Finite Group Databasebeta

← #34111926

Subgroups — #34111926

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#23758082C212yesmaximal, complemented
2#23758081C212yesmaximal, complemented
3#23757647C211yesmaximal, characteristic, complemented
4#23757539C212yesmaximal, complemented
5#23757538C212yesmaximal, complemented
6#23757455C211yesmaximal, characteristic, complemented
7#23757442C212yesmaximal, complemented
8#23757387C212yesmaximal, complemented
9#456×C157C211yesmaximal, characteristic, complemented
10#19097819C2211yescharacteristic
11#19097817C2212yescomplemented
12#19097803—22no—
13#19097802—22no—
14#19097802C2212yescomplemented
15#19097799C2212yes—
16#19097798—22no—
17#19097796—22no—
18#19097796C2212yes—
19#19097794—22no—
20#19097791—22no—
21#19097791C2212yes—
22C2×#14578941—22no—
23C2×#14578941C2212yescomplemented
24C2×#14578941C2212yescomplemented
25SD16×D157—22no—
26#19097779C2212yescomplemented
27#19097778C2212yescomplemented
28#19097768C2211yescharacteristic
29#19097706C2212yescomplemented
30#19097699C2212yescomplemented
31#19097690C2211yescharacteristic, complemented
32#19097687C2212yescomplemented
33C4×#11929434C2211yescharacteristic, complemented
342−1+4×C157C2211yescharacteristic
35#137×C157C2212yescomplemented
36C4⋊Q8×C157C2211yescharacteristic, complemented
37#104×C157C2212yescomplemented
38#101×C157C2211yescharacteristic, complemented
39#14578955—22no—
40#14578955—22no—
41#14578954—22no—
42#14578954—41no—
43#14578954C2312yes—
44Q8×D157—22no—
45C2×D628—22no—
46C2×D628—41no—
47C2×D628C2312yes—
48D4×D157—22no—
49D4×D157—42no—
50#14578944—22no—
51#14578944D412yescomplemented
52#14578943—22no—
53#14578942D412yescomplemented
54#14578941—22no—
55#14578941—22no—
56#14578941—22no—
57#14578941—22no—
58#14578941—22no—
59#14578941—22no—
60#14578941D412yescomplemented
61D1256—22no—
62#14578939—22no—
63#14578939D412yescomplemented
64#14578938—22no—
65#14578938C2312yes—
66#14578937—22no—
67C2×#11929434—22no—
68C2×#11929434C2312yes—
69C8×D157—22no—
70#41×C157—22no—
71#41×C157—41no—
72#41×C157C2312yes—
73C2×Q8×C157—22no—
74C2×Q8×C157C2311yescharacteristic
75C2×Q8×C157C2311yescharacteristic
76Q16×C157—22no—
77SD16×C157—22no—
78M16×C157C2312yes—
79C4⋊C4×C157—41no—
80C42×C157C2311yescharacteristic
81C22×D157—22no—
82C22×D157—42no—
83Dic314—41no—
84Dic314C2×D412yes—
85D628—22no—
86D628—22no—
87D628—42no—
88D628—41no—
89D628—82no—
90D628C2×D412yes—
91C22⋊D157—22no—
92C22⋊D157—42no—
93C4×D157—22no—
94C4×D157—42no—
95C4×D157—42no—
96#11929434—22no—
97#11929434C2×D412yes—
98#11929434C2×D412yes—
99Q8×C157—22no—
100Q8×C157—22no—
101Q8×C157—41no—
102Q8×C157C2×D412yes—
103D4×C157—22no—
104D4×C157—41no—
105D4×C157C2×D412yes—
106C2×C4×C157—21no—
107C2×C4×C157—22no—
108C2×C4×C157—21no—
109C2×C4×C157—21no—
110C2×C4×C157—42no—
111C2×C4×C157C2411yescharacteristic
112C8×C157—22no—
113C22×C157—22no—
114C22×C157H(5,𝔽2)11yescharacteristic
115C2×D157—22no—
116C2×D157—42no—
117C2×D157—82no—
118C2×D157—162no—
119C4×C157—22no—
120C4×C157—22no—
121C4×C157—41no—
122C4×C157—41no—
123C4×C157C22×D412yes—
124Dic157—22no—
125C2×C157—21no—
126C2×C157—42no—
127C2×C157D4211yescharacteristic
128D157—42no—
129D157—82no—
130C157#329111yescharacteristic, complemented, Sylow
131#3291—1571nomaximal, Sylow
132#577—1572no—
133#577—1572no—
134#492—1571no—
135#471—1572no—
136#471—1572no—
137#456—1571no—
138#456D15711yescharacteristic, complemented
139#453—1572no—
140#443—1572no—
1412−1+4—1571no—
1422−1+4C2×D15711yescharacteristic
143H(5,𝔽2)—1572no—
144#137—3142no—
145#137—3142no—
146#137—1572no—
147#137C2×D15712yescomplemented
148AGL(1,ℤ/8)—3142no—
149AGL(1,ℤ/8)—3142no—
150AGL(1,ℤ/8)—1572no—
151AGL(1,ℤ/8)—1572no—
152#135—3142no—
153#135—3142no—
154#135—1572no—
155C2×SD16—3142no—
156C2×SD16—3142no—
157C2×SD16—1572no—
158C2×SD16—1572no—
159#131—1572no—
160#131—1572no—
161C4⋊Q8C2×D15711yescharacteristic, complemented
162#127—1571no—
163#108—1572no—
164#104—1572no—
165#104C2×D15712yescomplemented
166#101—1571no—
167#101C2×D15711yescharacteristic, complemented
168#100—1572no—
169C4×C8—1571no—
170#41—22no—
171#41—41no—
172#41—3142no—
173#41—3142no—
174#41—3142no—
175#41—6281no—
176#41—1572no—
177#41C22×D15712yes—
178C2×Q8—22no—
179C2×Q8—3142no—
180C2×Q8C22×D15711yescharacteristic
181C2×Q8C22×D15711yescharacteristic
182C2×D4—3142no—
183C2×D4—3142no—
184C2×D4—6281no—
185C2×D4—6282no—
186C2×D4—1572no—
187Q16—22no—
188Q16—3142no—
189Q16—1572no—
190SD16—22no—
191SD16—3142no—
192SD16—3142no—
193SD16—3142no—
194SD16—3142no—
195SD16—3142no—
196SD16—3142no—
197SD16—3142no—
198SD16—1572no—
199SD16—1572no—
200D8—3142no—
201D8—3142no—
202D8—1572no—
203M16—3142no—
204M16—3142no—
205M16—1572no—
206M16C22×D15712yes—
207C2×C8—3142no—
208C2×C8—3142no—
209C2×C8—1572no—
210C4⋊C4—41no—
211C42C22×D15711yescharacteristic
212C23—3142no—
213C23—6282no—
214Q8—22no—
215Q8—22no—
216Q8—41no—
217Q8—6281no—
218Q8—1572no—
219Q8D4×D15712yes—
220D4—22no—
221D4—41no—
222D4—3142no—
223D4—3142no—
224D4—3142no—
225D4—6282no—
226D4—6281no—
227D4—6282no—
228D4—1572no—
229D4—12562no—
230D4D4×D15712yes—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—42no—
236C2×C4—3142no—
237C2×C4—6282no—
238C2×C4—6282no—
239C2×C4C23×D15711yescharacteristic
240C8—22no—
241C8—3142no—
242C8—1572no—
243C8—1572no—
244C22—22no—
245C22—3142no—
246C22—6282no—
247C22—25122no—
248C22—12562no—
249C22#1909781611yescharacteristic
250C4—22no—
251C4—22no—
252C4—41no—
253C4—41no—
254C4—3142no—
255C4C2×D4×D15712yes—
256C2—21no—
257C2—42no—
258C2—6282no—
259C2—12562no—
260C2#2375790511yescharacteristic
subgroupG/H
centerC2#23757905
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#456×C157C2
socleC2×C157D42
hypercenter#456D157
hypocenterC157#3291
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C2×C4 ◁ #456

