Finite Group Databasebeta

← #34111736

Subgroups — #34111736

Order 20096.

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

normal subgroups only

This group has 546 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×#19097796C212yesmaximal, complemented
2D4×D628C211yesmaximal, characteristic, complemented
3#23757647C211yesmaximal, characteristic, complemented
4#23757521C212yesmaximal, complemented
5#23757475C211yesmaximal, characteristic, complemented
6#23757468C211yesmaximal, characteristic, complemented
7#23757423C212yesmaximal, complemented
8#23757418C212yesmaximal, complemented
9C4×#14578939C211yesmaximal, characteristic, complemented
10#23757351C211yesmaximal, characteristic, complemented
11#547×C157C211yesmaximal, characteristic, complemented
12C22×D628C2212yes—
13C2×D4×D157—21no—
14#19097796—28no—
15C2×#14578941C2212yes—
16C2×#14578941C2212yescomplemented
17C2×#14578941C2212yescomplemented
18C2×#14578939C2211yescharacteristic
19C2×#14578939C2212yescomplemented
20C2×#14578938C2212yescomplemented
21#19097768C2211yescharacteristic
22#19097746—21no—
23#19097745C2212yes—
24#19097741—22no—
25C4×D628C2211yescharacteristic
26#19097704C2211yescharacteristic, complemented
27C4⋊#11929434C2211yescharacteristic, complemented
28#19097695C2212yescomplemented
29#19097692C2211yescharacteristic, complemented
30#19097691C2211yescharacteristic
31#19097691C2212yescomplemented
32C22⋊#11929434C2212yescomplemented
33C4×#11929434C2211yescharacteristic, complemented
34C2×#41×C157C2212yescomplemented
35C4⋊Q8×C157C2211yescharacteristic, complemented
36#123×C157—22no—
37C22⋊Q8×C157C2212yescomplemented
38#121×C157—21no—
39C4×D4×C157C2211yescharacteristic
40C4×D4×C157C2211yescharacteristic, complemented
41C2×C4⋊C4×C157—21no—
42C23×D157—22no—
43C2×D628—24no—
44C2×D628—22no—
45C2×D628—21no—
46C2×D628—21no—
47C2×D628—42no—
48C2×D628C2312yes—
49C2×D628C2311yescharacteristic
50C2×C22⋊D157—22no—
51D4×D157—44no—
52C2×C4×D157—21no—
53#14578941—24no—
54#14578941—24no—
55#14578941—42no—
56#14578939—24no—
57#14578939D414yescomplemented
58#14578938—24no—
59C2×#11929434C2311yescharacteristic
60C2×#11929434C2312yes—
61C2×#11929434C2311yescharacteristic
62C4⋊Dic157C2311yescharacteristic
63#14578931—21no—
64#14578931—22no—
65#41×C157—24no—
66#41×C157—42no—
67C2×Q8×C157C2312yes—
68C2×D4×C157—22no—
69C2×D4×C157D411yescharacteristic
70C2×D4×C157C2311yescharacteristic
71C22×C4×C157—21no—
72C22×C4×C157—22no—
73C22×C4×C157C2312yes—
74C4⋊C4×C157—21no—
75C4⋊C4×C157—22no—
76C4⋊C4×C157—21no—
77C4⋊C4×C157D411yescharacteristic
78C4⋊C4×C157C2311yescharacteristic
79C22⋊C4×C157—22no—
80C22⋊C4×C157—21no—
81C22⋊C4×C157D412yes—
82C42×C157C2311yescharacteristic
83C22×D157—21no—
84C22×D157—22no—
85C22×D157—44no—
86C22×D157—44no—
87C22×D157—82no—
88D628—22no—
