Finite Group Databasebeta

← #34109765

Subgroups — #34109765

Order 20096.

496 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19097803C211yesmaximal, characteristic, complemented
2C2×M16×D157C211yesmaximal, characteristic, complemented
3#23757786C211yesmaximal, characteristic
4C22⋊Q8×D157C211yesmaximal, characteristic, complemented
5#23757529C211yesmaximal, characteristic, complemented
6#23757521C211yesmaximal, characteristic, complemented
7#23757519C211yesmaximal, characteristic, complemented
8#23757513C211yesmaximal, characteristic, complemented
9#23757511C211yesmaximal, characteristic, complemented
10#23757505C211yesmaximal, characteristic, complemented
11#23757503C211yesmaximal, characteristic, complemented
12#23757312C211yesmaximal, characteristic, complemented
13#23757280C211yesmaximal, characteristic, complemented
14#23757273C211yesmaximal, characteristic, complemented
15#470×C157C211yesmaximal, characteristic, complemented
16C2×#14578955C2211yescharacteristic
17C2×Q8×D157C2211yescharacteristic
18C22×C4×D157C2211yescharacteristic
19#19097803—24no—
20C2×#14578944C2211yescharacteristic, complemented
21C2×#14578943C2211yescharacteristic, complemented
22C2×#14578941C2211yescharacteristic, complemented
23C2×#14578938C2211yescharacteristic
24C2×#14578937C2211yescharacteristic
25C2×#14578937C2211yescharacteristic
26M16×D157—24no—
27C2×C8×D157C2211yescharacteristic
28C2×C8×D157C2211yescharacteristic
29#19097756—21no—
30#19097751C2211yescharacteristic
31#19097750C2211yescharacteristic
32#19097749—21no—
33#19097748—21no—
34#19097747C2211yescharacteristic
35#19097747C2211yescharacteristic
36#19097746C2211yescharacteristic
37#19097745C2211yescharacteristic
38#19097741—21no—
39#19097733—21no—
40#19097731—21no—
41#19097729C2211yescharacteristic
42C4⋊C4×D157—21no—
43C4⋊C4×D157C2211yescharacteristic
44C22⋊C4×D157—21no—
45#19097705C2211yescharacteristic, complemented
46#19097704C2211yescharacteristic, complemented
47#19097702C2211yescharacteristic, complemented
48#19097696C2211yescharacteristic, complemented
49#19097695C2211yescharacteristic, complemented
50#19097695C2211yescharacteristic, complemented
51#19097694C2211yescharacteristic, complemented
52#19097693C2211yescharacteristic, complemented
53#19097691C2211yescharacteristic, complemented
54C2×Q16×C157C2211yescharacteristic, complemented
55C2×M16×C157C2211yescharacteristic
56C22⋊Q8×C157C2211yescharacteristic
57C22⋊Q8×C157C2211yescharacteristic
58#106×C157C2211yescharacteristic, complemented
59#103×C157C2211yescharacteristic, complemented
60#103×C157C2211yescharacteristic, complemented
61C23×D157D411yescharacteristic
62#14578955—22no—
63#14578955—22no—
64C2×Dic314—21no—
65C2×Dic314C2311yescharacteristic
66Q8×D157—22no—
67Q8×D157—22no—
68C2×D628—21no—
69C2×D628—21no—
70C2×D628C2311yescharacteristic
71C2×C22⋊D157—21no—
72C22×Dic157D411yescharacteristic
73C2×C4×D157—22no—
74C2×C4×D157—22no—
75C2×C4×D157—21no—
76C2×C4×D157—21no—
77C2×C4×D157—21no—
78C2×C4×D157D411yescharacteristic
79C2×C4×D157D411yescharacteristic
80C2×C4×D157C2311yescharacteristic
81C2×C4×D157C2311yescharacteristic
82#14578944—22no—
83#14578943—22no—
84#14578941—22no—
85#14578938—22no—
86#14578937—22no—
87#14578937D414yescomplemented
88C2×#11929434C2311yescharacteristic
89C2×#11929434C2311yescharacteristic
90C8×D157—22no—
91C8×D157—22no—
92C4⋊Dic157—21no—
93C4⋊Dic157C2311yescharacteristic
94C4⋊Dic157C2311yescharacteristic
95#14578933—21no—
96#14578933—21no—
97#14578933—21no—
98#14578933—21no—
99C22⋊Dic157—21no—
100#14578931—21no—
101#14578931—21no—
102#14578931—21no—
103#14578931—21no—
104#14578931—21no—
105#14578931—21no—
