Finite Group Databasebeta

← #34109747

Subgroups — #34109747

Order 20096.

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

normal subgroups only

This group has 500 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×#19097794C211yesmaximal, characteristic, complemented
2C2×M16×D157C211yesmaximal, characteristic, complemented
3#23757786C212yesmaximal, complemented
4#23757528C211yesmaximal, characteristic, complemented
5#23757527C211yesmaximal, characteristic, complemented
6#23757526C211yesmaximal, characteristic, complemented
7#23757522C212yesmaximal, complemented
8#23757521C212yesmaximal, complemented
9#23757308C211yesmaximal, characteristic, complemented
10#23757268C212yesmaximal, complemented
11#470×C157C211yesmaximal, characteristic, complemented
12C2×#14578955C2212yes—
13C22×C4×D157C2211yescharacteristic
14#19097794—24no—
15C2×#14578941C2212yescomplemented
16C2×D1256C2211yescharacteristic, complemented
17C2×#14578938C2211yescharacteristic
18C2×#14578937C2211yescharacteristic
19C2×#14578937C2211yescharacteristic
20M16×D157—24no—
21C2×C8×D157C2211yescharacteristic
22C2×C8×D157C2211yescharacteristic
23#19097756—22no—
24#19097747C2212yes—
25#19097746C2212yes—
26#19097745C2212yes—
27#19097741—22no—
28#19097733—22no—
29#19097731—22no—
30#19097729C2212yes—
31#19097704C2212yescomplemented
32#19097703C2211yescharacteristic, complemented
33#19097695C2212yescomplemented
34#19097693C2212yescomplemented
35#19097691C2212yescomplemented
36C2×Q16×C157C2211yescharacteristic, complemented
37C2×M16×C157C2211yescharacteristic
38C22⋊Q8×C157C2212yes—
39#106×C157C2211yescharacteristic, complemented
40#103×C157C2212yescomplemented
41C23×D157D411yescharacteristic
42#14578955—24no—
43#14578955—24no—
44C2×D628—22no—
45C2×D628—22no—
46C2×D628C2312yes—
47C2×C22⋊D157—22no—
48C22×Dic157D411yescharacteristic
49C2×C4×D157—22no—
50C2×C4×D157—22no—
51C2×C4×D157—22no—
52C2×C4×D157D411yescharacteristic
53C2×C4×D157D411yescharacteristic
54C2×C4×D157C2311yescharacteristic
55C2×C4×D157C2311yescharacteristic
56#14578941—24no—
57D1256—22no—
58#14578938—22no—
59#14578937—22no—
60#14578937—22no—
61C2×#11929434C2311yescharacteristic
62C2×#11929434C2311yescharacteristic
63C8×D157—22no—
64C8×D157D414yescomplemented
65C4⋊Dic157C2312yes—
66#14578933—22no—
67#14578931—22no—
68#14578931—22no—
69#14578931—22no—
70#14578931—22no—
71C4×Dic157—22no—
72C2×Q8×C157C2312yes—
73C22×C4×C157C2311yescharacteristic
74Q16×C157—22no—
75M16×C157—22no—
76C2×C8×C157C2311yescharacteristic
77C2×C8×C157C2311yescharacteristic
78C4⋊C4×C157—22no—
79C4⋊C4×C157C2312yes—
80C22⋊C4×C157—22no—
81C22×D157—22no—
82C22×D157—22no—
83C22×D157—42no—
84C22×D157C2×D411yescharacteristic
85C22×D157C2×D411yescharacteristic
86D628—22no—
87D628—22no—
88D628—44no—
89D628—44no—
90C22⋊D157—44no—
91C2×Dic157—22no—
92C2×Dic157—22no—
93C2×Dic157C2×D411yescharacteristic
94C2×Dic157C2×D411yescharacteristic
95C4×D157—22no—
96C4×D157—21no—
97C4×D157—21no—
98C4×D157—22no—
99C4×D157—44no—
100C4×D157C2×D412yes—
101C4×D157#4112yes—
102#11929434—21no—
103#11929434C2×D412yes—
104C23×C157C2×D411yescharacteristic
105Q8×C157—22no—
106Q8×C157—22no—
107C2×C4×C157—21no—
108C2×C4×C157—21no—
109C2×C4×C157—22no—
110C2×C4×C157—22no—
111C2×C4×C157C2×D411yescharacteristic
112C2×C4×C157C2411yescharacteristic
113C8×C157—21no—
114C8×C157C2×D412yes—
115C22×C157—21no—
116C22×C157—22no—
117C22×C157C22×D411yescharacteristic
118C2×D157—22no—
119C2×D157—21no—
120C2×D157—22no—
121C2×D157—44no—
122C2×D157—42no—
123C2×D157—42no—
124C2×D157—42no—
125C2×D157#12112yes—
126C4×C157—21no—
127C4×C157—42no—
128C4×C157—42no—
129C4×C157C22×D411yescharacteristic
130C4×C157C2×#4111yescharacteristic
131Dic157—21no—
132Dic157—42no—
133Dic157#12112yes—
134C2×C157—41no—
135C2×C157C2×#12111yescharacteristic
