Finite Group Databasebeta

← #34110613

Subgroups — #34110613

Order 20096.

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

normal subgroups only

This group has 564 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×#19097801C211yesmaximal, characteristic, complemented
2C2×#19097796C211yesmaximal, characteristic, complemented
3C22×#14578943C211yesmaximal, characteristic, complemented
4#23757844C211yesmaximal, characteristic, complemented
5#23757837C211yesmaximal, characteristic, complemented
6#23757666C212yesmaximal, complemented
7#23757661C212yesmaximal, complemented
8#23757654C212yesmaximal, complemented
9#23757644C212yesmaximal, complemented
10#23757228C211yesmaximal, characteristic, complemented
11C2×#135×C157C211yesmaximal, characteristic, complemented
12C22×Dic314C2211yescharacteristic
13C22×D628C2211yescharacteristic
14#19097801—24no—
15#19097796—24no—
16C2×#14578944C2212yescomplemented
17C2×#14578943—24no—
18C2×#14578943C2212yes—
19C2×#14578943C2212yes—
20C2×#14578942C2212yescomplemented
21C2×#14578941C2212yes—
22C2×#14578939C2212yes—
23C2×#14578938C2211yescharacteristic
24#19097769C2212yescomplemented
25#19097767C2212yes—
26#19097760C2212yes—
27#19097758C2212yescomplemented
28#19097752—22no—
29#19097750—22no—
30#19097745—22no—
31#19097739—22no—
32#19097730C2211yescharacteristic
33#19097681C2212yescomplemented
34C8×Dic157C2212yescomplemented
35C2×#41×C157C2211yescharacteristic
36C2×#41×C157C2211yescharacteristic
37#135×C157—24no—
38C2×Q16×C157C2211yescharacteristic, complemented
39C2×SD16×C157C2211yescharacteristic, complemented
40C2×SD16×C157C2211yescharacteristic, complemented
41C2×D8×C157C2211yescharacteristic, complemented
42C22×C8×C157C2211yescharacteristic
43C23×D157—21no—
44C2×Dic314—21no—
45C2×Dic314—22no—
46C2×Dic314—22no—
47C2×Dic314C2312yes—
48C2×D628—22no—
49C2×D628—22no—
50C2×D628—21no—
51C2×D628C2312yes—
52C2×C22⋊D157—24no—
53C22×Dic157—21no—
54#14578944—24no—
55#14578943—24no—
56#14578943—24no—
57#14578942—24no—
58#14578941—24no—
59#14578939—24no—
60#14578938D414yescomplemented
61C2×#11929434C2312yes—
62C4⋊Dic157D412yes—
63#14578933—24no—
64C22⋊Dic157—24no—
65C22⋊Dic157D412yes—
66#14578931—24no—
67C4×Dic157C2312yes—
68#41×C157—22no—
69#41×C157—22no—
70#41×C157—22no—
71#41×C157—22no—
72C2×Q8×C157C2311yescharacteristic
73C2×Q8×C157C2311yescharacteristic
74C2×D4×C157—21no—
75C2×D4×C157—21no—
76C2×D4×C157C2311yescharacteristic
77C2×D4×C157C2311yescharacteristic
78C22×C4×C157—21no—
79C22×C4×C157—21no—
80C22×C4×C157C2311yescharacteristic
81Q16×C157—22no—
82SD16×C157—22no—
83SD16×C157—22no—
84D8×C157—22no—
85C2×C8×C157D414yescomplemented
86C2×C8×C157C2311yescharacteristic
87C2×C8×C157C2311yescharacteristic
88C22×D157—22no—
89C22×D157—42no—
90C22×D157—82no—
91Dic314—22no—
92Dic314—22no—
93Dic314—42no—
94D628—22no—
95D628—22no—
96D628—42no—
97C22⋊D157—84no—
98C2×Dic157—22no—
99C2×Dic157—42no—
100C2×Dic157C2×D414yes—
101#11929434C2×D414yes—
102C23×C157—21no—
103C23×C157—21no—
104C23×C157C2×D411yescharacteristic
105Q8×C157—21no—
106Q8×C157—21no—
107Q8×C157—21no—
108Q8×C157—21no—
109D4×C157—21no—
110D4×C157—21no—
111D4×C157—21no—
112D4×C157—21no—
113D4×C157—42no—
114D4×C157—42no—
115C2×C4×C157—21no—
116C2×C4×C157—21no—
117C2×C4×C157—42no—
118C2×C4×C157—42no—
119C2×C4×C157C2×D411yescharacteristic
120C2×C4×C157C2×D412yes—
121C2×C4×C157C2×D412yes—
122C2×C4×C157C2411yescharacteristic
123C8×C157C2×D412yes—
124C8×C157C2×D412yes—
125C22×C157—22no—
126C22×C157—41no—
127C22×C157—41no—
128C22×C157—42no—
129C22×C157—42no—
130C22×C157#12712yes—
131C22×C157C22×D411yescharacteristic
132C2×D157—42no—
133C2×D157—44no—
134C2×D157—84no—
135C4×C157—41no—
136C4×C157—41no—
137C4×C157#12712yes—
138C4×C157C22×D411yescharacteristic
139C4×C157C22×D411yescharacteristic
140Dic157—24no—
141Dic157—42no—
142C2×C157—22no—
143C2×C157—81no—
144C2×C157—81no—
145C2×C157C2×#12711yescharacteristic
146C2×C157#57712yes—
147D157—82no—
148C157#315411yescharacteristic, complemented, Sylow
149#3154—1571nomaximal, Sylow
150C2×#137—1571no—
151C2×AGL(1,ℤ/8)—1571no—
