Finite Group Databasebeta

← #34109489

Subgroups — #34109489

Order 20096.

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

normal subgroups only

This group has 724 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#23758073C212yesmaximal, complemented
3#23757819C211yesmaximal, characteristic, complemented
4#23757458C212yesmaximal, complemented
5#23757453C212yesmaximal, complemented
6#23757448C212yesmaximal, complemented
7#23757443C212yesmaximal, complemented
8C2×#19097697C212yesmaximal, complemented
9#413×C157C211yesmaximal, characteristic, complemented
10#19097819C2212yes—
11#19097818—22no—
12#19097817—22no—
13#19097816—22no—
14#19097816C2212yes—
15C2×#14578954—22no—
16C2×#14578954C2212yes—
17C2×#14578954C2211yescharacteristic, complemented
18#41×D157—24no—
19#19097804—22no—
20#19097804C2214yescomplemented
21#19097799—22no—
22#19097799C2214yescomplemented
23#19097793—24no—
24#19097769C2211yescharacteristic, complemented
25#19097767C2211yescharacteristic, complemented
26#19097760C2211yescharacteristic, complemented
27#19097758C2211yescharacteristic, complemented
28#19097743—22no—
29C4×C22⋊D157—22no—
30C2×C4×Dic157C2211yescharacteristic, complemented
31#19097697C2214yescomplemented
32#19097697C2214yescomplemented
33C2×#41×C157C2211yescharacteristic
34C2×M16×C157C2212yescomplemented
35#101×C157C2212yescomplemented
36#100×C157C2212yescomplemented
37#14578955—24no—
38#14578954—24no—
39#14578954—24no—
40#14578954—22no—
41#14578954—24no—
42#14578954—22no—
43#14578954—22no—
44#14578954—22no—
45#14578954D414yescomplemented
46#14578954C2314yes—
47#14578953—28no—
48#14578953—24no—
49C2×Dic314—22no—
50C2×Dic314D412yescomplemented
51C2×Dic314C2311yescharacteristic
52Q8×D157—24no—
53C2×D628—22no—
54C2×D628D412yescomplemented
55C2×D628C2311yescharacteristic
56C2×C22⋊D157—24no—
57C2×C22⋊D157—22no—
58C2×C22⋊D157—22no—
59C2×C22⋊D157—21no—
60D4×D157—28no—
61D4×D157—24no—
62C22×Dic157—21no—
63C2×C4×D157—24no—
64C2×C4×D157—22no—
65C2×C4×D157—21no—
66Dic628—24no—
67#14578943—24no—
68#14578943—24no—
69D1256—24no—
70#14578933—22no—
71C22⋊Dic157—22no—
72#14578931—22no—
73C4×Dic157—21no—
74C4×Dic157C2311yescharacteristic
75C4×Dic157C2311yescharacteristic
76#41×C157—22no—
77#41×C157—22no—
78C2×Q8×C157C2311yescharacteristic
79C2×D4×C157D412yes—
80C2×D4×C157C2311yescharacteristic
81C22×C4×C157D412yes—
82C22×C4×C157C2311yescharacteristic
83M16×C157—22no—
84M16×C157C2314yes—
85C2×C8×C157—22no—
86C22×D157—24no—
87C22×D157—22no—
88C22×D157—21no—
89Dic314—24no—
90Dic314—22no—
91Dic314—22no—
92Dic314C2×D414yes—
93D628—24no—
94D628—22no—
95D628—22no—
96D628C2×D414yes—
97C22⋊D157—28no—
98C22⋊D157—24no—
99C22⋊D157—24no—
100C22⋊D157—44no—
101C22⋊D157—42no—
102C2×Dic157—24no—
103C2×Dic157—22no—
104C2×Dic157—21no—
105C2×Dic157—21no—
106C2×Dic157—21no—
107C2×Dic157—42no—
108C4×D157—28no—
109C4×D157—24no—
110C4×D157—24no—
111C4×D157—41no—
112C4×D157—41no—
113C23×C157C2×D412yes—
114C23×C157C2×D411yescharacteristic
115Q8×C157—22no—
116D4×C157—24no—
117D4×C157—22no—
118C2×C4×C157—22no—
119C2×C4×C157—22no—
120C2×C4×C157C2×D412yes—
121C2×C4×C157C2×D412yes—
122C2×C4×C157C2×D412yes—
123C2×C4×C157C2×D411yescharacteristic
124C2×C4×C157C2411yescharacteristic
125C8×C157—24no—
126C22×C157—22no—
127C22×C157—41no—
128C22×C157—42no—
129C22×C157#12012yes—
130C22×C157C22×D411yescharacteristic
131C2×D157—24no—
132C2×D157—42no—
133C2×D157—41no—
134C2×D157—82no—
135C2×D157—84no—
136C4×C157—22no—
137C4×C157#12011yescharacteristic
138C4×C157#12011yescharacteristic
139C4×C157C22×D412yes—
140Dic157—22no—
141Dic157—24no—
142Dic157—41no—
143Dic157—41no—
144C2×C157—21no—
145C2×C157—22no—
146C2×C157—42no—
147C2×C157C2×#12011yescharacteristic
148D157—44no—
149D157—81no—
150C157#301811yescharacteristic, complemented, Sylow
151#3018—1571nomaximal, Sylow
152#585—1571no—
153#575—1572no—
154#532—1571no—
155#456—1572no—
156#455—1572no—
157#454—1572no—
