Finite Group Databasebeta

← C2×#23757950 #34113634

Subgroups — C2×#23757950 #34113634

Order 20096.

1328 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23757950C2116yesmaximal, direct product factor
2C2×#19097766C211yesmaximal, characteristic, complemented
3C2×#19097766C211yesmaximal, characteristic, complemented
4C2×#19097760C211yesmaximal, characteristic, complemented
5C2×#19097756C211yesmaximal, characteristic, complemented
6C2×#19097756C211yesmaximal, characteristic, complemented
7C2×#19097755C211yesmaximal, characteristic, complemented
8C2×#19097747C211yesmaximal, characteristic, complemented
9C2×#19097741C211yesmaximal, characteristic
10C2×#19097733C211yesmaximal, characteristic, complemented
11C2×C4×C22⋊D157C211yesmaximal, characteristic, complemented
12C2×#19097729C211yesmaximal, characteristic, complemented
13C2×#19097727C211yesmaximal, characteristic, complemented
14C22×#14578931C211yesmaximal, characteristic, complemented
15C2×C22⋊C4×D157C211yesmaximal, characteristic, complemented
16C2×C22⋊Q8×C157C211yesmaximal, characteristic, complemented
17C24×D157C2211yescharacteristic
18C22×D628C2211yescharacteristic
19C22×D628C2211yescharacteristic
20C22×C22⋊D157C2211yescharacteristic
21C23×Dic157C2211yescharacteristic
22C22×C4×D157C2211yescharacteristic
23C22×C4×D157C2211yescharacteristic
24C22×C4×D157C2211yescharacteristic, complemented
25#19097766C2218yescomplemented
26#19097766C2218yescomplemented
27#19097760C2218yescomplemented
28#19097756C2218yescomplemented
29#19097756C2218yescomplemented
30#19097755C2218yescomplemented
31#19097747C2218yescomplemented
32#19097741C2218yes—
33#19097733C2218yescomplemented
34C4×C22⋊D157C2218yescomplemented
35#19097729C2218yescomplemented
36#19097727C2218yescomplemented
37C2×C4⋊Dic157C2211yescharacteristic, complemented
38C2×C4⋊Dic157C2211yescharacteristic, complemented
39C2×#14578933C2211yescharacteristic, complemented
40C2×#14578933C2211yescharacteristic, complemented
41C2×#14578933C2211yescharacteristic, complemented
42C2×C22⋊Dic157C2211yescharacteristic
43C2×#14578931C2211yescharacteristic
44C2×#14578931C2211yescharacteristic
45C2×#14578931C2211yescharacteristic
46C2×#14578931C2211yescharacteristic
47C2×#14578931C2211yescharacteristic
48C2×#14578931C2211yescharacteristic
49C2×#14578931C2211yescharacteristic
50C2×#14578931C2211yescharacteristic
51C2×#14578931C2218yescomplemented
52C2×#14578931C2211yescharacteristic, complemented
53C2×#14578931C2211yescharacteristic, complemented
54C2×#14578931C2211yescharacteristic, complemented
55C22⋊C4×D157C2218yescomplemented
56C2×C4×Dic157C2211yescharacteristic, complemented
57C2×C4×Dic157C2211yescharacteristic, complemented
58C2×C4×Dic157C2211yescharacteristic, complemented
59C22×Q8×C157C2211yescharacteristic, complemented
