Finite Group Databasebeta

← #34111985

Subgroups — #34111985

Order 20096.

420 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23757976C211yesmaximal, characteristic, complemented
2#23757916C211yesmaximal, characteristic, complemented
3#23757607C211yesmaximal, characteristic, complemented
4#23757595C211yesmaximal, characteristic, complemented
5#23757570C211yesmaximal, characteristic, complemented
6#23757508C211yesmaximal, characteristic, complemented
7#23757505C211yesmaximal, characteristic, complemented
8#23757485C211yesmaximal, characteristic, complemented
9#23757421C211yesmaximal, characteristic, complemented
10#23757408C211yesmaximal, characteristic, complemented
11C4×D1256C211yesmaximal, characteristic, complemented
12C8×D628C211yesmaximal, characteristic, complemented
13#23757268C211yesmaximal, characteristic, complemented
14#102×D157C211yesmaximal, characteristic, complemented
15#488×C157C211yesmaximal, characteristic, complemented
16C2×#14578955C2211yescharacteristic
17C2×D4×D157C2211yescharacteristic
18C2×#14578941C2211yescharacteristic, complemented
19C2×D1256C2211yescharacteristic
20C2×#14578939C2211yescharacteristic
21C2×C8×D157C2211yescharacteristic
22C2×C8×D157C2211yescharacteristic, complemented
23#19097766—21no—
24#19097756—21no—
25#19097753—21no—
26#19097747C2211yescharacteristic
27#19097746—21no—
28#19097746C2211yescharacteristic
29#19097746C2211yescharacteristic, complemented
30#19097744—21no—
31#19097742C2211yescharacteristic
32#19097741—21no—
33C4×D628C2211yescharacteristic
34C4×D628C2211yescharacteristic
35C4×D628C2211yescharacteristic, complemented
36#19097733—21no—
37#19097729C2211yescharacteristic, complemented
38C4⋊C4×D157C2211yescharacteristic
39C22⋊C4×D157—21no—
40#19097705C2211yescharacteristic, complemented
41#19097704C2211yescharacteristic, complemented
42#19097702C2211yescharacteristic, complemented
43C4⋊#11929434C2211yescharacteristic, complemented
44#19097695C2211yescharacteristic, complemented
45#19097693C2211yescharacteristic
46#19097693C2211yescharacteristic, complemented
47#19097692C2211yescharacteristic
48#19097691C2211yescharacteristic
49#19097691C2211yescharacteristic
50#19097683C2211yescharacteristic
51#19097683C2211yescharacteristic, complemented
52#125×C157C2211yescharacteristic, complemented
53#124×C157C2211yescharacteristic
54#103×C157C2211yescharacteristic, complemented
55#103×C157C2211yescharacteristic, complemented
56#102×C157C2211yescharacteristic
57#102×C157C2211yescharacteristic
58C4×C8×C157C2211yescharacteristic, complemented
59C23×D157—21no—
60#14578955—22no—
61#14578955—41no—
62C2×D628—21no—
63C2×D628—21no—
64C2×D628—21no—
65C2×D628D411yescharacteristic
66C2×D628C2311yescharacteristic
67C2×D628C2311yescharacteristic
68C2×C22⋊D157—21no—
69C2×C22⋊D157—21no—
70D4×D157—22no—
71D4×D157—41no—
72C2×C4×D157—21no—
73C2×C4×D157—21no—
74C2×C4×D157—21no—
75C2×C4×D157—21no—
76C2×C4×D157C2311yescharacteristic
77C2×C4×D157C2311yescharacteristic
78#14578941—41no—
79D1256—22no—
80#14578939—41no—
81C2×#11929434C2311yescharacteristic
82C2×#11929434C2311yescharacteristic
83C8×D157—41no—
84C8×D157—41no—
85C4⋊Dic157D411yescharacteristic
86C4⋊Dic157C2311yescharacteristic
87C4⋊Dic157C2311yescharacteristic
88#14578933—21no—
89#14578933—21no—
90C22⋊Dic157—21no—
91#14578931—21no—
92#14578931—21no—
93#14578931—21no—
94#14578931—21no—
95#14578931—21no—
96#14578931—21no—
97#14578931D411yescharacteristic
98#14578931D411yescharacteristic
99C4×Dic157—21no—
100C2×Q8×C157C2311yescharacteristic
101C2×D4×C157C2311yescharacteristic
102C2×C8×C157C2311yescharacteristic
103C2×C8×C157C2311yescharacteristic
104C4⋊C4×C157—21no—
105C4⋊C4×C157—21no—
106C4⋊C4×C157C2311yescharacteristic
107C4⋊C4×C157C2311yescharacteristic
108C22⋊C4×C157—21no—
109C22⋊C4×C157—21no—
110C42×C157C2311yescharacteristic
111C22×D157—21no—
112C22×D157—21no—
113C22×D157—42no—
114C22×D157—42no—
115C22×D157—81no—
116C22×D157C2×D411yescharacteristic
117C22×D157C2×D411yescharacteristic
118D628—22no—
119D628—21no—
120D628—21no—
121D628—21no—
122D628—42no—
123D628—42no—
124D628—81no—
125D628#4112yes—
126C22⋊D157—42no—
127C22⋊D157—81no—
128C2×Dic157—21no—
129C2×Dic157—21no—
130C2×Dic157C2×D411yescharacteristic
131C2×Dic157C2×D411yescharacteristic
132C4×D157—21no—
133C4×D157—21no—
134C4×D157—21no—
135C4×D157—21no—
136C4×D157—42no—
137C4×D157—81no—
138C4×D157—81no—
139C4×D157—81no—
140#11929434—21no—
141#11929434—21no—
142C23×C157—21no—
143Q8×C157—21no—
144Q8×C157—21no—
145D4×C157—21no—
146D4×C157—21no—
147C2×C4×C157—21no—
148C2×C4×C157—21no—
149C2×C4×C157—21no—
150C2×C4×C157C2×D411yescharacteristic
151C2×C4×C157C2×D411yescharacteristic
