Finite Group Databasebeta

← #34099119

Subgroups — #34099119

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#23757863C211yesmaximal, characteristic
2#23757428C211yesmaximal, characteristic, complemented
3#23757414C211yesmaximal, characteristic, complemented
4#23757186C211yesmaximal, characteristic, complemented
5#23757006C211yesmaximal, characteristic, complemented
6#23756996C211yesmaximal, characteristic, complemented
7#328×C157C211yesmaximal, characteristic, complemented
8C22×Dic314C2211yescharacteristic
9C2×#14578944—21no—
10C2×#14578943—21no—
11#19097766—21no—
12#19097760—21no—
13#19097756—21no—
14#19097746—21no—
15#19097745C2211yescharacteristic
16#19097741—21no—
17#19097731—21no—
18C2×#14578931—21no—
19C2×#14578931C2211yescharacteristic
20#19097696—21no—
21#19097694—21no—
22C22⋊#11929434C2211yescharacteristic
23#19097678—21no—
24C22⋊Q8×C157C2211yescharacteristic
25C22⋊C8×C157C2211yescharacteristic
26#95×C157C2211yescharacteristic
27C23×D157D411yescharacteristic
28C2×Dic314—21no—
29C2×Dic314—21no—
30C2×Dic314—42no—
31C2×D628—21no—
32C2×D628—21no—
33C2×D628D411yescharacteristic
34C2×C22⋊D157—21no—
35C22×Dic157—21no—
36C22×Dic157D411yescharacteristic
37C2×C4×D157—21no—
38#14578944—42no—
39#14578943—42no—
40C2×#11929434—21no—
41C4⋊Dic157D411yescharacteristic
42#14578933—21no—
43#14578931—21no—
44#14578931—21no—
45#14578931—21no—
46#14578931—21no—
47#14578931—21no—
48#14578931—21no—
49#14578931—42no—
50C4×Dic157—21no—
51C2×Q8×C157D411yescharacteristic
52C22×C4×C157—21no—
53C22×C4×C157C2311yescharacteristic
54C2×C8×C157—21no—
55C4⋊C4×C157—21no—
56C4⋊C4×C157D411yescharacteristic
57C22⋊C4×C157—21no—
58C22×D157—21no—
59C22×D157—21no—
60C22×D157—21no—
61C22×D157—21no—
62C22×D157—41no—
63C22×D157—41no—
64Dic314—22no—
65Dic314—22no—
66Dic314—41no—
67Dic314—41no—
68D628—41no—
69D628—81no—
70D628—81no—
71C22⋊D157—41no—
72C22⋊D157—41no—
73C2×Dic157—22no—
74C2×Dic157—21no—
75C2×Dic157—21no—
76C2×Dic157—41no—
77C2×Dic157—41no—
78C2×Dic157—41no—
79C4×D157—81no—
80#11929434—41no—
81C23×C157C2×D411yescharacteristic
82Q8×C157—41no—
83C2×C4×C157—21no—
84C2×C4×C157—21no—
85C2×C4×C157—41no—
86C2×C4×C157—41no—
87C2×C4×C157—41no—
88C2×C4×C157—41no—
89C2×C4×C157C2×D411yescharacteristic
90C2×C4×C157C2×D411yescharacteristic
91C8×C157—41no—
92C22×C157—21no—
93C22×C157—21no—
94C22×C157#12011yescharacteristic
95C22×C157#13511yescharacteristic
96C22×C157#13711yescharacteristic
97C2×D157—42no—
98C2×D157—41no—
99C2×D157—41no—
100C2×D157—41no—
101C2×D157—41no—
102C2×D157—41no—
103C2×D157—41no—
104C2×D157—81no—
105C4×C157—21no—
106C4×C157—21no—
107C4×C157—41no—
108C4×C157—41no—
109C4×C157—81no—
110Dic157—41no—
111Dic157—41no—
112Dic157—42no—
113C2×C157—21no—
114C2×C157—21no—
115C2×C157#45211yescharacteristic
116C2×C157#45511yescharacteristic
117C2×C157B(4,𝔽2)11yescharacteristic
118D157—81no—
119D157—81no—
