Finite Group Databasebeta

← #34109539

Subgroups — #34109539

Order 20096.

772 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 772 conjugacy classes of subgroups — too many to draw. Only its 145 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#23758109C211yesmaximal, characteristic, complemented
2#23757842C212yesmaximal, complemented
3#23757837C211yesmaximal, characteristic, complemented
4#23757461C218yesmaximal, complemented
5#23757246C212yesmaximal, complemented
6#410×C157C211yesmaximal, characteristic, complemented
7#19097819—21no—
8#19097818—22no—
9#19097817—22no—
10#19097816—21no—
11#19097816C2214yes—
12C2×#14578954—22no—
13C2×#14578954C2212yes—
14#41×D157—24no—
15C22×D628C2211yescharacteristic, complemented
16#19097768—22no—
17#19097767—21no—
18#19097760—21no—
19#19097759—22no—
20#19097746—24no—
21#19097745—22no—
22#19097741C2218yescomplemented
23#19097739C2214yescomplemented
24#19097730C2211yescharacteristic, complemented
25#19097685C2218yescomplemented
26C2×#41×C157C2211yescharacteristic
27#117×C157C2212yescomplemented
28#99×C157C2214yescomplemented
29C23×D157C2312yes—
30#14578955—24no—
31#14578954—24no—
32#14578954—24no—
33#14578954—22no—
34#14578954—22no—
35#14578954—24no—
36#14578954—22no—
37#14578954—22no—
38#14578953—24no—
39#14578953—28no—
40C2×Dic314—22no—
41C2×Dic314D412yescomplemented
42Q8×D157—24no—
43C2×D628—22no—
44C2×D628—28no—
45C2×D628—21no—
46C2×D628—22no—
47C2×D628—22no—
48C2×D628—21no—
49C2×D628D412yescomplemented
50C2×C22⋊D157—24no—
51C2×C22⋊D157—24no—
52C2×C22⋊D157C2314yes—
53D4×D157—24no—
54D4×D157—28no—
55C2×C4×D157—24no—
56C2×C4×D157D414yescomplemented
57C4⋊Dic157—21no—
58C22⋊Dic157C2312yes—
59#14578931—28no—
60#14578931—24no—
61C4×Dic157—21no—
62#41×C157—22no—
63#41×C157—22no—
64C2×Q8×C157D411yescharacteristic, complemented
65C2×D4×C157D411yescharacteristic, complemented
66C2×D4×C157C2312yes—
67C22×C4×C157D412yescomplemented
68C22×C4×C157C2311yescharacteristic
69C4⋊C4×C157—22no—
70C22⋊C4×C157C2314yes—
71C42×C157—22no—
72C22×D157—24no—
73C22×D157—22no—
74C22×D157—24no—
75C22×D157—82no—
76C22×D157C2×D414yes—
77Dic314—24no—
78Dic314—22no—
79Dic314—22no—
80D628—24no—
81D628—22no—
82D628—22no—
83D628—41no—
84D628—41no—
85D628—42no—
86D628—88no—
87C22⋊D157—28no—
88C22⋊D157—24no—
89C22⋊D157—24no—
90C22⋊D157—84no—
91C2×Dic157—24no—
92C2×Dic157—22no—
93C2×Dic157C2×D414yes—
94C4×D157—28no—
95C4×D157—24no—
96C4×D157—24no—
97C23×C157C2×D412yes—
98C23×C157C2411yescharacteristic
99Q8×C157—22no—
100D4×C157—24no—
101D4×C157—22no—
102C2×C4×C157—22no—
103C2×C4×C157—22no—
104C2×C4×C157—24no—
105C2×C4×C157C2×D412yes—
106C2×C4×C157C2×D412yes—
107C2×C4×C157C2×D412yes—
108C2×C4×C157C2×D411yescharacteristic
109C2×C4×C157C2×D411yescharacteristic
110C22×C157—22no—
111C22×C157—41no—
112C22×C157—42no—
113C22×C157C22×D412yes—
114C22×C157C22×D411yescharacteristic
115C2×D157—24no—
116C2×D157—44no—
117C2×D157—44no—
118C2×D157—48no—
119C2×D157—42no—
120C2×D157—84no—
121C4×C157—22no—
122C4×C157—44no—
123C4×C157#12011yescharacteristic
124C4×C157#12011yescharacteristic
125C4×C157#12012yes—
126Dic157—24no—
127Dic157—42no—
128C2×C157—21no—
129C2×C157—22no—
130C2×C157—42no—
131C2×C157C2×#12011yescharacteristic
132D157—44no—
133D157—82no—
134C157#302711yescharacteristic, complemented, Sylow
135#3027—1571nomaximal, Sylow
136#585—1571no—
137#535—1571no—
138#535—1572no—
139B(4,𝔽2)—1578no—
140#410—1572no—
141#410D15711yescharacteristic, complemented