152C2×C4×C157C2411yescharacteristic
153C8×C157—21no—
154C8×C157—21no—
155C22×C157—41no—
156C22×C157—41no—
157C22×C157—41no—
158C22×C157C22×D411yescharacteristic
159C2×D157—21no—
160C2×D157—21no—
161C2×D157—21no—
162C2×D157—22no—
163C2×D157—21no—
164C2×D157—21no—
165C2×D157—41no—
166C2×D157—41no—
167C2×D157—41no—
168C2×D157—41no—
169C2×D157—41no—
170C2×D157—84no—
171C2×D157#13512yes—
172C4×C157—21no—
173C4×C157—21no—
174C4×C157—41no—
175C4×C157—41no—
176C4×C157—41no—
177C4×C157C2×#4111yescharacteristic
178C4×C157H(5,𝔽2)11yescharacteristic
179Dic157—21no—
180Dic157—21no—
181Dic157—41no—
182Dic157—41no—
183C2×C157—81no—
184C2×C157#54611yescharacteristic
185C2×C157C2×#13511yescharacteristic
186C2×C157#57611yescharacteristic
187D157—22no—
188D157—41no—
189D157—42no—
190D157—81no—
191C157#329711yescharacteristic, complemented, Sylow
192#3297—1571nomaximal, Sylow
193#553—1571no—
194#546—1571no—
195#488D15711yescharacteristic, complemented
196#484—1571no—
197#482—1571no—
198#478—1571no—
199#466—1571no—
200#465—1571no—
201#463—1571no—
202#449—1571no—
203#447—1571no—
204C4×D8—1571no—
205C8×D4—1571no—
206#416—1571no—
207C2×#102—1571no—
208C2×#41—1571no—
209C22×D4—1571no—
210C2×SD16—1571no—
211C2×D8—1571no—
212C2×D8—1571no—
213C22×C8—1571no—
214C22×C8—1571no—
215#126—3141no—
216#125C2×D15711yescharacteristic, complemented
217#124C2×D15711yescharacteristic
218#123—3141no—
219#123—3141no—
220C22⋊Q8—1571no—
221#121—3141no—
222#121—3141no—
223#121—1571no—
224#121—1571no—
225#121—1571no—
226#120—3141no—
227C4×D4—3141no—
228C4×D4—1571no—
229C4×D4—1571no—
230C4×D4—1571no—
231#117—1571no—
232C2×C4⋊C4—1571no—
233C2×C22⋊C4—3141no—
234#107—1571no—
235#107—1571no—
236#106—1571no—
237C4⋊C8—1571no—
238#103—1571no—
239#103C2×D15711yescharacteristic, complemented
240#103C2×D15711yescharacteristic, complemented
241#102—1571no—
242#102—1571no—
243#102—1571no—
244#102—1571no—
245#102—1571no—
246#102C2×D15711yescharacteristic
247#102C2×D15711yescharacteristic
248C22⋊C8—1571no—
249C22⋊C8—1571no—
250C4×C8C2×D15711yescharacteristic, complemented
251C24—3141no—
252#41—3142no—
253#41—6281no—
254C2×Q8C22×D15711yescharacteristic
255C2×D4—3141no—
256C2×D4—3141no—
257C2×D4—3141no—
258C2×D4—3142no—
259C2×D4—3141no—
260C2×D4—3141no—
261C2×D4—6281no—
262C2×D4—1571no—
263C2×D4—1571no—
264C2×D4—1571no—
265C2×D4C22×D15711yescharacteristic
266C22×C4—3141no—
267C22×C4—3141no—
268C22×C4—3141no—
269C22×C4—3141no—
270C22×C4—1571no—
271C22×C4—1571no—
272SD16—6281no—
273D8—3142no—
274D8—6281no—
275C2×C8—6281no—
276C2×C8—6281no—
277C2×C8—1571no—
278C2×C8—1571no—
279C2×C8C22×D15711yescharacteristic
280C2×C8C22×D15711yescharacteristic
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—3141no—
284C4⋊C4—3141no—
285C4⋊C4—1571no—
286C4⋊C4—1571no—
287C4⋊C4—1571no—
288C4⋊C4C22×D15711yescharacteristic
289C4⋊C4C22×D15711yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—3141no—
293C22⋊C4—3141no—
294C22⋊C4—3141no—
295C22⋊C4—3141no—
296C22⋊C4—3141no—
297C22⋊C4—3141no—
298C22⋊C4—3141no—
299C22⋊C4—1571no—
300C22⋊C4—1571no—
301C42—3141no—
302C42C22×D15711yescharacteristic
303C23—21no—
304C23—3141no—
305C23—3141no—
306C23—6282no—
307C23—6282no—
308C23—1571no—
309C23—1571no—
310C23—12561no—
311Q8—21no—
312Q8—21no—
313D4—21no—
314D4—21no—
315D4—3142no—
316D4—3141no—
317D4—3141no—
318D4—3141no—
319D4—6282no—
320D4—6282no—
321D4—6282no—
322D4—1572no—
323D4—12561no—
324D4—12561no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—3141no—
329C2×C4—3141no—
330C2×C4—3141no—
331C2×C4—3141no—
332C2×C4—3141no—
333C2×C4—3141no—
334C2×C4—6282no—
335C2×C4—1571no—
336C2×C4—1571no—
337C2×C4—12561no—
338C2×C4—12561no—
339C2×C4—12561no—
340C2×C4D4×D15711yescharacteristic
341C2×C4D4×D15711yescharacteristic
342C2×C4C23×D15711yescharacteristic
343C8—21no—
344C8—21no—
345C8—3141no—
346C8—3141no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—3141no—
351C22—3141no—
352C22—3141no—
353C22—3142no—
354C22—3141no—
355C22—3141no—
356C22—6281no—
357C22—6281no—
358C22—6281no—
359C22—6281no—
360C22—6281no—
361C22—1572no—
362C22—12564no—
363C22C2×D4×D15711yescharacteristic
364C4—21no—
365C4—21no—
366C4—41no—
367C4—41no—
368C4—41no—
369C4—3141no—
370C4—3141no—
371C4—6281no—
372C4—6281no—
373C4#41×D15711yescharacteristic
374C4#1909781711yescharacteristic
375C2—81no—
376C2—3142no—
377C2—6281no—
378C2—6282no—
379C2—12561no—
380C2#2375791611yescharacteristic
381C2#135×D15711yescharacteristic
382C2#2375807511yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#488×C157C2
socleC22×C157C22×D4
hypercenter#488D157
hypocenterC157#3297
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #488

