Finite Group Databasebeta

← #34109927

Subgroups — #34109927

Order 20096.

388 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 131 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#23757828C211yesmaximal, characteristic, complemented
2C4×#14578955C211yesmaximal, characteristic
3#23757582C212yesmaximal, complemented
4#23757576C212yesmaximal, complemented
5#23757560C211yesmaximal, characteristic, complemented
6#23757558C211yesmaximal, characteristic, complemented
7#23757554C211yesmaximal, characteristic, complemented
8#23757310C212yesmaximal, complemented
9C4⋊C8×D157C211yesmaximal, characteristic, complemented
10#23757268C212yesmaximal
11#477×C157C211yesmaximal, characteristic, complemented
12C2×#14578955C2211yescharacteristic
13C2×C8×D157C2212yes—
14#19097771C2211yescharacteristic
15#19097762C2212yes—
16#19097751—21no—
17#19097747—21no—
18Q8×Dic157—21no—
19Q8×Dic157C2211yescharacteristic
20#19097736—21no—
21C4×D628—21no—
22C4×D628C2211yescharacteristic
23#19097733—21no—
24#19097733C2211yescharacteristic
25#19097729—21no—
26#19097729C2212yes—
27#19097729C2211yescharacteristic
28C42×D157—21no—
29C42×D157C2211yescharacteristic
30#19097705C2212yes—
31#19097702C2212yes—
32#19097702C2212yes—
33C4⋊#11929434C2211yescharacteristic
34#19097700C2212yes—
35#19097695C2212yes—
36#19097693C2212yes—
37#19097691C2212yes—
38C4⋊Q8×C157C2211yescharacteristic
39C4×Q8×C157C2211yescharacteristic
40#107×C157C2212yes—
41C4⋊C8×C157C2211yescharacteristic
42#103×C157C2212yes—
43#14578955—22no—
44#14578955Q814yescomplemented
45C2×Dic314—21no—
46C2×D628—21no—
47C2×D628C2311yescharacteristic
48C2×C4×D157—21no—
49C2×C4×D157—21no—
50C2×C4×D157—21no—
51C2×C4×D157D411yescharacteristic
52C2×C4×D157D411yescharacteristic
53C2×C4×D157C2311yescharacteristic
54C2×#11929434C2312yes—
55C8×D157—24no—
56C4⋊Dic157—21no—
57C4⋊Dic157—22no—
58C4⋊Dic157—21no—
59C4⋊Dic157C2312yes—
60C4⋊Dic157C2311yescharacteristic
61#14578933—22no—
62#14578931—21no—
63#14578931—22no—
64#14578931—21no—
65#14578931—21no—
66C4×Dic157—21no—
67C4×Dic157—21no—
68C4×Dic157—22no—
69C4×Dic157D411yescharacteristic
70C4×Dic157D411yescharacteristic
71C4×Dic157C2311yescharacteristic
72C2×Q8×C157—21no—
73C2×Q8×C157C2311yescharacteristic
74C2×C8×C157C2312yes—
75C4⋊C4×C157—21no—
76C4⋊C4×C157—21no—
77C4⋊C4×C157C2312yes—
78C4⋊C4×C157C2311yescharacteristic
79C42×C157—21no—
80C42×C157C2311yescharacteristic
81C22×D157—21no—
82C22×D157C2×D411yescharacteristic
83Dic314—42no—
84D628—24no—
85D628—21no—
86D628C2×Q812yes—
87C2×Dic157—21no—
88C2×Dic157—22no—
89C2×Dic157C2×D411yescharacteristic
90C2×Dic157C2×D411yescharacteristic
91C2×Dic157C2×D411yescharacteristic
92C4×D157—22no—
93C4×D157—24no—
94C4×D157—41no—
95C4×D157—42no—
96C4×D157—42no—
97C4×D157C2×Q812yes—
98C4×D157#4112yes—
99#11929434—22no—
100Q8×C157—21no—
101Q8×C157—42no—
102Q8×C157C2×Q812yes—
103C2×C4×C157—21no—
104C2×C4×C157—21no—
105C2×C4×C157—22no—
106C2×C4×C157C2×D411yescharacteristic
107C2×C4×C157C2×D411yescharacteristic
108C2×C4×C157C2411yescharacteristic
109C8×C157—22no—
110C22×C157C22×D411yescharacteristic
111C2×D157—21no—
112C2×D157—21no—
113C2×D157—22no—
114C2×D157—41no—
115C2×D157—41no—
116C2×D157C22⋊Q812yes—
117C4×C157—22no—
118C4×C157—21no—
119C4×C157—42no—
120C4×C157—41no—
121C4×C157#13512yes—
122C4×C157C22×Q811yescharacteristic
123C4×C157C2×#4111yescharacteristic
124Dic157—21no—
125Dic157—21no—
126Dic157—41no—
127Dic157—42no—
128Dic157C22⋊Q812yes—
129C2×C157C2×C22⋊Q811yescharacteristic
