Finite Group Databasebeta

← #34109779

Subgroups — #34109779

Order 20096.

524 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 127 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×D8×D157C211yesmaximal, characteristic, complemented
2C2×#19097777C211yesmaximal, characteristic, complemented
3#23757839C212yesmaximal, complemented
4#23757525C212yesmaximal, complemented
5#23757524C212yesmaximal, complemented
6#23757510C211yesmaximal, characteristic, complemented
7#23757509C211yesmaximal, characteristic, complemented
8#23757506C211yesmaximal, characteristic, complemented
9#23757310C211yesmaximal, characteristic, complemented
10#23757276C212yesmaximal, complemented
11#466×C157C211yesmaximal, characteristic, complemented
12C2×#14578954C2211yescharacteristic
13C2×D4×D157C2212yes—
14C2×D1256C2211yescharacteristic
15C2×#14578939C2212yes—
16D8×D157—24no—
17#19097777—24no—
18C2×#14578938C2211yescharacteristic, complemented
19C2×#14578937C2211yescharacteristic
20C2×#14578937C2211yescharacteristic, complemented
21C2×C8×D157C2211yescharacteristic
22C2×C8×D157C2211yescharacteristic, complemented
23#19097767—22no—
24#19097757—22no—
25#19097746C2212yescomplemented
26#19097745C2212yescomplemented
27#19097742C2212yescomplemented
28#19097741—22no—
29#19097739—22no—
30#19097729C2212yes—
31#19097705C2211yescharacteristic, complemented
32#19097702C2212yescomplemented
33#19097693C2212yescomplemented
34#19097692C2212yescomplemented
35#19097691C2212yescomplemented
36C2×D8×C157C2211yescharacteristic
37C22×C8×C157C2211yescharacteristic, complemented
38#121×C157C2212yescomplemented
39#107×C157C2211yescharacteristic, complemented
40#102×C157C2212yescomplemented
41C23×D157—22no—
42#14578954—22no—
43#14578954—22no—
44C2×Dic314D411yescharacteristic
45C2×D628—22no—
46C2×D628D411yescharacteristic
47C2×D628C2312yes—
48C2×C22⋊D157—22no—
49C2×C22⋊D157—22no—
50C2×C22⋊D157—22no—
51C2×C22⋊D157D411yescharacteristic
52C2×C22⋊D157D411yescharacteristic
53D4×D157—24no—
54D4×D157—24no—
55C2×C4×D157C2311yescharacteristic
56C2×C4×D157C2311yescharacteristic
57D1256—22no—
58#14578939—24no—
59#14578938—22no—
60#14578937—22no—
61#14578937—22no—
62C2×#11929434C2311yescharacteristic
63C2×#11929434C2311yescharacteristic
64C8×D157—22no—
65C8×D157D414yescomplemented
66C4⋊Dic157C2312yes—
67C22⋊Dic157—22no—
68#14578931—22no—
69#14578931—22no—
70#14578931—22no—
71C4×Dic157—22no—
72C2×D4×C157—22no—
73C2×D4×C157C2312yes—
74C22×C4×C157C2311yescharacteristic
75D8×C157—22no—
76C2×C8×C157—22no—
77C2×C8×C157C2311yescharacteristic
78C2×C8×C157C2311yescharacteristic
79C4⋊C4×C157C2312yes—
80C22⋊C4×C157—22no—
81C22×D157—22no—
82C22×D157—44no—
83C22×D157—82no—
84C22×D157—82no—
85C22×D157C2×D411yescharacteristic
86C22×D157C2×D411yescharacteristic
87Dic314—22no—
88D628—22no—
89D628—22no—
90D628—22no—
91D628—82no—
92C22⋊D157—22no—
93C22⋊D157—22no—
94C22⋊D157—44no—
95C22⋊D157—82no—
96C22⋊D157—82no—
97C2×Dic157—22no—
98C2×Dic157C2×D411yescharacteristic
99C2×Dic157C2×D411yescharacteristic
100C4×D157—21no—
101C4×D157—21no—
102C4×D157C2×D412yes—
103C4×D157#4112yes—
104#11929434—21no—
105#11929434C2×D412yes—
106C23×C157—22no—
107C23×C157C2×D411yescharacteristic
108D4×C157—22no—
109D4×C157—22no—
110D4×C157—82no—
111C2×C4×C157—21no—
112C2×C4×C157—21no—
113C2×C4×C157—22no—
114C2×C4×C157C2×D411yescharacteristic
115C2×C4×C157C2411yescharacteristic
116C8×C157—21no—
117C8×C157C2×D412yes—
118C22×C157—21no—
119C22×C157—21no—
120C22×C157—21no—
121C22×C157—42no—
122C22×C157—42no—
123C22×C157—42no—
124C22×C157C22×D411yescharacteristic
125C2×D157—21no—
126C2×D157—21no—
127C2×D157—21no—
128C2×D157—21no—
129C2×D157—21no—
130C2×D157—42no—
131C2×D157—42no—
132C2×D157—42no—
133C2×D157—88no—
134C2×D157#12112yes—
135C4×C157—21no—
136C4×C157—42no—
137C4×C157C22×D411yescharacteristic
138C4×C157C2×#4111yescharacteristic
139Dic157—21no—
140Dic157—42no—
141Dic157#12112yes—
142C2×C157—41no—
143C2×C157—82no—
144C2×C157C2×#12111yescharacteristic
145C2×C157#57611yescharacteristic
146C2×C157#57611yescharacteristic
147D157—22no—
148D157—41no—
149D157—82no—
150C157#305811yescharacteristic, complemented, Sylow
151#3058—1571nomaximal, Sylow
152C22×D8—1571no—
153C2×#131—1571no—
154#535—1572no—
155#469—1572no—
156#469—1572no—
157#466—1571no—
158#466—1571no—
159#466—1571no—
160#466D15711yescharacteristic, complemented
161#427—1571no—
162#418—1572no—
163C2×#41—1571no—
164C22×D4—1572no—
165C2×D8—3144no—
166C2×D8—1572no—
167C2×D8—1571no—
168C2×D8C2×D15711yescharacteristic
169#131—3144no—
170C2×M16—1571no—
171C2×M16—1571no—
172C2×M16—1571no—
173C22×C8—1571no—
174C22×C8—1571no—
175C22×C8C2×D15711yescharacteristic, complemented
176#127—3142no—
177#124—3142no—
178#121—1572no—
179#121—1572no—
180#121—1572no—
181#121C2×D15712yescomplemented
182#120—3142no—
183#120—3142no—
184#117—1572no—
185#107—1571no—
186#107C2×D15711yescharacteristic, complemented
187#106—1572no—
188#102—1572no—
189#102—1572no—
190#102—1572no—
191#102C2×D15712yescomplemented
192C24—3142no—
193#41—3142no—
194#41—3142no—
195C2×Q8—1571no—
196C2×D4—22no—
197C2×D4—3142no—
198C2×D4—3142no—
199C2×D4—3142no—
200C2×D4—3144no—
201C2×D4—3144no—
202C2×D4—3142no—
203C2×D4—1571no—
204C2×D4—1571no—
205C2×D4—1571no—
206C2×D4—1572no—
207C2×D4C22×D15712yes—
208C22×C4—1571no—
209C22×C4—1571no—
210C22×C4C22×D15711yescharacteristic
211D8—22no—
212D8—3144no—
213D8—3142no—
214M16—3142no—
215M16—3142no—
216M16—3142no—
217C2×C8—22no—
218C2×C8—3142no—
219C2×C8—1571no—
220C2×C8—1571no—
221C2×C8—1574no—
222C2×C8C22×D15711yescharacteristic
223C2×C8C22×D15711yescharacteristic
224C4⋊C4—1572no—
225C4⋊C4C22×D15712yes—
226C22⋊C4—22no—
227C22⋊C4—3142no—
228C22⋊C4—3142no—
229C22⋊C4—3142no—
230C22⋊C4—3142no—
231C42—3142no—
232C23—22no—
233C23—3142no—
234C23—6284no—
235C23—1571no—
236C23—1571no—
237C23—12562no—
238C23—12562no—
239C23D4×D15711yescharacteristic
240Q8—3142no—
241D4—22no—
242D4—22no—
243D4—82no—
244D4—3142no—
245D4—3142no—
246D4—3142no—
247D4—3142no—
248D4—3142no—
249D4—6284no—
250D4—12562no—
251D4—12562no—
252D4—12562no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—3141no—
257C2×C4—3141no—
258C2×C4—3142no—
259C2×C4—1571no—
260C2×C4—1571no—
261C2×C4—1572no—
262C2×C4—1572no—
263C2×C4D4×D15711yescharacteristic
264C2×C4C23×D15711yescharacteristic
265C8—21no—
266C8—3141no—
267C8—1572no—
268C8D4×D15712yes—
269C22—21no—
270C22—21no—
271C22—21no—
272C22—42no—
273C22—42no—
274C22—42no—
275C22—3141no—
276C22—3141no—
277C22—3141no—
278C22—3141no—
279C22—3141no—
280C22—6282no—
281C22—6282no—
282C22—6282no—
283C22—1572no—
284C22—12568no—
285C22C2×D4×D15711yescharacteristic
286C4—21no—
287C4—42no—
288C4—3141no—
289C4—6282no—
290C4—1572no—
291C4C2×D4×D15711yescharacteristic
292C4#41×D15711yescharacteristic
293C2—41no—
294C2—82no—
295C2—3142no—
296C2—6281no—
297C2—12562no—
298C2#121×D15711yescharacteristic
299C2#2375807411yescharacteristic
300C2#2375807511yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#466×C157C2
socleC22×C157C22×D4
hypercenter#466D157
hypocenterC157#3058
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #466

