Finite Group Databasebeta

← #34111836

Subgroups — #34111836

Order 20096.

504 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19097794C211yesmaximal, characteristic, complemented
2C22×#14578941C211yesmaximal, characteristic, complemented
3#23757934C211yesmaximal, characteristic, complemented
4#23757914C211yesmaximal, characteristic, complemented
5#23757658C211yesmaximal, characteristic, complemented
6#23757505C211yesmaximal, characteristic, complemented
7#23757504C211yesmaximal, characteristic, complemented
8#23757497C211yesmaximal, characteristic, complemented
9#23757491C211yesmaximal, characteristic, complemented
10#23757434C211yesmaximal, characteristic, complemented
11#23757421C211yesmaximal, characteristic, complemented
12#23757409C211yesmaximal, characteristic, complemented
13#23757358C211yesmaximal, characteristic, complemented
14#23757336C211yesmaximal, characteristic, complemented
15#451×C157C211yesmaximal, characteristic, complemented
16C2×#14578955C2211yescharacteristic
17C2×#14578955C2211yescharacteristic, complemented
18C22×D628C2211yescharacteristic
19#19097794—24no—
20C2×#14578941—24no—
21C2×#14578941C2211yescharacteristic
22C2×#14578941C2211yescharacteristic
23C2×#14578941C2211yescharacteristic
24C2×#14578941C2211yescharacteristic
25C2×#14578941C2211yescharacteristic, complemented
26C2×D1256C2211yescharacteristic
27C2×D1256C2211yescharacteristic
28C22×#11929434C2211yescharacteristic
29C2×C8×D157C2211yescharacteristic, complemented
30#19097760—21no—
31#19097760C2211yescharacteristic
32#19097760C2211yescharacteristic, complemented
33#19097756—21no—
34#19097747—21no—
35#19097747C2211yescharacteristic, complemented
36#19097746—21no—
37#19097746C2211yescharacteristic
38#19097745—21no—
39#19097745C2211yescharacteristic
40#19097741—21no—
41Q8×Dic157C2211yescharacteristic, complemented
42#19097733C2211yescharacteristic
43C4×C22⋊D157—21no—
44C4×C22⋊D157C2211yescharacteristic
45C22⋊C4×D157—21no—
46#19097702C2211yescharacteristic, complemented
47#19097700C2211yescharacteristic, complemented
48#19097695C2211yescharacteristic, complemented
49#19097693C2211yescharacteristic
50#19097693C2211yescharacteristic
51#19097691C2211yescharacteristic, complemented
52#19097683C2211yescharacteristic, complemented
53C8×Dic157C2211yescharacteristic, complemented
54C22×Q8×C157C2211yescharacteristic
55C2×Q16×C157C2211yescharacteristic, complemented
56C2×Q16×C157C2211yescharacteristic, complemented
57C22⋊Q8×C157C2211yescharacteristic
58#103×C157C2211yescharacteristic, complemented
59#103×C157C2211yescharacteristic, complemented
60C22⋊C8×C157C2211yescharacteristic
61C23×D157—21no—
62#14578955—22no—
63#14578955—22no—
64#14578955—41no—
65#14578955—41no—
66C2×D628—22no—
67C2×D628—21no—
68C2×D628—21no—
69C2×D628—21no—
70C2×D628—21no—
71C2×D628—41no—
