Finite Group Databasebeta

← #34111740

Subgroups — #34111740

Order 20096.

520 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19097798C211yesmaximal, characteristic, complemented
2C2×#19097796C211yesmaximal, characteristic, complemented
3#23757914C211yesmaximal, characteristic, complemented
4#23757911C211yesmaximal, characteristic, complemented
5#23757658C211yesmaximal, characteristic, complemented
6#23757522C211yesmaximal, characteristic, complemented
7#23757521C211yesmaximal, characteristic, complemented
8#23757491C211yesmaximal, characteristic, complemented
9#23757490C211yesmaximal, characteristic, complemented
10#23757434C211yesmaximal, characteristic, complemented
11#23757409C211yesmaximal, characteristic, complemented
12#23757408C211yesmaximal, characteristic, complemented
13#23757353C211yesmaximal, characteristic, complemented
14#23757348C211yesmaximal, characteristic, complemented
15#452×C157C211yesmaximal, characteristic, complemented
16C2×#14578955C2211yescharacteristic
17C22×D628C2211yescharacteristic, complemented
18C2×D4×D157C2211yescharacteristic, complemented
19#19097798—24no—
20#19097796—24no—
21C2×#14578941C2211yescharacteristic, complemented
22C2×#14578941C2211yescharacteristic, complemented
23C2×#14578941C2211yescharacteristic, complemented
24C2×D1256C2211yescharacteristic
25C2×D1256C2211yescharacteristic
26C2×#14578939C2211yescharacteristic
27C2×#14578939C2211yescharacteristic
28C2×#14578938C2211yescharacteristic, complemented
29C2×#14578937C2211yescharacteristic, complemented
30#19097760C2211yescharacteristic, complemented
31#19097760C2211yescharacteristic, complemented
32#19097756—21no—
33#19097754—21no—
34#19097747C2211yescharacteristic, complemented
35#19097746—21no—
36#19097746C2211yescharacteristic
37#19097745C2211yescharacteristic
38#19097743—21no—
39#19097741—21no—
40#19097739—21no—
41D4×Dic157C2211yescharacteristic, complemented
42#19097733C2211yescharacteristic, complemented
43C4×C22⋊D157C2211yescharacteristic
44C2×#14578931—21no—
45C22⋊C4×D157—21no—
46#19097704C2211yescharacteristic, complemented
47#19097700C2211yescharacteristic, complemented
48#19097695C2211yescharacteristic, complemented
49#19097693C2211yescharacteristic, complemented
50#19097693C2211yescharacteristic, complemented
51#19097691C2211yescharacteristic, complemented
52#19097683C2211yescharacteristic, complemented
53C8×Dic157C2211yescharacteristic, complemented
54C2×#41×C157C2211yescharacteristic
55C2×Q16×C157C2211yescharacteristic, complemented
56C2×SD16×C157C2211yescharacteristic, complemented
57C22⋊Q8×C157C2211yescharacteristic, complemented
58#103×C157C2211yescharacteristic, complemented
59#102×C157C2211yescharacteristic, complemented
60C22⋊C8×C157C2211yescharacteristic, complemented
61C23×D157—21no—
62C23×D157—21no—
63#14578955—22no—
64#14578955—41no—
65C2×D628—22no—
66C2×D628—21no—
67C2×D628—21no—
68C2×D628—21no—
69C2×D628—41no—
