Finite Group Databasebeta

← C2×#23758019 #34114261

Subgroups — C2×#23758019 #34114261

Order 20096.

1152 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#23758019C2116yesmaximal, direct product factor
2C2×#19097761C211yesmaximal, characteristic, complemented
3C2×#19097760C211yesmaximal, characteristic, complemented
4C2×#19097757C211yesmaximal, characteristic, complemented
5C2×#19097756C211yesmaximal, characteristic, complemented
6C2×#19097756C211yesmaximal, characteristic, complemented
7C2×#19097754C211yesmaximal, characteristic, complemented
8C2×#19097753C211yesmaximal, characteristic, complemented
9C2×#19097751C211yesmaximal, characteristic, complemented
10C2×#19097749C211yesmaximal, characteristic, complemented
11C2×#19097747C211yesmaximal, characteristic, complemented
12C2×#19097746C211yesmaximal, characteristic, complemented
13C2×#19097745C211yesmaximal, characteristic, complemented
14C2×#19097744C211yesmaximal, characteristic, complemented
15C2×#19097744C211yesmaximal, characteristic, complemented
16C2×C22⋊Q8×C157C211yesmaximal, characteristic, complemented
17C22×Dic314C2211yescharacteristic
18C22×D628C2211yescharacteristic
19C22×D628C2211yescharacteristic
20C22×D628C2211yescharacteristic
21C22×C22⋊D157C2211yescharacteristic
22C22×C22⋊D157C2211yescharacteristic
23C22×C22⋊D157C2211yescharacteristic
24C22×C4×D157C2211yescharacteristic
25C22×C4×D157C2211yescharacteristic
26C22×C4×D157C2211yescharacteristic
27#19097761C2218yescomplemented
28#19097760C2218yescomplemented
29#19097757C2218yescomplemented
30#19097756C2218yescomplemented
31#19097756C2218yescomplemented
32#19097754C2218yescomplemented
33#19097753C2218yescomplemented
34#19097751C2218yescomplemented
35#19097749C2218yescomplemented
36#19097747C2218yescomplemented
37#19097746C2218yescomplemented
38#19097745C2218yescomplemented
39#19097744C2218yescomplemented
40#19097744C2218yescomplemented
41C2×C4⋊Dic157C2211yescharacteristic
42C2×C4⋊Dic157C2211yescharacteristic
43C2×#14578933C2211yescharacteristic
44C2×#14578933C2211yescharacteristic
45C2×#14578933C2211yescharacteristic
46C2×#14578933C2211yescharacteristic
47C2×#14578933C2211yescharacteristic
48C2×C22⋊Dic157C2211yescharacteristic
49C2×#14578931C2211yescharacteristic
50C2×#14578931C2211yescharacteristic
51C2×#14578931C2211yescharacteristic
52C2×#14578931C2211yescharacteristic
53C2×#14578931C2211yescharacteristic
54C2×#14578931C2211yescharacteristic
55C2×#14578931C2211yescharacteristic
56C2×#14578931C2211yescharacteristic
57C2×#14578931C2211yescharacteristic
58C2×C4×Dic157C2211yescharacteristic
59C22×Q8×C157C2211yescharacteristic
60C23×C4×C157C2211yescharacteristic
61C22⋊Q8×C157C2218yescomplemented
62C2×C4⋊C4×C157C2211yescharacteristic
63C2×C4⋊C4×C157C2211yescharacteristic
