Finite Group Databasebeta

← D23×AΣL(1,𝔽16) #15675027

Subgroups — D23×AΣL(1,𝔽16) #15675027

Order 2944.

438 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 57 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#120×D23C211yesmaximal, characteristic
2#100×D23C211yesmaximal, characteristic, complemented
3#99×D23C211yesmaximal, characteristic, complemented
4#13092205C211yesmaximal, characteristic, complemented
5#13092204C211yesmaximal, characteristic, complemented
6#13092203C211yesmaximal, characteristic, complemented
7C23×AΣL(1,𝔽16)C211yesmaximal, characteristic, complemented
8C24×D23C411yescharacteristic, complemented
9C2×D4×D23—21no—
10C2×D4×D23C2211yescharacteristic
11C23×#120C2211yescharacteristic
12#10625124C411yescharacteristic, complemented
13#10625108—21no—
14#10625108C2211yescharacteristic
15M16×D23—21no—
16#10625053—21no—
17#10625053C411yescharacteristic, complemented
18C22⋊C4×D23—21no—
19C22⋊C4×D23—21no—
20C22⋊C4×D23C411yescharacteristic, complemented
21C23×#100C2211yescharacteristic, complemented
22C23×#99C2211yescharacteristic, complemented
23#10625016C2211yescharacteristic, complemented
24#10625015C2211yescharacteristic, complemented
25#10625005—21no—
26#10624988—21no—
27C24×C23C2×C411yescharacteristic, complemented
28C23×D23—21no—
29C23×D23—21no—
30C23×D23—44no—
31C23×D23—41no—
32C23×D23—41no—
33C23×D23C2×C411yescharacteristic
34C23×D23C2×C411yescharacteristic, complemented
35C23×D23D411yescharacteristic
36C2×D4×C23—21no—
37C2×D4×C23C2311yescharacteristic
38C2×C22⋊D23—21no—
39C2×C22⋊D23—21no—
40C2×C22⋊D23C2×C411yescharacteristic
41C2×C22⋊D23D411yescharacteristic
42D4×D23—41no—
43D4×D23—41no—
44D4×D23—44no—
45C2×D92—21no—
46C2×D92D411yescharacteristic
47C2×C4×D23—21no—
48C2×C4×D23—41no—
49C2×C4×D23D411yescharacteristic
50M16×C23—21no—
51C22⋊C4×C23—21no—
52C22⋊C4×C23—21no—
53C22⋊C4×C23C2×C411yescharacteristic, complemented
54C22⋊Dic23—21no—
55C22⋊Dic23—21no—
56C22⋊Dic23C2×C411yescharacteristic, complemented
57(C2×D23)⋊C4—21no—
58(C2×D23)⋊C4—21no—
59(C2×D23)⋊C4—21no—
60(C2×D23)⋊C4—41no—
61#67267—21no—
62#67262—41no—
63C8×D23—41no—
64C23×C23—21no—
65C23×C23—42no—
66C23×C23—41no—
67C23×C23C22×C411yescharacteristic
68C23×C23C2×D411yescharacteristic
69C22×D23—24no—
70C22×D23—21no—
71C22×D23—21no—
72C22×D23—21no—
73C22×D23—21no—
74C22×D23—21no—
75C22×D23—42no—
76C22×D23—42no—
77C22×D23—42no—
78C22×D23—42no—
79C22×D23—42no—
80C22×D23—42no—
81C22×D23—42no—
82C22×D23—44no—
83C22×D23—44no—
84C22×D23—44no—
85C22×D23—41no—
86C22×D23—41no—
87C22×D23—41no—
88C22×D23—44no—
89C22×D23C22⋊C411yescharacteristic
90C22×D23C22⋊C411yescharacteristic
91C22×D23C22⋊C411yescharacteristic
92D4×C23—41no—
93D4×C23—42no—
94C2×C4×C23—21no—
95C2×C4×C23—41no—
96C2×C4×C23C2×D411yescharacteristic
97C22⋊D23—41no—
98C22⋊D23—41no—
99C22⋊D23—42no—
100C22⋊D23—42no—
101C2×Dic23—21no—
102C2×Dic23—41no—
103C2×Dic23C22⋊C411yescharacteristic
104D92—41no—
105D92—42no—
106C4×D23—21no—
107C4×D23—21no—
108C4×D23—42no—
109C4×D23—82no—
110C8×C23—41no—