130C2×C157C2×#13511yescharacteristic
131C2×C157#57511yescharacteristic
132D157—22no—
133D157—42no—
134C157#307611yescharacteristic, complemented, Sylow
135#3076—1571nomaximal, Sylow
136#533—1571no—
137C4×#41—1571no—
138#479—1572no—
139#478—1572no—
140#477D15711yescharacteristic, complemented
141#476—1571no—
142#475—1571no—
143#474—1571no—
144#427—1572no—
145C2×C4⋊C8—1571no—
146#416—1572no—
147C2×#41—1571no—
148C22×C8—1572no—
149C4⋊Q8—1571no—
150C4⋊Q8C2×D15711yescharacteristic
151#125—1572no—
152C22⋊Q8—3141no—
153C22⋊Q8—3141no—
154C4×Q8—3141no—
155C4×Q8—3141no—
156C4×Q8—1571no—
157C4×Q8C2×D15711yescharacteristic
158C4×D4—3141no—
159C4×D4—3141no—
160C4×D4—1571no—
161C4×D4—1571no—
162#117—3141no—
163#117—1571no—
164#117—1572no—
165C2×C42—3141no—
166C2×C42—1571no—
167#107—1572no—
168#107C2×D15712yes—
169#106—1572no—
170#106—1572no—
171C4⋊C8—1572no—
172C4⋊C8—1571no—
173C4⋊C8C2×D15711yescharacteristic
174#103—1572no—
175#103C2×D15712yes—
176#102—1572no—
177#102—1572no—
178#41—3142no—
179#41—1574no—
180C2×Q8—21no—
181C2×Q8—3141no—
182C2×Q8C22×D15711yescharacteristic
183C2×D4—3141no—
184C2×D4—1571no—
185C22×C4—3141no—
186C22×C4—3141no—
187C22×C4—3141no—
188C22×C4—1571no—
189C22×C4—1571no—
190C22×C4—1571no—
191C2×C8—3144no—
192C2×C8—1572no—
193C2×C8C22×D15712yes—
194C4⋊C4—21no—
195C4⋊C4—21no—
196C4⋊C4—3141no—
197C4⋊C4—3142no—
198C4⋊C4—3142no—
199C4⋊C4—3141no—
200C4⋊C4—1572no—
201C4⋊C4—1571no—
202C4⋊C4C22×D15712yes—
203C4⋊C4C22×D15711yescharacteristic
204C22⋊C4—3141no—
205C22⋊C4—3142no—
206C22⋊C4—3141no—
207C22⋊C4—3141no—
208C42—21no—
209C42—3141no—
210C42—3141no—
211C42—3142no—
212C42—1571no—
213C42—1571no—
214C42—1571no—
215C42C22×D15711yescharacteristic
216C23—3141no—
217C23—1571no—
218Q8—21no—
219Q8—42no—
220Q8—6282no—
221Q8Q8×D15712yes—
222D4—3144no—
223D4—3141no—
224D4—1572no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—22no—
228C2×C4—3144no—
229C2×C4—3142no—
230C2×C4—3141no—
231C2×C4—3142no—
232C2×C4—6281no—
233C2×C4—6282no—
234C2×C4—6282no—
235C2×C4—1571no—
236C2×C4—1571no—
237C2×C4—1571no—
238C2×C4—1572no—
239C2×C4—1572no—
240C2×C4D4×D15711yescharacteristic
241C2×C4D4×D15711yescharacteristic
242C2×C4C23×D15711yescharacteristic
243C8—22no—
244C8—3142no—
245C22—3141no—
246C22—3141no—
247C22—3142no—
248C22—6281no—
249C22—6281no—
250C22—1572no—
251C22C2×D4×D15711yescharacteristic
252C4—22no—
253C4—21no—
254C4—41no—
255C4—42no—
256C4—3141no—
257C4—3141no—
258C4—6281no—
259C4—6282no—
260C4—1572no—
261C4#1909779412yes—
262C4C2×Q8×D15711yescharacteristic
263C4#41×D15711yescharacteristic
264C2—3142no—
265C2—6282no—
266C2C22⋊Q8×D15711yescharacteristic
267C2C2×#1909779411yescharacteristic
268C2#2375806911yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#477×C157C2
socleC22×C157C22×D4
prime radicalC2×#14578955C22
hypercenter#477D157
hypocenterC157#3076
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #477

Complements

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

NH ≅ G/Ntype
1. #23757828C2semidirect
3. #23757582C2semidirect
4. #23757576C2semidirect
5. #23757560C2semidirect
6. #23757558C2semidirect
7. #23757554C2semidirect
8. #23757310C2semidirect
9. C4⋊C8×D157C2semidirect
11. #477×C157C2semidirect
44. #14578955Q8semidirect
134. C157#3076semidirect
140. #477D157semidirect

Maximal subgroups

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

subgroupindex
#237578282
C4×#145789552
#237575822
#237575762
#237575602
#237575582
#237575542
#237573102
C4⋊C8×D1572
#237572682
#477×C1572
#3076157