60C23×C4×C157C2211yescharacteristic
61C22⋊Q8×C157C2218yescomplemented
62C2×C4⋊C4×C157C2211yescharacteristic, complemented
63C2×C4⋊C4×C157C2211yescharacteristic, complemented
64C2×C4⋊C4×C157C2211yescharacteristic, complemented
65C2×C22⋊C4×C157C2211yescharacteristic
66C2×C22⋊C4×C157C2211yescharacteristic
67C23×D157—24no—
68C23×D157—24no—
69C23×D157—42no—
70C23×D157C2311yescharacteristic
71C23×D157C2311yescharacteristic
72C23×D157C2314yes—
73C23×D157C2311yescharacteristic
74C2×D628—24no—
75C2×D628—42no—
76C2×D628C2314yes—
77C2×D628C2314yes—
78C2×C22⋊D157—24no—
79C2×C22⋊D157C2314yes—
80C22×Dic157—24no—
81C22×Dic157C2311yescharacteristic
82C22×Dic157C2311yescharacteristic
83C22×Dic157C2311yescharacteristic
84C22×Dic157C2311yescharacteristic
85C22×Dic157C2311yescharacteristic
86C22×Dic157C2314yes—
87C2×C4×D157—24no—
88C2×C4×D157—42no—
89C2×C4×D157—42no—
90C2×C4×D157C2314yes—
91C2×C4×D157C2314yes—
92C2×C4×D157C2314yescomplemented
93C4⋊Dic157C2314yescomplemented
94C4⋊Dic157C2314yescomplemented
95#14578933C2314yescomplemented
96#14578933C2314yescomplemented
97#14578933C2314yescomplemented
98C22⋊Dic157C2314yes—
99#14578931C2314yes—
100#14578931C2314yes—
101#14578931C2314yes—
102#14578931C2314yes—
103#14578931C2314yes—
104#14578931C2314yes—
105#14578931C2314yes—
106#14578931C2314yes—
107#14578931C2314yescomplemented
108#14578931C2314yescomplemented
109#14578931C2314yescomplemented
110C4×Dic157C2314yescomplemented
111C4×Dic157C2314yescomplemented
112C4×Dic157C2314yescomplemented
113C24×C157C2311yescharacteristic
114C2×Q8×C157—42no—
115C2×Q8×C157C2314yescomplemented
116C22×C4×C157—42no—
117C22×C4×C157C2311yescharacteristic
118C22×C4×C157C2311yescharacteristic
119C22×C4×C157C2311yescharacteristic
120C22×C4×C157C2311yescharacteristic
121C22×C4×C157C2311yescharacteristic
122C22×C4×C157C2311yescharacteristic
123C22×C4×C157C2314yes—
124C4⋊C4×C157C2314yescomplemented
125C4⋊C4×C157C2314yescomplemented
126C4⋊C4×C157C2314yescomplemented
127C22⋊C4×C157C2314yes—
128C22⋊C4×C157C2314yes—
129C22×D157—22no—
130C22×D157—22no—
131C22×D157—22no—
132C22×D157—22no—
133C22×D157—22no—
134C22×D157—28no—
135C22×D157—22no—
136C22×D157—22no—
137C22×D157—22no—
138C22×D157—28no—
139C22×D157—44no—
140C22×D157—416no—
141C22×D157#4114yes—
142C22×D157C2412yes—
143C22×D157C2412yes—
144C22×D157C2412yes—
145D628—28no—
146D628—44no—
147C22⋊D157—28no—
148C2×Dic157—22no—
149C2×Dic157—28no—
150C2×Dic157—22no—
151C2×Dic157—22no—
152C2×Dic157—22no—
153C2×Dic157—22no—
154C2×Dic157C2412yes—
155C2×Dic157C2412yes—
156C2×Dic157C2412yes—
157C2×Dic157C2412yes—
158C2×Dic157C2412yes—
159C4×D157—28no—
160C4×D157—44no—
161C4×D157—44no—
162C23×C157—22no—
163C23×C157—22no—
164C23×C157#4114yes—
165C23×C157C2412yes—
