Finite Group Databasebeta

← #34101148

Subgroups — #34101148

Order 20096.

478 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 75 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#23757837C211yesmaximal, characteristic, complemented
2C2×#19097767C211yesmaximal, characteristic, complemented
3C2×#19097698C211yesmaximal, characteristic, complemented
4#23757278C211yesmaximal, characteristic, complemented
5C2×#19097687C211yesmaximal, characteristic, complemented
6#23757034C211yesmaximal, characteristic, complemented
7C2×AGL(1,ℤ/8)×C157C211yesmaximal, characteristic, complemented
8C22×D628C2211yescharacteristic
9C22×C22⋊D157—22no—
10C2×#14578938C2211yescharacteristic
11#19097767—22no—
12#19097767—21no—
13#19097767—22no—
14#19097760—21no—
15#19097745—22no—
16#19097739—22no—
17#19097730C2211yescharacteristic
18C2×C4×Dic157C2211yescharacteristic
19#19097698—22no—
20#19097692—22no—
21#19097687—22no—
22C2×#41×C157C2211yescharacteristic
23C22×D4×C157C2211yescharacteristic
24AGL(1,ℤ/8)×C157—22no—
25AGL(1,ℤ/8)×C157—22no—
26C2×SD16×C157—21no—
27C2×D8×C157—21no—
28C2×M16×C157C2211yescharacteristic
29C23×D157D412yes—
30C2×D628—21no—
31C2×D628—21no—
32C2×D628—21no—
33C2×D628—21no—
34C2×D628—21no—
35C2×D628—21no—
36C2×C22⋊D157—28no—
37C2×C22⋊D157—24no—
38C2×C22⋊D157—48no—
39C22×Dic157—21no—
40#14578938—21no—
41#14578938—21no—
42C2×#11929434—21no—
43C4⋊Dic157D412yescomplemented
44C22⋊Dic157—21no—
45#14578931—24no—
46C4×Dic157—21no—
47C4×Dic157—21no—
48C4×Dic157D412yescomplemented
49C24×C157—21no—
50#41×C157—21no—
51#41×C157—21no—
52#41×C157—21no—
53#41×C157—21no—
54C2×Q8×C157D411yescharacteristic, complemented
55C2×D4×C157—21no—
56C2×D4×C157—21no—
57C2×D4×C157—21no—
58C2×D4×C157—21no—
59C2×D4×C157—21no—
60C2×D4×C157—21no—
61C2×D4×C157D411yescharacteristic, complemented
62C2×D4×C157D412yescomplemented
63C22×C4×C157—21no—
64C22×C4×C157C2311yescharacteristic
65SD16×C157—24no—
66D8×C157—24no—
67M16×C157—21no—
68M16×C157D412yescomplemented
69C2×C8×C157—21no—
70C22×D157—22no—
71C22×D157—22no—
72C22×D157—22no—
73C22×D157—44no—
74D628—41no—
75D628—41no—
76D628—41no—
77D628—41no—
78C22⋊D157—416no—
79C22⋊D157—84no—
80C2×Dic157—24no—
81C2×Dic157—22no—
82C2×Dic157—22no—
83#11929434—41no—
84C23×C157—21no—
85C23×C157—22no—
86C23×C157—44no—
87C23×C157—41no—
88C23×C157—41no—
89C23×C157C2×D411yescharacteristic
90Q8×C157—21no—
91Q8×C157—21no—
92D4×C157—21no—
93D4×C157—22no—
94D4×C157—22no—
95D4×C157—21no—
96D4×C157—41no—
97D4×C157—41no—
98D4×C157—41no—
99D4×C157—41no—
100C2×C4×C157—21no—
101C2×C4×C157—41no—
102C2×C4×C157—41no—
103C2×C4×C157C2×D411yescharacteristic
104C2×C4×C157C2×D411yescharacteristic
105C2×C4×C157C2×D411yescharacteristic
106C2×C4×C157C2×D411yescharacteristic
107C2×C4×C157C2×D411yescharacteristic
108C2×C4×C157#4111yescharacteristic
109C8×C157—22no—
110C22×C157—22no—
111C22×C157—44no—
112C22×C157—41no—
113C22×C157—42no—
114C22×C157—42no—
115C22×C157—82no—
116C22×C157—82no—
117C22×C157#12011yescharacteristic
118C22×C157#12011yescharacteristic
119C22×C157#12111yescharacteristic
120C2×D157—44no—
121C2×D157—42no—
122C2×D157—44no—
123C2×D157—82no—
124C4×C157—41no—
125C4×C157#12011yescharacteristic
126C4×C157#12111yescharacteristic
127C4×C157#12111yescharacteristic
128C4×C157#12711yescharacteristic
129Dic157—24no—
130Dic157—42no—
131C2×C157—21no—
132C2×C157—21no—
133C2×C157—81no—
134C2×C157—82no—
135C2×C157#39211yescharacteristic
136C2×C157#45312yes—
137D157—82no—
138C157#200211yescharacteristic, complemented, Sylow
139#2002—1571nomaximal, Sylow
140C2×AGL(1,ℤ/8)D15711yescharacteristic, complemented