64C2×C4⋊C4×C157C2211yescharacteristic
65C2×C22⋊C4×C157C2211yescharacteristic
66C2×C22⋊C4×C157C2211yescharacteristic
67C23×D157C2311yescharacteristic
68C23×D157C2311yescharacteristic
69C23×D157C2311yescharacteristic
70C2×Dic314—42no—
71C2×Dic314C2314yes—
72C2×D628—42no—
73C2×D628—42no—
74C2×D628—42no—
75C2×D628C2314yes—
76C2×D628C2314yes—
77C2×D628C2314yes—
78C2×C22⋊D157—42no—
79C2×C22⋊D157—42no—
80C2×C22⋊D157—42no—
81C2×C22⋊D157C2314yes—
82C2×C22⋊D157C2314yes—
83C2×C22⋊D157C2314yes—
84C22×Dic157C2311yescharacteristic
85C22×Dic157C2311yescharacteristic
86C22×Dic157C2311yescharacteristic
87C22×Dic157C2311yescharacteristic
88C22×Dic157C2311yescharacteristic
89C2×C4×D157—42no—
90C2×C4×D157—42no—
91C2×C4×D157—42no—
92C2×C4×D157C2314yes—
93C2×C4×D157C2314yes—
94C2×C4×D157C2314yes—
95C4⋊Dic157C2314yes—
96C4⋊Dic157C2314yes—
97#14578933C2314yes—
98#14578933C2314yes—
99#14578933C2314yes—
100#14578933C2314yes—
101#14578933C2314yes—
102C22⋊Dic157C2314yes—
103#14578931C2314yes—
104#14578931C2314yes—
105#14578931C2314yes—
106#14578931C2314yes—
107#14578931C2314yes—
108#14578931C2314yes—
109#14578931C2314yes—
110#14578931C2314yes—
111#14578931C2314yes—
112C4×Dic157C2314yes—
113C24×C157C2311yescharacteristic
114C2×Q8×C157—42no—
115C2×Q8×C157C2314yes—
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×C157C2314yes—
125C4⋊C4×C157C2314yes—
126C4⋊C4×C157C2314yes—
127C22⋊C4×C157C2314yes—
128C22⋊C4×C157C2314yes—
129C22×D157—22no—
130C22×D157—22no—
131C22×D157—22no—
132C22×D157—22no—
133C22×D157—22no—
134C22×D157—22no—
135C22×D157—22no—
136C22×D157—22no—
137C22×D157—22no—
138C22×D157C2412yes—
139C22×D157C2412yes—
140C22×D157C2412yes—
141Dic314—44no—
142D628—44no—
143D628—44no—
144D628—44no—
145C22⋊D157—44no—
146C22⋊D157—44no—
147C22⋊D157—44no—
148C2×Dic157—22no—
149C2×Dic157—22no—
150C2×Dic157—22no—
151C2×Dic157—22no—
152C2×Dic157—22no—
153C2×Dic157C2412yes—
154C2×Dic157C2412yes—
155C2×Dic157C2412yes—
156C2×Dic157C2412yes—
157C2×Dic157C2412yes—
158C4×D157—44no—
159C4×D157—44no—
160C4×D157—44no—
161C23×C157—22no—
162C23×C157—22no—
163C23×C157—22no—
164C23×C157C2412yes—
165C23×C157C2411yescharacteristic
166Q8×C157—44no—
167C2×C4×C157—22no—
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×C157C2412yes—
175C2×C4×C157C2412yes—
176C2×C4×C157C2412yes—
177C2×C4×C157C2412yes—
178C2×C4×C157C2412yes—
179C2×C4×C157C2412yes—
180C22×C157—22no—
181C22×C157—22no—
182C22×C157—22no—
183C22×C157—44no—
184C22×C157H(5,𝔽2)12yes—
185C22×C157H(5,𝔽2)12yes—
186C22×C1572−1+412yes—
187C22×C157C2511yescharacteristic
188C2×D157—22no—
189C2×D157—22no—
190C2×D157—22no—
191C2×D157—22no—
192C2×D157—22no—
193C2×D157—22no—
194C2×D157—22no—
195C2×D157—22no—
196C2×D157—22no—
197C2×D157—44no—
198C2×D157—44no—
199C2×D157—44no—
