Finite Group Databasebeta

← #34103425

Subgroups — #34103425

Order 20096.

568 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 143 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
1C2×#19097736C211yesmaximal, characteristic, complemented
2C2×#19097733C211yesmaximal, characteristic
3#23757211C211yesmaximal, characteristic, complemented
4#23757202C212yesmaximal, complemented
5#23757191C211yesmaximal, characteristic
6#23757190C211yesmaximal, characteristic
7#23757186C212yesmaximal, complemented
8#23757120C212yesmaximal, complemented
9#23757120C212yesmaximal, complemented
10#23757118C211yesmaximal, characteristic, complemented
11#400×C157C211yesmaximal, characteristic, complemented
12C22×Dic314C2211yescharacteristic
13C22×D628C2211yescharacteristic
14C22×C4×D157C2212yes—
15#19097736—24no—
16#19097733—24no—
17C2×#14578933C2211yescharacteristic
18C2×#14578933C2211yescharacteristic
19C2×#14578931C2212yes—
20C2×#14578931C2212yes—
21C2×#14578931C2212yes—
22C2×#14578931C2212yes—
23C2×#14578931C2212yes—
24C2×C4×Dic157C2212yes—
25C2×C4×Dic157C2211yescharacteristic
26C2×C4×Dic157C2211yescharacteristic
27C2×C4×Dic157C2211yescharacteristic
28#19097678C2212yes—
29#19097678C2211yescharacteristic
30#19097678C2212yes—
31#19097678C2212yes—
32C2×C4⋊C4×C157C2211yescharacteristic
33C2×C4⋊C4×C157C2211yescharacteristic
34#95×C157C2211yescharacteristic
35#95×C157C2212yes—
36#95×C157C2212yes—
37C23×D157C2312yes—
38C2×Dic314—22no—
39C2×Dic314—22no—
40C2×Dic314—22no—
41C2×D628—24no—
42C2×D628—22no—
43C22×Dic157C2312yes—
44C22×Dic157C2311yescharacteristic
45C22×Dic157C2311yescharacteristic
46C22×Dic157C2311yescharacteristic
47C22×Dic157C2311yescharacteristic
48C2×C4×D157—24no—
49C2×C4×D157—24no—
50C2×C4×D157—24no—
51#14578933—22no—
52#14578933—22no—
53#14578931—24no—
54#14578931—24no—
55#14578931—24no—
56#14578931—24no—
57#14578931—24no—
58C4×Dic157—22no—
59C4×Dic157—22no—
60C4×Dic157—24no—
61C4×Dic157D414yes—
62C22×C4×C157C2312yes—
63C22×C4×C157C2312yes—
64C22×C4×C157C2311yescharacteristic
65C22×C4×C157C2311yescharacteristic
66C22×C4×C157C2311yescharacteristic
67C4⋊C4×C157—22no—
68C4⋊C4×C157—22no—
69C22×D157—22no—
70C22×D157—22no—
71C22×D157—22no—
72C22×D157—22no—
73C22×D157—22no—
74C22×D157—22no—
75C22×D157#4114yes—
76Dic314—28no—
77D628—28no—
78C2×Dic157—21no—
79C2×Dic157—21no—
80C2×Dic157—21no—
81C2×Dic157—21no—
82C2×Dic157—21no—
83C2×Dic157—22no—
84C2×Dic157—22no—
85C2×Dic157—22no—
86C2×Dic157—22no—
87C2×Dic157C2×D412yes—
88C2×Dic157C2×D412yes—
89C2×Dic157#4112yes—
90C2×Dic157#4114yes—
91C4×D157—48no—
92C23×C157C2411yescharacteristic
93C2×C4×C157—21no—
94C2×C4×C157—22no—
95C2×C4×C157—21no—
96C2×C4×C157—22no—
97C2×C4×C157—21no—
98C2×C4×C157—22no—
99C2×C4×C157—22no—
100C2×C4×C157—22no—
101C2×C4×C157—22no—
102C2×C4×C157—22no—
103C2×C4×C157—22no—
104C2×C4×C157C2×D412yes—
105C2×C4×C157#4112yes—
106C22×C157C22×D411yescharacteristic
107C22×C157C2×#4112yes—
108C22×C157C2×#4111yescharacteristic
109C22×C157C2×#4112yes—
110C22×C157C2×#4111yescharacteristic
111C2×D157—24no—
112C2×D157—24no—
113C2×D157—24no—
114C2×D157—42no—
115C2×D157—42no—
116C2×D157—42no—
117C2×D157—42no—
118C4×C157—22no—
119C4×C157—22no—
120C4×C157—41no—
121C4×C157—42no—
122C4×C157—42no—
123Dic157—22no—
124Dic157—22no—
125Dic157—41no—
126Dic157—42no—
127Dic157#12414yes—
