Finite Group Databasebeta

← #34109375

Subgroups — #34109375

Order 20096.

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

normal subgroups only

This group has 758 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
1C22×#14578953C211yesmaximal, characteristic, complemented
2C2×#19097801C211yesmaximal, characteristic, complemented
3#23757780C211yesmaximal, characteristic, complemented
4#23757432C214yesmaximal, complemented
5#23757415C214yesmaximal, complemented
6#23757283C211yesmaximal, characteristic, complemented
7#23757278C211yesmaximal, characteristic, complemented
8#23757236C211yesmaximal, characteristic, complemented
9C2×AGL(1,ℤ/8)×C157C211yesmaximal, characteristic, complemented
10C2×#14578953—24no—
11C2×#14578953—24no—
12C2×#14578953—22no—
13C2×#14578953C2214yes—
14C22×Dic314C2211yescharacteristic
15C22×C22⋊D157—21no—
16C23×Dic157—21no—
17C22×C4×D157C2211yescharacteristic
18#19097801—24no—
19C2×#14578944C2212yescomplemented
20C2×#14578942C2212yescomplemented
21C2×#14578938C2211yescharacteristic
22#19097752—21no—
23#19097747C2212yescomplemented
24#19097742C2212yescomplemented
25#19097739—21no—
26C4×C22⋊D157—22no—
27#19097730C2211yescharacteristic
28#19097696C2214yescomplemented
29#19097692C2214yescomplemented
30#19097683C2214yescomplemented
31C2×#41×C157C2211yescharacteristic
32C22×D4×C157C2211yescharacteristic
33AGL(1,ℤ/8)×C157—24no—
34C2×SD16×C157C2212yescomplemented
35C2×D8×C157C2212yescomplemented
36C2×M16×C157C2211yescharacteristic
37C23×D157D411yescharacteristic
38#14578953—216no—
39#14578953—28no—
40#14578953—28no—
41C2×Dic314—21no—
42C2×Dic314—22no—
43C2×Dic314D414yescomplemented
44C2×Dic314C2312yes—
45C2×C22⋊D157—28no—
46C2×C22⋊D157—24no—
47C2×C22⋊D157—22no—
48C22×Dic157—28no—
49C22×Dic157—24no—
50C22×Dic157—21no—
51C22×Dic157D411yescharacteristic
52C2×C4×D157—22no—
53C2×C4×D157—22no—
54C2×C4×D157D412yes—
55C2×C4×D157C2312yes—
56#14578944—24no—
57#14578942—24no—
58#14578938—22no—
59C2×#11929434C2312yes—
60C4⋊Dic157C2312yes—
61#14578933—22no—
62C22⋊Dic157—22no—
63C22⋊Dic157—21no—
64#14578931—22no—
65C4×Dic157—21no—
66C24×C157—21no—
67#41×C157—22no—
68#41×C157—22no—
69C2×Q8×C157C2311yescharacteristic
70C2×D4×C157—21no—
71C2×D4×C157—21no—
72C2×D4×C157—22no—
73C2×D4×C157D414yescomplemented
74C2×D4×C157C2311yescharacteristic
75C2×D4×C157C2312yes—
76C22×C4×C157—21no—
77C22×C4×C157C2311yescharacteristic
78SD16×C157—24no—
79D8×C157—24no—
80M16×C157—22no—
81C2×C8×C157C2312yes—
82C22×D157—22no—
83C22×D157—42no—
84C22×D157C2×D412yes—
85Dic314—24no—
86Dic314—22no—
87Dic314C2×D414yes—
88C22⋊D157—216no—
89C22⋊D157—44no—
90C2×Dic157—216no—
91C2×Dic157—22no—
92C2×Dic157—24no—
93C2×Dic157—22no—
94C2×Dic157—22no—
95C2×Dic157C2×D412yes—
96C4×D157—24no—
97C4×D157—22no—
98C4×D157—22no—
99#11929434—22no—
100C23×C157—24no—
101C23×C157—22no—
102C23×C157—21no—
103C23×C157—44no—
104C23×C157C2×D411yescharacteristic
105Q8×C157—21no—
106Q8×C157—21no—
107D4×C157—24no—
108D4×C157—22no—
109D4×C157—21no—
110D4×C157—21no—
111D4×C157—42no—
112D4×C157C2×D414yes—
113C2×C4×C157—21no—
114C2×C4×C157—42no—
115C2×C4×C157C2×D411yescharacteristic
116C2×C4×C157C2×D412yes—
117C2×C4×C157C2×D412yes—
118C2×C4×C157C2411yescharacteristic
119C8×C157—22no—
120C22×C157—22no—
121C22×C157—24no—
122C22×C157—48no—
123C22×C157—44no—
124C22×C157—41no—
125C22×C157—42no—
126C22×C157#12012yes—
127C22×C157C22×D411yescharacteristic
128C2×D157—22no—
129C2×D157—22no—
130C2×D157—22no—
131C2×D157—44no—
132C4×C157—41no—
133C4×C157#12012yes—
134C4×C157C22×D411yescharacteristic
135C4×C157C22×D411yescharacteristic
136Dic157—22no—
137Dic157—24no—
138Dic157—42no—
139C2×C157—22no—
140C2×C157—44no—
141C2×C157—81no—
142C2×C157C2×#12011yescharacteristic
143C2×C157#57512yes—
144D157—42no—
