Finite Group Databasebeta

← #34112151

Subgroups — #34112151

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
1D4×D628C211yesmaximal, characteristic
2#23757639C211yesmaximal, characteristic, complemented
3#23757597C211yesmaximal, characteristic, complemented
4#23757586C211yesmaximal, characteristic, complemented
5#23757570C211yesmaximal, characteristic, complemented
6#23757521C211yesmaximal, characteristic, complemented
7#23757504C211yesmaximal, characteristic, complemented
8#23757473C211yesmaximal, characteristic, complemented
9#23757423C211yesmaximal, characteristic, complemented
10#23757423C211yesmaximal, characteristic, complemented
11#23757371C211yesmaximal, characteristic, complemented
12#23757345C211yesmaximal, characteristic, complemented
13#23757277C211yesmaximal, characteristic, complemented
14C2×#19097691C211yesmaximal, characteristic, complemented
15#552×C157C211yesmaximal, characteristic, complemented
16C22×D628C2211yescharacteristic
17C22×D628C2211yescharacteristic
18C2×D4×D157—21no—
19C2×#14578941C2211yescharacteristic
20C2×#14578941C2211yescharacteristic, complemented
21C2×#14578941C2211yescharacteristic, complemented
22C2×#14578938C2211yescharacteristic, complemented
23C22×#11929434C2211yescharacteristic, complemented
24#19097768C2211yescharacteristic
25#19097746—21no—
26#19097745C2211yescharacteristic
27#19097745C2211yescharacteristic
28#19097741—21no—
29#19097741—21no—
30C4×D628C2211yescharacteristic
31#19097704C2211yescharacteristic, complemented
32#19097704C2211yescharacteristic, complemented
33#19097702C2211yescharacteristic, complemented
34C4⋊#11929434C2211yescharacteristic, complemented
35#19097695C2211yescharacteristic, complemented
36#19097691C2211yescharacteristic
37#19097691C2211yescharacteristic
38#19097691C2211yescharacteristic
39#19097691C2211yescharacteristic, complemented
40#19097691C2211yescharacteristic, complemented
41#19097691C2211yescharacteristic, complemented
42#19097691C2211yescharacteristic, complemented
43#19097691C2211yescharacteristic, complemented
44#19097691C2211yescharacteristic, complemented
45C22⋊#11929434C2211yescharacteristic, complemented
46C22⋊#11929434C2211yescharacteristic, complemented
47#19097682C2211yescharacteristic, complemented
48#126×C157—21no—
49#125×C157—21no—
50#125×C157C2211yescharacteristic, complemented
51#123×C157—21no—
52C22⋊Q8×C157C2211yescharacteristic, complemented
53C22⋊Q8×C157C2211yescharacteristic, complemented
54C4×Q8×C157C2211yescharacteristic, complemented
55C4×D4×C157C2211yescharacteristic
56#117×C157—21no—
57#117×C157C2211yescharacteristic, complemented
58C2×C4⋊C4×C157C2211yescharacteristic, complemented
59C23×D157—21no—
60C23×D157—21no—
61C2×D628—22no—
62C2×D628—22no—
63C2×D628—21no—
64C2×D628—21no—
65C2×D628—21no—
66C2×D628—21no—
67C2×D628—41no—
68C2×D628—41no—
69C2×D628C2311yescharacteristic
70C2×D628C2311yescharacteristic
71C2×D628C2311yescharacteristic
72C2×C22⋊D157—21no—
73C2×C22⋊D157—21no—
74D4×D157—44no—
75C2×C4×D157—21no—
76#14578941—22no—
77#14578941—41no—
78#14578941—41no—
79#14578938—41no—
80C2×#11929434—41no—
81C2×#11929434C2311yescharacteristic
82C2×#11929434C2311yescharacteristic
83C2×#11929434C2311yescharacteristic
84C2×#11929434C2311yescharacteristic
85C4⋊Dic157C2311yescharacteristic
86#14578931—21no—
87#14578931—21no—
88#14578931—21no—
89C2×Q8×C157C2311yescharacteristic
90C2×D4×C157D411yescharacteristic
91C22×C4×C157—21no—
92C22×C4×C157C2311yescharacteristic
93C22×C4×C157C2311yescharacteristic
94C4⋊C4×C157—21no—
95C4⋊C4×C157—21no—
96C4⋊C4×C157—21no—
97C4⋊C4×C157—21no—
98C4⋊C4×C157—21no—
99C4⋊C4×C157—21no—
100C4⋊C4×C157D411yescharacteristic
101C4⋊C4×C157C2311yescharacteristic
102C4⋊C4×C157C2311yescharacteristic
103C4⋊C4×C157C2311yescharacteristic
104C22⋊C4×C157—21no—
105C22⋊C4×C157—21no—
106C22⋊C4×C157—21no—
107C22⋊C4×C157D411yescharacteristic
108C22⋊C4×C157D411yescharacteristic
109C42×C157—21no—
110C42×C157—21no—
111C42×C157C2311yescharacteristic
112C22×D157—21no—
113C22×D157—21no—
114C22×D157—21no—
115C22×D157—42no—
116C22×D157—42no—
117C22×D157—42no—
118C22×D157—42no—
119C22×D157—81no—
120C22×D157—81no—
121D628—21no—
122D628—21no—
123D628—21no—
124D628—21no—
125D628—21no—
126D628—42no—
127D628—42no—
128D628—42no—
129D628—81no—
130D628#4112yes—
131C22⋊D157—42no—
132C22⋊D157—42no—
133C2×Dic157—21no—
134C4×D157—42no—
135#11929434—21no—
136#11929434—21no—
137#11929434—21no—
138#11929434—21no—
139C23×C157C2×D411yescharacteristic
140C23×C157C2×D411yescharacteristic
141Q8×C157—21no—
142Q8×C157—21no—
