Finite Group Databasebeta

← #34101263

Subgroups — #34101263

Order 20096.

638 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 638 conjugacy classes of subgroups — too many to draw. Only its 75 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#23757835C211yesmaximal, characteristic, complemented
2C2×#19097741C212yesmaximal, complemented
3C2×#19097685C212yesmaximal, complemented
4#23757128C211yesmaximal, characteristic, complemented
5#342×C157C211yesmaximal, characteristic, complemented
6C24×D157C2211yescharacteristic
7C22×D628—22no—
8C22×C22⋊D157C2212yes—
9#19097755—22no—
10#19097745—22no—
11#19097741—24no—
12#19097741—22no—
13#19097741—24no—
14C2×C22⋊Dic157C2211yescharacteristic
15C2×#14578931—22no—
16C2×#14578931—22no—
17#19097685—24no—
18C2×C22⋊C4×C157C2211yescharacteristic
19C2×C22⋊C4×C157C2212yes—
20C23×D157—21no—
21C23×D157—22no—
22C23×D157—22no—
23C23×D157—21no—
24C23×D157—44no—
25C23×D157D412yescomplemented
26C23×D157D412yescomplemented
27C2×D628—28no—
28C2×D628—22no—
29C2×D628—48no—
30C2×C22⋊D157—22no—
31C2×C22⋊D157—22no—
32C2×C22⋊D157—22no—
33C2×C22⋊D157—22no—
34C2×C22⋊D157—22no—
35C2×C22⋊D157—22no—
36C22×Dic157—21no—
37C22×Dic157D412yescomplemented
38C4⋊Dic157—22no—
39C22⋊Dic157—22no—
40#14578931—28no—
41#14578931—24no—
42#14578931—44no—
43C24×C157C2311yescharacteristic
44C22×C4×C157—22no—
45C22×C4×C157D412yescomplemented
46C22⋊C4×C157—22no—
47C22⋊C4×C157—22no—
48C22⋊C4×C157D414yescomplemented
49C22×D157—28no—
50C22×D157—24no—
51C22×D157—24no—
52C22×D157—22no—
53C22×D157—22no—
54C22×D157—22no—
55C22×D157—22no—
56C22×D157—44no—
57C22×D157—48no—
58C22×D157—44no—
59C22×D157—42no—
60C22×D157—44no—
61C22×D157—41no—
62C22×D157—42no—
63C22×D157—41no—
64C22×D157—44no—
65D628—416no—
66D628—82no—
67C22⋊D157—42no—
68C22⋊D157—42no—
69C22⋊D157—42no—
70C22⋊D157—42no—
71C2×Dic157—22no—
72C2×Dic157—22no—
73C2×Dic157—22no—
74C2×Dic157—42no—
75C2×Dic157—41no—
76C23×C157—42no—
77C23×C157C2×D412yes—
78C23×C157C2×D411yescharacteristic
79C23×C157C2×D411yescharacteristic
80C23×C157C2×D412yes—
81C23×C157#4111yescharacteristic
82C2×C4×C157—24no—
83C2×C4×C157—22no—
84C2×C4×C157—44no—
85C2×C4×C157—42no—
86C22×C157—22no—
87C22×C157—24no—
88C22×C157—41no—
89C22×C157—42no—
90C22×C157—41no—
91C22×C157#12012yes—
92C22×C157#12011yescharacteristic
93C22×C157#12112yes—
94C22×C157#12111yescharacteristic
95C22×C157#12711yescharacteristic
96C2×D157—28no—
97C2×D157—24no—
98C2×D157—44no—
99C2×D157—44no—
100C2×D157—42no—
101C2×D157—48no—
102C2×D157—42no—
103C2×D157—44no—
104C2×D157—44no—
105C2×D157—42no—
106C2×D157—44no—
107C2×D157—82no—
108C4×C157—44no—
109C4×C157—42no—
110Dic157—42no—
111Dic157—81no—
112C2×C157—21no—
113C2×C157—22no—
114C2×C157—21no—
115C2×C157—22no—
116C2×C157#39211yescharacteristic
117C2×C157B(4,𝔽2)12yes—
118D157—44no—
119D157—82no—
120C157#202111yescharacteristic, complemented, Sylow
121#2021—1571nomaximal, Sylow
