Finite Group Databasebeta

← D157⋊AΣL(1,𝔽16) #34116382

Subgroups — D157⋊AΣL(1,𝔽16) #34116382

Order 20096.

438 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 54 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
1D157⋊#100C211yesmaximal, characteristic, complemented
2D157⋊#99C211yesmaximal, characteristic, complemented
3#23758174C212yesmaximal, complemented
4#23758173C212yesmaximal, complemented
5#120×D157C211yesmaximal, characteristic
6D157⋊M16—21no—
7C2×C22⋊#10597845—21no—
8#19097831—21no—
9#19097831C2212yescomplemented
10#19097830—21no—
11#19097829C2212yescomplemented
12C24×D157C411yescharacteristic, complemented
13C2×D4×D157—21no—
14C2×D4×D157C2211yescharacteristic
15#19097741—21no—
16#19097741C2211yescharacteristic
17#19097740C2211yescharacteristic
18#19097739—21no—
19#19097739C411yescharacteristic, complemented
20C22⋊C4×D157—21no—
21C22⋊C4×D157C411yescharacteristic, complemented
22#120×C157C411yescharacteristic, complemented
23C22×#10597845—41no—
24#14578964—41no—
25#14578963—22no—
26D157⋊C8—41no—
27C22⋊#10597845—22no—
28C22⋊#10597845—41no—
29C23×D157—21no—
30C23×D157—21no—
31C23×D157—44no—
32C23×D157—41no—
33C23×D157—41no—
34C23×D157C2×C411yescharacteristic
35C23×D157C2×C411yescharacteristic, complemented
36C23×D157D411yescharacteristic
37C2×D628—21no—
38C2×D628D411yescharacteristic
39C2×C22⋊D157—21no—
40C2×C22⋊D157—21no—
41C2×C22⋊D157D411yescharacteristic
42C2×C22⋊D157C2311yescharacteristic
43D4×D157—41no—
44D4×D157—41no—
45D4×D157—44no—
46C2×C4×D157—21no—
47C2×C4×D157D411yescharacteristic
48C22⋊Dic157—21no—
49C22⋊Dic157C2×C411yescharacteristic, complemented
50#14578931—21no—
51#14578931—21no—
52#14578931—21no—
53C24×C157C2×C411yescharacteristic, complemented
54C2×D4×C157—21no—
55C2×D4×C157C2×C411yescharacteristic
56C22⋊C4×C157—21no—
57C22⋊C4×C157C2×C411yescharacteristic, complemented
58C2×#10597845—42no—
59C2×#10597845—82no—
60#11929441—42no—
61C22×D157—24no—
62C22×D157—21no—
63C22×D157—21no—
64C22×D157—21no—
65C22×D157—21no—
66C22×D157—21no—
67C22×D157—42no—
68C22×D157—42no—
69C22×D157—42no—
70C22×D157—42no—
71C22×D157—42no—
72C22×D157—42no—
73C22×D157—42no—
74C22×D157—44no—
75C22×D157—44no—
76C22×D157—44no—
77C22×D157—41no—
78C22×D157—41no—
79C22×D157—41no—
80C22×D157—44no—
81C22×D157C22⋊C411yescharacteristic
82C22×D157C22⋊C411yescharacteristic
83C22×D157C2×D411yescharacteristic
84D628—41no—
85D628—42no—
86C22⋊D157—41no—
87C22⋊D157—41no—
88C22⋊D157—42no—
89C22⋊D157—42no—
90C2×Dic157—21no—
91C2×Dic157C2×D411yescharacteristic
92C4×D157—21no—
93C4×D157—21no—
94C4×D157—42no—
95C23×C157—21no—
96C23×C157—42no—
97C23×C157—41no—
98C23×C157C22⋊C411yescharacteristic
99C23×C157C22×C411yescharacteristic
100D4×C157—41no—
101D4×C157—42no—
102C2×C4×C157—21no—
103C2×C4×C157C22⋊C411yescharacteristic
104C22×C157—22no—
105C22×C157—21no—
106C22×C157—21no—
107C22×C157—41no—
108C22×C157—41no—
109C22×C157—41no—
110C22×C157—42no—
111C22×C157—42no—
112C22×C157—41no—
113C22×C157C2×C22⋊C411yescharacteristic
114C2×D157—22no—
115C2×D157—22no—
116C2×D157—21no—
117C2×D157—21no—
118C2×D157—44no—
119C2×D157—42no—
120C2×D157—41no—
121C2×D157—48no—
122C2×D157—42no—
123C2×D157—41no—
124C2×D157—44no—
125C2×D157—42no—
126C2×D157—42no—
127C2×D157—41no—
128C2×D157—41no—
129C2×D157—42no—
130C2×D157—81no—
131C2×D157#9911yescharacteristic
132C2×D157#9911yescharacteristic
133#10597845—82no—
134C4×C157—21no—
135C4×C157—41no—
136Dic157—21no—
137Dic157—41no—
138C2×C157—21no—
139C2×C157—42no—
140C2×C157—41no—
141C2×C157—41no—
142C2×C157C2×#9911yescharacteristic
143D157—21no—
144D157—41no—
145D157—42no—
146D157—41no—
147D157AΣL(1,𝔽16)12yescomplemented
148C157C2×AΣL(1,𝔽16)11yescharacteristic, complemented, Sylow