141#535—1571no—
142C2×#127—1571no—
143C2×#104—1571no—
144#418—1571no—
145C2×#100—1571no—
146#339—1571no—
147C2×#41C2×D15711yescharacteristic
148C22×D4—3142no—
149C22×D4—1571no—
150C22×D4C2×D15711yescharacteristic
151AGL(1,ℤ/8)—22no—
152AGL(1,ℤ/8)—22no—
153C2×SD16—21no—
154C2×D8—21no—
155C2×M16—1571no—
156C2×M16C2×D15711yescharacteristic
157#127—3142no—
158#127—3141no—
159#127—3142no—
160#124—3141no—
161#121—3142no—
162#120—3142no—
163#117—1571no—
164C2×C42—1571no—
165#104—3142no—
166#102—3142no—
167#100—3142no—
168C24—21no—
169C24—1572no—
170#41—21no—
171#41—21no—
172#41—21no—
173#41—21no—
174C2×Q8C22⋊D15711yescharacteristic, complemented
175C2×D4—21no—
176C2×D4—21no—
177C2×D4—21no—
178C2×D4—21no—
179C2×D4—21no—
180C2×D4—21no—
181C2×D4—3148no—
182C2×D4—3144no—
183C2×D4—3141no—
184C2×D4—3141no—
185C2×D4—3141no—
186C2×D4—3141no—
187C2×D4—3141no—
188C2×D4—3141no—
189C2×D4—6288no—
190C2×D4C22⋊D15711yescharacteristic, complemented
191C2×D4C22⋊D15712yescomplemented
192C22×C4—21no—
193C22×C4—3141no—
194C22×C4C22×D15711yescharacteristic
195SD16—24no—
196D8—24no—
197M16—21no—
198M16—3141no—
199M16—3141no—
200M16C22⋊D15712yescomplemented
201C2×C8—21no—
202C2×C8—3141no—
203C4⋊C4—1572no—
204C22⋊C4—3144no—
205C22⋊C4—3141no—
206C42—3141no—
207C42—3141no—
208C42—1572no—
209C23—22no—
210C23—21no—
211C23—44no—
212C23—41no—
213C23—41no—
214C23—3142no—
215C23—3142no—
216C23—3142no—
217C23—6284no—
218C23D4×D15711yescharacteristic
219Q8—21no—
220Q8—21no—
221D4—21no—
222D4—22no—
223D4—22no—
224D4—21no—
225D4—41no—
226D4—41no—
227D4—41no—
228D4—41no—
229D4—62816no—
230D4—6281no—
231D4—6281no—
232D4—6281no—
233D4—6281no—
234D4—12564no—
235C2×C4—21no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—3144no—
239C2×C4—3142no—
240C2×C4—3142no—
241C2×C4D4×D15711yescharacteristic
242C2×C4D4×D15711yescharacteristic
243C2×C4C2×C22⋊D15711yescharacteristic
244C2×C4C2×C22⋊D15711yescharacteristic
245C2×C4C2×C22⋊D15711yescharacteristic
246C2×C4#1457895311yescharacteristic
247C8—22no—
248C8—6281no—
249C22—22no—
250C22—44no—
251C22—41no—
252C22—42no—
253C22—42no—
254C22—82no—
255C22—82no—
256C22—6284no—
257C22—6282no—
258C22—6284no—
259C22—12562no—
260C22#1909773911yescharacteristic
261C22#1909773911yescharacteristic
262C22#1909774211yescharacteristic
263C4—41no—
264C4—3144no—
265C4—6282no—
266C4#1909774011yescharacteristic
267C4#1909774311yescharacteristic
268C4#1909774311yescharacteristic
269C4#1909776711yescharacteristic
270C2—21no—
271C2—21no—
272C2—82no—
273C2—81no—
274C2—12562no—
275C2#2375718111yescharacteristic
276C2#2375743912yes—
subgroupG/H
centerC22#19097739
commutator, cosocleC22×C4×C157C23
FrattiniC22×C4C22×D157
FittingC2×AGL(1,ℤ/8)×C157C2
socleC22×C157#120
hypercenterC2×AGL(1,ℤ/8)D157
hypocenterC157#2002
LCSG ▷ C22×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #23757837C2semidirect
2. C2×#19097767C2semidirect
3. C2×#19097698C2semidirect
4. #23757278C2semidirect
5. C2×#19097687C2semidirect
6. #23757034C2semidirect
7. C2×AGL(1,ℤ/8)×C157C2semidirect
43. C4⋊Dic157D4semidirect
48. C4×Dic157D4semidirect
54. C2×Q8×C157D4semidirect
61. C2×D4×C157D4semidirect
62. C2×D4×C157D4semidirect
68. M16×C157D4semidirect
138. C157#2002semidirect
140. C2×AGL(1,ℤ/8)D157semidirect
174. C2×Q8C22⋊D157semidirect
190. C2×D4C22⋊D157semidirect
191. C2×D4C22⋊D157semidirect
200. M16C22⋊D157semidirect

Maximal subgroups

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

subgroupindex
#237578372
C2×#190977672
C2×#190976982
#237572782
C2×#190976872
#237570342
C2×AGL(1,ℤ/8)×C1572
#2002157