122#534—1571no—
123C2×#120—1572no—
124C2×#99—1572no—
125#379—1571no—
126#342D15711yescharacteristic, complemented
127C25—1571no—
128C22×D4—3142no—
129C22×D4—1572no—
130#123—3142no—
131#121—3142no—
132#120—3144no—
133#120—3142no—
134#120—3144no—
135C2×C22⋊C4—3142no—
136C2×C22⋊C4—3142no—
137C2×C22⋊C4—1571no—
138C2×C22⋊C4C2×D15711yescharacteristic
139C2×C22⋊C4C2×D15712yes—
140#99—3144no—
141C24—3142no—
142C24—3141no—
143C24—3142no—
144C24—3141no—
145C24—6284no—
146C24—1572no—
147C24—1572no—
148C24C22×D15711yescharacteristic
149C2×D4—3148no—
150C2×D4—3142no—
151C2×D4—3142no—
152C2×D4—3142no—
153C2×D4—3142no—
154C2×D4—3142no—
155C2×D4—3142no—
156C2×D4—3142no—
157C2×D4—6288no—
158C22×C4—22no—
159C22×C4—3141no—
160C22×C4—1572no—
161C22×C4D62812yescomplemented
162C4⋊C4—3142no—
163C22⋊C4—22no—
164C22⋊C4—22no—
165C22⋊C4—3148no—
166C22⋊C4—3144no—
167C22⋊C4—3142no—
168C22⋊C4—6284no—
169C22⋊C4D62814yescomplemented
170C23—42no—
171C23—3144no—
172C23—3148no—
173C23—3144no—
174C23—3142no—
175C23—3142no—
176C23—3142no—
177C23—3142no—
178C23—6282no—
179C23—6284no—
180C23—6284no—
181C23—6284no—
182C23—6288no—
183C23—6284no—
184C23—6281no—
185C23—6282no—
186C23—6281no—
187C23D4×D15712yes—
188C23D4×D15711yescharacteristic
189C23C2×D62811yescharacteristic
190C23C2×D62812yes—
191C23#1457895511yescharacteristic
192D4—62816no—
193D4—6282no—
194D4—6282no—
195D4—6282no—
196D4—6282no—
197D4—12562no—
198C2×C4—24no—
199C2×C4—22no—
200C2×C4—44no—
201C2×C4—42no—
202C2×C4—3142no—
203C2×C4—3142no—
204C2×C4—3142no—
205C2×C4—6282no—
206C2×C4—6281no—
207C22—22no—
208C22—24no—
209C22—41no—
210C22—42no—
211C22—41no—
212C22—3148no—
213C22—3144no—
214C22—6284no—
215C22—6284no—
216C22—6282no—
217C22—6288no—
218C22—6284no—
219C22—6282no—
220C22—6284no—
221C22—6282no—
222C22—6284no—
223C22—12562no—
224C22#1909774112yes—
225C22#1909774111yescharacteristic
226C22#1909774612yes—
227C22#1909774611yescharacteristic
228C22#1909776811yescharacteristic
229C4—44no—
230C4—42no—
231C4—6282no—
232C4—12561no—
233C2—22no—
234C2—21no—
235C2—21no—
236C2—22no—
237C2—6284no—
238C2—12562no—
239C2#2375718311yescharacteristic
240C2#2375746112yes—
subgroupG/H
centerC22#19097741
commutator, cosocleC24×C157C23
FrattiniC24C22×D157
Fitting#342×C157C2
socleC22×C157#120
hypercenter#342D157
hypocenterC157#2021
LCSG ▷ C24×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
1. #23757835C2semidirect
2. C2×#19097741C2semidirect
3. C2×#19097685C2semidirect
4. #23757128C2semidirect
5. #342×C157C2semidirect
25. C23×D157D4semidirect
26. C23×D157D4semidirect
37. C22×Dic157D4semidirect
45. C22×C4×C157D4semidirect
48. C22⋊C4×C157D4semidirect
120. C157#2021semidirect
126. #342D157semidirect
161. C22×C4D628semidirect
169. C22⋊C4D628semidirect

Maximal subgroups

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

subgroupindex
#237578352
C2×#190977412
C2×#190976852
#237571282
#342×C1572
#2021157