Finite Group Databasebeta

← #34099085

Subgroups — #34099085

Order 20096.

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

normal subgroups only

This group has 320 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, complemented
2#23757423C211yesmaximal, characteristic, complemented
3#23757409C211yesmaximal, characteristic, complemented
4#23757183C211yesmaximal, characteristic, complemented
5#23757006C211yesmaximal, characteristic, complemented
6#23756996C211yesmaximal, characteristic, complemented
7#328×C157C211yesmaximal, characteristic, complemented
8C22×D628—21no—
9C22×D628C2211yescharacteristic
10C2×#14578941—21no—
11C2×D1256—21no—
12#19097766—21no—
13#19097760—21no—
14#19097756—21no—
15#19097746—21no—
16#19097745C2211yescharacteristic
17#19097741—21no—
18#19097731—21no—
19C2×#14578931—21no—
20C2×#14578931C2211yescharacteristic
21#19097693—21no—
22#19097691—21no—
23C22⋊#11929434C2211yescharacteristic
24C22⋊Q8×C157C2211yescharacteristic
25C22⋊C8×C157C2211yescharacteristic
26#95×C157C2211yescharacteristic
27C23×D157—21no—
28C23×D157D411yescharacteristic
29C2×D628—21no—
30C2×D628—21no—
31C2×D628—21no—
32C2×D628—21no—
33C2×D628—42no—
34C2×D628—42no—
35C2×D628—42no—
36C2×D628—42no—
37C2×D628D411yescharacteristic, complemented
38C2×C22⋊D157—21no—
39C22×Dic157D411yescharacteristic
40C2×C4×D157—21no—
41#14578941—42no—
42D1256—42no—
43C2×#11929434—21no—
44C4⋊Dic157D411yescharacteristic, complemented
45#14578933—21no—
46#14578931—21no—
47#14578931—21no—
48#14578931—21no—
49#14578931—21no—
50#14578931—21no—
51#14578931—21no—
52#14578931—42no—
53C4×Dic157—21no—
54C2×Q8×C157D411yescharacteristic, complemented
55C22×C4×C157—21no—
56C22×C4×C157C2311yescharacteristic
57C2×C8×C157—21no—
58C4⋊C4×C157—21no—
59C4⋊C4×C157D411yescharacteristic, complemented
60C22⋊C4×C157—21no—
61C22×D157—22no—
62C22×D157—21no—
63C22×D157—21no—
64C22×D157—21no—
65C22×D157—21no—
66C22×D157—41no—
67C22×D157—41no—
68C22×D157—41no—
69C22×D157—41no—
70C22×D157—41no—
71C22×D157—41no—
72C22×D157—41no—
73C22×D157—41no—
74D628—22no—
75D628—22no—
76D628—41no—
77D628—41no—
78D628—41no—
79D628—48no—
80D628—81no—
81D628—81no—
82C22⋊D157—41no—
83C22⋊D157—41no—
84C2×Dic157—21no—
85C2×Dic157—21no—
86C2×Dic157—41no—
87C4×D157—81no—
88#11929434—41no—
89C23×C157C2×D411yescharacteristic
90Q8×C157—41no—
91C2×C4×C157—22no—
92C2×C4×C157—21no—
93C2×C4×C157—21no—
94C2×C4×C157—41no—
95C2×C4×C157—41no—
96C2×C4×C157—41no—
97C2×C4×C157C2×D411yescharacteristic
98C2×C4×C157C2×D411yescharacteristic
99C8×C157—41no—
100C22×C157—21no—
101C22×C157—21no—
102C22×C157#12011yescharacteristic
103C22×C157#13511yescharacteristic
104C22×C157AGL(1,ℤ/8)11yescharacteristic
105C2×D157—42no—
106C2×D157—42no—
107C2×D157—42no—
108C2×D157—41no—
109C2×D157—41no—
110C2×D157—41no—
111C2×D157—41no—
112C2×D157—42no—
113C2×D157—41no—
114C2×D157—41no—
115C2×D157—81no—
116C2×D157—81no—
117C2×D157—81no—
118C2×D157—81no—
119C2×D157—81no—
120C4×C157—21no—
121C4×C157—21no—
122C4×C157—42no—
123C4×C157—41no—
124C4×C157—41no—
125Dic157—41no—
126Dic157—41no—
127C2×C157—21no—
128C2×C157—21no—
129C2×C157#44911yescharacteristic
130C2×C157#45311yescharacteristic
131C2×C157B(4,𝔽2)11yescharacteristic
132D157—81no—
133D157—81no—
134D157—82no—
