Finite Group Databasebeta

← #34109655

Subgroups — #34109655

Order 20096.

490 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19097803C211yesmaximal, characteristic, complemented
2C2×SD16×D157C211yesmaximal, characteristic, complemented
3#23757841C211yesmaximal, characteristic
4#23757749C211yesmaximal, characteristic, complemented
5#23757498C211yesmaximal, characteristic, complemented
6#23757497C211yesmaximal, characteristic, complemented
7#23757492C211yesmaximal, characteristic, complemented
8#23757474C211yesmaximal, characteristic, complemented
9#23757473C211yesmaximal, characteristic, complemented
10#23757472C211yesmaximal, characteristic, complemented
11#23757471C211yesmaximal, characteristic, complemented
12#23757298C211yesmaximal, characteristic, complemented
13#23757276C211yesmaximal, characteristic, complemented
14#23757268C211yesmaximal, characteristic, complemented
15#464×C157C211yesmaximal, characteristic, complemented
16C2×#14578955C2211yescharacteristic
17C2×Q8×D157C2211yescharacteristic
18C2×D4×D157C2211yescharacteristic
19#19097803—24no—
20C2×#14578944C2211yescharacteristic, complemented
21C2×#14578943C2211yescharacteristic
22C2×#14578943C2211yescharacteristic, complemented
23C2×#14578942C2211yescharacteristic
24C2×#14578941C2211yescharacteristic
25C2×#14578941C2211yescharacteristic, complemented
26SD16×D157—24no—
27C2×#14578937C2211yescharacteristic
28C2×C8×D157C2211yescharacteristic
29#19097768C2211yescharacteristic
30#19097767C2211yescharacteristic
31#19097760C2211yescharacteristic
32#19097744—21no—
33#19097744—21no—
34#19097741—21no—
35#19097741—21no—
36Q8×Dic157C2211yescharacteristic, complemented
37C4×Dic314—21no—
38C4×Dic314C2211yescharacteristic, complemented
39#19097736—21no—
40#19097736C2211yescharacteristic, complemented
41#19097729—21no—
42#19097729C2211yescharacteristic
43#19097728—21no—
44#19097728C2211yescharacteristic
45C4⋊#11929434C2211yescharacteristic, complemented
46#19097700C2211yescharacteristic, complemented
47#19097700C2211yescharacteristic, complemented
48#19097695C2211yescharacteristic, complemented
49#19097693C2211yescharacteristic
50#19097693C2211yescharacteristic, complemented
51#19097692C2211yescharacteristic
52#19097691C2211yescharacteristic
53#19097691C2211yescharacteristic, complemented
54C2×Q16×C157C2211yescharacteristic, complemented
55C2×SD16×C157C2211yescharacteristic
56#124×C157C2211yescharacteristic
57C4×Q8×C157C2211yescharacteristic, complemented
58C4⋊C8×C157C2211yescharacteristic, complemented
59#103×C157C2211yescharacteristic, complemented
60#102×C157C2211yescharacteristic
61C23×D157—21no—
62#14578955—22no—
63#14578955—22no—
64C2×Dic314—21no—
65C2×Dic314C2311yescharacteristic
66Q8×D157—22no—
67Q8×D157D414yescomplemented
68C2×D628—21no—
69C2×D628—21no—
70C2×D628—21no—
71C2×D628C2311yescharacteristic
72C2×D628C2311yescharacteristic
73C2×C22⋊D157—21no—
74C2×C22⋊D157—21no—
75C2×C22⋊D157—21no—
76D4×D157—22no—
77D4×D157—22no—
78C2×C4×D157—21no—
79C2×C4×D157—21no—
80C2×C4×D157C2311yescharacteristic
81#14578944—22no—
82#14578943—22no—
83#14578943—22no—
84#14578942—22no—
85#14578941—22no—
86#14578941—22no—
87#14578937—22no—
88C2×#11929434C2311yescharacteristic
89C2×#11929434C2311yescharacteristic
90C8×D157—22no—
91C4⋊Dic157—21no—
92C4⋊Dic157C2311yescharacteristic
93#14578933—21no—
94#14578933—21no—
95#14578933D411yescharacteristic
96#14578933D411yescharacteristic
97#14578931—21no—
98#14578931—21no—
99#14578931—21no—
100#14578931D411yescharacteristic
101#14578931D411yescharacteristic
102C4×Dic157—21no—
103C4×Dic157—21no—
104C4×Dic157—21no—
