Finite Group Databasebeta

← C2×#23757972 #34113658

Subgroups — C2×#23757972 #34113658

Order 20096.

1252 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23757972C2116yesmaximal, direct product factor
2C2×#19097765C211yesmaximal, characteristic, complemented
3C2×#19097764C211yesmaximal, characteristic, complemented
4C2×#19097761C211yesmaximal, characteristic, complemented
5C2×#19097756C211yesmaximal, characteristic, complemented
6C2×#19097754C211yesmaximal, characteristic, complemented
7C2×#19097745C211yesmaximal, characteristic, complemented
8C2×#19097744C211yesmaximal, characteristic, complemented
9C2×#19097743C211yesmaximal, characteristic, complemented
10C2×#19097743C211yesmaximal, characteristic, complemented
11C2×C4×C22⋊D157C211yesmaximal, characteristic, complemented
12C2×#19097731C211yesmaximal, characteristic, complemented
13C2×#19097731C211yesmaximal, characteristic, complemented
14C2×#19097728C211yesmaximal, characteristic, complemented
15C22×#14578933C211yesmaximal, characteristic, complemented
16C2×C4×D4×C157C211yesmaximal, characteristic, complemented
17C22×D628C2211yescharacteristic
18C22×C22⋊D157C2211yescharacteristic
19C22×C22⋊D157C2211yescharacteristic
20C22×C22⋊D157C2211yescharacteristic
21C22×C22⋊D157C2211yescharacteristic
22C23×Dic157C2211yescharacteristic
23C23×Dic157C2211yescharacteristic
24C22×C4×D157C2211yescharacteristic
25C22×C4×D157C2211yescharacteristic
26#19097765C2218yescomplemented
27#19097764C2218yescomplemented
28#19097761C2218yescomplemented
29#19097756C2218yescomplemented
30#19097754C2218yescomplemented
31#19097745C2218yescomplemented
32#19097744C2218yescomplemented
33#19097743C2218yescomplemented
34#19097743C2218yescomplemented
35C4×C22⋊D157C2218yescomplemented
36#19097731C2218yescomplemented
37#19097731C2218yescomplemented
38#19097728C2218yescomplemented
39C2×C4⋊Dic157C2211yescharacteristic
40C2×#14578933C2211yescharacteristic
41C2×#14578933C2211yescharacteristic
42C2×#14578933C2211yescharacteristic
43C2×#14578933C2211yescharacteristic
44C2×#14578933C2211yescharacteristic
45C2×#14578933C2211yescharacteristic
46C2×#14578933C2211yescharacteristic
47C2×#14578933C2211yescharacteristic
48C2×#14578933C2218yescomplemented
49C2×C22⋊Dic157C2211yescharacteristic
50C2×C22⋊Dic157C2211yescharacteristic
51C2×#14578931C2211yescharacteristic
52C2×#14578931C2211yescharacteristic
53C2×#14578931C2211yescharacteristic
54C2×#14578931C2211yescharacteristic
55C2×#14578931C2211yescharacteristic
56C2×#14578931C2211yescharacteristic
57C2×C4×Dic157C2211yescharacteristic
58C2×C4×Dic157C2211yescharacteristic
59C22×D4×C157C2211yescharacteristic
60C23×C4×C157C2211yescharacteristic
61C23×C4×C157C2211yescharacteristic
62C4×D4×C157C2218yescomplemented
63C2×C4⋊C4×C157C2211yescharacteristic
64C2×C22⋊C4×C157C2211yescharacteristic
65C2×C22⋊C4×C157C2211yescharacteristic
66C2×C42×C157C2211yescharacteristic
67C23×D157C2311yescharacteristic
68C23×D157C2311yescharacteristic
69C2×D628—42no—
70C2×D628C2314yes—
