Finite Group Databasebeta

← #34109753

Subgroups — #34109753

Order 20096.

548 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 135 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×#19097797C211yesmaximal, characteristic, complemented
2C2×M16×D157C211yesmaximal, characteristic, complemented
3#23757785C211yesmaximal, characteristic, complemented
4C2×#19097742C211yesmaximal, characteristic, complemented
5#23757525C212yesmaximal, complemented
6#23757518C212yesmaximal, complemented
7#23757508C212yesmaximal, complemented
8#23757501C212yesmaximal, complemented
9#23757316C211yesmaximal, characteristic, complemented
10#23757275C211yesmaximal, characteristic, complemented
11#418×C157C211yesmaximal, characteristic, complemented
12C2×#14578954C2211yescharacteristic
13C22×C22⋊D157—21no—
14C22×C4×D157C2211yescharacteristic
15#19097797—24no—
16C2×#14578942C2212yescomplemented
17C2×#14578939C2212yescomplemented
18C2×#14578938C2211yescharacteristic
19C2×#14578937C2212yescomplemented
20M16×D157—24no—
21C2×C8×D157C2212yescomplemented
22#19097755—21no—
23#19097751C2211yescharacteristic, complemented
24#19097751C2211yescharacteristic, complemented
25#19097746C2211yescharacteristic, complemented
26#19097746C2211yescharacteristic, complemented
27#19097742—21no—
28#19097742—21no—
29#19097742C2212yes—
30#19097742C2212yes—
31#19097741—21no—
32C4×D628—21no—
33C4×D628—21no—
34C2×C4⋊Dic157C2211yescharacteristic
35C2×C22⋊Dic157—21no—
36#19097704C2212yescomplemented
37#19097702C2212yescomplemented
38#19097696C2212yescomplemented
39#19097693C2212yescomplemented
40C22×D4×C157C2211yescharacteristic
41C2×M16×C157C2211yescharacteristic
42#117×C157C2211yescharacteristic
43#102×C157C2212yescomplemented
44#102×C157C2212yescomplemented
45C23×D157D411yescharacteristic
46#14578954—22no—
47#14578954—22no—
48C2×Dic314C2311yescharacteristic
49C2×D628—21no—
50C2×D628—21no—
51C2×D628C2311yescharacteristic
52C2×C22⋊D157—22no—
53C2×C22⋊D157—22no—
54C2×C22⋊D157—21no—
55C2×C22⋊D157—44no—
56C22×Dic157—21no—
57C22×Dic157D411yescharacteristic
58C2×C4×D157—22no—
59C2×C4×D157—21no—
60C2×C4×D157—22no—
61C2×C4×D157D411yescharacteristic
62C2×C4×D157D411yescharacteristic
63C2×C4×D157C2311yescharacteristic
64C2×C4×D157C2311yescharacteristic
65#14578942—24no—
66#14578939—24no—
67#14578938D414yescomplemented
68#14578937—24no—
69C2×#11929434C2312yes—
70C8×D157—24no—
71C4⋊Dic157—21no—
72C4⋊Dic157—21no—
73C4⋊Dic157—21no—
74C4⋊Dic157C2312yes—
75C22⋊Dic157—22no—
76C22⋊Dic157—22no—
77#14578931—21no—
78#14578931—21no—
79#14578931—21no—
80#14578931—21no—
81C24×C157—21no—
82C2×D4×C157—22no—
83C2×D4×C157—21no—
84C2×D4×C157—22no—
85C2×D4×C157C2312yes—
86C22×C4×C157C2311yescharacteristic
87M16×C157—22no—
88C2×C8×C157C2312yes—
89C4⋊C4×C157C2311yescharacteristic
90C4⋊C4×C157C2311yescharacteristic
91C22⋊C4×C157—21no—
92C42×C157—21no—
93C22×D157—22no—
94C22×D157—22no—
95C22×D157—21no—
96C22×D157—41no—
97C22×D157C2×D411yescharacteristic
98C22×D157C2×D411yescharacteristic