135C157#160011yescharacteristic, complemented, Sylow
136#1600—1571nomaximal, Sylow
137#538—1571no—
138#450—1571no—
139#447—1571no—
140#392—1571no—
141#328D15711yescharacteristic, complemented
142#327—1571no—
143#323—1571no—
144C22×D4—3141no—
145C22×D4—1571no—
146C2×SD16—3141no—
147C2×D8—3141no—
148#126—3141no—
149#124—3141no—
150#123—3141no—
151C22⋊Q8C2×D15711yescharacteristic
152#121—3141no—
153#121—1571no—
154#120—3141no—
155C4×D4—3141no—
156C2×C22⋊C4—3141no—
157C2×C22⋊C4—1571no—
158#102—3141no—
159#102—3141no—
160C22⋊C8—1571no—
161C22⋊C8C2×D15711yescharacteristic
162#95C2×D15711yescharacteristic
163C24—3141no—
164C24—1571no—
165C2×Q8D62811yescharacteristic, complemented
166C2×D4—3141no—
167C2×D4—3141no—
168C2×D4—3141no—
169C2×D4—3141no—
170C2×D4—3141no—
171C2×D4—6282no—
172C2×D4—6282no—
173C2×D4—6282no—
174C2×D4—6282no—
175C2×D4—1571no—
176C22×C4—21no—
177C22×C4—3141no—
178C22×C4—1571no—
179C22×C4C22×D15711yescharacteristic
180SD16—6282no—
181D8—6282no—
182C2×C8—21no—
183C2×C8—3141no—
184C4⋊C4—21no—
185C4⋊C4—3141no—
186C4⋊C4—1571no—
187C4⋊C4D62811yescharacteristic, complemented
188C22⋊C4—21no—
189C22⋊C4—3141no—
190C22⋊C4—3141no—
191C22⋊C4—3141no—
192C22⋊C4—3141no—
193C22⋊C4—3141no—
194C22⋊C4—3141no—
195C22⋊C4—6282no—
196C42—3141no—
197C23—3142no—
198C23—3141no—
199C23—3141no—
200C23—3141no—
201C23—3141no—
202C23—6281no—
203C23—6281no—
204C23—6281no—
205C23—6281no—
206C23—6281no—
207C23—6281no—
208C23—6281no—
209C23—6281no—
210C23D4×D15711yescharacteristic
211Q8—41no—
212D4—3142no—
213D4—3142no—
214D4—6281no—
215D4—6281no—
216D4—6281no—
217D4—6288no—
218D4—6281no—
219D4—6281no—
220D4—12561no—
221D4—12561no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—3141no—
229C2×C4—3141no—
230C2×C4—6281no—
231C2×C4—12561no—
232C2×C4D4×D15711yescharacteristic
233C2×C4C2×D62811yescharacteristic
234C8—41no—
235C8—6281no—
236C22—21no—
237C22—21no—
238C22—6282no—
239C22—6282no—
240C22—6282no—
241C22—6281no—
242C22—6281no—
243C22—6281no—
244C22—6281no—
245C22—6282no—
246C22—6281no—
247C22—6281no—
248C22—12561no—
249C22—12561no—
250C22—12561no—
251C22—12561no—
252C22—12561no—
253C22#1909774111yescharacteristic
254C22#1909779411yescharacteristic
255C22#1909779811yescharacteristic
256C4—21no—
257C4—21no—
258C4—42no—
259C4—41no—
260C4—41no—
261C4—6281no—
262C4—6281no—
263C2—21no—
264C2—21no—
265C2—12561no—
266C2—12562no—
267C2—12561no—
268C2#2375742111yescharacteristic
269C2#2375744211yescharacteristic
270C2#2375746111yescharacteristic
subgroupG/H
centerC22#19097741
commutator, cosocleC22×C4×C157C23
FrattiniC22×C4C22×D157
Fitting#328×C157C2
socleC22×C157#120
hypercenter#328D157
hypocenterC157#1600
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
1. #23757863C2semidirect
2. #23757423C2semidirect
3. #23757409C2semidirect
4. #23757183C2semidirect
5. #23757006C2semidirect
6. #23756996C2semidirect
7. #328×C157C2semidirect
37. C2×D628D4semidirect
44. C4⋊Dic157D4semidirect
54. C2×Q8×C157D4semidirect
59. C4⋊C4×C157D4semidirect
135. C157#1600semidirect
141. #328D157semidirect
165. C2×Q8D628semidirect
187. C4⋊C4D628semidirect

Maximal subgroups

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

subgroupindex
#237578632
#237574232
#237574092
#237571832
#237570062
#237569962
#328×C1572
#1600157