Finite Group Databasebeta

← #34098472

Subgroups — #34098472

Order 20096.

316 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 77 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#23757237C211yesmaximal, characteristic, complemented
3#23756998C212yesmaximal, complemented
4#23756996C212yesmaximal, complemented
5#407×C157C211yesmaximal, characteristic, complemented
6C22×C22⋊D157C411yescharacteristic
7C22×C4×D157C411yescharacteristic
8C2×#14578938—21no—
9C4×C22⋊D157—21no—
10C4×C22⋊D157—21no—
11C4×C22⋊D157—21no—
12C4×C22⋊D157—21no—
13C2×#14578933C2211yescharacteristic
14C2×C22⋊Dic157C411yescharacteristic
15C2×#14578931C2211yescharacteristic
16C2×C4×Dic157C411yescharacteristic
17C22⋊#11929434—21no—
18C22⋊#11929434C2212yes—
19C23×C4×C157C2211yescharacteristic
20C2×M16×C157—21no—
21C22⋊C8×C157—21no—
22C22⋊C8×C157C2212yes—
23C23×D157C2×C411yescharacteristic
24C2×C22⋊D157—22no—
25C2×C22⋊D157—22no—
26C2×C22⋊D157—21no—
27C2×C22⋊D157—21no—
28C22×Dic157C2×C411yescharacteristic
29C22×Dic157C2×C411yescharacteristic
30C22×Dic157C2×C411yescharacteristic
31C2×C4×D157—21no—
32C2×C4×D157—21no—
33C2×C4×D157—42no—
34#14578938—42no—
35C2×#11929434—22no—
36#14578933—21no—
37#14578933—21no—
38C22⋊Dic157—21no—
39C22⋊Dic157—21no—
40#14578931—21no—
41#14578931—21no—
42C4×Dic157—21no—
43C4×Dic157—21no—
44C24×C157C2×C411yescharacteristic
45C22×C4×C157—42no—
46C22×C4×C157C2×C411yescharacteristic
47C22×C4×C157D411yescharacteristic
48C22×C4×C157D411yescharacteristic
49C22×C4×C157D411yescharacteristic
50C22×C4×C157D411yescharacteristic
51C22×C4×C157C2311yescharacteristic
52M16×C157—42no—
53C2×C8×C157—22no—
54C22×D157—21no—
55C22×D157—21no—
56C22×D157—41no—
57C22×D157—41no—
58C22×D157M1612yescomplemented
59C22⋊D157—24no—
60C22⋊D157—24no—
61C2×Dic157—21no—
62C2×Dic157—21no—
63C2×Dic157—21no—
64C2×Dic157—21no—
65C2×Dic157—41no—
66C2×Dic157—41no—
67C2×Dic157M1612yescomplemented
68C4×D157—42no—
69C4×D157—42no—
70#11929434—42no—
71C23×C157—21no—
72C23×C157—21no—
73C23×C157—41no—
74C23×C157—41no—
75C23×C157C22⋊C411yescharacteristic
76C23×C157C22⋊C411yescharacteristic
77C23×C157C22×C411yescharacteristic
78C2×C4×C157—22no—
79C2×C4×C157—22no—
80C2×C4×C157—22no—
81C2×C4×C157—41no—
82C2×C4×C157—41no—
83C2×C4×C157—41no—
84C2×C4×C157C22⋊C411yescharacteristic
85C2×C4×C157C22⋊C411yescharacteristic
86C2×C4×C157C2×D411yescharacteristic
87C2×C4×C157C2×D411yescharacteristic
88C8×C157—42no—
89C22×C157—21no—
90C22×C157—21no—
91C22×C157—21no—
92C22×C157—21no—
93C22×C157—21no—
94C22×C157—21no—
95C22×C157—21no—
96C22×C157—21no—
97C22×C157—42no—
98C22×C157—42no—
99C22×C157C2×C22⋊C411yescharacteristic
100C22×C157C2×M1611yescharacteristic
101C22×C157C2×M1611yescharacteristic
102C2×D157—22no—
103C2×D157—42no—
104C2×D157—41no—
105C2×D157—41no—
106C2×D157—42no—
107C4×C157—21no—
108C4×C157—21no—
109C4×C157—21no—
110C4×C157#9912yes—
111Dic157—22no—
112Dic157—41no—
113Dic157—41no—
114C2×C157—21no—
115C2×C157—21no—
116C2×C157—41no—
117C2×C157—41no—
118C2×C157#40711yescharacteristic
119C2×C157C2×#9911yescharacteristic