105C4×Dic157—21no—
106C4×Dic157C2311yescharacteristic
107C2×Q8×C157C2311yescharacteristic
108C2×Q8×C157C2311yescharacteristic
109C2×D4×C157C2311yescharacteristic
110Q16×C157—22no—
111SD16×C157—22no—
112C2×C8×C157C2311yescharacteristic
113C2×C8×C157C2311yescharacteristic
114C4⋊C4×C157—21no—
115C4⋊C4×C157C2311yescharacteristic
116C22⋊C4×C157—21no—
117C22⋊C4×C157—21no—
118C42×C157—21no—
119C42×C157C2311yescharacteristic
120C22×D157—21no—
121C22×D157—21no—
122C22×D157—42no—
123C22×D157—81no—
124C22×D157—81no—
125C22×D157C2×D411yescharacteristic
126Dic314—21no—
127Dic314—24no—
128Dic314C2×D412yes—
129D628—21no—
130D628—21no—
131D628—21no—
132D628—21no—
133D628—42no—
134D628—81no—
135D628—81no—
136C22⋊D157—42no—
137C22⋊D157—81no—
138C22⋊D157—81no—
139C2×Dic157—21no—
140C2×Dic157—21no—
141C2×Dic157C2×D411yescharacteristic
142C2×Dic157C2×D411yescharacteristic
143C2×Dic157C2×D411yescharacteristic
144C4×D157—24no—
145C4×D157—42no—
146C4×D157C2×D412yes—
147C4×D157#4112yes—
148#11929434—21no—
149#11929434—21no—
150C23×C157—21no—
151Q8×C157—21no—
152Q8×C157—21no—
153Q8×C157—21no—
154Q8×C157C2×D412yes—
155D4×C157—21no—
156D4×C157—21no—
157C2×C4×C157—21no—
158C2×C4×C157—21no—
159C2×C4×C157—21no—
160C2×C4×C157C2×D411yescharacteristic
161C2×C4×C157C2×D411yescharacteristic
162C2×C4×C157C2411yescharacteristic
163C8×C157—21no—
164C8×C157—21no—
165C22×C157—41no—
166C22×C157—41no—
167C22×C157—41no—
168C22×C157C22×D411yescharacteristic
169C2×D157—21no—
170C2×D157—21no—
171C2×D157—41no—
172C2×D157—41no—
173C2×D157—41no—
174C2×D157—41no—
175C2×D157—41no—
176C2×D157—41no—
177C2×D157—84no—
178C2×D157#12112yes—
179C4×C157—22no—
180C4×C157—21no—
181C4×C157—21no—
182C4×C157—41no—
183C4×C157—41no—
184C4×C157C22×D411yescharacteristic
185C4×C157C2×#4111yescharacteristic
186Dic157—22no—
187Dic157—21no—
188Dic157—21no—
189Dic157—41no—
190Dic157#12112yes—
191C2×C157—81no—
192C2×C157C2×#12111yescharacteristic
193C2×C157#57711yescharacteristic
194C2×C157#57711yescharacteristic
195D157—22no—
196D157—81no—
197D157—81no—
198C157#304311yescharacteristic, complemented, Sylow
199#3043—1571nomaximal, Sylow
200C2×#137—1571no—
201C22×SD16—1571no—
202#535—1571no—
203#519—1571no—
204#464—1571no—
205#464—1571no—
206#464—1571no—
207#464D15711yescharacteristic, complemented
208#460—1571no—
209#460—1571no—
210#460—1571no—
211#460—1571no—
212#424—1571no—
213#418—1571no—
214#416—1571no—
215C2×#41—1571no—
216C22×Q8—1571no—
217C22×D4—1571no—
218#137—3144no—
219C2×Q16—1571no—
220C2×Q16C2×D15711yescharacteristic, complemented
221C2×SD16—3144no—
222C2×SD16—1571no—
223C2×SD16—1571no—
224C2×SD16—1571no—
225C2×SD16—1571no—
226C2×SD16—1571no—
227C2×SD16C2×D15711yescharacteristic
228C2×M16—1571no—
229C22×C8—1571no—
230#127—1571no—
231#127—1571no—
232#124—1571no—
233#124C2×D15711yescharacteristic
234#121—3141no—
235#121—3141no—
236#120—3141no—
237#120—3141no—
238C4×Q8—3141no—
239C4×Q8—3141no—
240C4×Q8—1571no—
241C4×Q8—1571no—
242C4×Q8—1571no—
243C4×Q8C2×D15711yescharacteristic, complemented
244#117—3141no—
245#117—3141no—
246#117—1571no—
247#117—1571no—
248C4⋊C8—1571no—
249C4⋊C8—1571no—
250C4⋊C8—1571no—
251C4⋊C8C2×D15711yescharacteristic, complemented
252#103—1571no—
253#103C2×D15711yescharacteristic, complemented
254#102—1571no—
255#102—1571no—
256#102—1571no—
257#102—1571no—
258#102—1571no—
259#102C2×D15711yescharacteristic
260C24—3141no—
261#41—3142no—
262#41—3142no—
263C2×Q8—3141no—
264C2×Q8—3142no—
265C2×Q8—1574no—
266C2×Q8—1571no—
267C2×Q8C22×D15711yescharacteristic
268C2×Q8C22×D15711yescharacteristic
269C2×D4—3141no—
270C2×D4—3142no—
271C2×D4—3142no—
272C2×D4—3141no—