128C2×C157#52511yescharacteristic
129C2×C157#52511yescharacteristic
130C2×C157C2×#12411yescharacteristic
131C2×C157#52912yes—
132C2×C157#52912yes—
133D157—44no—
134C157#239811yescharacteristic, complemented, Sylow
135#2398—1571nomaximal, Sylow
136C2×C4×Q8—1571no—
137C2×C4×D4—1571no—
138#400—1571no—
139#400D15711yescharacteristic, complemented
140#397—1572no—
141#394—1571no—
142#394—1571no—
143#393—1572no—
144C4×C22⋊C4—1572no—
145C4×C22⋊C4—1572no—
146#376—1571no—
147C22×Q8—1571no—
148C22×D4—1571no—
149C23×C4—1572no—
150C4×Q8—3144no—
151C4×D4—3144no—
152C2×C4⋊C4—1571no—
153C2×C4⋊C4—1571no—
154C2×C4⋊C4C2×D15711yescharacteristic
155C2×C4⋊C4C2×D15711yescharacteristic
156C2×C22⋊C4—1572no—
157C2×C22⋊C4—1572no—
158C2×C22⋊C4—1572no—
159C2×C22⋊C4—1572no—
160C2×C22⋊C4—1572no—
161C2×C42—1571no—
162C2×C42—1571no—
163C2×C42—1572no—
164C2×C42—1571no—
165#95—1572no—
166#95—1571no—
167#95—1572no—
168#95—1572no—
169#95C2×D15711yescharacteristic
170#95C2×D15712yes—
171#95C2×D15712yes—
172C24—1572no—
173C2×Q8—3142no—
174C2×Q8—3142no—
175C2×Q8—3142no—
176C2×D4—3144no—
177C2×D4—3142no—
178C22×C4—3144no—
179C22×C4—3144no—
180C22×C4—3144no—
181C22×C4—1572no—
182C22×C4—1571no—
183C22×C4—1571no—
184C22×C4—1571no—
185C22×C4—1571no—
186C22×C4C22×D15712yes—
187C22×C4C22×D15712yes—
188C22×C4C22×D15711yescharacteristic
189C22×C4C22×D15711yescharacteristic
190C22×C4C22×D15711yescharacteristic
191C4⋊C4—22no—
192C4⋊C4—22no—
193C4⋊C4—3142no—
194C4⋊C4—3142no—
195C22⋊C4—3144no—
196C22⋊C4—3144no—
197C22⋊C4—3144no—
198C22⋊C4—3144no—
199C22⋊C4—3144no—
200C42—3142no—
201C42—3142no—
202C42—3144no—
203C42—1574no—
204C23—3142no—
205C23—3142no—
206C23—3142no—
207C23—3142no—
208C23—3142no—
209C23—3142no—
210C23—1574no—
211C23C23×D15711yescharacteristic
212Q8—3148no—
213D4—3148no—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—3141no—
226C2×C4—3141no—
227C2×C4—3141no—
228C2×C4—3141no—
229C2×C4—3141no—
230C2×C4—3142no—
231C2×C4—3142no—
232C2×C4—3142no—
233C2×C4—3142no—
234C2×C4—6288no—
235C2×C4—1572no—
236C2×C4—1572no—
237C2×C4—1572no—
238C2×C4—1574no—
239C2×C4D4×D15712yes—
240C2×C4#1457895512yes—
241C22—3144no—
242C22—3144no—
243C22—3144no—
244C22—6282no—
245C22—6282no—
246C22—6282no—
247C22—6282no—
248C22C2×D4×D15711yescharacteristic
249C22#41×D15712yes—
250C22#41×D15711yescharacteristic
251C22#41×D15712yes—
252C22C2×#1457895511yescharacteristic
253C4—22no—
254C4—22no—
255C4—41no—
256C4—42no—
257C4—42no—
258C4—3142no—
259C4—3142no—
260C4—6281no—
261C4—6282no—
262C4—1574no—
263C2—6284no—
264C2#2375778111yescharacteristic
265C2#2375778611yescharacteristic
266C2#124×D15711yescharacteristic
267C2#2375780612yes—
268C2#2375780812yes—
subgroupG/H
center, FrattiniC23C23×D157
commutator, socle, cosocleC23×C157C24
Fitting#400×C157C2
prime radicalC22×D628C22
hypercenter#400D157
hypocenterC157#2398
LCSG ▷ C23×C157 ▷ C157
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
1. C2×#19097736C2semidirect
3. #23757211C2semidirect
4. #23757202C2semidirect
7. #23757186C2semidirect
8. #23757120C2semidirect
9. #23757120C2semidirect
10. #23757118C2semidirect
11. #400×C157C2semidirect
134. C157#2398semidirect
139. #400D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190977362
C2×#190977332
#237572112
#237572022
#237571912
#237571902
#237571862
#237571202
#237571182
#400×C1572
#2398157