Finite Group Databasebeta

← #34111979

Subgroups — #34111979

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#23757970C211yesmaximal, characteristic, complemented
2#23757911C211yesmaximal, characteristic, complemented
3#23757629C211yesmaximal, characteristic, complemented
4#23757607C211yesmaximal, characteristic, complemented
5#23757598C211yesmaximal, characteristic, complemented
6#23757575C211yesmaximal, characteristic, complemented
7#23757507C211yesmaximal, characteristic, complemented
8#23757499C211yesmaximal, characteristic, complemented
9#23757490C211yesmaximal, characteristic, complemented
10#23757405C211yesmaximal, characteristic, complemented
11D8×Dic157C211yesmaximal, characteristic, complemented
12#23757336C211yesmaximal, characteristic, complemented
13#23757269C211yesmaximal, characteristic, complemented
14#102×D157C211yesmaximal, characteristic, complemented
15#449×C157C211yesmaximal, characteristic, complemented
16C2×D4×D157C2211yescharacteristic
17C2×#14578939C2211yescharacteristic
18C22×#11929434C2211yescharacteristic, complemented
19C2×C8×D157C2211yescharacteristic
20#19097764—21no—
21#19097762C2211yescharacteristic, complemented
22#19097760C2211yescharacteristic
23#19097754—21no—
24#19097752—21no—
25#19097747C2211yescharacteristic
26#19097745C2211yescharacteristic
27#19097743—21no—
28#19097742—21no—
29#19097742C2211yescharacteristic
30#19097739—21no—
31D4×Dic157C2211yescharacteristic
32D4×Dic157C2211yescharacteristic, complemented
33C4×C22⋊D157C2211yescharacteristic
34#19097731—21no—
35#19097730C2211yescharacteristic, complemented
36C4⋊C4×D157C2211yescharacteristic
37C2×#14578931—21no—
38#19097705C2211yescharacteristic, complemented
39#19097704C2211yescharacteristic, complemented
40#19097703C2211yescharacteristic, complemented
41#19097700C2211yescharacteristic, complemented
42#19097695C2211yescharacteristic, complemented
43#19097695C2211yescharacteristic, complemented
44#19097693C2211yescharacteristic
45#19097693C2211yescharacteristic
46#19097692C2211yescharacteristic
47#19097692C2211yescharacteristic
48#19097692C2211yescharacteristic, complemented
49#19097691C2211yescharacteristic
50#19097683C2211yescharacteristic
51C8×Dic157C2211yescharacteristic, complemented
52C2×#41×C157C2211yescharacteristic
53C2×SD16×C157C2211yescharacteristic, complemented
54C2×D8×C157C2211yescharacteristic
55#121×C157C2211yescharacteristic, complemented
56#103×C157C2211yescharacteristic, complemented
57#102×C157C2211yescharacteristic
58C22⋊C8×C157C2211yescharacteristic, complemented
59C23×D157—21no—
60C2×D628C2311yescharacteristic
61C2×C22⋊D157—21no—
62C2×C22⋊D157—21no—
63C2×C22⋊D157—21no—
64C2×C22⋊D157D411yescharacteristic
65D4×D157—22no—
66D4×D157—41no—
67C22×Dic157—21no—
68C22×Dic157—21no—
69C2×C4×D157—21no—
70C2×C4×D157C2311yescharacteristic
71#14578939—41no—
72C2×#11929434—41no—
73C2×#11929434C2311yescharacteristic
74C2×#11929434C2311yescharacteristic
75C8×D157—41no—
76C4⋊Dic157—21no—
77C4⋊Dic157C2311yescharacteristic
78C4⋊Dic157C2311yescharacteristic
79C4⋊Dic157C2311yescharacteristic
80#14578933—21no—
81#14578933—21no—
82#14578933—21no—
83#14578933D411yescharacteristic
84C22⋊Dic157—21no—
85C22⋊Dic157—21no—