99Dic314—21no—
100Dic314—21no—
101D628—21no—
102D628—21no—
103D628—42no—
104D628—42no—
105C22⋊D157—44no—
106C22⋊D157—42no—
107C22⋊D157—44no—
108C2×Dic157—22no—
109C2×Dic157—21no—
110C2×Dic157—22no—
111C2×Dic157—42no—
112C2×Dic157C2×D411yescharacteristic
113C2×Dic157C2×D411yescharacteristic
114C4×D157—22no—
115C4×D157—22no—
116C4×D157—21no—
117C4×D157—21no—
118C4×D157—21no—
119C4×D157—21no—
120C4×D157—42no—
121#11929434C2×D414yescomplemented
122C23×C157—22no—
123C23×C157—42no—
124C23×C157—42no—
125C23×C157—42no—
126C23×C157C2×D411yescharacteristic
127D4×C157—22no—
128D4×C157—22no—
129D4×C157—42no—
130C2×C4×C157—21no—
131C2×C4×C157—21no—
132C2×C4×C157C2×D411yescharacteristic
133C2×C4×C157C2×D412yes—
134C2×C4×C157#4112yes—
135C2×C4×C157C2411yescharacteristic
136C8×C157—22no—
137C22×C157—21no—
138C22×C157—21no—
139C22×C157—42no—
140C22×C157—42no—
141C22×C157—42no—
142C22×C157—84no—
143C22×C157#12112yes—
144C22×C157C22×D411yescharacteristic
145C2×D157—21no—
146C2×D157—21no—
147C2×D157—21no—
148C2×D157—21no—
149C2×D157—21no—
150C2×D157—44no—
151C2×D157—41no—
152C2×D157—41no—
153C2×D157—41no—
154C2×D157AGL(1,ℤ/8)12yescomplemented
155C4×C157—41no—
156C4×C157—41no—
157C4×C157#12112yes—
158C4×C157C22×D411yescharacteristic
159C4×C157C2×#4111yescharacteristic
160Dic157—21no—
161Dic157—21no—
162Dic157—41no—
163Dic157—42no—
164C2×C157—22no—
165C2×C157—82no—
166C2×C157C2×#12111yescharacteristic
167C2×C157C2×AGL(1,ℤ/8)11yescharacteristic
168C2×C157#57511yescharacteristic
169D157—22no—
170D157—41no—
171D157—81no—
172C157#305511yescharacteristic, complemented, Sylow
173#3055—1571nomaximal, Sylow
174C2×AGL(1,ℤ/8)—1571no—
175C22×M16—1571no—
176#525—1571no—
177C2×#121—1571no—
178#469—1572no—
179#468—1572no—
180#466—1572no—
181#465—1572no—
182#428—1571no—
183#418D15711yescharacteristic, complemented
184#417—1571no—
185C2×#41—1571no—
186C22×D4—3141no—
187C22×D4C2×D15711yescharacteristic
188C23×C4—1571no—
189AGL(1,ℤ/8)—3144no—
190C2×SD16—1572no—
191C2×D8—1572no—
192C2×M16—3144no—
193C2×M16—1571no—
194C2×M16—1572no—
195C2×M16C2×D15711yescharacteristic
196C22×C8—1572no—
197#123—3141no—
198C22⋊Q8—1571no—
199C22⋊Q8—1571no—
200#121—3141no—
201#121—3141no—
202#121—1572no—
203#121—1572no—
204#121—1571no—
205#121—1571no—
206#120—3141no—
207C4×D4—3141no—
208C4×D4—3141no—
209#117C2×D15711yescharacteristic
210C2×C4⋊C4—1571no—
211C2×C22⋊C4—3141no—
212#107—1572no—
213#106—1572no—
214#103—1572no—
215#102—1572no—
216#102C2×D15712yescomplemented
217#102C2×D15712yescomplemented
218C24—21no—
219C24—1571no—
220#41—3142no—
221#41—3142no—
222C2×Q8—1571no—
223C2×D4—22no—
224C2×D4—21no—
225C2×D4—22no—
226C2×D4—3141no—
227C2×D4—3141no—
228C2×D4—3142no—
229C2×D4—3142no—
230C2×D4—3141no—
231C2×D4—6284no—
232C2×D4—1571no—
233C2×D4C22×D15712yes—
234C22×C4—3142no—
235C22×C4—3142no—
236C22×C4—3141no—
237C22×C4—3141no—
238C22×C4—1571no—
239C22×C4—1571no—
240C22×C4—1571no—
241C22×C4—1571no—
242C22×C4—1571no—
243C22×C4C22×D15711yescharacteristic
244SD16—3144no—
