Finite Group Databasebeta

← #34109416

Subgroups — #34109416

Order 20096.

752 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 752 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C2×#19097817C211yesmaximal, characteristic
2C2×#19097801C211yesmaximal, characteristic, complemented
3C22×#14578943C211yesmaximal, characteristic, complemented
4#23757849C211yesmaximal, characteristic, complemented
5#23757438C214yesmaximal, complemented
6#23757425C214yesmaximal, complemented
7#23757277C211yesmaximal, characteristic, complemented
8#23757239C211yesmaximal, characteristic, complemented
9#416×C157C211yesmaximal, characteristic, complemented
10#19097817—24no—
11#19097817—24no—
12C2×#14578955C2212yes—
13C2×#14578954—21no—
14C2×#14578954C2211yescharacteristic
15C22×Dic314C2211yescharacteristic, complemented
16C22×D628—21no—
17C22×D628C2211yescharacteristic
18C2×D4×D157—22no—
19C2×D4×D157C2212yes—
20#19097801—24no—
21C2×#14578944C2212yescomplemented
22C2×#14578943—24no—
23C2×#14578943C2214yescomplemented
24C2×#14578942C2212yescomplemented
25C2×#14578938C2211yescharacteristic, complemented
26#19097771—21no—
27#19097759—21no—
28#19097755—22no—
29#19097751C2214yescomplemented
30#19097691C2214yescomplemented
31#19097683C2214yescomplemented
32C2×#41×C157C2211yescharacteristic
33C22×C8×C157C2211yescharacteristic, complemented
34#117×C157C2211yescharacteristic, complemented
35#103×C157C2212yescomplemented
36#102×C157C2212yescomplemented
37C23×D157—22no—
38C23×D157—21no—
39#14578955—24no—
40#14578955—24no—
41#14578954—28no—
42#14578954—22no—
43#14578954—22no—
44C2×Dic314—21no—
45C2×Dic314—22no—
46C2×Dic314—22no—
47C2×Dic314D411yescharacteristic
48C2×Dic314C2312yes—
49C2×D628—28no—
50C2×D628—22no—
51C2×D628—21no—
52C2×D628—22no—
53C2×D628D411yescharacteristic
54C2×D628D414yes—
55C2×D628C2312yes—
56C2×C22⋊D157—22no—
57C2×C22⋊D157D412yes—
58D4×D157—216no—
59D4×D157—24no—
60D4×D157—24no—
61C22×Dic157—21no—
62C2×C4×D157—22no—
63C2×C4×D157C2312yes—
64#14578944—24no—
65#14578943—28no—
66#14578942—24no—
67#14578938—22no—
68C2×#11929434C2312yes—
69C4⋊Dic157—24no—
70#14578931—24no—
71#41×C157—22no—
72#41×C157D414yes—
73C2×Q8×C157C2311yescharacteristic
74C2×D4×C157—21no—
75C2×D4×C157C2311yescharacteristic
76C22×C4×C157—21no—
77C22×C4×C157C2311yescharacteristic
78C2×C8×C157—22no—
79C2×C8×C157C2312yes—
80C4⋊C4×C157C2312yes—
81C22⋊C4×C157—21no—
82C42×C157—21no—
83C22×D157—22no—
84C22×D157—28no—
85C22×D157—24no—
86C22×D157—84no—
87C22×D157—82no—
88C22×D157C2×D412yes—
89Dic314—22no—
90Dic314—22no—
91Dic314—22no—
92Dic314—42no—
93D628—28no—
94D628—22no—
95D628—24no—
96D628—22no—
97D628C2×D414yes—
98C22⋊D157—28no—
99C22⋊D157—24no—
100C2×Dic157—22no—
101C2×Dic157—42no—
102C2×Dic157C2×D412yes—
103C4×D157—28no—
104C4×D157—22no—
105C4×D157—22no—
106#11929434—22no—
107C23×C157—21no—
108C23×C157C2×D411yescharacteristic
109Q8×C157—21no—
110Q8×C157C2×D412yes—
111D4×C157—24no—
112D4×C157—21no—
113D4×C157C2×D412yes—
114C2×C4×C157—21no—
115C2×C4×C157—24no—
116C2×C4×C157—22no—
117C2×C4×C157C2×D411yescharacteristic
118C2×C4×C157C2×D412yes—
119C2×C4×C157C2×D412yes—
120C2×C4×C157C2411yescharacteristic
121C8×C157—22no—
122C22×C157—22no—
123C22×C157—22no—
124C22×C157—42no—
125C22×C157#12012yes—
126C22×C157C22×D411yescharacteristic
127C2×D157—22no—
128C2×D157—24no—
129C2×D157—24no—
130C2×D157—44no—
131C2×D157—84no—
132C2×D157—88no—
133C4×C157—22no—
134C4×C157—42no—
135C4×C157#12012yes—
136C4×C157C22×D411yescharacteristic
137C4×C157C22×D411yescharacteristic
138Dic157—22no—
139Dic157—42no—
140C2×C157—22no—
141C2×C157—42no—
142C2×C157C2×#12011yescharacteristic
143C2×C157#57712yes—
144D157—42no—
145D157—44no—
146C157#300911yescharacteristic, complemented, Sylow
147#3009—1571nomaximal, Sylow
148C2×H(5,𝔽2)—1571no—
