Finite Group Databasebeta

← #34110057

Subgroups — #34110057

Order 20096.

458 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 129 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
1#23757780C211yesmaximal, characteristic
2C2×C4⋊C4×D157C211yesmaximal, characteristic, complemented
3#23757607C212yesmaximal, complemented
4#23757602C212yesmaximal, complemented
5#23757590C212yesmaximal, complemented
6#23757585C212yesmaximal, complemented
7#23757315C211yesmaximal, characteristic, complemented
8#23757313C211yesmaximal, characteristic, complemented
9#23757274C211yesmaximal, characteristic, complemented
10#23757236C211yesmaximal, characteristic, complemented
11#417×C157C211yesmaximal, characteristic, complemented
12C22×C4×D157—21no—
13C22×C4×D157C2211yescharacteristic
14C2×#14578938C2211yescharacteristic
15#19097752—21no—
16#19097747C2211yescharacteristic
17#19097747C2211yescharacteristic
18#19097742C2211yescharacteristic
19#19097742C2211yescharacteristic
20#19097739—21no—
21C4×C22⋊D157—21no—
22C4×C22⋊D157—21no—
23#19097730C2211yescharacteristic
24C2×C4⋊Dic157C2211yescharacteristic
25C2×#14578933—21no—
26C4⋊C4×D157—21no—
27C4⋊C4×D157—21no—
28C4⋊C4×D157C2212yes—
29C4⋊C4×D157C2212yes—
30#19097705C2212yescomplemented
31#19097704C2212yescomplemented
32#19097703C2212yescomplemented
33#19097702C2212yescomplemented
34#19097695C2212yescomplemented
35#19097692C2212yescomplemented
36#19097683C2212yes—
37#19097683C2212yes—
38C2×#41×C157C2211yescharacteristic
39C2×M16×C157C2211yescharacteristic
40C2×C4⋊C4×C157C2211yescharacteristic
41#103×C157C2212yescomplemented
42#102×C157C2212yescomplemented
43C23×D157D411yescharacteristic
44C2×C22⋊D157—21no—
45C2×C22⋊D157—21no—
46C22×Dic157—21no—
47C22×Dic157D411yescharacteristic
48C2×C4×D157—22no—
49C2×C4×D157—22no—
50C2×C4×D157—22no—
51C2×C4×D157—22no—
52C2×C4×D157—44no—
53C2×C4×D157D411yescharacteristic
54C2×C4×D157D411yescharacteristic
55C2×C4×D157C2311yescharacteristic
56C2×C4×D157C2311yescharacteristic
57#14578938—22no—
58C2×#11929434C2312yes—
59C4⋊Dic157—21no—
60C4⋊Dic157C2312yes—
61C4⋊Dic157C2311yescharacteristic
62C4⋊Dic157C2311yescharacteristic
63#14578933—22no—
64#14578933—22no—
65#14578933—21no—
66#14578933—21no—
67C22⋊Dic157—21no—
68C22⋊Dic157—21no—
69C22⋊Dic157—21no—
70#14578931—21no—
71#14578931—21no—
72C4×Dic157—21no—
73#41×C157—22no—
74#41×C157—22no—
75C2×Q8×C157C2311yescharacteristic
76C2×D4×C157—21no—
77C2×D4×C157C2311yescharacteristic
78C22×C4×C157—21no—
79C22×C4×C157—21no—
80C22×C4×C157C2311yescharacteristic
81M16×C157—22no—
82C2×C8×C157C2312yes—
83C4⋊C4×C157—21no—
84C4⋊C4×C157C2312yes—
85C22×D157—22no—
86C22×D157—22no—
87C22×D157—22no—
88C22×D157C2×D411yescharacteristic
89C22×D157C2×D411yescharacteristic
90C22⋊D157—42no—
91C22⋊D157—42no—
92C2×Dic157—22no—
93C2×Dic157—22no—
94C2×Dic157—21no—
95C2×Dic157—21no—
96C2×Dic157—42no—
97C2×Dic157C2×D411yescharacteristic
98C2×Dic157C2×D411yescharacteristic
99C4×D157—22no—
100C4×D157—22no—
101C4×D157—21no—
102C4×D157—21no—
103C4×D157—21no—
104C4×D157—21no—
105C4×D157—44no—
106C4×D157—44no—
107#11929434—22no—
108C23×C157—21no—
109C23×C157C2×D411yescharacteristic
110Q8×C157—21no—
111Q8×C157—21no—
112D4×C157—21no—
113D4×C157—21no—
114D4×C157—42no—
115C2×C4×C157—22no—
116C2×C4×C157—21no—
117C2×C4×C157—42no—
118C2×C4×C157—42no—
119C2×C4×C157C2×D411yescharacteristic
120C2×C4×C157#4112yes—
121C2×C4×C157#4112yes—
122C2×C4×C157C2411yescharacteristic
123C8×C157—22no—
124C22×C157—21no—
125C22×C157—21no—
126C22×C157—41no—
127C22×C157—41no—
128C22×C157—41no—
129C22×C157#12312yes—
130C22×C157C22×D411yescharacteristic
131C2×D157—28no—
132C2×D157—21no—
133C2×D157—21no—
134C2×D157—21no—
135C2×D157—21no—
136C2×D157AGL(1,ℤ/8)12yes—
137C2×D157#13712yes—
138C4×C157—41no—
139C4×C157—42no—
140C4×C157#12312yes—
141C4×C157C2×#4111yescharacteristic
142C4×C157C2×#4111yescharacteristic
143Dic157—21no—
144Dic157—21no—
145Dic157—42no—
146Dic157—41no—
147Dic157—41no—
148C2×C157—22no—
149C2×C157—81no—
150C2×C157C2×#12311yescharacteristic