143D4×C157—22no—
144C2×C4×C157—21no—
145C2×C4×C157—21no—
146C2×C4×C157—21no—
147C2×C4×C157—21no—
148C2×C4×C157—21no—
149C2×C4×C157—21no—
150C2×C4×C157—21no—
151C2×C4×C157—21no—
152C2×C4×C157—42no—
153C2×C4×C157C2×D411yescharacteristic
154C2×C4×C157C2×D411yescharacteristic
155C2×C4×C157C2×D411yescharacteristic
156C2×C4×C157C2×D411yescharacteristic
157C2×C4×C157C2411yescharacteristic
158C22×C157—21no—
159C22×C157—21no—
160C22×C157—21no—
161C22×C157—21no—
162C22×C157—21no—
163C22×C157AGL(1,ℤ/8)12yes—
164C22×C157C22×D411yescharacteristic
165C2×D157—22no—
166C2×D157—41no—
167C2×D157—41no—
168C2×D157—41no—
169C2×D157—41no—
170C2×D157—41no—
171C2×D157—41no—
172C2×D157—41no—
173C2×D157—41no—
174C2×D157—84no—
175C2×D157—84no—
176C4×C157—21no—
177C4×C157—21no—
178C4×C157—21no—
179C4×C157—21no—
180C4×C157—41no—
181C4×C157—41no—
182C4×C157—41no—
183C4×C157—41no—
184C4×C157C2×#4111yescharacteristic
185C4×C157H(5,𝔽2)11yescharacteristic
186Dic157—41no—
187C2×C157—22no—
188C2×C157—41no—
189C2×C157#54611yescharacteristic
190C2×C157C2×AGL(1,ℤ/8)11yescharacteristic
191C2×C157#57711yescharacteristic
192D157—42no—
193D157—81no—
194D157—81no—
195C157#331011yescharacteristic, complemented, Sylow
196#3310—1571nomaximal, Sylow
197#552D15711yescharacteristic, complemented
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#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×SD16—1571no—
218C2×M16—1571no—
219C22×C8—1571no—
220#127—1571no—
221#126—21no—
222#125—21no—
223#125C2×D15711yescharacteristic, complemented
224#123—21no—
225C22⋊Q8C2×D15711yescharacteristic, complemented
226C22⋊Q8C2×D15711yescharacteristic, complemented
227#121—3141no—
228#121—1571no—
229#121—1571no—
230#120—3141no—
231#120—3141no—
232C4×Q8C2×D15711yescharacteristic, complemented
233C4×D4—1571no—
234C4×D4C2×D15711yescharacteristic
235#117—21no—
236#117C2×D15711yescharacteristic, complemented
237C2×C4⋊C4C2×D15711yescharacteristic, complemented
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#102—1571no—
251#102—1571no—
252C22⋊C8—1571no—
253C22⋊C8—1571no—
254#97—1571no—
255C24—3141no—
256C24—3141no—
257C2×Q8C22×D15711yescharacteristic
258C2×D4—3142no—
259C2×D4—3142no—
260C2×D4—3141no—
261C2×D4—3141no—
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—21no—
274C22×C4—3141no—
275C22×C4C22×D15711yescharacteristic
276C22×C4C22×D15711yescharacteristic
277SD16—3142no—
278SD16—6281no—
279SD16—6281no—
280M16—6281no—
281C2×C8—6281no—
282C2×C8—1571no—
283C2×C8—1571no—
284C2×C8—1571no—
285C2×C8—1571no—
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—1571no—
293C4⋊C4C22⋊D15711yescharacteristic
294C4⋊C4C22×D15711yescharacteristic
295C4⋊C4C22×D15711yescharacteristic
296C4⋊C4C22×D15711yescharacteristic
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—3141no—
301C22⋊C4—3141no—
302C22⋊C4—3141no—
303C22⋊C4C22⋊D15711yescharacteristic
304C22⋊C4C22⋊D15711yescharacteristic
305C42—21no—
306C42—21no—
307C42C22×D15711yescharacteristic
308C23—3141no—
309C23—3141no—
310C23—3141no—
311C23—6282no—
312C23—6282no—
313C23—6282no—
314C23—6282no—
315C23—12561no—
316C23—12561no—
317C23C2×C22⋊D15711yescharacteristic
318C23C2×C22⋊D15711yescharacteristic
319Q8—21no—
320Q8—21no—
321D4—22no—
322D4—3141no—
323D4—3141no—
324D4—3141no—
325D4—3141no—
326D4—3141no—
327D4—6282no—
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—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—42no—
343C2×C4—3141no—
344C2×C4—6282no—
345C2×C4C2×C22⋊D15711yescharacteristic
346C2×C4C2×C22⋊D15711yescharacteristic
347C2×C4C2×C22⋊D15711yescharacteristic
348C2×C4C2×C22⋊D15711yescharacteristic
349C2×C4C23×D15711yescharacteristic
350C8—3141no—
351C8—3141no—
352C8—3141no—
353C8—3141no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—3142no—
360C22—6281no—
361C22—6281no—
362C22—6281no—
363C22—6281no—
364C22—6281no—
365C22—6281no—
366C22—6281no—
367C22—6281no—
368C22—12564no—
369C22—12564no—
370C22#1909779612yes—
371C22C22×C22⋊D15711yescharacteristic
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—41no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—6281no—
381C4#41×D15711yescharacteristic
382C4#1909781711yescharacteristic
383C2—22no—
384C2—41no—
385C2—6282no—
386C2—12561no—
387C2—12561no—
388C2#2375791111yescharacteristic
389C2C2×#1909779611yescharacteristic
390C2#2375808111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#552×C157C2
socleC22×C157C22×D4
prime radicalD4×D628C2
hypercenter#552D157
hypocenterC157#3310
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C4×D4 ◁ #552

