Finite Group Databasebeta

← #34112152

Subgroups — #34112152

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
2#23757907C211yesmaximal, characteristic
3#23757636C211yesmaximal, characteristic, complemented
4#23757594C211yesmaximal, characteristic, complemented
5#23757588C211yesmaximal, characteristic, complemented
6#23757576C211yesmaximal, characteristic, complemented
7#23757505C211yesmaximal, characteristic, complemented
8#23757471C211yesmaximal, characteristic, complemented
9#23757423C211yesmaximal, characteristic, complemented
10#23757422C211yesmaximal, characteristic, complemented
11#23757376C211yesmaximal, characteristic, complemented
12#23757342C211yesmaximal, characteristic, complemented
13#23757279C211yesmaximal, characteristic, complemented
14#102×D157C211yesmaximal, characteristic, complemented
15#468×C157C211yesmaximal, characteristic, complemented
16C22×D628C2211yescharacteristic
17C2×D4×D157—21no—
18C2×D4×D157C2211yescharacteristic
19C2×#14578941C2211yescharacteristic
20C2×#14578941C2211yescharacteristic, complemented
21C2×C8×D157C2211yescharacteristic, complemented
22#19097767C2211yescharacteristic
23#19097764—21no—
24#19097762C2211yescharacteristic, complemented
25#19097761—21no—
26#19097753—21no—
27#19097747C2211yescharacteristic, complemented
28#19097746C2211yescharacteristic
29#19097744—21no—
30#19097741—21no—
31#19097739—21no—
32Q8×Dic157C2211yescharacteristic, complemented
33#19097733C2211yescharacteristic
34C4×C22⋊D157C2211yescharacteristic
35#19097730C2211yescharacteristic, complemented
36#19097729—21no—
37C4⋊C4×D157C2211yescharacteristic, complemented
38#19097704C2211yescharacteristic, complemented
39#19097702C2211yescharacteristic, complemented
40#19097700C2211yescharacteristic, complemented
41#19097693C2211yescharacteristic
42#19097693C2211yescharacteristic
43#19097693C2211yescharacteristic, complemented
44#19097693C2211yescharacteristic, complemented
45#19097692C2211yescharacteristic, complemented
46#19097692C2211yescharacteristic, complemented
47#19097691C2211yescharacteristic
48#19097691C2211yescharacteristic, complemented
49C22⋊#11929434C2211yescharacteristic, complemented
50#19097683C2211yescharacteristic, complemented
51#19097681C2211yescharacteristic, complemented
52C2×SD16×C157C2211yescharacteristic, complemented
53C2×M16×C157C2211yescharacteristic, complemented
54C22⋊Q8×C157C2211yescharacteristic, complemented
55#121×C157C2211yescharacteristic
56#107×C157C2211yescharacteristic, complemented
57#103×C157C2211yescharacteristic, complemented
58#102×C157C2211yescharacteristic, complemented
59C23×D157—21no—
60C23×D157—21no—
61C2×D628—22no—
62C2×D628—21no—
63C2×D628—21no—
64C2×D628—41no—
65C2×D628C2311yescharacteristic
66C2×D628C2311yescharacteristic
67C2×C22⋊D157—21no—
68C2×C22⋊D157—21no—
69C2×C22⋊D157—21no—
70C2×C22⋊D157D411yescharacteristic
71D4×D157—22no—
72D4×D157—41no—
73D4×D157—44no—
74C2×C4×D157—21no—
75C2×C4×D157—21no—
76C2×C4×D157C2311yescharacteristic
77#14578941—22no—
78#14578941—41no—
79C2×#11929434C2311yescharacteristic
80C2×#11929434C2311yescharacteristic
81C8×D157—41no—
82C4⋊Dic157—21no—
83C4⋊Dic157—21no—
84C4⋊Dic157C2311yescharacteristic
85C4⋊Dic157C2311yescharacteristic
86#14578933—21no—
87#14578933—21no—
88#14578933—21no—