120C2×C157#41311yescharacteristic
121D157—42no—
122C157#146511yescharacteristic, complemented, Sylow
123#1465—1571nomaximal, Sylow
124C2×C4×D4—1571no—
125#407—1571no—
126#407D15711yescharacteristic, complemented
127#324—1572no—
128#323—1572no—
129C22×D4—1571no—
130C23×C4—1571no—
131C23×C4C2×D15711yescharacteristic
132C2×M16—21no—
133C2×M16—3141no—
134C4×D4—3141no—
135C4×D4—3141no—
136C4×D4—3141no—
137C4×D4—3141no—
138C2×C4⋊C4—1571no—
139C2×C22⋊C4—1571no—
140C2×C22⋊C4—1571no—
141C2×C42—1571no—
142C22⋊C8—21no—
143C22⋊C8—3141no—
144C22⋊C8—1572no—
145C22⋊C8C2×D15712yes—
146C24—1571no—
147C24C4×D15711yescharacteristic
148C2×D4—3142no—
149C2×D4—3141no—
150C2×D4—3141no—
151C2×D4—3142no—
152C22×C4—42no—
153C22×C4—3141no—
154C22×C4—3141no—
155C22×C4—6282no—
156C22×C4—1571no—
157C22×C4—1571no—
158C22×C4—1571no—
159C22×C4C4×D15711yescharacteristic
160C22×C4C22⋊D15711yescharacteristic
161C22×C4C22⋊D15711yescharacteristic
162C22×C4D62811yescharacteristic
163C22×C4D62811yescharacteristic
164C22×C4C22×D15711yescharacteristic
165M16—42no—
166M16—6282no—
167C2×C8—22no—
168C2×C8—3142no—
169C4⋊C4—3141no—
170C4⋊C4—3141no—
171C22⋊C4—3141no—
172C22⋊C4—3141no—
173C22⋊C4—3141no—
174C22⋊C4—3141no—
175C42—3141no—
176C42—3141no—
177C23—21no—
178C23—21no—
179C23—41no—
180C23—41no—
181C23—3141no—
182C23—3141no—
183C23—6281no—
184C23—6281no—
185C23—1572no—
186C23#1457893111yescharacteristic
187C23#1457893111yescharacteristic
188C23C2×C4×D15711yescharacteristic
189D4—3144no—
190D4—3144no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—3141no—
198C2×C4—3141no—
199C2×C4—3141no—
200C2×C4—3141no—
201C2×C4—6282no—
202C2×C4—6282no—
203C2×C4—6281no—
204C2×C4—6281no—
205C2×C4—1572no—
206C2×C4#1457893111yescharacteristic
207C2×C4#1457893111yescharacteristic
208C2×C4C2×C22⋊D15711yescharacteristic
209C2×C4C2×D62811yescharacteristic
210C8—42no—
211C8—6282no—
212C22—21no—
213C22—21no—
214C22—21no—
215C22—21no—
216C22—21no—
217C22—21no—
218C22—21no—
219C22—21no—
220C22—42no—
221C22—42no—
222C22—3142no—
223C22—6282no—
224C22—6282no—
225C22—6281no—
226C22—6281no—
227C22C2×#1457893111yescharacteristic
228C22M16×D15711yescharacteristic
229C22M16×D15711yescharacteristic
230C4—21no—
231C4—21no—
232C4—21no—
233C4—3142no—
234C4—6281no—
235C4—6281no—
236C4#1909768512yes—
237C2—21no—
238C2—21no—
239C2—41no—
240C2—41no—
241C2—6282no—
242C2#2375723611yescharacteristic
243C2C2×#1909768511yescharacteristic
244C2#2375725511yescharacteristic
subgroupG/H
centerC22C2×#14578931
commutatorC23×C157C22×C4
FrattiniC22×C4C22×D157
Fitting#407×C157C2
socleC22×C157C2×C22⋊C4
prime radicalC22×C22⋊D157C4
cosocleC22×C4×C157C23
hypercenter#407D157
hypocenterC157#1465
LCSG ▷ C23×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
2. #23757237C2semidirect
3. #23756998C2semidirect
4. #23756996C2semidirect
5. #407×C157C2semidirect
58. C22×D157M16semidirect
67. C2×Dic157M16semidirect
122. C157#1465semidirect
126. #407D157semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊D1572
#237572372
#237569982
#237569962
#407×C1572
#1465157