Finite Group Databasebeta

← #34100052

Subgroups — #34100052

Order 20096.

612 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 101 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
1C22×#14578953C211yesmaximal, characteristic
2C2×#19097686C211yesmaximal, characteristic, complemented
3C2×#19097685C212yesmaximal, complemented
4#23757134C212yesmaximal, complemented
5C2×#99×C157C211yesmaximal, characteristic, complemented
6C2×#14578953—22no—
7C2×#14578953—22no—
8C2×#14578953—24no—
9C2×#14578953—24no—
10C22×Dic314C411yescharacteristic, complemented
11C22×C22⋊D157C411yescharacteristic, complemented
12C22×C22⋊D157C2211yescharacteristic
13C23×Dic157C411yescharacteristic, complemented
14C23×Dic157C2211yescharacteristic
15C22×C4×D157C411yescharacteristic, complemented
16C2×C22⋊Dic157C2212yes—
17#19097686—22no—
18#19097685—24no—
19#19097678—24no—
20C22×D4×C157C2211yescharacteristic
21C2×C22⋊C4×C157C2212yes—
22#99×C157—22no—
23C23×D157C2×C411yescharacteristic
24#14578953—216no—
25#14578953—216no—
26C2×Dic314—22no—
27C2×Dic314—22no—
28C2×Dic314—22no—
29C2×C22⋊D157—22no—
30C2×C22⋊D157—22no—
31C2×C22⋊D157—22no—
32C2×C22⋊D157—22no—
33C2×C22⋊D157—21no—
34C2×C22⋊D157—21no—
35C2×C22⋊D157D414yes—
36C22×Dic157—22no—
37C22×Dic157—22no—
38C22×Dic157—22no—
39C22×Dic157—22no—
40C22×Dic157—21no—
41C22×Dic157—21no—
42C22×Dic157—22no—
43C22×Dic157C2×C411yescharacteristic
44C22×Dic157C2×C411yescharacteristic
45C22×Dic157C2×C411yescharacteristic
46C22×Dic157D414yes—
47C2×C4×D157—22no—
48C2×C4×D157—22no—
49C2×C4×D157—22no—
50C22⋊Dic157—22no—
51C22⋊Dic157—22no—
52C24×C157C2×C411yescharacteristic
53C24×C157C2311yescharacteristic
54C2×D4×C157—22no—
55C2×D4×C157—22no—
56C2×D4×C157D412yes—
57C2×D4×C157D412yes—
58C22×C4×C157—22no—
59C22×C4×C157C2×C411yescharacteristic
60C22⋊C4×C157—22no—
61C22⋊C4×C157—22no—
62C22×D157—21no—
63C22×D157—21no—
64C22×D157—42no—
65C22×D157C22⋊C412yes—
66Dic314—28no—
67C22⋊D157—28no—
68C22⋊D157—24no—
69C22⋊D157—24no—
70C2×Dic157—28no—
71C2×Dic157—21no—
72C2×Dic157—21no—
73C2×Dic157—21no—
74C2×Dic157—21no—
75C2×Dic157—21no—
76C2×Dic157—21no—
77C2×Dic157—24no—
78C2×Dic157—24no—
79C2×Dic157—44no—
80C2×Dic157—42no—
81C2×Dic157C22⋊C412yes—
82C2×Dic157C22⋊C412yes—
83C2×Dic157C22⋊C412yes—
84C4×D157—28no—
85C23×C157—21no—
86C23×C157—21no—
87C23×C157—42no—
88C23×C157—42no—
89C23×C157C22⋊C412yes—
90C23×C157C22×C411yescharacteristic
91C23×C157C2×D412yes—
92C23×C157C2×D412yes—
93C23×C157C2×D411yescharacteristic
94C23×C157C2×D411yescharacteristic
95D4×C157—24no—
96D4×C157—24no—
97C2×C4×C157—21no—
98C2×C4×C157—21no—
99C2×C4×C157—44no—
100C2×C4×C157—42no—
101C2×C4×C157C22⋊C412yes—
102C22×C157—24no—
103C22×C157—22no—
104C22×C157—22no—
105C22×C157—42no—
106C22×C157—41no—
107C22×C157—41no—
108C22×C157—42no—
109C22×C157—42no—
110C22×C157—42no—
111C22×C157C2×C22⋊C411yescharacteristic
112C22×C157C2×C22⋊C411yescharacteristic
113C22×C157C2×C22⋊C411yescharacteristic
114C22×C157#12012yes—
115C22×C157#12012yes—
116C2×D157—22no—
117C2×D157—42no—
118C2×D157—42no—
119C2×D157—42no—
120C4×C157—22no—
121C4×C157—82no—
122Dic157—22no—
123Dic157—22no—
124Dic157—22no—
125Dic157—82no—
126C2×C157—22no—
127C2×C157—22no—
128C2×C157—21no—
129C2×C157—21no—
130C2×C157—42no—
131C2×C157#37911yescharacteristic
132C2×C157#41012yes—
133D157—42no—
