Finite Group Databasebeta

← #34109442

Subgroups — #34109442

Order 20096.

764 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 764 conjugacy classes of subgroups — too many to draw. Only its 149 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19097817C211yesmaximal, characteristic, complemented
2C2×#19097799C211yesmaximal, characteristic, complemented
3C2×#19097791C211yesmaximal, characteristic, complemented
4#23757854C211yesmaximal, characteristic, complemented
5#23757438C212yesmaximal, complemented
6#23757425C212yesmaximal, complemented
7#23757421C212yesmaximal, complemented
8#23757408C212yesmaximal, complemented
9C2×#19097691C211yesmaximal, characteristic, complemented
10#23757241C211yesmaximal, characteristic, complemented
11#417×C157C211yesmaximal, characteristic, complemented
12#19097817—24no—
13#19097817—24no—
14C2×#14578955C2212yes—
15C2×#14578954—21no—
16C2×#14578954C2211yescharacteristic
17C2×#14578954C2211yescharacteristic, complemented
18C22×D628—21no—
19C22×D628C2211yescharacteristic
20C2×D4×D157—22no—
21C2×D4×D157C2212yes—
22#19097799—24no—
23#19097791—24no—
24C2×#14578944C2211yescharacteristic, complemented
25C2×#14578943C2212yescomplemented
26C2×#14578942C2211yescharacteristic, complemented
27C2×#14578941C2211yescharacteristic, complemented
28C2×D1256C2212yescomplemented
29C2×#14578939C2211yescharacteristic, complemented
30C22×#11929434C2211yescharacteristic, complemented
31#19097751—21no—
32#19097751C2212yescomplemented
33#19097746—21no—
34#19097746C2212yescomplemented
35#19097745—22no—
36#19097691C2212yescomplemented
37#19097691C2212yescomplemented
38#19097683C2212yescomplemented
39#19097683C2212yescomplemented
40C2×#41×C157C2211yescharacteristic
41C2×M16×C157C2211yescharacteristic, complemented
42C2×C4⋊C4×C157C2211yescharacteristic, complemented
43#103×C157C2212yescomplemented
44#102×C157C2212yescomplemented
45C23×D157—22no—
46C23×D157—21no—
47#14578955—24no—
48#14578955—24no—
49#14578954—28no—
50#14578954—22no—
51#14578954—22no—
52#14578954—22no—
53#14578954—22no—
54C2×Dic314D411yescharacteristic
55C2×Dic314C2311yescharacteristic
56C2×D628—28no—
57C2×D628—22no—
58C2×D628—22no—
59C2×D628—21no—
60C2×D628—22no—
61C2×D628D411yescharacteristic
62C2×D628D414yescomplemented
63C2×D628C2311yescharacteristic
64C2×D628C2312yes—
65C2×C22⋊D157—22no—
66C2×C22⋊D157—21no—
67C2×C22⋊D157D412yes—
68D4×D157—216no—
69D4×D157—24no—
70D4×D157—24no—
71C2×C4×D157—22no—
72C2×C4×D157—21no—
73C2×C4×D157C2312yes—
74#14578944—22no—
75#14578943—24no—
76#14578942—22no—
77#14578941—22no—
78D1256—24no—
79#14578939—22no—
80C2×#11929434—22no—
81C2×#11929434C2311yescharacteristic
82C2×#11929434C2311yescharacteristic
83C4⋊Dic157—22no—
84#14578931—22no—
85#14578931—22no—
86#41×C157—22no—
87#41×C157D414yescomplemented
88C2×Q8×C157C2311yescharacteristic
89C2×D4×C157—21no—
90C2×D4×C157C2311yescharacteristic
91C22×C4×C157—21no—
92C22×C4×C157—21no—
93C22×C4×C157C2311yescharacteristic
94M16×C157—22no—
95C2×C8×C157C2312yes—
96C4⋊C4×C157—21no—
97C4⋊C4×C157C2312yes—
98C22×D157—22no—
99C22×D157—28no—
100C22×D157—24no—
101C22×D157—21no—
102C22×D157—82no—
103C22×D157—82no—
