Finite Group Databasebeta

← #34100438

Subgroups — #34100438

Order 20096.

376 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 97 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#23757784C211yesmaximal, characteristic
2C2×#19097697C212yesmaximal, complemented
3C4×C22⋊Dic157C211yesmaximal, characteristic, complemented
4#23757034C212yesmaximal, complemented
5C22×M16×C157C211yesmaximal, characteristic, complemented
6C2×#14578954C2211yescharacteristic
7#19097754—21no—
8#19097750C411yescharacteristic, complemented
9#19097750C411yescharacteristic, complemented
10#19097745C411yescharacteristic, complemented
11#19097745C411yescharacteristic, complemented
12#19097740—21no—
13C4×C22⋊D157—21no—
14C4×C22⋊D157—21no—
15#19097730C2211yescharacteristic
16C2×C22⋊Dic157—21no—
17C2×C4×Dic157C2212yes—
18#19097697—24no—
19#19097678—21no—
20C23×C4×C157C2211yescharacteristic
21C2×M16×C157—21no—
22C2×M16×C157—24no—
23C2×M16×C157C2212yes—
24C22×C8×C157—21no—
25#14578954—22no—
26#14578954—22no—
27C2×Dic314C2×C411yescharacteristic
28C2×D628C2×C411yescharacteristic
29C2×C22⋊D157—21no—
30C2×C22⋊D157—21no—
31C2×C22⋊D157—21no—
32C22×Dic157—22no—
33C2×C4×D157—21no—
34C4⋊Dic157C2×C411yescharacteristic
35C4⋊Dic157C2×C411yescharacteristic
36#14578933—21no—
37#14578933—21no—
38C22⋊Dic157—24no—
39C22⋊Dic157—21no—
40C22⋊Dic157—21no—
41C22⋊Dic157—21no—
42#14578931—21no—
43#14578931—21no—
44C4×Dic157—21no—
45C4×Dic157—22no—
46C4×Dic157—22no—
47C24×C157D411yescharacteristic
48C22×C4×C157—22no—
49C22×C4×C157—22no—
50C22×C4×C157C2×C411yescharacteristic
51C22×C4×C157C2×C411yescharacteristic
52C22×C4×C157D411yescharacteristic
53C22×C4×C157D411yescharacteristic
54C22×C4×C157D411yescharacteristic
55C22×C4×C157C2311yescharacteristic
56M16×C157—24no—
57M16×C157—22no—
58M16×C157D414yescomplemented
59C2×C8×C157—24no—
60C2×C8×C157—22no—
61C22×D157—21no—
62Dic314—22no—
63D628—22no—
64C22⋊D157—42no—
65C22⋊D157—42no—
66C22⋊D157—42no—
67C2×Dic157—24no—
68C2×Dic157—21no—
69C2×Dic157—21no—
70C2×Dic157—21no—
71C2×Dic157—42no—
72C2×Dic157—42no—
73C4×D157—41no—
74C4×D157—41no—
75C23×C157—22no—
76C23×C157—22no—
77C23×C157—41no—
78C23×C157C22⋊C411yescharacteristic
79C23×C157C22⋊C411yescharacteristic
80C23×C157C2×D411yescharacteristic
81C2×C4×C157—22no—
82C2×C4×C157—22no—
83C2×C4×C157—22no—
84C2×C4×C157—21no—
85C2×C4×C157—21no—
86C2×C4×C157—21no—
87C2×C4×C157—21no—
88C2×C4×C157C22⋊C411yescharacteristic
89C2×C4×C157C22⋊C411yescharacteristic
90C2×C4×C157C22×C411yescharacteristic
91C2×C4×C157C2×D412yes—
92C2×C4×C157C2×D411yescharacteristic
93C2×C4×C157#4112yes—
94C8×C157—24no—
95C22×C157—21no—
96C22×C157—21no—
97C22×C157—21no—
98C22×C157—21no—
99C22×C157—21no—
100C22×C157—21no—
101C22×C157—21no—
102C22×C157—44no—
103C22×C157#10412yescomplemented
104C22×C157C2×C22⋊C411yescharacteristic
105C22×C157#12112yes—
106C2×D157—41no—
107C2×D157—41no—
108C2×D157—41no—
109C4×C157—21no—
110C4×C157—21no—
111C4×C157C4×D412yes—
112C4×C157#12011yescharacteristic
113C4×C157#12311yescharacteristic
114Dic157—44no—
115Dic157—41no—
116Dic157—41no—
117Dic157—41no—
118C2×C157—22no—
119C2×C157—22no—
120C2×C157—41no—
121C2×C157#38611yescharacteristic
122C2×C157C2×#10411yescharacteristic
123C2×C157#42111yescharacteristic
124D157—81no—
125C157#187111yescharacteristic, complemented, Sylow
126#1871—1571nomaximal, Sylow