106C4×Dic157—21no—
107C2×Q8×C157C2311yescharacteristic
108C2×Q8×C157C2311yescharacteristic
109C22×C4×C157C2311yescharacteristic
110Q16×C157—22no—
111M16×C157—22no—
112C2×C8×C157C2311yescharacteristic
113C2×C8×C157C2311yescharacteristic
114C4⋊C4×C157—21no—
115C4⋊C4×C157—21no—
116C4⋊C4×C157C2311yescharacteristic
117C4⋊C4×C157C2311yescharacteristic
118C22⋊C4×C157—21no—
119C22⋊C4×C157—21no—
120C22×D157—22no—
121C22×D157—22no—
122C22×D157—21no—
123C22×D157—41no—
124C22×D157C2×D411yescharacteristic
125C22×D157C2×D411yescharacteristic
126Dic314—21no—
127Dic314—21no—
128Dic314—42no—
129D628—21no—
130D628—21no—
131D628—42no—
132D628—42no—
133C22⋊D157—42no—
134C2×Dic157—22no—
135C2×Dic157—21no—
136C2×Dic157—21no—
137C2×Dic157—21no—
138C2×Dic157C2×D411yescharacteristic
139C2×Dic157C2×D411yescharacteristic
140C4×D157—22no—
141C4×D157—21no—
142C4×D157—21no—
143C4×D157—22no—
144C4×D157—42no—
145C4×D157—42no—
146C4×D157—42no—
147C4×D157C2×D412yes—
148C4×D157#4112yes—
149#11929434—21no—
150#11929434C2×D412yes—
151C23×C157C2×D411yescharacteristic
152Q8×C157—21no—
153Q8×C157—21no—
154Q8×C157—21no—
155Q8×C157—21no—
156C2×C4×C157—21no—
157C2×C4×C157—21no—
158C2×C4×C157—21no—
159C2×C4×C157—21no—
160C2×C4×C157—21no—
161C2×C4×C157—21no—
162C2×C4×C157C2×D411yescharacteristic
163C2×C4×C157C2411yescharacteristic
164C8×C157—21no—
165C8×C157C2×D412yes—
166C22×C157—21no—
167C22×C157—21no—
168C22×C157—21no—
169C22×C157C22×D411yescharacteristic
170C2×D157—21no—
171C2×D157—21no—
172C2×D157—21no—
173C2×D157—21no—
174C2×D157—44no—
175C2×D157—41no—
176C2×D157—41no—
177C2×D157—41no—
178C2×D157#12112yes—
179C2×D157#13712yescomplemented
180C4×C157—21no—
181C4×C157—41no—
182C4×C157—41no—
183C4×C157—41no—
184C4×C157—41no—
185C4×C157C22×D411yescharacteristic
186C4×C157C2×#4111yescharacteristic
187Dic157—21no—
188Dic157—41no—
189Dic157—41no—
190Dic157—41no—
191Dic157#12112yes—
192C2×C157—41no—
193C2×C157C2×#12111yescharacteristic
194C2×C157C2×#13711yescharacteristic
195C2×C157#57511yescharacteristic
196D157—22no—
197D157—22no—
198D157—81no—
199C157#305611yescharacteristic, complemented, Sylow
200#3056—1571nomaximal, Sylow
201C2×#137—1571no—
202C22×M16—1571no—
203#525—1571no—
204C2×C22⋊Q8—1571no—
205#470—1571no—
206#470D15711yescharacteristic, complemented
207#468—1571no—
208#468—1571no—
209#467—1571no—
210#467—1571no—
211#465—1571no—
212#465—1571no—
213#428—1571no—
214#419—1571no—
215#417—1571no—
216C2×#41—1571no—
217C22×Q8—1571no—
218C23×C4—1571no—
219#137—3144no—
220C2×Q16—1571no—
221C2×Q16C2×D15711yescharacteristic, complemented
222C2×SD16—1571no—
223C2×SD16—1571no—
224C2×M16—3144no—
225C2×M16—1571no—
226C2×M16—1571no—
227C2×M16—1571no—
228C2×M16C2×D15711yescharacteristic
229C22×C8—1571no—
230C22×C8—1571no—
231#123—3141no—
232C22⋊Q8—3141no—
233C22⋊Q8—3141no—
234C22⋊Q8—1571no—
235C22⋊Q8—1571no—
236C22⋊Q8—1571no—
237C22⋊Q8—1571no—
238C22⋊Q8C2×D15711yescharacteristic
239C22⋊Q8C2×D15711yescharacteristic
240#121—1571no—
241#121—1571no—
242#120—3141no—
243C4×D4—3141no—
244C4×D4—3141no—
245#117—1571no—
246C2×C4⋊C4—3141no—
247C2×C4⋊C4—1571no—
248C2×C22⋊C4—3141no—
249#107—1571no—
250#107—1571no—
251#106—1571no—
252#106C2×D15711yescharacteristic, complemented
253#103—1571no—
254#103—1571no—
255#103—1571no—
256#103—1571no—
257#103C2×D15711yescharacteristic, complemented
258#103C2×D15711yescharacteristic, complemented
259#102—1571no—
260#102—1571no—
261C24—1571no—
262#41—3142no—
263#41—3142no—
264C2×Q8—3141no—
265C2×Q8—3142no—
266C2×Q8—3142no—
267C2×Q8—1571no—
268C2×Q8C22×D15711yescharacteristic
269C2×Q8C22×D15711yescharacteristic
270C2×D4—3141no—
271C2×D4—3141no—
272C2×D4—3141no—
