Finite Group Databasebeta

← #34111988

Subgroups — #34111988

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#23757969C211yesmaximal, characteristic, complemented
2#23757919C211yesmaximal, characteristic, complemented
3#23757628C211yesmaximal, characteristic, complemented
4#23757609C211yesmaximal, characteristic, complemented
5#23757598C211yesmaximal, characteristic, complemented
6#23757574C211yesmaximal, characteristic, complemented
7#23757509C211yesmaximal, characteristic, complemented
8#23757502C211yesmaximal, characteristic, complemented
9#23757489C211yesmaximal, characteristic, complemented
10#23757420C211yesmaximal, characteristic, complemented
11#23757409C211yesmaximal, characteristic, complemented
12C4×D1256C211yesmaximal, characteristic, complemented
13#23757270C211yesmaximal, characteristic, complemented
14C2×#19097693C211yesmaximal, characteristic, complemented
15C8×D4×C157C211yesmaximal, characteristic, complemented
16C2×#14578954C2211yescharacteristic
17C22×D628C2211yescharacteristic
18C2×#14578943C2211yescharacteristic, complemented
19C2×D1256C2211yescharacteristic
20C2×D1256C2211yescharacteristic
21#19097766—21no—
22#19097763C2211yescharacteristic, complemented
23#19097759C2211yescharacteristic
24#19097756—21no—
25#19097753—21no—
26#19097750C2211yescharacteristic
27#19097745C2211yescharacteristic
28#19097745C2211yescharacteristic
29#19097745C2211yescharacteristic, complemented
30#19097744—21no—
31#19097743—21no—
32#19097739—21no—
33C4×D628C2211yescharacteristic
34C4×D628C2211yescharacteristic, complemented
35#19097731—21no—
36#19097730C2211yescharacteristic, complemented
37C2×C4⋊Dic157C2211yescharacteristic
38C22⋊C4×D157—21no—
39#19097705C2211yescharacteristic, complemented
40#19097705C2211yescharacteristic, complemented
41#19097703C2211yescharacteristic, complemented
42#19097696C2211yescharacteristic, complemented
43#19097696C2211yescharacteristic, complemented
44#19097696C2211yescharacteristic, complemented
45#19097693C2211yescharacteristic
46#19097693C2211yescharacteristic
47#19097693C2211yescharacteristic
48#19097693C2211yescharacteristic
49#19097693C2211yescharacteristic
50#19097693C2211yescharacteristic
51#19097693C2211yescharacteristic, complemented
52C22×C8×C157C2211yescharacteristic
53C22×C8×C157C2211yescharacteristic, complemented
54C4×D4×C157C2211yescharacteristic
55C4⋊C8×C157C2211yescharacteristic, complemented
56C22⋊C8×C157C2211yescharacteristic
57C22⋊C8×C157C2211yescharacteristic, complemented
58C4×C8×C157C2211yescharacteristic, complemented
59C23×D157—21no—
60#14578954—22no—
61#14578954—41no—
62C2×Dic314C2311yescharacteristic
63C2×D628—22no—
64C2×D628—21no—
65C2×D628—41no—
66C2×D628C2311yescharacteristic
67C2×D628C2311yescharacteristic
68C2×D628C2311yescharacteristic
69C2×C22⋊D157—21no—
70C2×C22⋊D157—21no—
71C2×C22⋊D157—21no—
72C2×C22⋊D157—21no—
73C22×Dic157—21no—
74C2×C4×D157—21no—
75C2×C4×D157—21no—
76C2×C4×D157—21no—
77#14578943—41no—
78D1256—22no—
79D1256—41no—
80C4⋊Dic157—21no—
81C4⋊Dic157C2311yescharacteristic
82C4⋊Dic157C2311yescharacteristic
83C4⋊Dic157C2311yescharacteristic
84C4⋊Dic157C2311yescharacteristic
85#14578933—21no—
86#14578933—21no—
87#14578933—21no—
88C22⋊Dic157—21no—
89C22⋊Dic157—21no—
90#14578931—21no—
91#14578931—21no—
92#14578931—21no—
93#14578931—21no—
94#14578931—21no—
95#14578931—21no—
96#14578931—21no—
97C4×Dic157—21no—
98C2×D4×C157D411yescharacteristic