72C2×D628C2311yescharacteristic
73C2×D628C2311yescharacteristic
74C2×D628C2311yescharacteristic
75C2×C22⋊D157—21no—
76C2×C22⋊D157D411yescharacteristic
77C2×C4×D157—21no—
78C2×C4×D157—21no—
79C2×C4×D157—21no—
80C2×C4×D157C2311yescharacteristic
81#14578941—22no—
82#14578941—22no—
83#14578941—22no—
84#14578941—22no—
85#14578941D414yescomplemented
86D1256—22no—
87D1256—41no—
88C2×#11929434—22no—
89C2×#11929434C2311yescharacteristic
90C2×#11929434C2311yescharacteristic
91C8×D157—22no—
92C4⋊Dic157—21no—
93C4⋊Dic157C2311yescharacteristic
94#14578933—21no—
95#14578933D411yescharacteristic
96C22⋊Dic157D411yescharacteristic
97#14578931—21no—
98#14578931—21no—
99#14578931—21no—
100#14578931—21no—
101#14578931—21no—
102#14578931—21no—
103#14578931—21no—
104#14578931—21no—
105#14578931D411yescharacteristic
106C4×Dic157—21no—
107C4×Dic157C2311yescharacteristic
108C2×Q8×C157—21no—
109C2×Q8×C157—22no—
110C2×Q8×C157—41no—
111C2×Q8×C157C2311yescharacteristic
112C2×Q8×C157C2311yescharacteristic
113C2×Q8×C157C2311yescharacteristic
114C22×C4×C157—21no—
115C22×C4×C157C2311yescharacteristic
116Q16×C157—22no—
117Q16×C157—41no—
118C2×C8×C157C2311yescharacteristic
119C2×C8×C157C2311yescharacteristic
120C4⋊C4×C157—21no—
121C4⋊C4×C157C2311yescharacteristic
122C22⋊C4×C157—21no—
123C22×D157—21no—
124C22×D157—21no—
125C22×D157—21no—
126C22×D157—42no—
127C22×D157—42no—
128C22×D157—81no—
129C22×D157C2×D411yescharacteristic
130D628—21no—
131D628—21no—
132D628—21no—
133D628—21no—
134D628—21no—
135D628—42no—
136D628—42no—
137D628—42no—
138D628—81no—
139D628C2×D412yes—
140C22⋊D157—22no—
141C22⋊D157—42no—
142C2×Dic157—21no—
143C2×Dic157C2×D411yescharacteristic
144C2×Dic157C2×D411yescharacteristic
145C2×Dic157C2×D411yescharacteristic
146C4×D157—21no—
147C4×D157—21no—
148C4×D157—42no—
149C4×D157—42no—
150C4×D157—81no—
151#11929434—21no—
152#11929434C2×D412yes—
153C23×C157C2×D411yescharacteristic
154Q8×C157—21no—
155Q8×C157—21no—
156Q8×C157—21no—
157Q8×C157—21no—
158Q8×C157—21no—
159Q8×C157—42no—
160Q8×C157C2×D412yes—
161C2×C4×C157—21no—
162C2×C4×C157—21no—
163C2×C4×C157—21no—
164C2×C4×C157—21no—
165C2×C4×C157—21no—
166C2×C4×C157—21no—
167C2×C4×C157—42no—
168C2×C4×C157C2×D411yescharacteristic
169C2×C4×C157C2411yescharacteristic
170C8×C157—21no—
171C8×C157—21no—
172C22×C157—21no—
173C22×C157—21no—
174C22×C157#13512yescomplemented
175C22×C157C22×D411yescharacteristic
176C2×D157—21no—
177C2×D157—22no—
178C2×D157—21no—
179C2×D157—21no—
180C2×D157—41no—
181C2×D157—41no—
182C2×D157—41no—
183C2×D157—41no—
184C2×D157—41no—
185C2×D157—41no—
186C2×D157—41no—
187C2×D157—41no—
188C2×D157—84no—
189C4×C157—21no—
190C4×C157—22no—
191C4×C157—41no—
192C4×C157—41no—
193C4×C157—41no—
194C4×C157C22×D411yescharacteristic
195C4×C157H(5,𝔽2)11yescharacteristic
196Dic157—21no—
197Dic157—21no—
198Dic157—21no—
