Finite Group Databasebeta

← #34109412

Subgroups — #34109412

Order 20096.

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

normal subgroups only

This group has 752 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×#19097817C211yesmaximal, characteristic
2C22×#14578941C211yesmaximal, characteristic, complemented
3#23757847C211yesmaximal, characteristic, complemented
4#23757434C214yesmaximal, complemented
5#23757422C214yesmaximal, complemented
6#23757279C211yesmaximal, characteristic, complemented
7#23757269C211yesmaximal, characteristic, complemented
8#23757241C211yesmaximal, characteristic, complemented
9C2×#137×C157C211yesmaximal, characteristic, complemented
10#19097817—24no—
11#19097817—24no—
12C2×#14578955C2212yes—
13C2×#14578954—21no—
14C2×#14578954C2211yescharacteristic
15C22×D628—21no—
16C22×D628C2211yescharacteristic
17C2×D4×D157—22no—
18C2×D4×D157C2212yes—
19C2×#14578941—24no—
20C2×#14578941C2214yescomplemented
21C22×#11929434C2211yescharacteristic, complemented
22#19097769—21no—
23#19097760—21no—
24#19097754—22no—
25#19097747C2214yescomplemented
26#19097730C2211yescharacteristic, complemented
27#19097695C2212yescomplemented
28#19097693C2214yescomplemented
29#19097692C2212yescomplemented
30#19097683C2214yescomplemented
31C2×#41×C157C2211yescharacteristic
32C22×Q8×C157C2211yescharacteristic, complemented
33#137×C157—24no—
34C2×Q16×C157C2212yescomplemented
35C2×SD16×C157C2212yescomplemented
36C2×M16×C157C2211yescharacteristic, complemented
37C23×D157—22no—
38C23×D157—21no—
39#14578955—24no—
40#14578955—24no—
41#14578954—28no—
42#14578954—22no—
43#14578954—22no—
44C2×Dic314D411yescharacteristic
45C2×D628—28no—
46C2×D628—22no—
47C2×D628—21no—
48C2×D628—22no—
49C2×D628D411yescharacteristic
50C2×D628D414yes—
51C2×D628C2312yes—
52C2×C22⋊D157—22no—
53C2×C22⋊D157D412yes—
54D4×D157—216no—
55D4×D157—24no—
56D4×D157—24no—
57C2×C4×D157—22no—
58C2×C4×D157C2312yes—
59#14578941—28no—
60C2×#11929434—22no—
61C2×#11929434C2312yes—
62C4⋊Dic157C2312yes—
63#14578933—24no—
64C22⋊Dic157—21no—
65#14578931—24no—
66C4×Dic157—21no—
67#41×C157—22no—
68#41×C157D414yes—
69C2×Q8×C157—21no—
70C2×Q8×C157—22no—
71C2×Q8×C157—22no—
72C2×Q8×C157C2312yes—
73C2×Q8×C157C2311yescharacteristic
74C2×D4×C157—21no—
75C2×D4×C157C2311yescharacteristic
76C22×C4×C157—21no—
77C22×C4×C157—21no—
78C22×C4×C157C2311yescharacteristic
79Q16×C157—24no—
80SD16×C157—24no—
81M16×C157—22no—
82C2×C8×C157C2312yes—
83C22×D157—22no—
84C22×D157—28no—
85C22×D157—24no—
86C22×D157—84no—
87C22×D157—82no—
88C22×D157C2×D412yes—
89Dic314—22no—
90D628—28no—
91D628—22no—
92D628—24no—
93D628—22no—
94D628C2×D414yes—
95C22⋊D157—28no—
96C22⋊D157—24no—
97C2×Dic157—22no—
98C2×Dic157C2×D412yes—
99C4×D157—28no—
100C4×D157—22no—
101C4×D157—22no—
102#11929434—22no—
103C23×C157—21no—
104C23×C157C2×D411yescharacteristic
105Q8×C157—21no—
106Q8×C157—22no—
107Q8×C157—22no—
108Q8×C157—42no—
109Q8×C157C2×D412yes—
110D4×C157—24no—
111D4×C157—21no—
112D4×C157C2×D412yes—
113C2×C4×C157—21no—
114C2×C4×C157—24no—
115C2×C4×C157—22no—
116C2×C4×C157—42no—
117C2×C4×C157C2×D411yescharacteristic
118C2×C4×C157C2×D412yes—
119C2×C4×C157C2×D412yes—
120C2×C4×C157C2411yescharacteristic
121C8×C157—22no—
122C22×C157—22no—
123C22×C157—22no—
124C22×C157—42no—
125C22×C157#12012yes—
126C22×C157C22×D411yescharacteristic
127C2×D157—22no—
128C2×D157—24no—
129C2×D157—24no—
130C2×D157—44no—
131C2×D157—84no—
132C2×D157—88no—
133C4×C157—22no—
134C4×C157—42no—
135C4×C157#12012yes—
136C4×C157C22×D411yescharacteristic
137C4×C157C22×D411yescharacteristic
138Dic157—22no—
139Dic157—42no—
140C2×C157—22no—
141C2×C157—42no—
142C2×C157C2×#12011yescharacteristic
143C2×C157#57712yes—
144D157—42no—
145D157—44no—
146C157#300911yescharacteristic, complemented, Sylow
147#3009—1571nomaximal, Sylow
148C2×H(5,𝔽2)—1571no—
