Finite Group Databasebeta

← #34109759

Subgroups — #34109759

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×#19097800C211yesmaximal, characteristic, complemented
2C22×#14578937C211yesmaximal, characteristic, complemented
3#23757783C211yesmaximal, characteristic
4C22⋊Q8×D157C211yesmaximal, characteristic, complemented
5#23757532C211yesmaximal, characteristic, complemented
6#23757530C211yesmaximal, characteristic, complemented
7#23757518C211yesmaximal, characteristic, complemented
8#23757516C211yesmaximal, characteristic, complemented
9#23757514C211yesmaximal, characteristic, complemented
10#23757512C211yesmaximal, characteristic, complemented
11#23757500C211yesmaximal, characteristic, complemented
12#23757312C211yesmaximal, characteristic, complemented
13#23757280C211yesmaximal, characteristic, complemented
14#23757271C211yesmaximal, characteristic, complemented
15#465×C157C211yesmaximal, characteristic, complemented
16C2×#14578953C2211yescharacteristic
17C2×Q8×D157C2211yescharacteristic
18C22×C4×D157C2211yescharacteristic
19#19097800—24no—
20C2×Dic628C2211yescharacteristic, complemented
21C2×#14578944C2211yescharacteristic, complemented
22C2×#14578942C2211yescharacteristic, complemented
23C2×#14578937—24no—
24C2×#14578937C2211yescharacteristic
25C2×#14578937C2211yescharacteristic
26C2×#14578937C2211yescharacteristic
27C2×#14578937C2211yescharacteristic
28C22×#11929434C2211yescharacteristic
29#19097753—21no—
30#19097751C2211yescharacteristic
31#19097751C2211yescharacteristic
32#19097750C2211yescharacteristic
33#19097750C2211yescharacteristic
34#19097749—21no—
35#19097748—21no—
36#19097747C2211yescharacteristic
37#19097742C2211yescharacteristic
38#19097739—21no—
39D4×Dic157—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#19097696C2211yescharacteristic, complemented
50#19097695C2211yescharacteristic, complemented
51#19097694C2211yescharacteristic, complemented
52#19097694C2211yescharacteristic, complemented
53#19097692C2211yescharacteristic, complemented
54C2×SD16×C157C2211yescharacteristic, complemented
55C22×C8×C157C2211yescharacteristic
56C22⋊Q8×C157C2211yescharacteristic
57#121×C157C2211yescharacteristic
58#106×C157C2211yescharacteristic, complemented
59#103×C157C2211yescharacteristic, complemented
60#102×C157C2211yescharacteristic, complemented
61C23×D157D411yescharacteristic
62#14578953—22no—
63#14578953—22no—
64C2×Dic314—21no—
65C2×Dic314C2311yescharacteristic
66C2×Dic314C2311yescharacteristic
67Q8×D157—22no—
68Q8×D157—22no—
69C2×C22⋊D157—21no—
70C2×C22⋊D157—21no—
71C22×Dic157—21no—
72C22×Dic157D411yescharacteristic
73C2×C4×D157—22no—
74C2×C4×D157—22no—
75C2×C4×D157—21no—
76C2×C4×D157—21no—
77C2×C4×D157D411yescharacteristic
78C2×C4×D157D411yescharacteristic
79C2×C4×D157C2311yescharacteristic
80C2×C4×D157C2311yescharacteristic
81Dic628—22no—
82#14578944—22no—
83#14578942—22no—
84#14578937—22no—
85#14578937—22no—
86#14578937—22no—
87#14578937D414yescomplemented
88C2×#11929434—22no—
89C2×#11929434C2311yescharacteristic
90C2×#11929434C2311yescharacteristic
91C4⋊Dic157—21no—
92C4⋊Dic157—21no—
93C4⋊Dic157C2311yescharacteristic
94C4⋊Dic157C2311yescharacteristic
95#14578933—21no—
96#14578933—21no—
97#14578933—21no—
98#14578933—21no—
99C22⋊Dic157—21no—
100C22⋊Dic157—21no—
101C22⋊Dic157—21no—
102#14578931—21no—
103#14578931—21no—
104#14578931—21no—
105#14578931—21no—
106C4×Dic157—21no—
107C2×Q8×C157C2311yescharacteristic
108C2×D4×C157—21no—
109C2×D4×C157C2311yescharacteristic
110C22×C4×C157C2311yescharacteristic
111SD16×C157—22no—
112C2×C8×C157—22no—
113C2×C8×C157C2311yescharacteristic
114C2×C8×C157C2311yescharacteristic
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—41no—
123C22×D157C2×D411yescharacteristic
124C22×D157C2×D411yescharacteristic
125Dic314—21no—
126Dic314—21no—
127Dic314—21no—
128Dic314—21no—
129Dic314—42no—
130C22⋊D157—42no—
131C22⋊D157—42no—
132C2×Dic157—22no—
133C2×Dic157—21no—
134C2×Dic157—21no—
135C2×Dic157—21no—
136C2×Dic157—21no—
137C2×Dic157—42no—
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×D157C2×D412yes—
147C4×D157#4112yes—
148#11929434—21no—
149#11929434C2×D412yes—
150C23×C157—21no—
151C23×C157C2×D411yescharacteristic
152Q8×C157—21no—
153Q8×C157—21no—
154D4×C157—21no—
155D4×C157—21no—
156D4×C157—42no—
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—41no—
169C22×C157—41no—
170C22×C157—41no—
171C22×C157#13712yescomplemented
172C22×C157C22×D411yescharacteristic
173C2×D157—21no—
174C2×D157—21no—
175C2×D157—21no—
176C2×D157—21no—
177C2×D157—21no—
