Finite Group Databasebeta

← #34111918

Subgroups — #34111918

Order 20096.

454 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 125 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
1#23758087C211yesmaximal, characteristic, complemented
2#23758084C212yesmaximal, complemented
3#23757656C211yesmaximal, characteristic, complemented
4#23757548C211yesmaximal, characteristic, complemented
5#23757540C211yesmaximal, characteristic, complemented
6#23757534C212yesmaximal, complemented
7#23757451C212yesmaximal, complemented
8#23757444C212yesmaximal, complemented
9#23757390C211yesmaximal, characteristic, complemented
10#23757388C211yesmaximal, characteristic, complemented
11#576×C157C211yesmaximal, characteristic, complemented
12#19097818C2212yescomplemented
13#19097801—22no—
14#19097801C2212yes—
15#19097801C2212yescomplemented
16#19097800—22no—
17#19097797—22no—
18#19097797C2212yescomplemented
19#19097791—22no—
20#19097791C2211yescharacteristic
21#19097790—21no—
22C2×Dic628C2211yescharacteristic, complemented
23Q16×D157—21no—
24C2×#14578942—22no—
25C2×#14578942C2212yescomplemented
26#19097778C2212yescomplemented
27#19097777C2211yescharacteristic, complemented
28#19097758C2212yescomplemented
29#19097707C2211yescharacteristic, complemented
30#19097706C2211yescharacteristic, complemented
31#19097698C2211yescharacteristic
32#19097698C2212yescomplemented
33#19097697C2211yescharacteristic, complemented
34#19097689C2212yescomplemented
35#19097688C2212yescomplemented
36C8×Dic157C2211yescharacteristic, complemented
37H(5,𝔽2)×C157C2212yes—
38AGL(1,ℤ/8)×C157—22no—
39AGL(1,ℤ/8)×C157C2212yes—
40#135×C157—21no—
41#135×C157C2211yescharacteristic
42C2×D8×C157—21no—
43C2×D8×C157C2211yescharacteristic
44#131×C157C2211yescharacteristic
45#14578954—22no—
46#14578954C2311yescharacteristic
47#14578953—22no—
48#14578953—42no—
49C2×Dic314—22no—
50C2×Dic314C2312yes—
51Q8×D157—22no—
52Dic628—21no—
53Dic628—21no—
54#14578944—22no—
55#14578944—22no—
56#14578944D411yescharacteristic
57#14578942—22no—
58#14578942—22no—
59#14578942—22no—
60#14578942—22no—
61#14578942—22no—
62#14578942D411yescharacteristic
63#14578941D411yescharacteristic
64#14578939—22no—
65#14578939D411yescharacteristic
66#14578938—22no—
67#14578938C2311yescharacteristic
68#14578938C2312yes—
69#14578937—21no—
70C2×#11929434—22no—
71C2×#11929434C2311yescharacteristic
72C8×D157—21no—
73C22⋊Dic157—42no—
74C4×Dic157C2311yescharacteristic
75#41×C157—22no—
76#41×C157—22no—
77#41×C157C2311yescharacteristic
78#41×C157C2312yes—
79C2×D4×C157—22no—
80C2×D4×C157—22no—
81C2×D4×C157—42no—
82C2×D4×C157C2312yes—
83Q16×C157D411yescharacteristic
84SD16×C157—22no—
85SD16×C157D412yes—
86D8×C157—22no—
87D8×C157—21no—
88D8×C157—21no—
89D8×C157D411yescharacteristic
90M16×C157—21no—
91M16×C157C2311yescharacteristic
92C2×C8×C157—21no—
93C2×C8×C157C2311yescharacteristic
94Dic314—22no—
95Dic314—22no—
96Dic314—42no—
97Dic314C2×D411yescharacteristic
98D628C2×D411yescharacteristic
99C22⋊D157—21no—
100C22⋊D157—42no—
101C2×Dic157—22no—
102C2×Dic157—21no—
103C2×Dic157—42no—
104C4×D157—21no—
105C4×D157—42no—
106#11929434—21no—
107#11929434—22no—
108#11929434C2×D411yescharacteristic
109#11929434C2×D411yescharacteristic
110C23×C157—22no—
111C23×C157—42no—
112Q8×C157C2×D411yescharacteristic
113Q8×C157C2×D412yes—
114D4×C157—22no—
115D4×C157—21no—
116D4×C157—22no—
117D4×C157—22no—
118D4×C157—42no—
119D4×C157—42no—
120D4×C157C2×D411yescharacteristic
121D4×C157C2×D412yes—
122C2×C4×C157—22no—
123C2×C4×C157—21no—
124C2×C4×C157—42no—
125C2×C4×C157C2411yescharacteristic
126C8×C157—21no—
127C8×C157C2×D411yescharacteristic
128C8×C157C2×D411yescharacteristic
129C22×C157—21no—
130C22×C157—22no—
131C22×C157—42no—
132C22×C157—82no—
133C22×C157—162no—
134C22×C157H(5,𝔽2)11yescharacteristic
135C2×D157—21no—
136C4×C157—21no—
137C4×C157—22no—
138C4×C157C22×D411yescharacteristic
139C4×C157C22×D411yescharacteristic
140Dic157—21no—
141Dic157—41no—
142Dic157—42no—
143C2×C157—21no—
144C2×C157—42no—
145C2×C157—41no—
146C2×C157—82no—
147C2×C157D4211yescharacteristic