120C157#160611yescharacteristic, complemented, Sylow
121#1606—1571nomaximal, Sylow
122#538—1571no—
123#451—1571no—
124#448—1571no—
125#393—1571no—
126#328D15711yescharacteristic, complemented
127#327—1571no—
128#323—1571no—
129C22×Q8—1571no—
130C2×Q16—3141no—
131C2×SD16—3141no—
132#126—3141no—
133#124—3141no—
134#123—3141no—
135C22⋊Q8C2×D15711yescharacteristic
136#121—3141no—
137#121—1571no—
138#120—3141no—
139C4×D4—3141no—
140C2×C22⋊C4—3141no—
141C2×C22⋊C4—1571no—
142#103—3141no—
143#103—3141no—
144C22⋊C8—1571no—
145C22⋊C8C2×D15711yescharacteristic
146#95—3141no—
147#95C2×D15711yescharacteristic
148C24—1571no—
149C2×Q8—3141no—
150C2×Q8—3141no—
151C2×Q8—6282no—
152C2×Q8D62811yescharacteristic
153C2×D4—3141no—
154C2×D4—3141no—
155C2×D4—3141no—
156C2×D4—1571no—
157C22×C4—21no—
158C22×C4—3141no—
159C22×C4—3141no—
160C22×C4—1571no—
161C22×C4C22×D15711yescharacteristic
162Q16—6282no—
163SD16—6282no—
164C2×C8—21no—
165C2×C8—3141no—
166C4⋊C4—21no—
167C4⋊C4—3141no—
168C4⋊C4—1571no—
169C4⋊C4D62811yescharacteristic
170C22⋊C4—21no—
171C22⋊C4—3141no—
172C22⋊C4—3141no—
173C22⋊C4—3141no—
174C22⋊C4—3141no—
175C22⋊C4—3141no—
176C22⋊C4—3141no—
177C22⋊C4—6282no—
178C42—3141no—
179C23—3141no—
180C23—3141no—
181C23—3141no—
182C23—3141no—
183C23—6281no—
184C23—6281no—
185C23D4×D15711yescharacteristic
186Q8—41no—
187Q8—3142no—
188Q8—3142no—
189Q8—6281no—
190Q8—6281no—
191D4—6281no—
192D4—6281no—
193D4—6281no—
194D4—12561no—
195D4—12561no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—3142no—
203C2×C4—3141no—
204C2×C4—3141no—
205C2×C4—6281no—
206C2×C4—6281no—
207C2×C4—6281no—
208C2×C4—12561no—
209C2×C4D4×D15711yescharacteristic
210C2×C4C2×D62811yescharacteristic
211C8—41no—
212C8—6281no—
213C22—21no—
214C22—21no—
215C22—6282no—
216C22—6281no—
217C22—6281no—
218C22—6281no—
219C22—6281no—
220C22—6281no—
221C22—6281no—
222C22—12561no—
223C22#1909774111yescharacteristic
224C22#1909779211yescharacteristic
225C22#1909780311yescharacteristic
226C4—21no—
227C4—21no—
228C4—41no—
229C4—41no—
230C4—81no—
231C4—6281no—
232C4—6281no—
233C4—6282no—
234C2—21no—
235C2—21no—
236C2—12561no—
237C2—12561no—
238C2#2375743811yescharacteristic
239C2#2375745211yescharacteristic
240C2#2375746111yescharacteristic
subgroupG/H
centerC22#19097741
commutator, cosocleC22×C4×C157C23
FrattiniC22×C4C22×D157
Fitting#328×C157C2
socleC22×C157#120
prime radical#23757863C2
hypercenter#328D157
hypocenterC157#1606
LCSG ▷ C22×C4×C157 ▷ C22×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
2. #23757428C2semidirect
3. #23757414C2semidirect
4. #23757186C2semidirect
5. #23757006C2semidirect
6. #23756996C2semidirect
7. #328×C157C2semidirect
120. C157#1606semidirect
126. #328D157semidirect

Maximal subgroups

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

subgroupindex
#237578632
#237574282
#237574142
#237571862
#237570062
#237569962
#328×C1572
#1606157