158#453—1572no—
159C2×#104—1572no—
160#413D15711yescharacteristic, complemented
1612−1+4—3142no—
1622−1+4—1572no—
163H(5,𝔽2)—3142no—
164H(5,𝔽2)—3142no—
165H(5,𝔽2)—1572no—
166C2×#41—3144no—
167C2×#41—3142no—
168C2×#41—1571no—
169C2×#41—1572no—
170C2×#41C2×D15711yescharacteristic
171#137—3142no—
172#137—1574no—
173AGL(1,ℤ/8)—3142no—
174AGL(1,ℤ/8)—1574no—
175#135—3144no—
176C2×M16C2×D15712yescomplemented
177C4⋊Q8—1571no—
178#127—1571no—
179#124—1571no—
180#124—1571no—
181#121—3142no—
182C4×D4—3142no—
183C2×C42—1571no—
184#104—1574no—
185#104—1574no—
186#101C2×D15712yescomplemented
187#100C2×D15712yescomplemented
188#41—22no—
189#41—22no—
190#41—3144no—
191#41—3144no—
192#41—3148no—
193#41—3144no—
194#41—3144no—
195#41—3142no—
196#41—3142no—
197#41—3144no—
198#41—3142no—
199#41—3142no—
200#41—1574no—
201#41—1574no—
202C2×Q8—3142no—
203C2×Q8—3144no—
204C2×Q8—1572no—
205C2×Q8—1571no—
206C2×Q8C22×D15711yescharacteristic
207C2×D4—3142no—
208C2×D4—3148no—
209C2×D4—3144no—
210C2×D4—3144no—
211C2×D4—3142no—
212C2×D4—3142no—
213C2×D4—3141no—
214C2×D4—1572no—
215C2×D4—1571no—
216C2×D4D62812yes—
217C2×D4C22×D15711yescharacteristic
218C22×C4—3144no—
219C22×C4—3142no—
220C22×C4—3141no—
221C22×C4—3141no—
222C22×C4D62812yes—
223C22×C4C22×D15711yescharacteristic
224Q16—3144no—
225SD16—3144no—
226SD16—3144no—
227D8—3144no—
228M16—22no—
229M16C22×D15714yes—
230C2×C8—22no—
231C4⋊C4—3142no—
232C22⋊C4—3142no—
233C22⋊C4—3142no—
234C42—3141no—
235C42—1571no—
236C42—1571no—
237C23—3144no—
238C23—3142no—
239C23—3141no—
240C23C2×D62812yes—
241C23C2×D62811yescharacteristic
242Q8—22no—
243Q8—3144no—
244Q8—3142no—
245Q8—3142no—
246Q8—1574no—
247D4—24no—
248D4—22no—
249D4—3144no—
250D4—3148no—
251D4—3144no—
252D4—3144no—
253D4—3142no—
254D4—3142no—
255D4—6284no—
256D4—6282no—
257D4—1574no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—3148no—
261C2×C4—3144no—
262C2×C4—3142no—
263C2×C4—3141no—
264C2×C4—3144no—
265C2×C4—3144no—
266C2×C4—3141no—
267C2×C4—3141no—
268C2×C4—6282no—
269C2×C4—6281no—
270C2×C4—6281no—
271C2×C4D4×D15712yes—
272C2×C4D4×D15712yes—
273C2×C4C2×D62812yes—
274C2×C4C2×D62811yescharacteristic
275C2×C4C23×D15711yescharacteristic
276C8—24no—
277C22—22no—
278C22—41no—
279C22—42no—
280C22—3144no—
281C22—6282no—
282C22—6281no—
283C22—12562no—
284C22—12564no—
285C22#1909774112yes—
286C22C22×D62811yescharacteristic
287C4—22no—
288C4—3142no—
289C4—3144no—
290C4—6281no—
291C4—6281no—
292C4#1909774111yescharacteristic
293C4#1909774111yescharacteristic
294C4C2×D4×D15712yes—
295C2—21no—
296C2—22no—
297C2—42no—
298C2—6284no—
299C2—12561no—
300C2C2×#1909774111yescharacteristic
subgroupG/H
centerC4#19097741
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#413×C157C2
socleC2×C157C2×#120
hypercenter#413D157
hypocenterC157#3018
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C4 ◁ C22×C4 ◁ #413

Complements

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

NH ≅ G/Ntype
1. #23758109C2semidirect
2. #23758073C2semidirect
3. #23757819C2semidirect
4. #23757458C2semidirect
5. #23757453C2semidirect
6. #23757448C2semidirect
7. #23757443C2semidirect
8. C2×#19097697C2semidirect
9. #413×C157C2semidirect
17. C2×#14578954C22semidirect
20. #19097804C22semidirect
22. #19097799C22semidirect
24. #19097769C22semidirect
25. #19097767C22semidirect
26. #19097760C22semidirect
27. #19097758C22semidirect
30. C2×C4×Dic157C22semidirect
31. #19097697C22semidirect
32. #19097697C22semidirect
34. C2×M16×C157C22semidirect
35. #101×C157C22semidirect
36. #100×C157C22semidirect
45. #14578954D4semidirect
50. C2×Dic314D4semidirect
54. C2×D628D4semidirect
150. C157#3018semidirect
160. #413D157semidirect
176. C2×M16C2×D157semidirect
186. #101C2×D157semidirect
187. #100C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237581092
#237580732
#237578192
#237574582
#237574532
#237574482
#237574432
C2×#190976972
#413×C1572
#3018157