89D628—21no—
90D628—22no—
91D628—44no—
92D628—42no—
93D628—81no—
94D628C2×D412yes—
95C22⋊D157—44no—
96C2×Dic157—21no—
97C4×D157—42no—
98#11929434—22no—
99#11929434—21no—
100#11929434C2×D412yescomplemented
101C23×C157—21no—
102C23×C157C2×D412yes—
103Q8×C157—22no—
104Q8×C157—22no—
105D4×C157—22no—
106D4×C157—21no—
107D4×C157—44no—
108D4×C157C2×D412yescomplemented
109C2×C4×C157—22no—
110C2×C4×C157—22no—
111C2×C4×C157—21no—
112C2×C4×C157—22no—
113C2×C4×C157—44no—
114C2×C4×C157—42no—
115C2×C4×C157C2×D411yescharacteristic
116C2×C4×C157C2×D412yes—
117C2×C4×C157C2×D411yescharacteristic
118C2×C4×C157C2411yescharacteristic
119C22×C157—22no—
120C22×C157—22no—
121C22×C157—22no—
122C22×C157—22no—
123C22×C157—41no—
124C22×C157—41no—
125C22×C157C22×D411yescharacteristic
126C2×D157—22no—
127C2×D157—41no—
128C2×D157—41no—
129C2×D157—42no—
130C2×D157—42no—
131C2×D157—42no—
132C2×D157—88no—
133C4×C157—22no—
134C4×C157—21no—
135C4×C157—42no—
136C4×C157—41no—
137C4×C157AGL(1,ℤ/8)12yescomplemented
138C4×C157C22×D411yescharacteristic
139C4×C157H(5,𝔽2)11yescharacteristic
140Dic157—41no—
141C2×C157—42no—
142C2×C157—42no—
143C2×C157D4211yescharacteristic
144C2×C157C2×AGL(1,ℤ/8)11yescharacteristic
145C2×C157#57711yescharacteristic
146D157—42no—
147D157—82no—
148C157#327211yescharacteristic, complemented, Sylow
149#3272—1571nomaximal, Sylow
150C2×AGL(1,ℤ/8)—1572no—
151#547D15711yescharacteristic, complemented
152D42—1571no—
153#492—1571no—
154#468—1572no—
155#461—1571no—
156#459—1571no—
157#450—1572no—
158#449—1572no—
159C4×D8—1571no—
160#436—1571no—
161C2×#41C2×D15712yescomplemented
162C22×D4—3141no—
163C22×D4—1572no—
164AGL(1,ℤ/8)—3148no—
165C2×SD16—1572no—
166C2×SD16—1572no—
167C2×SD16—1572no—
168C2×D8—1572no—
169C2×D8—1571no—
170C2×M16—1572no—
171C4⋊Q8C2×D15711yescharacteristic, complemented
172#127—1571no—
173#123—22no—
174C22⋊Q8C2×D15712yescomplemented
175#121—21no—
176#121—3141no—
177#121—1572no—
178#120—3142no—
179C4×D4—1571no—
180C4×D4C2×D15711yescharacteristic
181C4×D4C2×D15711yescharacteristic, complemented
182C2×C4⋊C4—21no—
183#107—1571no—
184C4⋊C8—1571no—
185#103—1572no—
186#102—1571no—
187#102—1571no—
188#102—1572no—
189C22⋊C8—1572no—
190C4×C8—1571no—
191C24—3142no—
192#41—24no—
193#41—42no—
194C2×Q8C22×D15712yes—
195C2×D4—22no—
196C2×D4—3144no—
197C2×D4—3142no—
198C2×D4—3142no—
199C2×D4—3141no—
200C2×D4—3141no—
201C2×D4—6282no—
202C2×D4—6284no—
203C2×D4—1572no—
204C2×D4—1571no—
205C2×D4C22⋊D15711yescharacteristic
206C2×D4C22×D15711yescharacteristic
207C22×C4—21no—
208C22×C4—22no—
209C22×C4—3141no—
210C22×C4C22×D15712yes—
211SD16—3144no—
212SD16—3144no—
213SD16—6282no—
214D8—3144no—
215D8—1574no—
216M16—3144no—
217C2×C8—1571no—
218C2×C8—1572no—
219C2×C8—1571no—
220C4⋊C4—21no—
221C4⋊C4—22no—
222C4⋊C4—21no—
223C4⋊C4—1571no—
