Finite Group Databasebeta

← #34111735

Subgroups — #34111735

Order 20096.

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

normal subgroups only

This group has 546 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
1C2×#19097798C212yesmaximal, complemented
2#23757926C211yesmaximal, characteristic, complemented
3#23757908C211yesmaximal, characteristic, complemented
4#23757516C212yesmaximal, complemented
5#23757475C211yesmaximal, characteristic, complemented
6#23757465C211yesmaximal, characteristic, complemented
7#23757422C212yesmaximal, complemented
8#23757417C212yesmaximal, complemented
9C4×D1256C211yesmaximal, characteristic, complemented
10#23757352C211yesmaximal, characteristic, complemented
11#492×C157C211yesmaximal, characteristic, complemented
12C2×#14578955C2212yescomplemented
13C2×D4×D157—21no—
14C2×D4×D157C2212yes—
15#19097798—28no—
16C2×#14578941C2212yes—
17C2×D1256C2211yescharacteristic
18C2×#14578939C2212yescomplemented
19C2×#14578937C2212yescomplemented
20#19097756—22no—
21#19097747C2212yescomplemented
22#19097746—21no—
23#19097742—21no—
24#19097742C2212yes—
25#19097739—22no—
26C4×D628C2211yescharacteristic
27C4×D628C2211yescharacteristic
28C4×D628C2211yescharacteristic, complemented
29C4⋊C4×D157—21no—
30#19097705C2211yescharacteristic, complemented
31C4⋊#11929434C2211yescharacteristic, complemented
32#19097695C2212yescomplemented
33#19097693C2211yescharacteristic
34#19097693C2211yescharacteristic, complemented
35#19097692C2212yescomplemented
36#19097683C2212yescomplemented
37C2×SD16×C157C2212yescomplemented
38C2×SD16×C157C2212yescomplemented
39C4⋊Q8×C157C2211yescharacteristic, complemented
40#127×C157C2211yescharacteristic
41C4×C8×C157C2211yescharacteristic, complemented
42C23×D157—22no—
43#14578955—24no—
44#14578955—42no—
45C2×D628—22no—
46C2×D628D411yescharacteristic
47C2×D628C2311yescharacteristic
48C2×D628C2311yescharacteristic
49C2×C22⋊D157—22no—
50C2×C22⋊D157—22no—
51D4×D157—24no—
52D4×D157—42no—
53D4×D157—44no—
54C2×C4×D157—22no—
55C2×C4×D157—21no—
56C2×C4×D157—21no—
57C2×C4×D157C2312yes—
58#14578941—24no—
59D1256D414yescomplemented
60#14578939—24no—
61#14578937—24no—
62C2×#11929434C2312yes—
63C4⋊Dic157D411yescharacteristic
64C4⋊Dic157C2311yescharacteristic
65C4⋊Dic157C2311yescharacteristic
66#14578933—22no—
67C22⋊Dic157—22no—
68#14578931—22no—
69#14578931—21no—
70#14578931—21no—
71#14578931D412yes—
72C2×Q8×C157C2312yes—
73C2×D4×C157—21no—
74C2×D4×C157—21no—
75C2×D4×C157C2312yes—
76SD16×C157—24no—
77SD16×C157—42no—
78C2×C8×C157C2311yescharacteristic
79C2×C8×C157C2311yescharacteristic
80C4⋊C4×C157—21no—
81C4⋊C4×C157—21no—
82C42×C157C2311yescharacteristic
83C22×D157—21no—
84C22×D157—21no—
85C22×D157—44no—
86C22×D157—44no—
87C22×D157—82no—
88C22×D157C2×D412yes—
89D628—22no—
90D628—21no—
91D628—21no—
92D628—44no—
93D628C2×D412yes—
94D628C2×D412yescomplemented
95C22⋊D157—44no—
96C22⋊D157—44no—
97C2×Dic157—21no—
98C2×Dic157—21no—
99C2×Dic157C2×D412yes—
100C4×D157—22no—
101C4×D157—22no—
102C4×D157—44no—
103C4×D157—42no—
104C4×D157—42no—
105#11929434—22no—
106C23×C157—22no—
107Q8×C157—22no—
108Q8×C157—22no—
109D4×C157—22no—
110D4×C157—22no—
111D4×C157—42no—
112D4×C157—81no—
113C2×C4×C157—22no—
114C2×C4×C157C2×D411yescharacteristic
115C2×C4×C157C2×D411yescharacteristic
116C2×C4×C157C2411yescharacteristic
117C8×C157—21no—
118C8×C157C2×D412yescomplemented
119C22×C157—42no—
120C22×C157—42no—
121C22×C157—42no—
122C22×C157C22×D411yescharacteristic
123C2×D157—22no—
124C2×D157—22no—
125C2×D157—22no—
