Finite Group Databasebeta

← #34109744

Subgroups — #34109744

Order 20096.

500 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 133 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×#19097791C211yesmaximal, characteristic, complemented
2C22×#14578937C211yesmaximal, characteristic, complemented
3#23757785C211yesmaximal, characteristic, complemented
4#23757780C211yesmaximal, characteristic, complemented
5#23757530C212yesmaximal, complemented
6#23757525C212yesmaximal, complemented
7#23757522C212yesmaximal, complemented
8#23757518C212yesmaximal, complemented
9#23757309C211yesmaximal, characteristic, complemented
10#23757270C211yesmaximal, characteristic, complemented
11#416×C157C211yesmaximal, characteristic, complemented
12C2×#14578954C2211yescharacteristic
13C22×C4×D157C2211yescharacteristic
14#19097791—24no—
15C2×#14578944C2211yescharacteristic, complemented
16C2×#14578942C2211yescharacteristic, complemented
17C2×#14578941C2211yescharacteristic, complemented
18C2×#14578939C2211yescharacteristic, complemented
19C2×#14578937—24no—
20C2×#14578937C2212yes—
21C2×#14578937C2212yes—
22C22×#11929434C2211yescharacteristic
23#19097755—21no—
24#19097752—21no—
25#19097751C2212yes—
26#19097747C2212yes—
27#19097746C2212yes—
28#19097742C2212yes—
29#19097741—21no—
30#19097739—21no—
31C4×D628—22no—
32C4×C22⋊D157—22no—
33#19097730C2211yescharacteristic
34#19097704C2212yescomplemented
35#19097702C2212yescomplemented
36#19097696C2212yescomplemented
37#19097693C2212yescomplemented
38C2×#41×C157C2211yescharacteristic
39C22×C8×C157C2211yescharacteristic
40#117×C157C2211yescharacteristic
41#103×C157C2212yescomplemented
42#102×C157C2212yescomplemented
43C23×D157D411yescharacteristic
44#14578954—22no—
45#14578954—22no—
46C2×Dic314C2311yescharacteristic
47C2×D628—22no—
48C2×D628C2311yescharacteristic
49C2×C22⋊D157—22no—
50C2×C22⋊D157—21no—
51C22×Dic157D411yescharacteristic
52C2×C4×D157—22no—
53C2×C4×D157—21no—
54C2×C4×D157—22no—
55C2×C4×D157D412yes—
56C2×C4×D157C2312yes—
57#14578944—22no—
58#14578942—22no—
59#14578941—22no—
60#14578939—22no—
61#14578937—24no—
62#14578937—24no—
63C2×#11929434D414yescomplemented
64C2×#11929434C2311yescharacteristic
65C2×#11929434C2311yescharacteristic
66C4⋊Dic157—22no—
67C4⋊Dic157C2312yes—
68#14578933—22no—
69C22⋊Dic157—22no—
70C22⋊Dic157—21no—
71#14578931—22no—
72#14578931—22no—
73#14578931—22no—
74C4×Dic157—21no—
75#41×C157—22no—
76#41×C157—22no—
77C2×Q8×C157C2311yescharacteristic
78C2×D4×C157—21no—
79C2×D4×C157C2311yescharacteristic
80C22×C4×C157—21no—
81C22×C4×C157C2311yescharacteristic
82C2×C8×C157—22no—
83C2×C8×C157C2312yes—
84C4⋊C4×C157C2312yes—
85C22⋊C4×C157—21no—
86C42×C157—21no—
87C22×D157—22no—
88C22×D157—21no—
89C22×D157—42no—
90C22×D157C2×D412yes—
91Dic314—21no—
92Dic314—21no—
93D628—21no—
94D628—21no—
95D628—44no—
96C22⋊D157—44no—
97C22⋊D157—42no—
98C2×Dic157—22no—
99C2×Dic157—21no—
100C2×Dic157—22no—
101C2×Dic157C2×D412yes—
102C4×D157—24no—
103C4×D157—22no—
104C4×D157—22no—
105C4×D157—42no—
106#11929434C2×D412yes—
107#11929434C2×D412yes—
108C23×C157—21no—
109C23×C157C2×D411yescharacteristic
110Q8×C157—21no—
111Q8×C157—21no—
112D4×C157—21no—
113D4×C157—21no—
114D4×C157—42no—
115C2×C4×C157—22no—
116C2×C4×C157—21no—
117C2×C4×C157—42no—
118C2×C4×C157C2×D412yes—
119C2×C4×C157C2×D411yescharacteristic
120C2×C4×C157#4112yes—
121C2×C4×C157C2411yescharacteristic
122C8×C157—22no—
123C22×C157—22no—
124C22×C157—41no—
125C22×C157—42no—
126C22×C157#12112yes—
127C22×C157C22×D411yescharacteristic
128C2×D157—22no—
129C2×D157—22no—
130C2×D157—22no—
131C2×D157—44no—
132C2×D157—41no—
133C2×D157—42no—
134C4×C157—41no—
135C4×C157—42no—
136C4×C157#12112yes—
137C4×C157C22×D411yescharacteristic
138C4×C157C2×#4111yescharacteristic
139Dic157—22no—
140Dic157—41no—
141Dic157—42no—
142C2×C157—22no—
143C2×C157—81no—
144C2×C157C2×#12111yescharacteristic
145C2×C157#57512yes—
146D157—42no—
