Finite Group Databasebeta

← #34110055

Subgroups — #34110055

Order 20096.

458 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 123 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#23757783C211yesmaximal, characteristic
2C2×C4⋊C4×D157C211yesmaximal, characteristic, complemented
3#23757610C212yesmaximal, complemented
4#23757604C211yesmaximal, characteristic, complemented
5#23757603C211yesmaximal, characteristic, complemented
6#23757590C212yesmaximal, complemented
7#23757584C211yesmaximal, characteristic, complemented
8#23757314C212yesmaximal, complemented
9#23757271C212yesmaximal, complemented
10#23757235C211yesmaximal, characteristic, complemented
11#480×C157C211yesmaximal, characteristic, complemented
12C2×#14578953C2211yescharacteristic
13C22×C4×D157—21no—
14C22×C4×D157C2211yescharacteristic
15C2×#14578937C2212yes—
16#19097753—21no—
17#19097751C2211yescharacteristic
18#19097750C2211yescharacteristic
19#19097742C2211yescharacteristic
20#19097739—21no—
21D4×Dic157—21no—
22#19097731—21no—
23#19097729C2211yescharacteristic
24C2×C4⋊Dic157C2211yescharacteristic
25C2×#14578933—21no—
26C4⋊C4×D157—21no—
27C4⋊C4×D157—21no—
28C4⋊C4×D157C2212yes—
29C4⋊C4×D157C2212yes—
30#19097704C2212yescomplemented
31#19097703C2212yescomplemented
32#19097702C2212yescomplemented
33#19097696C2212yescomplemented
34#19097694C2212yescomplemented
35#19097692C2212yescomplemented
36C22⋊#11929434C2211yescharacteristic
37#19097683C2212yes—
38#121×C157C2211yescharacteristic
39C2×C4⋊C4×C157C2211yescharacteristic
40#107×C157C2212yescomplemented
41#102×C157C2212yescomplemented
42C22⋊C8×C157C2211yescharacteristic
43C23×D157D411yescharacteristic
44#14578953—24no—
45C2×Dic314C2311yescharacteristic
46C2×C22⋊D157—21no—
47C2×C22⋊D157—21no—
48C22×Dic157—21no—
49C22×Dic157—21no—
50C22×Dic157D411yescharacteristic
51C2×C4×D157—24no—
52C2×C4×D157—22no—
53C2×C4×D157—22no—
54C2×C4×D157—44no—
55C2×C4×D157D411yescharacteristic
56C2×C4×D157D411yescharacteristic
57C2×C4×D157C2311yescharacteristic
58C2×C4×D157C2311yescharacteristic
59#14578937—24no—
60C2×#11929434C2312yes—
61C4⋊Dic157—21no—
62C4⋊Dic157—21no—
63C4⋊Dic157C2312yes—
64C4⋊Dic157C2311yescharacteristic
65#14578933—22no—
66#14578933—22no—
67#14578933—21no—
68C22⋊Dic157—21no—
69C22⋊Dic157—21no—
70#14578931—21no—
71#14578931—21no—
72C4×Dic157—21no—
73C2×D4×C157—21no—
74C2×D4×C157C2311yescharacteristic
75C22×C4×C157—21no—
76C22×C4×C157C2311yescharacteristic
77C2×C8×C157C2312yes—
78C4⋊C4×C157—21no—
79C4⋊C4×C157C2312yes—
80C4⋊C4×C157C2311yescharacteristic
81C22⋊C4×C157—21no—
82C22×D157—22no—
83C22×D157—22no—
84C22×D157—22no—
85C22×D157C2×D411yescharacteristic
86C22×D157C2×D411yescharacteristic
87Dic314—22no—
88C22⋊D157—42no—
89C22⋊D157—42no—
90C2×Dic157—22no—
91C2×Dic157—22no—
92C2×Dic157—21no—
93C2×Dic157—21no—
94C2×Dic157—42no—
95C2×Dic157—42no—
96C2×Dic157C2×D411yescharacteristic
97C2×Dic157C2×D411yescharacteristic
98C4×D157—22no—
99C4×D157—22no—
100C4×D157—22no—
101C4×D157—44no—
102C4×D157—44no—
103C4×D157#4114yes—
104#11929434—22no—
105C23×C157—21no—
106C23×C157C2×D411yescharacteristic
107D4×C157—22no—
108D4×C157—42no—
109C2×C4×C157—22no—
110C2×C4×C157—22no—
111C2×C4×C157—21no—
112C2×C4×C157—42no—
113C2×C4×C157C2×D411yescharacteristic
114C2×C4×C157C2411yescharacteristic
115C8×C157—22no—
116C22×C157—21no—
117C22×C157—21no—
118C22×C157—41no—
119C22×C157—41no—
120C22×C157—41no—
121C22×C157AGL(1,ℤ/8)12yes—
122C22×C157C22×D411yescharacteristic
123C2×D157—28no—
124C2×D157—21no—
125C2×D157—21no—
126C2×D157—21no—
127C2×D157—21no—
128C2×D157#12312yes—
129C2×D157#13712yes—
130C4×C157—21no—
131C4×C157—42no—
132C4×C157—41no—
133C4×C157C2×#4112yes—
134Dic157—21no—
135Dic157—41no—
136Dic157—42no—
137Dic157—41no—
138Dic157#12312yes—
139C2×C157—22no—
140C2×C157—81no—
141C2×C157C2×#12311yescharacteristic