166C23×C157C2411yescharacteristic
167Q8×C157—44no—
168C2×C4×C157—22no—
169C2×C4×C157—22no—
170C2×C4×C157—22no—
171C2×C4×C157—22no—
172C2×C4×C157—22no—
173C2×C4×C157—22no—
174C2×C4×C157—44no—
175C2×C4×C157C2412yes—
176C2×C4×C157C2412yes—
177C2×C4×C157C2412yes—
178C2×C4×C157C2412yes—
179C2×C4×C157C2412yes—
180C2×C4×C157C2412yes—
181C22×C157—28no—
182C22×C157—22no—
183C22×C157—22no—
184C22×C157C2×#4112yes—
185C22×C157C2×#4114yes—
186C22×C157C2×#4112yes—
187C22×C157H(5,𝔽2)12yes—
188C22×C157C2511yescharacteristic
189C2×D157—28no—
190C2×D157—22no—
191C2×D157—22no—
192C2×D157—22no—
193C2×D157—22no—
194C2×D157—22no—
195C2×D157—22no—
196C2×D157—22no—
197C2×D157—22no—
198C2×D157—416no—
199C2×D157—44no—
200C2×D157—44no—
201C2×D157C2×#4114yes—
202C4×C157—22no—
203C4×C157—22no—
204C4×C157—22no—
205C4×C157—22no—
206C4×C157—22no—
207C4×C157—22no—
208Dic157—22no—
209Dic157—22no—
210Dic157—22no—
211Dic157—22no—
212Dic157—22no—
213C2×C157—24no—
214C2×C157#55114yescomplemented
215C2×C157C22×#4111yescharacteristic
216C2×C157C22×#4111yescharacteristic
217C2×C157C2×H(5,𝔽2)11yescharacteristic
218D157—24no—
219D157—42no—
220D157—42no—
221C157C2×#55111yescharacteristic, complemented, Sylow
222C2×#551—1571nomaximal, Sylow
223#551—15716no—
224C2×#126—1571no—
225C2×#126—1571no—
226C2×#124—1571no—
227C2×#123—1571no—
228C2×#123—1571no—
229C2×#123—1571no—
230C2×C22⋊Q8—1571no—
231C2×C22⋊Q8D15711yescharacteristic, complemented
232C2×#120—1571no—
233C2×C4×D4—1571no—
234C2×C4×D4—1571no—
235C2×#117—1571no—
236C2×#117—1571no—
237C22×C22⋊C4—1571no—
238C22×C22⋊C4—1571no—
239C25—1571no—
240C22×Q8C2×D15711yescharacteristic, complemented
241C22×D4—1571no—
242C22×D4—1571no—
243C22×D4—1571no—
244C23×C4—1571no—
245C23×C4—1571no—
246C23×C4—1571no—
247C23×C4—1571no—
248C23×C4C2×D15711yescharacteristic
249#126—1578no—
250#126—1578no—
251#124—1578no—
252#123—1578no—
253#123—1578no—
254#123—1578no—
255C22⋊Q8—1578no—
256C22⋊Q8C2×D15718yescomplemented
257#120—1578no—
258C4×D4—1578no—
259C4×D4—1578no—
260#117—1578no—
261#117—1578no—
262C2×C4⋊C4—1571no—
263C2×C4⋊C4—1571no—
264C2×C4⋊C4—1571no—
265C2×C4⋊C4—1571no—
266C2×C4⋊C4—1571no—
267C2×C4⋊C4C2×D15711yescharacteristic, complemented
268C2×C4⋊C4C2×D15711yescharacteristic, complemented
269C2×C4⋊C4C2×D15711yescharacteristic, complemented
270C2×C22⋊C4—1571no—
271C2×C22⋊C4—1571no—
272C2×C22⋊C4—1571no—
273C2×C22⋊C4—1571no—
274C2×C22⋊C4—1571no—
275C2×C22⋊C4—1571no—
276C2×C22⋊C4—1571no—
277C2×C22⋊C4—1578no—
278C2×C22⋊C4—1571no—
279C2×C22⋊C4—1571no—
280C2×C22⋊C4—1571no—
281C2×C22⋊C4—1571no—
282C2×C22⋊C4—1571no—
283C2×C22⋊C4—1578no—
284C2×C22⋊C4C2×D15711yescharacteristic
285C2×C22⋊C4C2×D15711yescharacteristic
286C2×C42—1571no—
287C2×C42—1571no—
288C2×C42—1571no—
289C24—3144no—
290C24—3144no—
291C24—6282no—
292C24—1571no—
