C22×D15714yescomplemented
364C42—1574no—
365C42—1574no—
366C23—22no—
367C23—22no—
368C23—22no—
369C23—22no—
370C23—22no—
371C23—22no—
372C23—3148no—
373C23—3142no—
374C23—3142no—
375C23—3142no—
376C23—3142no—
377C23—3142no—
378C23—3142no—
379C23—3142no—
380C23—3142no—
381C23—6284no—
382C23—6284no—
383C23—62816no—
384C23—1572no—
385C23—1572no—
386C23—1574no—
387C23—1572no—
388C23C23×D15712yes—
389C23C23×D15712yes—
390C23C23×D15711yescharacteristic
391Q8—6284no—
392D4—44no—
393D4—3148no—
394D4—3148no—
395D4—6284no—
396D4—6284no—
397D4—6284no—
398D4—6284no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—44no—
405C2×C4—3142no—
406C2×C4—3142no—
407C2×C4—3142no—
408C2×C4—3142no—
409C2×C4—3142no—
410C2×C4—6284no—
411C2×C4—6284no—
412C2×C4—6284no—
413C2×C4—1572no—
414C2×C4—1572no—
415C2×C4—1572no—
416C2×C4—1572no—
417C2×C4—1572no—
418C2×C4C23×D15712yes—
419C2×C4C23×D15712yes—
420C2×C4C23×D15712yes—
421C2×C4C23×D15712yes—
422C2×C4C23×D15712yes—
423C22—22no—
424C22—22no—
425C22—22no—
426C22—22no—
427C22—22no—
428C22—22no—
429C22—44no—
430C22—44no—
431C22—3142no—
432C22—3142no—
433C22—3148no—
434C22—3142no—
435C22—3142no—
436C22—3142no—
437C22—3142no—
438C22—3142no—
439C22—3142no—
440C22—62816no—
441C22—6284no—
442C22—6284no—
443C22—1574no—
444C22#41×D15712yes—
445C22#1909781612yes—
446C22#1909781712yes—
447C22C24×D15711yescharacteristic
448C4—22no—
449C4—22no—
450C4—22no—
451C4—22no—
452C4—22no—
453C4—3142no—
454C4—3142no—
455C4—3142no—
456C4—3142no—
457C4—3142no—
458C2—42no—
459C2—42no—
460C2—3144no—
461C2—6282no—
462C2—6282no—
463C2#2375785914yesdirect product factor
464C2C2×#41×D15711yescharacteristic
465C2C2×#1909781611yescharacteristic
466C2C2×#1909781711yescharacteristic
subgroupG/H
centerC23C23×D157
commutator, cosocleC22×C157C25
FrattiniC22C24×D157
FittingC2×#123×C157C2
socleC23×C157C24
hypercenterC2×#123D157
hypocenterC157C2×#538
LCSG ▷ C22×C157 ▷ C157
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. #23757859C2direct (N × H)
2. C2×#19097766C2semidirect
3. C2×#19097764C2semidirect
4. C2×#19097757C2semidirect
5. C2×#19097757C2semidirect
6. C2×#19097756C2semidirect
7. C2×#19097751C2semidirect
8. C2×#19097745C2semidirect
9. C2×#19097744C2semidirect
10. C2×#19097742C2semidirect
11. C2×#19097741C2semidirect
12. C2×#19097739C2semidirect
13. C2×#19097733C2semidirect
14. C2×C4×C22⋊D157C2semidirect
15. C2×C22⋊C4×D157C2semidirect
16. C2×#123×C157C2semidirect
18. C22×Dic314C22semidirect
24. C22×C22⋊D157C22semidirect
27. C22×C4×D157C22semidirect
28. #19097766C22semidirect
29. #19097764C22semidirect
30. #19097757C22semidirect
31. #19097757C22semidirect
32. #19097756C22semidirect
33. #19097751C22semidirect
34. #19097745C22semidirect
35. #19097744C22semidirect
36. #19097742C22semidirect
37. #19097741C22semidirect
38. #19097739C22semidirect
39. #19097733C22semidirect
40. C4×C22⋊D157C22semidirect
41. C2×C4⋊Dic157C22semidirect
42. C2×C4⋊Dic157C22semidirect
43. C2×#14578933C22semidirect
44. C2×#14578933C22semidirect
47. C2×C22⋊Dic157C22semidirect
54. C2×#14578931C22semidirect
55. C2×#14578931C22semidirect
56. C22⋊C4×D157C22semidirect
57. C2×C4×Dic157C22semidirect
58. C2×C4×Dic157C22semidirect
60. C23×C4×C157C22semidirect
61. #123×C157C22semidirect
62. C2×C4⋊C4×C157C22semidirect
63. C2×C4⋊C4×C157C22semidirect
66. C2×C22⋊C4×C157C22semidirect
75. C2×Dic314C23semidirect
87. C2×C22⋊D157C23semidirect
98. C2×C4×D157C23semidirect
99. C4⋊Dic157C23semidirect
100. C4⋊Dic157C23semidirect
101. #14578933C23semidirect
102. #14578933C23semidirect
105. C22⋊Dic157C23semidirect
112. #14578931C23semidirect
113. #14578931C23semidirect
114. C4×Dic157C23semidirect
115. C4×Dic157C23semidirect
126. C22×C4×C157C23semidirect
127. C4⋊C4×C157C23semidirect
128. C4⋊C4×C157C23semidirect
131. C22⋊C4×C157C23semidirect
226. C2×C157#538semidirect
233. C157C2×#538semidirect
241. C2×#123D157semidirect
263. C23×C4C2×D157semidirect
269. #123C2×D157semidirect
282. C2×C4⋊C4C2×D157semidirect
283. C2×C4⋊C4C2×D157semidirect
298. C2×C22⋊C4C2×D157semidirect
343. C22×C4C22×D157semidirect
348. C4⋊C4C22×D157semidirect
349. C4⋊C4C22×D157semidirect
363. C22⋊C4C22×D157semidirect
463. C2#23757859direct (N × H)

Maximal subgroups

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

subgroupindex
#237578592
C2×#190977662
C2×#190977642
C2×#190977572
C2×#190977562
C2×#190977512
C2×#190977452
C2×#190977442
C2×#190977422
C2×#190977412
C2×#190977392
C2×#190977332
C2×C4×C22⋊D1572
C2×C22⋊C4×D1572
C2×#123×C1572
C2×#538157