147D157—81no—
148C157#305411yescharacteristic, complemented, Sylow
149#3054—1571nomaximal, Sylow
150C2×#135—1571no—
151C22×M16—1571no—
152#525—1571no—
153#525—1571no—
154#470—1572no—
155#469—1572no—
156#468—1572no—
157#468—1572no—
158#427—1571no—
159#416—1571no—
160#416D15711yescharacteristic, complemented
161C2×#41—1571no—
162C2×#41C2×D15711yescharacteristic
163C23×C4—1571no—
164#135—3144no—
165C2×Q16—1571no—
166C2×SD16—1571no—
167C2×SD16—1571no—
168C2×D8—1571no—
169C2×M16—3144no—
170C2×M16—1572no—
171C2×M16—1572no—
172C22×C8—1571no—
173C22×C8C2×D15711yescharacteristic
174#123—3141no—
175#123—3141no—
176C22⋊Q8—1572no—
177C22⋊Q8—1572no—
178#121—1572no—
179#121—1572no—
180#120—3141no—
181#120—3141no—
182C4×D4—3142no—
183C4×D4—3142no—
184#117—1571no—
185#117C2×D15711yescharacteristic
186#107—1572no—
187#106—1572no—
188#103—1572no—
189#103C2×D15712yescomplemented
190#102—1572no—
191#102C2×D15712yescomplemented
192C24—1571no—
193#41—22no—
194#41—22no—
195#41—3142no—
196#41—3142no—
197C2×Q8—1571no—
198C2×Q8C22×D15711yescharacteristic
199C2×D4—21no—
200C2×D4—3142no—
201C2×D4—3142no—
202C2×D4—3141no—
203C2×D4—1571no—
204C2×D4C22×D15711yescharacteristic
205C22×C4—21no—
206C22×C4—3142no—
207C22×C4—3142no—
208C22×C4—3141no—
209C22×C4—1571no—
210C22×C4—1572no—
211C22×C4—1572no—
212C22×C4C22×D15711yescharacteristic
213Q16—3142no—
214SD16—3142no—
215SD16—3142no—
216D8—3142no—
217M16—3144no—
218M16—3144no—
219C2×C8—22no—
220C2×C8—1574no—
221C2×C8—1571no—
222C2×C8—1571no—
223C2×C8C22×D15712yes—
224C4⋊C4—3142no—
225C4⋊C4—3142no—
226C4⋊C4—1572no—
227C4⋊C4C22×D15712yes—
228C22⋊C4—21no—
229C22⋊C4—3142no—
230C22⋊C4—3142no—
231C22⋊C4—3142no—
232C22⋊C4—3141no—
233C22⋊C4—3142no—
234C42—21no—
235C42—3141no—
236C23—21no—
237C23—3142no—
238C23—3141no—
239C23—6282no—
240C23—1572no—
241C23D4×D15711yescharacteristic
242Q8—21no—
243Q8—21no—
244Q8—3141no—
245Q8—3141no—
246D4—21no—
247D4—21no—
248D4—42no—
249D4—3141no—
250D4—3141no—
251D4—6284no—
252D4—6284no—
253D4—6282no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—42no—
257C2×C4—3142no—
258C2×C4—3144no—
259C2×C4—3142no—
260C2×C4—3142no—
261C2×C4—3142no—
262C2×C4—3141no—
263C2×C4—6282no—
264C2×C4—1572no—
265C2×C4D4×D15712yes—
266C2×C4D4×D15711yescharacteristic
267C2×C4#1457895412yes—
268C2×C4C23×D15711yescharacteristic
269C8—22no—
270C8—1572no—
271C8—1572no—
272C22—22no—
273C22—41no—
274C22—42no—
275C22—3142no—
276C22—3142no—
277C22—3142no—
278C22—6284no—
279C22—6281no—
280C22—6282no—
281C22#1909774412yes—
282C22C2×D4×D15711yescharacteristic
283C4—41no—
284C4—42no—
285C4—3142no—
286C4—6281no—
287C4—6282no—
288C4#1909774412yes—
289C4C2×D4×D15711yescharacteristic
290C4C2×#1457895411yescharacteristic
291C2—22no—
292C2—81no—
293C2—6282no—
294C2—12561no—
295C2C2×#1909774411yescharacteristic
296C2#2375807012yes—
subgroupG/H
centerC2×C4D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#416×C157C2
socleC22×C157C22×D4
hypercenter#416D157
hypocenterC157#3054
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. C2×#19097791C2semidirect
2. C22×#14578937C2semidirect
3. #23757785C2semidirect
4. #23757780C2semidirect
5. #23757530C2semidirect
6. #23757525C2semidirect
7. #23757522C2semidirect
8. #23757518C2semidirect
9. #23757309C2semidirect
10. #23757270C2semidirect
11. #416×C157C2semidirect
15. C2×#14578944C22semidirect
16. C2×#14578942C22semidirect
17. C2×#14578941C22semidirect
18. C2×#14578939C22semidirect
34. #19097704C22semidirect
35. #19097702C22semidirect
36. #19097696C22semidirect
37. #19097693C22semidirect
41. #103×C157C22semidirect
42. #102×C157C22semidirect
63. C2×#11929434D4semidirect
148. C157#3054semidirect
160. #416D157semidirect
189. #103C2×D157semidirect
191. #102C2×D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190977912
C22×#145789372
#237577852
#237577802
#237575302
#237575252
#237575222
#237575182
#237573092
#237572702
#416×C1572
#3054157