Finite Group Databasebeta

← #34100551

Subgroups — #34100551

Order 20096.

342 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 79 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#23758039C211yesmaximal, characteristic, complemented
2#23757784C211yesmaximal, characteristic
3#23757255C211yesmaximal, characteristic, complemented
4#23757237C211yesmaximal, characteristic, complemented
5#23757045C211yesmaximal, characteristic, complemented
6#23757044C211yesmaximal, characteristic, complemented
7#407×C157C211yesmaximal, characteristic, complemented
8C2×#14578954C2211yescharacteristic
9#19097779—22no—
10#19097777—22no—
11C2×#14578938C2211yescharacteristic
12C2×#14578938C2211yescharacteristic
13M16×D157—22no—
14#19097754C412yes—
15#19097750—21no—
16#19097745—21no—
17#19097740C412yes—
18C4×C22⋊D157—22no—
19#19097730C2211yescharacteristic
20#19097689—21no—
21#19097687—21no—
22C22⋊#11929434—22no—
23C23×C4×C157C2211yescharacteristic
24C2×M16×C157C2211yescharacteristic
25C2×M16×C157C2211yescharacteristic
26C22⋊C8×C157—22no—
27#14578954—21no—
28#14578954—21no—
29#14578954—21no—
30#14578954—21no—
31C2×Dic314D411yescharacteristic
32C2×D628D411yescharacteristic
33C2×C22⋊D157—22no—
34C2×C22⋊D157C2×C412yes—
35C2×C4×D157D412yes—
36#14578938—21no—
37#14578938—21no—
38#14578938—21no—
39#14578938—21no—
40#14578937—24no—
41C2×#11929434—21no—
42C2×#11929434D412yescomplemented
43C8×D157—24no—
44C4⋊Dic157—21no—
45#14578933—22no—
46C22⋊Dic157—22no—
47C22⋊Dic157C2×C412yes—
48#14578931—22no—
49C4×Dic157—21no—
50C24×C157C2×C411yescharacteristic
51C22×C4×C157—21no—
52C22×C4×C157—21no—
53C22×C4×C157—21no—
54C22×C4×C157—21no—
55C22×C4×C157—21no—
56C22×C4×C157—21no—
57C22×C4×C157C2×C411yescharacteristic
58C22×C4×C157C2311yescharacteristic
59M16×C157—21no—
60M16×C157—21no—
61M16×C157—21no—
62M16×C157—21no—
63C2×C8×C157—21no—
64C2×C8×C157D412yescomplemented
65C22×D157C22⋊C412yes—
66Dic314—21no—
67Dic314—21no—
68D628—21no—
69D628—21no—
70C22⋊D157—22no—
71C22⋊D157—22no—
72C22⋊D157—44no—
73C2×Dic157—22no—
74C2×Dic157C22⋊C412yes—
75C4×D157—22no—
76C4×D157—22no—
77#11929434—22no—
78#11929434—41no—
79C23×C157—21no—
80C23×C157—21no—
81C23×C157—21no—
82C23×C157—42no—
83C23×C157C22×C411yescharacteristic
84C2×C4×C157—21no—
85C2×C4×C157—21no—
86C2×C4×C157—21no—
87C2×C4×C157—41no—
88C2×C4×C157—41no—
89C2×C4×C157—41no—
90C2×C4×C157—41no—
91C2×C4×C157C2×D411yescharacteristic
92C2×C4×C157C2×D411yescharacteristic
93C2×C4×C157C2×D411yescharacteristic
94C2×C4×C157C2×D411yescharacteristic
95C2×C4×C157#4111yescharacteristic
96C2×C4×C157#4111yescharacteristic
97C8×C157—22no—
98C8×C157—41no—
99C22×C157—22no—
100C22×C157—22no—
101C22×C157—22no—
102C22×C157—41no—
103C22×C157—41no—
104C22×C157—41no—
105C22×C157—41no—
106C22×C157—41no—
107C22×C157C2×C22⋊C411yescharacteristic
108C22×C157C4×D411yescharacteristic
109C22×C157C4×D411yescharacteristic
110C2×D157—22no—
111C2×D157—42no—
112C4×C157—41no—
113C4×C157#12011yescharacteristic
114C4×C157#12111yescharacteristic
115C4×C157#12111yescharacteristic
116C4×C157#12311yescharacteristic
117Dic157—22no—
118Dic157—42no—
119C2×C157—21no—
120C2×C157—21no—
121C2×C157—21no—
122C2×C157—42no—
