Finite Group Databasebeta

← #34100361

Subgroups — #34100361

Order 20096.

302 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 83 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#23757754C211yesmaximal, characteristic
2#23757313C211yesmaximal, characteristic, complemented
3C2×#19097699C211yesmaximal, characteristic
4#23757272C211yesmaximal, characteristic
5#23757042C211yesmaximal, characteristic, complemented
6#23757033C211yesmaximal, characteristic, complemented
7#384×C157C211yesmaximal, characteristic, complemented
8C2×#14578954C2211yescharacteristic
9C2×#14578938C2211yescharacteristic
10C2×#14578938C2211yescharacteristic
11#19097736—21no—
12#19097733—21no—
13C4×C22⋊D157—22no—
14#19097730C2211yescharacteristic
15#19097729—21no—
16C4⋊C4×D157—21no—
17#19097704—21no—
18#19097702—21no—
19#19097699C414yescomplemented
20#19097694—21no—
21#19097691—21no—
22C2×C4⋊C4×C157—22no—
23C2×C4⋊C4×C157C2211yescharacteristic
24C2×C4⋊C4×C157C2211yescharacteristic
25C2×C42×C157C2211yescharacteristic
26#14578954—21no—
27#14578954—21no—
28#14578954C2×C412yes—
29#14578954Q812yes—
30C2×Dic314D411yescharacteristic
31C2×D628D411yescharacteristic
32C2×C22⋊D157—21no—
33C2×C4×D157—22no—
34C2×C4×D157—21no—
35#14578938—21no—
36#14578938—21no—
37#14578938—21no—
38#14578938C2×C412yescomplemented
39C2×#11929434—21no—
40C2×#11929434—21no—
41C4⋊Dic157D411yescharacteristic
42C4⋊Dic157D411yescharacteristic
43#14578933—22no—
44C22⋊Dic157—21no—
45#14578931—22no—
46C4×Dic157—22no—
47C4×Dic157—21no—
48C22×C4×C157—21no—
49C22×C4×C157—21no—
50C22×C4×C157D412yes—
51C22×C4×C157C2311yescharacteristic
52C4⋊C4×C157—21no—
53C4⋊C4×C157—21no—
54C4⋊C4×C157—21no—
55C4⋊C4×C157—21no—
56C4⋊C4×C157—44no—
57C42×C157—21no—
58C42×C157C2×C412yescomplemented
59C22×D157—21no—
60Dic314—21no—
61Dic314C4⋊C412yescomplemented
62D628—21no—
63D628C4⋊C412yescomplemented
64C22⋊D157—22no—
65C22⋊D157—22no—
66C2×Dic157—21no—
67C2×Dic157—21no—
68C2×Dic157—21no—
69C4×D157—22no—
70C4×D157—22no—
71C4×D157—82no—
72#11929434—22no—
73#11929434—41no—
74C23×C157C2×D411yescharacteristic
75C2×C4×C157—22no—
76C2×C4×C157—22no—
77C2×C4×C157—22no—
78C2×C4×C157—22no—
79C2×C4×C157—42no—
80C2×C4×C157—42no—
81C2×C4×C157—41no—
82C2×C4×C157C22×C411yescharacteristic
83C2×C4×C157C2×D411yescharacteristic
84C2×C4×C157C2×D411yescharacteristic
85C2×C4×C157C2×Q811yescharacteristic
86C2×C4×C157#4111yescharacteristic
87C2×C4×C157#4111yescharacteristic
88C22×C157—21no—
89C22×C157—21no—
90C22×C157C4×D411yescharacteristic
91C22×C157#12011yescharacteristic
92C22×C157C22⋊Q811yescharacteristic
93C2×D157—22no—
94C2×D157—41no—
95C2×D157—41no—
96C4×C157—24no—
97C4×C157—42no—
98C4×C157—81no—
99C4×C157C2×C4⋊C411yescharacteristic
100C4×C157C4×D411yescharacteristic
101C4×C157C22⋊Q811yescharacteristic
102C4×C157#12311yescharacteristic
103Dic157—22no—
104Dic157—41no—
105Dic157—41no—
106C2×C157—21no—
107C2×C157—21no—
108C2×C157#38511yescharacteristic
109C2×C157#45311yescharacteristic
110C2×C157#45611yescharacteristic
111D157—42no—
112C157#185911yescharacteristic, complemented, Sylow
113#1859—1571nomaximal, Sylow
