Finite Group Databasebeta

← #34111830

Subgroups — #34111830

Order 20096.

546 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14578942C211yesmaximal, characteristic, complemented
2C2×SD16×D157C211yesmaximal, characteristic, complemented
3#23757936C211yesmaximal, characteristic, complemented
4D4×C22⋊D157C211yesmaximal, characteristic
5#23757644C211yesmaximal, characteristic, complemented
6#23757504C211yesmaximal, characteristic, complemented
7#23757501C211yesmaximal, characteristic, complemented
8#23757476C211yesmaximal, characteristic, complemented
9#23757472C211yesmaximal, characteristic, complemented
10#23757425C211yesmaximal, characteristic, complemented
11#23757414C211yesmaximal, characteristic, complemented
12#23757410C211yesmaximal, characteristic, complemented
13#23757364C211yesmaximal, characteristic, complemented
14#23757336C211yesmaximal, characteristic, complemented
15#450×C157C211yesmaximal, characteristic, complemented
16C22×Dic314C2211yescharacteristic, complemented
17C2×Q8×D157C2211yescharacteristic, complemented
18C22×C22⋊D157—21no—
19C2×D4×D157C2211yescharacteristic
20C2×#14578943C2211yescharacteristic, complemented
21C2×#14578943C2211yescharacteristic, complemented
22C2×#14578942—24no—
23C2×#14578942C2211yescharacteristic
24C2×#14578942C2211yescharacteristic
25C2×#14578942C2211yescharacteristic, complemented
26C2×#14578942C2211yescharacteristic, complemented
27C2×#14578941C2211yescharacteristic, complemented
28SD16×D157—24no—
29C22×#11929434C2211yescharacteristic, complemented
30C2×C8×D157C2211yescharacteristic, complemented
31#19097770—21no—
32#19097769C2211yescharacteristic, complemented
33#19097767C2211yescharacteristic
34#19097751C2211yescharacteristic, complemented
35#19097749—21no—
36#19097748—21no—
37#19097745C2211yescharacteristic
38#19097743—21no—
39#19097742C2211yescharacteristic
40#19097740—21no—
41#19097739—21no—
42#19097736C2211yescharacteristic, complemented
43D4×Dic157C2211yescharacteristic
44C4×C22⋊D157C2211yescharacteristic
45#19097731—21no—
46#19097702C2211yescharacteristic, complemented
47#19097700C2211yescharacteristic, complemented
48#19097696C2211yescharacteristic, complemented
49#19097696C2211yescharacteristic, complemented
50#19097695C2211yescharacteristic, complemented
51#19097691C2211yescharacteristic, complemented
52#19097683C2211yescharacteristic, complemented
53C8×Dic157C2211yescharacteristic, complemented
54C22×D4×C157C2211yescharacteristic
55C2×SD16×C157C2211yescharacteristic
56C2×SD16×C157C2211yescharacteristic, complemented
57C22⋊Q8×C157C2211yescharacteristic, complemented
58#102×C157C2211yescharacteristic
59#102×C157C2211yescharacteristic, complemented
60C22⋊C8×C157C2211yescharacteristic, complemented
61C23×D157—21no—
62C2×Dic314—21no—
63C2×Dic314—22no—
64C2×Dic314—21no—
65C2×Dic314—41no—
66C2×Dic314C2311yescharacteristic
67C2×Dic314C2311yescharacteristic
68Q8×D157—22no—
69Q8×D157—41no—
70C2×D628C2311yescharacteristic
71C2×C22⋊D157—21no—
72C2×C22⋊D157—21no—
73C2×C22⋊D157—21no—