126C2×D157—22no—
127C2×D157—22no—
128C2×D157—41no—
129C2×D157—41no—
130C2×D157—41no—
131C2×D157—41no—
132C2×D157—88no—
133C4×C157—21no—
134C4×C157—42no—
135C4×C157AGL(1,ℤ/8)12yescomplemented
136C4×C157C22×D411yescharacteristic
137C4×C157H(5,𝔽2)11yescharacteristic
138Dic157—22no—
139Dic157—41no—
140Dic157—41no—
141C2×C157—82no—
142C2×C157D4211yescharacteristic
143C2×C157C2×AGL(1,ℤ/8)11yescharacteristic
144C2×C157#57711yescharacteristic
145D157—42no—
146D157—42no—
147D157—42no—
148C157#327211yescharacteristic, complemented, Sylow
149#3272—1571nomaximal, Sylow
150C2×AGL(1,ℤ/8)—1572no—
151#547—1571no—
152D42—1571no—
153#492D15711yescharacteristic, complemented
154#468—1572no—
155#461—1571no—
156#459—1571no—
157#450—1572no—
158#449—1572no—
159C4×D8—1571no—
160#436—1571no—
161C2×#41—1572no—
162C22×D4—3141no—
163C22×D4—1572no—
164AGL(1,ℤ/8)—3148no—
165C2×SD16—1572no—
166C2×SD16C2×D15712yescomplemented
167C2×SD16C2×D15712yescomplemented
168C2×D8—1572no—
169C2×D8—1571no—
170C2×M16—1572no—
171C4⋊Q8C2×D15711yescharacteristic, complemented
172#127C2×D15711yescharacteristic
173#123—3142no—
174C22⋊Q8—1572no—
175#121—3141no—
176#121—3141no—
177#121—1572no—
178#120—3142no—
179C4×D4—1571no—
180C4×D4—1571no—
181C4×D4—1571no—
182C2×C4⋊C4—3141no—
183#107—1571no—
184C4⋊C8—1571no—
185#103—1572no—
186#102—1572no—
187#102—1571no—
188#102—1571no—
189C22⋊C8—1572no—
190C4×C8C2×D15711yescharacteristic, complemented
191C24—3142no—
192#41—3144no—
193#41—6282no—
194C2×Q8C22×D15712yes—
195C2×D4—21no—
196C2×D4—21no—
197C2×D4—3142no—
198C2×D4—3144no—
199C2×D4—3142no—
200C2×D4—3142no—
201C2×D4—6282no—
202C2×D4—6284no—
203C2×D4—1571no—
204C2×D4—1571no—
205C2×D4—1571no—
206C2×D4C22×D15712yes—
207C22×C4—3142no—
208C22×C4—3141no—
209C22×C4—3141no—
210C22×C4—1572no—
211SD16—24no—
212SD16—42no—
213SD16—3144no—
214D8—3144no—
215D8—1574no—
216M16—3144no—
217C2×C8—1572no—
218C2×C8C22×D15711yescharacteristic
219C2×C8C22×D15711yescharacteristic
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—3142no—
223C4⋊C4—1571no—
224C4⋊C4—1571no—
225C4⋊C4—1571no—
226C22⋊C4—3142no—
227C22⋊C4—3142no—
228C22⋊C4—3141no—
229C22⋊C4—3141no—
230C22⋊C4—1572no—
231C42C22×D15711yescharacteristic
232C23—22no—
233C23—3141no—
234C23—3141no—
235C23—6284no—
236C23—6284no—
237C23—1572no—
238C23—12562no—
239Q8—22no—
240Q8—22no—
241D4—22no—
242D4—22no—
243D4—42no—
244D4—81no—
245D4—3142no—
246D4—3141no—
247D4—3141no—
248D4—6284no—
249D4—6284no—
250D4—6284no—
251D4—1572no—
252D4—1572no—
253C2×C4—22no—
254C2×C4—3142no—
255C2×C4—3142no—
256C2×C4—3141no—
257C2×C4—3141no—
258C2×C4—6284no—
259C2×C4—6282no—
260C2×C4—6282no—
261C2×C4—1572no—
262C2×C4D4×D15711yescharacteristic
263C2×C4D4×D15711yescharacteristic
264C2×C4C23×D15711yescharacteristic
265C8—21no—
266C8—3142no—
267C8D4×D15712yescomplemented
268C22—42no—
269C22—42no—
270C22—42no—
271C22—3142no—
272C22—3142no—
273C22—3142no—
274C22—3142no—
275C22—3142no—
276C22—6281no—
277C22—6281no—
278C22—6281no—
279C22—6281no—
280C22—12568no—
281C22C2×D4×D15711yescharacteristic
282C4—21no—
283C4—42no—
284C4—3142no—
285C4—6281no—
286C4—6281no—
287C4#1909779812yescomplemented
288C4C2×D4×D15711yescharacteristic
289C4#1909781611yescharacteristic
290C2—82no—
291C2—6282no—
292C2—6282no—
293C2—6282no—
294C2#2375790811yescharacteristic
295C2C2×#1909779811yescharacteristic
296C2#2375807911yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#492×C157C2
socleC22×C157C22×D4
hypercenter#492D157
hypocenterC157#3272
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #492

