Finite Group Databasebeta

← #34109782

Subgroups — #34109782

Order 20096.

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

normal subgroups only

This group has 524 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×D8×D157C211yesmaximal, characteristic, complemented
2C2×#19097779C211yesmaximal, characteristic, complemented
3#23757839C212yesmaximal, complemented
4#23757527C211yesmaximal, characteristic, complemented
5#23757526C211yesmaximal, characteristic, complemented
6#23757523C211yesmaximal, characteristic, complemented
7#23757508C212yesmaximal, complemented
8#23757507C212yesmaximal, complemented
9#23757308C211yesmaximal, characteristic, complemented
10#23757276C212yesmaximal, complemented
11#469×C157C211yesmaximal, characteristic, complemented
12C2×#14578954C2211yescharacteristic
13C2×D4×D157C2212yes—
14C2×D1256C2211yescharacteristic
15C2×#14578939C2212yes—
16D8×D157—24no—
17#19097779—24no—
18C2×#14578937C2211yescharacteristic
19C2×#14578937C2211yescharacteristic, complemented
20C22×#11929434C2211yescharacteristic, complemented
21C2×C8×D157C2211yescharacteristic
22C2×C8×D157C2211yescharacteristic, complemented
23#19097767—22no—
24#19097757—22no—
25#19097746C2212yescomplemented
26#19097745C2212yescomplemented
27#19097742C2212yescomplemented
28#19097741—22no—
29#19097739—22no—
30#19097729C2212yes—
31#19097704C2212yescomplemented
32#19097703C2211yescharacteristic, complemented
33#19097693C2212yescomplemented
34#19097692C2212yescomplemented
35#19097691C2212yescomplemented
36C2×D8×C157C2211yescharacteristic
37C2×M16×C157C2211yescharacteristic, complemented
38#121×C157C2212yescomplemented
39#106×C157C2211yescharacteristic, complemented
40#102×C157C2212yescomplemented
41C23×D157—22no—
42#14578954—22no—
43#14578954—22no—
44C2×Dic314D411yescharacteristic
45C2×D628—22no—
46C2×D628D411yescharacteristic
47C2×D628C2312yes—
48C2×C22⋊D157—22no—
49C2×C22⋊D157—22no—
50C2×C22⋊D157—22no—
51C2×C22⋊D157D411yescharacteristic
52C2×C22⋊D157D411yescharacteristic
53D4×D157—24no—
54D4×D157—24no—
55C2×C4×D157C2311yescharacteristic
56C2×C4×D157C2311yescharacteristic
57D1256—22no—
58#14578939—24no—
59#14578937—22no—
60#14578937—22no—
61C2×#11929434—22no—
62C2×#11929434C2311yescharacteristic
63C2×#11929434C2311yescharacteristic
64C8×D157—22no—
65C8×D157D414yescomplemented
66C4⋊Dic157C2312yes—
67C22⋊Dic157—22no—
68#14578931—22no—
69#14578931—22no—
70#14578931—22no—
71C4×Dic157—22no—
72C2×D4×C157—22no—
73C2×D4×C157C2312yes—
74C22×C4×C157C2311yescharacteristic
75D8×C157—22no—
76M16×C157—22no—
77C2×C8×C157C2311yescharacteristic
78C2×C8×C157C2311yescharacteristic
79C4⋊C4×C157C2312yes—
80C22⋊C4×C157—22no—
81C22×D157—22no—
82C22×D157—44no—
83C22×D157—82no—
84C22×D157—82no—
85C22×D157C2×D411yescharacteristic
86C22×D157C2×D411yescharacteristic
87Dic314—22no—
88D628—22no—
89D628—22no—
90D628—22no—
91D628—82no—
92C22⋊D157—22no—
93C22⋊D157—22no—
94C22⋊D157—44no—
95C22⋊D157—82no—
96C22⋊D157—82no—
97C2×Dic157—22no—
98C2×Dic157C2×D411yescharacteristic
99C2×Dic157C2×D411yescharacteristic
100C4×D157—21no—
101C4×D157—21no—
102C4×D157C2×D412yes—
103C4×D157#4112yes—
104#11929434—21no—
105#11929434C2×D412yes—
106C23×C157—22no—
107C23×C157C2×D411yescharacteristic
108D4×C157—22no—
109D4×C157—22no—
110D4×C157—82no—
111C2×C4×C157—21no—
112C2×C4×C157—21no—
113C2×C4×C157—22no—
114C2×C4×C157C2×D411yescharacteristic
115C2×C4×C157C2411yescharacteristic
116C8×C157—21no—
117C8×C157C2×D412yes—
118C22×C157—21no—
119C22×C157—22no—
120C22×C157—42no—
121C22×C157—42no—
122C22×C157—42no—
123C22×C157C22×D411yescharacteristic
124C2×D157—22no—
125C2×D157—21no—
126C2×D157—22no—
127C2×D157—42no—
128C2×D157—42no—
129C2×D157—42no—
130C2×D157—88no—
131C2×D157#12112yes—
132C4×C157—21no—
133C4×C157—42no—
134C4×C157C22×D411yescharacteristic
135C4×C157C2×#4111yescharacteristic
136Dic157—21no—
137Dic157—42no—
138Dic157#12112yes—
139C2×C157—41no—
140C2×C157—82no—
141C2×C157C2×#12111yescharacteristic
142C2×C157#57612yes—
143D157—22no—
144D157—41no—
145D157—82no—