123C2×C157#38611yescharacteristic
124D157—42no—
125C157#189111yescharacteristic, complemented, Sylow
126#1891—1571nomaximal, Sylow
127#568—1571no—
128#525—1571no—
129#413—1571no—
130#407—1571no—
131#407D15711yescharacteristic, complemented
132#344—1571no—
133#344—1571no—
134C2×#41—1571no—
135C23×C4C2×D15711yescharacteristic
136#131—3142no—
137#131—3142no—
138C2×M16—3142no—
139C2×M16—1571no—
140C2×M16—1571no—
141C2×M16C2×D15711yescharacteristic
142C2×M16C2×D15711yescharacteristic
143#123—1572no—
144C22⋊Q8—3141no—
145#121—3141no—
146#120—1572no—
147C4×D4—3142no—
148#117—1571no—
149#101—3141no—
150#100—3141no—
151C22⋊C8—22no—
152C22⋊C8—3142no—
153C24C4×D15711yescharacteristic
154#41—3141no—
155#41—3141no—
156#41—3141no—
157#41—3141no—
158C2×Q8—1571no—
159C2×D4—3142no—
160C2×D4—1572no—
161C2×D4—1571no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—1572no—
169C22×C4C4×D15711yescharacteristic
170C22×C4C22×D15711yescharacteristic
171M16—21no—
172M16—21no—
173M16—21no—
174M16—21no—
175M16—3144no—
176M16—3141no—
177M16—3141no—
178M16—3141no—
179M16—3141no—
180C2×C8—21no—
181C2×C8—3144no—
182C2×C8—3141no—
183C2×C8—1572no—
184C2×C8C22⋊D15712yescomplemented
185C4⋊C4—3142no—
186C4⋊C4—3141no—
187C22⋊C4—3142no—
188C22⋊C4—3142no—
189C22⋊C4—1572no—
190C42—3141no—
191C23—21no—
192C23—21no—
193C23—21no—
194C23—42no—
195C23—1572no—
196C23C2×C4×D15711yescharacteristic
197Q8—3141no—
198Q8—3141no—
199D4—3142no—
200D4—3142no—
201D4—3141no—
202D4—3141no—
203D4—6284no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—3142no—
212C2×C4—3142no—
213C2×C4—3142no—
214C2×C4—1572no—
215C2×C4D4×D15711yescharacteristic
216C2×C4D4×D15711yescharacteristic
217C2×C4D4×D15711yescharacteristic
218C2×C4C2×C22⋊D15711yescharacteristic
219C2×C4#1457895311yescharacteristic
220C2×C4#1457895411yescharacteristic
221C8—22no—
222C8—41no—
223C8—3142no—
224C8—6281no—
225C22—22no—
226C22—22no—
227C22—22no—
228C22—41no—
229C22—41no—
230C22—41no—
231C22—41no—
232C22—41no—
233C22—3142no—
234C22—6282no—
235C22C22⋊C4×D15711yescharacteristic
236C22#1909773111yescharacteristic
237C22C4×C22⋊D15711yescharacteristic
238C4—41no—
239C4—3142no—
240C4—6282no—
241C4#1909773911yescharacteristic
242C4#1909774211yescharacteristic
243C4#1909774411yescharacteristic
244C4#1909775311yescharacteristic
245C2—21no—
246C2—21no—
247C2—21no—
248C2—42no—
249C2—6282no—
250C2#2375714911yescharacteristic
subgroupG/H
centerC4#19097739
commutatorC23×C157C22×C4
FrattiniC22×C4C22×D157
Fitting#407×C157C2
socleC2×C157#386
prime radical#23757784C2
cosocleC22×C4×C157C23
hypercenter#407D157
hypocenterC157#1891
LCSG ▷ C23×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C4 ◁ C22×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
1. #23758039C2semidirect
3. #23757255C2semidirect
4. #23757237C2semidirect
5. #23757045C2semidirect
6. #23757044C2semidirect
7. #407×C157C2semidirect
42. C2×#11929434D4semidirect
64. C2×C8×C157D4semidirect
125. C157#1891semidirect
131. #407D157semidirect
184. C2×C8C22⋊D157semidirect

Maximal subgroups

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

subgroupindex
#237580392
#237577842
#237572552
#237572372
#237570452
#237570442
#407×C1572
#1891157