145C157#300511yescharacteristic, complemented, Sylow
146#3005—1571nomaximal, Sylow
147C22×#41—1571no—
148C2×#137—1571no—
149C2×AGL(1,ℤ/8)D15711yescharacteristic, complemented
150#525—1571no—
151#452—1574no—
152#449—1574no—
153#419—1571no—
154#418—1571no—
155#407—1571no—
156C2×#41—3144no—
157C2×#41—3144no—
158C2×#41—3142no—
159C2×#41—1574no—
160C2×#41C2×D15711yescharacteristic
161C22×Q8—1571no—
162C22×D4—3141no—
163C22×D4C2×D15711yescharacteristic
164C23×C4—3141no—
165C23×C4—1571no—
166#137—3144no—
167AGL(1,ℤ/8)—24no—
168C2×Q16—1572no—
169C2×SD16—1572no—
170C2×SD16C2×D15712yescomplemented
171C2×D8C2×D15712yescomplemented
172C2×M16—1571no—
173C2×M16C2×D15711yescharacteristic
174#123—3141no—
175C22⋊Q8—1572no—
176#121—1572no—
177#120—3141no—
178C4×D4—3142no—
179#117—1571no—
180#103—1574no—
181#102—1574no—
182C22⋊C8—1574no—
183C24—21no—
184C24—1571no—
185#41—22no—
186#41—22no—
187#41—31416no—
188#41—3148no—
189#41—3148no—
190C2×Q8—3141no—
191C2×Q8—3142no—
192C2×Q8—1574no—
193C2×Q8—1572no—
194C2×Q8C22×D15711yescharacteristic
195C2×D4—21no—
196C2×D4—21no—
197C2×D4—22no—
198C2×D4—3148no—
199C2×D4—3144no—
200C2×D4—3142no—
201C2×D4C22⋊D15714yescomplemented
202C2×D4C22×D15711yescharacteristic
203C2×D4C22×D15712yes—
204C22×C4—21no—
205C22×C4—3148no—
206C22×C4—3141no—
207C22×C4—3144no—
208C22×C4—3142no—
209C22×C4—3142no—
210C22×C4—1571no—
211C22×C4—1572no—
212C22×C4—1572no—
213C22×C4C22×D15711yescharacteristic
214Q16—3144no—
215SD16—24no—
216SD16—3144no—
217D8—24no—
218M16—22no—
219M16—3142no—
220C2×C8—1572no—
221C2×C8C22×D15712yes—
222C4⋊C4—3142no—
223C4⋊C4—1572no—
224C22⋊C4—3142no—
225C22⋊C4—3141no—
226C22⋊C4—3142no—
227C42—3141no—
228C23—24no—
229C23—22no—
230C23—21no—
231C23—44no—
232C23—3142no—
233C23—6282no—
234C23—1572no—
235C23D4×D15711yescharacteristic
236Q8—21no—
237Q8—21no—
238Q8—3144no—
239Q8—3142no—
240Q8—1574no—
241D4—24no—
242D4—22no—
243D4—21no—
244D4—21no—
245D4—42no—
246D4—31416no—
247D4—6284no—
248D4C2×C22⋊D15714yes—
249C2×C4—21no—
250C2×C4—42no—
251C2×C4—31416no—
252C2×C4—3142no—
253C2×C4—3144no—
254C2×C4—3142no—
255C2×C4—3144no—
256C2×C4—3142no—
257C2×C4—3142no—
258C2×C4—3142no—
259C2×C4—1572no—
260C2×C4D4×D15711yescharacteristic
261C2×C4D4×D15712yes—
262C2×C4C2×C22⋊D15712yes—
263C2×C4C23×D15711yescharacteristic
264C8—22no—
265C8—3142no—
266C22—22no—
267C22—24no—
268C22—48no—
269C22—44no—
270C22—41no—
271C22—42no—
272C22—3142no—
273C22—3142no—
274C22—3142no—
275C22—6284no—
276C22#1909773912yes—
277C22C2×D4×D15711yescharacteristic
278C4—41no—
279C4—3142no—
280C4—3144no—
281C4—6282no—
282C4#1909773912yes—
283C4C2×D4×D15711yescharacteristic
284C4C22×C22⋊D15711yescharacteristic
285C2—22no—
286C2—44no—
287C2—81no—
288C2—6282no—
289C2C2×#1909773911yescharacteristic
290C2#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#3005
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×#14578953C2semidirect
2. C2×#19097801C2semidirect
3. #23757780C2semidirect
4. #23757432C2semidirect
5. #23757415C2semidirect
6. #23757283C2semidirect
7. #23757278C2semidirect
8. #23757236C2semidirect
9. C2×AGL(1,ℤ/8)×C157C2semidirect
19. C2×#14578944C22semidirect
20. C2×#14578942C22semidirect
23. #19097747C22semidirect
24. #19097742C22semidirect
28. #19097696C22semidirect
29. #19097692C22semidirect
30. #19097683C22semidirect
34. C2×SD16×C157C22semidirect
35. C2×D8×C157C22semidirect
43. C2×Dic314D4semidirect
73. C2×D4×C157D4semidirect
145. C157#3005semidirect
149. C2×AGL(1,ℤ/8)D157semidirect
170. C2×SD16C2×D157semidirect
171. C2×D8C2×D157semidirect
201. C2×D4C22⋊D157semidirect

Maximal subgroups

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

subgroupindex
C22×#145789532
C2×#190978012
#237577802
#237574322
#237574152
#237572832
#237572782
#237572362
C2×AGL(1,ℤ/8)×C1572
#3005157