70C2×D628C2311yescharacteristic
71C2×D628C2311yescharacteristic
72C2×D628C2311yescharacteristic
73C2×C22⋊D157—21no—
74C2×C22⋊D157—21no—
75C2×C22⋊D157D411yescharacteristic
76D4×D157—22no—
77D4×D157—41no—
78C22×Dic157—21no—
79C2×C4×D157—21no—
80C2×C4×D157—21no—
81C2×C4×D157C2311yescharacteristic
82#14578941—22no—
83#14578941—22no—
84#14578941—22no—
85D1256—22no—
86D1256—41no—
87#14578939—22no—
88#14578939D414yescomplemented
89#14578938—22no—
90#14578937—22no—
91C2×#11929434C2311yescharacteristic
92C2×#11929434C2311yescharacteristic
93C4⋊Dic157C2311yescharacteristic
94#14578933—21no—
95#14578933D411yescharacteristic
96C22⋊Dic157—21no—
97C22⋊Dic157D411yescharacteristic
98#14578931—21no—
99#14578931—21no—
100#14578931—21no—
101#14578931—21no—
102#14578931—21no—
103#14578931—21no—
104#14578931—21no—
105#14578931—21no—
106#14578931D411yescharacteristic
107C4×Dic157C2311yescharacteristic
108#41×C157—22no—
109#41×C157—41no—
110C2×Q8×C157C2311yescharacteristic
111C2×Q8×C157C2311yescharacteristic
112C2×D4×C157—21no—
113C2×D4×C157C2311yescharacteristic
114C22×C4×C157—21no—
115C22×C4×C157C2311yescharacteristic
116Q16×C157—41no—
117SD16×C157—22no—
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—42no—
129C22×D157—42no—
130C22×D157—81no—
131C22×D157—81no—
132C22×D157C2×D411yescharacteristic
133D628—21no—
134D628—21no—
135D628—21no—
136D628—21no—
137D628—21no—
138D628—42no—
139D628—42no—
140D628—81no—
141D628C2×D412yes—
142C22⋊D157—41no—
143C22⋊D157—42no—
144C22⋊D157—81no—
145C2×Dic157—21no—
146C2×Dic157—81no—
147C2×Dic157C2×D411yescharacteristic
148C2×Dic157C2×D411yescharacteristic
149C2×Dic157C2×D411yescharacteristic
150C4×D157—21no—
151C4×D157—21no—
152C4×D157—42no—
153C4×D157—81no—
154#11929434—21no—
155#11929434C2×D412yes—
156C23×C157—21no—
157C23×C157C2×D411yescharacteristic
158Q8×C157—21no—
159Q8×C157—21no—
160Q8×C157—21no—
161Q8×C157—21no—
162D4×C157—21no—
163D4×C157—42no—
164D4×C157C2×D412yes—
165C2×C4×C157—21no—
166C2×C4×C157—21no—
167C2×C4×C157—21no—
168C2×C4×C157—21no—
169C2×C4×C157—21no—
170C2×C4×C157—42no—
171C2×C4×C157C2×D411yescharacteristic
172C2×C4×C157C2411yescharacteristic
173C8×C157—21no—
174C8×C157—21no—
175C22×C157—21no—
176C22×C157—22no—
177C22×C157—21no—
178C22×C157—21no—
179C22×C157—41no—
180C22×C157—41no—
181C22×C157C22×D411yescharacteristic
182C2×D157—21no—
183C2×D157—22no—
184C2×D157—21no—
185C2×D157—21no—
186C2×D157—41no—
187C2×D157—41no—
188C2×D157—41no—
189C2×D157—41no—
190C2×D157—41no—
191C2×D157—41no—
192C2×D157—41no—
193C2×D157—41no—
194C2×D157—84no—
195C2×D157—84no—
196C4×C157—21no—
197C4×C157—41no—
198C4×C157—41no—
199C4×C157—41no—
200C4×C157C22×D411yescharacteristic
201C4×C157H(5,𝔽2)11yescharacteristic
202Dic157—21no—
203Dic157—21no—
204Dic157—21no—
205Dic157—41no—
206C2×C157—41no—
207C2×C157—42no—
208C2×C157D4211yescharacteristic
209C2×C157#57511yescharacteristic
210C2×C157#57611yescharacteristic
211D157—41no—
