Finite Group Databasebeta

← #34109677

Subgroups — #34109677

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×#19097793C211yesmaximal, characteristic, complemented
3#23757842C211yesmaximal, characteristic, complemented
4#23757753C211yesmaximal, characteristic, complemented
5#23757496C212yesmaximal, complemented
6#23757489C212yesmaximal, complemented
7#23757474C212yesmaximal, complemented
8#23757469C212yesmaximal, complemented
9#23757275C211yesmaximal, characteristic, complemented
10C2×#19097693C211yesmaximal, characteristic, complemented
11#424×C157C211yesmaximal, characteristic, complemented
12C2×#14578954C2211yescharacteristic
13C2×#14578954C2211yescharacteristic
14C22×D628C2211yescharacteristic
15#19097799—24no—
16#19097793—24no—
17C2×Dic628C2211yescharacteristic, complemented
18C2×#14578943C2212yes—
19C2×#14578943C2211yescharacteristic, complemented
20C2×#14578943C2211yescharacteristic, complemented
21C2×D1256C2212yes—
22C2×D1256C2211yescharacteristic, complemented
23#19097768C2212yes—
24#19097759C2212yes—
25#19097746—21no—
26#19097746—21no—
27#19097741—22no—
28C4×Dic314C2212yescomplemented
29C4×D628C2212yescomplemented
30#19097731—22no—
31#19097729—21no—
32C2×C4⋊Dic157C2211yescharacteristic
33C4⋊C4×D157—21no—
34#19097696C2212yescomplemented
35#19097693C2212yes—
36#19097693C2212yes—
37#19097693C2212yescomplemented
38C2×M16×C157C2211yescharacteristic
39C22×C8×C157C2211yescharacteristic
40#117×C157C2211yescharacteristic
41C4⋊C8×C157C2212yescomplemented
42C4⋊C8×C157C2212yescomplemented
43C23×D157—21no—
44#14578954—22no—
45#14578954—22no—
46#14578954—22no—
47#14578954D414yescomplemented
48C2×Dic314C2311yescharacteristic
49C2×Dic314C2311yescharacteristic
50C2×D628—22no—
51C2×D628—21no—
52C2×D628—22no—
53C2×D628—22no—
54C2×D628—22no—
55C2×D628C2312yes—
56C2×D628C2311yescharacteristic
57C2×D628C2311yescharacteristic
58C2×C22⋊D157—21no—
59C2×C22⋊D157—21no—
60C22×Dic157—21no—
61C2×C4×D157—21no—
62C2×C4×D157—21no—
63C2×C4×D157—22no—
64Dic628—22no—
65#14578943—24no—
66#14578943—22no—
67#14578943—22no—
68D1256—24no—
69D1256—22no—
70C4⋊Dic157—21no—
71C4⋊Dic157C2312yes—
72#14578933—22no—
73#14578931—22no—
74#14578931—22no—
75#14578931—22no—
76C4×Dic157—22no—
77C22×C4×C157C2311yescharacteristic
78M16×C157—22no—
79C2×C8×C157—22no—
80C2×C8×C157C2311yescharacteristic
81C2×C8×C157C2311yescharacteristic
82C2×C8×C157C2312yes—
83C4⋊C4×C157D411yescharacteristic
84C4⋊C4×C157D411yescharacteristic
85C22⋊C4×C157D412yes—
86C42×C157C2312yes—
87C22×D157—21no—
88C22×D157—21no—
89C22×D157—22no—
90C22×D157—42no—
91C22×D157—81no—
92C22×D157—81no—
93Dic314—21no—
94Dic314—21no—
95Dic314—21no—
96Dic314C2×D412yes—
97D628—22no—
98D628—21no—
99D628—21no—
100D628—22no—
101D628—21no—
102D628—42no—
103D628—82no—
104D628—82no—
105D628C2×D412yes—
106C22⋊D157—24no—
107C22⋊D157—42no—
108C2×Dic157—21no—
109C2×Dic157—22no—
110C2×Dic157—21no—
111C2×Dic157—81no—
112C4×D157—24no—
113C4×D157—42no—
114C4×D157—82no—
115C23×C157C2×D411yescharacteristic
116C2×C4×C157C2×D412yes—
117C2×C4×C157C2×D412yes—
118C2×C4×C157C2×D412yes—
119C2×C4×C157C2×D411yescharacteristic
120C2×C4×C157#4112yes—
121C2×C4×C157C2411yescharacteristic
122C8×C157—21no—
123C8×C157—21no—
124C8×C157—22no—
125C22×C157—21no—
126C22×C157—21no—
127C22×C157#12112yes—
128C22×C157C22×D411yescharacteristic
129C2×D157—22no—
130C2×D157—41no—
131C2×D157—41no—
132C2×D157—41no—
133C2×D157—42no—
134C2×D157—41no—
135C2×D157—42no—
136C2×D157—41no—
137C2×D157—42no—
138C2×D157—84no—
139C4×C157—22no—
140C4×C157—22no—
141C4×C157#12112yes—
142C4×C157C22×D411yescharacteristic
143C4×C157C2×#4111yescharacteristic
144Dic157—22no—
145Dic157—41no—
146Dic157—42no—
147C2×C157—22no—
148C2×C157C2×#12111yescharacteristic
149C2×C157#57611yescharacteristic
150C2×C157#57711yescharacteristic
151D157—42no—
152D157—81no—
153D157—82no—
154C157#304511yescharacteristic, complemented, Sylow
155#3045—1571nomaximal, Sylow
156C2×AGL(1,ℤ/8)—1571no—
157C2×#135—1571no—
158#535—1571no—
159#520—1571no—
160#464—1572no—