71C2×C22⋊D157—24no—
72C2×C22⋊D157—24no—
73C2×C22⋊D157—24no—
74C2×C22⋊D157—42no—
75C2×C22⋊D157C2314yes—
76C2×C22⋊D157C2314yes—
77C2×C22⋊D157C2314yes—
78C2×C22⋊D157C2314yes—
79C22×Dic157—24no—
80C22×Dic157—42no—
81C22×Dic157C2311yescharacteristic
82C22×Dic157C2311yescharacteristic
83C22×Dic157C2311yescharacteristic
84C22×Dic157C2311yescharacteristic
85C22×Dic157C2311yescharacteristic
86C22×Dic157C2314yes—
87C22×Dic157C2314yes—
88C22×Dic157C2311yescharacteristic
89C2×C4×D157—24no—
90C2×C4×D157—42no—
91C2×C4×D157C2314yes—
92C2×C4×D157C2314yes—
93C4⋊Dic157C2314yes—
94#14578933C2314yes—
95#14578933C2314yes—
96#14578933C2314yes—
97#14578933C2314yes—
98#14578933C2314yes—
99#14578933C2314yes—
100#14578933C2314yes—
101#14578933C2314yes—
102C22⋊Dic157C2314yes—
103C22⋊Dic157C2314yes—
104#14578931C2314yes—
105#14578931C2314yes—
106#14578931C2314yes—
107#14578931C2314yes—
108#14578931C2314yes—
109#14578931C2314yes—
110C4×Dic157C2314yes—
111C4×Dic157C2314yes—
112C24×C157C2311yescharacteristic
113C24×C157C2311yescharacteristic
114C2×D4×C157—24no—
115C2×D4×C157C2314yes—
116C22×C4×C157—42no—
117C22×C4×C157—42no—
118C22×C4×C157C2311yescharacteristic
119C22×C4×C157C2311yescharacteristic
120C22×C4×C157C2311yescharacteristic
121C22×C4×C157C2311yescharacteristic
122C22×C4×C157C2311yescharacteristic
123C22×C4×C157C2314yes—
124C22×C4×C157C2314yes—
125C4⋊C4×C157C2314yes—
126C22⋊C4×C157C2314yes—
127C22⋊C4×C157C2314yes—
128C42×C157C2314yes—
129C22×D157—22no—
130C22×D157—22no—
131C22×D157—22no—
132C22×D157—22no—
133C22×D157—22no—
134C22×D157—22no—
135C22×D157C2412yes—
136C22×D157C2412yes—
137D628—44no—
138C22⋊D157—28no—
139C22⋊D157—28no—
140C22⋊D157—28no—
141C22⋊D157—44no—
142C2×Dic157—22no—
143C2×Dic157—22no—
144C2×Dic157—22no—
145C2×Dic157—22no—
146C2×Dic157—22no—
147C2×Dic157—28no—
148C2×Dic157—44no—
149C2×Dic157#4114yes—
150C2×Dic157C2412yes—
151C2×Dic157C2412yes—
152C2×Dic157C2412yes—
153C2×Dic157C2412yes—
154C2×Dic157C2412yes—
155C2×Dic157C2412yes—
156C4×D157—28no—
157C4×D157—44no—
158C23×C157—22no—
159C23×C157—22no—
160C23×C157—22no—
161C23×C157—22no—
162C23×C157—22no—
163C23×C157#4114yes—
164C23×C157C2412yes—
165C23×C157C2412yes—
166C23×C157C2411yescharacteristic
167D4×C157—28no—
168C2×C4×C157—22no—
169C2×C4×C157—22no—
170C2×C4×C157—22no—
171C2×C4×C157—22no—
172C2×C4×C157—22no—
173C2×C4×C157—44no—
174C2×C4×C157—44no—
175C2×C4×C157C2412yes—
176C2×C4×C157C2412yes—
177C2×C4×C157C2412yes—
178C2×C4×C157C2412yes—
179C2×C4×C157C2412yes—
180C22×C157—28no—
181C22×C157—22no—
182C22×C157—22no—
183C22×C157—22no—
184C22×C157—22no—
185C22×C157—22no—
186C22×C157—44no—
187C22×C157C2×#4112yes—
188C22×C157C2×#4114yes—
189C22×C157C2×#4112yes—
190C22×C157H(5,𝔽2)12yes—
191C22×C157C2511yescharacteristic
192C2×D157—22no—
193C2×D157—22no—
194C2×D157—22no—
195C2×D157—22no—
196C2×D157—22no—
197C2×D157—22no—
198C2×D157—44no—
199C2×D157—44no—
200C4×C157—22no—
201C4×C157—22no—
202C4×C157—22no—
203C4×C157—22no—
204C4×C157—22no—
205Dic157—22no—
206Dic157—22no—
207Dic157—22no—
