Finite Group Databasebeta

← #34109625

Subgroups — #34109625

Order 20096.

500 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×#19097800C211yesmaximal, characteristic, complemented
2C2×#19097795C211yesmaximal, characteristic
3#124×D157C211yesmaximal, characteristic, complemented
4C4×#14578953C211yesmaximal, characteristic, complemented
5#23757498C211yesmaximal, characteristic, complemented
6#23757496C211yesmaximal, characteristic, complemented
7#23757494C211yesmaximal, characteristic, complemented
8#23757492C211yesmaximal, characteristic, complemented
9#23757490C211yesmaximal, characteristic, complemented
10#23757488C211yesmaximal, characteristic, complemented
11#23757486C211yesmaximal, characteristic, complemented
12#23757295C211yesmaximal, characteristic, complemented
13#23757280C211yesmaximal, characteristic, complemented
14#23757276C211yesmaximal, characteristic, complemented
15#463×C157C211yesmaximal, characteristic, complemented
16C2×#14578953C2211yescharacteristic
17C2×Q8×D157C2211yescharacteristic
18C2×D4×D157C2211yescharacteristic
19#19097800—24no—
20#19097795—24no—
21C2×Dic628C2211yescharacteristic, complemented
22C2×#14578944C2211yescharacteristic, complemented
23C2×#14578943C2211yescharacteristic
24C2×#14578942C2211yescharacteristic
25C2×#14578942C2211yescharacteristic, complemented
26C2×#14578939C2211yescharacteristic
27C2×#14578937C2211yescharacteristic
28C2×#14578937C2211yescharacteristic
29#19097760C2211yescharacteristic
30#19097759C2211yescharacteristic
31#19097758C2211yescharacteristic
32#19097757—21no—
33#19097757—21no—
34C4×Dic314C2211yescharacteristic, complemented
35#19097736C2211yescharacteristic, complemented
36D4×Dic157C2211yescharacteristic, complemented
37C4×C22⋊D157—21no—
38#19097731—21no—
39#19097729C2211yescharacteristic
40#19097727—21no—
41C22⋊C4×D157—21no—
42C22⋊C4×D157—21no—
43C2×C4×Dic157—21no—
44C42×D157C2211yescharacteristic
45C4⋊#11929434C2211yescharacteristic, complemented
46#19097700C2211yescharacteristic, complemented
47#19097700C2211yescharacteristic, complemented
48#19097696C2211yescharacteristic, complemented
49#19097695C2211yescharacteristic, complemented
50#19097694C2211yescharacteristic, complemented
51#19097693C2211yescharacteristic
52#19097692C2211yescharacteristic
53#19097691C2211yescharacteristic
54C2×SD16×C157C2211yescharacteristic, complemented
55C2×D8×C157C2211yescharacteristic
56#124×C157C2211yescharacteristic
57C4×D4×C157C2211yescharacteristic, complemented
58C4⋊C8×C157C2211yescharacteristic, complemented
59#103×C157C2211yescharacteristic, complemented
60#102×C157C2211yescharacteristic
61C23×D157—21no—
62#14578953—22no—
63#14578953D414yes—
64C2×Dic314—21no—
65C2×Dic314C2311yescharacteristic
66C2×Dic314C2311yescharacteristic
67Q8×D157—22no—
68Q8×D157—22no—
69C2×D628C2311yescharacteristic
70C2×C22⋊D157—21no—
71C2×C22⋊D157—21no—
72D4×D157—22no—
73D4×D157—22no—
74C22×Dic157—21no—
75C22×Dic157—21no—
76C2×C4×D157—21no—
77C2×C4×D157D411yescharacteristic
78C2×C4×D157D411yescharacteristic
79C2×C4×D157C2311yescharacteristic
80Dic628—22no—
81#14578944—22no—
82#14578943—22no—
83#14578942—22no—
84#14578942—22no—
85#14578939—22no—
86#14578937—22no—
87#14578937—22no—
88C2×#11929434C2311yescharacteristic
89C2×#11929434C2311yescharacteristic
90C4⋊Dic157C2311yescharacteristic
91#14578933—21no—
92#14578933—21no—
93C22⋊Dic157—21no—
94C22⋊Dic157—21no—
95C22⋊Dic157—21no—
96#14578931—21no—
97#14578931—21no—
98#14578931—21no—
99#14578931—21no—
100#14578931—21no—
101C4×Dic157—21no—
102C4×Dic157—21no—
