Finite Group Databasebeta

← #34109398

Subgroups — #34109398

Order 20096.

754 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 754 conjugacy classes of subgroups — too many to draw. Only its 153 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
1C22×#14578954C211yesmaximal, characteristic
2C2×#19097804C212yesmaximal, complemented
3C2×#19097750C211yesmaximal, characteristic, complemented
4#23757437C214yesmaximal, complemented
5#23757431C212yesmaximal, complemented
6#23757426C212yesmaximal, complemented
7#23757275C212yesmaximal, complemented
8#407×C157C211yesmaximal, characteristic, complemented
9C2×#14578954—21no—
10C2×#14578954—21no—
11C2×#14578954—28no—
12C2×#14578954C2212yes—
13C2×#14578954C2212yes—
14C22×Dic314C2211yescharacteristic
15C22×Dic314C2211yescharacteristic, complemented
16C22×D628C2211yescharacteristic
17C22×C22⋊D157—21no—
18C22×C4×D157—21no—
19#19097804—28no—
20C2×Dic628C2214yescomplemented
21C2×#14578943C2214yescomplemented
22#19097750—21no—
23#19097750—21no—
24#19097750C2212yes—
25#19097750C2212yescomplemented
26C2×C4⋊Dic157C2211yescharacteristic, complemented
27C2×#14578933—21no—
28C2×C22⋊Dic157—21no—
29#19097696C2214yescomplemented
30#19097693C2214yescomplemented
31C23×C4×C157C2211yescharacteristic
32C2×M16×C157C2212yescomplemented
33C22⋊C8×C157C2212yes—
34C22⋊C8×C157C2212yescomplemented
35C23×D157—21no—
36#14578954—28no—
37#14578954—28no—
38#14578954—24no—
39#14578954—24no—
40#14578954—24no—
41#14578954D418yes—
42C2×Dic314—24no—
43C2×Dic314—21no—
44C2×Dic314—21no—
45C2×Dic314—24no—
46C2×Dic314C2312yes—
47C2×Dic314C2312yes—
48C2×D628—24no—
49C2×D628—21no—
50C2×D628C2312yes—
51C2×C22⋊D157—28no—
52C2×C22⋊D157—22no—
53C2×C22⋊D157—22no—
54C22×Dic157—21no—
55C22×Dic157—21no—
56C22×Dic157—21no—
57C2×C4×D157—28no—
58C2×C4×D157—22no—
59C2×C4×D157—22no—
60Dic628—28no—
61#14578943—28no—
62C4⋊Dic157—21no—
63C4⋊Dic157C2312yes—
64#14578933—22no—
65#14578933—22no—
66C22⋊Dic157—22no—
67C22⋊Dic157—22no—
68C24×C157D411yescharacteristic
69C22×C4×C157—24no—
70C22×C4×C157D411yescharacteristic
71C22×C4×C157D411yescharacteristic
72C22×C4×C157D411yescharacteristic
73C22×C4×C157C2311yescharacteristic
74C22×C4×C157C2311yescharacteristic
75C22×C4×C157C2311yescharacteristic
76M16×C157—24no—
77C2×C8×C157C2314yes—
78C22×D157—24no—
79C22×D157—22no—
80C22×D157—42no—
81C22×D157—42no—
82Dic314—24no—
83Dic314—22no—
84Dic314—24no—
85Dic314—42no—
86Dic314C2×D414yes—
87D628—24no—
88D628—22no—
89D628C2×D414yes—
90C22⋊D157—28no—
91C22⋊D157—28no—
92C2×Dic157—24no—
93C2×Dic157—22no—
94C2×Dic157—22no—
95C2×Dic157—22no—
96C2×Dic157—42no—
97C2×Dic157—42no—
98C4×D157—28no—
99C4×D157—28no—
100C23×C157—22no—
101C23×C157—22no—
102C23×C157—41no—
103C23×C157C2×D411yescharacteristic
104C23×C157C2×D411yescharacteristic
105C23×C157C2×D411yescharacteristic
106C2×C4×C157—22no—
107C2×C4×C157—22no—
108C2×C4×C157—22no—
109C2×C4×C157—22no—
110C2×C4×C157—22no—
111C2×C4×C157C2×D414yes—
112C2×C4×C157C2×D411yescharacteristic
113C2×C4×C157C2×D411yescharacteristic
114C2×C4×C157C2×D411yescharacteristic
115C2×C4×C157C2411yescharacteristic
116C8×C157—24no—
117C22×C157—21no—
118C22×C157—21no—
119C22×C157—21no—
120C22×C157—21no—
121C22×C157—21no—
122C22×C157—21no—
123C22×C157—21no—
124C22×C157—44no—
125C22×C157#12012yes—
126C22×C157#13712yescomplemented
127C22×C157C22×D411yescharacteristic
128C2×D157—24no—
129C2×D157—48no—
130C2×D157—42no—
131C2×D157—42no—
132C4×C157—21no—
133C4×C157—21no—
134C4×C157#12012yes—
135C4×C157C22×D412yes—
136Dic157—24no—
137Dic157—42no—
138Dic157—42no—
139C2×C157—22no—
140C2×C157—22no—
141C2×C157—41no—
142C2×C157C2×#12011yescharacteristic
143C2×C157C2×#13711yescharacteristic
144C2×C157#57511yescharacteristic
145D157—44no—
146C157#300711yescharacteristic, complemented, Sylow
147#3007—1571nomaximal, Sylow
