Finite Group Databasebeta

← #34099118

Subgroups — #34099118

Order 20096.

260 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 49 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#23757861C211yesmaximal, characteristic
2#23757431C211yesmaximal, characteristic, complemented
3#23757414C211yesmaximal, characteristic, complemented
4#23757185C211yesmaximal, characteristic, complemented
5#23757011C211yesmaximal, characteristic, complemented
6#23757008C211yesmaximal, characteristic, complemented
7#323×C157C211yesmaximal, characteristic, complemented
8C22×Dic314C2211yescharacteristic
9C2×Dic628—21no—
10C2×#14578943—21no—
11#19097764—21no—
12#19097757—21no—
13#19097753—21no—
14#19097750C2211yescharacteristic
15#19097745C2211yescharacteristic
16#19097744—21no—
17#19097740—21no—
18C4×C22⋊D157—21no—
19C2×C22⋊Dic157—21no—
20#19097696—21no—
21#19097696—21no—
22#19097678—21no—
23#19097678C2211yescharacteristic
24C2×C22⋊C4×C157C2211yescharacteristic
25C22⋊C8×C157C2211yescharacteristic
26C22⋊C8×C157C2211yescharacteristic
27C2×Dic314—21no—
28C2×Dic314—21no—
29C2×Dic314—42no—
30C2×Dic314D411yescharacteristic
31C2×D628D411yescharacteristic
32C2×C22⋊D157—21no—
33C2×C22⋊D157—21no—
34C2×C22⋊D157—21no—
35C22×Dic157—21no—
36C22×Dic157—21no—
37C2×C4×D157—21no—
38Dic628—42no—
39#14578943—42no—
40C4⋊Dic157D411yescharacteristic
41C4⋊Dic157D411yescharacteristic
42#14578933—21no—
43#14578933—21no—
44C22⋊Dic157—21no—
45C22⋊Dic157—21no—
46C22⋊Dic157—21no—
47C22⋊Dic157—42no—
48#14578931—21no—
49#14578931—21no—
50C4×Dic157—21no—
51C24×C157D411yescharacteristic
52C22×C4×C157D411yescharacteristic
53C22×C4×C157C2311yescharacteristic
54C2×C8×C157—21no—
55C2×C8×C157—21no—
56C22⋊C4×C157—21no—
57C22⋊C4×C157—21no—
58C22×D157—21no—
59Dic314—22no—
60Dic314—22no—
61Dic314—41no—
62Dic314—41no—
63Dic314—41no—
64D628—41no—
65C22⋊D157—41no—
66C22⋊D157—41no—
67C22⋊D157—81no—
68C22⋊D157—81no—
69C2×Dic157—22no—
70C2×Dic157—21no—
71C2×Dic157—21no—
72C2×Dic157—21no—
73C2×Dic157—41no—
74C2×Dic157—41no—
75C2×Dic157—41no—
76C2×Dic157—41no—
77C2×Dic157—41no—
78C4×D157—81no—
79C23×C157—21no—
80C23×C157—21no—
81C23×C157—21no—
82C23×C157—41no—
83C23×C157—41no—
84C23×C157C2×D411yescharacteristic
85C2×C4×C157—21no—
86C2×C4×C157—41no—
87C2×C4×C157—41no—
88C2×C4×C157C2×D411yescharacteristic
89C2×C4×C157C2×D411yescharacteristic
90C8×C157—41no—
91C8×C157—41no—
92C22×C157—21no—
93C22×C157—21no—
94C22×C157—42no—
95C22×C157—41no—
96C22×C157—41no—
97C22×C157—41no—
98C22×C157—81no—
99C22×C157#12011yescharacteristic
100C22×C157#13511yescharacteristic
101C22×C157#13711yescharacteristic
102C2×D157—41no—
103C2×D157—41no—
104C2×D157—41no—
105C4×C157—21no—
106C4×C157—21no—
107C4×C157—41no—
108Dic157—41no—
109Dic157—41no—
110Dic157—42no—
111Dic157—41no—
112Dic157—81no—
113C2×C157—21no—
114C2×C157—21no—
115C2×C157—81no—
116C2×C157#45211yescharacteristic
117C2×C157#45511yescharacteristic
118C2×C157B(4,𝔽2)11yescharacteristic
