Finite Group Databasebeta

← #34101198

Subgroups — #34101198

Order 20096.

348 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 69 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#23758073C211yesmaximal, characteristic
2C2×#19097802C211yesmaximal, characteristic, complemented
3#23757847C211yesmaximal, characteristic, complemented
4#23757291C211yesmaximal, characteristic, complemented
5#23757246C211yesmaximal, characteristic, complemented
6#23757044C211yesmaximal, characteristic, complemented
7#419×C157C211yesmaximal, characteristic, complemented
8C2×#14578954C2211yescharacteristic
9C2×#14578954C2211yescharacteristic
10#19097804—21no—
11#19097804—21no—
12#19097802—22no—
13#19097802—22no—
14#19097799—21no—
15#19097799—21no—
16#19097793—22no—
17C2×#14578944—21no—
18C2×#14578941—21no—
19C2×#14578938C2211yescharacteristic
20#19097769—21no—
21#19097760—21no—
22#19097754—22no—
23#19097747—22no—
24#19097730C2211yescharacteristic
25#19097699—22no—
26#19097685—22no—
27C22×Q8×C157C2211yescharacteristic
28C2×M16×C157C2211yescharacteristic
29#117×C157C2211yescharacteristic
30#103×C157—22no—
31#14578954—21no—
32#14578954—21no—
33#14578954—21no—
34#14578954—21no—
35#14578954—21no—
36#14578954—21no—
37#14578954—21no—
38#14578954—21no—
39C2×Dic314D411yescharacteristic
40C2×Dic314D411yescharacteristic
41C2×D628D411yescharacteristic
42C2×D628D411yescharacteristic
43C2×C22⋊D157—21no—
44C2×C22⋊D157—21no—
45C2×C4×D157—21no—
46C2×C4×D157—21no—
47Dic628—22no—
48#14578944—24no—
49#14578943—22no—
50#14578943—22no—
51#14578941—24no—
52D1256—22no—
53#14578938—21no—
54#14578938D412yes—
55C2×#11929434—21no—
56C4⋊Dic157—21no—
57#14578933—24no—
58C22⋊Dic157D412yes—
59#14578931—24no—
60C4×Dic157—21no—
61C2×Q8×C157—21no—
62C2×Q8×C157—21no—
63C2×Q8×C157—21no—
64C2×Q8×C157—21no—
65C2×Q8×C157—21no—
66C2×Q8×C157D412yes—
67C22×C4×C157—21no—
68C22×C4×C157C2311yescharacteristic
69M16×C157—21no—
70M16×C157—21no—
71C2×C8×C157D412yes—
72C4⋊C4×C157—21no—
73C22⋊C4×C157—21no—
74C42×C157—21no—
75C22×D157—21no—
76C22×D157—21no—
77Dic314—21no—
78Dic314—21no—
79Dic314—21no—
80Dic314—21no—
81D628—21no—
82D628—21no—
83D628—21no—
84D628—21no—
85C22⋊D157—41no—
86C22⋊D157—41no—
87C22⋊D157—41no—
88C22⋊D157—41no—
89C2×Dic157—21no—
90C2×Dic157—22no—
91C2×Dic157—21no—
92C4×D157—41no—
93C4×D157—41no—
94C4×D157—41no—
95C4×D157—41no—
96#11929434—22no—
97C23×C157C2×D411yescharacteristic
98Q8×C157—22no—
99Q8×C157—22no—
100Q8×C157—41no—
101Q8×C157—41no—
102C2×C4×C157—22no—
103C2×C4×C157—41no—
104C2×C4×C157—41no—
105C2×C4×C157—41no—
106C2×C4×C157C2×D411yescharacteristic
107C2×C4×C157C2×D411yescharacteristic
108C2×C4×C157C2×D411yescharacteristic
109C2×C4×C157C2×D411yescharacteristic
110C2×C4×C157C2×D411yescharacteristic
111C2×C4×C157#4111yescharacteristic
112C8×C157—22no—
113C22×C157—41no—
114C22×C157#12011yescharacteristic
115C22×C157#12111yescharacteristic
116C22×C157#12711yescharacteristic
117C2×D157—42no—
118C2×D157—41no—
119C2×D157—41no—
120C2×D157—81no—
121C4×C157—42no—
122C4×C157—81no—
123C4×C157#12011yescharacteristic
124C4×C157#12011yescharacteristic
125C4×C157#12111yescharacteristic
126C4×C157#12111yescharacteristic
127Dic157—42no—
128Dic157—41no—
129Dic157—41no—
130C2×C157—21no—
131C2×C157—21no—
132C2×C157—21no—
133C2×C157#39211yescharacteristic