148C22×#41—1571no—
149C2×#137—1572no—
150C2×C22⋊Q8—1571no—
151#452—1574no—
152#451—1572no—
153#450—1572no—
154#417—1572no—
155#407D15711yescharacteristic, complemented
156C2×#41—3148no—
157C2×#41—3141no—
158C2×#41—3141no—
159C2×#41—1572no—
160C2×#41—1572no—
161C22×Q8—1571no—
162C22×Q8—1571no—
163C22×D4—3141no—
164C22×D4—1571no—
165C23×C4—3141no—
166C23×C4C2×D15711yescharacteristic
167#137—3148no—
168C2×Q16—1574no—
169C2×SD16—1574no—
170C2×M16C2×D15712yescomplemented
171C22⋊Q8—3141no—
172C22⋊Q8—3141no—
173C22⋊Q8—1572no—
174C22⋊Q8—1572no—
175C2×C4⋊C4—3141no—
176C2×C4⋊C4—1571no—
177C2×C22⋊C4—3141no—
178#103—1574no—
179#102—1574no—
180C22⋊C8C2×D15712yes—
181C22⋊C8C2×D15712yescomplemented
182C24—3141no—
183C24D62811yescharacteristic
184#41—3148no—
185#41—3148no—
186#41—3144no—
187#41—3144no—
188#41—3144no—
189#41—1578no—
190C2×Q8—3144no—
191C2×Q8—3141no—
192C2×Q8—3141no—
193C2×Q8—3144no—
194C2×Q8—1572no—
195C2×Q8—1572no—
196C2×D4—3144no—
197C2×D4—3148no—
198C2×D4—3142no—
199C2×D4—3142no—
200C2×D4—3141no—
201C2×D4—1572no—
202C22×C4—24no—
203C22×C4—3148no—
204C22×C4—3141no—
205C22×C4—3142no—
206C22×C4—3142no—
207C22×C4—3141no—
208C22×C4—3141no—
209C22×C4D62811yescharacteristic
210C22×C4D62811yescharacteristic
211C22×C4D62811yescharacteristic
212C22×C4C22×D15711yescharacteristic
213C22×C4C22×D15711yescharacteristic
214C22×C4C22×D15711yescharacteristic
215Q16—3148no—
216SD16—3148no—
217M16—24no—
218C2×C8C22×D15714yes—
219C4⋊C4—3142no—
220C4⋊C4—3142no—
221C4⋊C4—3141no—
222C4⋊C4—1572no—
223C22⋊C4—3142no—
224C22⋊C4—3142no—
225C23—22no—
226C23—22no—
227C23—41no—
228C23—3144no—
229C23—3142no—
230C23—6282no—
231C23—6282no—
232C23C2×D62811yescharacteristic
233C23C2×D62811yescharacteristic
234C23C2×D62811yescharacteristic
235Q8—3144no—
236Q8—3142no—
237Q8—3144no—
238Q8—6282no—
239Q8—1574no—
240D4—3144no—
241D4—3148no—
242D4—3148no—
243D4—3142no—
244D4—1574no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—3148no—
251C2×C4—3144no—
252C2×C4—3142no—
253C2×C4—3148no—
254C2×C4—3142no—
255C2×C4—3142no—
256C2×C4—6282no—
257C2×C4—6282no—
258C2×C4D4×D15714yes—
259C2×C4C2×D62811yescharacteristic
260C2×C4C2×D62811yescharacteristic
261C2×C4C2×D62811yescharacteristic
262C2×C4C23×D15711yescharacteristic
263C8—24no—
264C22—21no—
265C22—21no—
266C22—21no—
267C22—21no—
268C22—21no—
269C22—21no—
270C22—21no—
271C22—44no—
272C22—3144no—
273C22—6288no—
274C22—6282no—
275C22—6282no—
276C22#1909774112yes—
277C22#1909780412yescomplemented
278C22C22×D62811yescharacteristic
279C4—21no—
280C4—21no—
281C4—3144no—
282C4—6282no—
283C4—6282no—
284C4#1909774112yes—
285C4C2×D4×D15712yes—
286C2—22no—
287C2—22no—
288C2—41no—
289C2—6284no—
290C2C2×#1909774111yescharacteristic
291C2C2×#1909780411yescharacteristic
292C2#2375807311yescharacteristic
subgroupG/H
centerC22C22×D628
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#407×C157C2
socleC22×C157C22×D4
prime radicalC22×#14578954C2
hypercenter#407D157
hypocenterC157#3007
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
2. C2×#19097804C2semidirect
3. C2×#19097750C2semidirect
4. #23757437C2semidirect
5. #23757431C2semidirect
6. #23757426C2semidirect
7. #23757275C2semidirect
8. #407×C157C2semidirect
15. C22×Dic314C22semidirect
20. C2×Dic628C22semidirect
21. C2×#14578943C22semidirect
25. #19097750C22semidirect
26. C2×C4⋊Dic157C22semidirect
29. #19097696C22semidirect
30. #19097693C22semidirect
32. C2×M16×C157C22semidirect
34. C22⋊C8×C157C22semidirect
126. C22×C157#137semidirect
146. C157#3007semidirect
155. #407D157semidirect
170. C2×M16C2×D157semidirect
181. C22⋊C8C2×D157semidirect
277. C22#19097804semidirect

Maximal subgroups

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

subgroupindex
C22×#145789542
C2×#190978042
C2×#190977502
#237574372
#237574312
#237574262
#237572752
#407×C1572
#3007157