Complements

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

NH ≅ G/Ntype
2. #23757639C2semidirect
3. #23757597C2semidirect
4. #23757586C2semidirect
5. #23757570C2semidirect
6. #23757521C2semidirect
7. #23757504C2semidirect
8. #23757473C2semidirect
9. #23757423C2semidirect
10. #23757423C2semidirect
11. #23757371C2semidirect
12. #23757345C2semidirect
13. #23757277C2semidirect
14. C2×#19097691C2semidirect
15. #552×C157C2semidirect
20. C2×#14578941C22semidirect
21. C2×#14578941C22semidirect
22. C2×#14578938C22semidirect
23. C22×#11929434C22semidirect
31. #19097704C22semidirect
32. #19097704C22semidirect
33. #19097702C22semidirect
34. C4⋊#11929434C22semidirect
35. #19097695C22semidirect
39. #19097691C22semidirect
40. #19097691C22semidirect
41. #19097691C22semidirect
42. #19097691C22semidirect
43. #19097691C22semidirect
44. #19097691C22semidirect
45. C22⋊#11929434C22semidirect
46. C22⋊#11929434C22semidirect
47. #19097682C22semidirect
50. #125×C157C22semidirect
52. C22⋊Q8×C157C22semidirect
53. C22⋊Q8×C157C22semidirect
54. C4×Q8×C157C22semidirect
57. #117×C157C22semidirect
58. C2×C4⋊C4×C157C22semidirect
195. C157#3310semidirect
197. #552D157semidirect
223. #125C2×D157semidirect
225. C22⋊Q8C2×D157semidirect
226. C22⋊Q8C2×D157semidirect
232. C4×Q8C2×D157semidirect
236. #117C2×D157semidirect
237. C2×C4⋊C4C2×D157semidirect

Maximal subgroups

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

subgroupindex
D4×D6282
#237576392
#237575972
#237575862
#237575702
#237575212
#237575042
#237574732
#237574232
#237573712
#237573452
#237572772
C2×#190976912
#552×C1572
#3310157