293C24—1571no—
294C24—1574no—
295C24—1571no—
296C24C22×D15711yescharacteristic
297C2×Q8—42no—
298C2×Q8C22×D15714yescomplemented
299C2×D4—3144no—
300C2×D4—3144no—
301C2×D4—6282no—
302C2×D4—1574no—
303C2×D4—1574no—
304C2×D4—1574no—
305C22×C4—42no—
306C22×C4—3144no—
307C22×C4—3144no—
308C22×C4—6282no—
309C22×C4—6282no—
310C22×C4—1574no—
311C22×C4—1571no—
312C22×C4—1571no—
313C22×C4—1571no—
314C22×C4—1571no—
315C22×C4—1571no—
316C22×C4—1574no—
317C22×C4—1574no—
318C22×C4—1574no—
319C22×C4C22×D15711yescharacteristic
320C22×C4C22×D15711yescharacteristic
321C22×C4C22×D15711yescharacteristic
322C22×C4C22×D15711yescharacteristic
323C22×C4C22×D15711yescharacteristic
324C22×C4C22×D15711yescharacteristic
325C22×C4C22×D15714yes—
326C4⋊C4—1574no—
327C4⋊C4—1574no—
328C4⋊C4—1574no—
329C4⋊C4—1574no—
330C4⋊C4—1574no—
331C4⋊C4C22×D15714yescomplemented
332C4⋊C4C22×D15714yescomplemented
333C4⋊C4C22×D15714yescomplemented
334C22⋊C4—1574no—
335C22⋊C4—1574no—
336C22⋊C4—1574no—
337C22⋊C4—1574no—
338C22⋊C4—1574no—
339C22⋊C4—1574no—
340C22⋊C4—1574no—
341C22⋊C4—1574no—
342C22⋊C4—1574no—
343C22⋊C4—1574no—
344C22⋊C4—1574no—
345C22⋊C4—1574no—
346C22⋊C4C22×D15714yes—
347C22⋊C4C22×D15714yes—
348C42—1574no—
349C42—1574no—
350C42—1574no—
351C23—22no—
352C23—22no—
353C23—3148no—
354C23—3142no—
355C23—3142no—
356C23—3142no—
357C23—3142no—
358C23—3142no—
359C23—3148no—
360C23—3142no—
361C23—3142no—
362C23—3142no—
363C23—6284no—
364C23—62816no—
365C23—1572no—
366C23—1572no—
367C23—1574no—
368C23—1572no—
369C23#1457895514yes—
370C23C23×D15712yes—
371C23C23×D15711yescharacteristic
372Q8—44no—
373D4—3148no—
374D4—3148no—
375D4—6284no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—44no—
383C2×C4—3148no—
384C2×C4—3142no—
385C2×C4—3148no—
386C2×C4—3142no—
387C2×C4—3142no—
388C2×C4—3142no—
389C2×C4—3142no—
390C2×C4—6284no—
391C2×C4—6284no—
392C2×C4—1572no—
393C2×C4—1572no—
394C2×C4—1572no—
395C2×C4—1572no—
396C2×C4—1572no—
397C2×C4C23×D15712yes—
398C2×C4C23×D15712yes—
399C2×C4C23×D15712yes—
400C2×C4C23×D15712yes—
401C2×C4C23×D15712yes—
402C2×C4C23×D15712yes—
403C22—22no—
404C22—22no—
405C22—28no—
406C22—3142no—
407C22—3142no—
408C22—3148no—
409C22—3142no—
410C22—3142no—
411C22—3142no—
412C22—3142no—
413C22—3142no—
414C22—3142no—
415C22—62816no—
416C22—6284no—
417C22—6284no—
418C22—1574no—
419C22#41×D15712yes—
420C22C2×#1457895514yes—
421C22C2×#1457895512yes—
422C22#1909781612yes—
423C22C24×D15711yescharacteristic
424C4—22no—
425C4—22no—
426C4—22no—
427C4—22no—
428C4—22no—
429C4—22no—
430C4—3142no—
431C4—3142no—
432C4—3142no—
433C4—3142no—
434C4—3142no—
435C2—24no—
436C2—3144no—
437C2—6282no—
438C2—6282no—
439C2#2375795014yesdirect product factor
440C2C2×#41×D15711yescharacteristic
441C2C22×#1457895511yescharacteristic