89#14578933D411yescharacteristic
90C22⋊Dic157—21no—
91C22⋊Dic157D411yescharacteristic
92#14578931—21no—
93#14578931—21no—
94#14578931—21no—
95#14578931D411yescharacteristic
96C4×Dic157—21no—
97C4×Dic157—21no—
98C4×Dic157C2311yescharacteristic
99C2×Q8×C157C2311yescharacteristic
100C2×D4×C157—21no—
101C2×D4×C157C2311yescharacteristic
102C22×C4×C157C2311yescharacteristic
103SD16×C157—41no—
104M16×C157—41no—
105C2×C8×C157C2311yescharacteristic
106C2×C8×C157C2311yescharacteristic
107C4⋊C4×C157—21no—
108C4⋊C4×C157C2311yescharacteristic
109C4⋊C4×C157C2311yescharacteristic
110C22⋊C4×C157—21no—
111C22⋊C4×C157—21no—
112C22×D157—21no—
113C22×D157—21no—
114C22×D157—42no—
115C22×D157—42no—
116C22×D157—42no—
117C22×D157—42no—
118C22×D157—81no—
119C22×D157—81no—
120C22×D157C2×D411yescharacteristic
121D628—21no—
122D628—21no—
123D628—21no—
124D628—42no—
125D628—42no—
126D628#4112yes—
127C22⋊D157—22no—
128C22⋊D157—42no—
129C22⋊D157—42no—
130C22⋊D157—81no—
131C2×Dic157—21no—
132C2×Dic157—21no—
133C2×Dic157C2×D411yescharacteristic
134C2×Dic157C2×D411yescharacteristic
135C2×Dic157C2×D411yescharacteristic
136C4×D157—21no—
137C4×D157—21no—
138C4×D157—42no—
139C4×D157—42no—
140#11929434—21no—
141#11929434—21no—
142C23×C157—21no—
143C23×C157C2×D411yescharacteristic
144Q8×C157—21no—
145Q8×C157—21no—
146D4×C157—21no—
147D4×C157—21no—
148D4×C157—42no—
149C2×C4×C157—21no—
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—21no—
160C22×C157—21no—
161C22×C157—41no—
162C22×C157—41no—
163C22×C157—41no—
164C22×C157C22×D411yescharacteristic
165C2×D157—21no—
166C2×D157—21no—
167C2×D157—22no—
168C2×D157—41no—
169C2×D157—41no—
170C2×D157—41no—
171C2×D157—41no—
172C2×D157—41no—
173C2×D157—84no—
174C2×D157—84no—
175C2×D157AGL(1,ℤ/8)12yes—
176C4×C157—21no—
177C4×C157—41no—
178C4×C157—41no—
179C4×C157—41no—
180C4×C157C2×#4111yescharacteristic
181C4×C157H(5,𝔽2)11yescharacteristic
182Dic157—21no—
183Dic157—21no—
184Dic157—21no—
185Dic157—41no—
186Dic157—41no—
187C2×C157—41no—
188C2×C157—81no—
189C2×C157#54611yescharacteristic
190C2×C157C2×AGL(1,ℤ/8)11yescharacteristic
191C2×C157#57711yescharacteristic
192D157—22no—
193D157—42no—
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#468D15711yescharacteristic, complemented
204#465—1571no—
205#460—1571no—
206#450—1571no—
207#450—1571no—
208#440—1571no—
209#435—1571no—
210#418—1571no—
211C2×#102—1571no—
212C22×D4—3141no—
213C22×D4—1571no—
214C22×D4—1571no—
215C2×SD16—1571no—
216C2×SD16—1571no—
217C2×SD16C2×D15711yescharacteristic, complemented
218C2×M16C2×D15711yescharacteristic, complemented
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#121C2×D15711yescharacteristic
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#107C2×D15711yescharacteristic, complemented
240#106—1571no—
241C4⋊C8—1571no—
242#103C2×D15711yescharacteristic, complemented
243#102—1571no—
244#102—1571no—
245#102—1571no—
246#102—1571no—
247#102—1571no—
248#102—1571no—
249#102—1571no—
250#102—1571no—
251#102C2×D15711yescharacteristic, complemented
252C22⋊C8—1571no—
253C22⋊C8—1571no—
254#97—1571no—
255C24—3141no—
256C24—3141no—
257C2×Q8C22×D15711yescharacteristic
258C2×D4—21no—
259C2×D4—3142no—
260C2×D4—3141no—
261C2×D4—3142no—
262C2×D4—3141no—
263C2×D4—3141no—
264C2×D4—3141no—
265C2×D4—3141no—
