Finite Group Databasebeta

← #34109817

Subgroups — #34109817

Order 20096.

504 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 141 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×#19097799C211yesmaximal, characteristic, complemented
2C2×#19097779C211yesmaximal, characteristic, complemented
3#23757854C211yesmaximal, characteristic, complemented
4#23757842C211yesmaximal, characteristic, complemented
5#23757527C212yesmaximal, complemented
6#23757520C212yesmaximal, complemented
7#23757510C212yesmaximal, complemented
8#23757503C212yesmaximal, complemented
9#23757267C211yesmaximal, characteristic, complemented
10C2×#19097691C211yesmaximal, characteristic, complemented
11#428×C157C211yesmaximal, characteristic, complemented
12C2×#14578954C2211yescharacteristic
13C2×#14578954C2211yescharacteristic
14C22×D628C2211yescharacteristic
15#19097799—24no—
16C2×#14578943C2212yes—
17C2×D1256C2212yes—
18#19097779—24no—
19C2×#14578937C2212yescomplemented
20C22×#11929434C2211yescharacteristic
21C2×C8×D157C2212yescomplemented
22#19097768—21no—
23#19097759—21no—
24#19097751—21no—
25#19097751C2212yescomplemented
26#19097746—21no—
27#19097746C2212yes—
28#19097746C2212yes—
29#19097746C2212yescomplemented
30#19097745—22no—
31#19097741—22no—
32#19097694C2211yescharacteristic, complemented
33#19097694C2211yescharacteristic, complemented
34#19097691C2212yes—
35#19097691C2212yes—
36#19097691C2211yescharacteristic, complemented
37#19097691C2211yescharacteristic, complemented
38C2×M16×C157C2211yescharacteristic
39#117×C157C2211yescharacteristic
40C2×C4⋊C4×C157C2211yescharacteristic
41#107×C157C2212yescomplemented
42#106×C157C2212yescomplemented
43C23×D157—21no—
44#14578954—22no—
45#14578954—22no—
46#14578954—22no—
47#14578954—22no—
48C2×Dic314D411yescharacteristic
49C2×Dic314C2311yescharacteristic
50C2×D628—22no—
51C2×D628—22no—
52C2×D628—22no—
53C2×D628—22no—
54C2×D628—21no—
55C2×D628—22no—
56C2×D628D411yescharacteristic
57C2×D628C2312yes—
58C2×D628C2311yescharacteristic
59C2×C22⋊D157—21no—
60C2×C22⋊D157D412yes—
61C2×C4×D157—21no—
62C2×C4×D157C2312yes—
63#14578943—24no—
64D1256—24no—
65#14578937—24no—
66C2×#11929434—22no—
67C2×#11929434C2311yescharacteristic
68C2×#11929434C2311yescharacteristic
69C8×D157—24no—
70C4⋊Dic157—22no—
71#14578931—22no—
72#14578931—22no—
73#14578931—22no—
74#14578931—22no—
75C22×C4×C157—21no—
76C22×C4×C157C2311yescharacteristic
77M16×C157D414yescomplemented
78C2×C8×C157C2312yes—
79C4⋊C4×C157—21no—
80C4⋊C4×C157C2311yescharacteristic
81C4⋊C4×C157C2312yes—
82C4⋊C4×C157C2311yescharacteristic
83C22⋊C4×C157—21no—
84C42×C157—21no—
85C22×D157—21no—
86C22×D157—22no—
87C22×D157—42no—
88C22×D157—81no—
89C22×D157—81no—
90C22×D157C2×D412yes—
91Dic314—22no—
92Dic314—21no—
93Dic314—21no—
94D628—22no—
95D628—22no—
96D628—21no—
97D628—22no—
98D628—21no—
99D628—42no—
100D628—82no—
101D628—82no—
102D628—82no—
103C22⋊D157—24no—
104C22⋊D157—42no—
105C2×Dic157—21no—
106C2×Dic157C2×D412yes—
107C4×D157—22no—
108C4×D157—22no—
109C4×D157—42no—
110#11929434—21no—
111#11929434—21no—
112C23×C157C2×D411yescharacteristic
113C2×C4×C157—21no—
114C2×C4×C157—21no—
115C2×C4×C157—22no—
116C2×C4×C157—81no—
117C2×C4×C157C2×D411yescharacteristic
118C2×C4×C157C2×D412yes—
119C2×C4×C157#4112yes—
120C2×C4×C157C2411yescharacteristic
121C8×C157C2×D414yes—
122C22×C157—21no—
123C22×C157—21no—
124C22×C157#12112yes—
125C22×C157C22×D411yescharacteristic
126C2×D157—22no—
127C2×D157—22no—
128C2×D157—22no—
129C2×D157—41no—
130C2×D157—42no—
131C2×D157—41no—
132C2×D157—42no—
133C2×D157—42no—
134C2×D157—41no—
135C2×D157—84no—
136C4×C157—42no—
137C4×C157—41no—
138C4×C157—41no—
139C4×C157#12112yes—
140C4×C157C22×D411yescharacteristic
141C4×C157C2×#4111yescharacteristic
142Dic157—22no—
143Dic157—41no—
144C2×C157—22no—
145C2×C157C2×#12111yescharacteristic