99C22×C4×C157C2311yescharacteristic
100C22×C4×C157C2311yescharacteristic
101C2×C8×C157—41no—
102C2×C8×C157—41no—
103C2×C8×C157C2311yescharacteristic
104C2×C8×C157C2311yescharacteristic
105C2×C8×C157C2311yescharacteristic
106C2×C8×C157C2311yescharacteristic
107C4⋊C4×C157D411yescharacteristic
108C22⋊C4×C157D411yescharacteristic
109C22⋊C4×C157D411yescharacteristic
110C42×C157C2311yescharacteristic
111C22×D157—21no—
112C22×D157—21no—
113C22×D157—21no—
114C22×D157—42no—
115C22×D157—42no—
116C22×D157—81no—
117Dic314—21no—
118Dic314—21no—
119D628—21no—
120D628—21no—
121D628—21no—
122D628—21no—
123D628—21no—
124D628—42no—
125D628#4112yes—
126C22⋊D157—42no—
127C22⋊D157—42no—
128C22⋊D157—81no—
129C22⋊D157—81no—
130C2×Dic157—21no—
131C2×Dic157—21no—
132C2×Dic157—21no—
133C2×Dic157—21no—
134C2×Dic157—21no—
135C2×Dic157—81no—
136C4×D157—42no—
137C4×D157—81no—
138C4×D157—81no—
139C23×C157C2×D411yescharacteristic
140C23×C157C2×D411yescharacteristic
141D4×C157—22no—
142C2×C4×C157—21no—
143C2×C4×C157—21no—
144C2×C4×C157—21no—
145C2×C4×C157—21no—
146C2×C4×C157C2×D411yescharacteristic
147C2×C4×C157C2×D411yescharacteristic
148C2×C4×C157C2×D411yescharacteristic
149C2×C4×C157C2×D411yescharacteristic
150C2×C4×C157C2411yescharacteristic
151C8×C157—21no—
152C8×C157—21no—
153C8×C157—21no—
154C8×C157—21no—
155C22×C157—21no—
156C22×C157—21no—
157C22×C157—21no—
158C22×C157—21no—
159C22×C157—21no—
160C22×C157#13512yes—
161C22×C157C22×D411yescharacteristic
162C2×D157—22no—
163C2×D157—41no—
164C2×D157—41no—
165C2×D157—41no—
166C2×D157—41no—
167C2×D157—41no—
168C2×D157—41no—
169C2×D157—41no—
170C2×D157—41no—
171C2×D157—84no—
172C4×C157—21no—
173C4×C157—21no—
174C4×C157—21no—
175C4×C157—21no—
176C4×C157C2×#4111yescharacteristic
177C4×C157H(5,𝔽2)11yescharacteristic
178Dic157—41no—
179Dic157—41no—
180Dic157—41no—
181Dic157—41no—
182Dic157—41no—
183C2×C157—22no—
184C2×C157—41no—
185C2×C157#54611yescharacteristic
186C2×C157C2×#13511yescharacteristic
187C2×C157#57611yescharacteristic
188D157—42no—
189D157—81no—
190D157—81no—
191C157#329711yescharacteristic, complemented, Sylow
192#3297—1571nomaximal, Sylow
193#553—1571no—
194#546—1571no—
195#488—1571no—
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×D4D15711yescharacteristic, complemented
206#416—1571no—
207C2×#102—1571no—
208C2×#41—1571no—
209C22×D4—1571no—
210C2×SD16—1571no—
211C2×D8—1571no—
212C2×D8—1571no—
213C22×C8C2×D15711yescharacteristic
214C22×C8C2×D15711yescharacteristic, complemented
215#126—3141no—
216#125—1571no—
217#124—1571no—
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×D4C2×D15711yescharacteristic
231#117—1571no—
232C2×C4⋊C4—1571no—
233C2×C22⋊C4—3141no—
234#107—1571no—
235#107—1571no—
236#106—1571no—
237C4⋊C8C2×D15711yescharacteristic, complemented
238#103—1571no—
239#103—1571no—
240#103—1571no—
241#102—1571no—
242#102—1571no—
243#102—1571no—
244#102—1571no—
245#102—1571no—
246#102—1571no—
247#102—1571no—
248C22⋊C8C2×D15711yescharacteristic
249C22⋊C8C2×D15711yescharacteristic, complemented
250C4×C8C2×D15711yescharacteristic, complemented
251C24—3141no—
252#41—3142no—
253#41—6281no—
254C2×Q8—1571no—
255C2×D4—3142no—
256C2×D4—3141no—
257C2×D4—3141no—
258C2×D4—3141no—
259C2×D4—3141no—
260C2×D4—3141no—
261C2×D4—6281no—
262C2×D4—1571no—
263C2×D4—1571no—
264C2×D4—1571no—
265C2×D4D62811yescharacteristic
266C22×C4—3141no—
267C22×C4—3141no—
268C22×C4—3141no—
269C22×C4—3141no—
