Finite Group Databasebeta

← #34112150

Subgroups — #34112150

Order 20096.

450 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23757957C211yesmaximal, characteristic, complemented
2D4×C22⋊D157C211yesmaximal, characteristic
3#23757636C211yesmaximal, characteristic, complemented
4#23757598C211yesmaximal, characteristic, complemented
5#23757585C211yesmaximal, characteristic, complemented
6#23757575C211yesmaximal, characteristic, complemented
7#23757521C211yesmaximal, characteristic, complemented
8#23757499C211yesmaximal, characteristic, complemented
9#23757471C211yesmaximal, characteristic, complemented
10#23757422C211yesmaximal, characteristic, complemented
11#23757374C211yesmaximal, characteristic, complemented
12#23757347C211yesmaximal, characteristic, complemented
13#23757278C211yesmaximal, characteristic, complemented
14#102×D157C211yesmaximal, characteristic, complemented
15#450×C157C211yesmaximal, characteristic, complemented
16C22×C22⋊D157—21no—
17C2×D4×D157C2211yescharacteristic
18C2×#14578941C2211yescharacteristic, complemented
19C2×#14578938C2211yescharacteristic, complemented
20C2×C8×D157C2211yescharacteristic, complemented
21#19097767C2211yescharacteristic
22#19097764—21no—
23#19097762C2211yescharacteristic, complemented
24#19097761—21no—
25#19097753—21no—
26#19097747C2211yescharacteristic, complemented
27#19097745C2211yescharacteristic
28#19097743—21no—
29#19097742C2211yescharacteristic
30#19097740—21no—
31#19097739—21no—
32Q8×Dic157C2211yescharacteristic, complemented
33D4×Dic157C2211yescharacteristic
34C4×C22⋊D157C2211yescharacteristic
35#19097730C2211yescharacteristic, complemented
36#19097729—21no—
37C4⋊C4×D157C2211yescharacteristic, complemented
38#19097705C2211yescharacteristic, complemented
39#19097704C2211yescharacteristic, complemented
40#19097703C2211yescharacteristic, complemented
41#19097700C2211yescharacteristic, complemented
42#19097695C2211yescharacteristic, complemented
43#19097693C2211yescharacteristic, complemented
44#19097693C2211yescharacteristic, complemented
45#19097692C2211yescharacteristic
46#19097692C2211yescharacteristic
47#19097692C2211yescharacteristic, complemented
48#19097692C2211yescharacteristic, complemented
49#19097691C2211yescharacteristic, complemented
50#19097683C2211yescharacteristic, complemented
51#19097681C2211yescharacteristic, complemented
52C22×D4×C157C2211yescharacteristic
53C2×SD16×C157C2211yescharacteristic
54C2×SD16×C157C2211yescharacteristic, complemented
55C22⋊Q8×C157C2211yescharacteristic, complemented
56#102×C157C2211yescharacteristic
57#102×C157C2211yescharacteristic, complemented
58C22⋊C8×C157C2211yescharacteristic, complemented
59C23×D157—21no—
60C2×D628C2311yescharacteristic
61C2×C22⋊D157—21no—
62C2×C22⋊D157—21no—
63C2×C22⋊D157—21no—
64C2×C22⋊D157—21no—
65C2×C22⋊D157—21no—
66C2×C22⋊D157—44no—
67C2×C22⋊D157D411yescharacteristic
68D4×D157—22no—
69D4×D157—41no—
70C22×Dic157—21no—
71C2×C4×D157—21no—
72C2×C4×D157C2311yescharacteristic
73#14578941—41no—
74#14578938—41no—
75C2×#11929434C2311yescharacteristic
76C2×#11929434C2311yescharacteristic
77C8×D157—41no—
78C4⋊Dic157—21no—
79C4⋊Dic157—21no—
80C4⋊Dic157C2311yescharacteristic
81C4⋊Dic157C2311yescharacteristic
82C4⋊Dic157C2311yescharacteristic
83#14578933—21no—
84#14578933—21no—
85#14578933—21no—
86#14578933D411yescharacteristic
87C22⋊Dic157—21no—
88C22⋊Dic157—21no—
89C22⋊Dic157—21no—
90C22⋊Dic157D411yescharacteristic
91#14578931—21no—
92#14578931—21no—
93#14578931D411yescharacteristic
94C4×Dic157—21no—
95C4×Dic157—21no—
96C4×Dic157C2311yescharacteristic
97C24×C157—21no—
98C2×Q8×C157C2311yescharacteristic
99C2×D4×C157—22no—