199Dic157—41no—
200C2×C157—22no—
201C2×C157D4211yescharacteristic
202C2×C157C2×#13511yescharacteristic
203C2×C157#57711yescharacteristic
204D157—41no—
205D157—42no—
206D157—81no—
207D157—81no—
208C157#328211yescharacteristic, complemented, Sylow
209#3282—1571nomaximal, Sylow
210C2×#135—1571no—
211C22×SD16—1571no—
212#548—1571no—
213#546—1571no—
214#495—1571no—
215#465—1571no—
216#465—1571no—
217#464—1571no—
218#463—1571no—
219#452—1571no—
220#451D15711yescharacteristic, complemented
221#449—1571no—
222#447—1571no—
223C4×SD16—1571no—
224C8×D4—1571no—
225C2×#41—1571no—
226C2×#41—1571no—
227C22×Q8C2×D15711yescharacteristic
228C22×D4—1571no—
229#135—3144no—
230C2×Q16C2×D15711yescharacteristic, complemented
231C2×Q16C2×D15711yescharacteristic, complemented
232C2×SD16—3144no—
233C2×SD16—1571no—
234C2×SD16—1571no—
235C2×SD16—1571no—
236C2×SD16—1571no—
237C2×SD16—1571no—
238C2×D8—1571no—
239C2×D8—1571no—
240C22×C8—1571no—
241C22×C8—1571no—
242#124—3141no—
243#124—1571no—
244#124—1571no—
245#123—3141no—
246C22⋊Q8—3141no—
247C22⋊Q8—1571no—
248C22⋊Q8C2×D15711yescharacteristic
249#121—3141no—
250#121—3141no—
251#121—1571no—
252#121—1571no—
253#120—3141no—
254C4×Q8—1571no—
255C4×D4—3141no—
256C4×D4—1571no—
257C4×D4—1571no—
258C2×C22⋊C4—3141no—
259#106—1571no—
260C4⋊C8—1571no—
261#103—1571no—
262#103C2×D15711yescharacteristic, complemented
263#103C2×D15711yescharacteristic, complemented
264#102—1571no—
265#102—1571no—
266#102—1571no—
267C22⋊C8—1571no—
268C22⋊C8C2×D15711yescharacteristic
269C4×C8—1571no—
270C24—3141no—
271#41—3142no—
272#41—3142no—
273#41—6281no—
274#41—6281no—
275C2×Q8—21no—
276C2×Q8—22no—
277C2×Q8—41no—
278C2×Q8C22×D15711yescharacteristic
279C2×Q8C22×D15711yescharacteristic
280C2×Q8C22×D15711yescharacteristic
281C2×D4—3142no—
282C2×D4—3141no—
283C2×D4—3141no—
284C2×D4—3141no—
285C2×D4—3141no—
286C2×D4—3141no—
287C2×D4—6281no—
288C2×D4—1571no—
289C2×D4—1571no—
290C2×D4—1571no—
291C2×D4—1571no—
292C22×C4—21no—
293C22×C4—3141no—
294C22×C4—3141no—
295C22×C4—3141no—
296C22×C4—1571no—
297C22×C4C22×D15711yescharacteristic
298Q16—22no—
299Q16—41no—
300SD16—3142no—
301SD16—3142no—
302SD16—3142no—
303SD16—3142no—
304SD16—1574no—
305D8—3142no—
306D8—6281no—
307C2×C8—3142no—
308C2×C8—3142no—
309C2×C8—1571no—
310C2×C8—1571no—
311C2×C8C22×D15711yescharacteristic
312C2×C8C22×D15711yescharacteristic
313C4⋊C4—21no—
314C4⋊C4—3141no—
315C4⋊C4—3141no—
316C4⋊C4—1571no—
317C4⋊C4—1571no—
318C4⋊C4C22×D15711yescharacteristic
319C22⋊C4—21no—
320C22⋊C4—3141no—
321C22⋊C4—3141no—
322C22⋊C4—3141no—
323C22⋊C4—3141no—
324C22⋊C4—3141no—
325C22⋊C4—3141no—
326C22⋊C4—3141no—
327C22⋊C4—3141no—
328C22⋊C4—1571no—
329C22⋊C4—1571no—
330C42—3141no—
331C42—1571no—
332C23—3141no—
333C23—3141no—
334C23—3141no—
335C23—6282no—
336C23—6282no—
337C23—1571no—
338C23—12561no—
