Finite Group Databasebeta

← #34109387

Subgroups — #34109387

Order 20096.

806 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 806 conjugacy classes of subgroups — too many to draw. Only its 141 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
1C22×#14578955C211yesmaximal, characteristic, complemented
2C2×#19097798C212yesmaximal, complemented
3#121×D157C211yesmaximal, characteristic, complemented
4#23757421C212yesmaximal, complemented
5#23757417C212yesmaximal, complemented
6#23757412C211yesmaximal, characteristic, complemented
7#23757409C211yesmaximal, characteristic, complemented
8#23757406C211yesmaximal, characteristic, complemented
9#23757273C212yesmaximal, complemented
10#23757235C211yesmaximal, characteristic, complemented
11#448×C157C211yesmaximal, characteristic, complemented
12C2×#14578955—28no—
13C2×#14578955—21no—
14C2×#14578955—21no—
15C2×#14578955C2212yes—
16C2×#14578955C2212yescomplemented
17C22×D628—21no—
18C22×D628C2211yescharacteristic, complemented
19C2×D4×D157—21no—
20C2×D4×D157C2211yescharacteristic, complemented
21C22×C4×D157—21no—
22C22×C4×D157C2211yescharacteristic
23#19097798—28no—
24C2×#14578941C2212yescomplemented
25C2×D1256C2212yescomplemented
26C2×#14578939C2212yescomplemented
27C2×#14578937C2212yescomplemented
28#19097746C2211yescharacteristic, complemented
29#19097745C2211yescharacteristic
30#19097744—21no—
31#19097743—21no—
32#19097742C2211yescharacteristic, complemented
33C4⋊C4×D157C2211yescharacteristic, complemented
34C22⋊C4×D157—21no—
35#19097695C2212yescomplemented
36#19097693C2212yescomplemented
37#19097691C2212yescomplemented
38C22⋊#11929434C2211yescharacteristic, complemented
39#19097683C2212yescomplemented
40C22×Q8×C157C2211yescharacteristic, complemented
41C2×SD16×C157C2212yescomplemented
42#121×C157C2211yescharacteristic
43#103×C157C2212yescomplemented
44C22⋊C8×C157C2211yescharacteristic, complemented
45C23×D157—21no—
46C23×D157—21no—
47C23×D157D411yescharacteristic
48#14578955—28no—
49#14578955—28no—
50#14578955—24no—
51#14578955—24no—
52#14578955—24no—
53#14578955D418yescomplemented
54C2×D628—28no—
55C2×D628—22no—
56C2×D628—22no—
57C2×D628—24no—
58C2×D628—21no—
59C2×D628—21no—
60C2×D628C2311yescharacteristic
61C2×D628C2312yes—
62C2×C22⋊D157—21no—
63C2×C22⋊D157—21no—
64C2×C22⋊D157—21no—
65D4×D157—24no—
66D4×D157—44no—
67C22×Dic157D411yescharacteristic
68C2×C4×D157—28no—
69C2×C4×D157—22no—
70C2×C4×D157—22no—
71C2×C4×D157—24no—
72C2×C4×D157—21no—
73C2×C4×D157D411yescharacteristic
74C2×C4×D157D411yescharacteristic
75C2×C4×D157C2311yescharacteristic
76C2×C4×D157C2311yescharacteristic
77#14578941—24no—
78D1256—24no—
79#14578939—24no—
80#14578937—24no—
81C2×#11929434C2312yes—
82C4⋊Dic157C2311yescharacteristic
83#14578933—21no—
84C22⋊Dic157—21no—
85#14578931—21no—
86#14578931—21no—
87C2×Q8×C157—24no—
88C2×Q8×C157—21no—
89C2×Q8×C157C2312yes—
90C2×D4×C157—21no—
91C2×D4×C157C2311yescharacteristic
92C22×C4×C157—21no—
93C22×C4×C157C2311yescharacteristic
94SD16×C157—24no—
95C2×C8×C157C2312yes—
96C4⋊C4×C157C2311yescharacteristic
97C22⋊C4×C157—21no—
98C22×D157—24no—
99C22×D157—22no—
100C22×D157—22no—
101C22×D157—22no—
102C22×D157—21no—
103C22×D157—41no—
104C22×D157—42no—
105C22×D157—42no—
106C22×D157—42no—
107C22×D157—42no—
108C22×D157—42no—
