Finite Group Databasebeta

← #34109814

Subgroups — #34109814

Order 20096.

504 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 133 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×#19097797C211yesmaximal, characteristic, complemented
2C2×#19097777C211yesmaximal, characteristic, complemented
3#23757854C211yesmaximal, characteristic, complemented
4#23757840C211yesmaximal, characteristic, complemented
5#23757525C212yesmaximal, complemented
6#23757518C212yesmaximal, complemented
7#23757508C212yesmaximal, complemented
8#23757501C212yesmaximal, complemented
9#23757313C211yesmaximal, characteristic, complemented
10#23757270C211yesmaximal, characteristic, complemented
11C2×#102×C157C211yesmaximal, characteristic, complemented
12C2×#14578954C2211yescharacteristic
13C2×#14578954C2211yescharacteristic
14#19097797—24no—
15C2×#14578942C2212yes—
16C2×#14578939C2212yes—
17#19097777—24no—
18C2×#14578938C2211yescharacteristic
19C2×#14578937C2212yescomplemented
20C2×C8×D157C2212yescomplemented
21#19097767—21no—
22#19097758—21no—
23#19097751—21no—
24#19097751C2212yescomplemented
25#19097746—21no—
26#19097746C2212yescomplemented
27#19097745—22no—
28#19097742C2212yes—
29#19097742C2212yes—
30#19097740—22no—
31#19097730C2211yescharacteristic
32#19097704C2212yescomplemented
33#19097702C2212yescomplemented
34#19097696C2211yescharacteristic, complemented
35#19097696C2211yescharacteristic, complemented
36#19097693C2211yescharacteristic, complemented
37#19097693C2211yescharacteristic, complemented
38C22×D4×C157C2211yescharacteristic
39C22×C8×C157C2211yescharacteristic
40C2×C4⋊C4×C157C2211yescharacteristic
41#102×C157C2212yes—
42#102×C157C2212yes—
43#14578954—22no—
44#14578954—22no—
45#14578954—22no—
46#14578954—22no—
47C2×Dic314D411yescharacteristic
48C2×Dic314C2311yescharacteristic
49C2×D628—22no—
50C2×D628D411yescharacteristic
51C2×D628C2311yescharacteristic
52C2×C22⋊D157—22no—
53C2×C22⋊D157—22no—
54C2×C22⋊D157—21no—
55C2×C22⋊D157D412yes—
56C2×C4×D157—21no—
57C2×C4×D157C2312yes—
58#14578942—24no—
59#14578939—24no—
60#14578938D414yescomplemented
61#14578937—24no—
62C2×#11929434C2312yes—
63C8×D157—24no—
64C4⋊Dic157—22no—
65C4⋊Dic157C2311yescharacteristic
66C4⋊Dic157C2311yescharacteristic
67C22⋊Dic157—22no—
68C22⋊Dic157—22no—
69C22⋊Dic157—21no—
70#14578931—22no—
71#14578931—22no—
72C4×Dic157—21no—
73C24×C157—21no—
74C2×D4×C157—22no—
75C2×D4×C157—21no—
76C2×D4×C157—22no—
77C2×D4×C157C2312yes—
78C22×C4×C157—21no—
79C22×C4×C157C2311yescharacteristic
80C2×C8×C157—22no—
81C2×C8×C157C2311yescharacteristic
82C2×C8×C157C2311yescharacteristic
83C4⋊C4×C157—21no—
84C4⋊C4×C157C2312yes—
85C22×D157—21no—
86C22×D157C2×D412yes—
87Dic314—22no—
88Dic314—21no—
89Dic314—21no—
90D628—22no—
91D628—21no—
92D628—21no—
93D628—82no—
94C22⋊D157—24no—
95C22⋊D157—42no—
96C22⋊D157—82no—
97C22⋊D157—82no—
98C2×Dic157—21no—
99C2×Dic157—21no—
100C2×Dic157—21no—
101C2×Dic157C2×D412yes—
102C4×D157—22no—
103C4×D157—22no—
104C4×D157—42no—
105#11929434C2×D414yes—
106C23×C157—22no—
107C23×C157—42no—
108C23×C157—81no—
109C23×C157—81no—
110C23×C157C2×D411yescharacteristic
111D4×C157—22no—
112D4×C157—22no—
113D4×C157—42no—
114C2×C4×C157—22no—
115C2×C4×C157—81no—
116C2×C4×C157C2×D411yescharacteristic
117C2×C4×C157C2×D412yes—
118C2×C4×C157#4112yes—
119C2×C4×C157C2411yescharacteristic
120C8×C157—21no—
121C8×C157—21no—
122C22×C157—21no—
123C22×C157—21no—
124C22×C157—42no—
125C22×C157—42no—
126C22×C157—42no—
127C22×C157—84no—
128C22×C157#12112yes—
129C22×C157C22×D411yescharacteristic
130C2×D157—22no—
131C2×D157—22no—
132C2×D157—22no—
133C2×D157—41no—
134C2×D157—41no—
135C2×D157—41no—
136C4×C157—42no—
137C4×C157#12112yes—
138C4×C157C22×D411yescharacteristic
139C4×C157C2×#4111yescharacteristic
140Dic157—22no—
141Dic157—41no—
142Dic157—41no—
143Dic157—41no—
144C2×C157—22no—
145C2×C157—82no—
146C2×C157C2×#12111yescharacteristic
147C2×C157#57611yescharacteristic
