Finite Group Databasebeta

← #34116186

Subgroups — #34116186

Order 20096.

380 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 66 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#23758340C211yesmaximal, characteristic, complemented
2C2×#19097830C212yesmaximal, complemented
3D157⋊#99C211yesmaximal, characteristic, complemented
4#23758150C212yesmaximal, complemented
5#23757834C211yesmaximal, characteristic
6D4×#10597845—24no—
7#19097850—21no—
8C2×C22⋊#10597845—21no—
9C2×C22⋊#10597845C2212yes—
10C2×C22⋊#10597845C2212yes—
11#19097830—24no—
12#19097830—21no—
13#19097829—21no—
14C22×D628C2211yescharacteristic
15C2×D4×D157C2211yescharacteristic
16#19097756—21no—
17#19097754C411yescharacteristic, complemented
18#19097746—21no—
19#19097744—21no—
20#19097741—21no—
21#19097741C411yescharacteristic, complemented
22#19097739C411yescharacteristic, complemented
23C22⋊C4×D157C2211yescharacteristic
24#123×C157C411yescharacteristic, complemented
25C22×#10597845—22no—
26C22×#10597845—22no—
27C22×#10597845—22no—
28C4⋊#10597845—22no—
29C22⋊#10597845—22no—
30C22⋊#10597845—22no—
31C22⋊#10597845—22no—
32C22⋊#10597845—22no—
33C22⋊#10597845D414yescomplemented
34C4×#10597845—22no—
35C23×D157C2×C411yescharacteristic, complemented
36C23×D157D411yescharacteristic
37C23×D157C2311yescharacteristic
38C2×D628—21no—
39C2×D628—21no—
40C2×D628—22no—
41C2×D628—22no—
42C2×D628—21no—
43C2×D628D411yescharacteristic
44C2×C22⋊D157—21no—
45C2×C22⋊D157C2×C411yescharacteristic
46C2×C22⋊D157D411yescharacteristic
47D4×D157—22no—
48D4×D157—22no—
49C2×C4×D157—21no—
50C2×C4×D157D411yescharacteristic
51#14578933—21no—
52C22⋊Dic157C2×C411yescharacteristic, complemented
53#14578931—21no—
54#14578931—21no—
55#14578931—21no—
56#14578931—21no—
57C2×D4×C157C2×C411yescharacteristic
58C22×C4×C157C2×C411yescharacteristic, complemented
59C4⋊C4×C157—21no—
60C22⋊C4×C157—21no—
61C22⋊C4×C157C2×C411yescharacteristic, complemented
62C2×#10597845—24no—
63C2×#10597845—42no—
64C2×#10597845—82no—
65C2×#10597845—82no—
66C2×#10597845—82no—
67C22×D157—21no—
68C22×D157—21no—
69C22×D157—21no—
70C22×D157—22no—
71C22×D157—81no—
72C22×D157—81no—
73C22×D157—81no—
74C22×D157C22⋊C411yescharacteristic
75C22×D157C2×D412yes—
76C22×D157C2×D411yescharacteristic
77C22×D157C2×D411yescharacteristic
78C22×D157#4112yes—
79D628—22no—
80D628—41no—
81D628—42no—
82D628—41no—
83D628—81no—
84C22⋊D157—22no—
85C22⋊D157—22no—
86C22⋊D157—81no—
87C2×Dic157—21no—
88C2×Dic157C22⋊C411yescharacteristic
89C4×D157—21no—
90C4×D157—21no—
91C4×D157—81no—
92C23×C157C22⋊C411yescharacteristic
93C23×C157C22×C411yescharacteristic
94D4×C157—22no—
95C2×C4×C157—21no—
96C2×C4×C157—21no—
97C2×C4×C157—22no—
98C2×C4×C157C22⋊C411yescharacteristic
99C22×C157—21no—
100C22×C157—41no—
101C22×C157—41no—
102C22×C157C2×C22⋊C411yescharacteristic
103C22×C157C4×D412yes—
104C2×D157—21no—
105C2×D157—41no—
106C2×D157—41no—
107C2×D157—42no—
108C2×D157—42no—
109C2×D157—42no—
110C2×D157—41no—
111C2×D157—41no—
112C2×D157—41no—
113C2×D157—41no—
114C2×D157—81no—
115C2×D157—82no—
116C2×D157#12011yescharacteristic
117C2×D157#12112yes—
118C2×D157#12311yescharacteristic
119#10597845—44no—
120#10597845—82no—
121C4×C157—21no—
122C4×C157—41no—
123C4×C157—41no—
124Dic157—21no—
125Dic157—41no—
126C2×C157—21no—
127C2×C157—22no—
128C2×C157—41no—
129C2×C157#38611yescharacteristic
130D157—21no—
131D157—22no—
132D157—21no—
133D157—41no—
134D157—81no—
135C157#191311yescharacteristic, complemented, Sylow