109C22×D157C2×D411yescharacteristic
110C22×D157C2×D411yescharacteristic
111D628—28no—
112D628—24no—
113D628—28no—
114D628—22no—
115D628—22no—
116D628—42no—
117D628C2×D414yescomplemented
118C22⋊D157—42no—
119C22⋊D157—42no—
120C22⋊D157—42no—
121C2×Dic157—22no—
122C2×Dic157—21no—
123C2×Dic157C2×D411yescharacteristic
124C2×Dic157C2×D411yescharacteristic
125C4×D157—28no—
126C4×D157—22no—
127C4×D157—22no—
128C4×D157—28no—
129C4×D157—22no—
130C4×D157—42no—
131C4×D157C2×D414yes—
132#11929434—22no—
133C23×C157—21no—
134C23×C157C2×D411yescharacteristic
135Q8×C157—24no—
136Q8×C157—22no—
137Q8×C157C2×D414yescomplemented
138D4×C157—22no—
139D4×C157—42no—
140C2×C4×C157—24no—
141C2×C4×C157—22no—
142C2×C4×C157—22no—
143C2×C4×C157—21no—
144C2×C4×C157C2×D411yescharacteristic
145C2×C4×C157C2411yescharacteristic
146C8×C157—22no—
147C22×C157—21no—
148C22×C157—21no—
149C22×C157—41no—
150C22×C157—41no—
151C22×C157—41no—
152C22×C157AGL(1,ℤ/8)12yescomplemented
153C22×C157C22×D411yescharacteristic
154C2×D157—21no—
155C2×D157—21no—
156C2×D157—21no—
157C2×D157—24no—
158C2×D157—21no—
159C2×D157—21no—
160C2×D157—44no—
161C2×D157—48no—
162C2×D157—42no—
163C2×D157—42no—
164C2×D157—41no—
165C2×D157—41no—
166C2×D157—41no—
167C2×D157—84no—
168C2×D157#12012yes—
169C4×C157—24no—
170C4×C157—21no—
171C4×C157—41no—
172C4×C157C22×D412yes—
173Dic157—21no—
174Dic157—41no—
175Dic157#12012yes—
176C2×C157—22no—
177C2×C157—81no—
178C2×C157C2×#12011yescharacteristic
179C2×C157C2×AGL(1,ℤ/8)11yescharacteristic
180C2×C157#57511yescharacteristic
181D157—22no—
182D157—41no—
183D157—44no—
184D157—81no—
185C157#300611yescharacteristic, complemented, Sylow
186#3006—1571nomaximal, Sylow
187C22×#41—1571no—
188C2×AGL(1,ℤ/8)—1572no—
189C2×#121—1571no—
190#449—1572no—
191#449—1572no—
192#448—1571no—
193#448D15711yescharacteristic, complemented
194#447—1571no—
195#447—1571no—
196#417—1572no—
197#407—1571no—
198C2×#41—3148no—
199C2×#41—3141no—
200C2×#41—3141no—
201C2×#41—1572no—
202C2×#41—1572no—
203C22×Q8C2×D15711yescharacteristic, complemented
204C22×D4—3141no—
205C22×D4—3141no—
206C22×D4—1571no—
207C22×D4—1571no—
208C23×C4—3141no—
209C23×C4—1571no—
210AGL(1,ℤ/8)—3148no—
211C2×SD16—1572no—
212C2×SD16C2×D15712yescomplemented
213C2×D8—1572no—
214C2×D8—1572no—
215C2×M16—1572no—
216#121—3141no—
217#121—3141no—
218#121—1571no—
219#121—1571no—
220#121—1571no—
221#121C2×D15711yescharacteristic
222C2×C4⋊C4—1571no—
223C2×C22⋊C4—3141no—
224#103—1572no—
225#103C2×D15712yescomplemented
226#102—1572no—
227#102—1572no—
228C22⋊C8—1571no—
229C22⋊C8—1572no—
230C22⋊C8C2×D15711yescharacteristic, complemented
231C24—3141no—
232C24—3141no—
233C24—1571no—
234#41—3148no—
235#41—3148no—
236#41—3144no—
237#41—3144no—
238#41—3144no—
239#41—1578no—
240C2×Q8—24no—
241C2×Q8—21no—
242C2×Q8C22×D15712yes—
243C2×D4—21no—
244C2×D4—3148no—
245C2×D4—3142no—
246C2×D4—3142no—
247C2×D4—3144no—
248C2×D4—3141no—
249C2×D4—3141no—
250C2×D4—3141no—
251C2×D4—3144no—
252C2×D4—3141no—
253C2×D4—3141no—
254C2×D4—6284no—
255C2×D4—1571no—
256C2×D4—1572no—
257C2×D4C22×D15711yescharacteristic
258C22×C4—21no—
259C22×C4—3148no—
260C22×C4—3142no—
261C22×C4—3142no—
262C22×C4—3144no—
263C22×C4—3141no—
264C22×C4—1571no—
265C22×C4—1571no—
266C22×C4—1571no—
267C22×C4—1571no—
268C22×C4—1571no—
269C22×C4C22×D15711yescharacteristic