86C22⋊Dic157—21no—
87C22⋊Dic157—21no—
88C22⋊Dic157D411yescharacteristic
89#14578931—21no—
90#14578931—21no—
91#14578931—21no—
92#14578931—21no—
93#14578931D411yescharacteristic
94C4×Dic157—21no—
95C4×Dic157C2311yescharacteristic
96#41×C157—22no—
97#41×C157—41no—
98C2×Q8×C157C2311yescharacteristic
99C2×D4×C157—21no—
100C2×D4×C157—21no—
101C2×D4×C157C2311yescharacteristic
102C2×D4×C157C2311yescharacteristic
103C22×C4×C157—21no—
104C22×C4×C157C2311yescharacteristic
105SD16×C157—41no—
106D8×C157—22no—
107C2×C8×C157C2311yescharacteristic
108C2×C8×C157C2311yescharacteristic
109C4⋊C4×C157C2311yescharacteristic
110C22⋊C4×C157—21no—
111C22×D157—21no—
112C22×D157—42no—
113C22×D157—42no—
114C22×D157—81no—
115C22×D157C2×D411yescharacteristic
116D628—21no—
117D628—21no—
118C22⋊D157—22no—
119C22⋊D157—42no—
120C22⋊D157—42no—
121C22⋊D157—81no—
122C2×Dic157—21no—
123C2×Dic157—21no—
124C2×Dic157—21no—
125C2×Dic157—81no—
126C2×Dic157—81no—
127C2×Dic157C2×D411yescharacteristic
128C2×Dic157C2×D411yescharacteristic
129C2×Dic157C2×D411yescharacteristic
130C4×D157—21no—
131C4×D157—21no—
132C4×D157—81no—
133#11929434—21no—
134#11929434—21no—
135C23×C157—21no—
136C23×C157—21no—
137C23×C157C2×D411yescharacteristic
138Q8×C157—21no—
139Q8×C157—21no—
140D4×C157—21no—
141D4×C157—21no—
142D4×C157—21no—
143D4×C157—42no—
144D4×C157—81no—
145D4×C157#4112yes—
146C2×C4×C157—21no—
147C2×C4×C157—21no—
148C2×C4×C157—21no—
149C2×C4×C157—21no—
150C2×C4×C157—42no—
151C2×C4×C157C2×D411yescharacteristic
152C2×C4×C157C2411yescharacteristic
153C8×C157—21no—
154C8×C157—21no—
155C22×C157—21no—
156C22×C157—22no—
157C22×C157—21no—
158C22×C157—21no—
159C22×C157—41no—
160C22×C157—41no—
161C22×C157—41no—
162C22×C157—41no—
163C22×C157—41no—
164C22×C157C22×D411yescharacteristic
165C2×D157—21no—
166C2×D157—21no—
167C2×D157—41no—
168C2×D157—41no—
169C2×D157—41no—
170C2×D157—84no—
171C2×D157#13512yes—
172C4×C157—21no—
173C4×C157—41no—
174C4×C157—41no—
175C4×C157C2×#4111yescharacteristic
176C4×C157H(5,𝔽2)11yescharacteristic
177Dic157—21no—
178Dic157—21no—
179Dic157—21no—
180Dic157—41no—
181Dic157—41no—
182Dic157—41no—
183C2×C157—42no—
184C2×C157—41no—
185C2×C157—81no—
186C2×C157#54611yescharacteristic
187C2×C157C2×#13511yescharacteristic
188C2×C157#57611yescharacteristic
189D157—22no—
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#449D15711yescharacteristic, complemented
203#447—1571no—
204C4×D8—1571no—
205C8×D4—1571no—
206#416—1571no—
207C2×#102—1571no—
208C2×#41C2×D15711yescharacteristic
209C22×D4—1571no—
210C2×SD16C2×D15711yescharacteristic, complemented
211C2×D8—1571no—
212C2×D8C2×D15711yescharacteristic
213C22×C8—1571no—
214C22×C8—1571no—
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#121C2×D15711yescharacteristic, complemented
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#103—1571no—
240#103C2×D15711yescharacteristic, complemented
241#102—1571no—
242#102—1571no—
243#102—1571no—
244#102—1571no—
245#102—1571no—
246#102—1571no—
247#102C2×D15711yescharacteristic
248C22⋊C8—1571no—
249C22⋊C8C2×D15711yescharacteristic, complemented
250C4×C8—1571no—
251C24—3141no—
252#41—22no—
253#41—41no—
254C2×Q8C22×D15711yescharacteristic
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—3142no—