212D157—42no—
213D157—81no—
214D157—81no—
215C157#327311yescharacteristic, complemented, Sylow
216#3273—1571nomaximal, Sylow
217C2×AGL(1,ℤ/8)—1571no—
218C2×AGL(1,ℤ/8)—1571no—
219#546—1571no—
220#546—1571no—
221#495—1571no—
222#468—1571no—
223#468—1571no—
224#463—1571no—
225#463—1571no—
226#452—1571no—
227#452D15711yescharacteristic, complemented
228#447—1571no—
229#447—1571no—
230C4×D8—1571no—
231#436—1571no—
232C2×#41—1571no—
233C2×#41C2×D15711yescharacteristic
234C22×D4—1571no—
235C22×D4—1571no—
236AGL(1,ℤ/8)—3144no—
237AGL(1,ℤ/8)—3144no—
238C2×Q16C2×D15711yescharacteristic, complemented
239C2×SD16—1571no—
240C2×SD16—1571no—
241C2×SD16—1571no—
242C2×SD16C2×D15711yescharacteristic, complemented
243C2×D8—1571no—
244C2×D8—1571no—
245C2×D8—1571no—
246C2×D8—1571no—
247C2×M16—1571no—
248C2×M16—1571no—
249#124—1571no—
250#124—1571no—
251#123—3141no—
252#123—3141no—
253C22⋊Q8—1571no—
254C22⋊Q8C2×D15711yescharacteristic, complemented
255#121—3141no—
256#121—3141no—
257#121—1571no—
258#121—1571no—
259#120—3141no—
260#120—3141no—
261C4×D4—1571no—
262C4×D4—1571no—
263C4×D4—1571no—
264C2×C22⋊C4—3141no—
265C2×C22⋊C4—3141no—
266#107—1571no—
267C4⋊C8—1571no—
268#103—1571no—
269#103C2×D15711yescharacteristic, complemented
270#102—1571no—
271#102—1571no—
272#102—1571no—
273#102C2×D15711yescharacteristic, complemented
274C22⋊C8—1571no—
275C22⋊C8C2×D15711yescharacteristic, complemented
276C4×C8—1571no—
277C24—3141no—
278C24—3141no—
279#41—22no—
280#41—41no—
281#41—3142no—
282#41—6281no—
283C2×Q8C22×D15711yescharacteristic
284C2×Q8C22×D15711yescharacteristic
285C2×D4—21no—
286C2×D4—3142no—
287C2×D4—3141no—
288C2×D4—3141no—
289C2×D4—3142no—
290C2×D4—3141no—
291C2×D4—3141no—
292C2×D4—3141no—
293C2×D4—6281no—
294C2×D4—6281no—
295C2×D4—1571no—
296C2×D4—1571no—
297C2×D4—1571no—
298C2×D4—1571no—
299C2×D4C22×D15711yescharacteristic
300C22×C4—21no—
301C22×C4—3141no—
302C22×C4—3141no—
303C22×C4—3141no—
304C22×C4—1571no—
305C22×C4C22×D15711yescharacteristic
306Q16—41no—
307SD16—22no—
308SD16—3142no—
309SD16—3142no—
310SD16—3142no—
311D8—3142no—
312D8—3142no—
313D8—6281no—
314D8—1574no—
315M16—3142no—
316M16—3142no—
317C2×C8—1571no—
318C2×C8—1571no—
319C2×C8C22×D15711yescharacteristic
320C2×C8C22×D15711yescharacteristic
321C4⋊C4—21no—
322C4⋊C4—3141no—
323C4⋊C4—1571no—
324C4⋊C4—1571no—
325C4⋊C4C22×D15711yescharacteristic
326C22⋊C4—21no—
327C22⋊C4—3141no—
328C22⋊C4—3141no—
329C22⋊C4—3141no—
330C22⋊C4—3141no—
331C22⋊C4—3141no—
332C22⋊C4—3141no—
333C22⋊C4—3141no—
334C22⋊C4—3141no—
335C22⋊C4—3141no—
336C22⋊C4—1571no—
337C22⋊C4—1571no—
338C42—1571no—
339C23—21no—
340C23—3141no—
341C23—3141no—
342C23—3141no—
343C23—6282no—
344C23—6282no—
345C23—6282no—
346C23—6282no—
347C23—1571no—
348C23—12561no—
349C23—12561no—
350C23D4×D15711yescharacteristic
351Q8—21no—
352Q8—21no—
353Q8—21no—
354Q8—21no—
355D4—21no—
356D4—42no—
357D4—3141no—
358D4—3141no—
359D4—3141no—
360D4—3141no—
