Finite Group Databasebeta

← #34115412

Subgroups — #34115412

Order 20096.

322 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 64 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#23758243C211yesmaximal, characteristic, complemented
2C2×C22⋊#11929441C211yesmaximal, characteristic, complemented
3#23758117C212yesmaximal, complemented
4#23758116C212yesmaximal, complemented
5C2×#19097743C211yesmaximal, characteristic
6C2×#14578963—21no—
7C22×#11929441—21no—
8C22⋊#11929441—21no—
9C22⋊#11929441—21no—
10C22⋊#11929441C2212yes—
11C22⋊#11929441C2212yes—
12C22×C22⋊D157C411yescharacteristic
13C22×C22⋊D157C411yescharacteristic
14C23×Dic157C2211yescharacteristic
15#19097743—21no—
16#19097743—21no—
17#19097743—21no—
18#19097743—21no—
19C2×#14578933C411yescharacteristic
20C2×C22⋊Dic157C2211yescharacteristic
21C2×#14578931C2211yescharacteristic
22C22×D4×C157C411yescharacteristic
23#14578963—42no—
24C2×#11929441—22no—
25C2×#11929441—22no—
26C2×#11929441—42no—
27C23×D157C2×C411yescharacteristic
28C2×C22⋊D157—21no—
29C2×C22⋊D157—21no—
30C2×C22⋊D157—21no—
31C2×C22⋊D157—21no—
32C2×C22⋊D157—42no—
33C2×C22⋊D157—42no—
34C22×Dic157—42no—
35C22×Dic157C2×C411yescharacteristic
36C22×Dic157C2×C411yescharacteristic
37C22×Dic157D411yescharacteristic
38C22×Dic157D411yescharacteristic
39C22×Dic157D411yescharacteristic
40C22×Dic157D411yescharacteristic
41C22×Dic157C2311yescharacteristic
42#14578933—21no—
43#14578933—21no—
44C22⋊Dic157—21no—
45C22⋊Dic157—21no—
46#14578931—21no—
47#14578931—21no—
48C24×C157C2×C411yescharacteristic
49C24×C157C2×C411yescharacteristic
50C2×D4×C157—21no—
51C2×D4×C157—21no—
52C2×D4×C157—42no—
53C22×C4×C157C2×C411yescharacteristic
54#11929441—42no—
55#11929441—42no—
56C22×D157—21no—
57C22×D157—22no—
58C22×D157—41no—
59C22×D157—41no—
60C22⋊D157—42no—
61C22⋊D157—42no—
62C22⋊D157—42no—
63C22⋊D157—42no—
64C2×Dic157—22no—
65C2×Dic157—22no—
66C2×Dic157—22no—
67C2×Dic157—21no—
68C2×Dic157—41no—
69C2×Dic157—41no—
70C2×Dic157—41no—
71C2×Dic157—41no—
72C2×Dic157C22⋊C411yescharacteristic
73C2×Dic157C22⋊C411yescharacteristic
74C2×Dic157C2×D411yescharacteristic
75C2×Dic157C2×D411yescharacteristic
76C23×C157—22no—
77C23×C157—22no—
78C23×C157—21no—
79C23×C157—22no—
80C23×C157—41no—
81C23×C157—41no—
82C23×C157—41no—
83C23×C157C22⋊C411yescharacteristic
84C23×C157C22⋊C411yescharacteristic
85C23×C157C22×C411yescharacteristic
86D4×C157—42no—
87D4×C157—42no—
88C2×C4×C157—21no—
89C2×C4×C157—41no—
90C22×C157—21no—
91C22×C157—21no—
92C22×C157—21no—
93C22×C157—21no—
94C22×C157—21no—
95C22×C157—21no—
96C22×C157—21no—
97C22×C157—44no—
98C22×C157—41no—
99C22×C157—41no—
100C22×C157—41no—
101C22×C157—82no—
102C22×C157#10012yes—
103C22×C157C2×C22⋊C411yescharacteristic
104C22×C157#13112yes—
105C2×D157—41no—
106C2×D157—41no—
107C2×D157—41no—
108C2×D157—82no—
109C4×C157—41no—
110Dic157—21no—
111Dic157—21no—
112Dic157—21no—
113Dic157—41no—
114Dic157#9912yes—
115C2×C157—22no—
116C2×C157—22no—
117C2×C157—41no—
118C2×C157—81no—
119C2×C157#40811yescharacteristic