Complements

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

NH ≅ G/Ntype
1. C2×#19097798C2semidirect
2. #23757926C2semidirect
3. #23757908C2semidirect
4. #23757516C2semidirect
5. #23757475C2semidirect
6. #23757465C2semidirect
7. #23757422C2semidirect
8. #23757417C2semidirect
9. C4×D1256C2semidirect
10. #23757352C2semidirect
11. #492×C157C2semidirect
12. C2×#14578955C22semidirect
18. C2×#14578939C22semidirect
19. C2×#14578937C22semidirect
21. #19097747C22semidirect
28. C4×D628C22semidirect
30. #19097705C22semidirect
31. C4⋊#11929434C22semidirect
32. #19097695C22semidirect
34. #19097693C22semidirect
35. #19097692C22semidirect
36. #19097683C22semidirect
37. C2×SD16×C157C22semidirect
38. C2×SD16×C157C22semidirect
39. C4⋊Q8×C157C22semidirect
41. C4×C8×C157C22semidirect
59. D1256D4semidirect
94. D628C2×D4semidirect
118. C8×C157C2×D4semidirect
135. C4×C157AGL(1,ℤ/8)semidirect
148. C157#3272semidirect
153. #492D157semidirect
166. C2×SD16C2×D157semidirect
167. C2×SD16C2×D157semidirect
171. C4⋊Q8C2×D157semidirect
190. C4×C8C2×D157semidirect
267. C8D4×D157semidirect
287. C4#19097798semidirect

Maximal subgroups

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

subgroupindex
C2×#190977982
#237579262
#237579082
#237575162
#237574752
#237574652
#237574222
#237574172
C4×D12562
#237573522
#492×C1572
#3272157