119D157—81no—
120C157#160611yescharacteristic, complemented, Sylow
121#1606—1571nomaximal, Sylow
122#538—1571no—
123#451—1571no—
124#448—1571no—
125#393—1571no—
126#328—1571no—
127#327—1571no—
128#323D15711yescharacteristic, complemented
129C22×Q8—1571no—
130C2×Q16—3141no—
131C2×SD16—3141no—
132#126—3141no—
133#124—3141no—
134#123—3141no—
135C22⋊Q8—1571no—
136#121—3141no—
137#121—1571no—
138#120—3141no—
139C4×D4—3141no—
140C2×C22⋊C4—3141no—
141C2×C22⋊C4C2×D15711yescharacteristic
142#103—3141no—
143#103—3141no—
144C22⋊C8C2×D15711yescharacteristic
145C22⋊C8C2×D15711yescharacteristic
146#95—3141no—
147#95—1571no—
148C24D62811yescharacteristic
149C2×Q8—3141no—
150C2×Q8—3141no—
151C2×Q8—6282no—
152C2×Q8—1571no—
153C2×D4—3141no—
154C2×D4—3141no—
155C2×D4—3141no—
156C2×D4—1571no—
157C22×C4—3141no—
158C22×C4—3141no—
159C22×C4—3141no—
160C22×C4D62811yescharacteristic
161C22×C4C22×D15711yescharacteristic
162Q16—6282no—
163SD16—6282no—
164C2×C8—21no—
165C2×C8—21no—
166C4⋊C4—3141no—
167C4⋊C4—3141no—
168C4⋊C4—1571no—
169C4⋊C4—1571no—
170C22⋊C4—21no—
171C22⋊C4—21no—
172C22⋊C4—3141no—
173C22⋊C4—3141no—
174C22⋊C4—3141no—
175C22⋊C4—3141no—
176C22⋊C4—3141no—
177C22⋊C4—6282no—
178C42—3141no—
179C23—21no—
180C23—21no—
181C23—21no—
182C23—41no—
183C23—41no—
184C23—3141no—
185C23C2×D62811yescharacteristic
186Q8—3142no—
187Q8—3142no—
188Q8—6281no—
189Q8—6281no—
190Q8—6281no—
191D4—6281no—
192D4—6281no—
193D4—6281no—
194D4—12561no—
195D4—12561no—
196C2×C4—21no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—3142no—
200C2×C4—3141no—
201C2×C4—3141no—
202C2×C4—3141no—
203C2×C4—6281no—
204C2×C4—6281no—
205C2×C4—6281no—
206C2×C4—6281no—
207C2×C4—6281no—
208C2×C4—12561no—
209C2×C4D4×D15711yescharacteristic
210C2×C4D4×D15711yescharacteristic
211C8—41no—
212C8—41no—
213C22—21no—
214C22—21no—
215C22—42no—
216C22—41no—
217C22—41no—
218C22—41no—
219C22—81no—
220C22—6281no—
221C22—6281no—
222C22—6281no—
223C22#1909774111yescharacteristic
224C22#1909779311yescharacteristic
225C22#1909780411yescharacteristic
226C4—21no—
227C4—21no—
228C4—41no—
229C4—6281no—
230C4—6281no—
231C4—6282no—
232C4—6281no—
233C4—12561no—
234C2—21no—
235C2—21no—
236C2—81no—
237C2—12561no—
238C2#2375743711yescharacteristic
239C2#2375745311yescharacteristic
240C2#2375746111yescharacteristic
subgroupG/H
centerC22#19097741
commutator, cosocleC22×C4×C157C23
FrattiniC22×C4C22×D157
Fitting#323×C157C2
socleC22×C157#120
prime radical#23757861C2
hypercenter#323D157
hypocenterC157#1606
LCSG ▷ C22×C4×C157 ▷ C22×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. #23757431C2semidirect
3. #23757414C2semidirect
4. #23757185C2semidirect
5. #23757011C2semidirect
6. #23757008C2semidirect
7. #323×C157C2semidirect
120. C157#1606semidirect
128. #323D157semidirect

Maximal subgroups

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

subgroupindex
#237578612
#237574312
#237574142
#237571852
#237570112
#237570082
#323×C1572
#1606157