442C2C2×#1909781611yescharacteristic
subgroupG/H
centerC23C23×D157
commutator, cosocleC22×C157C25
FrattiniC22C24×D157
FittingC2×C22⋊Q8×C157C2
socleC23×C157C24
prime radicalC2×#19097741C2
hypercenterC2×C22⋊Q8D157
hypocenterC157C2×#551
LCSG ▷ C22×C157 ▷ C157
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. #23757950C2direct (N × H)
2. C2×#19097766C2semidirect
3. C2×#19097766C2semidirect
4. C2×#19097760C2semidirect
5. C2×#19097756C2semidirect
6. C2×#19097756C2semidirect
7. C2×#19097755C2semidirect
8. C2×#19097747C2semidirect
10. C2×#19097733C2semidirect
11. C2×C4×C22⋊D157C2semidirect
12. C2×#19097729C2semidirect
13. C2×#19097727C2semidirect
14. C22×#14578931C2semidirect
15. C2×C22⋊C4×D157C2semidirect
16. C2×C22⋊Q8×C157C2semidirect
24. C22×C4×D157C22semidirect
25. #19097766C22semidirect
26. #19097766C22semidirect
27. #19097760C22semidirect
28. #19097756C22semidirect
29. #19097756C22semidirect
30. #19097755C22semidirect
31. #19097747C22semidirect
33. #19097733C22semidirect
34. C4×C22⋊D157C22semidirect
35. #19097729C22semidirect
36. #19097727C22semidirect
37. C2×C4⋊Dic157C22semidirect
38. C2×C4⋊Dic157C22semidirect
39. C2×#14578933C22semidirect
40. C2×#14578933C22semidirect
41. C2×#14578933C22semidirect
51. C2×#14578931C22semidirect
52. C2×#14578931C22semidirect
53. C2×#14578931C22semidirect
54. C2×#14578931C22semidirect
55. C22⋊C4×D157C22semidirect
56. C2×C4×Dic157C22semidirect
57. C2×C4×Dic157C22semidirect
58. C2×C4×Dic157C22semidirect
59. C22×Q8×C157C22semidirect
61. C22⋊Q8×C157C22semidirect
62. C2×C4⋊C4×C157C22semidirect
63. C2×C4⋊C4×C157C22semidirect
64. C2×C4⋊C4×C157C22semidirect
92. C2×C4×D157C23semidirect
93. C4⋊Dic157C23semidirect
94. C4⋊Dic157C23semidirect
95. #14578933C23semidirect
96. #14578933C23semidirect
97. #14578933C23semidirect
107. #14578931C23semidirect
108. #14578931C23semidirect
109. #14578931C23semidirect
110. C4×Dic157C23semidirect
111. C4×Dic157C23semidirect
112. C4×Dic157C23semidirect
115. C2×Q8×C157C23semidirect
124. C4⋊C4×C157C23semidirect
125. C4⋊C4×C157C23semidirect
126. C4⋊C4×C157C23semidirect
214. C2×C157#551semidirect
221. C157C2×#551semidirect
231. C2×C22⋊Q8D157semidirect
240. C22×Q8C2×D157semidirect
256. C22⋊Q8C2×D157semidirect
267. C2×C4⋊C4C2×D157semidirect
268. C2×C4⋊C4C2×D157semidirect
269. C2×C4⋊C4C2×D157semidirect
298. C2×Q8C22×D157semidirect
331. C4⋊C4C22×D157semidirect
332. C4⋊C4C22×D157semidirect
333. C4⋊C4C22×D157semidirect
439. C2#23757950direct (N × H)

Maximal subgroups

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

subgroupindex
#237579502
C2×#190977662
C2×#190977602
C2×#190977562
C2×#190977552
C2×#190977472
C2×#190977412
C2×#190977332
C2×C4×C22⋊D1572
C2×#190977292
C2×#190977272
C22×#145789312
C2×C22⋊C4×D1572
C2×C22⋊Q8×C1572
C2×#551157