104C22×D157—81no—
105C22×D157—81no—
106C22×D157C2×D412yes—
107Dic314—22no—
108Dic314—21no—
109Dic314—21no—
110D628—28no—
111D628—22no—
112D628—24no—
113D628—22no—
114D628—21no—
115D628—21no—
116D628—82no—
117D628C2×D414yes—
118C22⋊D157—28no—
119C22⋊D157—24no—
120C22⋊D157—42no—
121C2×Dic157—21no—
122C2×Dic157C2×D412yes—
123C4×D157—28no—
124C4×D157—22no—
125C4×D157—22no—
126C4×D157—42no—
127#11929434—21no—
128#11929434—21no—
129C23×C157—21no—
130C23×C157C2×D411yescharacteristic
131Q8×C157—21no—
132Q8×C157C2×D412yes—
133D4×C157—24no—
134D4×C157—21no—
135D4×C157C2×D412yes—
136C2×C4×C157—21no—
137C2×C4×C157—24no—
138C2×C4×C157—22no—
139C2×C4×C157—81no—
140C2×C4×C157C2×D411yescharacteristic
141C2×C4×C157C2×D412yes—
142C2×C4×C157C2×D412yes—
143C2×C4×C157C2411yescharacteristic
144C8×C157—22no—
145C22×C157—21no—
146C22×C157—21no—
147C22×C157—22no—
148C22×C157—41no—
149C22×C157—41no—
150C22×C157#12012yes—
151C22×C157C22×D411yescharacteristic
152C2×D157—22no—
153C2×D157—24no—
154C2×D157—22no—
155C2×D157—22no—
156C2×D157—42no—
157C2×D157—42no—
158C2×D157—41no—
159C2×D157—41no—
160C2×D157—41no—
161C2×D157—84no—
162C2×D157—88no—
163C4×C157—22no—
164C4×C157—42no—
165C4×C157#12012yes—
166C4×C157C22×D411yescharacteristic
167C4×C157C22×D411yescharacteristic
168Dic157—22no—
169Dic157—41no—
170C2×C157—22no—
171C2×C157—42no—
172C2×C157C2×#12011yescharacteristic
173C2×C157#57611yescharacteristic
174C2×C157#57711yescharacteristic
175D157—42no—
176D157—44no—
177D157—81no—
178C157#301211yescharacteristic, complemented, Sylow
179#3012—1571nomaximal, Sylow
180C2×H(5,𝔽2)—1571no—
181C2×AGL(1,ℤ/8)—1571no—
182C2×#135—1571no—
183#537—1571no—
184#452—1572no—
185#450—1572no—
186#449—1572no—
187#447—1572no—
188#417D15711yescharacteristic, complemented
189C2×#102—1571no—
190#408—1571no—
191H(5,𝔽2)—3144no—
192H(5,𝔽2)—3144no—
193C2×#41—3141no—
194C2×#41—1572no—
195C2×#41—1571no—
196C2×#41—1571no—
197C2×#41C2×D15711yescharacteristic
198C22×D4—3141no—
199C22×D4—3142no—
200C22×D4—1572no—
201C22×D4—1571no—
202AGL(1,ℤ/8)—3144no—
203#135—3144no—
204C2×Q16—1571no—
205C2×SD16—1571no—
206C2×SD16—1571no—
207C2×SD16—1572no—
208C2×D8—1571no—
209C2×D8—1572no—
210C2×M16C2×D15711yescharacteristic, complemented
211C22×C8—1571no—
212C22⋊Q8—3141no—
213C22⋊Q8—1572no—
214#121—3141no—
215#121—3142no—
216#121—1572no—
217C2×C4⋊C4C2×D15711yescharacteristic, complemented
218#103C2×D15712yescomplemented
219#102—1572no—
220#102—1572no—
221#102C2×D15712yescomplemented
222C22⋊C8—1572no—
223C22⋊C8—1572no—
224C24—3142no—
225C24—3141no—
226#41—22no—
227#41—3148no—
228#41—3144no—
229#41—3144no—
230#41—3142no—
231#41—3142no—
232#41—3142no—
233#41—3142no—
234#41D62814yescomplemented
235C2×Q8—1571no—
236C2×Q8—1571no—
237C2×Q8C22×D15711yescharacteristic
238C2×D4—21no—
239C2×D4—3148no—
240C2×D4—3142no—
241C2×D4—3142no—
242C2×D4—31416no—
243C2×D4—3144no—
244C2×D4—3144no—
245C2×D4—3141no—
246C2×D4—3142no—
247C2×D4—3141no—
248C2×D4—3142no—
249C2×D4—1572no—
250C2×D4—1571no—
251C2×D4—1574no—
252C2×D4—1571no—
253C2×D4—1572no—
254C2×D4C22×D15711yescharacteristic
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—3142no—
258C22×C4—3141no—
