Finite Group Databasebeta

← #34109810

Subgroups — #34109810

Order 20096.

504 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 141 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×#19097779C211yesmaximal, characteristic, complemented
2#23757851C211yesmaximal, characteristic, complemented
3#23757840C211yesmaximal, characteristic, complemented
4#23757523C212yesmaximal, complemented
5#23757516C212yesmaximal, complemented
6#23757506C212yesmaximal, complemented
7#23757499C212yesmaximal, complemented
8#23757315C211yesmaximal, characteristic, complemented
9#23757269C211yesmaximal, characteristic, complemented
10C2×#19097692C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C157C211yesmaximal, characteristic, complemented
12C2×#14578954C2211yescharacteristic
13#19097779—24no—
14C2×#14578937C2212yescomplemented
15C22×#11929434C2211yescharacteristic
16C2×C8×D157C2212yescomplemented
17#19097767—21no—
18#19097758—21no—
19#19097750—21no—
20#19097747C2212yescomplemented
21#19097745—21no—
22#19097743—22no—
23#19097742C2212yes—
24#19097742C2212yes—
25#19097742C2212yescomplemented
26#19097740—22no—
27#19097730C2211yescharacteristic
28C2×C4⋊Dic157C2211yescharacteristic
29#19097705C2212yescomplemented
30#19097703C2212yescomplemented
31#19097695C2211yescharacteristic, complemented
32#19097695C2211yescharacteristic, complemented
33#19097692C2212yes—
34#19097692C2212yes—
35#19097692C2211yescharacteristic, complemented
36#19097692C2211yescharacteristic, complemented
37C2×#41×C157C2211yescharacteristic
38C22×D4×C157C2211yescharacteristic
39AGL(1,ℤ/8)×C157—24no—
40C2×SD16×C157C2212yes—
41C2×D8×C157C2212yes—
42C2×M16×C157C2211yescharacteristic
43#14578954—22no—
44#14578954—22no—
45C2×Dic314D411yescharacteristic
46C2×D628D411yescharacteristic
47C2×C22⋊D157—22no—
48C2×C22⋊D157—22no—
49C2×C22⋊D157—22no—
50C2×C22⋊D157D412yes—
51C22×Dic157—21no—
52C2×C4×D157C2312yes—
53#14578937—24no—
54C2×#11929434—22no—
55C2×#11929434C2311yescharacteristic
56C2×#11929434C2311yescharacteristic
57C8×D157—24no—
58C4⋊Dic157—21no—
59C4⋊Dic157C2312yes—
60C4⋊Dic157C2311yescharacteristic
61C4⋊Dic157C2311yescharacteristic
62#14578933—22no—
63C22⋊Dic157—22no—
64C22⋊Dic157—22no—
65C22⋊Dic157—22no—
66C22⋊Dic157—21no—
67#14578931—22no—
68C4×Dic157—21no—
69C24×C157—21no—
70#41×C157—22no—
71#41×C157—22no—
72C2×Q8×C157C2311yescharacteristic
73C2×D4×C157—21no—
74C2×D4×C157—22no—
75C2×D4×C157—21no—
76C2×D4×C157—22no—
77C2×D4×C157C2311yescharacteristic
78C2×D4×C157C2312yes—
79C22×C4×C157—21no—
80C22×C4×C157C2311yescharacteristic
81SD16×C157—24no—
82D8×C157—24no—
83M16×C157D414yescomplemented
84C2×C8×C157C2312yes—
85C22×D157C2×D412yes—
86Dic314—22no—
87D628—22no—
88C22⋊D157—24no—
89C22⋊D157—82no—
90C22⋊D157—82no—
91C22⋊D157—82no—
92C2×Dic157—22no—
93C2×Dic157—21no—
94C2×Dic157—21no—
95C2×Dic157—81no—
96C2×Dic157C2×D412yes—
97C4×D157—22no—
98C4×D157—22no—
99#11929434—21no—
100#11929434—21no—
101C23×C157—21no—
102C23×C157—22no—
103C23×C157—42no—
104C23×C157—81no—
105C23×C157—81no—
106C23×C157C2×D411yescharacteristic
107Q8×C157—21no—
108Q8×C157—21no—
109D4×C157—21no—
110D4×C157—22no—
111D4×C157—22no—
112D4×C157—21no—
113D4×C157—42no—
114D4×C157—42no—
115C2×C4×C157—21no—
116C2×C4×C157—42no—
117C2×C4×C157C2×D411yescharacteristic
118C2×C4×C157C2×D412yes—
119C2×C4×C157#4112yes—
120C2×C4×C157C2411yescharacteristic
121C8×C157C2×D414yes—
122C22×C157—21no—
123C22×C157—21no—
124C22×C157—41no—
125C22×C157—42no—
126C22×C157—42no—
127C22×C157—41no—
128C22×C157—41no—
129C22×C157—42no—
130C22×C157—84no—
131C22×C157#12112yes—
132C22×C157C22×D411yescharacteristic
133C2×D157—22no—
134C2×D157—22no—
135C2×D157—22no—
136C4×C157—41no—
137C4×C157#12112yes—
138C4×C157C22×D411yescharacteristic
139C4×C157C2×#4111yescharacteristic
140Dic157—22no—
141Dic157—42no—
142Dic157—41no—
143Dic157—41no—
144C2×C157—22no—
145C2×C157—81no—
146C2×C157—82no—
147C2×C157C2×#12111yescharacteristic
148C2×C157#57611yescharacteristic