224C4⋊C4C22⋊D15711yescharacteristic
225C4⋊C4C22×D15711yescharacteristic
226C22⋊C4—22no—
227C22⋊C4—21no—
228C22⋊C4—3141no—
229C22⋊C4—3142no—
230C22⋊C4C22⋊D15712yes—
231C42C22×D15711yescharacteristic
232C23—21no—
233C23—3141no—
234C23—3142no—
235C23—6284no—
236C23—6284no—
237C23—12562no—
238C23C2×C22⋊D15712yes—
239Q8—22no—
240Q8—22no—
241D4—22no—
242D4—21no—
243D4—44no—
244D4—3142no—
245D4—3141no—
246D4—3142no—
247D4—6284no—
248D4—6284no—
249D4—6282no—
250D4—1572no—
251D4—12561no—
252D4D4×D15712yescomplemented
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—44no—
258C2×C4—42no—
259C2×C4—3141no—
260C2×C4—6282no—
261C2×C4C2×C22⋊D15711yescharacteristic
262C2×C4C2×C22⋊D15712yes—
263C2×C4C2×C22⋊D15711yescharacteristic
264C2×C4C23×D15711yescharacteristic
265C8—3142no—
266C8—3141no—
267C8—1572no—
268C22—22no—
269C22—22no—
270C22—22no—
271C22—22no—
272C22—41no—
273C22—41no—
274C22—3142no—
275C22—6281no—
276C22—6281no—
277C22—6282no—
278C22—6282no—
279C22—6282no—
280C22—12568no—
281C22C22×C22⋊D15711yescharacteristic
282C4—21no—
283C4—22no—
284C4—42no—
285C4—41no—
286C4—6281no—
287C4#1909779612yescomplemented
288C4C2×D4×D15711yescharacteristic
289C4#1909781611yescharacteristic
290C2—42no—
291C2—42no—
292C2—6282no—
293C2—12562no—
294C2D4×C22⋊D15711yescharacteristic
295C2C2×#1909779611yescharacteristic
296C2#2375808111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#547×C157C2
socleC22×C157C22×D4
hypercenter#547D157
hypocenterC157#3272
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. C2×#19097796C2semidirect
2. D4×D628C2semidirect
3. #23757647C2semidirect
4. #23757521C2semidirect
5. #23757475C2semidirect
6. #23757468C2semidirect
7. #23757423C2semidirect
8. #23757418C2semidirect
9. C4×#14578939C2semidirect
10. #23757351C2semidirect
11. #547×C157C2semidirect
16. C2×#14578941C22semidirect
17. C2×#14578941C22semidirect
19. C2×#14578939C22semidirect
20. C2×#14578938C22semidirect
26. #19097704C22semidirect
27. C4⋊#11929434C22semidirect
28. #19097695C22semidirect
29. #19097692C22semidirect
31. #19097691C22semidirect
32. C22⋊#11929434C22semidirect
33. C4×#11929434C22semidirect
34. C2×#41×C157C22semidirect
35. C4⋊Q8×C157C22semidirect
37. C22⋊Q8×C157C22semidirect
40. C4×D4×C157C22semidirect
57. #14578939D4semidirect
100. #11929434C2×D4semidirect
108. D4×C157C2×D4semidirect
137. C4×C157AGL(1,ℤ/8)semidirect
148. C157#3272semidirect
151. #547D157semidirect
161. C2×#41C2×D157semidirect
171. C4⋊Q8C2×D157semidirect
174. C22⋊Q8C2×D157semidirect
181. C4×D4C2×D157semidirect
252. D4D4×D157semidirect
287. C4#19097796semidirect

Maximal subgroups

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

subgroupindex
C2×#190977962
D4×D6282
#237576472
#237575212
#237574752
#237574682
#237574232
#237574182
C4×#145789392
#237573512
#547×C1572
#3272157