Finite Group Databasebeta

← #34100578

Subgroups — #34100578

Order 20096.

442 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 87 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#23757785C211yesmaximal, characteristic
2C2×C42×D157C211yesmaximal, characteristic, complemented
3C2×#19097699C211yesmaximal, characteristic, complemented
4C2×#19097697C211yesmaximal, characteristic, complemented
5#23757236C211yesmaximal, characteristic, complemented
6#23757032C211yesmaximal, characteristic, complemented
7#339×C157C211yesmaximal, characteristic, complemented
8C2×#14578954C2211yescharacteristic
9C22×C4×D157—21no—
10C22×C4×D157C2211yescharacteristic
11C2×#14578938C2211yescharacteristic
12#19097755—21no—
13#19097751C412yescomplemented
14#19097746C412yescomplemented
15#19097741—21no—
16C4×D628—22no—
17C2×C4×Dic157—21no—
18C2×C4×Dic157C2211yescharacteristic
19C42×D157—22no—
20C42×D157—22no—
21#19097699—22no—
22#19097697—22no—
23#19097683—22no—
24C2×M16×C157C2211yescharacteristic
25#117×C157C2211yescharacteristic
26C2×C42×C157C2211yescharacteristic
27C23×D157D411yescharacteristic
28#14578954—21no—
29#14578954—21no—
30#14578954—21no—
31#14578954—21no—
32C2×Dic314C2×C411yescharacteristic, complemented
33C2×D628—22no—
34C2×D628C2×C411yescharacteristic, complemented
35C2×C22⋊D157—21no—
36C22×Dic157—21no—
37C22×Dic157D411yescharacteristic
38C2×C4×D157—24no—
39C2×C4×D157—28no—
40C2×C4×D157—21no—
41C2×C4×D157—21no—
42C2×C4×D157—21no—
43C2×C4×D157—21no—
44C2×C4×D157—21no—
45C2×C4×D157C2×C412yes—
46C2×C4×D157D412yes—
47#14578938—21no—
48#14578938—21no—
49C2×#11929434—21no—
50C4⋊Dic157—22no—
51#14578931—22no—
52#14578931—22no—
53C4×Dic157—24no—
54C4×Dic157—21no—
55C4×Dic157D412yescomplemented
56C22×C4×C157—21no—
57C22×C4×C157C2311yescharacteristic
58M16×C157—21no—
59M16×C157—21no—
60C2×C8×C157—21no—
61C4⋊C4×C157C2×C412yescomplemented
62C22⋊C4×C157—21no—
63C42×C157—21no—
64C42×C157—21no—
65C42×C157D412yescomplemented
66C22×D157—21no—
67C22×D157—21no—
68C22×D157—24no—
69C22×D157—21no—
70C22×D157C22⋊C412yes—
71Dic314—21no—
72Dic314—21no—
73D628—21no—
74D628—21no—
75D628—44no—
76C22⋊D157—41no—
77C22⋊D157—41no—
78C2×Dic157—24no—
79C2×Dic157—22no—
80C2×Dic157—21no—
81C2×Dic157—21no—
82C2×Dic157—21no—
83C2×Dic157C22⋊C412yes—
84C4×D157—216no—
85C4×D157—22no—
86C4×D157—22no—
87C4×D157—22no—
88C4×D157—22no—
89C4×D157—41no—
90C4×D157—41no—
91#11929434—41no—
92C23×C157C2×D411yescharacteristic
93C2×C4×C157—24no—
94C2×C4×C157—22no—
95C2×C4×C157—22no—
96C2×C4×C157C22×C411yescharacteristic
97C2×C4×C157C2×D411yescharacteristic
98C2×C4×C157C2×D411yescharacteristic
99C2×C4×C157C2×D411yescharacteristic
100C2×C4×C157#4111yescharacteristic
101C2×C4×C157#4111yescharacteristic
102C8×C157—41no—
103C22×C157—22no—
104C22×C157C2×C22⋊C411yescharacteristic
105C22×C157#12011yescharacteristic
106C22×C157#12311yescharacteristic
107C2×D157—24no—
108C2×D157—24no—
109C2×D157—22no—
110C2×D157—22no—
111C2×D157—41no—
112C2×D157—42no—
113C2×D157#10414yescomplemented
114C4×C157—24no—
115C4×C157—42no—
116C4×C157C4×D411yescharacteristic
117C4×C157C4×D411yescharacteristic
118C4×C157#12111yescharacteristic
119C4×C157#12111yescharacteristic
120Dic157—24no—
121Dic157—22no—
122Dic157—41no—
123C2×C157—21no—
124C2×C157—21no—
125C2×C157#38611yescharacteristic
126C2×C157C2×#10412yes—
127D157—24no—
128D157—81no—
129C157#189611yescharacteristic, complemented, Sylow
130#1896—1571nomaximal, Sylow
