Finite Group Databasebeta

← #34111931

Subgroups — #34111931

Order 20096.

460 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 117 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#23758084C212yesmaximal, complemented
2#23758083C212yesmaximal, complemented
3#23757645C211yesmaximal, characteristic, complemented
4#23757535C212yesmaximal, complemented
5#23757534C212yesmaximal, complemented
6#23757457C212yesmaximal, complemented
7#23757440C211yesmaximal, characteristic, complemented
8#23757387C212yesmaximal, complemented
9#453×C157C211yesmaximal, characteristic, complemented
10#19097818C2212yes—
11#19097816C2211yescharacteristic, complemented
12#19097804C2212yescomplemented
13#19097801—22no—
14#19097801C2212yescomplemented
15#19097800—22no—
16#19097797—22no—
17#19097797C2212yes—
18#19097795—22no—
19#19097791—22no—
20#19097791C2212yes—
21#19097790—22no—
22C2×#14578942—22no—
23C2×#14578942C2212yescomplemented
24C2×#14578942C2212yescomplemented
25SD16×D157—22no—
26#19097779C2212yescomplemented
27#19097778C2212yescomplemented
28#19097771C2211yescharacteristic, complemented
29#19097706C2212yescomplemented
30#19097699C2212yescomplemented
31#19097689C2212yescomplemented
32#19097688C2211yescharacteristic, complemented
33C4×#11929434C2211yescharacteristic, complemented
34H(5,𝔽2)×C157C2211yescharacteristic, complemented
35AGL(1,ℤ/8)×C157C2212yes—
36#127×C157C2211yescharacteristic
37#104×C157C2212yescomplemented
38#100×C157C2211yescharacteristic, complemented
39#14578954—22no—
40#14578954C2312yes—
41#14578953—22no—
42#14578953—22no—
43#14578953—42no—
44C2×Dic314—22no—
45C2×Dic314C2312yes—
46Q8×D157—22no—
47C2×C22⋊D157—41no—
48D4×D157—22no—
49Dic628—22no—
50#14578944—22no—
51#14578944D412yescomplemented
52#14578943—22no—
53#14578942—22no—
54#14578942—22no—
55#14578942—22no—
56#14578942—22no—
57#14578942—22no—
58#14578942—22no—
59#14578942D412yescomplemented
60#14578941D412yescomplemented
61#14578939—22no—
62#14578939D412yescomplemented
63#14578938—22no—
64#14578938C2312yes—
65#14578937—22no—
66C2×#11929434—22no—
67C2×#11929434C2312yes—
68C8×D157—22no—
69C4⋊Dic157—41no—
70#41×C157—22no—
71#41×C157C2312yes—
72C2×D4×C157—22no—
73C2×D4×C157—41no—
74C2×D4×C157—41no—
75C2×D4×C157C2311yescharacteristic
76C2×D4×C157C2311yescharacteristic
77SD16×C157—22no—
78D8×C157—22no—
79M16×C157C2312yes—
80C42×C157C2311yescharacteristic
81C22×D157—41no—
82Dic314—22no—
83Dic314—22no—
84Dic314—42no—
85Dic314C2×D412yes—
86D628C2×D412yes—
87C22⋊D157—22no—
88C22⋊D157—42no—
89C22⋊D157—42no—
90C2×Dic157—22no—
91C2×Dic157—41no—
92C2×Dic157—42no—
93C4×D157—22no—
94C4×D157—42no—
95#11929434—22no—
96#11929434C2×D412yes—
97#11929434C2×D412yes—
98C23×C157—21no—
99C23×C157—21no—
100C23×C157—41no—
101Q8×C157C2×D412yes—
102D4×C157—22no—
103D4×C157—22no—
104D4×C157—22no—
105D4×C157—42no—
106D4×C157—82no—
107D4×C157C2×D412yes—
108C2×C4×C157—21no—
109C2×C4×C157—22no—
110C2×C4×C157—41no—
111C2×C4×C157C2411yescharacteristic
112C8×C157—22no—
113C22×C157—22no—
114C22×C157—41no—
115C22×C157—41no—
116C22×C157—81no—
117C22×C157—81no—
118C22×C157—161no—
119C22×C157H(5,𝔽2)11yescharacteristic
120C2×D157—22no—
121C2×D157—161no—
122C4×C157—22no—
123C4×C157—22no—
124C4×C157C22×D412yes—
125Dic157—22no—
126Dic157—42no—
127C2×C157—21no—
128C2×C157—42no—
129C2×C157—81no—
130C2×C157—81no—
131C2×C157D4211yescharacteristic
132D157—42no—
133C157#329111yescharacteristic, complemented, Sylow
134#3291—1571nomaximal, Sylow
135#577—1572no—
136#577—1572no—
137#492—1571no—
138#471—1572no—
139#471—1572no—
140#456—1572no—
141#453—1571no—
142#453D15711yescharacteristic, complemented
143#443—1572no—
1442−1+4—1572no—
145H(5,𝔽2)—1571no—
146H(5,𝔽2)C2×D15711yescharacteristic, complemented
147#137—3142no—
148#137—3142no—
149#137—1572no—
150#137—1572no—
151AGL(1,ℤ/8)—3142no—
152AGL(1,ℤ/8)—3142no—
153AGL(1,ℤ/8)—1572no—
154AGL(1,ℤ/8)C2×D15712yes—
155#135—3142no—
156#135—3142no—
157#135—1572no—
158C2×SD16—3142no—
159C2×SD16—3142no—
160C2×SD16—1572no—
161C2×SD16—1572no—
162#131—1572no—
163#131—1572no—
164C4⋊Q8—1571no—
165#127C2×D15711yescharacteristic
166#108—1572no—
167#104—1572no—
168#104C2×D15712yescomplemented
169#101—1572no—
170#100—1571no—
171#100C2×D15711yescharacteristic, complemented
172C4×C8—1571no—
173#41—22no—
174#41—3142no—
175#41—3142no—
176#41—3142no—
177#41—6282no—
178#41—1572no—
179#41C22×D15712yes—
180C2×Q8—3142no—
181C2×Q8—3142no—
182C2×Q8—1572no—
183C2×D4—22no—
184C2×D4—41no—
185C2×D4—41no—
186C2×D4—3142no—
187C2×D4—6281no—
188C2×D4C22×D15711yescharacteristic
189C2×D4C22×D15711yescharacteristic
190Q16—3142no—
191Q16—3142no—
192Q16—1572no—
193SD16—22no—
194SD16—3142no—
195SD16—3142no—
196SD16—3142no—
197SD16—3142no—
198SD16—3142no—
199SD16—3142no—
200SD16—3142no—
201SD16—1572no—
202SD16—1572no—
203D8—22no—
204D8—3142no—
205D8—1572no—
206M16—3142no—
207M16—3142no—
208M16—1572no—
209M16C22×D15712yes—
210C2×C8—3142no—
211C2×C8—3142no—
212C2×C8—1572no—
213C4⋊C4—6281no—
214C42C22×D15711yescharacteristic
215C23—21no—
216C23—21no—
217C23—41no—
218C23—6281no—
219Q8—3142no—
220Q8—3142no—
221Q8—6282no—
222Q8—1572no—
223Q8D4×D15712yes—
224D4—22no—
225D4—22no—
226D4—22no—
227D4—42no—
228D4—82no—
229D4—3142no—
230D4—6282no—
231D4—6282no—
232D4—1572no—
233D4D4×D15712yes—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—41no—
237C2×C4—3142no—
238C2×C4—3142no—
239C2×C4—6282no—
240C2×C4—6281no—
241C2×C4—6282no—
242C2×C4C23×D15711yescharacteristic
243C8—22no—
244C8—3142no—
245C8—1572no—
246C8—1572no—
247C22—22no—
248C22—41no—
249C22—41no—
250C22—81no—
251C22—81no—
252C22—161no—
253C22—3142no—
254C22—25121no—
255C22#1909781611yescharacteristic
256C4—22no—
257C4—22no—
258C4—3142no—
259C4—6282no—
260C4C2×D4×D15712yes—
261C2—21no—
262C2—42no—
263C2—81no—
264C2—81no—
265C2—6282no—
266C2#2375790511yescharacteristic
subgroupG/H
centerC2#23757905
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#453×C157C2
socleC2×C157D42
hypercenter#453D157
hypocenterC157#3291
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C2×C4 ◁ #453