200C4×C157—22no—
201C4×C157—22no—
202C4×C157—22no—
203C4×C157—22no—
204C4×C157—22no—
205C4×C157—22no—
206Dic157—22no—
207Dic157—22no—
208Dic157—22no—
209Dic157—22no—
210Dic157—22no—
211C2×C157—42no—
212C2×C157#56214yescomplemented
213C2×C157C2×H(5,𝔽2)11yescharacteristic
214C2×C157C2×H(5,𝔽2)11yescharacteristic
215C2×C157C2×2−1+411yescharacteristic
216D157—42no—
217D157—42no—
218D157—42no—
219C157C2×#56211yescharacteristic, complemented, Sylow
220C2×#562—1571nomaximal, Sylow
221#562—15716no—
222C2×#125—1571no—
223C2×#124—1571no—
224C2×#124—1571no—
225C2×#123—1571no—
226C2×#123—1571no—
227C2×#123—1571no—
228C2×#123—1571no—
229C2×C22⋊Q8—1571no—
230C2×C22⋊Q8—1571no—
231C2×C22⋊Q8—1571no—
232C2×C22⋊Q8D15711yescharacteristic, complemented
233C2×#121—1571no—
234C2×#121—1571no—
235C2×#121—1571no—
236C2×#121—1571no—
237C22×Q8—1571no—
238C22×Q8C2×D15711yescharacteristic
239C22×D4—1571no—
240C22×D4—1571no—
241C22×D4—1571no—
242C22×D4—1571no—
243C22×D4—1571no—
244C22×D4—1571no—
245C23×C4—1571no—
246C23×C4—1571no—
247C23×C4—1571no—
248C23×C4C2×D15711yescharacteristic
249#125—1578no—
250#124—1578no—
251#124—1578no—
252#123—1578no—
253#123—1578no—
254#123—1578no—
255#123—1578no—
256C22⋊Q8—1578no—
257C22⋊Q8—1578no—
258C22⋊Q8—1578no—
259C22⋊Q8C2×D15718yescomplemented
260#121—1578no—
261#121—1578no—
262#121—1578no—
263#121—1578no—
264C2×C4⋊C4—1571no—
265C2×C4⋊C4—1571no—
266C2×C4⋊C4—1571no—
267C2×C4⋊C4—1571no—
268C2×C4⋊C4—1571no—
269C2×C4⋊C4—1571no—
270C2×C4⋊C4—1571no—
271C2×C4⋊C4C2×D15711yescharacteristic
272C2×C4⋊C4C2×D15711yescharacteristic
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⋊C4—1571no—
283C2×C22⋊C4—1571no—
284C2×C22⋊C4C2×D15711yescharacteristic
285C2×C22⋊C4C2×D15711yescharacteristic
286C2×C42—1571no—
287C24—1571no—
288C24—1571no—
289C24—1571no—
290C24C22×D15711yescharacteristic
291C2×Q8—42no—
292C2×Q8—6282no—
293C2×Q8—1574no—
294C2×Q8C22×D15714yes—
295C2×D4—6282no—
296C2×D4—6282no—
297C2×D4—6282no—
298C2×D4—6282no—
299C2×D4—6282no—
300C2×D4—6282no—
301C2×D4—1574no—
302C2×D4—1574no—
303C2×D4—1574no—
304C2×D4—1574no—
305C2×D4—1574no—
306C2×D4—1574no—
307C22×C4—42no—
308C22×C4—6282no—
309C22×C4—6282no—
310C22×C4—6282no—
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⋊C4—1574no—
332C4⋊C4—1574no—
333C4⋊C4C22×D15714yes—
334C4⋊C4C22×D15714yes—
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⋊C4—1574no—
345C22⋊C4—1574no—
346C22⋊C4C22×D15714yes—
347C22⋊C4C22×D15714yes—
348C42—1574no—
349C23—22no—
350C23—22no—
351C23—22no—
352C23—3142no—
353C23—3142no—
354C23—3142no—
355C23—3142no—
356C23—3142no—
357C23—3142no—
358C23—3142no—
359C23—3142no—
360C23—3142no—
361C23—1572no—
362C23—1572no—
363C23—1572no—
364C23C23×D15712yes—
365C23C23×D15711yescharacteristic
366Q8—44no—
367Q8—6284no—
