Finite Group Databasebeta

← #34113279

Subgroups — #34113279

Order 20096.

360 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 113 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#23758075C211yesmaximal, characteristic, complemented
3#23757715C212yesmaximal, complemented
4#23757713C211yesmaximal, characteristic, complemented
5#23757708C211yesmaximal, characteristic, complemented
6#23757707C212yesmaximal, complemented
7#23757704C212yesmaximal, complemented
8C2×#19097716C211yesmaximal, characteristic, complemented
9SD32×D157C212yesmaximal, complemented
10#23757689C211yesmaximal, characteristic, complemented
11#508×C157C211yesmaximal, characteristic, complemented
12#19097818—21no—
13#19097817—21no—
14#19097801—21no—
15#19097800—22no—
16#19097798—22no—
17#19097796—21no—
18#19097794C2212yes—
19#19097790C2212yescomplemented
20C2×Dic628C2211yescharacteristic, complemented
21Q16×D157C2212yescomplemented
22C2×D1256C2211yescharacteristic
23D8×D157C2212yes—
24#19097777C2211yescharacteristic
25#19097717C2212yescomplemented
26#19097716C2212yescomplemented
27#19097716C2212yescomplemented
28#19097715C2212yescomplemented
29#19097714C2212yescomplemented
30#19097712C2212yescomplemented
31#19097711C2211yescharacteristic, complemented
32#19097710C2212yescomplemented
33C16×D157C2212yescomplemented
34#135×C157C2211yescharacteristic
35#135×C157C2211yescharacteristic, complemented
36Q32×C157C2211yescharacteristic, complemented
37SD32×C157C2211yescharacteristic, complemented
38SD32×C157C2211yescharacteristic, complemented
39D16×C157C2211yescharacteristic, complemented
40C2×C16×C157C2211yescharacteristic, complemented
41#14578955—22no—
42#14578954—41no—
43#14578954—41no—
44#14578954D411yescharacteristic
45#14578953—22no—
46#14578953—42no—
47C2×Dic314—21no—
48C2×Dic314—41no—
49Q8×D157—22no—
50C2×D628—21no—
51C2×D628—41no—
52D4×D157—22no—
53D4×D157—42no—
54Dic628—21no—
55Dic628C2312yes—
56#14578944—22no—
57#14578942—22no—
58#14578941—22no—
59D1256—21no—
60D1256C2312yes—
61#14578939—22no—
62#14578938D411yescharacteristic
63#14578937D412yes—
64C8×D157C2312yes—
65#14578929C2312yes—
66#41×C157—21no—
67#41×C157—21no—
68Q16×C157C2311yescharacteristic
69Q16×C157C2311yescharacteristic
70SD16×C157—21no—
71SD16×C157—21no—
72D8×C157C2311yescharacteristic
73D8×C157C2311yescharacteristic
74C2×C8×C157C2311yescharacteristic
75C16×C157C2311yescharacteristic
76C16×C157C2311yescharacteristic
77C22×D157—41no—
78C22×D157—42no—
79Dic314—22no—
80Dic314—41no—
81Dic314—42no—
82Dic314D811yescharacteristic
83D628—22no—
84D628—42no—
85D628—41no—
86D628D811yescharacteristic
87C22⋊D157—42no—
88C22⋊D157—42no—
89C22⋊D157D812yes—
90C2×Dic157—41no—
91C2×Dic157—42no—
92C4×D157—42no—
93C4×D157—42no—
94C4×D157C2×D412yes—
95#11929434C2×D412yes—
96Q8×C157—21no—
97Q8×C157—21no—
98D4×C157—21no—
99D4×C157—21no—
100D4×C157—41no—
101D4×C157—41no—
102C2×C4×C157—41no—
103C2×C4×C157—41no—
104C2×C4×C157C2×D411yescharacteristic
105C8×C157C2×D411yescharacteristic
106C8×C157C2411yescharacteristic
107C22×C157—41no—
108C22×C157—41no—
109C22×C157C2×D811yescharacteristic
110C2×D157—42no—
111C2×D157—162no—
112C2×D157—161no—
113C2×D157C2×D812yes—
114C4×C157—41no—
115C4×C157—41no—
116C4×C157C2×D811yescharacteristic
117C4×C157C22×D411yescharacteristic
118Dic157—42no—
119Dic157C2×D812yes—
120C2×C157—21no—
121C2×C157—81no—
122C2×C157—81no—
123C2×C157C22×D811yescharacteristic
124D157—22no—
125D157—82no—
126C157#341611yescharacteristic, complemented, Sylow
127#3416—1571nomaximal, Sylow
128#578—1571no—
129#576—1571no—
130#510—1571no—
131#510—1572no—
132#509—1571no—
133#509—1572no—
134#508—1572no—
135#508D15711yescharacteristic, complemented
136C2×SD32—1572no—
137C2×SD32—1571no—
138#504—1571no—
1392−1+4—3141no—
140H(5,𝔽2)—3141no—
141#137—3142no—
142#137—3141no—
143AGL(1,ℤ/8)—3142no—
144AGL(1,ℤ/8)—3141no—
145#135—1572no—
146#135—1572no—
147#135C2×D15711yescharacteristic
148#135C2×D15711yescharacteristic, complemented
149C2×Q16—1572no—
150C2×Q16—1571no—
151C2×D8—1572no—
152C2×D8—1571no—
153#131—1571no—
154Q32—1572no—
155Q32C2×D15711yescharacteristic, complemented
156SD32—1572no—
157SD32—1572no—
158SD32—1572no—
159SD32—1572no—
160SD32C2×D15711yescharacteristic, complemented
161SD32C2×D15711yescharacteristic, complemented
162D16—1572no—
163D16C2×D15711yescharacteristic, complemented
164M32—1571no—
165M32—1572no—
166C2×C16—1572no—
167C2×C16C2×D15711yescharacteristic, complemented
168#41—21no—
169#41—21no—
170#41—3142no—
171#41—3142no—
172#41—6281no—
173#41—6281no—
174#41—6282no—
175#41—1571no—
176C2×Q8—3142no—
177C2×Q8—3141no—
178C2×Q8—6281no—
179C2×D4—3142no—
180C2×D4—3141no—
181C2×D4—6281no—
182C2×D4—6282no—
183Q16—3142no—
184Q16—3141no—
185Q16—1572no—
186Q16C22×D15711yescharacteristic
187Q16C22×D15711yescharacteristic
188SD16—21no—
189SD16—21no—
190SD16—3142no—
191SD16—3142no—
192D8—3142no—
193D8—3141no—
194D8—1572no—
195D8C22×D15711yescharacteristic
196D8C22×D15711yescharacteristic
197M16—1571no—
198M16—1572no—
199C2×C8—1572no—
200C2×C8C22×D15711yescharacteristic
201C16—1572no—
202C16C22×D15711yescharacteristic
203C16C22×D15711yescharacteristic
204C23—6282no—
205C23—6281no—
206Q8—21no—
207Q8—21no—
208Q8—3142no—
209Q8—6282no—
210Q8—6281no—
211Q8—1571no—
212D4—21no—
213D4—21no—
214D4—41no—
215D4—41no—
216D4—3142no—
217D4—6282no—
218D4—6282no—
219D4—6282no—
220D4—6281no—
221D4—1572no—
222D4—1571no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—6282no—
226C2×C4—6282no—
227C2×C4—6281no—
228C2×C4—6282no—
229C2×C4—1572no—
230C2×C4D4×D15711yescharacteristic
231C8—1572no—
232C8D4×D15711yescharacteristic
233C8C23×D15711yescharacteristic
234C22—41no—
235C22—41no—
236C22—6282no—
237C22—1572no—
238C22—25121no—
239C22—25122no—
240C22D8×D15711yescharacteristic
241C4—41no—
242C4—41no—
243C4—6282no—
244C4—1572no—
245C4D8×D15711yescharacteristic
246C4C2×D4×D15711yescharacteristic
247C2—21no—
248C2—81no—
249C2—81no—
250C2—3142no—
251C2—12562no—
252C2C2×D8×D15711yescharacteristic
subgroupG/H
centerC2C2×D8×D157
commutator, cosocleC8×C157C24
FrattiniC8C23×D157
Fitting#508×C157C2
socleC2×C157C22×D8
hypercenter#508D157
hypocenterC157#3416
LCSG ▷ C8×C157 ▷ C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #508

