Finite Group Databasebeta

← #34103393

Subgroups — #34103393

Order 20096.

560 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 157 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×C4×C22⋊D157C211yesmaximal, characteristic
2#23757211C211yesmaximal, characteristic, complemented
3#23757208C211yesmaximal, characteristic, complemented
4#23757204C211yesmaximal, characteristic, complemented
5#23757201C212yesmaximal, complemented
6#23757199C211yesmaximal, characteristic, complemented
7#23757188C212yesmaximal
8C4×#14578933C212yesmaximal, complemented
9C4×#14578931C212yesmaximal, complemented
10C4×C22⋊Dic157C211yesmaximal, characteristic, complemented
11C2×#117×C157C211yesmaximal, characteristic, complemented
12C22×C22⋊D157C2211yescharacteristic
13C22×C4×D157C2211yescharacteristic
14C4×C22⋊D157—24no—
15C2×#14578933C2212yes—
16C2×#14578933C2212yes—
17C2×#14578933C2211yescharacteristic
18C2×C22⋊Dic157C2212yes—
19C2×C22⋊Dic157C2211yescharacteristic
20C2×#14578931C2212yes—
21C2×#14578931C2212yes—
22C2×#14578931C2211yescharacteristic
23C2×C4×Dic157C2212yes—
24C2×C4×Dic157C2211yescharacteristic
25#19097678C2211yescharacteristic
26#19097678C2211yescharacteristic
27#19097678C2212yes—
28#19097678C2212yes—
29#19097678C2212yes—
30#19097678C2212yes—
31C23×C4×C157C2211yescharacteristic
32#117×C157—24no—
33C2×C4⋊C4×C157C2211yescharacteristic
34C2×C4⋊C4×C157C2211yescharacteristic
35C2×C22⋊C4×C157C2212yes—
36C2×C42×C157C2212yes—
37C23×D157C2311yescharacteristic
38C2×C22⋊D157—22no—
39C2×C22⋊D157—22no—
40C2×C22⋊D157—22no—
41C22×Dic157C2312yes—
42C22×Dic157C2311yescharacteristic
43C22×Dic157C2311yescharacteristic
44C22×Dic157C2312yes—
45C22×Dic157C2311yescharacteristic
46C2×C4×D157—22no—
47C2×C4×D157—24no—
48#14578933—24no—
49#14578933—24no—
50#14578933—22no—
51C22⋊Dic157—24no—
52C22⋊Dic157—22no—
53#14578931—24no—
54#14578931—24no—
55#14578931—22no—
56C4×Dic157—22no—
57C4×Dic157—24no—
58C24×C157C2311yescharacteristic
59C22×C4×C157—24no—
60C22×C4×C157D414yes—
61C22×C4×C157C2312yes—
62C22×C4×C157C2312yes—
63C22×C4×C157C2311yescharacteristic
64C22×C4×C157C2311yescharacteristic
65C4⋊C4×C157—22no—
66C4⋊C4×C157—22no—
67C22⋊C4×C157—24no—
68C42×C157—24no—
69C22×D157—21no—
70C22×D157—22no—
71C22×D157—21no—
72C22×D157—22no—
73C22×D157#4112yes—
74C22⋊D157—28no—
75C2×Dic157—21no—
76C2×Dic157—21no—
77C2×Dic157—22no—
78C2×Dic157—22no—
79C2×Dic157—21no—
80C2×Dic157—21no—
81C2×Dic157—22no—
82C2×Dic157—22no—
83C2×Dic157—22no—
84C2×Dic157—22no—
85C2×Dic157—22no—
86C2×Dic157#4112yes—
87C2×Dic157#4114yes—
88C4×D157—44no—
89C23×C157—21no—
90C23×C157—21no—
91C23×C157—22no—
92C23×C157—22no—
93C23×C157C2×D412yes—
94C23×C157C2411yescharacteristic
95C2×C4×C157—22no—
96C2×C4×C157—28no—
97C2×C4×C157—22no—
98C2×C4×C157—22no—
99C2×C4×C157—22no—
100C2×C4×C157—22no—
101C2×C4×C157—22no—
102C2×C4×C157—22no—
103C2×C4×C157C2×D412yes—
104C2×C4×C157C2×D412yes—
105C2×C4×C157#4114yes—
106C22×C157—22no—
107C22×C157—22no—
108C22×C157—22no—
109C22×C157—42no—
110C22×C157—42no—
111C22×C157C22×D411yescharacteristic
112C22×C157C2×#4111yescharacteristic
113C22×C157C2×#4111yescharacteristic
114C22×C157C2×#4112yes—
115C22×C157C2×#4112yes—
116C2×D157—22no—
117C2×D157—24no—
118C2×D157—41no—
119C2×D157—42no—
120C2×D157—41no—
121C4×C157—22no—
122C4×C157—42no—
123C4×C157—42no—
124C4×C157#12314yes—
125Dic157—22no—
126Dic157—24no—
127Dic157—41no—
128Dic157—42no—
129Dic157—41no—
130C2×C157—42no—
131C2×C157C2×#12311yescharacteristic
132C2×C157#52511yescharacteristic
