Finite Group Databasebeta

← #34109485

Subgroups — #34109485

Order 20096.

724 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 724 conjugacy classes of subgroups — too many to draw. Only its 161 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
1#23758071C212yesmaximal, complemented
2#23757819C211yesmaximal, characteristic, complemented
3#23757456C212yesmaximal, complemented
4#23757451C212yesmaximal, complemented
5#23757446C212yesmaximal, complemented
6#23757441C212yesmaximal, complemented
7C2×#19097698C212yesmaximal, complemented
8#23757257C211yesmaximal, characteristic, complemented
9#585×C157C211yesmaximal, characteristic, complemented
10C2×#14578954C2211yescharacteristic, complemented
11#19097802—22no—
12#19097801C2214yescomplemented
13#19097797—22no—
14#19097796C2214yescomplemented
15#19097791—24no—
16C2×#14578938C2212yescomplemented
17#19097769C2211yescharacteristic, complemented
18#19097767C2211yescharacteristic, complemented
19#19097760C2211yescharacteristic, complemented
20#19097758C2211yescharacteristic, complemented
21#19097743—22no—
22C4×C22⋊D157—22no—
23C2×C4×Dic157C2211yescharacteristic, complemented
24#19097698C2214yescomplemented
25#19097698C2214yescomplemented
26#19097690C2212yescomplemented
27#19097688C2212yescomplemented
282−1+4×C157—22no—
292−1+4×C157C2212yes—
30H(5,𝔽2)×C157—22no—
31H(5,𝔽2)×C157—22no—
32H(5,𝔽2)×C157C2212yes—
33C2×#41×C157—24no—
34C2×#41×C157—22no—
35C2×#41×C157C2212yes—
36C2×#41×C157C2211yescharacteristic
37#14578954—22no—
38#14578954—22no—
39C2×Dic314C2311yescharacteristic
40C2×D628C2311yescharacteristic
41C2×C22⋊D157—22no—
42C2×C22⋊D157—21no—
43C22×Dic157—21no—
44C2×C4×D157—21no—
45#14578944—24no—
46#14578942—24no—
47#14578941—24no—
48#14578939—24no—
49#14578938—22no—
50#14578938C2314yes—
51C2×#11929434—22no—
52#14578933—22no—
53C22⋊Dic157—22no—
54#14578931—22no—
55C4×Dic157—21no—
56C4×Dic157C2311yescharacteristic
57C4×Dic157C2311yescharacteristic
58#41×C157—24no—
59#41×C157—28no—
60#41×C157—24no—
61#41×C157—24no—
62#41×C157—24no—
63#41×C157—22no—
64#41×C157—22no—
65#41×C157—24no—
66#41×C157—22no—
67#41×C157—22no—
68#41×C157D414yescomplemented
69#41×C157C2314yes—
70C2×Q8×C157—22no—
71C2×Q8×C157—24no—
72C2×Q8×C157D412yescomplemented
73C2×Q8×C157C2311yescharacteristic
74C2×D4×C157—28no—
75C2×D4×C157—22no—
76C2×D4×C157—24no—
77C2×D4×C157—22no—
78C2×D4×C157—24no—
79C2×D4×C157D412yes—
80C2×D4×C157D412yescomplemented
81C2×D4×C157C2311yescharacteristic
82C22×C4×C157—24no—
83C22×C4×C157—22no—
84C22×C4×C157D412yes—
85C22×C4×C157C2311yescharacteristic
86C22×D157—21no—
87Dic314—22no—
88D628—22no—
89C22⋊D157—44no—
90C22⋊D157—42no—
91C2×Dic157—21no—
92C2×Dic157—21no—
93C2×Dic157—21no—
94C2×Dic157—42no—
95C4×D157—41no—
96C4×D157—41no—
97#11929434—24no—
98C23×C157—24no—
99C23×C157—22no—
100C23×C157C2×D412yes—
101C23×C157C2×D411yescharacteristic
102Q8×C157—24no—
103Q8×C157—22no—
104Q8×C157—22no—
105Q8×C157C2×D414yes—
106D4×C157—28no—
107D4×C157—24no—
108D4×C157—24no—
109D4×C157—24no—
110D4×C157—24no—
111D4×C157—22no—
112D4×C157—22no—
113D4×C157C2×D414yes—
114C2×C4×C157—24no—
115C2×C4×C157—28no—
116C2×C4×C157—22no—
117C2×C4×C157—24no—
118C2×C4×C157—22no—
119C2×C4×C157—22no—
120C2×C4×C157—24no—
121C2×C4×C157C2×D412yes—
122C2×C4×C157C2×D411yescharacteristic
123C2×C4×C157C2×D412yes—
124C2×C4×C157C2×D412yes—
125C2×C4×C157C2411yescharacteristic
126C22×C157—22no—
127C22×C157—24no—
128C22×C157—41no—
129C22×C157—42no—
130C22×C157—82no—
131C22×C157—84no—
132C22×C157#12012yes—
133C22×C157C22×D411yescharacteristic
134C2×D157—42no—
135C2×D157—41no—
136C4×C157—24no—
137C4×C157—22no—
138C4×C157#12011yescharacteristic
139C4×C157#12011yescharacteristic
140C4×C157C22×D412yes—
141Dic157—22no—
142Dic157—41no—
143Dic157—41no—
144C2×C157—21no—
145C2×C157—22no—
146C2×C157—42no—
147C2×C157—44no—
148C2×C157C2×#12011yescharacteristic
149D157—81no—
150C157#301811yescharacteristic, complemented, Sylow
151#3018—1571nomaximal, Sylow
152#585D15711yescharacteristic, complemented
153#575—1572no—
154#532—1571no—
155#456—1572no—
156#455—1572no—
157#454—1572no—
158#453—1572no—
159C2×#104—1572no—
160#413—1571no—
1612−1+4—22no—
1622−1+4C2×D15712yes—
163H(5,𝔽2)—22no—
164H(5,𝔽2)—22no—
165H(5,𝔽2)C2×D15712yes—
166C2×#41—24no—
167C2×#41—22no—
168C2×#41—1571no—
169C2×#41C2×D15712yes—
170C2×#41C2×D15711yescharacteristic
171#137—3142no—
172#137—1574no—
173AGL(1,ℤ/8)—3142no—
174AGL(1,ℤ/8)—1574no—
175#135—3144no—
176C2×M16—1572no—
177C4⋊Q8—1571no—
178#127—1571no—
179#124—1571no—
180#124—1571no—
181#121—3142no—
182C4×D4—3142no—
183C2×C42—1571no—
184#104—1574no—
185#104—1574no—
186#101—1572no—
187#100—1572no—
188#41—24no—
189#41—28no—
190#41—24no—
191#41—24no—
192#41—24no—
193#41—22no—
194#41—22no—
195#41—24no—
196#41—22no—
197#41—22no—
198#41—3142no—
199#41—3142no—
200#41C22⋊D15714yescomplemented
201#41C22×D15714yes—
202C2×Q8—22no—
203C2×Q8—24no—
204C2×Q8—1571no—
205C2×Q8C22⋊D15712yescomplemented
206C2×Q8C22×D15711yescharacteristic
207C2×D4—28no—
208C2×D4—22no—
209C2×D4—24no—
210C2×D4—22no—
211C2×D4—24no—
212C2×D4—3142no—
213C2×D4—3141no—
214C2×D4—1571no—
215C2×D4C22⋊D15712yes—
216C2×D4C22⋊D15712yescomplemented
217C2×D4C22×D15711yescharacteristic
218C22×C4—24no—
219C22×C4—22no—
220C22×C4—3141no—
221C22×C4—3141no—
222C22×C4C22⋊D15712yes—
223C22×C4C22×D15711yescharacteristic
224Q16—3144no—
225SD16—3144no—
226SD16—3144no—
227D8—3144no—
228M16—3142no—
229M16—1574no—
230C2×C8—3142no—
231C4⋊C4—3142no—
232C22⋊C4—3142no—
233C22⋊C4—3142no—
234C42—3141no—
235C42—1571no—
236C42—1571no—
237C23—24no—
238C23—22no—
239C23—3141no—
240C23C2×C22⋊D15712yes—
241C23C2×C22⋊D15711yescharacteristic
242Q8—24no—
243Q8—22no—
244Q8—22no—
245Q8—3142no—
246Q8C2×C22⋊D15714yes—
247D4—28no—
248D4—24no—
249D4—24no—
250D4—24no—
251D4—24no—
252D4—22no—
253D4—22no—
254D4—3142no—
255D4—6284no—
256D4—6282no—
257D4C2×C22⋊D15714yes—
258C2×C4—24no—
259C2×C4—28no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—24no—
263C2×C4—22no—
264C2×C4—24no—
265C2×C4—3141no—
266C2×C4—3141no—
267C2×C4—3141no—
268C2×C4—6282no—
269C2×C4—6281no—
270C2×C4—6281no—
271C2×C4C2×C22⋊D15712yes—
272C2×C4C2×C22⋊D15711yescharacteristic
273C2×C4C2×C22⋊D15712yes—
274C2×C4C2×C22⋊D15712yes—
275C2×C4C23×D15711yescharacteristic
276C8—3144no—
277C22—22no—
278C22—24no—
279C22—41no—
280C22—42no—
281C22—82no—
282C22—84no—
283C22—6282no—
284C22—6281no—
285C22#1909774012yes—
286C22C22×C22⋊D15711yescharacteristic
287C4—24no—
288C4—22no—
289C4—3142no—
290C4—6281no—
291C4—6281no—
292C4#1909774011yescharacteristic
293C4#1909774011yescharacteristic
294C4C22×C22⋊D15712yes—
295C2—21no—
296C2—22no—
297C2—42no—
298C2—44no—
299C2—12561no—
300C2C2×#1909774011yescharacteristic
subgroupG/H
centerC4#19097740
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#585×C157C2
socleC2×C157C2×#120
hypercenter#585D157
hypocenterC157#3018
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C4 ◁ C22×C4 ◁ #585