134C157#178211yescharacteristic, complemented, Sylow
135#1782—1571nomaximal, Sylow
136C22×#41—1571no—
137C2×#99—1571no—
138C2×#99—1572no—
139C2×#99D15711yescharacteristic, complemented
140#381—1572no—
141C2×#41—3144no—
142C2×#41—3142no—
143C2×#41—3142no—
144C2×#41—3144no—
145C22×Q8—1571no—
146C22×D4—1571no—
147C22×D4—1571no—
148C22×D4C2×D15711yescharacteristic
149C23×C4—1571no—
150C23×C4—1571no—
151C23×C4—1571no—
152C2×C22⋊C4—1572no—
153C2×C22⋊C4C2×D15712yes—
154#99—22no—
155#99—3144no—
156#99—3142no—
157#95—3144no—
158C24—1571no—
159C24C4×D15711yescharacteristic
160C24C22×D15711yescharacteristic
161#41—31416no—
162#41—31416no—
163C2×Q8—3142no—
164C2×Q8—3142no—
165C2×Q8—3142no—
166C2×D4—22no—
167C2×D4—22no—
168C2×D4—3142no—
169C2×D4—3142no—
170C2×D4—3141no—
171C2×D4—3141no—
172C2×D4—3142no—
173C2×D4—3142no—
174C2×D4—1574no—
175C2×D4C22⋊D15712yes—
176C2×D4D62812yes—
177C22×C4—22no—
178C22×C4—3142no—
179C22×C4—3142no—
180C22×C4—3142no—
181C22×C4—3141no—
182C22×C4—3141no—
183C22×C4—3142no—
184C22×C4—3142no—
185C22×C4—3142no—
186C22×C4—3142no—
187C22×C4—3142no—
188C22×C4—1571no—
189C22×C4—1571no—
190C22×C4—1571no—
191C22×C4—1574no—
192C22×C4C4×D15711yescharacteristic
193C22⋊C4—22no—
194C22⋊C4—22no—
195C22⋊C4—3142no—
196C22⋊C4—3142no—
197C23—21no—
198C23—21no—
199C23—42no—
200C23—42no—
201C23—3141no—
202C23—3141no—
203C23—6282no—
204C23—1572no—
205C23#1457893112yes—
206C23C2×C4×D15711yescharacteristic
207C23D4×D15712yes—
208C23D4×D15712yes—
209C23C2×C22⋊D15711yescharacteristic
210C23C2×D62811yescharacteristic
211Q8—3148no—
212D4—24no—
213D4—24no—
214D4—3148no—
215D4—3144no—
216D4—3144no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—44no—
220C2×C4—42no—
221C2×C4—3148no—
222C2×C4—3148no—
223C2×C4—3141no—
224C2×C4—3141no—
225C2×C4—3141no—
226C2×C4—3141no—
227C2×C4—3141no—
228C2×C4—3141no—
229C2×C4—3144no—
230C2×C4—3144no—
231C2×C4—6284no—
232C2×C4—6282no—
233C2×C4—1572no—
234C2×C4—1572no—
235C2×C4—1572no—
236C2×C4#1457893112yes—
237C22—24no—
238C22—22no—
239C22—22no—
240C22—42no—
241C22—41no—
242C22—41no—
243C22—42no—
244C22—42no—
245C22—42no—
246C22—3142no—
247C22—6282no—
248C22—6282no—
249C22—6282no—
250C22C22⋊C4×D15711yescharacteristic
251C22C22⋊C4×D15711yescharacteristic
252C22C2×#1457893111yescharacteristic
253C22#1909773912yes—
254C22#1909774112yes—
255C4—22no—
256C4—82no—
257C4—3142no—
258C4—3142no—
259C4—3142no—
260C4—12562no—
261C2—22no—
262C2—22no—
263C2—21no—
264C2—21no—
265C2—42no—
266C2—6282no—
267C2#2375712811yescharacteristic
268C2#2375724512yes—
subgroupG/H
centerC22C22⋊C4×D157
commutatorC23×C157C22×C4
FrattiniC24C22×D157
FittingC2×#99×C157C2
socleC22×C157C2×C22⋊C4
prime radicalC22×#14578953C2
cosocleC24×C157C23
hypercenterC2×#99D157
hypocenterC157#1782
LCSG ▷ C23×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C24 ◁ C2×#99

Complements

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

NH ≅ G/Ntype
2. C2×#19097686C2semidirect
3. C2×#19097685C2semidirect
4. #23757134C2semidirect
5. C2×#99×C157C2semidirect
10. C22×Dic314C4semidirect
11. C22×C22⋊D157C4semidirect
13. C23×Dic157C4semidirect
15. C22×C4×D157C4semidirect
134. C157#1782semidirect
139. C2×#99D157semidirect

Maximal subgroups

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

subgroupindex
C22×#145789532
C2×#190976862
C2×#190976852
#237571342
C2×#99×C1572
#1782157