361D4—3141no—
362D4—6282no—
363D4—6282no—
364D4—6281no—
365D4—6282no—
366D4—1572no—
367D4—12561no—
368D4—12561no—
369D4D4×D15712yes—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—42no—
376C2×C4—3141no—
377C2×C4—3141no—
378C2×C4—3141no—
379C2×C4—6282no—
380C2×C4—1571no—
381C2×C4—1571no—
382C2×C4—1571no—
383C2×C4—12561no—
384C2×C4—12561no—
385C2×C4D4×D15711yescharacteristic
386C2×C4C23×D15711yescharacteristic
387C8—21no—
388C8—21no—
389C8—3141no—
390C8—1572no—
391C22—21no—
392C22—22no—
393C22—21no—
394C22—21no—
395C22—41no—
396C22—41no—
397C22—3141no—
398C22—3142no—
399C22—3141no—
400C22—3141no—
401C22—6281no—
402C22—6281no—
403C22—6281no—
404C22—6281no—
405C22—6281no—
406C22—6281no—
407C22—6281no—
408C22—6281no—
409C22—12564no—
410C22—12564no—
411C22C2×D4×D15711yescharacteristic
412C4—21no—
413C4—41no—
414C4—41no—
415C4—41no—
416C4—3141no—
417C4—3141no—
418C4—3141no—
419C4—6281no—
420C4C2×D4×D15711yescharacteristic
421C4#1909781611yescharacteristic
422C2—41no—
423C2—42no—
424C2—6281no—
425C2—6282no—
426C2—12561no—
427C2—12561no—
428C2#2375790511yescharacteristic
429C2#2375807211yescharacteristic
430C2#2375807511yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#452×C157C2
socleC22×C157C22×D4
hypercenter#452D157
hypocenterC157#3273
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
1. C2×#19097798C2semidirect
2. C2×#19097796C2semidirect
3. #23757914C2semidirect
4. #23757911C2semidirect
5. #23757658C2semidirect
6. #23757522C2semidirect
7. #23757521C2semidirect
8. #23757491C2semidirect
9. #23757490C2semidirect
10. #23757434C2semidirect
11. #23757409C2semidirect
12. #23757408C2semidirect
13. #23757353C2semidirect
14. #23757348C2semidirect
15. #452×C157C2semidirect
17. C22×D628C22semidirect
18. C2×D4×D157C22semidirect
21. C2×#14578941C22semidirect
22. C2×#14578941C22semidirect
23. C2×#14578941C22semidirect
28. C2×#14578938C22semidirect
29. C2×#14578937C22semidirect
30. #19097760C22semidirect
31. #19097760C22semidirect
34. #19097747C22semidirect
41. D4×Dic157C22semidirect
42. #19097733C22semidirect
46. #19097704C22semidirect
47. #19097700C22semidirect
48. #19097695C22semidirect
49. #19097693C22semidirect
50. #19097693C22semidirect
51. #19097691C22semidirect
52. #19097683C22semidirect
53. C8×Dic157C22semidirect
55. C2×Q16×C157C22semidirect
56. C2×SD16×C157C22semidirect
57. C22⋊Q8×C157C22semidirect
58. #103×C157C22semidirect
59. #102×C157C22semidirect
60. C22⋊C8×C157C22semidirect
88. #14578939D4semidirect
215. C157#3273semidirect
227. #452D157semidirect
238. C2×Q16C2×D157semidirect
242. C2×SD16C2×D157semidirect
254. C22⋊Q8C2×D157semidirect
269. #103C2×D157semidirect
273. #102C2×D157semidirect
275. C22⋊C8C2×D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190977982
C2×#190977962
#237579142
#237579112
#237576582
#237575222
#237575212
#237574912
#237574902
#237574342
#237574092
#237574082
#237573532
#237573482
#452×C1572
#3273157