1422−1+4—3142no—
1432−1+4—3141no—
144H(5,𝔽2)—3142no—
145H(5,𝔽2)—3141no—
146H(5,𝔽2)—1574no—
147C2×#41—3144no—
148C2×#41—3142no—
149C2×#41—1572no—
150C2×#41C2×D15711yescharacteristic
151C22×D4—1571no—
152#127—3142no—
153#127—3141no—
154#124—3142no—
155#124—3141no—
156#121—3144no—
157#121—3142no—
158#120—1574no—
159#120—1578no—
160#117—1571no—
161#117C2×D15712yescomplemented
162#99—1578no—
163#99C2×D15714yescomplemented
164C24—1572no—
165#41—22no—
166#41—22no—
167#41—3144no—
168#41—3144no—
169#41—3144no—
170#41—3144no—
171#41—3148no—
172#41—3142no—
173#41—3142no—
174#41—3142no—
175#41—3144no—
176#41—3142no—
177C2×Q8—3142no—
178C2×Q8—3144no—
179C2×Q8—1572no—
180C2×Q8D62811yescharacteristic, complemented
181C2×D4—3142no—
182C2×D4—3148no—
183C2×D4—3144no—
184C2×D4—3148no—
185C2×D4—3144no—
186C2×D4—3144no—
187C2×D4—3142no—
188C2×D4—3142no—
189C2×D4—3141no—
190C2×D4—3141no—
191C2×D4—1574no—
192C2×D4—1572no—
193C2×D4D62811yescharacteristic, complemented
194C2×D4C22×D15712yes—
195C22×C4—3144no—
196C22×C4—1574no—
197C22×C4D62812yescomplemented
198C22×C4C22×D15711yescharacteristic
199C4⋊C4—22no—
200C4⋊C4—3141no—
201C22⋊C4—3148no—
202C22⋊C4—3144no—
203C22⋊C4—1572no—
204C22⋊C4C22×D15714yes—
205C42—22no—
206C42—3141no—
207C23—3144no—
208C23—3144no—
209C23—3142no—
210C23—1574no—
211C23—12562no—
212C23C2×D62812yes—
213C23C23×D15711yescharacteristic
214Q8—22no—
215Q8—3144no—
216Q8—3142no—
217Q8—3142no—
218D4—24no—
219D4—22no—
220D4—3144no—
221D4—3148no—
222D4—3144no—
223D4—3144no—
224D4—3142no—
225D4—3142no—
226D4—6281no—
227D4—6281no—
228D4—6282no—
229D4—12568no—
230D4—12564no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—24no—
234C2×C4—3148no—
235C2×C4—3144no—
236C2×C4—3144no—
237C2×C4—3144no—
238C2×C4—3142no—
239C2×C4—1574no—
240C2×C4D4×D15712yes—
241C2×C4D4×D15712yes—
242C2×C4C2×D62812yes—
243C2×C4C2×D62811yescharacteristic
244C2×C4C2×D62811yescharacteristic
245C22—22no—
246C22—41no—
247C22—42no—
248C22—3144no—
249C22—6284no—
250C22—6288no—
251C22—6284no—
252C22—6282no—
253C22—12564no—
254C22C2×D4×D15712yes—
255C22C22×D62811yescharacteristic
256C4—22no—
257C4—44no—
258C4—3144no—
259C4—6282no—
260C4#1909774111yescharacteristic
261C4#1909774111yescharacteristic
262C4#1909774112yes—
263C2—22no—
264C2—21no—
265C2—42no—
266C2—6284no—
267C2—12562no—
268C2C2×#1909774111yescharacteristic
subgroupG/H
centerC2C2×#19097741
commutator, cosocleC23×C157C24
FrattiniC23C23×D157
Fitting#410×C157C2
socleC2×C157C2×#120
hypercenter#410D157
hypocenterC157#3027
LCSG ▷ C23×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
1. #23758109C2semidirect
2. #23757842C2semidirect
3. #23757837C2semidirect
4. #23757461C2semidirect
5. #23757246C2semidirect
6. #410×C157C2semidirect
15. C22×D628C22semidirect
22. #19097741C22semidirect
23. #19097739C22semidirect
24. #19097730C22semidirect
25. #19097685C22semidirect
27. #117×C157C22semidirect
28. #99×C157C22semidirect
41. C2×Dic314D4semidirect
49. C2×D628D4semidirect
56. C2×C4×D157D4semidirect
64. C2×Q8×C157D4semidirect
65. C2×D4×C157D4semidirect
67. C22×C4×C157D4semidirect
134. C157#3027semidirect
141. #410D157semidirect
161. #117C2×D157semidirect
163. #99C2×D157semidirect
180. C2×Q8D628semidirect
193. C2×D4D628semidirect
197. C22×C4D628semidirect

Maximal subgroups

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

subgroupindex
#237581092
#237578422
#237578372
#237574612
#237572462
#410×C1572
#3027157