149C2×#137D15711yescharacteristic, complemented
150C22×SD16—1571no—
151#536—1571no—
152#452—1574no—
153#450—1574no—
154#418—1571no—
155#416—1571no—
156#408—1571no—
157H(5,𝔽2)—3144no—
158H(5,𝔽2)—3144no—
159C2×#41—3141no—
160C2×#41—1572no—
161C2×#41—1571no—
162C2×#41C2×D15711yescharacteristic
163C22×Q8C2×D15711yescharacteristic, complemented
164C22×D4—3141no—
165C22×D4—3142no—
166C22×D4—1572no—
167C22×D4—1571no—
168#137—24no—
169C2×Q16C2×D15712yescomplemented
170C2×SD16—3144no—
171C2×SD16—1574no—
172C2×SD16C2×D15712yescomplemented
173C2×M16C2×D15711yescharacteristic, complemented
174C22×C8—1571no—
175C4⋊Q8—3141no—
176#124—3141no—
177#123—3142no—
178C22⋊Q8—1574no—
179#117—1571no—
180#103—1572no—
181#102—1572no—
182#102—1574no—
183C22⋊C8—1574no—
184C24—3142no—
185C24—3141no—
186#41—22no—
187#41—3148no—
188#41—3144no—
189#41—3144no—
190#41—3142no—
191#41—3142no—
192#41C22⋊D15714yes—
193C2×Q8—21no—
194C2×Q8—22no—
195C2×Q8—22no—
196C2×Q8—1571no—
197C2×Q8C22×D15712yes—
198C2×Q8C22×D15711yescharacteristic
199C2×D4—21no—
200C2×D4—3148no—
201C2×D4—3142no—
202C2×D4—31416no—
203C2×D4—3144no—
204C2×D4—3144no—
205C2×D4—3142no—
206C2×D4—3141no—
207C2×D4—3142no—
208C2×D4—1572no—
209C2×D4—1571no—
210C2×D4—1574no—
211C2×D4—1572no—
212C2×D4C22×D15711yescharacteristic
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—3142no—
216C22×C4—1572no—
217C22×C4C22×D15711yescharacteristic
218Q16—24no—
219SD16—24no—
220SD16—3148no—
221M16—22no—
222C2×C8—3142no—
223C2×C8—1572no—
224C2×C8C22×D15712yes—
225C4⋊C4—3144no—
226C4⋊C4—1572no—
227C22⋊C4—3144no—
228C22⋊C4—3141no—
229C42—3141no—
230C23—21no—
231C23—3148no—
232C23—3142no—
233C23—3144no—
234C23—1572no—
235C23—12562no—
236C23—12564no—
237C23D4×D15711yescharacteristic
238Q8—21no—
239Q8—22no—
240Q8—22no—
241Q8—42no—
242Q8—3142no—
243Q8C2×C22⋊D15712yes—
244D4—24no—
245D4—21no—
246D4—3148no—
247D4—3148no—
248D4—3144no—
249D4—3142no—
250D4—3144no—
251D4—3142no—
252D4—1574no—
253D4C2×C22⋊D15712yes—
254C2×C4—21no—
255C2×C4—24no—
256C2×C4—22no—
257C2×C4—42no—
258C2×C4—3148no—
259C2×C4—3142no—
260C2×C4—3142no—
261C2×C4—3142no—
262C2×C4—1572no—
263C2×C4D4×D15711yescharacteristic
264C2×C4D4×D15712yes—
265C2×C4C2×C22⋊D15712yes—
266C2×C4C23×D15711yescharacteristic
267C8—22no—
268C8—3142no—
269C22—22no—
270C22—22no—
271C22—42no—
272C22—3144no—
273C22—3142no—
274C22—3144no—
275C22—6284no—
276C22—12564no—
277C22—12568no—
278C22#1909773912yes—
279C22C2×D4×D15711yescharacteristic
280C4—22no—
281C4—42no—
282C4—3142no—
283C4—6282no—
284C4#1909773912yes—
285C4C2×D4×D15711yescharacteristic
286C4C22×C22⋊D15711yescharacteristic
287C2—22no—
288C2—42no—
289C2—6282no—
290C2—6284no—
291C2C2×#1909773911yescharacteristic
292C2#2375808212yes—
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
FittingC2×#137×C157C2
socleC22×C157C22×D4
prime radicalC2×#19097817C2
hypercenterC2×#137D157
hypocenterC157#3009
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
2. C22×#14578941C2semidirect
3. #23757847C2semidirect
4. #23757434C2semidirect
5. #23757422C2semidirect
6. #23757279C2semidirect
7. #23757269C2semidirect
8. #23757241C2semidirect
9. C2×#137×C157C2semidirect
20. C2×#14578941C22semidirect
21. C22×#11929434C22semidirect
25. #19097747C22semidirect
26. #19097730C22semidirect
27. #19097695C22semidirect
28. #19097693C22semidirect
29. #19097692C22semidirect
30. #19097683C22semidirect
32. C22×Q8×C157C22semidirect
34. C2×Q16×C157C22semidirect
35. C2×SD16×C157C22semidirect
36. C2×M16×C157C22semidirect
146. C157#3009semidirect
149. C2×#137D157semidirect
163. C22×Q8C2×D157semidirect
169. C2×Q16C2×D157semidirect
172. C2×SD16C2×D157semidirect
173. C2×M16C2×D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190978172
C22×#145789412
#237578472
#237574342
#237574222
#237572792
#237572692
#237572412
C2×#137×C1572
#3009157