258C2×D4—3141no—
259C2×D4—3141no—
260C2×D4—3141no—
261C2×D4—6281no—
262C2×D4—1571no—
263C2×D4—1571no—
264C2×D4C22×D15711yescharacteristic
265C2×D4C22×D15711yescharacteristic
266C22×C4—21no—
267C22×C4—3141no—
268C22×C4—3141no—
269C22×C4—3141no—
270C22×C4—1571no—
271C22×C4C22×D15711yescharacteristic
272SD16—41no—
273D8—22no—
274D8—6281no—
275C2×C8—6281no—
276C2×C8—6281no—
277C2×C8—1571no—
278C2×C8—1571no—
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⋊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—21no—
304C23—21no—
305C23—3141no—
306C23—6282no—
307C23—6282no—
308C23—1571no—
309C23—12561no—
310C23D4×D15711yescharacteristic
311Q8—21no—
312Q8—21no—
313D4—21no—
314D4—21no—
315D4—21no—
316D4—42no—
317D4—81no—
318D4—3142no—
319D4—3141no—
320D4—3141no—
321D4—6282no—
322D4—6282no—
323D4—12561no—
324D4#1457895312yes—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—42no—
330C2×C4—3141no—
331C2×C4—3141no—
332C2×C4—3141no—
333C2×C4—3141no—
334C2×C4—3141no—
335C2×C4—1571no—
336C2×C4—1571no—
337C2×C4—1571no—
338C2×C4—12561no—
339C2×C4—12561no—
340C2×C4—12561no—
341C2×C4D4×D15711yescharacteristic
342C2×C4C23×D15711yescharacteristic
343C8—21no—
344C8—21no—
345C8—3141no—
346C8—3141no—
347C22—21no—
348C22—22no—
349C22—21no—
350C22—21no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—3141no—
357C22—3141no—
358C22—6281no—
359C22—6281no—
360C22—6281no—
361C22—1572no—
362C22—12564no—
363C22C2×D4×D15711yescharacteristic
364C4—21no—
365C4—41no—
366C4—41no—
367C4—3141no—
368C4—3141no—
369C4—3141no—
370C4—6281no—
371C4—6281no—
372C4—6281no—
373C4C2×#1457895311yescharacteristic
374C4#1909781611yescharacteristic
375C2—41no—
376C2—42no—
377C2—81no—
378C2—3142no—
379C2—12561no—
380C2#2375791011yescharacteristic
381C2#135×D15711yescharacteristic
382C2#2375807711yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#449×C157C2
socleC22×C157C22×D4
hypercenter#449D157
hypocenterC157#3297
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. #23757970C2semidirect
2. #23757911C2semidirect
3. #23757629C2semidirect
4. #23757607C2semidirect
5. #23757598C2semidirect
6. #23757575C2semidirect
7. #23757507C2semidirect
8. #23757499C2semidirect
9. #23757490C2semidirect
10. #23757405C2semidirect
11. D8×Dic157C2semidirect
12. #23757336C2semidirect
13. #23757269C2semidirect
14. #102×D157C2semidirect
15. #449×C157C2semidirect
18. C22×#11929434C22semidirect
21. #19097762C22semidirect
32. D4×Dic157C22semidirect
35. #19097730C22semidirect
38. #19097705C22semidirect
39. #19097704C22semidirect
40. #19097703C22semidirect
41. #19097700C22semidirect
42. #19097695C22semidirect
43. #19097695C22semidirect
48. #19097692C22semidirect
51. C8×Dic157C22semidirect
53. C2×SD16×C157C22semidirect
55. #121×C157C22semidirect
56. #103×C157C22semidirect
58. C22⋊C8×C157C22semidirect
191. C157#3297semidirect
202. #449D157semidirect
210. C2×SD16C2×D157semidirect
225. #121C2×D157semidirect
240. #103C2×D157semidirect
249. C22⋊C8C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237579702
#237579112
#237576292
#237576072
#237575982
#237575752
#237575072
#237574992
#237574902
#237574052
D8×Dic1572
#237573362
#237572692
#102×D1572
#449×C1572
#3297157