103C4×Dic157—21no—
104C4×Dic157D411yescharacteristic
105C4×Dic157D411yescharacteristic
106C4×Dic157C2311yescharacteristic
107C2×Q8×C157C2311yescharacteristic
108C2×D4×C157C2311yescharacteristic
109C2×D4×C157C2311yescharacteristic
110C22×C4×C157—21no—
111SD16×C157—22no—
112D8×C157—22no—
113C2×C8×C157C2311yescharacteristic
114C2×C8×C157C2311yescharacteristic
115C4⋊C4×C157C2311yescharacteristic
116C22⋊C4×C157—21no—
117C22⋊C4×C157—21no—
118C22⋊C4×C157—21no—
119C42×C157C2311yescharacteristic
120C22×D157—21no—
121C22×D157—42no—
122C22×D157—42no—
123C22×D157—42no—
124C22×D157C2×D411yescharacteristic
125Dic314—21no—
126Dic314—21no—
127Dic314—21no—
128Dic314—42no—
129Dic314C2×D412yes—
130D628—21no—
131D628—21no—
132C22⋊D157—24no—
133C22⋊D157—42no—
134C2×Dic157—21no—
135C2×Dic157—24no—
136C2×Dic157—21no—
137C2×Dic157—21no—
138C2×Dic157—42no—
139C2×Dic157C2×D411yescharacteristic
140C2×Dic157C2×D411yescharacteristic
141C2×Dic157C2×D411yescharacteristic
142C4×D157—41no—
143C4×D157—41no—
144C4×D157—42no—
145C4×D157C2×D412yes—
146C4×D157#4112yes—
147#11929434—21no—
148#11929434—21no—
149C23×C157—21no—
150C23×C157—21no—
151Q8×C157—21no—
152Q8×C157—21no—
153D4×C157—21no—
154D4×C157—21no—
155D4×C157—21no—
156D4×C157C2×D412yes—
157C2×C4×C157—21no—
158C2×C4×C157—21no—
159C2×C4×C157—42no—
160C2×C4×C157C2×D411yescharacteristic
161C2×C4×C157C2×D411yescharacteristic
162C2×C4×C157C2411yescharacteristic
163C8×C157—21no—
164C8×C157—21no—
165C22×C157—22no—
166C22×C157—41no—
167C22×C157—41no—
168C22×C157—41no—
169C22×C157—41no—
170C22×C157—41no—
171C22×C157C22×D411yescharacteristic
172C2×D157—21no—
173C2×D157—21no—
174C2×D157—41no—
175C2×D157—41no—
176C2×D157—41no—
177C2×D157—84no—
178C2×D157#12112yes—
179C4×C157—21no—
180C4×C157—21no—
181C4×C157—41no—
182C4×C157—41no—
183C4×C157C22×D411yescharacteristic
184C4×C157C2×#4111yescharacteristic
185Dic157—22no—
186Dic157—21no—
187Dic157—21no—
188Dic157—41no—
189Dic157—41no—
190Dic157#12112yes—
191C2×C157—42no—
192C2×C157—81no—
193C2×C157C2×#12111yescharacteristic
194C2×C157#57511yescharacteristic
195C2×C157#57511yescharacteristic
196D157—22no—
197D157—81no—
198C157#304011yescharacteristic, complemented, Sylow
199#3040—1571nomaximal, Sylow
200C2×#137—1571no—
201C2×AGL(1,ℤ/8)—1571no—
202C2×#124—1571no—
203C4×#41—1571no—
204#464—1571no—
205#464—1571no—
206#464—1571no—
207#464—1571no—
208#463—1571no—
209#463—1571no—
210#463—1571no—
211#463D15711yescharacteristic, complemented
212#423—1571no—
213#419—1571no—
214#418—1571no—
215C2×#41—1571no—
216C22×Q8—1571no—
217C22×D4—1571no—
218#137—3144no—
219AGL(1,ℤ/8)—3144no—
220C2×Q16—1571no—
221C2×Q16—1571no—
222C2×SD16—1571no—
223C2×SD16—1571no—
224C2×SD16—1571no—
225C2×SD16C2×D15711yescharacteristic, complemented
226C2×D8—1571no—
227C2×D8C2×D15711yescharacteristic
228C2×M16—1571no—
229C2×M16—1571no—
230#124—3141no—
231#124—3141no—
232#124—1571no—
233#124—1571no—
234#124—1571no—
235#124C2×D15711yescharacteristic
236C4×Q8—1571no—
237C4×Q8—1571no—
238C4×D4—3141no—
239C4×D4—3141no—
240C4×D4—1571no—
241C4×D4C2×D15711yescharacteristic, complemented
242#117—3141no—
243#117—1571no—
244C2×C22⋊C4—3141no—
245C2×C22⋊C4—3141no—
246C2×C42—3141no—
247C2×C42—1571no—
248C4⋊C8—1571no—
249C4⋊C8—1571no—
250C4⋊C8—1571no—
251C4⋊C8C2×D15711yescharacteristic, complemented
252#103—1571no—
253#103—1571no—
254#103—1571no—
255#103C2×D15711yescharacteristic, complemented
256#102—1571no—
257#102—1571no—
258#102—1571no—
259#102C2×D15711yescharacteristic
260C24—3141no—
261#41—3142no—
262#41—1574no—
263C2×Q8—3141no—