149C2×#137—1571no—
150C22×SD16—1571no—
151#536—1571no—
152#452—1574no—
153#450—1574no—
154#418—1571no—
155#416D15711yescharacteristic, complemented
156#408—1571no—
157H(5,𝔽2)—3144no—
158H(5,𝔽2)—3144no—
159C2×#41—3141no—
160C2×#41—1572no—
161C2×#41—1571no—
162C2×#41C2×D15711yescharacteristic
163C22×Q8—1571no—
164C22×D4—3141no—
165C22×D4—3142no—
166C22×D4—1572no—
167C22×D4—1571no—
168#137—3144no—
169C2×Q16—1572no—
170C2×SD16—3144no—
171C2×SD16—1572no—
172C2×SD16—1574no—
173C2×M16—1571no—
174C22×C8C2×D15711yescharacteristic, complemented
175C4⋊Q8—3141no—
176#124—3141no—
177#123—3142no—
178C22⋊Q8—1574no—
179#117C2×D15711yescharacteristic, complemented
180#103C2×D15712yescomplemented
181#102—1574no—
182#102C2×D15712yescomplemented
183C22⋊C8—1574no—
184C24—3142no—
185C24—3141no—
186#41—22no—
187#41—3148no—
188#41—3144no—
189#41—3144no—
190#41—3142no—
191#41—3142no—
192#41D62814yes—
193C2×Q8—3141no—
194C2×Q8—3142no—
195C2×Q8—3142no—
196C2×Q8—1572no—
197C2×Q8—1571no—
198C2×Q8C22×D15711yescharacteristic
199C2×D4—21no—
200C2×D4—3148no—
201C2×D4—3142no—
202C2×D4—31416no—
203C2×D4—3144no—
204C2×D4—3144no—
205C2×D4—3142no—
206C2×D4—3141no—
207C2×D4—3142no—
208C2×D4—1572no—
209C2×D4—1571no—
210C2×D4—1574no—
211C2×D4—1572no—
212C2×D4C22×D15711yescharacteristic
213C22×C4—21no—
214C22×C4—3142no—
215C22×C4—3141no—
216C22×C4—1572no—
217C22×C4C22×D15711yescharacteristic
218Q16—3144no—
219SD16—3148no—
220SD16—3144no—
221M16—3142no—
222C2×C8—22no—
223C2×C8—1572no—
224C2×C8C22×D15712yes—
225C4⋊C4—3144no—
226C4⋊C4C22×D15712yes—
227C22⋊C4—21no—
228C22⋊C4—3144no—
229C42—21no—
230C23—21no—
231C23—3148no—
232C23—3142no—
233C23—3144no—
234C23—1572no—
235C23—12562no—
236C23—12564no—
237C23D4×D15711yescharacteristic
238Q8—21no—
239Q8—3142no—
240Q8—3142no—
241Q8—3142no—
242Q8—6282no—
243Q8C2×D62812yes—
244D4—24no—
245D4—21no—
246D4—3148no—
247D4—3148no—
248D4—3144no—
249D4—3142no—
250D4—3144no—
251D4—3142no—
252D4—1574no—
253D4C2×D62812yes—
254C2×C4—21no—
255C2×C4—24no—
256C2×C4—22no—
257C2×C4—3148no—
258C2×C4—3142no—
259C2×C4—3142no—
260C2×C4—3142no—
261C2×C4—6282no—
262C2×C4—1572no—
263C2×C4D4×D15711yescharacteristic
264C2×C4D4×D15712yes—
265C2×C4C2×D62812yes—
266C2×C4C23×D15711yescharacteristic
267C8—22no—
268C8—3142no—
269C22—22no—
270C22—22no—
271C22—42no—
272C22—3144no—
273C22—3142no—
274C22—3144no—
275C22—6284no—
276C22—12564no—
277C22—12568no—
278C22#1909774112yes—
279C22C2×D4×D15711yescharacteristic
280C4—22no—
281C4—42no—
282C4—3142no—
283C4—6282no—
284C4#1909774112yes—
285C4C2×D4×D15711yescharacteristic
286C4C22×D62811yescharacteristic
287C2—22no—
288C2—42no—
289C2—6282no—
290C2—6284no—
291C2C2×#1909774111yescharacteristic
292C2#2375808012yes—
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#416×C157C2
socleC22×C157C22×D4
prime radicalC2×#19097817C2
hypercenter#416D157
hypocenterC157#3009
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
2. C2×#19097801C2semidirect
3. C22×#14578943C2semidirect
4. #23757849C2semidirect
5. #23757438C2semidirect
6. #23757425C2semidirect
7. #23757277C2semidirect
8. #23757239C2semidirect
9. #416×C157C2semidirect
15. C22×Dic314C22semidirect
21. C2×#14578944C22semidirect
23. C2×#14578943C22semidirect
24. C2×#14578942C22semidirect
25. C2×#14578938C22semidirect
29. #19097751C22semidirect
30. #19097691C22semidirect
31. #19097683C22semidirect
33. C22×C8×C157C22semidirect
34. #117×C157C22semidirect
35. #103×C157C22semidirect
36. #102×C157C22semidirect
146. C157#3009semidirect
155. #416D157semidirect
174. C22×C8C2×D157semidirect
179. #117C2×D157semidirect
180. #103C2×D157semidirect
182. #102C2×D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190978172
C2×#190978012
C22×#145789432
#237578492
#237574382
#237574252
#237572772
#237572392
#416×C1572
#3009157