146C2×C157#57611yescharacteristic
147C2×C157#57711yescharacteristic
148D157—42no—
149D157—81no—
150D157—82no—
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#428D15711yescharacteristic, complemented
162#416—1571no—
163C2×#102—1571no—
164C2×#41—1571no—
165C2×#41—1571no—
166C22×D4—1571no—
167AGL(1,ℤ/8)—3144no—
168C2×SD16—1572no—
169C2×D8—1572no—
170#131—3144no—
171C2×M16—1572no—
172C2×M16C2×D15711yescharacteristic
173C22×C8—1571no—
174C22×C8—1572no—
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#117C2×D15711yescharacteristic
186C2×C4⋊C4C2×D15711yescharacteristic
187#107C2×D15712yescomplemented
188#106C2×D15712yescomplemented
189#103—1571no—
190#103—1571no—
191#102—1572no—
192#102—1572no—
193#102—1571no—
194#102—1571no—
195C24—3141no—
196#41—3142no—
197#41—3142no—
198#41—3142no—
199#41—3142no—
200C2×Q8—1571no—
201C2×Q8—1571no—
202C2×D4—3142no—
203C2×D4—3142no—
204C2×D4—3142no—
205C2×D4—3141no—
206C2×D4—3141no—
207C2×D4—3142no—
208C2×D4—3142no—
209C2×D4—1572no—
210C2×D4—1571no—
211C2×D4—1571no—
212C2×D4—1572no—
213C22×C4—21no—
214C22×C4—3141no—
215C22×C4—1572no—
216C22×C4C22×D15711yescharacteristic
217SD16—3144no—
218D8—3144no—
219M16—3144no—
220M16D62814yescomplemented
221C2×C8—3142no—
222C2×C8—3144no—
223C2×C8—1571no—
224C2×C8—1571no—
225C2×C8C22×D15712yes—
226C4⋊C4—21no—
227C4⋊C4—3142no—
228C4⋊C4C22×D15712yes—
229C4⋊C4C22×D15711yescharacteristic
230C4⋊C4C22×D15711yescharacteristic
231C22⋊C4—21no—
232C22⋊C4—3142no—
233C22⋊C4—3142no—
234C22⋊C4—3142no—
235C22⋊C4—3142no—
236C42—21no—
237C23—3141no—
238C23—3142no—
239C23—6282no—
240C23—1572no—
241C23—12561no—
242C23—12561no—
243C23D4×D15711yescharacteristic
244Q8—3142no—
245Q8—3141no—
246Q8—3141no—
247D4—3144no—
248D4—3142no—
249D4—3141no—
250D4—3142no—
251D4—3142no—
252D4—3141no—
253D4—6282no—
254D4—6282no—
255D4—12562no—
256D4—12562no—
257D4—12562no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—81no—
262C2×C4—3142no—
263C2×C4—3142no—
264C2×C4—3141no—
265C2×C4—6282no—
266C2×C4—1572no—
267C2×C4D4×D15711yescharacteristic
268C2×C4C2×D62812yes—
269C2×C4#1457895512yes—
270C2×C4C23×D15711yescharacteristic
271C8—3141no—
272C8—3141no—
273C8C2×D62814yes—
274C22—21no—
275C22—21no—
276C22—3142no—
277C22—3142no—
278C22—3142no—
279C22—6282no—
280C22—6281no—
281C22—6281no—
282C22—6282no—
283C22—6281no—
284C22—6282no—
285C22—12564no—
286C22#1909774612yes—
287C22C2×D4×D15711yescharacteristic
288C4—41no—
289C4—42no—
290C4—41no—
291C4—3142no—
292C4—6281no—
293C4#1909774612yes—
294C4C22×D62811yescharacteristic
295C4C2×#1457895511yescharacteristic
296C2—22no—
297C2—6282no—
298C2—12562no—
299C2—12561no—
300C2C2×#1909774611yescharacteristic
301C2#2375807711yescharacteristic
302C2#2375808211yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#428×C157C2
socleC22×C157C22×D4
hypercenter#428D157
hypocenterC157#3062
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
1. C2×#19097799C2semidirect
2. C2×#19097779C2semidirect
3. #23757854C2semidirect
4. #23757842C2semidirect
5. #23757527C2semidirect
6. #23757520C2semidirect
7. #23757510C2semidirect
8. #23757503C2semidirect
9. #23757267C2semidirect
10. C2×#19097691C2semidirect
11. #428×C157C2semidirect
19. C2×#14578937C22semidirect
21. C2×C8×D157C22semidirect
25. #19097751C22semidirect
29. #19097746C22semidirect
32. #19097694C22semidirect
33. #19097694C22semidirect
36. #19097691C22semidirect
37. #19097691C22semidirect
41. #107×C157C22semidirect
42. #106×C157C22semidirect
77. M16×C157D4semidirect
151. C157#3062semidirect
161. #428D157semidirect
187. #107C2×D157semidirect
188. #106C2×D157semidirect
220. M16D628semidirect

Maximal subgroups

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

subgroupindex
C2×#190977992
C2×#190977792
#237578542
#237578422
#237575272
#237575202
#237575102
#237575032
#237572672
C2×#190976912
#428×C1572
#3062157