74C2×C22⋊D157—21no—
75C2×C22⋊D157—21no—
76C2×C22⋊D157—44no—
77C2×C22⋊D157D411yescharacteristic
78D4×D157—22no—
79D4×D157—41no—
80C22×Dic157—21no—
81C22×Dic157—21no—
82C2×C4×D157—21no—
83C2×C4×D157C2311yescharacteristic
84#14578943—22no—
85#14578943—41no—
86#14578942—22no—
87#14578942—22no—
88#14578942—22no—
89#14578942D414yescomplemented
90#14578941—22no—
91C2×#11929434—22no—
92C2×#11929434C2311yescharacteristic
93C2×#11929434C2311yescharacteristic
94C8×D157—22no—
95C4⋊Dic157—21no—
96C4⋊Dic157C2311yescharacteristic
97#14578933—21no—
98#14578933—21no—
99#14578933—21no—
100#14578933D411yescharacteristic
101C22⋊Dic157—21no—
102C22⋊Dic157—21no—
103C22⋊Dic157D411yescharacteristic
104#14578931—21no—
105#14578931—21no—
106#14578931D411yescharacteristic
107C4×Dic157—21no—
108C4×Dic157C2311yescharacteristic
109C24×C157—21no—
110C2×Q8×C157C2311yescharacteristic
111C2×D4×C157—22no—
112C2×D4×C157—21no—
113C2×D4×C157—41no—
114C2×D4×C157C2311yescharacteristic
115C2×D4×C157C2311yescharacteristic
116C22×C4×C157C2311yescharacteristic
117SD16×C157—22no—
118SD16×C157—41no—
119C2×C8×C157C2311yescharacteristic
120C2×C8×C157C2311yescharacteristic
121C4⋊C4×C157—21no—
122C4⋊C4×C157C2311yescharacteristic
123C22⋊C4×C157—21no—
124C22×D157—21no—
125C22×D157—42no—
126C22×D157—42no—
127C22×D157—81no—
128C22×D157C2×D411yescharacteristic
129Dic314—21no—
130Dic314—21no—
131Dic314—21no—
132Dic314—42no—
133Dic314—42no—
134Dic314C2×D412yescomplemented
135D628—21no—
136D628—21no—
137C22⋊D157—42no—
138C22⋊D157—42no—
139C22⋊D157—42no—
140C22⋊D157—42no—
141C22⋊D157—81no—
142C22⋊D157SD1614yescomplemented
143C2×Dic157—21no—
144C2×Dic157—21no—
145C2×Dic157—21no—
146C2×Dic157—42no—
147C2×Dic157—42no—
148C2×Dic157C2×D411yescharacteristic
149C2×Dic157C2×D411yescharacteristic
150C2×Dic157C2×D411yescharacteristic
151C4×D157—21no—
152C4×D157—21no—
153C4×D157—42no—
154#11929434—21no—
155#11929434C2×D412yescomplemented
156C23×C157—21no—
157C23×C157—21no—
158C23×C157—42no—
159C23×C157—42no—
160C23×C157—81no—
161C23×C157C2×D411yescharacteristic
162Q8×C157—21no—
163Q8×C157—21no—
164D4×C157—21no—
165D4×C157—21no—
166D4×C157—21no—
167D4×C157—42no—
168D4×C157C2×D412yes—
169C2×C4×C157—21no—
170C2×C4×C157—21no—
171C2×C4×C157—21no—
172C2×C4×C157—21no—
173C2×C4×C157C2×D411yescharacteristic
174C2×C4×C157C2411yescharacteristic
175C8×C157—21no—
176C8×C157—21no—
177C22×C157—21no—
178C22×C157—21no—
179C22×C157—22no—
180C22×C157—41no—
181C22×C157—41no—
182C22×C157—41no—
183C22×C157—41no—
184C22×C157—41no—
185C22×C157—84no—
186C22×C157C2×SD1612yescomplemented
187C22×C157C22×D411yescharacteristic
188C2×D157—21no—
189C2×D157—21no—
190C2×D157—41no—
191C2×D157—41no—
192C2×D157—41no—
193C2×D157—84no—
194C2×D157C2×SD1612yescomplemented
195C4×C157—21no—
196C4×C157—41no—
197C4×C157—41no—
198C4×C157C22×D411yescharacteristic
199C4×C157H(5,𝔽2)11yescharacteristic
200Dic157—21no—
201Dic157—22no—
202Dic157—21no—
203Dic157—41no—
204Dic157—41no—
205Dic157C2×SD1612yescomplemented
206C2×C157—22no—
207C2×C157—42no—
208C2×C157—81no—
