Finite Group Databasebeta

← #34099064

Subgroups — #34099064

Order 20096.

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

normal subgroups only

This group has 272 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#23757872C211yesmaximal, characteristic, complemented
2#23757407C211yesmaximal, characteristic
3#23757187C211yesmaximal, characteristic, complemented
4#23757010C211yesmaximal, characteristic, complemented
5#23757008C211yesmaximal, characteristic, complemented
6#23756998C211yesmaximal, characteristic, complemented
7#450×C157C211yesmaximal, characteristic, complemented
8C2×#14578939—21no—
9#19097766—21no—
10#19097762—21no—
11#19097756—21no—
12#19097755—21no—
13#19097747—21no—
14#19097745C2211yescharacteristic
15#19097731—21no—
16C2×#14578933C2211yescharacteristic
17C2×C22⋊Dic157—21no—
18C2×C22⋊Dic157—21no—
19#19097692—21no—
20C22⋊#11929434C2211yescharacteristic
21#19097678C2211yescharacteristic
22C22×D4×C157C2211yescharacteristic
23C2×SD16×C157—21no—
24C22⋊Q8×C157C2211yescharacteristic
25#102×C157—21no—
26C22⋊C8×C157C2211yescharacteristic
27C2×D628D411yescharacteristic
28C2×C22⋊D157—21no—
29C22×Dic157—21no—
30C22×Dic157D411yescharacteristic
31C22×Dic157D411yescharacteristic
32C2×C4×D157—21no—
33#14578939—42no—
34C2×#11929434—21no—
35C4⋊Dic157—21no—
36C4⋊Dic157—21no—
37C4⋊Dic157D411yescharacteristic
38#14578933—21no—
39#14578933—21no—
40#14578933—21no—
41C22⋊Dic157—42no—
42C22⋊Dic157—42no—
43#14578931—21no—
44#14578931—21no—
45#14578931—21no—
46#14578931—21no—
47C4×Dic157—21no—
48C24×C157—21no—
49C2×Q8×C157D411yescharacteristic
50C2×D4×C157—21no—
51C2×D4×C157—21no—
52C2×D4×C157—42no—
53C22×C4×C157C2311yescharacteristic
54SD16×C157—42no—
55C2×C8×C157—21no—
56C4⋊C4×C157—21no—
57C4⋊C4×C157D411yescharacteristic
58C22⋊C4×C157—21no—
59C22×D157—21no—
60D628—41no—
61C22⋊D157—41no—
62C22⋊D157—41no—
63C2×Dic157—21no—
64C2×Dic157—21no—
65C2×Dic157—21no—
66C2×Dic157—21no—
67C2×Dic157—21no—
68C2×Dic157—41no—
69C2×Dic157—41no—
70C2×Dic157—41no—
71C2×Dic157—41no—
72C4×D157—81no—
73#11929434—41no—
74C23×C157—22no—
75C23×C157—41no—
76C23×C157—41no—
77C23×C157—41no—
78C23×C157—41no—
79C23×C157—41no—
80C23×C157—41no—
81C23×C157C2×D411yescharacteristic
82Q8×C157—41no—
83D4×C157—22no—
84D4×C157—22no—
85D4×C157—41no—
86D4×C157—41no—
87C2×C4×C157—21no—
88C2×C4×C157—21no—
89C2×C4×C157—41no—
90C2×C4×C157C2×D411yescharacteristic
91C2×C4×C157C2×D411yescharacteristic
92C8×C157—41no—
93C22×C157—21no—
94C22×C157—21no—
95C22×C157—42no—
96C22×C157—42no—
97C22×C157—42no—
98C22×C157—81no—
99C22×C157—81no—
100C22×C157—81no—
101C22×C157—81no—
102C22×C157#12011yescharacteristic
103C22×C157#13511yescharacteristic
104C22×C157AGL(1,ℤ/8)11yescharacteristic
105C2×D157—41no—
106C2×D157—41no—
107C2×D157—41no—
108C4×C157—21no—
109C4×C157—21no—
110C4×C157—41no—
111C4×C157—41no—
112Dic157—41no—
113Dic157—41no—
114Dic157—41no—
115Dic157—81no—
116C2×C157—21no—
117C2×C157—21no—
118C2×C157—82no—
119C2×C157#44911yescharacteristic
120C2×C157#45411yescharacteristic
