Finite Group Databasebeta

← #34110598

Subgroups — #34110598

Order 20096.

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

normal subgroups only

This group has 572 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×#19097804C211yesmaximal, characteristic, complemented
2C2×#19097801C211yesmaximal, characteristic, complemented
3C2×#19097797C211yesmaximal, characteristic, complemented
4#23757819C211yesmaximal, characteristic, complemented
5C2×#19097758C211yesmaximal, characteristic, complemented
6#23757664C212yesmaximal, complemented
7#23757661C212yesmaximal, complemented
8#23757656C212yesmaximal, complemented
9#23757654C212yesmaximal, complemented
10M16×Dic157C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C157C211yesmaximal, characteristic, complemented
12C2×#14578954C2211yescharacteristic
13C22×Dic314C2211yescharacteristic
14#19097804—24no—
15#19097801—24no—
16#19097797—24no—
17C2×Dic628C2212yescomplemented
18C2×#14578944C2212yescomplemented
19C2×#14578943C2212yescomplemented
20C2×#14578942C2212yes—
21C2×#14578942C2212yescomplemented
22C2×#14578939C2212yes—
23C2×#14578938C2211yescharacteristic
24#19097769C2211yescharacteristic, complemented
25#19097767C2211yescharacteristic, complemented
26#19097760C2211yescharacteristic, complemented
27#19097758—22no—
28#19097758C2212yes—
29#19097758C2212yes—
30#19097758C2211yescharacteristic, complemented
31#19097743—22no—
32C4×C22⋊D157—22no—
33C2×C22⋊Dic157—22no—
34C2×C4×Dic157C2211yescharacteristic
35#19097681C2212yescomplemented
36C8×Dic157C2212yescomplemented
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×Dic314—21no—
46C2×Dic314—22no—
47C2×Dic314—22no—
48C2×Dic314C2312yes—
49C2×Dic314C2311yescharacteristic
50C2×D628C2311yescharacteristic
51C2×C22⋊D157—22no—
52C2×C22⋊D157—21no—
53C22×Dic157—21no—
54C22×Dic157D412yes—
55C2×C4×D157—21no—
56Dic628—24no—
57#14578944—24no—
58#14578943—24no—
59#14578942—24no—
60#14578942—24no—
61#14578939—24no—
62#14578938D414yescomplemented
63C2×#11929434C2312yes—
64#14578933—22no—
65C22⋊Dic157—24no—
66C22⋊Dic157—24no—
67C22⋊Dic157—22no—
68#14578931—22no—
69C4×Dic157D412yes—
70C4×Dic157C2311yescharacteristic
71C4×Dic157C2311yescharacteristic
72C24×C157—21no—
73#41×C157—22no—
74#41×C157—22no—
75C2×Q8×C157C2311yescharacteristic
76C2×D4×C157—21no—
77C2×D4×C157—22no—
78C2×D4×C157—22no—
79C2×D4×C157—21no—
80C2×D4×C157C2311yescharacteristic
81C2×D4×C157C2312yes—
82C22×C4×C157—21no—
83C22×C4×C157C2311yescharacteristic
84SD16×C157—24no—
85D8×C157—24no—
86M16×C157D414yescomplemented
87C2×C8×C157C2312yes—
88C22×D157—21no—
89Dic314—22no—
90Dic314—21no—
91Dic314—22no—
92Dic314—21no—
93Dic314—42no—
94D628—21no—
95D628—21no—
96C22⋊D157—44no—
97C22⋊D157—42no—
98C2×Dic157—21no—
99C2×Dic157—22no—
100C2×Dic157—42no—
101C2×Dic157—42no—
102C2×Dic157C2×D412yes—
103C2×Dic157C2×D412yes—
104C4×D157—42no—
105#11929434C2×D414yes—
106C23×C157—21no—
107C23×C157—22no—
108C23×C157—44no—
109C23×C157—42no—
110C23×C157C2×D411yescharacteristic
111Q8×C157—21no—
112Q8×C157—21no—
113D4×C157—21no—
114D4×C157—22no—
115D4×C157—22no—
116D4×C157—21no—
117D4×C157—42no—
118D4×C157—42no—
119C2×C4×C157—21no—
120C2×C4×C157—42no—
121C2×C4×C157C2×D411yescharacteristic
122C2×C4×C157C2×D412yes—
123C2×C4×C157C2×D412yes—
124C2×C4×C157C2411yescharacteristic
125C8×C157C2×D414yes—
126C22×C157—22no—
127C22×C157—41no—
128C22×C157—42no—
129C22×C157—44no—
130C22×C157—42no—
131C22×C157—84no—
132C22×C157#12712yes—
133C22×C157C22×D411yescharacteristic
134C2×D157—41no—
135C2×D157—42no—
136C4×C157—41no—
137C4×C157#12712yes—
138C4×C157C22×D411yescharacteristic
139C4×C157C22×D411yescharacteristic
140Dic157—22no—
141Dic157—22no—
142Dic157—41no—
143Dic157—42no—
144C2×C157—22no—
145C2×C157—81no—
146C2×C157—82no—
147C2×C157C2×#12711yescharacteristic
148C2×C157#57512yes—
149D157—81no—
