Finite Group Databasebeta

← #34109400

Subgroups — #34109400

Order 20096.

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

normal subgroups only

This group has 784 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×#19097816C211yesmaximal, characteristic, complemented
2C2×#19097797C211yesmaximal, characteristic, complemented
3#23757840C211yesmaximal, characteristic, complemented
4#23757415C214yesmaximal, complemented
5#23757405C214yesmaximal, complemented
6#23757278C211yesmaximal, characteristic, complemented
7#23757270C211yesmaximal, characteristic, complemented
8#23757239C211yesmaximal, characteristic, complemented
9C22×D8×C157C211yesmaximal, characteristic, complemented
10#19097816—24no—
11#19097816—24no—
12C2×#14578954C2211yescharacteristic
13C2×#14578954C2211yescharacteristic
14C2×#14578953—21no—
15C2×#14578953C2212yescomplemented
16C22×C22⋊D157—22no—
17C2×D4×D157—21no—
18C2×D4×D157C2212yescomplemented
19#19097797—24no—
20C2×#14578942C2212yescomplemented
21C2×#14578939C2212yescomplemented
22C2×#14578938C2211yescharacteristic, complemented
23#19097767—21no—
24#19097758—21no—
25#19097742C2214yescomplemented
26#19097740—22no—
27#19097730C2211yescharacteristic, complemented
28#19097696C2212yescomplemented
29#19097693C2212yescomplemented
30#19097692C2214yescomplemented
31#19097683C2214yescomplemented
32C22×D4×C157C2211yescharacteristic
33C22×D4×C157C2211yescharacteristic, complemented
34C2×D8×C157—24no—
35C2×D8×C157C2214yescomplemented
36C22×C8×C157C2211yescharacteristic, complemented
37C23×D157—22no—
38#14578954—22no—
39#14578954—22no—
40#14578954—22no—
41#14578954D414yescomplemented
42#14578953—28no—
43#14578953—24no—
44#14578953—24no—
45C2×Dic314D411yescharacteristic
46C2×Dic314C2311yescharacteristic
47C2×D628D411yescharacteristic
48C2×D628C2311yescharacteristic
49C2×C22⋊D157—216no—
50C2×C22⋊D157—22no—
51C2×C22⋊D157—24no—
52C2×C22⋊D157—22no—
53C2×C22⋊D157—21no—
54C2×C22⋊D157D412yes—
55D4×D157—28no—
56D4×D157—24no—
57D4×D157—24no—
58C22×Dic157—22no—
59C2×C4×D157—21no—
60C2×C4×D157C2312yes—
61#14578942—24no—
62#14578939—24no—
63#14578938—22no—
64C2×#11929434C2312yes—
65C4⋊Dic157C2312yes—
66C22⋊Dic157—24no—
67C22⋊Dic157—21no—
68C4×Dic157—21no—
69C24×C157—21no—
70C24×C157—21no—
71C2×D4×C157—21no—
72C2×D4×C157—22no—
73C2×D4×C157—22no—
74C2×D4×C157—22no—
75C2×D4×C157—21no—
76C2×D4×C157D414yescomplemented
77C2×D4×C157C2312yes—
78C2×D4×C157C2312yes—
79C22×C4×C157C2311yescharacteristic
80D8×C157—28no—
81C2×C8×C157—22no—
82C2×C8×C157C2312yes—
83C22×D157—21no—
84C22×D157—28no—
85C22×D157—84no—
86C22×D157C2×D412yes—
87Dic314—22no—
88Dic314—21no—
89Dic314C2×D412yes—
90D628—22no—
91D628—21no—
92D628C2×D412yes—
93C22⋊D157—28no—
94C22⋊D157—28no—
95C22⋊D157—24no—
96C22⋊D157—24no—
97C22⋊D157—84no—
98C2×Dic157—28no—
99C2×Dic157—21no—
100C2×Dic157—22no—
101C2×Dic157C2×D412yes—
102C4×D157—24no—
103C4×D157—22no—
104C4×D157—22no—
105#11929434—22no—
106C23×C157—22no—
107C23×C157—24no—
108C23×C157—22no—
109C23×C157—42no—
110C23×C157—82no—
111C23×C157—82no—
112C23×C157C2×D411yescharacteristic
113D4×C157—24no—
114D4×C157—22no—
115D4×C157—22no—
116D4×C157—22no—
117D4×C157—42no—
118D4×C157C2×D414yes—
119C2×C4×C157C2×D411yescharacteristic
120C2×C4×C157C2×D412yes—
121C2×C4×C157C2×D412yes—
122C2×C4×C157C2411yescharacteristic
123C8×C157—22no—
124C22×C157—22no—
125C22×C157—24no—
126C22×C157—44no—
127C22×C157—42no—
128C22×C157—44no—
129C22×C157—84no—
130C22×C157—84no—
131C22×C157#12012yes—
132C22×C157C22×D411yescharacteristic
133C2×D157—22no—
134C2×D157—22no—
135C2×D157—24no—
136C2×D157—42no—
137C2×D157—88no—
138C4×C157#12012yes—
139C4×C157C22×D411yescharacteristic
140C4×C157C22×D411yescharacteristic
141Dic157—22no—
142Dic157—22no—
143Dic157—42no—
144C2×C157—22no—
145C2×C157—44no—
146C2×C157—82no—
147C2×C157C2×#12011yescharacteristic
148C2×C157#57612yes—
149D157—42no—
150D157—42no—
151C157#300811yescharacteristic, complemented, Sylow
152#3008—1571nomaximal, Sylow
153C2×H(5,𝔽2)—1571no—
154C2×AGL(1,ℤ/8)—1571no—
155C22×D8D15711yescharacteristic, complemented
156#535—1571no—
157#449—1574no—
158#447—1574no—
159#418—1571no—
160#416—1571no—
161#408—1571no—
162H(5,𝔽2)—3144no—
163H(5,𝔽2)—3144no—
164C2×#41—3141no—
165C2×#41—1572no—
166C2×#41—1571no—
167C2×#41—1571no—
168C22×D4—3141no—
169C22×D4—3142no—
170C22×D4—1572no—
171C22×D4C2×D15711yescharacteristic
172C22×D4C2×D15711yescharacteristic, complemented
173AGL(1,ℤ/8)—3144no—
174C2×SD16—1572no—
175C2×D8—24no—
176C2×D8—1572no—
177C2×D8C2×D15714yescomplemented
178C2×M16—1571no—
179C22×C8C2×D15711yescharacteristic, complemented
180#127—3141no—
181#124—3141no—
182#121—1574no—
183#120—3142no—
184#117—1571no—
185#103—1572no—
186#102—1574no—
187#102—1572no—
188C22⋊C8—1574no—
189C24—21no—
190C24—21no—
191C24—3142no—
192#41—3148no—
193#41—3144no—
194#41—3144no—
195#41—3142no—
196#41—3142no—
197#41—3142no—
198#41—1574no—
199C2×Q8—1571no—
200C2×Q8—1571no—
201C2×D4—21no—
202C2×D4—22no—
203C2×D4—22no—
204C2×D4—21no—
205C2×D4—22no—
206C2×D4—3148no—
207C2×D4—3144no—
208C2×D4—31416no—
209C2×D4—3142no—
210C2×D4—3141no—
211C2×D4—3144no—
212C2×D4—3144no—
213C2×D4—3142no—
214C2×D4—1572no—
215C2×D4—1571no—
216C2×D4—1571no—
217C2×D4C22⋊D15714yescomplemented
218C2×D4C22×D15712yes—
219C2×D4C22×D15712yes—
220C22×C4—3142no—
221C22×C4—3141no—
222C22×C4—1572no—
223C22×C4C22×D15711yescharacteristic
224SD16—3144no—
225D8—28no—
226D8—3144no—
227M16—3142no—
228C2×C8—22no—
229C2×C8—1572no—
230C2×C8C22×D15712yes—
231C4⋊C4—1572no—
232C22⋊C4—3141no—
233C22⋊C4—3144no—
234C42—3141no—
235C23—22no—
236C23—24no—
237C23—22no—
238C23—42no—
239C23—82no—
240C23—82no—
241C23—3148no—
242C23—3141no—
243C23—1572no—
244C23—12564no—
245C23D4×D15711yescharacteristic
246Q8—3142no—
247Q8—3141no—
248Q8—1572no—
249D4—24no—
250D4—22no—
251D4—22no—
252D4—22no—
253D4—42no—
254D4—3148no—
255D4—3148no—
256D4—3144no—
257D4—3144no—
258D4—3142no—
259D4—3141no—
260D4—1572no—
261D4—12564no—
262D4C2×C22⋊D15714yes—
263C2×C4—3148no—
264C2×C4—3141no—
265C2×C4—3144no—
266C2×C4—3142no—
267C2×C4—3142no—
268C2×C4—3142no—
269C2×C4—1572no—
270C2×C4D4×D15711yescharacteristic
271C2×C4D4×D15712yes—
272C2×C4C2×C22⋊D15712yes—
273C2×C4C23×D15711yescharacteristic
274C8—22no—
275C8—3142no—
276C22—22no—
277C22—24no—
278C22—44no—
279C22—42no—
280C22—44no—
281C22—84no—
282C22—84no—
283C22—3142no—
284C22—3142no—
285C22—3144no—
286C22—6282no—
287C22—12568no—
288C22#1909773912yes—
289C22C2×D4×D15711yescharacteristic
290C4—3142no—
291C4—3142no—
292C4—6282no—
293C4#1909773912yes—
294C4C2×D4×D15711yescharacteristic
295C4C22×C22⋊D15711yescharacteristic
296C2—22no—
297C2—44no—
298C2—82no—
299C2—6282no—
300C2—6282no—
301C2C2×#1909773911yescharacteristic
302C2#2375807412yes—
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
FittingC22×D8×C157C2
socleC22×C157C22×D4
hypercenterC22×D8D157
hypocenterC157#3008
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×D8