161#463—1572no—
162#460—1572no—
163#459—1572no—
164#424D15711yescharacteristic, complemented
165#417—1571no—
166C2×#102—1571no—
167C2×#41—1571no—
168C2×#41—1571no—
169C22×D4—1571no—
170AGL(1,ℤ/8)—3144no—
171#135—3144no—
172C2×Q16—1571no—
173C2×SD16—1571no—
174C2×SD16—1571no—
175C2×SD16—1572no—
176C2×D8—1571no—
177C2×D8—1572no—
178C2×M16C2×D15711yescharacteristic
179C22×C8C2×D15711yescharacteristic
180#127—1572no—
181#124—1572no—
182#121—3141no—
183#121—3141no—
184#120—3142no—
185C4×Q8—1572no—
186C4×D4—3142no—
187C4×D4—1572no—
188#117—3141no—
189#117C2×D15711yescharacteristic
190C2×C4⋊C4—3141no—
191C2×C4⋊C4—1571no—
192C4⋊C8C2×D15712yescomplemented
193C4⋊C8C2×D15712yescomplemented
194#103—1572no—
195#102—1572no—
196#102—1572no—
197#102—1572no—
198C24—3141no—
199#41—3142no—
200#41—3142no—
201#41—3142no—
202#41—1574no—
203C2×Q8—1571no—
204C2×Q8—1571no—
205C2×D4—3141no—
206C2×D4—3141no—
207C2×D4—3142no—
208C2×D4—3141no—
209C2×D4—3142no—
210C2×D4—3142no—
211C2×D4—3142no—
212C2×D4—1572no—
213C2×D4—1571no—
214C2×D4—1571no—
215C22×C4—3141no—
216C22×C4—3141no—
217C22×C4—3141no—
218C22×C4—3142no—
219C22×C4C22×D15711yescharacteristic
220Q16—3142no—
221SD16—3142no—
222SD16—3144no—
223SD16—3142no—
224D8—3142no—
225D8—3144no—
226M16—22no—
227C2×C8—22no—
228C2×C8C22×D15711yescharacteristic
229C2×C8C22×D15711yescharacteristic
230C2×C8C22×D15712yes—
231C4⋊C4—3141no—
232C4⋊C4—3142no—
233C4⋊C4—1572no—
234C4⋊C4D62811yescharacteristic
235C4⋊C4D62811yescharacteristic
236C22⋊C4—3142no—
237C22⋊C4—3142no—
238C22⋊C4—3142no—
239C22⋊C4D62812yes—
240C42—3142no—
241C42C22×D15712yes—
242C23—3141no—
243C23—3142no—
244C23—3141no—
245C23—6282no—
246C23—12561no—
247C23—12561no—
248C23C2×D62811yescharacteristic
249Q8—3141no—
250Q8—3141no—
251Q8—3141no—
252Q8—1572no—
253D4—3144no—
254D4—3142no—
255D4—3141no—
256D4—3141no—
257D4—3142no—
258D4—3141no—
259D4—6282no—
260D4—6282no—
261D4—1572no—
262D4—12562no—
263D4—12562no—
264C2×C4—3141no—
265C2×C4—3144no—
266C2×C4—3142no—
267C2×C4—3141no—
268C2×C4—6282no—
269C2×C4—12562no—
270C2×C4—12561no—
271C2×C4D4×D15712yes—
272C2×C4C2×D62812yes—
273C2×C4C2×D62812yes—
274C2×C4C2×D62811yescharacteristic
275C2×C4#1457895512yes—
276C2×C4C23×D15711yescharacteristic
277C8—21no—
278C8—21no—
279C8—22no—
280C22—21no—
281C22—21no—
282C22—3142no—
283C22—6281no—
284C22—6281no—
285C22—6281no—
286C22—6282no—
287C22—6281no—
288C22—6282no—
289C22—6281no—
290C22—6282no—
291C22—12564no—
292C22#1909774612yes—
293C22C22×D62811yescharacteristic
294C4—22no—
295C4—22no—
296C4—3142no—
297C4—6281no—
298C4—6282no—
299C4#1909774612yes—
300C4C2×D4×D15711yescharacteristic
301C4C2×#1457895511yescharacteristic
302C2—22no—
303C2—6282no—
304C2—12562no—
305C2—12561no—
306C2C2×#1909774611yescharacteristic
307C2#2375807811yescharacteristic
308C2#2375808511yescharacteristic
subgroupG/H
centerC22C22×D628
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#424×C157C2
socleC22×C157C22×D4
hypercenter#424D157
hypocenterC157#3045
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ #117 ◁ #424

Complements

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

NH ≅ G/Ntype
1. C2×#19097799C2semidirect
2. C2×#19097793C2semidirect
3. #23757842C2semidirect
4. #23757753C2semidirect
5. #23757496C2semidirect
6. #23757489C2semidirect
7. #23757474C2semidirect
8. #23757469C2semidirect
9. #23757275C2semidirect
10. C2×#19097693C2semidirect
11. #424×C157C2semidirect
17. C2×Dic628C22semidirect
19. C2×#14578943C22semidirect
20. C2×#14578943C22semidirect
22. C2×D1256C22semidirect
28. C4×Dic314C22semidirect
29. C4×D628C22semidirect
34. #19097696C22semidirect
37. #19097693C22semidirect
41. C4⋊C8×C157C22semidirect
42. C4⋊C8×C157C22semidirect
47. #14578954D4semidirect
154. C157#3045semidirect
164. #424D157semidirect
192. C4⋊C8C2×D157semidirect
193. C4⋊C8C2×D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190977992
C2×#190977932
#237578422
#237577532
#237574962
#237574892
#237574742
#237574692
#237572752
C2×#190976932
#424×C1572
#3045157