100C2×D4×C157—21no—
101C2×D4×C157—41no—
102C2×D4×C157C2311yescharacteristic
103C2×D4×C157C2311yescharacteristic
104C22×C4×C157C2311yescharacteristic
105SD16×C157—22no—
106SD16×C157—41no—
107C2×C8×C157C2311yescharacteristic
108C2×C8×C157C2311yescharacteristic
109C4⋊C4×C157—21no—
110C4⋊C4×C157C2311yescharacteristic
111C22⋊C4×C157—21no—
112C22×D157—21no—
113C22×D157—42no—
114C22×D157—42no—
115C22×D157—81no—
116C22×D157C2×D411yescharacteristic
117D628—21no—
118D628—21no—
119C22⋊D157—22no—
120C22⋊D157—42no—
121C22⋊D157—42no—
122C22⋊D157—42no—
123C22⋊D157—42no—
124C22⋊D157—81no—
125C2×Dic157—21no—
126C2×Dic157—21no—
127C2×Dic157—21no—
128C2×Dic157—42no—
129C2×Dic157C2×D411yescharacteristic
130C2×Dic157C2×D411yescharacteristic
131C2×Dic157C2×D411yescharacteristic
132C4×D157—21no—
133C4×D157—21no—
134C4×D157—42no—
135#11929434—21no—
136#11929434—21no—
137C23×C157—21no—
138C23×C157—21no—
139C23×C157—42no—
140C23×C157—42no—
141C23×C157—81no—
142C23×C157C2×D411yescharacteristic
143Q8×C157—21no—
144Q8×C157—21no—
145D4×C157—21no—
146D4×C157—21no—
147D4×C157—21no—
148D4×C157—42no—
149D4×C157#4112yes—
150C2×C4×C157—21no—
151C2×C4×C157—21no—
152C2×C4×C157—21no—
153C2×C4×C157—21no—
154C2×C4×C157C2×D411yescharacteristic
155C2×C4×C157C2411yescharacteristic
156C8×C157—21no—
157C8×C157—21no—
158C22×C157—21no—
159C22×C157—22no—
160C22×C157—21no—
161C22×C157—21no—
162C22×C157—41no—
163C22×C157—41no—
164C22×C157—41no—
165C22×C157—41no—
166C22×C157—41no—
167C22×C157—84no—
168C22×C157C22×D411yescharacteristic
169C2×D157—21no—
170C2×D157—21no—
171C2×D157—41no—
172C2×D157—41no—
173C2×D157—41no—
174C2×D157—84no—
175C2×D157AGL(1,ℤ/8)12yes—
176C4×C157—21no—
177C4×C157—41no—
178C4×C157—41no—
179C4×C157C2×#4111yescharacteristic
180C4×C157H(5,𝔽2)11yescharacteristic
181Dic157—21no—
182Dic157—21no—
183Dic157—21no—
184Dic157—41no—
185Dic157—41no—
186Dic157—41no—
187C2×C157—42no—
188C2×C157—41no—
189C2×C157—81no—
190C2×C157#54611yescharacteristic
191C2×C157C2×AGL(1,ℤ/8)11yescharacteristic
192C2×C157#57711yescharacteristic
193D157—22no—
194D157—81no—
195C157#331011yescharacteristic, complemented, Sylow
196#3310—1571nomaximal, Sylow
197#552—1571no—
198D42—1571no—
199#490—1571no—
200#482—1571no—
201#480—1571no—
202#478—1571no—
203#468—1571no—
204#465—1571no—
205#460—1571no—
206#450—1571no—
207#450D15711yescharacteristic, complemented
208#440—1571no—
209#435—1571no—
210#418—1571no—
211C2×#102—1571no—
212C22×D4—3141no—
213C22×D4—1571no—
214C22×D4C2×D15711yescharacteristic
215C2×SD16—1571no—
216C2×SD16C2×D15711yescharacteristic
217C2×SD16C2×D15711yescharacteristic, complemented
218C2×M16—1571no—
219C22×C8—1571no—
220#127—1571no—
221#126—3141no—
222#125—3141no—
223#125—1571no—
224#123—3141no—
225C22⋊Q8—1571no—
226C22⋊Q8C2×D15711yescharacteristic, complemented
227#121—3141no—
228#121—1571no—
229#121—1571no—
230#120—3141no—
231#120—3141no—
232C4×Q8—1571no—
233C4×D4—1571no—
234C4×D4—1571no—
235#117—3141no—
236#117—1571no—
237C2×C4⋊C4—1571no—
238#107—1571no—
239#107—1571no—
240#106—1571no—
241C4⋊C8—1571no—
242#103—1571no—
243#102—1571no—
244#102—1571no—
245#102—1571no—
246#102—1571no—
247#102—1571no—
248#102—1571no—
249#102—1571no—
250#102C2×D15711yescharacteristic
251#102C2×D15711yescharacteristic, complemented
252C22⋊C8—1571no—
253C22⋊C8C2×D15711yescharacteristic, complemented
254#97—1571no—
255C24—21no—
256C24—3141no—
257C2×Q8C22×D15711yescharacteristic
258C2×D4—22no—
259C2×D4—21no—
260C2×D4—41no—
261C2×D4—3142no—
262C2×D4—3141no—
263C2×D4—3141no—
264C2×D4—3141no—
265C2×D4—3141no—
266C2×D4—3141no—
267C2×D4—6281no—
268C2×D4—6284no—
269C2×D4—1571no—
270C2×D4—1571no—
271C2×D4C22×D15711yescharacteristic