209C2×C157D4211yescharacteristic
210C2×C157C22×SD1611yescharacteristic
211C2×C157#57711yescharacteristic
212D157—22no—
213D157—81no—
214C157D4×SD1611yescharacteristic, complemented, Sylow
215D4×SD16—1571nomaximal, Sylow
216C22×SD16—1571no—
217C22×SD16—1571no—
218D4×Q8—1571no—
219D42—1571no—
220#492—1571no—
221#465—1571no—
222#465—1571no—
223#461—1571no—
224#460—1571no—
225#450—1571no—
226#450D15711yescharacteristic, complemented
227#448—1571no—
228#448—1571no—
229C4×SD16—1571no—
230C8×D4—1571no—
231C22×Q8—1571no—
232C22×Q8—1571no—
233C22×D4—3141no—
234C22×D4—1571no—
235C22×D4C2×D15711yescharacteristic
236C2×SD16—3144no—
237C2×SD16—3144no—
238C2×SD16—1571no—
239C2×SD16—1571no—
240C2×SD16—1571no—
241C2×SD16—1571no—
242C2×SD16—1571no—
243C2×SD16—1571no—
244C2×SD16—1571no—
245C2×SD16C2×D15711yescharacteristic
246C2×SD16C2×D15711yescharacteristic, complemented
247C22×C8—1571no—
248C22×C8—1571no—
249C4⋊Q8—3141no—
250C4⋊Q8—1571no—
251#127—1571no—
252C22⋊Q8—3141no—
253C22⋊Q8—3141no—
254C22⋊Q8—1571no—
255C22⋊Q8C2×D15711yescharacteristic, complemented
256#121—3141no—
257#121—1571no—
258#121—1571no—
259#120—3141no—
260#120—3141no—
261C4×Q8—1571no—
262C4×D4—3141no—
263C4×D4—1571no—
264C4×D4—1571no—
265#106—1571no—
266C4⋊C8—1571no—
267#103—1571no—
268#103—1571no—
269#103—1571no—
270#102—1571no—
271#102C2×D15711yescharacteristic
272#102C2×D15711yescharacteristic, complemented
273C22⋊C8—1571no—
274C22⋊C8C2×D15711yescharacteristic, complemented
275C4×C8—1571no—
276C24—21no—
277C24—3141no—
278C2×Q8—3141no—
279C2×Q8—3142no—
280C2×Q8—3141no—
281C2×Q8—3142no—
282C2×Q8—6281no—
283C2×Q8—6281no—
284C2×Q8—1571no—
285C2×Q8—1571no—
286C2×Q8C22×D15711yescharacteristic
287C2×D4—22no—
288C2×D4—21no—
289C2×D4—41no—
290C2×D4—3142no—
291C2×D4—3141no—
292C2×D4—3141no—
293C2×D4—3141no—
294C2×D4—3141no—
295C2×D4—3141no—
296C2×D4—6281no—
297C2×D4—6284no—
298C2×D4—1571no—
299C2×D4—1571no—
300C2×D4C22×D15711yescharacteristic
301C2×D4C22×D15711yescharacteristic
302C22×C4—3141no—
303C22×C4—3141no—
304C22×C4—3141no—
305C22×C4—1571no—
306C22×C4C22×D15711yescharacteristic
307SD16—22no—
308SD16—41no—
309SD16—3142no—
310SD16—3142no—
311SD16—3142no—
312SD16—3142no—
313SD16—3142no—
314SD16—6281no—
315SD16—1574no—
316C2×C8—3142no—
317C2×C8—3142no—
318C2×C8—1571no—
319C2×C8—1571no—
320C2×C8C22×D15711yescharacteristic
321C2×C8C22×D15711yescharacteristic
322C4⋊C4—21no—
323C4⋊C4—3141no—
324C4⋊C4—3141no—
325C4⋊C4—3141no—
326C4⋊C4—3141no—
327C4⋊C4—1571no—
328C4⋊C4—1571no—
329C4⋊C4C22×D15711yescharacteristic
330C22⋊C4—21no—
331C22⋊C4—3141no—
332C22⋊C4—3141no—
333C22⋊C4—3141no—
334C22⋊C4—3141no—
335C22⋊C4—1571no—
336C22⋊C4—1571no—
337C42—3141no—
338C42—1571no—
339C23—21no—
340C23—21no—
341C23—42no—
342C23—42no—
343C23—81no—
344C23—3141no—
345C23—6282no—
346C23—6282no—
347C23—1571no—
348C23—12561no—
349C23D4×D15711yescharacteristic
350Q8—21no—
351Q8—21no—
352Q8—3141no—
353Q8—3141no—
354Q8—3141no—
355Q8—6282no—
356Q8—6282no—
357Q8—1572no—
358D4—21no—
359D4—21no—
360D4—21no—
361D4—42no—
362D4—3141no—