149C2×AΣL(1,𝔽16)—1571nomaximal, Sylow
150C2×#120—1571no—
151C2×#100—1571no—
152C2×#99—1571no—
153AΣL(1,𝔽16)—1572no—
154AΣL(1,𝔽16)—1572no—
155C25—1571no—
156C22×D4—3141no—
157C22×D4—1571no—
158C2×M16—3141no—
159#120—3141no—
160#120—3141no—
161#120—1571no—
162#120—1571no—
163#120—1571no—
164#120#1059784511yescharacteristic, complemented
165C2×C22⋊C4—3141no—
166C2×C22⋊C4—3141no—
167C2×C22⋊C4—1571no—
168#100—3141no—
169#100—1572no—
170#99—3141no—
171#99—1572no—
172C24—3141no—
173C24—3141no—
174C24—6284no—
175C24—6281no—
176C24—6281no—
177C24—1571no—
178C24—1571no—
179C24—1571no—
180C24C2×#1059784511yescharacteristic, complemented
181C2×D4—21no—
182C2×D4—3141no—
183C2×D4—3141no—
184C2×D4—3141no—
185C2×D4—6281no—
186C2×D4—6281no—
187C2×D4—6284no—
188C2×D4—1571no—
189C2×D4—1571no—
190C2×D4—1571no—
191C2×D4C2×#1059784511yescharacteristic
192C22×C4—3141no—
193C22×C4—6281no—
194C22×C4—1571no—
195M16—3142no—
196M16—6281no—
197C2×C8—6281no—
198C22⋊C4—21no—
199C22⋊C4—3141no—
200C22⋊C4—3141no—
201C22⋊C4—3141no—
202C22⋊C4—3141no—
203C22⋊C4—3142no—
204C22⋊C4—6281no—
205C22⋊C4—1571no—
206C22⋊C4C2×#1059784511yescharacteristic, complemented
207C23—21no—
208C23—42no—
209C23—41no—
210C23—3144no—
211C23—3141no—
212C23—3141no—
213C23—3141no—
214C23—3141no—
215C23—3141no—
216C23—6281no—
217C23—6282no—
218C23—6282no—
219C23—6282no—
220C23—6282no—
221C23—6282no—
222C23—6282no—
223C23—6282no—
224C23—6284no—
225C23—6284no—
226C23—6284no—
227C23—6284no—
228C23—6281no—
229C23—6281no—
230C23—1571no—
231C23—1571no—
232C23—1571no—
233C23C22⋊#1059784511yescharacteristic
234C23C22×#1059784511yescharacteristic
235D4—41no—
236D4—42no—
237D4—6281no—
238D4—6282no—
239D4—6281no—
240D4—6281no—
241D4—6282no—
242D4—6282no—
243C2×C4—21no—
244C2×C4—3141no—
245C2×C4—3141no—
246C2×C4—3141no—
247C2×C4—6282no—
248C2×C4—6282no—
249C2×C4—1571no—
250C2×C4—12562no—
251C2×C4C22⋊#1059784511yescharacteristic
252C8—6282no—
253C22—22no—
254C22—21no—
255C22—21no—
256C22—41no—
257C22—41no—
258C22—41no—
259C22—42no—
260C22—42no—
261C22—41no—
262C22—3142no—
263C22—3142no—
264C22—3141no—
265C22—3141no—
266C22—6281no—
267C22—6281no—
268C22—6281no—
269C22—6284no—
270C22—6282no—
271C22—6282no—
272C22—6282no—
273C22—6288no—
274C22—6282no—
275C22—6284no—
276C22—6282no—
277C22—6281no—
278C22—1571no—
279C22—1571no—
280C22—12561no—
281C22C2×C22⋊#1059784511yescharacteristic
282C4—21no—
283C4—41no—
284C4—3141no—
285C4—6281no—
286C4—12562no—
287C2—21no—
288C2—42no—
289C2—41no—
290C2—41no—
291C2—3141no—
292C2—6281no—
293C2—6282no—
294C2—6281no—
295C2—1572no—
296C2D157⋊#9911yescharacteristic
subgroupG/H
centerC2D157⋊#99
commutatorC23×C157C22×C4
FrattiniC23C22×#10597845
Fitting#120×C157C4
socleC2×C157C2×#99
prime radical#120×D157C2
cosocleC2×C22⋊D157C23
hypercenter#120#10597845
hypocenterC157C2×AΣL(1,𝔽16)
LCSG ▷ C23×C157 ▷ C22×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #120

Complements

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

NH ≅ G/Ntype
1. D157⋊#100C2semidirect
2. D157⋊#99C2semidirect
3. #23758174C2semidirect
4. #23758173C2semidirect
9. #19097831C22semidirect
11. #19097829C22semidirect
12. C24×D157C4semidirect
19. #19097739C4semidirect
21. C22⋊C4×D157C4semidirect
22. #120×C157C4semidirect
35. C23×D157C2×C4semidirect
49. C22⋊Dic157C2×C4semidirect
53. C24×C157C2×C4semidirect
57. C22⋊C4×C157C2×C4semidirect
147. D157AΣL(1,𝔽16)semidirect
148. C157C2×AΣL(1,𝔽16)semidirect
164. #120#10597845semidirect
180. C24C2×#10597845semidirect
206. C22⋊C4C2×#10597845semidirect

Maximal subgroups

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

subgroupindex
D157⋊#1002
D157⋊#992
#237581742
#237581732
#120×D1572
C2×AΣL(1,𝔽16)157