Finite Group Databasebeta

← #34109361

Subgroups — #34109361

Order 20096.

994 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 994 conjugacy classes of subgroups — too many to draw. Only its 153 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
1C22×D4×D157C211yesmaximal, characteristic, complemented
2C2×#19097796C211yesmaximal, characteristic, complemented
3#23757780C211yesmaximal, characteristic, complemented
4#23757422C214yesmaximal, complemented
5#23757405C214yesmaximal, complemented
6#23757279C211yesmaximal, characteristic, complemented
7#23757278C211yesmaximal, characteristic, complemented
8#23757236C211yesmaximal, characteristic, complemented
9C2×AGL(1,ℤ/8)×C157C211yesmaximal, characteristic, complemented
10C24×D157—21no—
11C22×D628C2211yescharacteristic
12C22×C22⋊D157—21no—
13C2×D4×D157—24no—
14C2×D4×D157—24no—
15C2×D4×D157—22no—
16C2×D4×D157C2214yes—
17C22×C4×D157C2211yescharacteristic
18#19097796—24no—
19C2×#14578941C2212yescomplemented
20C2×#14578939C2212yescomplemented
21C2×#14578938C2211yescharacteristic, complemented
22#19097752—21no—
23#19097747C2212yescomplemented
24#19097742C2212yescomplemented
25#19097739—21no—
26C4×C22⋊D157—22no—
27#19097730C2211yescharacteristic, complemented
28#19097693C2214yescomplemented
29#19097692C2214yescomplemented
30#19097683C2214yescomplemented
31C2×#41×C157C2211yescharacteristic, complemented
32C22×D4×C157C2211yescharacteristic
33AGL(1,ℤ/8)×C157—24no—
34C2×SD16×C157C2212yescomplemented
35C2×D8×C157C2212yescomplemented
36C2×M16×C157C2211yescharacteristic, complemented
37C23×D157—28no—
38C23×D157—24no—
39C23×D157—21no—
40C23×D157—48no—
41C23×D157D411yescharacteristic
42C2×D628—21no—
43C2×D628—22no—
44C2×D628D414yescomplemented
45C2×D628C2312yes—
46C2×C22⋊D157—28no—
47C2×C22⋊D157—22no—
48C2×C22⋊D157—24no—
49D4×D157—216no—
50D4×D157—28no—
51D4×D157—28no—
52C22×Dic157D411yescharacteristic
53C2×C4×D157—22no—
54C2×C4×D157—22no—
55C2×C4×D157D412yes—
56C2×C4×D157C2312yes—
57#14578941—24no—
58#14578939—24no—
59#14578938—22no—
60C2×#11929434C2312yescomplemented
61C4⋊Dic157C2312yescomplemented
62#14578933—22no—
63C22⋊Dic157—22no—
64C22⋊Dic157—21no—
65#14578931—22no—
66C4×Dic157—21no—
67C24×C157—21no—
68#41×C157—22no—
69#41×C157—22no—
70C2×Q8×C157C2311yescharacteristic, complemented
71C2×D4×C157—21no—
72C2×D4×C157—21no—
73C2×D4×C157—22no—
74C2×D4×C157D414yescomplemented
75C2×D4×C157C2312yes—
76C2×D4×C157C2311yescharacteristic, complemented
77C22×C4×C157—21no—
78C22×C4×C157C2311yescharacteristic
79SD16×C157—24no—
80D8×C157—24no—
81M16×C157—22no—
82C2×C8×C157C2312yescomplemented
83C22×D157—22no—
84C22×D157—24no—
85C22×D157—216no—
86C22×D157—24no—
87C22×D157—22no—
88C22×D157—432no—
89C22×D157—48no—
90C22×D157—44no—
91C22×D157—48no—
92C22×D157C2×D412yes—
93D628—24no—
94D628—22no—