Complements

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

NH ≅ G/Ntype
1. #23758084C2semidirect
2. #23758083C2semidirect
3. #23757645C2semidirect
4. #23757535C2semidirect
5. #23757534C2semidirect
6. #23757457C2semidirect
7. #23757440C2semidirect
8. #23757387C2semidirect
9. #453×C157C2semidirect
11. #19097816C22semidirect
12. #19097804C22semidirect
14. #19097801C22semidirect
23. C2×#14578942C22semidirect
24. C2×#14578942C22semidirect
26. #19097779C22semidirect
27. #19097778C22semidirect
28. #19097771C22semidirect
29. #19097706C22semidirect
30. #19097699C22semidirect
31. #19097689C22semidirect
32. #19097688C22semidirect
33. C4×#11929434C22semidirect
34. H(5,𝔽2)×C157C22semidirect
37. #104×C157C22semidirect
38. #100×C157C22semidirect
51. #14578944D4semidirect
59. #14578942D4semidirect
60. #14578941D4semidirect
62. #14578939D4semidirect
133. C157#3291semidirect
142. #453D157semidirect
146. H(5,𝔽2)C2×D157semidirect
168. #104C2×D157semidirect
171. #100C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237580842
#237580832
#237576452
#237575352
#237575342
#237574572
#237574402
#237573872
#453×C1572
#3291157