264C2×Q8—3142no—
265C2×Q8—3142no—
266C2×Q8—1571no—
267C2×Q8—1571no—
268C2×Q8C22×D15711yescharacteristic
269C2×D4—3141no—
270C2×D4—3142no—
271C2×D4—3142no—
272C2×D4—3141no—
273C2×D4—1571no—
274C2×D4C22×D15711yescharacteristic
275C2×D4C22×D15711yescharacteristic
276C22×C4—21no—
277C22×C4—3141no—
278C22×C4—3141no—
279C22×C4—3141no—
280C22×C4—1571no—
281C22×C4—1571no—
282C22×C4—1571no—
283Q16—3142no—
284Q16—3142no—
285SD16—22no—
286SD16—3142no—
287SD16—3142no—
288SD16—3142no—
289D8—22no—
290D8—3142no—
291M16—3142no—
292M16—3142no—
293C2×C8—1571no—
294C2×C8—1571no—
295C2×C8C22×D15711yescharacteristic
296C2×C8C22×D15711yescharacteristic
297C4⋊C4—3141no—
298C4⋊C4—3141no—
299C4⋊C4—1571no—
300C4⋊C4C22×D15711yescharacteristic
301C22⋊C4—21no—
302C22⋊C4—21no—
303C22⋊C4—21no—
304C22⋊C4—3141no—
305C22⋊C4—3141no—
306C22⋊C4—3141no—
307C22⋊C4—3141no—
308C22⋊C4—3141no—
309C22⋊C4—3141no—
310C22⋊C4—3141no—
311C22⋊C4—3141no—
312C42—3141no—
313C42—3141no—
314C42—3141no—
315C42—1571no—
316C42—1571no—
317C42—1571no—
318C42C22×D15711yescharacteristic
319C23—21no—
320C23—21no—
321C23—3141no—
322C23—6282no—
323C23—6282no—
324C23—6282no—
325C23—1571no—
326Q8—21no—
327Q8—21no—
328Q8—3141no—
329Q8—3141no—
330Q8—3141no—
331Q8—6282no—
332Q8—1572no—
333D4—21no—
334D4—21no—
335D4—21no—
336D4—3144no—
337D4—3141no—
338D4—3141no—
339D4—6282no—
340D4D4×D15712yes—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—42no—
344C2×C4—3141no—
345C2×C4—3144no—
346C2×C4—3141no—
347C2×C4—3141no—
348C2×C4—6281no—
349C2×C4—6281no—
350C2×C4—6282no—
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—22no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—3141no—
371C22—3141no—
372C22—6281no—
373C22—6281no—
374C22—6281no—
375C22—1572no—
376C22—12564no—
377C22C2×D4×D15711yescharacteristic
378C4—21no—
379C4—21no—
380C4—41no—
381C4—41no—
382C4—3142no—
383C4—3141no—
384C4—3141no—
385C4—6281no—
386C4—6281no—
387C4—1572no—
388C4C2×D4×D15711yescharacteristic
389C4#41×D15711yescharacteristic
390C2—42no—
391C2—81no—
392C2—3142no—
393C2—12561no—
394C2#121×D15711yescharacteristic
395C2#2375806911yescharacteristic
396C2#2375807011yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#463×C157C2
socleC22×C157C22×D4
prime radicalC2×#19097795C2
hypercenter#463D157
hypocenterC157#3040
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. C2×#19097800C2semidirect
3. #124×D157C2semidirect
4. C4×#14578953C2semidirect
5. #23757498C2semidirect
6. #23757496C2semidirect
7. #23757494C2semidirect
8. #23757492C2semidirect
9. #23757490C2semidirect
10. #23757488C2semidirect
11. #23757486C2semidirect
12. #23757295C2semidirect
13. #23757280C2semidirect
14. #23757276C2semidirect
15. #463×C157C2semidirect
21. C2×Dic628C22semidirect
22. C2×#14578944C22semidirect
25. C2×#14578942C22semidirect
34. C4×Dic314C22semidirect
35. #19097736C22semidirect
36. D4×Dic157C22semidirect
45. C4⋊#11929434C22semidirect
46. #19097700C22semidirect
47. #19097700C22semidirect
48. #19097696C22semidirect
49. #19097695C22semidirect
50. #19097694C22semidirect
54. C2×SD16×C157C22semidirect
57. C4×D4×C157C22semidirect
58. C4⋊C8×C157C22semidirect
59. #103×C157C22semidirect
198. C157#3040semidirect
211. #463D157semidirect
225. C2×SD16C2×D157semidirect
241. C4×D4C2×D157semidirect
251. C4⋊C8C2×D157semidirect
255. #103C2×D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190978002
C2×#190977952
#124×D1572
C4×#145789532
#237574982
#237574962
#237574942
#237574922
#237574902
#237574882
#237574862
#237572952
#237572802
#237572762
#463×C1572
#3040157