133C2×C157#52912yes—
134C2×C157#52912yes—
135C2×C157#53211yescharacteristic
136D157—42no—
137C157#239511yescharacteristic, complemented, Sylow
138#2395—1571nomaximal, Sylow
139C2×C4×D4—1571no—
140C2×#117D15711yescharacteristic, complemented
141#400—1571no—
142#399—1571no—
143#398—1571no—
144#397—1571no—
145#397—1572no—
146#394—1572no—
147C4×C4⋊C4—1572no—
148C4×C22⋊C4—1571no—
149C4×C22⋊C4—1572no—
150C22×D4—1571no—
151C23×C4—1571no—
152C23×C4C2×D15711yescharacteristic
153C4×D4—3144no—
154#117—24no—
155C2×C4⋊C4—1571no—
156C2×C4⋊C4—1572no—
157C2×C4⋊C4—1572no—
158C2×C4⋊C4C2×D15711yescharacteristic
159C2×C4⋊C4C2×D15711yescharacteristic
160C2×C22⋊C4—1571no—
161C2×C22⋊C4—1572no—
162C2×C22⋊C4—1571no—
163C2×C22⋊C4—1572no—
164C2×C22⋊C4—1572no—
165C2×C22⋊C4C2×D15712yes—
166C2×C42—1571no—
167C2×C42—1572no—
168C2×C42C2×D15712yes—
169#95—1572no—
170#95—1571no—
171#95—1572no—
172#95—1571no—
173#95—1572no—
174#95—1572no—
175C24—1571no—
176C24C22×D15711yescharacteristic
177C2×D4—3142no—
178C2×D4—3142no—
179C2×D4—3142no—
180C22×C4—24no—
181C22×C4—3142no—
182C22×C4—3144no—
183C22×C4—1572no—
184C22×C4—1571no—
185C22×C4—1571no—
186C22×C4—1572no—
187C22×C4—1571no—
188C22×C4C22⋊D15714yes—
189C22×C4C22×D15712yes—
190C22×C4C22×D15712yes—
191C22×C4C22×D15711yescharacteristic
192C22×C4C22×D15711yescharacteristic
193C4⋊C4—22no—
194C4⋊C4—22no—
195C4⋊C4—3144no—
196C4⋊C4—3144no—
197C4⋊C4—3142no—
198C22⋊C4—24no—
199C22⋊C4—3144no—
200C22⋊C4—3144no—
201C22⋊C4—3142no—
202C22⋊C4—3144no—
203C22⋊C4—3142no—
204C42—24no—
205C42—3142no—
206C42—3144no—
207C23—22no—
208C23—21no—
209C23—21no—
210C23—22no—
211C23—3141no—
212C23—3142no—
213C23—3141no—
214C23—3142no—
215C23—1572no—
216C23C2×C22⋊D15712yes—
217C23C23×D15711yescharacteristic
218D4—3148no—
219C2×C4—22no—
220C2×C4—28no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—3141no—
228C2×C4—3142no—
229C2×C4—3141no—
230C2×C4—3142no—
231C2×C4—3141no—
232C2×C4—3141no—
233C2×C4—3142no—
234C2×C4—3142no—
235C2×C4—3142no—
236C2×C4—3142no—
237C2×C4—3142no—
238C2×C4—6284no—
239C2×C4—1574no—
240C2×C4—1572no—
241C2×C4C2×C22⋊D15712yes—
242C2×C4C2×C22⋊D15712yes—
243C2×C4#1457895414yes—
244C22—22no—
245C22—22no—
246C22—22no—
247C22—42no—
248C22—42no—
249C22—3142no—
250C22—3144no—
251C22—6281no—
252C22—6282no—
253C22—6281no—
254C22C22×C22⋊D15711yescharacteristic
255C22#41×D15711yescharacteristic
256C22#41×D15711yescharacteristic
257C22#41×D15712yes—
258C22C2×#1457895412yes—
259C4—22no—
260C4—42no—
261C4—42no—
262C4—3142no—
263C4—3144no—
264C4—6281no—
265C4—6282no—
266C4—6281no—
267C4#1909775414yes—
268C2—42no—
269C2—6282no—
270C2C2×#1909775411yescharacteristic
271C2#2375778011yescharacteristic
272C2#2375780112yes—
273C2#2375780112yes—
274C2#2375781911yescharacteristic
subgroupG/H
centerC22×C4C22×D157
commutator, socle, cosocleC23×C157C24
FrattiniC23C23×D157
FittingC2×#117×C157C2
prime radicalC22×C22⋊D157C22
hypercenterC2×#117D157
hypocenterC157#2395
LCSG ▷ C23×C157 ▷ C157
UCSC1 ◁ C22×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. #23757211C2semidirect
3. #23757208C2semidirect
4. #23757204C2semidirect
5. #23757201C2semidirect
6. #23757199C2semidirect
8. C4×#14578933C2semidirect
9. C4×#14578931C2semidirect
10. C4×C22⋊Dic157C2semidirect
11. C2×#117×C157C2semidirect
137. C157#2395semidirect
140. C2×#117D157semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊D1572
#237572112
#237572082
#237572042
#237572012
#237571992
#237571882
C4×#145789332
C4×#145789312
C4×C22⋊Dic1572
C2×#117×C1572
#2395157