Finite Group Databasebeta

← #34110138

Subgroups — #34110138

Order 20096.

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

normal subgroups only

This group has 492 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
1#23757824C211yesmaximal, characteristic, complemented
2C4×D4×D157C211yesmaximal, characteristic, complemented
3#23757588C212yesmaximal, complemented
4#23757550C211yesmaximal, characteristic, complemented
5#23757469C211yesmaximal, characteristic, complemented
6#23757468C211yesmaximal, characteristic, complemented
7#23757408C212yesmaximal, complemented
8#23757355C212yesmaximal, complemented
9#23757353C212yesmaximal, complemented
10#23757325C211yesmaximal, characteristic, complemented
11#474×C157C211yesmaximal, characteristic, complemented
12C2×#14578955—21no—
13C2×D4×D157C2211yescharacteristic
14C22×C4×D157—21no—
15C2×D1256C2212yes—
16C2×#14578939C2212yes—
17#19097768C2211yescharacteristic, complemented
18#19097760—21no—
19#19097746—21no—
20#19097746C2212yes—
21C4×D628C2211yescharacteristic
22D4×Dic157C2211yescharacteristic
23#19097733—21no—
24#19097733C2211yescharacteristic
25#19097733C2212yescomplemented
26C4×C22⋊D157—21no—
27#19097731—21no—
28C4⋊C4×D157C2211yescharacteristic
29C22⋊C4×D157—21no—
30C42×D157C2211yescharacteristic
31#19097704C2212yescomplemented
32C4⋊#11929434C2211yescharacteristic, complemented
33#19097693C2212yescomplemented
34#19097691C2212yes—
35#19097691C2212yescomplemented
36#19097683C2212yes—
37C8×Dic157C2212yescomplemented
38C4⋊Q8×C157C2211yescharacteristic, complemented
39C4×D4×C157C2211yescharacteristic
40#107×C157C2212yescomplemented
41C4⋊C8×C157C2211yescharacteristic, complemented
42#102×C157C2212yes—
43C23×D157—21no—
44#14578955—44no—
45C2×D628—22no—
46C2×D628—21no—
47C2×D628—21no—
48C2×D628C2312yes—
49C2×D628C2311yescharacteristic
50C2×C22⋊D157—21no—
51D4×D157—22no—
52D4×D157—22no—
53C22×Dic157—21no—
54C2×C4×D157—21no—
55C2×C4×D157—21no—
56C2×C4×D157—22no—
57C2×C4×D157—21no—
58C2×C4×D157—44no—
59C2×C4×D157D411yescharacteristic
60C2×C4×D157D411yescharacteristic
61C2×C4×D157C2311yescharacteristic
62D1256—24no—
63#14578939—24no—
64C2×#11929434C2312yes—
65C4⋊Dic157C2311yescharacteristic
66#14578933—21no—
67C22⋊Dic157—21no—
68#14578931—21no—
69#14578931—22no—
70#14578931—22no—
71#14578931—21no—
72C4×Dic157D411yescharacteristic
73C4×Dic157D411yescharacteristic
74C4×Dic157C2311yescharacteristic
75C2×Q8×C157—21no—
76C2×D4×C157C2311yescharacteristic
77C22×C4×C157—21no—
78C2×C8×C157C2312yes—
79C4⋊C4×C157—21no—
80C4⋊C4×C157C2312yes—
81C4⋊C4×C157C2311yescharacteristic
82C22⋊C4×C157—21no—
83C42×C157C2311yescharacteristic
84C22×D157—24no—
85C22×D157—21no—
86C22×D157—22no—
87C22×D157—42no—
88C22×D157—42no—
89C22×D157C2×D411yescharacteristic