114#520—1571no—
115#428—1571no—
116C2×#104—1571no—
117#417—1571no—
118#384D15711yescharacteristic, complemented
119#343—1571no—
120#339—1571no—
121C2×#41—1571no—
122C2×M16—1571no—
123C2×M16—1571no—
124C4×Q8—3141no—
125C4×D4—3141no—
126C4×D4—3142no—
127#117—3141no—
128#117—1571no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—3141no—
131C2×C4⋊C4C2×D15711yescharacteristic
132C2×C4⋊C4C2×D15711yescharacteristic
133C2×C42C2×D15711yescharacteristic
134#107—3141no—
135#106—3141no—
136#104—1574no—
137#103—3141no—
138#102—3141no—
139#41—3141no—
140#41—3141no—
141#41—1572no—
142#41—1572no—
143C2×Q8—1571no—
144C2×D4—3141no—
145C2×D4—1571no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—3142no—
149C22×C4—3141no—
150C22×C4C22⋊D15712yes—
151C22×C4C22×D15711yescharacteristic
152M16—3141no—
153M16—3141no—
154M16—3141no—
155M16—1572no—
156C2×C8—3141no—
157C2×C8—3141no—
158C4⋊C4—21no—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—21no—
162C4⋊C4—44no—
163C4⋊C4—3142no—
164C4⋊C4—1571no—
165C4⋊C4—1571no—
166C22⋊C4—3142no—
167C22⋊C4—3141no—
168C42—21no—
169C42—3142no—
170C42—3141no—
171C42C4×D15712yescomplemented
172C23—3141no—
173C23C2×C22⋊D15711yescharacteristic
174Q8—3141no—
175Q8—1572no—
176D4—3142no—
177D4—3142no—
178D4—3141no—
179D4—1572no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—41no—
187C2×C4—3141no—
188C2×C4—3142no—
189C2×C4—3142no—
190C2×C4—3141no—
191C2×C4—3141no—
192C2×C4—12562no—
193C2×C4C2×C4×D15711yescharacteristic
194C2×C4D4×D15711yescharacteristic
195C2×C4D4×D15711yescharacteristic
196C2×C4Q8×D15711yescharacteristic
197C2×C4#1457895411yescharacteristic
198C2×C4#1457895511yescharacteristic
199C8—3142no—
200C8—6281no—
201C22—21no—
202C22—21no—
203C22—3142no—
204C22—6281no—
205C22—6281no—
206C22C4×C22⋊D15711yescharacteristic
207C22#1909773911yescharacteristic
208C22#1909774711yescharacteristic
209C4—24no—
210C4—42no—
211C4—81no—
212C4—3142no—
213C4—6281no—
214C4—6281no—
215C4C4⋊C4×D15711yescharacteristic
216C4#1909773311yescharacteristic
217C4#1909774911yescharacteristic
218C4#1909775611yescharacteristic
219C2—21no—
220C2—21no—
221C2—6282no—
222C2#2375714411yescharacteristic
223C2#2375744011yescharacteristic
224C2#2375745511yescharacteristic
subgroupG/H
centerC22#19097739
commutatorC2×C4×C157C22×C4
FrattiniC22×C4C22×D157
Fitting#384×C157C2
socleC22×C157#120
prime radicalC2×#14578954C22
cosocleC22×C4×C157C23
hypercenter#384D157
hypocenterC157#1859
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #384

Complements

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

NH ≅ G/Ntype
2. #23757313C2semidirect
5. #23757042C2semidirect
6. #23757033C2semidirect
7. #384×C157C2semidirect
19. #19097699C4semidirect
38. #14578938C2×C4semidirect
58. C42×C157C2×C4semidirect
61. Dic314C4⋊C4semidirect
63. D628C4⋊C4semidirect
112. C157#1859semidirect
118. #384D157semidirect
171. C42C4×D157semidirect

Maximal subgroups

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

subgroupindex
#237577542
#237573132
C2×#190976992
#237572722
#237570422
#237570332
#384×C1572
#1859157