245D8—3144no—
246M16—22no—
247M16—3144no—
248M16—1574no—
249C2×C8—3144no—
250C2×C8—1572no—
251C2×C8C22×D15712yes—
252C4⋊C4—3141no—
253C4⋊C4—3141no—
254C4⋊C4—3141no—
255C4⋊C4—1572no—
256C4⋊C4C22×D15711yescharacteristic
257C4⋊C4C22×D15711yescharacteristic
258C22⋊C4—21no—
259C22⋊C4—3141no—
260C22⋊C4—3141no—
261C22⋊C4—3141no—
262C22⋊C4—3141no—
263C22⋊C4—3142no—
264C22⋊C4—3142no—
265C42—21no—
266C23—22no—
267C23—42no—
268C23—42no—
269C23—42no—
270C23—3142no—
271C23—3142no—
272C23—3141no—
273C23—6281no—
274C23—1571no—
275C23—1571no—
276C23D4×D15711yescharacteristic
277Q8—3141no—
278Q8—3141no—
279D4—22no—
280D4—22no—
281D4—42no—
282D4—3141no—
283D4—3141no—
284D4—6282no—
285D4—6282no—
286D4—6284no—
287D4—6284no—
288D4—6282no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—3142no—
292C2×C4—3142no—
293C2×C4—3142no—
294C2×C4—3141no—
295C2×C4—3141no—
296C2×C4—3141no—
297C2×C4—3141no—
298C2×C4—3141no—
299C2×C4—3142no—
300C2×C4—6282no—
301C2×C4—6282no—
302C2×C4—1571no—
303C2×C4—1571no—
304C2×C4D4×D15711yescharacteristic
305C2×C4D4×D15712yes—
306C2×C4#1457895412yes—
307C2×C4C23×D15711yescharacteristic
308C8—22no—
309C8—1574no—
310C22—21no—
311C22—21no—
312C22—42no—
313C22—42no—
314C22—42no—
315C22—84no—
316C22—3141no—
317C22—3141no—
318C22—3141no—
319C22—3141no—
320C22—3141no—
321C22—6284no—
322C22—6281no—
323C22—6281no—
324C22—6281no—
325C22—1572no—
326C22#1909774412yes—
327C22C2×D4×D15711yescharacteristic
328C4—41no—
329C4—41no—
330C4—3141no—
331C4—3141no—
332C4—6281no—
333C4—6282no—
334C4#1909774412yes—
335C4C2×D4×D15711yescharacteristic
336C4C2×#1457895411yescharacteristic
337C2—22no—
338C2—82no—
339C2—3142no—
340C2—6281no—
341C2—12561no—
342C2C2×#1909774411yescharacteristic
343C2AGL(1,ℤ/8)×D15711yescharacteristic
344C2#2375806911yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#418×C157C2
socleC22×C157C22×D4
hypercenter#418D157
hypocenterC157#3055
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. C2×#19097797C2semidirect
2. C2×M16×D157C2semidirect
3. #23757785C2semidirect
4. C2×#19097742C2semidirect
5. #23757525C2semidirect
6. #23757518C2semidirect
7. #23757508C2semidirect
8. #23757501C2semidirect
9. #23757316C2semidirect
10. #23757275C2semidirect
11. #418×C157C2semidirect
16. C2×#14578942C22semidirect
17. C2×#14578939C22semidirect
19. C2×#14578937C22semidirect
21. C2×C8×D157C22semidirect
23. #19097751C22semidirect
24. #19097751C22semidirect
25. #19097746C22semidirect
26. #19097746C22semidirect
36. #19097704C22semidirect
37. #19097702C22semidirect
38. #19097696C22semidirect
39. #19097693C22semidirect
43. #102×C157C22semidirect
44. #102×C157C22semidirect
67. #14578938D4semidirect
121. #11929434C2×D4semidirect
154. C2×D157AGL(1,ℤ/8)semidirect
172. C157#3055semidirect
183. #418D157semidirect
216. #102C2×D157semidirect
217. #102C2×D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190977972
C2×M16×D1572
#237577852
C2×#190977422
#237575252
#237575182
#237575082
#237575012
#237573162
#237572752
#418×C1572
#3055157