95D628C2×D414yes—
96C22⋊D157—216no—
97C22⋊D157—44no—
98C2×Dic157—22no—
99C2×Dic157—22no—
100C2×Dic157C2×D412yes—
101C4×D157—24no—
102C4×D157—22no—
103C4×D157—22no—
104#11929434—22no—
105C23×C157—24no—
106C23×C157—22no—
107C23×C157—21no—
108C23×C157—44no—
109C23×C157C2×D411yescharacteristic
110Q8×C157—21no—
111Q8×C157—21no—
112D4×C157—24no—
113D4×C157—22no—
114D4×C157—21no—
115D4×C157—21no—
116D4×C157—42no—
117D4×C157C2×D414yes—
118C2×C4×C157—21no—
119C2×C4×C157—42no—
120C2×C4×C157C2×D411yescharacteristic
121C2×C4×C157C2×D412yes—
122C2×C4×C157C2×D412yes—
123C2×C4×C157C2411yescharacteristic
124C8×C157—22no—
125C22×C157—22no—
126C22×C157—24no—
127C22×C157—48no—
128C22×C157—44no—
129C22×C157—41no—
130C22×C157—42no—
131C22×C157#12012yes—
132C22×C157C22×D411yescharacteristic
133C2×D157—28no—
134C2×D157—22no—
135C2×D157—22no—
136C2×D157—24no—
137C2×D157—432no—
138C2×D157—48no—
139C2×D157—44no—
140C2×D157AGL(1,ℤ/8)14yescomplemented
141C4×C157—41no—
142C4×C157#12012yes—
143C4×C157C22×D411yescharacteristic
144C4×C157C22×D411yescharacteristic
145Dic157—22no—
146Dic157—42no—
147C2×C157—22no—
148C2×C157—44no—
149C2×C157—81no—
150C2×C157C2×#12011yescharacteristic
151C2×C157C2×AGL(1,ℤ/8)12yes—
152D157—24no—
153D157—44no—
154C157#300311yescharacteristic, complemented, Sylow
155#3003—1571nomaximal, Sylow
156C23×D4—1571no—
157C2×AGL(1,ℤ/8)—1571no—
158C2×AGL(1,ℤ/8)D15711yescharacteristic, complemented
159#525—1571no—
160#450—1574no—
161#447—1574no—
162#418—1571no—
163#418—1571no—
164#407—1571no—
165C25—3141no—
166C2×#41C2×D15711yescharacteristic, complemented
167C22×D4—3144no—
168C22×D4—3144no—
169C22×D4—3142no—
170C22×D4—3141no—
171C22×D4—1574no—
172C22×D4—1571no—
173C22×D4C2×D15711yescharacteristic
174C23×C4—1571no—
175AGL(1,ℤ/8)—24no—
176AGL(1,ℤ/8)—3144no—
177C2×SD16—1572no—
178C2×SD16C2×D15712yescomplemented
179C2×D8—1572no—
180C2×D8C2×D15712yescomplemented
181C2×M16—1571no—
182C2×M16C2×D15711yescharacteristic, complemented
183#123—3141no—
184C22⋊Q8—1572no—
185#121—1572no—
186#120—3141no—
187C4×D4—3142no—
188#117—1571no—
189#102—1574no—
190#102—1574no—
191C22⋊C8—1574no—
192C24—21no—
193C24—3148no—
194C24—3144no—
195C24—3141no—
196C24—6288no—
197C24—1571no—
198#41—22no—
199#41—22no—
200C2×Q8C22×D15711yescharacteristic, complemented
201C2×D4—21no—
202C2×D4—21no—
203C2×D4—22no—
204C2×D4—31416no—
205C2×D4—3148no—
206C2×D4—3148no—
207C2×D4—3148no—
208C2×D4—3142no—
209C2×D4—3144no—
210C2×D4—3141no—
211C2×D4—3142no—
212C2×D4—1574no—
213C2×D4—1572no—
214C2×D4C22⋊D15714yescomplemented
215C2×D4C22×D15712yes—
216C2×D4C22×D15711yescharacteristic, complemented
217C22×C4—21no—
218C22×C4—3142no—
219C22×C4—3142no—
220C22×C4—1571no—
221C22×C4—1572no—
222C22×C4—1572no—
223C22×C4C22×D15711yescharacteristic
224SD16—24no—
225SD16—3144no—
226D8—24no—
227D8—3144no—