Complements

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

NH ≅ G/Ntype
1. #23758087C2semidirect
2. #23758075C2semidirect
3. #23757715C2semidirect
4. #23757713C2semidirect
5. #23757708C2semidirect
6. #23757707C2semidirect
7. #23757704C2semidirect
8. C2×#19097716C2semidirect
9. SD32×D157C2semidirect
10. #23757689C2semidirect
11. #508×C157C2semidirect
19. #19097790C22semidirect
20. C2×Dic628C22semidirect
21. Q16×D157C22semidirect
25. #19097717C22semidirect
26. #19097716C22semidirect
27. #19097716C22semidirect
28. #19097715C22semidirect
29. #19097714C22semidirect
30. #19097712C22semidirect
31. #19097711C22semidirect
32. #19097710C22semidirect
33. C16×D157C22semidirect
35. #135×C157C22semidirect
36. Q32×C157C22semidirect
37. SD32×C157C22semidirect
38. SD32×C157C22semidirect
39. D16×C157C22semidirect
40. C2×C16×C157C22semidirect
126. C157#3416semidirect
135. #508D157semidirect
148. #135C2×D157semidirect
155. Q32C2×D157semidirect
160. SD32C2×D157semidirect
161. SD32C2×D157semidirect
163. D16C2×D157semidirect
167. C2×C16C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237580872
#237580752
#237577152
#237577132
#237577082
#237577072
#237577042
C2×#190977162
SD32×D1572
#237576892
#508×C1572
#3416157