266C2×D4—6281no—
267C2×D4—6281no—
268C2×D4—6284no—
269C2×D4—1571no—
270C2×D4—1571no—
271C2×D4—1571no—
272C2×D4C22×D15711yescharacteristic
273C22×C4—3141no—
274C22×C4—3141no—
275C22×C4—1571no—
276C22×C4C22×D15711yescharacteristic
277SD16—41no—
278SD16—3142no—
279SD16—6281no—
280M16—41no—
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⋊C4C22×D15711yescharacteristic
296C4⋊C4C22×D15711yescharacteristic
297C22⋊C4—21no—
298C22⋊C4—21no—
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—3141no—
310C23—3141no—
311C23—6282no—
312C23—6282no—
313C23—6282no—
314C23—6282no—
315C23—1571no—
316C23—12561no—
317C23—12561no—
318C23D4×D15711yescharacteristic
319Q8—21no—
320Q8—21no—
321D4—21no—
322D4—21no—
323D4—42no—
324D4—3142no—
325D4—3141no—
326D4—3141no—
327D4—3141no—
328D4—6282no—
329D4—6282no—
330D4—6282no—
331D4—6282no—
332D4—1572no—
333D4—12561no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
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—21no—
356C22—21no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—3141no—
361C22—3141no—
362C22—3142no—
363C22—6281no—
364C22—6281no—
365C22—6281no—
366C22—6281no—
367C22—6281no—
368C22—1572no—
369C22—12564no—
370C22—12564no—
371C22C2×D4×D15711yescharacteristic
372C4—21no—
373C4—41no—
374C4—41no—
375C4—41no—
376C4—3141no—
377C4—3141no—
378C4—3141no—
379C4—6281no—
380C4—6281no—
381C4#41×D15711yescharacteristic
382C4#1909781611yescharacteristic
383C2—41no—
384C2—81no—
385C2—3142no—
386C2—6282no—
387C2—12561no—
388C2#2375791211yescharacteristic
389C2AGL(1,ℤ/8)×D15711yescharacteristic
390C2#2375808211yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#468×C157C2
socleC22×C157C22×D4
prime radical#23757907C2
hypercenter#468D157
hypocenterC157#3310
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. #23757957C2semidirect
3. #23757636C2semidirect
4. #23757594C2semidirect
5. #23757588C2semidirect
6. #23757576C2semidirect
7. #23757505C2semidirect
8. #23757471C2semidirect
9. #23757423C2semidirect
10. #23757422C2semidirect
11. #23757376C2semidirect
12. #23757342C2semidirect
13. #23757279C2semidirect
14. #102×D157C2semidirect
15. #468×C157C2semidirect
20. C2×#14578941C22semidirect
21. C2×C8×D157C22semidirect
24. #19097762C22semidirect
27. #19097747C22semidirect
32. Q8×Dic157C22semidirect
35. #19097730C22semidirect
37. C4⋊C4×D157C22semidirect
38. #19097704C22semidirect
39. #19097702C22semidirect
40. #19097700C22semidirect
43. #19097693C22semidirect
44. #19097693C22semidirect
45. #19097692C22semidirect
46. #19097692C22semidirect
48. #19097691C22semidirect
49. C22⋊#11929434C22semidirect
50. #19097683C22semidirect
51. #19097681C22semidirect
52. C2×SD16×C157C22semidirect
53. C2×M16×C157C22semidirect
54. C22⋊Q8×C157C22semidirect
56. #107×C157C22semidirect
57. #103×C157C22semidirect
58. #102×C157C22semidirect
195. C157#3310semidirect
203. #468D157semidirect
217. C2×SD16C2×D157semidirect
218. C2×M16C2×D157semidirect
226. C22⋊Q8C2×D157semidirect
239. #107C2×D157semidirect
242. #103C2×D157semidirect
251. #102C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237579572
#237579072
#237576362
#237575942
#237575882
#237575762
#237575052
#237574712
#237574232
#237574222
#237573762
#237573422
#237572792
#102×D1572
#468×C1572
#3310157