228M16—22no—
229M16—3142no—
230C2×C8—1572no—
231C2×C8C22×D15712yescomplemented
232C4⋊C4—3142no—
233C4⋊C4—1572no—
234C22⋊C4—3142no—
235C22⋊C4—3141no—
236C22⋊C4—3142no—
237C42—3141no—
238C23—24no—
239C23—22no—
240C23—21no—
241C23—44no—
242C23—3144no—
243C23—31416no—
244C23—3142no—
245C23—3144no—
246C23—3142no—
247C23—6284no—
248C23—62832no—
249C23—6288no—
250C23—6288no—
251C23—1572no—
252C23D4×D15711yescharacteristic
253Q8—21no—
254Q8—21no—
255D4—24no—
256D4—22no—
257D4—21no—
258D4—21no—
259D4—42no—
260D4—31416no—
261D4—3144no—
262D4—3142no—
263D4—6284no—
264D4—1574no—
265D4C2×C22⋊D15714yes—
266C2×C4—21no—
267C2×C4—42no—
268C2×C4—3142no—
269C2×C4—3144no—
270C2×C4—3142no—
271C2×C4—3142no—
272C2×C4—3142no—
273C2×C4—1572no—
274C2×C4D4×D15711yescharacteristic
275C2×C4D4×D15712yes—
276C2×C4C2×C22⋊D15712yes—
277C2×C4C23×D15711yescharacteristic
278C8—22no—
279C8—3142no—
280C22—22no—
281C22—24no—
282C22—48no—
283C22—44no—
284C22—41no—
285C22—42no—
286C22—3148no—
287C22—3142no—
288C22—3142no—
289C22—3144no—
290C22—6288no—
291C22—62832no—
292C22—6284no—
293C22—1574no—
294C22#1909773912yes—
295C22C2×D4×D15711yescharacteristic
296C4—41no—
297C4—3142no—
298C4—6282no—
299C4#1909773912yes—
300C4C2×D4×D15711yescharacteristic
301C4C22×C22⋊D15711yescharacteristic
302C2—22no—
303C2—44no—
304C2—81no—
305C2—3144no—
306C2—6284no—
307C2C2×#1909773911yescharacteristic
308C2AGL(1,ℤ/8)×D15712yes—
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#3003
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. C22×D4×D157C2semidirect
2. C2×#19097796C2semidirect
3. #23757780C2semidirect
4. #23757422C2semidirect
5. #23757405C2semidirect
6. #23757279C2semidirect
7. #23757278C2semidirect
8. #23757236C2semidirect
9. C2×AGL(1,ℤ/8)×C157C2semidirect
19. C2×#14578941C22semidirect
20. C2×#14578939C22semidirect
21. C2×#14578938C22semidirect
23. #19097747C22semidirect
24. #19097742C22semidirect
27. #19097730C22semidirect
28. #19097693C22semidirect
29. #19097692C22semidirect
30. #19097683C22semidirect
31. C2×#41×C157C22semidirect
34. C2×SD16×C157C22semidirect
35. C2×D8×C157C22semidirect
36. C2×M16×C157C22semidirect
44. C2×D628D4semidirect
60. C2×#11929434C23semidirect
61. C4⋊Dic157C23semidirect
70. C2×Q8×C157C23semidirect
74. C2×D4×C157D4semidirect
76. C2×D4×C157C23semidirect
82. C2×C8×C157C23semidirect
140. C2×D157AGL(1,ℤ/8)semidirect
154. C157#3003semidirect
158. C2×AGL(1,ℤ/8)D157semidirect
166. C2×#41C2×D157semidirect
178. C2×SD16C2×D157semidirect
180. C2×D8C2×D157semidirect
182. C2×M16C2×D157semidirect
200. C2×Q8C22×D157semidirect
214. C2×D4C22⋊D157semidirect
216. C2×D4C22×D157semidirect
231. C2×C8C22×D157semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D1572
C2×#190977962
#237577802
#237574222
#237574052
#237572792
#237572782
#237572362
C2×AGL(1,ℤ/8)×C1572
#3003157