121C2×C157B(4,𝔽2)11yescharacteristic
122D157—81no—
123C157#159711yescharacteristic, complemented, Sylow
124#1597—1571nomaximal, Sylow
125#539—1571no—
126#450D15711yescharacteristic, complemented
127#447—1571no—
128#394—1571no—
129#328—1571no—
130#327—1571no—
131#324—1571no—
132C22×D4C2×D15711yescharacteristic
133C2×SD16—21no—
134C2×D8—3141no—
135#126—3141no—
136#125—3141no—
137#123—3141no—
138#123—3141no—
139C22⋊Q8—3141no—
140C22⋊Q8C2×D15711yescharacteristic
141#121—1571no—
142C4×D4—3141no—
143C2×C4⋊C4—1571no—
144C2×C22⋊C4—3141no—
145C2×C22⋊C4—3141no—
146#102—21no—
147#102—3141no—
148C22⋊C8—1571no—
149C22⋊C8C2×D15711yescharacteristic
150#95—1571no—
151C24—21no—
152C2×Q8C22⋊D15711yescharacteristic
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—42no—
156C2×D4—3141no—
157C2×D4—1571no—
158C22×C4—3141no—
159C22×C4—3141no—
160C22×C4—1571no—
161C22×C4—1571no—
162C22×C4C22×D15711yescharacteristic
163SD16—42no—
164D8—6282no—
165C2×C8—21no—
166C2×C8—3141no—
167C4⋊C4—21no—
168C4⋊C4—3141no—
169C4⋊C4—3141no—
170C4⋊C4—3141no—
171C4⋊C4—3141no—
172C4⋊C4—3141no—
173C4⋊C4—1571no—
174C4⋊C4C22⋊D15711yescharacteristic
175C22⋊C4—21no—
176C22⋊C4—3141no—
177C22⋊C4—3141no—
178C22⋊C4—3141no—
179C22⋊C4—3141no—
180C22⋊C4—6282no—
181C22⋊C4—6282no—
182C42—3141no—
183C23—22no—
184C23—41no—
185C23—41no—
186C23—41no—
187C23—41no—
188C23—41no—
189C23—41no—
190C23—3141no—
191C23D4×D15711yescharacteristic
192Q8—41no—
193D4—22no—
194D4—22no—
195D4—41no—
196D4—41no—
197D4—6281no—
198D4—6281no—
199D4—6281no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—41no—
203C2×C4—3141no—
204C2×C4—3141no—
205C2×C4—3141no—
206C2×C4—3141no—
207C2×C4—3141no—
208C2×C4—6281no—
209C2×C4—6281no—
210C2×C4—6281no—
211C2×C4—6281no—
212C2×C4—12561no—
213C2×C4D4×D15711yescharacteristic
214C2×C4C2×C22⋊D15711yescharacteristic
215C8—41no—
216C8—6281no—
217C22—21no—
218C22—21no—
219C22—42no—
220C22—42no—
221C22—42no—
222C22—81no—
223C22—81no—
224C22—81no—
225C22—81no—
226C22—6281no—
227C22—6281no—
228C22—6281no—
229C22#1909773911yescharacteristic
230C22#1909779211yescharacteristic
231C22#1909779811yescharacteristic
232C4—21no—
233C4—21no—
234C4—41no—
235C4—41no—
236C4—6281no—
237C4—6281no—
238C4—6281no—
239C4—12561no—
240C2—21no—
241C2—21no—
242C2—82no—
243C2—12561no—
244C2#2375741711yescharacteristic
245C2#2375744411yescharacteristic
246C2#2375745911yescharacteristic
subgroupG/H
centerC22#19097739
commutator, cosocleC22×C4×C157C23
FrattiniC22×C4C22×D157
Fitting#450×C157C2
socleC22×C157#120
prime radical#23757407C2
hypercenter#450D157
hypocenterC157#1597
LCSG ▷ C22×C4×C157 ▷ C22×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #23757872C2semidirect
3. #23757187C2semidirect
4. #23757010C2semidirect
5. #23757008C2semidirect
6. #23756998C2semidirect
7. #450×C157C2semidirect
123. C157#1597semidirect
126. #450D157semidirect

Maximal subgroups

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

subgroupindex
#237578722
#237574072
#237571872
#237570102
#237570082
#237569982
#450×C1572
#1597157