Finite Group Databasebeta

← #34109413

Subgroups — #34109413

Order 20096.

752 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 752 conjugacy classes of subgroups — too many to draw. Only its 165 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
2C22×#14578942C211yesmaximal, characteristic, complemented
3#23757844C211yesmaximal, characteristic, complemented
4#23757435C214yesmaximal, complemented
5#23757424C214yesmaximal, complemented
6#23757278C211yesmaximal, characteristic, complemented
7#23757269C211yesmaximal, characteristic, complemented
8#23757240C211yesmaximal, characteristic, complemented
9C2×H(5,𝔽2)×C157C211yesmaximal, characteristic
10C22×Dic314C2211yescharacteristic, complemented
11#19097801—24no—
12C2×#14578944C2212yescomplemented
13C2×#14578942—24no—
14C2×#14578942C2212yescomplemented
15C2×#14578942C2214yescomplemented
16C2×#14578938C2211yescharacteristic, complemented
17C22×#11929434C2211yescharacteristic, complemented
18#19097769—21no—
19#19097758—21no—
20#19097752—22no—
21#19097750C2214yescomplemented
22#19097730C2211yescharacteristic, complemented
23#19097695C2212yescomplemented
24#19097692C2212yescomplemented
25#19097692C2214yescomplemented
26C22⋊#11929434C2214yescomplemented
27H(5,𝔽2)×C157—24no—
28H(5,𝔽2)×C157—24no—
29C2×#41×C157—21no—
30C2×#41×C157C2212yes—
31C2×#41×C157C2211yescharacteristic
32C2×#41×C157C2211yescharacteristic
33C22×D4×C157—22no—
34C22×D4×C157—21no—
35C22×D4×C157C2212yes—
36C22×D4×C157C2211yescharacteristic
37C2×Dic314—21no—
38C2×Dic314—22no—
39C2×Dic314—22no—
40C2×Dic314C2312yes—
41C22×Dic157—21no—
42#14578944—24no—
43#14578942—28no—
44#14578942—24no—
45#14578938—22no—
46C2×#11929434—22no—
47C2×#11929434C2312yes—
48C2×#11929434C2312yes—
49C4⋊Dic157C2312yes—
50#14578933—24no—
51C22⋊Dic157—21no—
52C22⋊Dic157—24no—
53C4×Dic157—21no—
54C24×C157—22no—
55C24×C157—21no—
56#41×C157—28no—
57#41×C157—24no—
58#41×C157—24no—
59#41×C157—22no—
60#41×C157—22no—
61#41×C157—22no—
62#41×C157D414yes—
63C2×Q8×C157D411yescharacteristic
64C2×Q8×C157C2311yescharacteristic
65C2×D4×C157—216no—
66C2×D4×C157—28no—
67C2×D4×C157—22no—
68C2×D4×C157—21no—
69C2×D4×C157—24no—
70C2×D4×C157—24no—
71C2×D4×C157—22no—
72C2×D4×C157—21no—
73C2×D4×C157—22no—
74C2×D4×C157D412yes—
75C2×D4×C157D411yescharacteristic
76C2×D4×C157D414yes—
77C2×D4×C157C2311yescharacteristic
78C2×D4×C157C2312yes—
79C22×C4×C157—22no—
80C22×C4×C157—21no—
81C22×C4×C157C2312yes—
82C22×C4×C157C2311yescharacteristic
83Dic314—22no—
84Dic314—22no—
85Dic314—42no—
86C2×Dic157—22no—
87C2×Dic157—22no—
88C2×Dic157—42no—
89#11929434—22no—
90#11929434—22no—
91C23×C157—28no—
92C23×C157—22no—
93C23×C157—21no—
94C23×C157—24no—
95C23×C157—82no—
96C23×C157—84no—
97C23×C157C2×D412yes—
98C23×C157C2×D411yescharacteristic
99Q8×C157—22no—
100Q8×C157—21no—
101Q8×C157C2×D412yes—
102D4×C157—28no—
103D4×C157—28no—
104D4×C157—24no—
105D4×C157—24no—
106D4×C157—22no—
107D4×C157—24no—
108D4×C157—22no—
109D4×C157—21no—
110D4×C157C2×D412yes—
111D4×C157C2×D414yes—
112C2×C4×C157—28no—
113C2×C4×C157—21no—
114C2×C4×C157—24no—
115C2×C4×C157—22no—
116C2×C4×C157—22no—
117C2×C4×C157C2×D412yes—
118C2×C4×C157C2×D411yescharacteristic
119C2×C4×C157C2×D412yes—
120C2×C4×C157C2×D412yes—
121C2×C4×C157C2411yescharacteristic
122C22×C157—22no—
123C22×C157—22no—
124C22×C157—24no—
125C22×C157—22no—
126C22×C157—24no—
127C22×C157—44no—
128C22×C157—42no—
129C22×C157—84no—
130C22×C157—88no—
131C22×C157#12012yes—
132C22×C157C22×D411yescharacteristic
133C4×C157—22no—
134C4×C157—22no—
135C4×C157#12012yes—
136C4×C157C22×D411yescharacteristic
137C4×C157C22×D411yescharacteristic
138Dic157—42no—
139Dic157—42no—
140C2×C157—22no—
141C2×C157—42no—
142C2×C157—44no—
143C2×C157—42no—
144C2×C157C2×#12011yescharacteristic
145C2×C157#57712yes—