Complements

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

NH ≅ G/Ntype
1. #23758082C2semidirect
2. #23758081C2semidirect
3. #23757647C2semidirect
4. #23757539C2semidirect
5. #23757538C2semidirect
6. #23757455C2semidirect
7. #23757442C2semidirect
8. #23757387C2semidirect
9. #456×C157C2semidirect
11. #19097817C22semidirect
14. #19097802C22semidirect
23. C2×#14578941C22semidirect
24. C2×#14578941C22semidirect
26. #19097779C22semidirect
27. #19097778C22semidirect
29. #19097706C22semidirect
30. #19097699C22semidirect
31. #19097690C22semidirect
32. #19097687C22semidirect
33. C4×#11929434C22semidirect
35. #137×C157C22semidirect
36. C4⋊Q8×C157C22semidirect
37. #104×C157C22semidirect
38. #101×C157C22semidirect
51. #14578944D4semidirect
53. #14578942D4semidirect
60. #14578941D4semidirect
63. #14578939D4semidirect
130. C157#3291semidirect
138. #456D157semidirect
147. #137C2×D157semidirect
161. C4⋊Q8C2×D157semidirect
165. #104C2×D157semidirect
167. #101C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237580822
#237580812
#237576472
#237575392
#237575382
#237574552
#237574422
#237573872
#456×C1572
#3291157