178C2×D157—44no—
179C2×D157#12112yes—
180C4×C157—21no—
181C4×C157—41no—
182C4×C157—41no—
183C4×C157—41no—
184C4×C157C22×D411yescharacteristic
185C4×C157C2×#4111yescharacteristic
186Dic157—21no—
187Dic157—41no—
188Dic157—41no—
189Dic157—41no—
190Dic157—41no—
191Dic157#12112yes—
192C2×C157—22no—
193C2×C157—81no—
194C2×C157C2×#12111yescharacteristic
195C2×C157C2×#13711yescharacteristic
196C2×C157#57511yescharacteristic
197D157—22no—
198D157—41no—
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#470—1571no—
207#468—1571no—
208#468—1571no—
209#467—1571no—
210#467—1571no—
211#465—1571no—
212#465D15711yescharacteristic, complemented
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×Q16—1571no—
222C2×SD16—1571no—
223C2×SD16C2×D15711yescharacteristic, complemented
224C2×M16—3144no—
225C2×M16—1571no—
226C2×M16—1571no—
227C2×M16—1571no—
228C2×M16—1571no—
229C22×C8—1571no—
230C22×C8C2×D15711yescharacteristic
231#123—3141no—
232C22⋊Q8—3141no—
233C22⋊Q8—3141no—
234C22⋊Q8—1571no—
235C22⋊Q8—1571no—
236C22⋊Q8—1571no—
237C22⋊Q8—1571no—
238C22⋊Q8—1571no—
239C22⋊Q8C2×D15711yescharacteristic
240#121—1571no—
241#121C2×D15711yescharacteristic
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#103—1571no—
258#103C2×D15711yescharacteristic, complemented
259#102—1571no—
260#102C2×D15711yescharacteristic, complemented
261C24—1571no—
262#41—3142no—
263#41—3142no—
264C2×Q8—3141no—
265C2×Q8—3142no—
266C2×Q8—3142no—
267C2×Q8—1571no—
268C2×Q8—1571no—
269C2×Q8C22×D15711yescharacteristic
270C2×D4—21no—
271C2×D4—3141no—
272C2×D4—3141no—
273C2×D4C22×D15711yescharacteristic
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—3142no—
286Q16—3142no—
287SD16—22no—
288SD16—3142no—
289M16—3142no—
290M16—3142no—
291M16—3142no—
292M16—1574no—
293C2×C8—22no—
294C2×C8—3142no—
295C2×C8—1571no—
296C2×C8—1571no—
297C2×C8C22×D15711yescharacteristic
298C2×C8C22×D15711yescharacteristic
299C4⋊C4—21no—
300C4⋊C4—3141no—
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—21no—
321C23—3142no—
322C23—3142no—
323C23—6281no—
324C23—1571no—
325C23—1571no—
326C23D4×D15711yescharacteristic
327Q8—21no—
328Q8—21no—
329Q8—3141no—
330Q8—3141no—
331Q8—3141no—
332Q8—3141no—
333Q8—6282no—
334D4—21no—
335D4—21no—
336D4—42no—
337D4—6282no—
338D4—6282no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—3142no—
345C2×C4—3142no—
346C2×C4—3142no—
347C2×C4—3141no—
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—41no—
369C22—41no—
370C22—41no—
371C22—3141no—
372C22—3141no—
373C22—3141no—
374C22—3141no—
375C22—3141no—
376C22—6284no—
377C22—1572no—
378C22#1909780012yescomplemented
379C22C2×D4×D15711yescharacteristic
380C4—21no—
381C4—41no—
382C4—41no—
383C4—41no—
384C4—3141no—
385C4—6281no—
386C4—6281no—
387C4—6281no—
388C4—6281no—
389C4—1572no—
390C4C2×D4×D15711yescharacteristic
391C4#41×D15711yescharacteristic
392C2—22no—
393C2—81no—
394C2—3142no—
395C2—6281no—
396C2#121×D15711yescharacteristic
397C2C2×#1909780011yescharacteristic
398C2#2375807011yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#465×C157C2
socleC22×C157C22×D4
prime radical#23757783C2
hypercenter#465D157
hypocenterC157#3056
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. C2×#19097800C2semidirect
2. C22×#14578937C2semidirect
4. C22⋊Q8×D157C2semidirect
5. #23757532C2semidirect
6. #23757530C2semidirect
7. #23757518C2semidirect
8. #23757516C2semidirect
9. #23757514C2semidirect
10. #23757512C2semidirect
11. #23757500C2semidirect
12. #23757312C2semidirect
13. #23757280C2semidirect
14. #23757271C2semidirect
15. #465×C157C2semidirect
20. C2×Dic628C22semidirect
21. C2×#14578944C22semidirect
22. C2×#14578942C22semidirect
45. #19097705C22semidirect
46. #19097704C22semidirect
47. #19097702C22semidirect
48. #19097696C22semidirect
49. #19097696C22semidirect
50. #19097695C22semidirect
51. #19097694C22semidirect
52. #19097694C22semidirect
53. #19097692C22semidirect
54. C2×SD16×C157C22semidirect
58. #106×C157C22semidirect
59. #103×C157C22semidirect
60. #102×C157C22semidirect
87. #14578937D4semidirect
171. C22×C157#137semidirect
199. C157#3056semidirect
212. #465D157semidirect
223. C2×SD16C2×D157semidirect
252. #106C2×D157semidirect
258. #103C2×D157semidirect
260. #102C2×D157semidirect
378. C22#19097800semidirect

Maximal subgroups

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

subgroupindex
C2×#190978002
C22×#145789372
#237577832
C22⋊Q8×D1572
#237575322
#237575302
#237575182
#237575162
#237575142
#237575122
#237575002
#237573122
#237572802
#237572712
#465×C1572
#3056157