208Dic157—22no—
209Dic157—22no—
210Dic157C2×#4114yes—
211C2×C157—24no—
212C2×C157—42no—
213C2×C157#55314yescomplemented
214C2×C157C22×#4111yescharacteristic
215C2×C157C22×#4111yescharacteristic
216C2×C157C2×H(5,𝔽2)11yescharacteristic
217D157—42no—
218D157—42no—
219C157C2×#55311yescharacteristic, complemented, Sylow
220C2×#553—1571nomaximal, Sylow
221#553—15716no—
222C2×#126—1571no—
223C2×#126—1571no—
224C2×#125—1571no—
225C2×#123—1571no—
226C2×#123—1571no—
227C2×#121—1571no—
228C2×#121—1571no—
229C2×#121—1571no—
230C2×#121—1571no—
231C2×C4×D4—1571no—
232C2×C4×D4—1571no—
233C2×C4×D4—1571no—
234C2×C4×D4D15711yescharacteristic, complemented
235C2×#117—1571no—
236C22×C4⋊C4—1571no—
237C22×D4—1571no—
238C22×D4—1571no—
239C22×D4—1571no—
240C22×D4—1571no—
241C22×D4—1571no—
242C22×D4C2×D15711yescharacteristic
243C23×C4—1571no—
244C23×C4—1571no—
245C23×C4—1571no—
246C23×C4—1571no—
247C23×C4C2×D15711yescharacteristic
248C23×C4C2×D15711yescharacteristic
249#126—1578no—
250#126—1578no—
251#125—1578no—
252#123—1578no—
253#123—1578no—
254#121—1578no—
255#121—1578no—
256#121—1578no—
257#121—1578no—
258C4×D4—1578no—
259C4×D4—1578no—
260C4×D4—1578no—
261C4×D4C2×D15718yescomplemented
262#117—1578no—
263C2×C4⋊C4—1571no—
264C2×C4⋊C4—1571no—
265C2×C4⋊C4—1571no—
266C2×C4⋊C4—1571no—
267C2×C4⋊C4—1578no—
268C2×C4⋊C4—1571no—
269C2×C4⋊C4—1571no—
270C2×C4⋊C4—1571no—
271C2×C4⋊C4—1571no—
272C2×C4⋊C4—1571no—
273C2×C4⋊C4C2×D15711yescharacteristic
274C2×C22⋊C4—1571no—
275C2×C22⋊C4—1571no—
276C2×C22⋊C4—1571no—
277C2×C22⋊C4—1571no—
278C2×C22⋊C4—1571no—
279C2×C22⋊C4—1571no—
280C2×C22⋊C4—1571no—
281C2×C22⋊C4—1571no—
282C2×C22⋊C4C2×D15711yescharacteristic
283C2×C22⋊C4C2×D15711yescharacteristic
284C2×C42—1571no—
285C2×C42—1571no—
286C2×C42C2×D15711yescharacteristic
287C24—1571no—
288C24—1571no—
289C24C22×D15711yescharacteristic
290C24C22×D15711yescharacteristic
291C2×D4—24no—
292C2×D4—3144no—
293C2×D4—3144no—
294C2×D4—3144no—
295C2×D4—6282no—
296C2×D4—6282no—
297C2×D4—1574no—
298C2×D4—1574no—
299C2×D4—1574no—
300C2×D4—1574no—
301C2×D4—1574no—
302C2×D4C22×D15714yes—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—3144no—
306C22×C4—3144no—
307C22×C4—6282no—
308C22×C4—6282no—
309C22×C4—1574no—
310C22×C4—1571no—
311C22×C4—1571no—
312C22×C4—1571no—
313C22×C4—1571no—
314C22×C4—1571no—
315C22×C4—1574no—
316C22×C4—1574no—
317C22×C4—1574no—
318C22×C4—1571no—
319C22×C4C22×D15711yescharacteristic
320C22×C4C22×D15711yescharacteristic
321C22×C4C22×D15711yescharacteristic
322C22×C4C22×D15711yescharacteristic
323C22×C4C22×D15711yescharacteristic
324C22×C4C22×D15714yes—
325C22×C4C22×D15714yes—
326C4⋊C4—1574no—
327C4⋊C4—1574no—
328C4⋊C4—1574no—
329C4⋊C4—1574no—
330C4⋊C4—1574no—
331C4⋊C4—1574no—
332C4⋊C4—1574no—
333C4⋊C4—1574no—
334C4⋊C4—1574no—
335C4⋊C4C22×D15714yes—
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⋊C4C22×D15714yes—
345C22⋊C4C22×D15714yes—
346C42—1574no—
347C42—1574no—
348C42C22×D15714yes—
349C23—22no—
350C23—22no—
351C23—22no—
352C23—22no—
353C23—22no—