148C2×C157#57711yescharacteristic
149D157—42no—
150D157—81no—
151C157#306211yescharacteristic, complemented, Sylow
152#3062—1571nomaximal, Sylow
153C2×AGL(1,ℤ/8)—1571no—
154C2×#131—1571no—
155#537—1571no—
156#535—1571no—
157#469—1572no—
158#468—1572no—
159#466—1572no—
160#465—1572no—
161#428—1571no—
162#416—1571no—
163C2×#102D15711yescharacteristic, complemented
164C2×#41—1571no—
165C2×#41—1571no—
166C22×D4C2×D15711yescharacteristic
167AGL(1,ℤ/8)—3144no—
168C2×SD16—1572no—
169C2×D8—1572no—
170#131—3144no—
171C2×M16—1571no—
172C2×M16—1572no—
173C22×C8—1572no—
174C22×C8C2×D15711yescharacteristic
175#127—3141no—
176#124—3141no—
177C22⋊Q8—3141no—
178C22⋊Q8—1572no—
179#121—3141no—
180#121—3142no—
181#121—1572no—
182#121—1572no—
183#121—1572no—
184#120—3142no—
185#117—1571no—
186C2×C4⋊C4C2×D15711yescharacteristic
187#107—1572no—
188#106—1572no—
189#103—1571no—
190#103—1571no—
191#102—1571no—
192#102—1571no—
193#102C2×D15712yes—
194#102C2×D15712yes—
195C24—21no—
196#41—3142no—
197#41—3142no—
198#41—3142no—
199#41—3142no—
200C2×Q8—1571no—
201C2×Q8—1571no—
202C2×D4—22no—
203C2×D4—21no—
204C2×D4—22no—
205C2×D4—3142no—
206C2×D4—3141no—
207C2×D4—3142no—
208C2×D4—3142no—
209C2×D4—1572no—
210C2×D4—1571no—
211C2×D4—1571no—
212C2×D4C22×D15712yes—
213C22×C4—21no—
214C22×C4—3141no—
215C22×C4—1572no—
216C22×C4C22×D15711yescharacteristic
217SD16—3144no—
218D8—3144no—
219M16—3144no—
220M16—1574no—
221C2×C8—22no—
222C2×C8—3144no—
223C2×C8—1572no—
224C2×C8C22×D15711yescharacteristic
225C2×C8C22×D15711yescharacteristic
226C4⋊C4—21no—
227C4⋊C4—3142no—
228C4⋊C4—1571no—
229C4⋊C4—1571no—
230C4⋊C4C22×D15712yes—
231C22⋊C4—3142no—
232C22⋊C4—3142no—
233C22⋊C4—3141no—
234C22⋊C4—3142no—
235C22⋊C4—3142no—
236C42—3141no—
237C23—22no—
238C23—42no—
239C23—81no—
240C23—81no—
241C23—3141no—
242C23—1572no—
243C23D4×D15711yescharacteristic
244Q8—3142no—
245Q8—3141no—
246Q8—3141no—
247D4—22no—
248D4—22no—
249D4—42no—
250D4—3144no—
251D4—3142no—
252D4—3141no—
253D4—3141no—
254D4—6282no—
255D4—12562no—
256D4—12562no—
257D4—12562no—
258C2×C4—22no—
259C2×C4—81no—
260C2×C4—3142no—
261C2×C4—3142no—
262C2×C4—3141no—
263C2×C4—3141no—
264C2×C4—3141no—
265C2×C4—6282no—
266C2×C4—1572no—
267C2×C4D4×D15711yescharacteristic
268C2×C4D4×D15712yes—
269C2×C4#1457895412yes—
270C2×C4C23×D15711yescharacteristic
271C8—21no—
272C8—21no—
273C8—1574no—
274C22—21no—
275C22—21no—
276C22—42no—
277C22—42no—
278C22—42no—
279C22—84no—
280C22—3142no—
281C22—3142no—
282C22—3142no—
283C22—6281no—
284C22—6281no—
285C22—6281no—
286C22#1909774412yes—
287C22C2×D4×D15711yescharacteristic
288C4—42no—
289C4—3142no—
290C4—6281no—
291C4—6281no—
292C4—6281no—
293C4#1909774412yes—
294C4C2×D4×D15711yescharacteristic
295C4C2×#1457895411yescharacteristic
296C2—22no—
297C2—82no—
298C2—6282no—
299C2—12561no—
300C2C2×#1909774411yescharacteristic
301C2#2375807411yescharacteristic
302C2#2375807911yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
FittingC2×#102×C157C2
socleC22×C157C22×D4
hypercenterC2×#102D157
hypocenterC157#3062
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
1. C2×#19097797C2semidirect
2. C2×#19097777C2semidirect
3. #23757854C2semidirect
4. #23757840C2semidirect
5. #23757525C2semidirect
6. #23757518C2semidirect
7. #23757508C2semidirect
8. #23757501C2semidirect
9. #23757313C2semidirect
10. #23757270C2semidirect
11. C2×#102×C157C2semidirect
19. C2×#14578937C22semidirect
20. C2×C8×D157C22semidirect
24. #19097751C22semidirect
26. #19097746C22semidirect
32. #19097704C22semidirect
33. #19097702C22semidirect
34. #19097696C22semidirect
35. #19097696C22semidirect
36. #19097693C22semidirect
37. #19097693C22semidirect
60. #14578938D4semidirect
151. C157#3062semidirect
163. C2×#102D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190977972
C2×#190977772
#237578542
#237578402
#237575252
#237575182
#237575082
#237575012
#237573132
#237572702
C2×#102×C1572
#3062157