273C2×D4—3141no—
274C2×D4—3141no—
275C2×D4—3141no—
276C2×D4—3141no—
277C2×D4—1571no—
278C2×D4—1571no—
279C2×D4C22×D15711yescharacteristic
280C22×C4—3141no—
281C22×C4—3141no—
282C22×C4—1571no—
283Q16—22no—
284Q16—3142no—
285SD16—22no—
286SD16—3142no—
287SD16—3142no—
288SD16—3142no—
289SD16—3142no—
290SD16—3142no—
291M16—3142no—
292C2×C8—3142no—
293C2×C8—1571no—
294C2×C8—1571no—
295C2×C8C22×D15711yescharacteristic
296C2×C8C22×D15711yescharacteristic
297C4⋊C4—21no—
298C4⋊C4—3141no—
299C4⋊C4—3141no—
300C4⋊C4—3141no—
301C4⋊C4—1571no—
302C4⋊C4—1571no—
303C4⋊C4—1571no—
304C4⋊C4C22×D15711yescharacteristic
305C22⋊C4—21no—
306C22⋊C4—21no—
307C22⋊C4—3141no—
308C22⋊C4—3141no—
309C22⋊C4—3141no—
310C22⋊C4—1571no—
311C22⋊C4—1571no—
312C42—21no—
313C42—3141no—
314C42—3141no—
315C42—3141no—
316C42—3141no—
317C42—1571no—
318C42C22×D15711yescharacteristic
319C23—21no—
320C23—3141no—
321C23—3141no—
322C23—6282no—
323C23—1571no—
324C23—12561no—
325C23—12561no—
326Q8—21no—
327Q8—21no—
328Q8—21no—
329Q8—3144no—
330Q8—3141no—
331Q8—1572no—
332Q8D4×D15712yes—
333D4—21no—
334D4—21no—
335D4—3141no—
336D4—3141no—
337D4—3141no—
338D4—3141no—
339D4—6282no—
340D4—6282no—
341D4—12561no—
342D4—12561no—
343D4—12561no—
344D4—12561no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—3144no—
349C2×C4—3141no—
350C2×C4—3141no—
351C2×C4—6282no—
352C2×C4—1571no—
353C2×C4—1571no—
354C2×C4—1571no—
355C2×C4—1572no—
356C2×C4—1572no—
357C2×C4D4×D15711yescharacteristic
358C2×C4D4×D15711yescharacteristic
359C2×C4C23×D15711yescharacteristic
360C8—21no—
361C8—21no—
362C8—3141no—
363C8—3141no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—3141no—
368C22—3141no—
369C22—6281no—
370C22—6281no—
371C22—6281no—
372C22—6281no—
373C22—6281no—
374C22—6281no—
375C22—1572no—
376C22—12564no—
377C22C2×D4×D15711yescharacteristic
378C4—21no—
379C4—21no—
380C4—22no—
381C4—41no—
382C4—41no—
383C4—3141no—
384C4—3141no—
385C4—3142no—
386C4—6281no—
387C4—1572no—
388C4C2×D4×D15711yescharacteristic
389C4#41×D15711yescharacteristic
390C2—81no—
391C2—3142no—
392C2—12561no—
393C2—12561no—
394C2#121×D15711yescharacteristic
395C2#2375808011yescharacteristic
396C2#2375808211yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#464×C157C2
socleC22×C157C22×D4
prime radical#23757841C2
hypercenter#464D157
hypocenterC157#3043
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
1. C2×#19097803C2semidirect
2. C2×SD16×D157C2semidirect
4. #23757749C2semidirect
5. #23757498C2semidirect
6. #23757497C2semidirect
7. #23757492C2semidirect
8. #23757474C2semidirect
9. #23757473C2semidirect
10. #23757472C2semidirect
11. #23757471C2semidirect
12. #23757298C2semidirect
13. #23757276C2semidirect
14. #23757268C2semidirect
15. #464×C157C2semidirect
20. C2×#14578944C22semidirect
22. C2×#14578943C22semidirect
25. C2×#14578941C22semidirect
36. Q8×Dic157C22semidirect
38. C4×Dic314C22semidirect
40. #19097736C22semidirect
45. C4⋊#11929434C22semidirect
46. #19097700C22semidirect
47. #19097700C22semidirect
48. #19097695C22semidirect
50. #19097693C22semidirect
53. #19097691C22semidirect
54. C2×Q16×C157C22semidirect
57. C4×Q8×C157C22semidirect
58. C4⋊C8×C157C22semidirect
59. #103×C157C22semidirect
67. Q8×D157D4semidirect
198. C157#3043semidirect
207. #464D157semidirect
220. C2×Q16C2×D157semidirect
243. C4×Q8C2×D157semidirect
251. C4⋊C8C2×D157semidirect
253. #103C2×D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190978032
C2×SD16×D1572
#237578412
#237577492
#237574982
#237574972
#237574922
#237574742
#237574732
#237574722
#237574712
#237572982
#237572762
#237572682
#464×C1572
#3043157