150C157#315311yescharacteristic, complemented, Sylow
151#3153—1571nomaximal, Sylow
152C2×#137—1571no—
153C2×#137—1571no—
154C2×AGL(1,ℤ/8)—1571no—
155C2×AGL(1,ℤ/8)D15711yescharacteristic, complemented
156#532—1571no—
157C2×#124—1571no—
158#497—1572no—
159#496—1572no—
160#495—1572no—
161#495—1572no—
162C4×M16—1571no—
163C2×#41—1571no—
164C2×#41C2×D15711yescharacteristic
165C22×Q8—1571no—
166C22×D4C2×D15711yescharacteristic
167#137—3144no—
168#137—3144no—
169AGL(1,ℤ/8)—24no—
170AGL(1,ℤ/8)—3144no—
171C2×Q16—1572no—
172C2×Q16—1572no—
173C2×SD16—1572no—
174C2×SD16—1572no—
175C2×SD16—1572no—
176C2×SD16C2×D15712yes—
177C2×D8—1572no—
178C2×D8C2×D15712yes—
179C2×M16—1571no—
180C2×M16C2×D15711yescharacteristic
181C4⋊Q8—1571no—
182#127—1571no—
183#124—3142no—
184#124—1571no—
185#124—1572no—
186#124—1572no—
187#124—1571no—
188#121—3142no—
189C4×D4—3142no—
190C2×C22⋊C4—3142no—
191C2×C42—1571no—
192#97—1572no—
193C4×C8—1572no—
194C24—21no—
195#41—22no—
196#41—22no—
197#41—3142no—
198#41—3142no—
199C2×Q8—3141no—
200C2×Q8—3142no—
201C2×Q8—3142no—
202C2×Q8—1572no—
203C2×Q8—1571no—
204C2×Q8C22×D15711yescharacteristic
205C2×D4—21no—
206C2×D4—22no—
207C2×D4—21no—
208C2×D4—22no—
209C2×D4—3141no—
210C2×D4—3142no—
211C2×D4—1571no—
212C2×D4C22×D15712yes—
213C2×D4C22×D15711yescharacteristic
214C22×C4—21no—
215C22×C4—3141no—
216C22×C4—3141no—
217C22×C4—1572no—
218C22×C4C22×D15711yescharacteristic
219Q16—3144no—
220Q16—3144no—
221SD16—24no—
222SD16—3144no—
223SD16—3144no—
224SD16—3144no—
225D8—24no—
226D8—3144no—
227M16—1574no—
228M16C22⋊D15714yescomplemented
229C2×C8—1572no—
230C2×C8C22×D15712yes—
231C4⋊C4—3142no—
232C22⋊C4—3142no—
233C22⋊C4—3144no—
234C22⋊C4—3144no—
235C22⋊C4—3142no—
236C42—1572no—
237C42—1571no—
238C42—1571no—
239C23—22no—
240C23—21no—
241C23—44no—
242C23—42no—
243C23—3141no—
244C23D4×D15711yescharacteristic
245Q8—21no—
246Q8—21no—
247Q8—3141no—
248Q8—3142no—
249Q8—3142no—
250Q8—3141no—
251Q8—6282no—
252D4—22no—
253D4—21no—
254D4—21no—
255D4—22no—
256D4—42no—
257D4—42no—
258D4—3141no—
259D4—3141no—
260D4—6284no—
261D4—6282no—
262C2×C4—21no—
263C2×C4—42no—
264C2×C4—3142no—
265C2×C4—3141no—
266C2×C4—6282no—
267C2×C4—6282no—
268C2×C4—6282no—
269C2×C4—1572no—
270C2×C4—1572no—
271C2×C4D4×D15711yescharacteristic
272C2×C4D4×D15712yes—
273C2×C4C2×C22⋊D15712yes—
274C2×C4C23×D15711yescharacteristic
275C8—1574no—
276C8C2×C22⋊D15714yes—
277C22—22no—
278C22—41no—
279C22—42no—
280C22—42no—
281C22—44no—
282C22—84no—
283C22—6281no—
284C22—6282no—
285C22#1909776712yes—
286C22C2×D4×D15711yescharacteristic
287C4—41no—
288C4—3142no—
289C4—3142no—
290C4—6281no—
291C4—6282no—
292C4#1909776712yes—
293C4C2×D4×D15711yescharacteristic
294C4C22×C22⋊D15711yescharacteristic
295C2—22no—
296C2—82no—
297C2—81no—
298C2—12561no—
299C2C2×#1909776711yescharacteristic
300C2#2375806912yes—
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#3153
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×#19097804C2semidirect
2. C2×#19097801C2semidirect
3. C2×#19097797C2semidirect
4. #23757819C2semidirect
5. C2×#19097758C2semidirect
6. #23757664C2semidirect
7. #23757661C2semidirect
8. #23757656C2semidirect
9. #23757654C2semidirect
10. M16×Dic157C2semidirect
11. C2×AGL(1,ℤ/8)×C157C2semidirect
17. C2×Dic628C22semidirect
18. C2×#14578944C22semidirect
19. C2×#14578943C22semidirect
21. C2×#14578942C22semidirect
24. #19097769C22semidirect
25. #19097767C22semidirect
26. #19097760C22semidirect
30. #19097758C22semidirect
35. #19097681C22semidirect
36. C8×Dic157C22semidirect
62. #14578938D4semidirect
86. M16×C157D4semidirect
150. C157#3153semidirect
155. C2×AGL(1,ℤ/8)D157semidirect
228. M16C22⋊D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190978042
C2×#190978012
C2×#190977972
#237578192
C2×#190977582
#237576642
#237576612
#237576562
#237576542
M16×Dic1572
C2×AGL(1,ℤ/8)×C1572
#3153157