134D157—81no—
135D157—81no—
136C157#200911yescharacteristic, complemented, Sylow
137#2009—1571nomaximal, Sylow
138#575—1571no—
139C2×#137—1571no—
140#536—1571no—
141#421—1571no—
142#419D15711yescharacteristic, complemented
143#410—1571no—
144#344—1571no—
145C2×#41—1571no—
146C2×#41—1571no—
147C22×Q8C2×D15711yescharacteristic
148#137—3141no—
149#137—3141no—
150#137—3142no—
151#137—3142no—
152AGL(1,ℤ/8)—3141no—
153AGL(1,ℤ/8)—3141no—
154#135—3142no—
155C2×Q16—3141no—
156C2×SD16—3141no—
157C2×M16—1571no—
158C2×M16C2×D15711yescharacteristic
159C4⋊Q8—3141no—
160#124—3141no—
161#123—3142no—
162C22⋊Q8—3142no—
163#117—1571no—
164#117C2×D15711yescharacteristic
165#104—3142no—
166#103—22no—
167#99—3142no—
168#41—3141no—
169#41—3141no—
170#41—3141no—
171#41—3141no—
172#41—3141no—
173#41—3141no—
174#41—3141no—
175#41—3141no—
176C2×Q8—21no—
177C2×Q8—21no—
178C2×Q8—21no—
179C2×Q8—21no—
180C2×Q8—21no—
181C2×Q8—1571no—
182C2×Q8—1571no—
183C2×Q8D62812yes—
184C2×D4—3141no—
185C2×D4—3141no—
186C2×D4—1571no—
187C2×D4—1571no—
188C22×C4—21no—
189C22×C4—3141no—
190C22×C4—3141no—
191C22×C4C22×D15711yescharacteristic
192Q16—3142no—
193Q16—3144no—
194SD16—3142no—
195SD16—3142no—
196SD16—3144no—
197D8—3142no—
198M16—21no—
199M16—21no—
200M16—3141no—
201M16—1572no—
202C2×C8—3141no—
203C2×C8C22⋊D15712yes—
204C4⋊C4—21no—
205C4⋊C4—3144no—
206C4⋊C4—3141no—
207C22⋊C4—21no—
208C22⋊C4—3144no—
209C22⋊C4—1572no—
210C42—21no—
211C42—3141no—
212C23—3141no—
213C23—3141no—
214C23D4×D15711yescharacteristic
215Q8—22no—
216Q8—22no—
217Q8—41no—
218Q8—41no—
219Q8—3141no—
220Q8—3141no—
221Q8—3141no—
222Q8—3141no—
223D4—3141no—
224D4—3141no—
225D4—3141no—
226D4—3141no—
227D4—6281no—
228D4—6281no—
229D4—6281no—
230D4—6281no—
231C2×C4—22no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—3142no—
236C2×C4—3141no—
237C2×C4—3141no—
238C2×C4—6281no—
239C2×C4—6281no—
240C2×C4—6281no—
241C2×C4—6281no—
242C2×C4D4×D15711yescharacteristic
243C2×C4D4×D15711yescharacteristic
244C2×C4D4×D15711yescharacteristic
245C2×C4C2×C22⋊D15711yescharacteristic
246C2×C4C2×D62811yescharacteristic
247C2×C4#1457895411yescharacteristic
248C8—22no—
249C8—3142no—
250C22—41no—
251C22—6282no—
252C22—6281no—
253C22—6281no—
254C22—12561no—
255C22#1909774111yescharacteristic
256C22#1909774411yescharacteristic
257C22#1909776711yescharacteristic
258C4—42no—
259C4—81no—
260C4—6282no—
261C4—6281no—
262C4—6281no—
263C4#1909773911yescharacteristic
264C4#1909774111yescharacteristic
265C4#1909774411yescharacteristic
266C4#1909774511yescharacteristic
267C2—21no—
268C2—21no—
269C2—21no—
270C2—12561no—
271C2—12561no—
272C2#2375718211yescharacteristic
subgroupG/H
centerC2#23757182
commutator, cosocleC22×C4×C157C23
FrattiniC22×C4C22×D157
Fitting#419×C157C2
socleC2×C157#392
prime radical#23758073C2
hypercenter#419D157
hypocenterC157#2009
LCSG ▷ C22×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
2. C2×#19097802C2semidirect
3. #23757847C2semidirect
4. #23757291C2semidirect
5. #23757246C2semidirect
6. #23757044C2semidirect
7. #419×C157C2semidirect
136. C157#2009semidirect
142. #419D157semidirect

Maximal subgroups

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

subgroupindex
#237580732
C2×#190978022
#237578472
#237572912
#237572462
#237570442
#419×C1572
#2009157