270C22×C4C22×D15711yescharacteristic
271C22×C4C22×D15711yescharacteristic
272SD16—6281no—
273D8—3142no—
274D8—6281no—
275C2×C8—41no—
276C2×C8—41no—
277C2×C8C22×D15711yescharacteristic
278C2×C8C22×D15711yescharacteristic
279C2×C8C22×D15711yescharacteristic
280C2×C8C22×D15711yescharacteristic
281C4⋊C4—3141no—
282C4⋊C4—3141no—
283C4⋊C4—3141no—
284C4⋊C4—3141no—
285C4⋊C4—1571no—
286C4⋊C4—1571no—
287C4⋊C4—1571no—
288C4⋊C4—1571no—
289C4⋊C4D62811yescharacteristic
290C22⋊C4—3141no—
291C22⋊C4—3141no—
292C22⋊C4—3141no—
293C22⋊C4—3141no—
294C22⋊C4—3141no—
295C22⋊C4—3141no—
296C22⋊C4—3141no—
297C22⋊C4—3141no—
298C22⋊C4—3141no—
299C22⋊C4D62811yescharacteristic
300C22⋊C4D62811yescharacteristic
301C42—3141no—
302C42C22×D15711yescharacteristic
303C23—3141no—
304C23—3141no—
305C23—3141no—
306C23—6282no—
307C23—6282no—
308C23—12561no—
309C23C2×D62811yescharacteristic
310C23C2×D62811yescharacteristic
311Q8—3141no—
312Q8—3141no—
313D4—22no—
314D4—3141no—
315D4—3141no—
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—21no—
329C2×C4—3141no—
330C2×C4—3141no—
331C2×C4—3141no—
332C2×C4—3141no—
333C2×C4—3141no—
334C2×C4—6282no—
335C2×C4—12561no—
336C2×C4—12561no—
337C2×C4—12561no—
338C2×C4C2×D62811yescharacteristic
339C2×C4C2×D62811yescharacteristic
340C2×C4C2×D62811yescharacteristic
341C2×C4C2×D62811yescharacteristic
342C2×C4C23×D15711yescharacteristic
343C8—21no—
344C8—21no—
345C8—21no—
346C8—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—3142no—
353C22—6281no—
354C22—6281no—
355C22—6281no—
356C22—6281no—
357C22—6281no—
358C22—6281no—
359C22—6281no—
360C22—6281no—
361C22—12564no—
362C22#1909779312yes—
363C22C22×D62811yescharacteristic
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—6281no—
369C4—6281no—
370C4—6281no—
371C4—6281no—
372C4—6281no—
373C4#41×D15711yescharacteristic
374C4#1909781611yescharacteristic
375C2—22no—
376C2—41no—
377C2—6282no—
378C2—12561no—
379C2—12561no—
380C2#2375791811yescharacteristic
381C2C2×#1909779311yescharacteristic
382C2#2375807811yescharacteristic
subgroupG/H
centerC22C22×D628
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
FittingC8×D4×C157C2
socleC22×C157C22×D4
hypercenterC8×D4D157
hypocenterC157#3297
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. #23757969C2semidirect
2. #23757919C2semidirect
3. #23757628C2semidirect
4. #23757609C2semidirect
5. #23757598C2semidirect
6. #23757574C2semidirect
7. #23757509C2semidirect
8. #23757502C2semidirect
9. #23757489C2semidirect
10. #23757420C2semidirect
11. #23757409C2semidirect
12. C4×D1256C2semidirect
13. #23757270C2semidirect
14. C2×#19097693C2semidirect
15. C8×D4×C157C2semidirect
18. C2×#14578943C22semidirect
22. #19097763C22semidirect
29. #19097745C22semidirect
34. C4×D628C22semidirect
36. #19097730C22semidirect
39. #19097705C22semidirect
40. #19097705C22semidirect
41. #19097703C22semidirect
42. #19097696C22semidirect
43. #19097696C22semidirect
44. #19097696C22semidirect
51. #19097693C22semidirect
53. C22×C8×C157C22semidirect
55. C4⋊C8×C157C22semidirect
57. C22⋊C8×C157C22semidirect
58. C4×C8×C157C22semidirect
191. C157#3297semidirect
205. C8×D4D157semidirect
214. C22×C8C2×D157semidirect
237. C4⋊C8C2×D157semidirect
249. C22⋊C8C2×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
#237579692
#237579192
#237576282
#237576092
#237575982
#237575742
#237575092
#237575022
#237574892
#237574202
#237574092
C4×D12562
#237572702
C2×#190976932
C8×D4×C1572
#3297157