Finite Group Databasebeta

← #34111984

Subgroups — #34111984

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#23757971C211yesmaximal, characteristic, complemented
2#23757914C211yesmaximal, characteristic, complemented
3#23757629C211yesmaximal, characteristic, complemented
4#23757600C211yesmaximal, characteristic, complemented
5#23757576C211yesmaximal, characteristic, complemented
6#23757510C211yesmaximal, characteristic, complemented
7#23757504C211yesmaximal, characteristic, complemented
8#23757491C211yesmaximal, characteristic, complemented
9#23757421C211yesmaximal, characteristic, complemented
10#23757409C211yesmaximal, characteristic, complemented
11#23757355C211yesmaximal, characteristic, complemented
12#23757336C211yesmaximal, characteristic, complemented
13#23757268C211yesmaximal, characteristic, complemented
14C2×#19097691C211yesmaximal, characteristic, complemented
15#484×C157C211yesmaximal, characteristic, complemented
16C2×#14578955C2211yescharacteristic
17C22×D628C2211yescharacteristic
18C2×#14578941C2211yescharacteristic, complemented
19C2×D1256C2211yescharacteristic
20C2×D1256C2211yescharacteristic
21C22×#11929434C2211yescharacteristic
22C2×C8×D157C2211yescharacteristic, complemented
23#19097766—21no—
24#19097762C2211yescharacteristic, complemented
25#19097760C2211yescharacteristic
26#19097756—21no—
27#19097756—21no—
28#19097746—21no—
29#19097746C2211yescharacteristic
30#19097746C2211yescharacteristic, complemented
31#19097745—21no—
32#19097745C2211yescharacteristic
33#19097741—21no—
34#19097733C2211yescharacteristic
35#19097733C2211yescharacteristic, complemented
36C4×C22⋊D157—21no—
37C4×C22⋊D157C2211yescharacteristic
38#19097729C2211yescharacteristic, complemented
39C22⋊C4×D157—21no—
40#19097702C2211yescharacteristic, complemented
41#19097700C2211yescharacteristic, complemented
42#19097695C2211yescharacteristic, complemented
43#19097693C2211yescharacteristic
44#19097693C2211yescharacteristic
45#19097691C2211yescharacteristic
46#19097691C2211yescharacteristic
47#19097691C2211yescharacteristic
48#19097691C2211yescharacteristic
49#19097691C2211yescharacteristic, complemented
50#19097683C2211yescharacteristic, complemented
51C8×Dic157C2211yescharacteristic, complemented
52C22⋊Q8×C157C2211yescharacteristic
53C2×C4⋊C4×C157C2211yescharacteristic
54#107×C157C2211yescharacteristic, complemented
55#107×C157C2211yescharacteristic, complemented
56#103×C157C2211yescharacteristic, complemented
57#103×C157C2211yescharacteristic, complemented
58C22⋊C8×C157C2211yescharacteristic
59C23×D157—21no—
60#14578955—22no—
61#14578955—41no—
62C2×D628—22no—
63C2×D628—21no—
64C2×D628—21no—
65C2×D628—21no—
66C2×D628—21no—
67C2×D628—41no—
68C2×D628C2311yescharacteristic
69C2×D628C2311yescharacteristic
70C2×D628C2311yescharacteristic
71C2×C22⋊D157—21no—
72C2×C22⋊D157D411yescharacteristic
73C2×C4×D157—21no—
74C2×C4×D157—21no—
75C2×C4×D157—21no—
76C2×C4×D157C2311yescharacteristic
77#14578941—41no—
78D1256—22no—
79D1256—41no—
80C2×#11929434—41no—
81C2×#11929434C2311yescharacteristic
82C2×#11929434C2311yescharacteristic
83C8×D157—41no—
84C4⋊Dic157—21no—
85C4⋊Dic157C2311yescharacteristic
86#14578933—21no—
87#14578933D411yescharacteristic
88C22⋊Dic157D411yescharacteristic
89#14578931—21no—
90#14578931—21no—
91#14578931—21no—
92#14578931—21no—
93#14578931—21no—
94#14578931—21no—
95#14578931—21no—
96#14578931—21no—
97#14578931D411yescharacteristic
98C4×Dic157—21no—
99C4×Dic157C2311yescharacteristic
100C2×Q8×C157C2311yescharacteristic
101C22×C4×C157—21no—
102C22×C4×C157C2311yescharacteristic
103C2×C8×C157C2311yescharacteristic
104C2×C8×C157C2311yescharacteristic
105C4⋊C4×C157—21no—
106C4⋊C4×C157—21no—
107C4⋊C4×C157C2311yescharacteristic
108C4⋊C4×C157C2311yescharacteristic
109C4⋊C4×C157C2311yescharacteristic
110C22⋊C4×C157—21no—
111C22×D157—21no—
112C22×D157—21no—
113C22×D157—21no—
114C22×D157—42no—
115C22×D157—42no—
116C22×D157—81no—
117C22×D157C2×D411yescharacteristic
118D628—21no—
119D628—21no—
120D628—21no—
121D628—21no—
122D628—21no—
123D628—42no—
124D628—42no—
125D628—81no—
126D628—81no—
127D628#4112yes—
128C22⋊D157—22no—
129C22⋊D157—42no—
130C2×Dic157—21no—
131C2×Dic157C2×D411yescharacteristic
132C2×Dic157C2×D411yescharacteristic
133C2×Dic157C2×D411yescharacteristic
134C4×D157—21no—
135C4×D157—21no—
136C4×D157—42no—
137C4×D157—81no—
138C4×D157—81no—
139#11929434—21no—
140#11929434—21no—
141C23×C157C2×D411yescharacteristic
142Q8×C157—21no—
143Q8×C157—21no—
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—81no—