142C2×C157C2×AGL(1,ℤ/8)11yescharacteristic
143C2×C157C2×#13711yescharacteristic
144D157—22no—
145D157—22no—
146C157#309211yescharacteristic, complemented, Sylow
147#3092—1571nomaximal, Sylow
148#525—1571no—
149C22×C4⋊C4—1571no—
150#484—1572no—
151#483—1571no—
152#483—1571no—
153#481—1572no—
154#480—1571no—
155#480D15711yescharacteristic, complemented
156#428—1572no—
157#417—1572no—
158#407—1571no—
159C2×#41—1571no—
160C23×C4—3141no—
161C23×C4—1571no—
162C2×M16—1572no—
163#123—3141no—
164C22⋊Q8—1571no—
165C22⋊Q8—1571no—
166#121—1571no—
167#121C2×D15711yescharacteristic
168#120—3141no—
169C4×D4—3141no—
170C4×D4—3141no—
171#117—1571no—
172C2×C4⋊C4—3141no—
173C2×C4⋊C4—3141no—
174C2×C4⋊C4—3141no—
175C2×C4⋊C4—1571no—
176C2×C4⋊C4—1572no—
177C2×C4⋊C4—1572no—
178C2×C4⋊C4C2×D15711yescharacteristic
179#107—1572no—
180#107C2×D15712yescomplemented
181#106—1572no—
182#106—1572no—
183#103—1572no—
184#103—1572no—
185#102—1572no—
186#102C2×D15712yescomplemented
187C22⋊C8—1571no—
188C22⋊C8—1572no—
189C22⋊C8C2×D15711yescharacteristic
190C24—1571no—
191#41—3144no—
192C2×Q8—1571no—
193C2×D4—21no—
194C2×D4—3141no—
195C2×D4—3141no—
196C2×D4C22×D15711yescharacteristic
197C22×C4—21no—
198C22×C4—3144no—
199C22×C4—3142no—
200C22×C4—3142no—
201C22×C4—3141no—
202C22×C4—3141no—
203C22×C4—6284no—
204C22×C4—1571no—
205C22×C4—1571no—
206C22×C4—1571no—
207C22×C4—1571no—
208C22×C4—1571no—
209C22×C4C22×D15711yescharacteristic
210M16—3144no—
211C2×C8—1572no—
212C2×C8C22×D15712yes—
213C4⋊C4—21no—
214C4⋊C4—3142no—
215C4⋊C4—3142no—
216C4⋊C4—3141no—
217C4⋊C4—3141no—
218C4⋊C4—3141no—
219C4⋊C4—1572no—
220C4⋊C4—1571no—
221C4⋊C4C22×D15712yes—
222C4⋊C4C22×D15711yescharacteristic
223C22⋊C4—21no—
224C22⋊C4—3141no—
225C22⋊C4—3141no—
226C22⋊C4—3141no—
227C22⋊C4—3141no—
228C42—3141no—
229C23—21no—
230C23—3142no—
231C23—3142no—
232C23—3142no—
233C23—1571no—
234C23—1571no—
235C23D4×D15711yescharacteristic
236Q8—3142no—
237D4—22no—
238D4—42no—
239D4—6282no—
240D4—6282no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—42no—
245C2×C4—3142no—
246C2×C4—3142no—
247C2×C4—3142no—
248C2×C4—3142no—
249C2×C4—3141no—
250C2×C4—3141no—
251C2×C4—3142no—
252C2×C4—6284no—
253C2×C4—6284no—
254C2×C4—6282no—
255C2×C4—6282no—
256C2×C4—1571no—
257C2×C4—1571no—
258C2×C4—1574no—
259C2×C4D4×D15711yescharacteristic
260C2×C4C23×D15711yescharacteristic
261C8—22no—
262C8—3142no—
263C22—21no—
264C22—21no—
265C22—41no—
266C22—41no—
267C22—41no—
268C22—3148no—
269C22—3141no—
270C22—3141no—
271C22—3141no—
272C22—3141no—
273C22—1572no—
274C22—1572no—
275C22#1909779512yes—
276C22C2×D4×D15711yescharacteristic
277C4—21no—
278C4—42no—
279C4—41no—
280C4—3141no—
281C4—6281no—
282C4—6282no—
283C4—6281no—
284C4—1572no—
285C4#41×D15712yes—
286C2—22no—
287C2—81no—
288C2—3142no—
289C2—3142no—
290C2#123×D15711yescharacteristic
291C2C2×#1909779511yescharacteristic
292C2#137×D15711yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#480×C157C2
socleC22×C157C22×D4
prime radical#23757783C2
hypercenter#480D157
hypocenterC157#3092
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×D157C2semidirect
3. #23757610C2semidirect
4. #23757604C2semidirect
5. #23757603C2semidirect
6. #23757590C2semidirect
7. #23757584C2semidirect
8. #23757314C2semidirect
9. #23757271C2semidirect
10. #23757235C2semidirect
11. #480×C157C2semidirect
30. #19097704C22semidirect
31. #19097703C22semidirect
32. #19097702C22semidirect
33. #19097696C22semidirect
34. #19097694C22semidirect
35. #19097692C22semidirect
40. #107×C157C22semidirect
41. #102×C157C22semidirect
146. C157#3092semidirect
155. #480D157semidirect
180. #107C2×D157semidirect
186. #102C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237577832
C2×C4⋊C4×D1572
#237576102
#237576042
#237576032
#237575902
#237575842
#237573142
#237572712
#237572352
#480×C1572
#3092157