146C157#305811yescharacteristic, complemented, Sylow
147#3058—1571nomaximal, Sylow
148C22×D8—1571no—
149C2×#131—1571no—
150#535—1572no—
151#469—1571no—
152#469—1571no—
153#469—1571no—
154#469D15711yescharacteristic, complemented
155#466—1572no—
156#466—1572no—
157#427—1571no—
158#418—1572no—
159C2×#41—1571no—
160C22×D4—1572no—
161C2×D8—3144no—
162C2×D8—1572no—
163C2×D8—1571no—
164C2×D8C2×D15711yescharacteristic
165#131—3144no—
166C2×M16—1571no—
167C2×M16—1571no—
168C2×M16C2×D15711yescharacteristic, complemented
169C22×C8—1571no—
170C22×C8—1571no—
171C22×C8—1571no—
172#127—3142no—
173#124—3142no—
174#121—1572no—
175#121—1572no—
176#121—1572no—
177#121C2×D15712yescomplemented
178#120—3142no—
179#120—3142no—
180#117—1572no—
181#107—1572no—
182#106—1571no—
183#106C2×D15711yescharacteristic, complemented
184#102—1572no—
185#102—1572no—
186#102—1572no—
187#102C2×D15712yescomplemented
188C24—3142no—
189#41—3142no—
190#41—3142no—
191C2×Q8—1571no—
192C2×D4—22no—
193C2×D4—3142no—
194C2×D4—3142no—
195C2×D4—3142no—
196C2×D4—3144no—
197C2×D4—3144no—
198C2×D4—3142no—
199C2×D4—1571no—
200C2×D4—1571no—
201C2×D4—1571no—
202C2×D4—1572no—
203C2×D4C22×D15712yes—
204C22×C4—1571no—
205C22×C4—1571no—
206C22×C4C22×D15711yescharacteristic
207D8—22no—
208D8—3144no—
209D8—3142no—
210M16—22no—
211M16—3142no—
212M16—3142no—
213C2×C8—3142no—
214C2×C8—3142no—
215C2×C8—1571no—
216C2×C8—1571no—
217C2×C8—1574no—
218C2×C8C22×D15711yescharacteristic
219C2×C8C22×D15711yescharacteristic
220C4⋊C4—1572no—
221C4⋊C4C22×D15712yes—
222C22⋊C4—22no—
223C22⋊C4—3142no—
224C22⋊C4—3142no—
225C22⋊C4—3142no—
226C22⋊C4—3142no—
227C42—3142no—
228C23—22no—
229C23—3142no—
230C23—6284no—
231C23—1571no—
232C23—1571no—
233C23—12562no—
234C23—12562no—
235C23D4×D15711yescharacteristic
236Q8—3142no—
237D4—22no—
238D4—22no—
239D4—82no—
240D4—3142no—
241D4—3142no—
242D4—3142no—
243D4—3142no—
244D4—3142no—
245D4—6284no—
246D4—12562no—
247D4—12562no—
248D4—12562no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—3141no—
253C2×C4—3141no—
254C2×C4—3142no—
255C2×C4—1571no—
256C2×C4—1571no—
257C2×C4—1572no—
258C2×C4—1572no—
259C2×C4D4×D15711yescharacteristic
260C2×C4C23×D15711yescharacteristic
261C8—21no—
262C8—3141no—
263C8—1572no—
264C8D4×D15712yes—
265C22—21no—
266C22—22no—
267C22—42no—
268C22—42no—
269C22—42no—
270C22—3142no—
271C22—3141no—
272C22—3142no—
273C22—6282no—
274C22—6282no—
275C22—6282no—
276C22—1572no—
277C22—12568no—
278C22C2×D4×D15711yescharacteristic
279C4—21no—
280C4—42no—
281C4—3141no—
282C4—6282no—
283C4—1572no—
284C4C2×D4×D15711yescharacteristic
285C4#41×D15711yescharacteristic
286C2—41no—
287C2—82no—
288C2—3142no—
289C2—6281no—
290C2—12562no—
291C2#121×D15711yescharacteristic
292C2#2375807712yes—
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#469×C157C2
socleC22×C157C22×D4
hypercenter#469D157
hypocenterC157#3058
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. C2×D8×D157C2semidirect
2. C2×#19097779C2semidirect
3. #23757839C2semidirect
4. #23757527C2semidirect
5. #23757526C2semidirect
6. #23757523C2semidirect
7. #23757508C2semidirect
8. #23757507C2semidirect
9. #23757308C2semidirect
10. #23757276C2semidirect
11. #469×C157C2semidirect
19. C2×#14578937C22semidirect
20. C22×#11929434C22semidirect
22. C2×C8×D157C22semidirect
25. #19097746C22semidirect
26. #19097745C22semidirect
27. #19097742C22semidirect
31. #19097704C22semidirect
32. #19097703C22semidirect
33. #19097693C22semidirect
34. #19097692C22semidirect
35. #19097691C22semidirect
37. C2×M16×C157C22semidirect
38. #121×C157C22semidirect
39. #106×C157C22semidirect
40. #102×C157C22semidirect
65. C8×D157D4semidirect
146. C157#3058semidirect
154. #469D157semidirect
168. C2×M16C2×D157semidirect
177. #121C2×D157semidirect
183. #106C2×D157semidirect
187. #102C2×D157semidirect

Maximal subgroups

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

subgroupindex
C2×D8×D1572
C2×#190977792
#237578392
#237575272
#237575262
#237575232
#237575082
#237575072
#237573082
#237572762
#469×C1572
#3058157