90D628—21no—
91D628—22no—
92D628—22no—
93D628—42no—
94D628—44no—
95D628—42no—
96D628#4112yescomplemented
97C22⋊D157—24no—
98C2×Dic157—21no—
99C2×Dic157—42no—
100C2×Dic157C2×D411yescharacteristic
101C2×Dic157C2×D411yescharacteristic
102C2×Dic157C2×D411yescharacteristic
103C4×D157—21no—
104C4×D157—21no—
105C4×D157—41no—
106C4×D157—42no—
107C4×D157—42no—
108C4×D157—42no—
109C4×D157—44no—
110C4×D157D814yescomplemented
111#11929434—22no—
112C23×C157—21no—
113Q8×C157—42no—
114D4×C157—21no—
115D4×C157#4112yescomplemented
116C2×C4×C157—21no—
117C2×C4×C157—22no—
118C2×C4×C157—42no—
119C2×C4×C157C2×D411yescharacteristic
120C2×C4×C157C2×D411yescharacteristic
121C2×C4×C157C2411yescharacteristic
122C8×C157—22no—
123C22×C157—22no—
124C22×C157—41no—
125C22×C157—41no—
126C22×C157C22×D411yescharacteristic
127C2×D157—21no—
128C2×D157—21no—
129C2×D157—22no—
130C2×D157—48no—
131C2×D157—41no—
132C2×D157—41no—
133C2×D157—42no—
134C2×D157—42no—
135C2×D157—42no—
136C2×D157C2×D812yes—
137C4×C157—21no—
138C4×C157—41no—
139C4×C157—42no—
140C4×C157C2×D812yes—
141C4×C157C2×#4111yescharacteristic
142C4×C157C2×#4111yescharacteristic
143Dic157—21no—
144Dic157—21no—
145Dic157—41no—
146Dic157C2×D812yes—
147C2×C157—42no—
148C2×C157#52511yescharacteristic
149C2×C157C22×D811yescharacteristic
150C2×C157#57511yescharacteristic
151D157—22no—
152D157—42no—
153D157—82no—
154C157#310011yescharacteristic, complemented, Sylow
155#3100—1571nomaximal, Sylow
156#532—1571no—
157C2×C4×D4—1571no—
158#480—1572no—
159#474—1571no—
160#474D15711yescharacteristic, complemented
161#459—1571no—
162#459—1571no—
163#447—1572no—
164C4×D8—1572no—
165C4×D8—1572no—
166#431—1571no—
167C2×#41—3141no—
168C22×D4—1571no—
169C23×C4—3141no—
170C2×D8—1572no—
171C2×D8—1572no—
172C4⋊Q8C2×D15711yescharacteristic, complemented
173#127—1571no—
174#124—3141no—
175#121—3141no—
176#121—1572no—
177C4×D4—3141no—
178C4×D4—3141no—
179C4×D4—3141no—
180C4×D4—1571no—
181C4×D4—1571no—
182C4×D4—1572no—
183C4×D4—1571no—
184C4×D4C2×D15711yescharacteristic
185C2×C4⋊C4—1571no—
186C2×C22⋊C4—3141no—
187C2×C42—1571no—
188#107—1572no—
189#107C2×D15712yescomplemented
190C4⋊C8—1571no—
191C4⋊C8C2×D15711yescharacteristic, complemented
192#102—1572no—
193#102—1572no—
194#102—1572no—
195#102C2×D15712yes—
196C22⋊C8—1572no—
197C4×C8—1572no—
198C24—3141no—
199#41—6284no—
200C2×Q8—21no—
201C2×D4—3142no—
202C2×D4—3142no—
203C2×D4—3142no—
204C2×D4—3141no—
205C2×D4—3141no—
206C2×D4—3141no—
207C2×D4—1572no—
208C2×D4—1571no—
209C2×D4C22×D15711yescharacteristic
210C22×C4—21no—
211C22×C4—3141no—
212C22×C4—3141no—
213C22×C4—3142no—
214C22×C4—3141no—
215C22×C4—3141no—
216C22×C4—6284no—
217C22×C4—1571no—
218C22×C4—1571no—
219C22×C4—1571no—
220D8—3144no—
221D8—3144no—
222C2×C8—1572no—