Complements

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

NH ≅ G/Ntype
1. #23757976C2semidirect
2. #23757916C2semidirect
3. #23757607C2semidirect
4. #23757595C2semidirect
5. #23757570C2semidirect
6. #23757508C2semidirect
7. #23757505C2semidirect
8. #23757485C2semidirect
9. #23757421C2semidirect
10. #23757408C2semidirect
11. C4×D1256C2semidirect
12. C8×D628C2semidirect
13. #23757268C2semidirect
14. #102×D157C2semidirect
15. #488×C157C2semidirect
18. C2×#14578941C22semidirect
22. C2×C8×D157C22semidirect
29. #19097746C22semidirect
35. C4×D628C22semidirect
37. #19097729C22semidirect
40. #19097705C22semidirect
41. #19097704C22semidirect
42. #19097702C22semidirect
43. C4⋊#11929434C22semidirect
44. #19097695C22semidirect
46. #19097693C22semidirect
51. #19097683C22semidirect
52. #125×C157C22semidirect
54. #103×C157C22semidirect
55. #103×C157C22semidirect
58. C4×C8×C157C22semidirect
191. C157#3297semidirect
195. #488D157semidirect
216. #125C2×D157semidirect
239. #103C2×D157semidirect
240. #103C2×D157semidirect
250. C4×C8C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237579762
#237579162
#237576072
#237575952
#237575702
#237575082
#237575052
#237574852
#237574212
#237574082
C4×D12562
C8×D6282
#237572682
#102×D1572
#488×C1572
#3297157