131#525—1571no—
132C22×C42—1571no—
133C2×#104—1571no—
134C2×#104—1571no—
135#407—1571no—
136#339—1571no—
137#339D15711yescharacteristic, complemented
138C2×#41—1571no—
139C23×C4—3141no—
140C23×C4—1571no—
141C2×M16—1571no—
142C2×M16C2×D15711yescharacteristic
143#123—3141no—
144C22⋊Q8—1572no—
145#121—1572no—
146#120—3141no—
147C4×D4—3142no—
148#117C2×D15711yescharacteristic
149C2×C42—3142no—
150C2×C42—3141no—
151C2×C42—3142no—
152C2×C42—1571no—
153C2×C42C2×D15711yescharacteristic
154#104—3142no—
155#104—3142no—
156C22⋊C8—3142no—
157C24—1571no—
158#41—3141no—
159#41—3141no—
160#41—3141no—
161#41—3141no—
162C2×Q8—1571no—
163C2×D4—3142no—
164C2×D4—3141no—
165C2×D4—1571no—
166C22×C4—21no—
167C22×C4—3144no—
168C22×C4—3148no—
169C22×C4—3141no—
170C22×C4—3141no—
171C22×C4—3141no—
172C22×C4—3141no—
173C22×C4—3141no—
174C22×C4—3141no—
175C22×C4—1571no—
176C22×C4—1572no—
177C22×C4—1572no—
178C22×C4C22×D15711yescharacteristic
179M16—21no—
180M16—21no—
181M16—3141no—
182M16—3141no—
183C2×C8—21no—
184C2×C8—3141no—
185C4⋊C4—3142no—
186C4⋊C4C4×D15712yescomplemented
187C22⋊C4—21no—
188C22⋊C4—3142no—
189C22⋊C4—3142no—
190C42—21no—
191C42—21no—
192C42—3144no—
193C42—3141no—
194C42—1572no—
195C42D62812yescomplemented
196C23—3144no—
197C23—3141no—
198C23—3141no—
199C23—3141no—
200C23—1572no—
201C23D4×D15711yescharacteristic
202Q8—3141no—
203Q8—3141no—
204D4—3141no—
205D4—3141no—
206D4—6284no—
207D4—6281no—
208D4—6281no—
209C2×C4—24no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—31416no—
213C2×C4—3144no—
214C2×C4—3141no—
215C2×C4—3142no—
216C2×C4—3141no—
217C2×C4—3142no—
218C2×C4—3142no—
219C2×C4—3142no—
220C2×C4—3142no—
221C2×C4—3141no—
222C2×C4—6281no—
223C2×C4—6281no—
224C2×C4—1572no—
225C2×C4C2×C4×D15711yescharacteristic
226C2×C4D4×D15711yescharacteristic
227C2×C4D4×D15711yescharacteristic
228C2×C4C2×D62811yescharacteristic
229C2×C4#1457895411yescharacteristic
230C2×C4#1457895511yescharacteristic
231C8—41no—
232C8—6281no—
233C22—22no—
234C22—3144no—
235C22—3144no—
236C22—3142no—
237C22—3142no—
238C22—6281no—
239C22—6282no—
240C22—1574no—
241C22C22⋊C4×D15711yescharacteristic
242C22#1909774111yescharacteristic
243C22#1909775611yescharacteristic
244C4—24no—
245C4—42no—
246C4—3144no—
247C4—3142no—
248C4—6281no—
249C4#1909773311yescharacteristic
250C4C4×D62811yescharacteristic
251C4#1909774411yescharacteristic
252C4#1909774611yescharacteristic
253C2—21no—
254C2—21no—
255C2—3144no—
256C2—12561no—
257C2#2375715111yescharacteristic
258C2#104×D15712yes—
subgroupG/H
centerC2×C4D4×D157
commutatorC2×C4×C157C22×C4
FrattiniC22×C4C22×D157
Fitting#339×C157C2
socleC22×C157C2×C22⋊C4
prime radical#23757785C2
cosocleC22×C4×C157C23
hypercenter#339D157
hypocenterC157#1896
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
2. C2×C42×D157C2semidirect
3. C2×#19097699C2semidirect
4. C2×#19097697C2semidirect
5. #23757236C2semidirect
6. #23757032C2semidirect
7. #339×C157C2semidirect
13. #19097751C4semidirect
14. #19097746C4semidirect
32. C2×Dic314C2×C4semidirect
34. C2×D628C2×C4semidirect
55. C4×Dic157D4semidirect
61. C4⋊C4×C157C2×C4semidirect
65. C42×C157D4semidirect
113. C2×D157#104semidirect
129. C157#1896semidirect
137. #339D157semidirect
186. C4⋊C4C4×D157semidirect
195. C42D628semidirect

Maximal subgroups

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

subgroupindex
#237577852
C2×C42×D1572
C2×#190976992
C2×#190976972
#237572362
#237570322
#339×C1572
#1896157