223C2×C8C22×D15712yes—
224C4⋊C4—21no—
225C4⋊C4—3141no—
226C4⋊C4—1571no—
227C4⋊C4C22×D15711yescharacteristic
228C4⋊C4C22×D15712yes—
229C22⋊C4—21no—
230C22⋊C4—3141no—
231C22⋊C4—3142no—
232C22⋊C4—3142no—
233C22⋊C4—3141no—
234C22⋊C4—3141no—
235C42—1571no—
236C42—1571no—
237C42—1571no—
238C42C22×D15711yescharacteristic
239C23—21no—
240C23—3144no—
241C23—3141no—
242C23—3142no—
243C23—6282no—
244C23—6282no—
245C23—1571no—
246Q8—42no—
247D4—21no—
248D4—3144no—
249D4—3142no—
250D4—3141no—
251D4—3142no—
252D4—6282no—
253D4—6284no—
254D4—6282no—
255D4—1572no—
256D4#1457895512yescomplemented
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—42no—
260C2×C4—3141no—
261C2×C4—3141no—
262C2×C4—3141no—
263C2×C4—6281no—
264C2×C4—6282no—
265C2×C4—6282no—
266C2×C4—6284no—
267C2×C4—6282no—
268C2×C4—6282no—
269C2×C4—1571no—
270C2×C4—1571no—
271C2×C4—1571no—
272C2×C4—1574no—
273C2×C4D4×D15711yescharacteristic
274C2×C4D4×D15711yescharacteristic
275C2×C4C23×D15711yescharacteristic
276C8—22no—
277C8—3142no—
278C22—22no—
279C22—41no—
280C22—41no—
281C22—3141no—
282C22—3141no—
283C22—3142no—
284C22—6288no—
285C22—6281no—
286C22—6281no—
287C22—6282no—
288C22—6282no—
289C22—6282no—
290C22—1572no—
291C22C2×D4×D15711yescharacteristic
292C4—21no—
293C4—41no—
294C4—42no—
295C4—3141no—
296C4—3141no—
297C4—6281no—
298C4—1572no—
299C4D8×D15712yes—
300C4#41×D15711yescharacteristic
301C4C2×#1457895511yescharacteristic
302C2—42no—
303C2—3142no—
304C2—6282no—
305C2—12562no—
306C2#2375778611yescharacteristic
307C2C2×D8×D15711yescharacteristic
308C2#2375807211yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#474×C157C2
socleC22×C157C22×D4
hypercenter#474D157
hypocenterC157#3100
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
1. #23757824C2semidirect
2. C4×D4×D157C2semidirect
3. #23757588C2semidirect
4. #23757550C2semidirect
5. #23757469C2semidirect
6. #23757468C2semidirect
7. #23757408C2semidirect
8. #23757355C2semidirect
9. #23757353C2semidirect
10. #23757325C2semidirect
11. #474×C157C2semidirect
17. #19097768C22semidirect
25. #19097733C22semidirect
31. #19097704C22semidirect
32. C4⋊#11929434C22semidirect
33. #19097693C22semidirect
35. #19097691C22semidirect
37. C8×Dic157C22semidirect
38. C4⋊Q8×C157C22semidirect
40. #107×C157C22semidirect
41. C4⋊C8×C157C22semidirect
96. D628#41semidirect
110. C4×D157D8semidirect
115. D4×C157#41semidirect
154. C157#3100semidirect
160. #474D157semidirect
172. C4⋊Q8C2×D157semidirect
189. #107C2×D157semidirect
191. C4⋊C8C2×D157semidirect
256. D4#14578955semidirect

Maximal subgroups

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

subgroupindex
#237578242
C4×D4×D1572
#237575882
#237575502
#237574692
#237574682
#237574082
#237573552
#237573532
#237573252
#474×C1572
#3100157