148D157—41no—
149C157#329011yescharacteristic, complemented, Sylow
150#3290—1571nomaximal, Sylow
151#578—1571no—
152#577—1572no—
153#576D15711yescharacteristic, complemented
154#495—1571no—
155#473—1571no—
156#472—1571no—
157#471—1572no—
158#455—1572no—
159#454—1572no—
160#443—1571no—
161#443—1571no—
1622−1+4—1572no—
163H(5,𝔽2)C2×D15712yes—
164#137—3142no—
165#137—3142no—
166#137—1572no—
167#137—1572no—
168AGL(1,ℤ/8)—22no—
169AGL(1,ℤ/8)—3142no—
170AGL(1,ℤ/8)—1572no—
171AGL(1,ℤ/8)C2×D15712yes—
172#135—21no—
173#135—3142no—
174#135—3141no—
175#135—1571no—
176#135C2×D15711yescharacteristic
177C2×Q16—3141no—
178C2×Q16—1571no—
179C2×SD16—3142no—
180C2×SD16—1572no—
181C2×D8—21no—
182C2×D8C2×D15711yescharacteristic
183#131—1572no—
184#131—1571no—
185#131C2×D15711yescharacteristic
186#124—1572no—
187#108—1571no—
188#108—1571no—
189#104—1571no—
190#104—1572no—
191#104—1571no—
192#101—1572no—
193#100—1572no—
194C4×C8—1571no—
195#41—22no—
196#41—22no—
197#41—3142no—
198#41—3142no—
199#41—6282no—
200#41—1571no—
201#41C22×D15711yescharacteristic
202#41C22×D15712yes—
203C2×Q8—3142no—
204C2×Q8—3142no—
205C2×Q8—1572no—
206C2×D4—22no—
207C2×D4—22no—
208C2×D4—42no—
209C2×D4C22×D15712yes—
210Q16—3142no—
211Q16—3142no—
212Q16—3141no—
213Q16—3141no—
214Q16—1571no—
215Q16C22⋊D15711yescharacteristic
216SD16—22no—
217SD16—3142no—
218SD16—3142no—
219SD16—3142no—
220SD16—3142no—
221SD16—3142no—
222SD16—1571no—
223SD16—1571no—
224SD16C22⋊D15712yes—
225D8—22no—
226D8—21no—
227D8—21no—
228D8—3142no—
229D8—1571no—
230D8C22⋊D15711yescharacteristic
231M16—21no—
232M16—3142no—
233M16—3141no—
234M16—1571no—
235M16—1572no—
236M16C22×D15711yescharacteristic
237C2×C8—21no—
238C2×C8—3142no—
239C2×C8—3141no—
240C2×C8—1571no—
241C2×C8C22×D15711yescharacteristic
242C22⋊C4—6282no—
243C42—1571no—
244C23—22no—
245C23—42no—
246Q8—3142no—
247Q8—3142no—
248Q8—6282no—
249Q8—1571no—
250Q8D4×D15711yescharacteristic
251Q8C2×C22⋊D15712yes—
252D4—21no—
253D4—22no—
254D4—22no—
255D4—22no—
256D4—42no—
257D4—42no—
258D4—3141no—
259D4—6282no—
260D4—1571no—
261D4D4×D15711yescharacteristic
262D4C2×C22⋊D15712yes—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—42no—
266C2×C4—3141no—
267C2×C4—3141no—
268C2×C4—3142no—
269C2×C4—6282no—
270C2×C4—6282no—
271C2×C4C23×D15711yescharacteristic
272C8—21no—
273C8—3142no—
274C8—3141no—
275C8—1571no—
276C8—1571no—
277C8C2×C22⋊D15711yescharacteristic
278C8C2×C22⋊D15711yescharacteristic
279C22—22no—
280C22—21no—
281C22—42no—
282C22—82no—
283C22—162no—
284C22—3141no—
285C22#1909781611yescharacteristic
286C4—21no—
287C4—22no—
288C4—3141no—
289C4—6281no—
290C4—6282no—
291C4C2×D4×D15711yescharacteristic
292C4C22×C22⋊D15711yescharacteristic
293C2—21no—
294C2—41no—
295C2—42no—
296C2—82no—
297C2—6281no—
298C2D4×C22⋊D15711yescharacteristic
subgroupG/H
centerC2D4×C22⋊D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#576×C157C2
socleC2×C157D42
hypercenter#576D157
hypocenterC157#3290
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C2×C4 ◁ #576

Complements

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

NH ≅ G/Ntype
1. #23758087C2semidirect
2. #23758084C2semidirect
3. #23757656C2semidirect
4. #23757548C2semidirect
5. #23757540C2semidirect
6. #23757534C2semidirect
7. #23757451C2semidirect
8. #23757444C2semidirect
9. #23757390C2semidirect
10. #23757388C2semidirect
11. #576×C157C2semidirect
12. #19097818C22semidirect
15. #19097801C22semidirect
18. #19097797C22semidirect
22. C2×Dic628C22semidirect
25. C2×#14578942C22semidirect
26. #19097778C22semidirect
27. #19097777C22semidirect
28. #19097758C22semidirect
29. #19097707C22semidirect
30. #19097706C22semidirect
32. #19097698C22semidirect
33. #19097697C22semidirect
34. #19097689C22semidirect
35. #19097688C22semidirect
36. C8×Dic157C22semidirect
149. C157#3290semidirect
153. #576D157semidirect

Maximal subgroups

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

subgroupindex
#237580872
#237580842
#237576562
#237575482
#237575402
#237575342
#237574512
#237574442
#237573902
#237573882
#576×C1572
#3290157