136#1913—1571nomaximal, Sylow
137#534—1571no—
138#518—1571no—
139C2×#99—1571no—
140C2×#99—1572no—
141#342—1572no—
142C22×D4—1571no—
143C22×D4—1571no—
144#123—3141no—
145#123—1571no—
146#123#1059784511yescharacteristic, complemented
147#121—3141no—
148#121—3141no—
149#120—3141no—
150#120—1571no—
151#120—1571no—
152C4×D4—3144no—
153#117—3141no—
154C2×C22⋊C4—3141no—
155C2×C22⋊C4—1572no—
156C2×C22⋊C4—1572no—
157C2×C22⋊C4—1571no—
158#99—3141no—
159#99—3141no—
160#99—3144no—
161C24—1571no—
162C24—1571no—
163C24—1571no—
164C2×D4—3142no—
165C2×D4—3141no—
166C2×D4—3141no—
167C2×D4—3142no—
168C2×D4—3141no—
169C2×D4—3142no—
170C2×D4—3142no—
171C2×D4—3141no—
172C2×D4—1571no—
173C2×D4—1571no—
174C2×D4—1571no—
175C2×D4C2×#1059784511yescharacteristic
176C22×C4—3141no—
177C22×C4—3142no—
178C22×C4—3142no—
179C22×C4—3142no—
180C22×C4—1571no—
181C22×C4C2×#1059784511yescharacteristic, complemented
182C4⋊C4—21no—
183C4⋊C4—3141no—
184C4⋊C4—3142no—
185C22⋊C4—21no—
186C22⋊C4—3141no—
187C22⋊C4—3141no—
188C22⋊C4—3141no—
189C22⋊C4—3142no—
190C22⋊C4—3142no—
191C22⋊C4—3141no—
192C22⋊C4—3142no—
193C22⋊C4—3142no—
194C22⋊C4—1574no—
195C22⋊C4—1571no—
196C22⋊C4C2×#1059784511yescharacteristic, complemented
197C42—3142no—
198C23—3141no—
199C23—3141no—
200C23—3141no—
201C23—3142no—
202C23—1571no—
203C23—1571no—
204C23—1571no—
205C23—1572no—
206C23—1572no—
207C23—12561no—
208C23—12561no—
209C23—12561no—
210C23C22⋊#1059784511yescharacteristic
211C23C22×#1059784511yescharacteristic
212D4—22no—
213D4—3142no—
214D4—3142no—
215D4—3142no—
216D4—6281no—
217D4—6281no—
218D4—6282no—
219D4—12561no—
220D4—12561no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—3141no—
225C2×C4—3141no—
226C2×C4—3144no—
227C2×C4—3141no—
228C2×C4—6282no—
229C2×C4—1571no—
230C2×C4—12561no—
231C2×C4—12562no—
232C2×C4—12562no—
233C2×C4—12562no—
234C2×C4C22⋊#1059784511yescharacteristic
235C22—21no—
236C22—41no—
237C22—41no—
238C22—3141no—
239C22—6281no—
240C22—6282no—
241C22—6282no—
242C22—6281no—
243C22—6281no—
244C22—6281no—
245C22—6281no—
246C22—6282no—
247C22—6281no—
248C22—1571no—
249C22—1571no—
250C22—1572no—
251C22—12562no—
252C22—12561no—
253C22C2×C22⋊#1059784511yescharacteristic
254C22D4×#1059784512yes—
255C4—21no—
256C4—41no—
257C4—41no—
258C4—3141no—
259C4—6281no—
260C4—6284no—
261C4—12562no—
262C2—21no—
263C2—22no—
264C2—41no—
265C2—3141no—
266C2—3142no—
267C2—3141no—
268C2—6281no—
269C2—12561no—
270C2#2375822111yescharacteristic
subgroupG/H
centerC2#23758221
commutatorC23×C157C22×C4
FrattiniC23C22×#10597845
Fitting#123×C157C4
socleC2×C157#386
prime radical#23757834C2
cosocleC23×D157C23
hypercenter#123#10597845
hypocenterC157#1913
LCSG ▷ C23×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C23 ◁ #123

Complements

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

NH ≅ G/Ntype
1. #23758340C2semidirect
2. C2×#19097830C2semidirect
3. D157⋊#99C2semidirect
4. #23758150C2semidirect
17. #19097754C4semidirect
21. #19097741C4semidirect
22. #19097739C4semidirect
24. #123×C157C4semidirect
33. C22⋊#10597845D4semidirect
35. C23×D157C2×C4semidirect
52. C22⋊Dic157C2×C4semidirect
58. C22×C4×C157C2×C4semidirect
61. C22⋊C4×C157C2×C4semidirect
135. C157#1913semidirect
146. #123#10597845semidirect
181. C22×C4C2×#10597845semidirect
196. C22⋊C4C2×#10597845semidirect

Maximal subgroups

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

subgroupindex
#237583402
C2×#190978302
D157⋊#992
#237581502
#237578342
#1913157