146C157#300911yescharacteristic, complemented, Sylow
147#3009—1571nomaximal, Sylow
148C2×H(5,𝔽2)D15711yescharacteristic
149C2×#137—1571no—
150C22×SD16—1571no—
151#536—1571no—
152#452—1574no—
153#450—1574no—
154#418—1571no—
155#416—1571no—
156#408—1571no—
157H(5,𝔽2)—24no—
158H(5,𝔽2)—24no—
159C2×#41—21no—
160C2×#41C2×D15712yes—
161C2×#41C2×D15711yescharacteristic
162C2×#41C2×D15711yescharacteristic
163C22×Q8—1571no—
164C22×D4—22no—
165C22×D4—21no—
166C22×D4C2×D15712yes—
167C22×D4C2×D15711yescharacteristic
168#137—3144no—
169C2×Q16—1572no—
170C2×SD16—3144no—
171C2×SD16—1574no—
172C2×SD16—1572no—
173C2×M16—1571no—
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—22no—
185C24—21no—
186#41—28no—
187#41—24no—
188#41—24no—
189#41—22no—
190#41—22no—
191#41—22no—
192#41C22⋊D15714yes—
193C2×Q8—3141no—
194C2×Q8—3142no—
195C2×Q8—3142no—
196C2×Q8—1572no—
197C2×Q8C22⋊D15711yescharacteristic
198C2×Q8C22×D15711yescharacteristic
199C2×D4—216no—
200C2×D4—28no—
201C2×D4—22no—
202C2×D4—21no—
203C2×D4—24no—
204C2×D4—24no—
205C2×D4—22no—
206C2×D4—21no—
207C2×D4—22no—
208C2×D4C22⋊D15712yes—
209C2×D4C22⋊D15711yescharacteristic
210C2×D4C22⋊D15714yes—
211C2×D4C22×D15711yescharacteristic
212C2×D4C22×D15712yes—
213C22×C4—22no—
214C22×C4—21no—
215C22×C4—3141no—
216C22×C4C22×D15712yes—
217C22×C4C22×D15711yescharacteristic
218Q16—3144no—
219SD16—3148no—
220SD16—3144no—
221M16—3142no—
222C2×C8—3142no—
223C2×C8—1572no—
224C2×C8—1572no—
225C4⋊C4—3144no—
226C4⋊C4—1572no—
227C22⋊C4—3141no—
228C22⋊C4—3144no—
229C42—3141no—
230C23—28no—
231C23—22no—
232C23—21no—
233C23—24no—
234C23—82no—
235C23—84no—
236C23C2×C22⋊D15712yes—
237C23C2×C22⋊D15711yescharacteristic
238Q8—22no—
239Q8—21no—
240Q8—3142no—
241Q8—3142no—
242Q8—6282no—
243Q8C2×C22⋊D15712yes—
244D4—28no—
245D4—28no—
246D4—24no—
247D4—24no—
248D4—22no—
249D4—24no—
250D4—22no—
251D4—21no—
252D4C2×C22⋊D15712yes—
253D4C2×C22⋊D15714yes—
254C2×C4—28no—
255C2×C4—21no—
256C2×C4—24no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—3142no—
260C2×C4—3142no—
261C2×C4—6282no—
262C2×C4C2×C22⋊D15712yes—
263C2×C4C2×C22⋊D15711yescharacteristic
264C2×C4C2×C22⋊D15712yes—
265C2×C4C2×C22⋊D15712yes—
266C2×C4C23×D15711yescharacteristic
267C8—3142no—
268C8—3142no—
269C22—22no—
270C22—24no—
271C22—22no—
272C22—24no—
273C22—22no—
274C22—44no—
275C22—42no—
276C22—84no—
277C22—88no—
278C22#1909774012yes—
279C22C22×C22⋊D15711yescharacteristic
280C4—22no—
281C4—22no—
282C4—6282no—
283C4—6282no—
284C4#1909774012yes—
285C4C22×C22⋊D15711yescharacteristic
286C4C22×C22⋊D15711yescharacteristic
287C2—22no—
288C2—42no—
289C2—44no—
290C2—42no—
291C2C2×#1909774011yescharacteristic
292C2#2375808412yes—
subgroupG/H
centerC22C22×C22⋊D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting, prime radicalC2×H(5,𝔽2)×C157C2
socleC22×C157C22×D4
hypercenterC2×H(5,𝔽2)D157
hypocenterC157#3009
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C2×#41 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. C2×#19097801C2semidirect
2. C22×#14578942C2semidirect
3. #23757844C2semidirect
4. #23757435C2semidirect
5. #23757424C2semidirect
6. #23757278C2semidirect
7. #23757269C2semidirect
8. #23757240C2semidirect
10. C22×Dic314C22semidirect
12. C2×#14578944C22semidirect
14. C2×#14578942C22semidirect
15. C2×#14578942C22semidirect
16. C2×#14578938C22semidirect
17. C22×#11929434C22semidirect
21. #19097750C22semidirect
22. #19097730C22semidirect
23. #19097695C22semidirect
24. #19097692C22semidirect
25. #19097692C22semidirect
26. C22⋊#11929434C22semidirect
146. C157#3009semidirect

Maximal subgroups

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

subgroupindex
C2×#190978012
C22×#145789422
#237578442
#237574352
#237574242
#237572782
#237572692
#237572402
C2×H(5,𝔽2)×C1572
#3009157