339C23D4×D15711yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—21no—
343Q8—21no—
344Q8—21no—
345Q8—42no—
346Q8D4×D15712yes—
347D4—3142no—
348D4—3141no—
349D4—3141no—
350D4—3141no—
351D4—3141no—
352D4—3141no—
353D4—6282no—
354D4—6282no—
355D4—6282no—
356D4—6282no—
357D4—1572no—
358D4—12561no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—42no—
366C2×C4—3141no—
367C2×C4—3141no—
368C2×C4—3141no—
369C2×C4—6282no—
370C2×C4—6282no—
371C2×C4—1571no—
372C2×C4—1571no—
373C2×C4—1571no—
374C2×C4—12561no—
375C2×C4D4×D15711yescharacteristic
376C2×C4C23×D15711yescharacteristic
377C8—21no—
378C8—21no—
379C8—3141no—
380C8—1572no—
381C22—21no—
382C22—21no—
383C22—3141no—
384C22—3141no—
385C22—3142no—
386C22—3141no—
387C22—6281no—
388C22—6281no—
389C22—6281no—
390C22—6281no—
391C22—6281no—
392C22—6281no—
393C22—6281no—
394C22—6281no—
395C22—12564no—
396C22#1909779412yescomplemented
397C22C2×D4×D15711yescharacteristic
398C4—21no—
399C4—22no—
400C4—41no—
401C4—41no—
402C4—41no—
403C4—3141no—
404C4—3141no—
405C4—3141no—
406C4—6281no—
407C4C2×D4×D15711yescharacteristic
408C4#1909781611yescharacteristic
409C2—22no—
410C2—6281no—
411C2—6282no—
412C2—12561no—
413C2—12561no—
414C2#2375790511yescharacteristic
415C2C2×#1909779411yescharacteristic
416C2#2375808211yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#451×C157C2
socleC22×C157C22×D4
hypercenter#451D157
hypocenterC157#3282
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
1. C2×#19097794C2semidirect
2. C22×#14578941C2semidirect
3. #23757934C2semidirect
4. #23757914C2semidirect
5. #23757658C2semidirect
6. #23757505C2semidirect
7. #23757504C2semidirect
8. #23757497C2semidirect
9. #23757491C2semidirect
10. #23757434C2semidirect
11. #23757421C2semidirect
12. #23757409C2semidirect
13. #23757358C2semidirect
14. #23757336C2semidirect
15. #451×C157C2semidirect
17. C2×#14578955C22semidirect
25. C2×#14578941C22semidirect
29. C2×C8×D157C22semidirect
32. #19097760C22semidirect
35. #19097747C22semidirect
41. Q8×Dic157C22semidirect
46. #19097702C22semidirect
47. #19097700C22semidirect
48. #19097695C22semidirect
51. #19097691C22semidirect
52. #19097683C22semidirect
53. C8×Dic157C22semidirect
55. C2×Q16×C157C22semidirect
56. C2×Q16×C157C22semidirect
58. #103×C157C22semidirect
59. #103×C157C22semidirect
85. #14578941D4semidirect
174. C22×C157#135semidirect
208. C157#3282semidirect
220. #451D157semidirect
230. C2×Q16C2×D157semidirect
231. C2×Q16C2×D157semidirect
262. #103C2×D157semidirect
263. #103C2×D157semidirect
396. C22#19097794semidirect

Maximal subgroups

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

subgroupindex
C2×#190977942
C22×#145789412
#237579342
#237579142
#237576582
#237575052
#237575042
#237574972
#237574912
#237574342
#237574212
#237574092
#237573582
#237573362
#451×C1572
#3282157