Complements

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

NH ≅ G/Ntype
1. #23758071C2semidirect
2. #23757819C2semidirect
3. #23757456C2semidirect
4. #23757451C2semidirect
5. #23757446C2semidirect
6. #23757441C2semidirect
7. C2×#19097698C2semidirect
8. #23757257C2semidirect
9. #585×C157C2semidirect
10. C2×#14578954C22semidirect
12. #19097801C22semidirect
14. #19097796C22semidirect
16. C2×#14578938C22semidirect
17. #19097769C22semidirect
18. #19097767C22semidirect
19. #19097760C22semidirect
20. #19097758C22semidirect
23. C2×C4×Dic157C22semidirect
24. #19097698C22semidirect
25. #19097698C22semidirect
26. #19097690C22semidirect
27. #19097688C22semidirect
68. #41×C157D4semidirect
72. C2×Q8×C157D4semidirect
80. C2×D4×C157D4semidirect
150. C157#3018semidirect
152. #585D157semidirect
200. #41C22⋊D157semidirect
205. C2×Q8C22⋊D157semidirect
216. C2×D4C22⋊D157semidirect

Maximal subgroups

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

subgroupindex
#237580712
#237578192
#237574562
#237574512
#237574462
#237574412
C2×#190976982
#237572572
#585×C1572
#3018157