151C2×C157C2×AGL(1,ℤ/8)11yescharacteristic
152C2×C157C2×#13711yescharacteristic
153D157—22no—
154D157—22no—
155C157#309211yescharacteristic, complemented, Sylow
156#3092—1571nomaximal, Sylow
157#525—1571no—
158C22×C4⋊C4—1571no—
159#484—1572no—
160#483—1572no—
161#481—1572no—
162#480—1572no—
163#428—1571no—
164#428—1571no—
165#417—1571no—
166#417D15711yescharacteristic, complemented
167#407—1571no—
168C2×#41C2×D15711yescharacteristic
169C23×C4—3141no—
170C23×C4—1571no—
171C2×M16—1571no—
172C2×M16C2×D15711yescharacteristic
173#123—3141no—
174C22⋊Q8—1571no—
175C22⋊Q8—1571no—
176#121—1571no—
177#121—1571no—
178#120—3141no—
179C4×D4—3141no—
180C4×D4—3141no—
181#117—1571no—
182C2×C4⋊C4—3141no—
183C2×C4⋊C4—3141no—
184C2×C4⋊C4—3141no—
185C2×C4⋊C4—1571no—
186C2×C4⋊C4—1572no—
187C2×C4⋊C4—1572no—
188C2×C4⋊C4C2×D15711yescharacteristic
189#107—1572no—
190#107—1572no—
191#106—1572no—
192#106—1572no—
193#103—1572no—
194#103C2×D15712yescomplemented
195#102—1572no—
196#102C2×D15712yescomplemented
197C22⋊C8—1572no—
198C22⋊C8—1572no—
199C24—1571no—
200#41—22no—
201#41—22no—
202C2×Q8C22×D15711yescharacteristic
203C2×D4—21no—
204C2×D4—3141no—
205C2×D4—3141no—
206C2×D4C22×D15711yescharacteristic
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—3142no—
210C22×C4—3142no—
211C22×C4—3142no—
212C22×C4—3142no—
213C22×C4—3141no—
214C22×C4—6284no—
215C22×C4—1571no—
216C22×C4—1571no—
217C22×C4—1571no—
218C22×C4—1571no—
219C22×C4—1571no—
220C22×C4C22×D15711yescharacteristic
221M16—22no—
222M16—3142no—
223C2×C8—1572no—
224C2×C8C22×D15712yes—
225C4⋊C4—21no—
226C4⋊C4—3142no—
227C4⋊C4—3142no—
228C4⋊C4—3141no—
229C4⋊C4—3141no—
230C4⋊C4—3141no—
231C4⋊C4—1572no—
232C4⋊C4—1571no—
233C4⋊C4—1571no—
234C4⋊C4C22×D15712yes—
235C22⋊C4—3141no—
236C22⋊C4—3141no—
237C22⋊C4—3141no—
238C22⋊C4—3141no—
239C22⋊C4—3141no—
240C42—3141no—
241C23—21no—
242C23—3142no—
243C23—3142no—
244C23—3142no—
245C23—1571no—
246C23—1571no—
247C23D4×D15711yescharacteristic
248Q8—21no—
249Q8—21no—
250D4—21no—
251D4—21no—
252D4—42no—
253D4—6282no—
254D4—6282no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—3142no—
260C2×C4—3142no—
261C2×C4—3142no—
262C2×C4—3141no—
263C2×C4—3141no—
264C2×C4—3141no—
265C2×C4—3141no—
266C2×C4—3141no—
267C2×C4—3141no—
268C2×C4—3142no—
269C2×C4—6284no—
270C2×C4—6284no—
271C2×C4—6282no—
272C2×C4—1571no—
273C2×C4—1571no—
274C2×C4D4×D15711yescharacteristic
275C2×C4#1457895312yes—
276C2×C4#1457895412yes—
277C2×C4C23×D15711yescharacteristic
278C8—22no—
279C8—3142no—
280C22—21no—
281C22—21no—
282C22—41no—
283C22—41no—
284C22—41no—
285C22—3148no—
286C22—3141no—
287C22—3141no—
288C22—3141no—
289C22—3141no—
290C22—1572no—
291C22—1572no—
292C22#1909775312yes—
293C22C2×D4×D15711yescharacteristic
294C4—41no—
295C4—42no—
296C4—3141no—
297C4—3141no—
298C4—6282no—
299C4—6281no—
300C4—6281no—
301C4#1909775312yes—
302C4C2×#1457895311yescharacteristic
303C4C2×#1457895411yescharacteristic
304C2—22no—
305C2—81no—
306C2—3142no—
307C2—3142no—
308C2C2×#1909775311yescharacteristic
309C2AGL(1,ℤ/8)×D15711yescharacteristic
310C2#137×D15711yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#417×C157C2
socleC22×C157C22×D4
prime radical#23757780C2
hypercenter#417D157
hypocenterC157#3092
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×D157C2semidirect
3. #23757607C2semidirect
4. #23757602C2semidirect
5. #23757590C2semidirect
6. #23757585C2semidirect
7. #23757315C2semidirect
8. #23757313C2semidirect
9. #23757274C2semidirect
10. #23757236C2semidirect
11. #417×C157C2semidirect
30. #19097705C22semidirect
31. #19097704C22semidirect
32. #19097703C22semidirect
33. #19097702C22semidirect
34. #19097695C22semidirect
35. #19097692C22semidirect
41. #103×C157C22semidirect
42. #102×C157C22semidirect
155. C157#3092semidirect
166. #417D157semidirect
194. #103C2×D157semidirect
196. #102C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237577802
C2×C4⋊C4×D1572
#237576072
#237576022
#237575902
#237575852
#237573152
#237573132
#237572742
#237572362
#417×C1572
#3092157