363D4—3141no—
364D4—6282no—
365D4—6282no—
366D4—6282no—
367D4—6282no—
368D4—1574no—
369D4—12561no—
370D4D4×D15712yes—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—3141no—
376C2×C4—3141no—
377C2×C4—3141no—
378C2×C4—3141no—
379C2×C4—3141no—
380C2×C4—6282no—
381C2×C4—6282no—
382C2×C4—6282no—
383C2×C4—1571no—
384C2×C4—1571no—
385C2×C4—1571no—
386C2×C4D4×D15711yescharacteristic
387C2×C4C23×D15711yescharacteristic
388C8—21no—
389C8—21no—
390C8—3141no—
391C8—1572no—
392C22—21no—
393C22—21no—
394C22—22no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—84no—
401C22—3141no—
402C22—3141no—
403C22—6281no—
404C22—6281no—
405C22—6281no—
406C22—1572no—
407C22—12564no—
408C22SD16×D15712yescomplemented
409C22C2×D4×D15711yescharacteristic
410C4—21no—
411C4—41no—
412C4—41no—
413C4—3141no—
414C4—3141no—
415C4—3142no—
416C4—6281no—
417C4—6281no—
418C4—1572no—
419C4C2×D4×D15711yescharacteristic
420C4#1909781611yescharacteristic
421C2—22no—
422C2—42no—
423C2—81no—
424C2—3142no—
425C2—12561no—
426C2#2375790511yescharacteristic
427C2C2×SD16×D15711yescharacteristic
428C2#2375808311yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#450×C157C2
socleC22×C157C22×D4
prime radicalD4×C22⋊D157C2
hypercenter#450D157
hypocenterC157D4×SD16
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. C22×#14578942C2semidirect
2. C2×SD16×D157C2semidirect
3. #23757936C2semidirect
5. #23757644C2semidirect
6. #23757504C2semidirect
7. #23757501C2semidirect
8. #23757476C2semidirect
9. #23757472C2semidirect
10. #23757425C2semidirect
11. #23757414C2semidirect
12. #23757410C2semidirect
13. #23757364C2semidirect
14. #23757336C2semidirect
15. #450×C157C2semidirect
16. C22×Dic314C22semidirect
17. C2×Q8×D157C22semidirect
20. C2×#14578943C22semidirect
21. C2×#14578943C22semidirect
25. C2×#14578942C22semidirect
26. C2×#14578942C22semidirect
27. C2×#14578941C22semidirect
29. C22×#11929434C22semidirect
30. C2×C8×D157C22semidirect
32. #19097769C22semidirect
34. #19097751C22semidirect
42. #19097736C22semidirect
46. #19097702C22semidirect
47. #19097700C22semidirect
48. #19097696C22semidirect
49. #19097696C22semidirect
50. #19097695C22semidirect
51. #19097691C22semidirect
52. #19097683C22semidirect
53. C8×Dic157C22semidirect
56. C2×SD16×C157C22semidirect
57. C22⋊Q8×C157C22semidirect
59. #102×C157C22semidirect
60. C22⋊C8×C157C22semidirect
89. #14578942D4semidirect
134. Dic314C2×D4semidirect
142. C22⋊D157SD16semidirect
155. #11929434C2×D4semidirect
186. C22×C157C2×SD16semidirect
194. C2×D157C2×SD16semidirect
205. Dic157C2×SD16semidirect
214. C157D4×SD16semidirect
226. #450D157semidirect
246. C2×SD16C2×D157semidirect
255. C22⋊Q8C2×D157semidirect
272. #102C2×D157semidirect
274. C22⋊C8C2×D157semidirect
408. C22SD16×D157semidirect

Maximal subgroups

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

subgroupindex
C22×#145789422
C2×SD16×D1572
#237579362
D4×C22⋊D1572
#237576442
#237575042
#237575012
#237574762
#237574722
#237574252
#237574142
#237574102
#237573642
#237573362
#450×C1572
D4×SD16157