152C2×#135D15711yescharacteristic, complemented
153C22×SD16—1571no—
154#536—1571no—
155#535—1571no—
156#497—1572no—
157#496—1572no—
158#495—1572no—
159#492—1572no—
160#405—1571no—
161C2×#41C2×D15711yescharacteristic
162C2×#41C2×D15711yescharacteristic
163C22×Q8—1571no—
164C22×D4—1571no—
165#137—3144no—
166AGL(1,ℤ/8)—3144no—
167#135—24no—
168C2×Q16—1572no—
169C2×Q16C2×D15711yescharacteristic, complemented
170C2×SD16—3144no—
171C2×SD16—1572no—
172C2×SD16—1572no—
173C2×SD16—1572no—
174C2×SD16—1572no—
175C2×SD16C2×D15711yescharacteristic, complemented
176C2×SD16C2×D15711yescharacteristic, complemented
177C2×D8—1572no—
178C2×D8C2×D15711yescharacteristic, complemented
179C2×M16—1571no—
180C22×C8C2×D15711yescharacteristic
181C4⋊Q8—1572no—
182#127—1572no—
183#124—1572no—
184#124—1572no—
185#123—3142no—
186C22⋊Q8—3142no—
187#121—3142no—
188#120—3142no—
189#117—1571no—
190#97—1572no—
191C4×C8—1572no—
192C24—3141no—
193#41—22no—
194#41—22no—
195#41—22no—
196#41—22no—
197C2×Q8—3141no—
198C2×Q8—3142no—
199C2×Q8—3142no—
200C2×Q8—1572no—
201C2×Q8C22×D15711yescharacteristic
202C2×Q8C22×D15711yescharacteristic
203C2×D4—21no—
204C2×D4—21no—
205C2×D4—3144no—
206C2×D4—3142no—
207C2×D4—3141no—
208C2×D4—3142no—
209C2×D4—1572no—
210C2×D4C22×D15711yescharacteristic
211C2×D4C22×D15711yescharacteristic
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—3141no—
215C22×C4C22×D15711yescharacteristic
216Q16—22no—
217Q16—3144no—
218SD16—22no—
219SD16—22no—
220SD16—3144no—
221SD16—3144no—
222SD16—3144no—
223SD16—3144no—
224D8—22no—
225D8—3144no—
226M16—1574no—
227C2×C8—1572no—
228C2×C8C22⋊D15714yescomplemented
229C2×C8C22×D15711yescharacteristic
230C2×C8C22×D15711yescharacteristic
231C4⋊C4—3144no—
232C4⋊C4—1572no—
233C22⋊C4—3144no—
234C22⋊C4—3144no—
235C22⋊C4—1572no—
236C42—1572no—
237C23—21no—
238C23—21no—
239C23—3142no—
240C23—6282no—
241C23—12562no—
242C23D4×D15711yescharacteristic
243Q8—21no—
244Q8—21no—
245Q8—21no—
246Q8—21no—
247Q8—3142no—
248Q8—3142no—
249Q8—6282no—
250D4—21no—
251D4—21no—
252D4—21no—
253D4—21no—
254D4—42no—
255D4—42no—
256D4—3142no—
257D4—3142no—
258D4—6282no—
259D4—12564no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—3142no—
265C2×C4—6282no—
266C2×C4—1574no—
267C2×C4D4×D15711yescharacteristic
268C2×C4D4×D15712yes—
269C2×C4C2×C22⋊D15712yes—
270C2×C4C23×D15711yescharacteristic
271C8—1574no—
272C8C2×C22⋊D15712yes—
273C8C2×C22⋊D15712yes—
274C22—22no—
275C22—42no—
276C22—42no—
277C22—41no—
278C22—41no—
279C22—6284no—
280C22—6282no—
281C22—12564no—
282C22#1909776712yes—
283C22C2×D4×D15711yescharacteristic
284C4—41no—
285C4—41no—
286C4—3144no—
287C4—6282no—
288C4#1909776712yes—
289C4C2×D4×D15711yescharacteristic
290C4C22×C22⋊D15711yescharacteristic
291C2—22no—
292C2—81no—
293C2—81no—
294C2—12562no—
295C2C2×#1909776711yescharacteristic
296C2#2375808012yes—
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
FittingC2×#135×C157C2
socleC22×C157C22×D4
hypercenterC2×#135D157
hypocenterC157#3154
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. C2×#19097801C2semidirect
2. C2×#19097796C2semidirect
3. C22×#14578943C2semidirect
4. #23757844C2semidirect
5. #23757837C2semidirect
6. #23757666C2semidirect
7. #23757661C2semidirect
8. #23757654C2semidirect
9. #23757644C2semidirect
10. #23757228C2semidirect
11. C2×#135×C157C2semidirect
16. C2×#14578944C22semidirect
20. C2×#14578942C22semidirect
24. #19097769C22semidirect
27. #19097758C22semidirect
33. #19097681C22semidirect
34. C8×Dic157C22semidirect
38. C2×Q16×C157C22semidirect
39. C2×SD16×C157C22semidirect
40. C2×SD16×C157C22semidirect
41. C2×D8×C157C22semidirect
60. #14578938D4semidirect
85. C2×C8×C157D4semidirect
148. C157#3154semidirect
152. C2×#135D157semidirect
169. C2×Q16C2×D157semidirect
175. C2×SD16C2×D157semidirect
176. C2×SD16C2×D157semidirect
178. C2×D8C2×D157semidirect
228. C2×C8C22⋊D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190978012
C2×#190977962
C22×#145789432
#237578442
#237578372
#237576662
#237576612
#237576542
#237576442
#237572282
C2×#135×C1572
#3154157