Complements

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

NH ≅ G/Ntype
1. C2×#19097816C2semidirect
2. C2×#19097797C2semidirect
3. #23757840C2semidirect
4. #23757415C2semidirect
5. #23757405C2semidirect
6. #23757278C2semidirect
7. #23757270C2semidirect
8. #23757239C2semidirect
9. C22×D8×C157C2semidirect
15. C2×#14578953C22semidirect
18. C2×D4×D157C22semidirect
20. C2×#14578942C22semidirect
21. C2×#14578939C22semidirect
22. C2×#14578938C22semidirect
25. #19097742C22semidirect
27. #19097730C22semidirect
28. #19097696C22semidirect
29. #19097693C22semidirect
30. #19097692C22semidirect
31. #19097683C22semidirect
33. C22×D4×C157C22semidirect
35. C2×D8×C157C22semidirect
36. C22×C8×C157C22semidirect
41. #14578954D4semidirect
76. C2×D4×C157D4semidirect
151. C157#3008semidirect
155. C22×D8D157semidirect
172. C22×D4C2×D157semidirect
177. C2×D8C2×D157semidirect
179. C22×C8C2×D157semidirect
217. C2×D4C22⋊D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190978162
C2×#190977972
#237578402
#237574152
#237574052
#237572782
#237572702
#237572392
C22×D8×C1572
#3008157