120C2×C157C2×#9911yescharacteristic
121C2×C157C2×#10011yescharacteristic
122D157—81no—
123C157#147111yescharacteristic, complemented, Sylow
124#1471—1571nomaximal, Sylow
125C2×#121—1571no—
126#407—1571no—
127C2×C22⋊C8—1571no—
128#323—1572no—
129#323—1572no—
130C22×D4—1571no—
131C22×D4—1571no—
132C22×D4#1059784511yescharacteristic
133C23×C4—1571no—
134C2×M16—3141no—
135C22×C8—3141no—
136#121—3141no—
137#121—3141no—
138#121—3141no—
139#121—3141no—
140C2×C4⋊C4—1571no—
141C2×C22⋊C4—1571no—
142C2×C22⋊C4—1571no—
143C22⋊C8—3141no—
144C22⋊C8—3141no—
145C22⋊C8—1572no—
146C22⋊C8—1572no—
147C24—1571no—
148C24C2×#1059784511yescharacteristic
149C24C2×#1059784511yescharacteristic
150C2×D4—21no—
151C2×D4—21no—
152C2×D4—42no—
153C2×D4—3141no—
154C2×D4—3141no—
155C2×D4—3141no—
156C2×D4—3141no—
157C2×D4—6282no—
158C2×D4—6282no—
159C22×C4—6282no—
160C22×C4—1571no—
161C22×C4—1571no—
162C22×C4—1571no—
163C22×C4—1571no—
164C22×C4—1571no—
165C22×C4—1571no—
166C22×C4—1571no—
167C22×C4C2×#1059784511yescharacteristic
168M16—6282no—
169C2×C8—3142no—
170C2×C8—3142no—
171C2×C8—6282no—
172C4⋊C4—3141no—
173C4⋊C4—3141no—
174C22⋊C4—3141no—
175C22⋊C4—3141no—
176C22⋊C4—3141no—
177C22⋊C4—3141no—
178C23—22no—
179C23—22no—
180C23—21no—
181C23—22no—
182C23—41no—
183C23—41no—
184C23—41no—
185C23—3141no—
186C23—3142no—
187C23—6281no—
188C23—6281no—
189C23C22⋊#1059784511yescharacteristic
190C23C22⋊#1059784511yescharacteristic
191C23C22×#1059784511yescharacteristic
192D4—42no—
193D4—42no—
194D4—6282no—
195D4—6282no—
196D4—6282no—
197D4—6282no—
198C2×C4—21no—
199C2×C4—41no—
200C2×C4—3142no—
201C2×C4—3142no—
202C2×C4—3142no—
203C2×C4—3141no—
204C2×C4—6281no—
205C2×C4—6281no—
206C2×C4—6281no—
207C2×C4—6281no—
208C2×C4—1571no—
209C2×C4—1571no—
210C2×C4—1571no—
211C2×C4—1571no—
212C8—6282no—
213C8—6282no—
214C22—21no—
215C22—21no—
216C22—21no—
217C22—21no—
218C22—21no—
219C22—21no—
220C22—21no—
221C22—44no—
222C22—41no—
223C22—41no—
224C22—41no—
225C22—82no—
226C22—6281no—
227C22—6281no—
228C22—6281no—
229C22—12562no—
230C22#1909783112yes—
231C22C2×C22⋊#1059784511yescharacteristic
232C22#1909785812yes—
233C4—41no—
234C4—3141no—
235C4—3141no—
236C4—3141no—
237C4—6281no—
238C4—1572no—
239C2—22no—
240C2—22no—
241C2—41no—
242C2—81no—
243C2—12561no—
244C2#2375824611yescharacteristic
245C2D157⋊#9911yescharacteristic
246C2C2×#1909783111yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10597845
commutatorC23×C157C22×C4
FrattiniC23C22×#10597845
FittingC22×D4×C157C4
socleC22×C157C2×C22⋊C4
prime radicalC2×#19097743C2
cosocleC22×Dic157C23
hypercenterC22×D4#10597845
hypocenterC157#1471
LCSG ▷ C23×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C24 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. #23758243C2semidirect
2. C2×C22⋊#11929441C2semidirect
3. #23758117C2semidirect
4. #23758116C2semidirect
123. C157#1471semidirect

Maximal subgroups

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

subgroupindex
#237582432
C2×C22⋊#119294412
#237581172
#237581162
C2×#190977432
#1471157