136C2×C157#57512yes—
137D157—22no—
138D157—41no—
139D157—82no—
140C157#305411yescharacteristic, complemented, Sylow
141#3054—1571nomaximal, Sylow
142C2×#135—1571no—
143C22×M16—1571no—
144#525—1572no—
145#470—1571no—
146#470D15711yescharacteristic, complemented
147#469—1571no—
148#469—1571no—
149#468—1572no—
150#468—1572no—
151#427—1571no—
152#416—1572no—
153C2×#41—1572no—
154C23×C4—1571no—
155#135—3144no—
156C2×Q16C2×D15711yescharacteristic, complemented
157C2×SD16—1572no—
158C2×D8—1571no—
159C2×M16—3144no—
160C2×M16—1571no—
161C2×M16—1571no—
162C2×M16—1571no—
163C2×M16C2×D15711yescharacteristic
164C22×C8—1571no—
165C22×C8—1571no—
166#123—3142no—
167C22⋊Q8—1572no—
168C22⋊Q8C2×D15712yes—
169#121—1572no—
170#121—1572no—
171#120—3142no—
172C4×D4—3142no—
173C4×D4—3142no—
174#117—1572no—
175#107—1572no—
176#106—1571no—
177#106C2×D15711yescharacteristic, complemented
178#103—1572no—
179#103C2×D15712yescomplemented
180#102—1572no—
181#102—1572no—
182C24—1571no—
183#41—3144no—
184#41—3144no—
185C2×Q8C22×D15712yes—
186C2×D4—3142no—
187C2×D4—3142no—
188C2×D4—3142no—
189C2×D4—1572no—
190C22×C4—3142no—
191C22×C4—3142no—
192C22×C4—3142no—
193C22×C4—1571no—
194C22×C4—1571no—
195C22×C4—1571no—
196C22×C4—1571no—
197C22×C4—1571no—
198C22×C4C22×D15711yescharacteristic
199Q16—22no—
200SD16—3144no—
201D8—3142no—
202M16—22no—
203M16—3142no—
204M16—3142no—
205M16—3142no—
206C2×C8—3142no—
207C2×C8—1571no—
208C2×C8—1571no—
209C2×C8—1574no—
210C2×C8C22×D15711yescharacteristic
211C2×C8C22×D15711yescharacteristic
212C4⋊C4—22no—
213C4⋊C4—3142no—
214C4⋊C4—1572no—
215C4⋊C4C22×D15712yes—
216C22⋊C4—22no—
217C22⋊C4—3142no—
218C22⋊C4—3142no—
219C22⋊C4—3142no—
220C22⋊C4—3142no—
221C42—3142no—
222C23—3142no—
223C23—3142no—
224C23—6282no—
225C23—1571no—
226C23—1571no—
227C23D4×D15711yescharacteristic
228Q8—22no—
229Q8—22no—
230D4—3142no—
231D4—3142no—
232D4—6284no—
233D4—6284no—
234D4—6284no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—3142no—
240C2×C4—3142no—
241C2×C4—3142no—
242C2×C4—3141no—
243C2×C4—3141no—
244C2×C4—3142no—
245C2×C4—6284no—
246C2×C4—1571no—
247C2×C4—1571no—
248C2×C4—1572no—
249C2×C4—1572no—
250C2×C4D4×D15711yescharacteristic
251C2×C4C23×D15711yescharacteristic
252C8—21no—
253C8—3141no—
254C8—1572no—
255C8D4×D15712yes—
256C22—21no—
257C22—22no—
258C22—3142no—
259C22—3141no—
260C22—3142no—
261C22—6284no—
262C22—6282no—
263C22—6282no—
264C22—6282no—
265C22—1572no—
266C22C2×D4×D15711yescharacteristic
267C4—21no—
268C4—42no—
269C4—42no—
270C4—3141no—
271C4—6282no—
272C4—1572no—
273C4C2×D4×D15711yescharacteristic
274C4#41×D15711yescharacteristic
275C2—41no—
276C2—3142no—
277C2—6281no—
278C2—12562no—
279C2#121×D15711yescharacteristic
280C2#2375807212yes—
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#470×C157C2
socleC22×C157C22×D4
hypercenter#470D157
hypocenterC157#3054
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
1. C2×#19097794C2semidirect
2. C2×M16×D157C2semidirect
3. #23757786C2semidirect
4. #23757528C2semidirect
5. #23757527C2semidirect
6. #23757526C2semidirect
7. #23757522C2semidirect
8. #23757521C2semidirect
9. #23757308C2semidirect
10. #23757268C2semidirect
11. #470×C157C2semidirect
15. C2×#14578941C22semidirect
16. C2×D1256C22semidirect
31. #19097704C22semidirect
32. #19097703C22semidirect
33. #19097695C22semidirect
34. #19097693C22semidirect
35. #19097691C22semidirect
36. C2×Q16×C157C22semidirect
39. #106×C157C22semidirect
40. #103×C157C22semidirect
64. C8×D157D4semidirect
140. C157#3054semidirect
146. #470D157semidirect
156. C2×Q16C2×D157semidirect
177. #106C2×D157semidirect
179. #103C2×D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190977942
C2×M16×D1572
#237577862
#237575282
#237575272
#237575262
#237575222
#237575212
#237573082
#237572682
#470×C1572
#3054157