368D4—6284no—
369D4—6284no—
370D4—6284no—
371D4—6284no—
372D4—6284no—
373D4—6284no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—44no—
381C2×C4—3142no—
382C2×C4—3142no—
383C2×C4—3142no—
384C2×C4—3142no—
385C2×C4—3142no—
386C2×C4—6284no—
387C2×C4—6284no—
388C2×C4—6284no—
389C2×C4—1572no—
390C2×C4—1572no—
391C2×C4—1572no—
392C2×C4—1572no—
393C2×C4—1572no—
394C2×C4C23×D15712yes—
395C2×C4C23×D15712yes—
396C2×C4C23×D15712yes—
397C2×C4C23×D15712yes—
398C2×C4C23×D15712yes—
399C2×C4C23×D15712yes—
400C22—22no—
401C22—22no—
402C22—22no—
403C22—44no—
404C22—3142no—
405C22—3142no—
406C22—3142no—
407C22—3142no—
408C22—3142no—
409C22—3142no—
410C22—3142no—
411C22—3142no—
412C22—3142no—
413C22—6284no—
414C22—6284no—
415C22—6284no—
416C22#1909781612yes—
417C22#1909781712yes—
418C22#1909781912yes—
419C22C24×D15711yescharacteristic
420C4—22no—
421C4—22no—
422C4—22no—
423C4—22no—
424C4—22no—
425C4—22no—
426C4—3142no—
427C4—3142no—
428C4—3142no—
429C4—3142no—
430C4—3142no—
431C2—42no—
432C2—6282no—
433C2—6282no—
434C2—6282no—
435C2#2375801914yesdirect product factor
436C2C2×#1909781611yescharacteristic
437C2C2×#1909781711yescharacteristic
438C2C2×#1909781911yescharacteristic
subgroupG/H
centerC23C23×D157
commutator, cosocleC22×C157C25
FrattiniC22C24×D157
FittingC2×C22⋊Q8×C157C2
socleC23×C157C24
hypercenterC2×C22⋊Q8D157
hypocenterC157C2×#562
LCSG ▷ C22×C157 ▷ C157
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. #23758019C2direct (N × H)
2. C2×#19097761C2semidirect
3. C2×#19097760C2semidirect
4. C2×#19097757C2semidirect
5. C2×#19097756C2semidirect
6. C2×#19097756C2semidirect
7. C2×#19097754C2semidirect
8. C2×#19097753C2semidirect
9. C2×#19097751C2semidirect
10. C2×#19097749C2semidirect
11. C2×#19097747C2semidirect
12. C2×#19097746C2semidirect
13. C2×#19097745C2semidirect
14. C2×#19097744C2semidirect
15. C2×#19097744C2semidirect
16. C2×C22⋊Q8×C157C2semidirect
27. #19097761C22semidirect
28. #19097760C22semidirect
29. #19097757C22semidirect
30. #19097756C22semidirect
31. #19097756C22semidirect
32. #19097754C22semidirect
33. #19097753C22semidirect
34. #19097751C22semidirect
35. #19097749C22semidirect
36. #19097747C22semidirect
37. #19097746C22semidirect
38. #19097745C22semidirect
39. #19097744C22semidirect
40. #19097744C22semidirect
61. C22⋊Q8×C157C22semidirect
212. C2×C157#562semidirect
219. C157C2×#562semidirect
232. C2×C22⋊Q8D157semidirect
259. C22⋊Q8C2×D157semidirect
435. C2#23758019direct (N × H)

Maximal subgroups

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

subgroupindex
#237580192
C2×#190977612
C2×#190977602
C2×#190977572
C2×#190977562
C2×#190977542
C2×#190977532
C2×#190977512
C2×#190977492
C2×#190977472
C2×#190977462
C2×#190977452
C2×#190977442
C2×C22⋊Q8×C1572
C2×#562157