127C22×M16D15711yescharacteristic, complemented
128#525—1571no—
129C2×#104—1572no—
130C4×C22⋊C4—1571no—
131#339—1572no—
132C2×#41—1571no—
133C23×C4C2×D15711yescharacteristic
134C2×M16—21no—
135C2×M16—24no—
136C2×M16C2×D15712yes—
137C22×C8—21no—
138#123—3141no—
139C22⋊Q8—1571no—
140C22⋊Q8—1571no—
141#121—1571no—
142#121—1571no—
143#120—3141no—
144C4×D4—3141no—
145C4×D4—3141no—
146#117—1571no—
147C2×C22⋊C4—3141no—
148C2×C42—1572no—
149#104—3144no—
150#95—3141no—
151C24D62811yescharacteristic
152#41—3142no—
153#41—3142no—
154C2×Q8—1571no—
155C2×D4—3141no—
156C2×D4—3141no—
157C2×D4—3141no—
158C2×D4—1571no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—3142no—
162C22×C4—3141no—
163C22×C4C4×D15711yescharacteristic
164C22×C4C4×D15711yescharacteristic
165C22×C4C22⋊D15711yescharacteristic
166C22×C4C22⋊D15711yescharacteristic
167C22×C4D62811yescharacteristic
168C22×C4C22×D15711yescharacteristic
169M16—24no—
170M16—22no—
171M16C22⋊D15714yescomplemented
172C2×C8—24no—
173C2×C8—22no—
174C4⋊C4—3141no—
175C4⋊C4—3141no—
176C4⋊C4—1571no—
177C4⋊C4—1571no—
178C22⋊C4—3144no—
179C22⋊C4—3141no—
180C22⋊C4—3141no—
181C22⋊C4—3141no—
182C22⋊C4—3141no—
183C22⋊C4—3141no—
184C42—3141no—
185C42—3142no—
186C42—3142no—
187C23—22no—
188C23—22no—
189C23—41no—
190C23—3141no—
191C23#1457893111yescharacteristic
192C23#1457893111yescharacteristic
193C23C2×D62811yescharacteristic
194Q8—3142no—
195D4—3142no—
196D4—6282no—
197D4—6282no—
198D4—6282no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—3144no—
207C2×C4—3141no—
208C2×C4—3141no—
209C2×C4—3141no—
210C2×C4—6282no—
211C2×C4—6282no—
212C2×C4—6281no—
213C2×C4—6281no—
214C2×C4#1457893111yescharacteristic
215C2×C4#1457893111yescharacteristic
216C2×C4C2×C4×D15711yescharacteristic
217C2×C4C2×C22⋊D15712yes—
218C2×C4C2×C22⋊D15711yescharacteristic
219C2×C4#1457895412yes—
220C8—24no—
221C22—21no—
222C22—21no—
223C22—21no—
224C22—21no—
225C22—21no—
226C22—21no—
227C22—21no—
228C22—44no—
229C22—6281no—
230C22—6281no—
231C22—6281no—
232C22#1909769712yescomplemented
233C22C2×#1457893111yescharacteristic
234C22#1909774512yes—
235C4—21no—
236C4—21no—
237C4—6284no—
238C4—6281no—
239C4—6281no—
240C4—6281no—
241C4C4×C22⋊D15712yes—
242C4#1909774011yescharacteristic
243C4#1909775411yescharacteristic
244C2—22no—
245C2—22no—
246C2—41no—
247C2—12561no—
248C2#2375715211yescharacteristic
249C2C2×#1909769711yescharacteristic
250C2#2375728911yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊D157
commutatorC2×C4×C157C22×C4
FrattiniC22×C4C22×D157
FittingC22×M16×C157C2
socleC22×C157C2×C22⋊C4
prime radical#23757784C2
cosocleC22×C4×C157C23
hypercenterC22×M16D157
hypocenterC157#1871
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C22×M16

Complements

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

NH ≅ G/Ntype
2. C2×#19097697C2semidirect
3. C4×C22⋊Dic157C2semidirect
4. #23757034C2semidirect
5. C22×M16×C157C2semidirect
8. #19097750C4semidirect
9. #19097750C4semidirect
10. #19097745C4semidirect
11. #19097745C4semidirect
58. M16×C157D4semidirect
103. C22×C157#104semidirect
125. C157#1871semidirect
127. C22×M16D157semidirect
171. M16C22⋊D157semidirect
232. C22#19097697semidirect

Maximal subgroups

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

subgroupindex
#237577842
C2×#190976972
C4×C22⋊Dic1572
#237570342
C22×M16×C1572
#1871157