270SD16—24no—
271SD16—3144no—
272D8—3144no—
273D8—3144no—
274M16—3144no—
275C2×C8—1572no—
276C2×C8C22×D15712yes—
277C4⋊C4—3141no—
278C4⋊C4—1571no—
279C4⋊C4C22×D15711yescharacteristic
280C22⋊C4—21no—
281C22⋊C4—3141no—
282C22⋊C4—3141no—
283C22⋊C4—3141no—
284C23—21no—
285C23—3144no—
286C23—3142no—
287C23—3142no—
288C23—3142no—
289C23—3141no—
290C23—6281no—
291C23—6282no—
292C23—6282no—
293C23—6282no—
294C23—6282no—
295C23—6282no—
296C23—1571no—
297C23—1571no—
298C23D4×D15711yescharacteristic
299Q8—24no—
300Q8—22no—
301Q8D4×D15714yescomplemented
302D4—22no—
303D4—42no—
304D4—3148no—
305D4—3144no—
306D4—3148no—
307D4—3142no—
308D4—3142no—
309D4—6282no—
310D4—6282no—
311D4—6282no—
312D4—6282no—
313D4—1574no—
314C2×C4—24no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—21no—
318C2×C4—3148no—
319C2×C4—3142no—
320C2×C4—3148no—
321C2×C4—3142no—
322C2×C4—3142no—
323C2×C4—3142no—
324C2×C4—3141no—
325C2×C4—6282no—
326C2×C4—1571no—
327C2×C4—1571no—
328C2×C4—1574no—
329C2×C4D4×D15711yescharacteristic
330C2×C4C23×D15711yescharacteristic
331C8—22no—
332C8—3142no—
333C22—21no—
334C22—21no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—3141no—
339C22—3141no—
340C22—3141no—
341C22—3144no—
342C22—3141no—
343C22—3141no—
344C22—6284no—
345C22—6288no—
346C22—6282no—
347C22—6282no—
348C22—6281no—
349C22—6281no—
350C22—6281no—
351C22—1572no—
352C22—12564no—
353C22#1909779812yescomplemented
354C22C2×D4×D15711yescharacteristic
355C4—24no—
356C4—21no—
357C4—41no—
358C4—3141no—
359C4—6281no—
360C4—1572no—
361C4C2×D4×D15712yes—
362C2—22no—
363C2—81no—
364C2—3142no—
365C2—6281no—
366C2—6284no—
367C2—12561no—
368C2#120×D15711yescharacteristic
369C2C2×#1909779811yescharacteristic
370C2#2375807211yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#448×C157C2
socleC22×C157C22×D4
hypercenter#448D157
hypocenterC157#3006
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. C22×#14578955C2semidirect
2. C2×#19097798C2semidirect
3. #121×D157C2semidirect
4. #23757421C2semidirect
5. #23757417C2semidirect
6. #23757412C2semidirect
7. #23757409C2semidirect
8. #23757406C2semidirect
9. #23757273C2semidirect
10. #23757235C2semidirect
11. #448×C157C2semidirect
16. C2×#14578955C22semidirect
18. C22×D628C22semidirect
20. C2×D4×D157C22semidirect
24. C2×#14578941C22semidirect
25. C2×D1256C22semidirect
26. C2×#14578939C22semidirect
27. C2×#14578937C22semidirect
28. #19097746C22semidirect
32. #19097742C22semidirect
33. C4⋊C4×D157C22semidirect
35. #19097695C22semidirect
36. #19097693C22semidirect
37. #19097691C22semidirect
38. C22⋊#11929434C22semidirect
39. #19097683C22semidirect
40. C22×Q8×C157C22semidirect
41. C2×SD16×C157C22semidirect
43. #103×C157C22semidirect
44. C22⋊C8×C157C22semidirect
53. #14578955D4semidirect
117. D628C2×D4semidirect
137. Q8×C157C2×D4semidirect
152. C22×C157AGL(1,ℤ/8)semidirect
185. C157#3006semidirect
193. #448D157semidirect
203. C22×Q8C2×D157semidirect
212. C2×SD16C2×D157semidirect
225. #103C2×D157semidirect
230. C22⋊C8C2×D157semidirect
301. Q8D4×D157semidirect
353. C22#19097798semidirect

Maximal subgroups

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

subgroupindex
C22×#145789552
C2×#190977982
#121×D1572
#237574212
#237574172
#237574122
#237574092
#237574062
#237572732
#237572352
#448×C1572
#3006157