272C2×D4C22×D15711yescharacteristic
273C22×C4—3141no—
274C22×C4—3141no—
275C22×C4—1571no—
276C22×C4C22×D15711yescharacteristic
277SD16—22no—
278SD16—41no—
279SD16—6281no—
280M16—6281no—
281C2×C8—6281no—
282C2×C8—1571no—
283C2×C8—1571no—
284C2×C8C22×D15711yescharacteristic
285C2×C8C22×D15711yescharacteristic
286C4⋊C4—21no—
287C4⋊C4—3141no—
288C4⋊C4—3141no—
289C4⋊C4—3141no—
290C4⋊C4—3141no—
291C4⋊C4—3141no—
292C4⋊C4—1571no—
293C4⋊C4—1571no—
294C4⋊C4—1571no—
295C4⋊C4—1571no—
296C4⋊C4C22×D15711yescharacteristic
297C22⋊C4—21no—
298C22⋊C4—3141no—
299C22⋊C4—3141no—
300C22⋊C4—3141no—
301C22⋊C4—3141no—
302C22⋊C4—3141no—
303C22⋊C4—1571no—
304C22⋊C4—1571no—
305C42—3141no—
306C42—3141no—
307C42—1571no—
308C23—21no—
309C23—21no—
310C23—42no—
311C23—42no—
312C23—81no—
313C23—3141no—
314C23—6282no—
315C23—6282no—
316C23—1571no—
317C23—12561no—
318C23D4×D15711yescharacteristic
319Q8—21no—
320Q8—21no—
321D4—21no—
322D4—21no—
323D4—21no—
324D4—42no—
325D4—3142no—
326D4—3141no—
327D4—3141no—
328D4—6282no—
329D4—6282no—
330D4—6282no—
331D4—6282no—
332D4—12561no—
333D4#1457895312yes—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—3141no—
339C2×C4—3141no—
340C2×C4—3141no—
341C2×C4—3141no—
342C2×C4—3141no—
343C2×C4—6282no—
344C2×C4—6282no—
345C2×C4—1571no—
346C2×C4—1571no—
347C2×C4—1571no—
348C2×C4D4×D15711yescharacteristic
349C2×C4C23×D15711yescharacteristic
350C8—21no—
351C8—21no—
352C8—3141no—
353C8—3141no—
354C22—21no—
355C22—22no—
356C22—21no—
357C22—21no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—84no—
364C22—3141no—
365C22—3141no—
366C22—6281no—
367C22—6281no—
368C22—6281no—
369C22—1572no—
370C22—12564no—
371C22C2×D4×D15711yescharacteristic
372C4—21no—
373C4—41no—
374C4—41no—
375C4—3141no—
376C4—3141no—
377C4—3141no—
378C4—6281no—
379C4—6281no—
380C4—6281no—
381C4C2×#1457895311yescharacteristic
382C4#1909781611yescharacteristic
383C2—41no—
384C2—42no—
385C2—81no—
386C2—3142no—
387C2—12561no—
388C2#2375791011yescharacteristic
389C2AGL(1,ℤ/8)×D15711yescharacteristic
390C2#2375807911yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#450×C157C2
socleC22×C157C22×D4
prime radicalD4×C22⋊D157C2
hypercenter#450D157
hypocenterC157#3310
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. #23757957C2semidirect
3. #23757636C2semidirect
4. #23757598C2semidirect
5. #23757585C2semidirect
6. #23757575C2semidirect
7. #23757521C2semidirect
8. #23757499C2semidirect
9. #23757471C2semidirect
10. #23757422C2semidirect
11. #23757374C2semidirect
12. #23757347C2semidirect
13. #23757278C2semidirect
14. #102×D157C2semidirect
15. #450×C157C2semidirect
18. C2×#14578941C22semidirect
19. C2×#14578938C22semidirect
20. C2×C8×D157C22semidirect
23. #19097762C22semidirect
26. #19097747C22semidirect
32. Q8×Dic157C22semidirect
35. #19097730C22semidirect
37. C4⋊C4×D157C22semidirect
38. #19097705C22semidirect
39. #19097704C22semidirect
40. #19097703C22semidirect
41. #19097700C22semidirect
42. #19097695C22semidirect
43. #19097693C22semidirect
44. #19097693C22semidirect
47. #19097692C22semidirect
48. #19097692C22semidirect
49. #19097691C22semidirect
50. #19097683C22semidirect
51. #19097681C22semidirect
54. C2×SD16×C157C22semidirect
55. C22⋊Q8×C157C22semidirect
57. #102×C157C22semidirect
58. C22⋊C8×C157C22semidirect
195. C157#3310semidirect
207. #450D157semidirect
217. C2×SD16C2×D157semidirect
226. C22⋊Q8C2×D157semidirect
251. #102C2×D157semidirect
253. C22⋊C8C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237579572
D4×C22⋊D1572
#237576362
#237575982
#237575852
#237575752
#237575212
#237574992
#237574712
#237574222
#237573742
#237573472
#237572782
#102×D1572
#450×C1572
#3310157