354C23—3142no—
355C23—3142no—
356C23—3142no—
357C23—3142no—
358C23—3142no—
359C23—3142no—
360C23—1572no—
361C23—1572no—
362C23#1457895414yes—
363C23C23×D15712yes—
364C23C23×D15712yes—
365C23C23×D15711yescharacteristic
366D4—28no—
367D4—3148no—
368D4—3148no—
369D4—3148no—
370D4—6284no—
371D4—6284no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—3148no—
380C2×C4—3142no—
381C2×C4—3142no—
382C2×C4—3142no—
383C2×C4—3142no—
384C2×C4—3142no—
385C2×C4—3148no—
386C2×C4—6284no—
387C2×C4—6284no—
388C2×C4—1572no—
389C2×C4—1572no—
390C2×C4—1572no—
391C2×C4—1572no—
392C2×C4—1572no—
393C2×C4—1574no—
394C2×C4—1572no—
395C2×C4C23×D15712yes—
396C2×C4C23×D15712yes—
397C2×C4C23×D15712yes—
398C2×C4C23×D15712yes—
399C2×C4C23×D15712yes—
400C22—22no—
401C22—22no—
402C22—28no—
403C22—22no—
404C22—22no—
405C22—22no—
406C22—44no—
407C22—3142no—
408C22—3142no—
409C22—3142no—
410C22—3142no—
411C22—3142no—
412C22—3142no—
413C22—6284no—
414C22—6284no—
415C22#41×D15712yes—
416C22C2×#1457895414yes—
417C22C2×#1457895412yes—
418C22#1909781612yes—
419C22C24×D15711yescharacteristic
420C4—22no—
421C4—22no—
422C4—22no—
423C4—22no—
424C4—22no—
425C4—3142no—
426C4—3142no—
427C4—3142no—
428C4—3142no—
429C4—3142no—
430C4—1574no—
431C2—24no—
432C2—42no—
433C2—6282no—
434C2—6282no—
435C2#2375797214yesdirect product factor
436C2C2×#41×D15711yescharacteristic
437C2C22×#1457895411yescharacteristic
438C2C2×#1909781611yescharacteristic
subgroupG/H
centerC23C23×D157
commutator, cosocleC22×C157C25
FrattiniC22C24×D157
FittingC2×C4×D4×C157C2
socleC23×C157C24
hypercenterC2×C4×D4D157
hypocenterC157C2×#553
LCSG ▷ C22×C157 ▷ C157
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. #23757972C2direct (N × H)
2. C2×#19097765C2semidirect
3. C2×#19097764C2semidirect
4. C2×#19097761C2semidirect
5. C2×#19097756C2semidirect
6. C2×#19097754C2semidirect
7. C2×#19097745C2semidirect
8. C2×#19097744C2semidirect
9. C2×#19097743C2semidirect
10. C2×#19097743C2semidirect
11. C2×C4×C22⋊D157C2semidirect
12. C2×#19097731C2semidirect
13. C2×#19097731C2semidirect
14. C2×#19097728C2semidirect
15. C22×#14578933C2semidirect
16. C2×C4×D4×C157C2semidirect
26. #19097765C22semidirect
27. #19097764C22semidirect
28. #19097761C22semidirect
29. #19097756C22semidirect
30. #19097754C22semidirect
31. #19097745C22semidirect
32. #19097744C22semidirect
33. #19097743C22semidirect
34. #19097743C22semidirect
35. C4×C22⋊D157C22semidirect
36. #19097731C22semidirect
37. #19097731C22semidirect
38. #19097728C22semidirect
48. C2×#14578933C22semidirect
62. C4×D4×C157C22semidirect
213. C2×C157#553semidirect
219. C157C2×#553semidirect
234. C2×C4×D4D157semidirect
261. C4×D4C2×D157semidirect
435. C2#23757972direct (N × H)

Maximal subgroups

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

subgroupindex
#237579722
C2×#190977652
C2×#190977642
C2×#190977612
C2×#190977562
C2×#190977542
C2×#190977452
C2×#190977442
C2×#190977432
C2×C4×C22⋊D1572
C2×#190977312
C2×#190977282
C22×#145789332
C2×C4×D4×C1572
C2×#553157