149C2×C157#57711yescharacteristic
150D157—42no—
151C157#306211yescharacteristic, complemented, Sylow
152#3062—1571nomaximal, Sylow
153C2×AGL(1,ℤ/8)D15711yescharacteristic, complemented
154C2×#131—1571no—
155#537—1571no—
156#535—1571no—
157#469—1572no—
158#468—1572no—
159#466—1572no—
160#465—1572no—
161#428—1571no—
162#416—1571no—
163C2×#102—1571no—
164C2×#41—1571no—
165C2×#41C2×D15711yescharacteristic
166C22×D4C2×D15711yescharacteristic
167AGL(1,ℤ/8)—24no—
168C2×SD16C2×D15712yes—
169C2×D8C2×D15712yes—
170#131—3144no—
171C2×M16—1572no—
172C2×M16C2×D15711yescharacteristic
173C22×C8—1571no—
174C22×C8—1572no—
175#127—3141no—
176#124—3141no—
177C22⋊Q8—3141no—
178C22⋊Q8—1572no—
179#121—3141no—
180#121—3142no—
181#121—1572no—
182#121—1572no—
183#121—1572no—
184#120—3142no—
185#117—1571no—
186C2×C4⋊C4—1571no—
187#107—1572no—
188#106—1572no—
189#103—1571no—
190#103—1571no—
191#102—1572no—
192#102—1572no—
193#102—1571no—
194#102—1571no—
195C24—21no—
196#41—22no—
197#41—22no—
198#41—3142no—
199#41—3142no—
200C2×Q8—1571no—
201C2×Q8C22×D15711yescharacteristic
202C2×D4—21no—
203C2×D4—21no—
204C2×D4—22no—
205C2×D4—22no—
206C2×D4—3142no—
207C2×D4—3142no—
208C2×D4—3142no—
209C2×D4—1572no—
210C2×D4—1571no—
211C2×D4C22×D15711yescharacteristic
212C2×D4C22×D15712yes—
213C22×C4—21no—
214C22×C4—3141no—
215C22×C4—1572no—
216C22×C4C22×D15711yescharacteristic
217SD16—24no—
218D8—24no—
219M16—3144no—
220M16C22⋊D15714yescomplemented
221C2×C8—3142no—
222C2×C8—3144no—
223C2×C8—1571no—
224C2×C8—1571no—
225C2×C8C22×D15712yes—
226C4⋊C4—3142no—
227C4⋊C4—3141no—
228C4⋊C4—1572no—
229C4⋊C4—1571no—
230C4⋊C4—1571no—
231C22⋊C4—3142no—
232C22⋊C4—3141no—
233C22⋊C4—3142no—
234C22⋊C4—3142no—
235C22⋊C4—3142no—
236C42—3141no—
237C23—21no—
238C23—22no—
239C23—42no—
240C23—81no—
241C23—81no—
242C23—1572no—
243C23D4×D15711yescharacteristic
244Q8—21no—
245Q8—21no—
246Q8—3142no—
247D4—21no—
248D4—22no—
249D4—22no—
250D4—21no—
251D4—42no—
252D4—42no—
253D4—3144no—
254D4—3142no—
255D4—12562no—
256D4—12562no—
257D4—12562no—
258C2×C4—21no—
259C2×C4—42no—
260C2×C4—3142no—
261C2×C4—3142no—
262C2×C4—3141no—
263C2×C4—3141no—
264C2×C4—3142no—
265C2×C4—1572no—
266C2×C4—12561no—
267C2×C4D4×D15711yescharacteristic
268C2×C4C2×C22⋊D15712yes—
269C2×C4#1457895312yes—
270C2×C4C23×D15711yescharacteristic
271C8—3141no—
272C8—3141no—
273C8C2×C22⋊D15714yes—
274C22—21no—
275C22—21no—
276C22—42no—
277C22—41no—
278C22—41no—
279C22—42no—
280C22—41no—
281C22—42no—
282C22—84no—
283C22—3142no—
284C22—3142no—
285C22—3142no—
286C22#1909774212yes—
287C22C2×D4×D15711yescharacteristic
288C4—41no—
289C4—3142no—
290C4—6281no—
291C4—6282no—
292C4—6281no—
293C4#1909774212yes—
294C4C22×C22⋊D15711yescharacteristic
295C4C2×#1457895311yescharacteristic
296C2—22no—
297C2—82no—
298C2—81no—
299C2—6282no—
300C2C2×#1909774211yescharacteristic
301C2#2375807711yescharacteristic
302C2#2375808311yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
FittingC2×AGL(1,ℤ/8)×C157C2
socleC22×C157C22×D4
hypercenterC2×AGL(1,ℤ/8)D157
hypocenterC157#3062
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#19097779C2semidirect
2. #23757851C2semidirect
3. #23757840C2semidirect
4. #23757523C2semidirect
5. #23757516C2semidirect
6. #23757506C2semidirect
7. #23757499C2semidirect
8. #23757315C2semidirect
9. #23757269C2semidirect
10. C2×#19097692C2semidirect
11. C2×AGL(1,ℤ/8)×C157C2semidirect
14. C2×#14578937C22semidirect
16. C2×C8×D157C22semidirect
20. #19097747C22semidirect
25. #19097742C22semidirect
29. #19097705C22semidirect
30. #19097703C22semidirect
31. #19097695C22semidirect
32. #19097695C22semidirect
35. #19097692C22semidirect
36. #19097692C22semidirect
83. M16×C157D4semidirect
151. C157#3062semidirect
153. C2×AGL(1,ℤ/8)D157semidirect
220. M16C22⋊D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190977792
#237578512
#237578402
#237575232
#237575162
#237575062
#237574992
#237573152
#237572692
C2×#190976922
C2×AGL(1,ℤ/8)×C1572
#3062157