273C2×D4—1571no—
274C22×C4—3142no—
275C22×C4—3142no—
276C22×C4—3141no—
277C22×C4—3141no—
278C22×C4—3141no—
279C22×C4—1571no—
280C22×C4—1571no—
281C22×C4—1571no—
282C22×C4—1571no—
283C22×C4—1571no—
284C22×C4C22×D15711yescharacteristic
285Q16—22no—
286Q16—3142no—
287SD16—3142no—
288SD16—3142no—
289M16—22no—
290M16—3142no—
291M16—3142no—
292M16—1574no—
293C2×C8—3142no—
294C2×C8—3142no—
295C2×C8—1571no—
296C2×C8—1571no—
297C2×C8C22×D15711yescharacteristic
298C2×C8C22×D15711yescharacteristic
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—3141no—
302C4⋊C4—3141no—
303C4⋊C4—3141no—
304C4⋊C4—3141no—
305C4⋊C4—3141no—
306C4⋊C4—1571no—
307C4⋊C4—1571no—
308C4⋊C4C22×D15711yescharacteristic
309C4⋊C4C22×D15711yescharacteristic
310C22⋊C4—21no—
311C22⋊C4—21no—
312C22⋊C4—3141no—
313C22⋊C4—3141no—
314C22⋊C4—3141no—
315C22⋊C4—3141no—
316C22⋊C4—3141no—
317C22⋊C4—3141no—
318C22⋊C4—3141no—
319C42—3141no—
320C23—3142no—
321C23—3142no—
322C23—3141no—
323C23—6281no—
324C23—1571no—
325C23—1571no—
326C23D4×D15711yescharacteristic
327Q8—21no—
328Q8—21no—
329Q8—21no—
330Q8—21no—
331Q8—3141no—
332Q8—3141no—
333Q8—6282no—
334D4—3141no—
335D4—3141no—
336D4—6282no—
337D4—6282no—
338D4—6282no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—3142no—
346C2×C4—3142no—
347C2×C4—3142no—
348C2×C4—3141no—
349C2×C4—3141no—
350C2×C4—3141no—
351C2×C4—3141no—
352C2×C4—3141no—
353C2×C4—6282no—
354C2×C4—6282no—
355C2×C4—6282no—
356C2×C4—1571no—
357C2×C4—1571no—
358C2×C4—1572no—
359C2×C4—1572no—
360C2×C4D4×D15711yescharacteristic
361C2×C4C23×D15711yescharacteristic
362C8—21no—
363C8—3141no—
364C8—1572no—
365C8D4×D15712yes—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—3141no—
370C22—3141no—
371C22—3141no—
372C22—3141no—
373C22—6284no—
374C22—6281no—
375C22—6281no—
376C22—6281no—
377C22—1572no—
378C22—1572no—
379C22C2×D4×D15711yescharacteristic
380C4—21no—
381C4—41no—
382C4—41no—
383C4—41no—
384C4—41no—
385C4—3141no—
386C4—6281no—
387C4—6281no—
388C4—6281no—
389C4—1572no—
390C4C2×D4×D15711yescharacteristic
391C4#41×D15711yescharacteristic
392C2—41no—
393C2—3142no—
394C2—3142no—
395C2—12561no—
396C2#121×D15711yescharacteristic
397C2#137×D15711yescharacteristic
398C2#2375807211yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#470×C157C2
socleC22×C157C22×D4
prime radical#23757786C2
hypercenter#470D157
hypocenterC157#3056
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
1. C2×#19097803C2semidirect
2. C2×M16×D157C2semidirect
4. C22⋊Q8×D157C2semidirect
5. #23757529C2semidirect
6. #23757521C2semidirect
7. #23757519C2semidirect
8. #23757513C2semidirect
9. #23757511C2semidirect
10. #23757505C2semidirect
11. #23757503C2semidirect
12. #23757312C2semidirect
13. #23757280C2semidirect
14. #23757273C2semidirect
15. #470×C157C2semidirect
20. C2×#14578944C22semidirect
21. C2×#14578943C22semidirect
22. C2×#14578941C22semidirect
45. #19097705C22semidirect
46. #19097704C22semidirect
47. #19097702C22semidirect
48. #19097696C22semidirect
49. #19097695C22semidirect
50. #19097695C22semidirect
51. #19097694C22semidirect
52. #19097693C22semidirect
53. #19097691C22semidirect
54. C2×Q16×C157C22semidirect
58. #106×C157C22semidirect
59. #103×C157C22semidirect
60. #103×C157C22semidirect
87. #14578937D4semidirect
179. C2×D157#137semidirect
199. C157#3056semidirect
206. #470D157semidirect
221. C2×Q16C2×D157semidirect
252. #106C2×D157semidirect
257. #103C2×D157semidirect
258. #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×M16×D1572
#237577862
C22⋊Q8×D1572
#237575292
#237575212
#237575192
#237575132
#237575112
#237575052
#237575032
#237573122
#237572802
#237572732
#470×C1572
#3056157