151C2×C4×C157C2×D411yescharacteristic
152C2×C4×C157C2411yescharacteristic
153C8×C157—21no—
154C8×C157—21no—
155C22×C157—21no—
156C22×C157—21no—
157C22×C157#13512yes—
158C22×C157C22×D411yescharacteristic
159C2×D157—21no—
160C2×D157—22no—
161C2×D157—21no—
162C2×D157—21no—
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—41no—
174C4×C157—41no—
175C4×C157—41no—
176C4×C157—41no—
177C4×C157C2×#4111yescharacteristic
178C4×C157H(5,𝔽2)11yescharacteristic
179Dic157—21no—
180Dic157—21no—
181Dic157—21no—
182Dic157—41no—
183C2×C157—22no—
184C2×C157#54611yescharacteristic
185C2×C157C2×#13511yescharacteristic
186C2×C157#57611yescharacteristic
187D157—41no—
188D157—42no—
189D157—81no—
190D157—81no—
191C157#329711yescharacteristic, complemented, Sylow
192#3297—1571nomaximal, Sylow
193#553—1571no—
194#546—1571no—
195#488—1571no—
196#484D15711yescharacteristic, complemented
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#125—1571no—
217#124—1571no—
218#123—3141no—
219#123—3141no—
220C22⋊Q8C2×D15711yescharacteristic
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⋊C4C2×D15711yescharacteristic
233C2×C22⋊C4—3141no—
234#107C2×D15711yescharacteristic, complemented
235#107C2×D15711yescharacteristic, complemented
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#102—1571no—
247#102—1571no—
248C22⋊C8—1571no—
249C22⋊C8C2×D15711yescharacteristic
250C4×C8—1571no—
251C24—3141no—
252#41—3142no—
253#41—6281no—
254C2×Q8C22×D15711yescharacteristic
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×D4—1571no—
266C22×C4—21no—
267C22×C4—3141no—
268C22×C4—3141no—
269C22×C4—3141no—
270C22×C4—1571no—
271C22×C4C22×D15711yescharacteristic
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⋊C4C22×D15711yescharacteristic
288C4⋊C4C22×D15711yescharacteristic
289C4⋊C4C22×D15711yescharacteristic
290C22⋊C4—21no—
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⋊C4—1571no—
300C22⋊C4—1571no—
301C42—3141no—
302C42—1571no—
303C23—3141no—
304C23—3141no—
305C23—3141no—
306C23—6282no—
307C23—6282no—
308C23—1571no—
309C23—12561no—
310C23D4×D15711yescharacteristic
311Q8—21no—
312Q8—21no—
313D4—3142no—
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—21no—
330C2×C4—21no—
331C2×C4—81no—
332C2×C4—3141no—
333C2×C4—3141no—
334C2×C4—3141no—
335C2×C4—6282no—
336C2×C4—1571no—
337C2×C4—1571no—
338C2×C4—1571no—
339C2×C4—12561no—
340C2×C4—12561no—
341C2×C4D4×D15711yescharacteristic
342C2×C4C23×D15711yescharacteristic
343C8—21no—
344C8—21no—
345C8—3141no—
346C8—3141no—
347C22—21no—
348C22—21no—
349C22—3141no—
350C22—3142no—
351C22—3141no—
352C22—3141no—
353C22—6281no—
354C22—6281no—
355C22—6281no—
356C22—6281no—
357C22—6281no—
358C22—6281no—
359C22—6281no—
360C22—6281no—
361C22—12564no—
362C22#1909779412yes—
363C22C2×D4×D15711yescharacteristic
364C4—21no—
365C4—41no—
366C4—41no—
367C4—41no—
368C4—41no—
369C4—3141no—
370C4—3141no—
371C4—3141no—
372C4—6281no—
373C4#41×D15711yescharacteristic
374C4#1909781711yescharacteristic
375C2—22no—
376C2—6281no—
377C2—6282no—
378C2—12561no—
379C2—12561no—
380C2#2375791511yescharacteristic
381C2C2×#1909779411yescharacteristic
382C2#2375807711yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#484×C157C2
socleC22×C157C22×D4
hypercenter#484D157
hypocenterC157#3297
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
1. #23757971C2semidirect
2. #23757914C2semidirect
3. #23757629C2semidirect
4. #23757600C2semidirect
5. #23757576C2semidirect
6. #23757510C2semidirect
7. #23757504C2semidirect
8. #23757491C2semidirect
9. #23757421C2semidirect
10. #23757409C2semidirect
11. #23757355C2semidirect
12. #23757336C2semidirect
13. #23757268C2semidirect
14. C2×#19097691C2semidirect
15. #484×C157C2semidirect
18. C2×#14578941C22semidirect
22. C2×C8×D157C22semidirect
24. #19097762C22semidirect
30. #19097746C22semidirect
35. #19097733C22semidirect
38. #19097729C22semidirect
40. #19097702C22semidirect
41. #19097700C22semidirect
42. #19097695C22semidirect
49. #19097691C22semidirect
50. #19097683C22semidirect
51. C8×Dic157C22semidirect
54. #107×C157C22semidirect
55. #107×C157C22semidirect
56. #103×C157C22semidirect
57. #103×C157C22semidirect
191. C157#3297semidirect
196. #484D157semidirect
234. #107C2×D157semidirect
235. #107C2×D157semidirect
239. #103C2×D157semidirect
240. #103C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237579712
#237579142
#237576292
#237576002
#237575762
#237575102
#237575042
#237574912
#237574212
#237574092
#237573552
#237573362
#237572682
C2×#190976912
#484×C1572
#3297157