Complements

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

NH ≅ G/Ntype
1. C2×D8×D157C2semidirect
2. C2×#19097777C2semidirect
3. #23757839C2semidirect
4. #23757525C2semidirect
5. #23757524C2semidirect
6. #23757510C2semidirect
7. #23757509C2semidirect
8. #23757506C2semidirect
9. #23757310C2semidirect
10. #23757276C2semidirect
11. #466×C157C2semidirect
18. C2×#14578938C22semidirect
20. C2×#14578937C22semidirect
22. C2×C8×D157C22semidirect
25. #19097746C22semidirect
26. #19097745C22semidirect
27. #19097742C22semidirect
31. #19097705C22semidirect
32. #19097702C22semidirect
33. #19097693C22semidirect
34. #19097692C22semidirect
35. #19097691C22semidirect
37. C22×C8×C157C22semidirect
38. #121×C157C22semidirect
39. #107×C157C22semidirect
40. #102×C157C22semidirect
65. C8×D157D4semidirect
150. C157#3058semidirect
160. #466D157semidirect
175. C22×C8C2×D157semidirect
181. #121C2×D157semidirect
186. #107C2×D157semidirect
191. #102C2×D157semidirect

Maximal subgroups

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

subgroupindex
C2×D8×D1572
C2×#190977772
#237578392
#237575252
#237575242
#237575102
#237575092
#237575062
#237573102
#237572762
#466×C1572
#3058157