259C22×C4—1572no—
260C22×C4C22×D15711yescharacteristic
261Q16—3142no—
262SD16—3144no—
263SD16—3142no—
264SD16—3142no—
265D8—3144no—
266D8—3142no—
267M16—22no—
268C2×C8—3142no—
269C2×C8—1571no—
270C2×C8—1571no—
271C2×C8C22×D15712yes—
272C4⋊C4—21no—
273C4⋊C4—3142no—
274C4⋊C4C22×D15712yes—
275C22⋊C4—3142no—
276C22⋊C4—3142no—
277C23—21no—
278C23—3148no—
279C23—3142no—
280C23—3144no—
281C23—3141no—
282C23—1572no—
283C23—12561no—
284C23—12561no—
285C23—12562no—
286C23—12562no—
287C23D4×D15711yescharacteristic
288Q8—21no—
289Q8—3142no—
290Q8—3141no—
291Q8—3141no—
292Q8C2×D62812yes—
293D4—24no—
294D4—21no—
295D4—3148no—
296D4—3148no—
297D4—3144no—
298D4—3142no—
299D4—3144no—
300D4—3141no—
301D4—3142no—
302D4—3141no—
303D4—6282no—
304D4—1574no—
305D4—12562no—
306D4C2×D62812yes—
307C2×C4—21no—
308C2×C4—24no—
309C2×C4—22no—
310C2×C4—81no—
311C2×C4—3148no—
312C2×C4—3142no—
313C2×C4—3142no—
314C2×C4—3141no—
315C2×C4—6282no—
316C2×C4—1572no—
317C2×C4D4×D15711yescharacteristic
318C2×C4D4×D15712yes—
319C2×C4C2×D62812yes—
320C2×C4C23×D15711yescharacteristic
321C8—22no—
322C8—3141no—
323C8—3141no—
324C22—21no—
325C22—21no—
326C22—22no—
327C22—41no—
328C22—41no—
329C22—3142no—
330C22—3142no—
331C22—3144no—
332C22—3142no—
333C22—6282no—
334C22—6282no—
335C22—6281no—
336C22—6281no—
337C22—6281no—
338C22—12564no—
339C22—12568no—
340C22#1909774112yes—
341C22C2×D4×D15711yescharacteristic
342C4—22no—
343C4—42no—
344C4—3142no—
345C4—6281no—
346C4#1909774112yes—
347C4C2×D4×D15711yescharacteristic
348C4C22×D62811yescharacteristic
349C2—22no—
350C2—42no—
351C2—6282no—
352C2—6284no—
353C2—12561no—
354C2C2×#1909774111yescharacteristic
355C2#2375807711yescharacteristic
356C2#2375808211yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#417×C157C2
socleC22×C157C22×D4
hypercenter#417D157
hypocenterC157#3012
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. C2×#19097817C2semidirect
2. C2×#19097799C2semidirect
3. C2×#19097791C2semidirect
4. #23757854C2semidirect
5. #23757438C2semidirect
6. #23757425C2semidirect
7. #23757421C2semidirect
8. #23757408C2semidirect
9. C2×#19097691C2semidirect
10. #23757241C2semidirect
11. #417×C157C2semidirect
17. C2×#14578954C22semidirect
24. C2×#14578944C22semidirect
25. C2×#14578943C22semidirect
26. C2×#14578942C22semidirect
27. C2×#14578941C22semidirect
28. C2×D1256C22semidirect
29. C2×#14578939C22semidirect
30. C22×#11929434C22semidirect
32. #19097751C22semidirect
34. #19097746C22semidirect
36. #19097691C22semidirect
37. #19097691C22semidirect
38. #19097683C22semidirect
39. #19097683C22semidirect
41. C2×M16×C157C22semidirect
42. C2×C4⋊C4×C157C22semidirect
43. #103×C157C22semidirect
44. #102×C157C22semidirect
62. C2×D628D4semidirect
87. #41×C157D4semidirect
178. C157#3012semidirect
188. #417D157semidirect
210. C2×M16C2×D157semidirect
217. C2×C4⋊C4C2×D157semidirect
218. #103C2×D157semidirect
221. #102C2×D157semidirect
234. #41D628semidirect

Maximal subgroups

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

subgroupindex
C2×#190978172
C2×#190977992
C2×#190977912
#237578542
#237574382
#237574252
#237574212
#237574082
C2×#190976912
#237572412
#417×C1572
#3012157