111C23⋊C8—41no—
112C2×AΣL(1,𝔽16)—231nomaximal, Sylow
113C22×C23—22no—
114C22×C23—21no—
115C22×C23—21no—
116C22×C23—41no—
117C22×C23—41no—
118C22×C23—41no—
119C22×C23—42no—
120C22×C23—42no—
121C22×C23—41no—
122C22×C23C2×C22⋊C411yescharacteristic
123C2×D23—22no—
124C2×D23—22no—
125C2×D23—21no—
126C2×D23—21no—
127C2×D23—44no—
128C2×D23—42no—
129C2×D23—41no—
130C2×D23—48no—
131C2×D23—42no—
132C2×D23—41no—
133C2×D23—42no—
134C2×D23—44no—
135C2×D23—42no—
136C2×D23—41no—
137C2×D23—41no—
138C2×D23—42no—
139C2×D23—81no—
140C2×D23#9911yescharacteristic
141C2×D23#9911yescharacteristic
142C4×C23—21no—
143C4×C23—41no—
144C4×C23—81no—
145Dic23—21no—
146Dic23—41no—
147Dic23—81no—
148C2×#120—231no—
149C2×#100—231no—
150C2×#99—231no—
151AΣL(1,𝔽16)—231no—
152AΣL(1,𝔽16)—231no—
153AΣL(1,𝔽16)—231no—
154AΣL(1,𝔽16)D2311yescharacteristic, direct product factor
155C2×C23—21no—
156C2×C23—41no—
157C2×C23—42no—
158C2×C23—41no—
159C2×C23C2×#9911yescharacteristic
160D23—21no—
161D23—42no—
162D23—41no—
163D23—41no—
164D23AΣL(1,𝔽16)12yesdirect product factor
165C25—231no—
166C22×D4—231no—
167C22×D4—461no—
168C2×M16—461no—
169#120—231no—
170#120—231no—
171#120—231no—
172#120—461no—
173#120—461no—
174#120C2×D2311yescharacteristic
175C2×C22⋊C4—231no—
176C2×C22⋊C4—461no—
177C2×C22⋊C4—461no—
178#100—231no—
179#100—461no—
180#100C2×D2311yescharacteristic, complemented
181#99—231no—
182#99—461no—
183#99C2×D2311yescharacteristic, complemented
184C23C2×AΣL(1,𝔽16)11yescharacteristic, complemented, Sylow
185C24—231no—
186C24—231no—
187C24—231no—
188C24—461no—
189C24—461no—
190C24—921no—
191C24—921no—
192C24—924no—
193C24C4×D2311yescharacteristic, complemented
194C2×D4—21no—
195C2×D4—231no—
196C2×D4—231no—
197C2×D4—231no—
198C2×D4—461no—
199C2×D4—461no—
200C2×D4—461no—
201C2×D4—921no—
202C2×D4—921no—
203C2×D4—924no—
204C2×D4C22×D2311yescharacteristic
205C22×C4—231no—
206C22×C4—461no—
207C22×C4—921no—
208M16—21no—
209M16—461no—
210M16—921no—
211C2×C8—921no—
212C22⋊C4—21no—
213C22⋊C4—21no—
214C22⋊C4—231no—
215C22⋊C4—461no—
216C22⋊C4—461no—
217C22⋊C4—461no—
218C22⋊C4—461no—
219C22⋊C4—461no—
220C22⋊C4—921no—
221C22⋊C4C4×D2311yescharacteristic, complemented
222C23—21no—
223C23—42no—
224C23—41no—
225C23—231no—
226C23—231no—
227C23—231no—
228C23—461no—
229C23—461no—
230C23—461no—
231C23—461no—
232C23—464no—
233C23—461no—
234C23—921no—
235C23—922no—
236C23—921no—
237C23—921no—
238C23—922no—
239C23—922no—
240C23—922no—
241C23—922no—
242C23—922no—
243C23—922no—
244C23—924no—
245C23—924no—
246C23—924no—
247C23—924no—
248C23C2×C4×D2311yescharacteristic
249C23D4×D2311yescharacteristic
250D4—41no—
251D4—42no—
252D4—921no—
253D4—922no—
254D4—921no—
255D4—921no—
256D4—922no—
257D4—922no—
258C2×C4—21no—
259C2×C4—41no—
260C2×C4—231no—
261C2×C4—461no—
262C2×C4—461no—
263C2×C4—461no—
264C2×C4—922no—
265C2×C4—921no—
266C2×C4—1842no—
267C2×C4D4×D2311yescharacteristic
268C8—41no—
269C8—921no—
270C22—22no—
271C22—21no—
272C22—21no—
273C22—41no—
274C22—41no—
275C22—41no—
276C22—42no—
277C22—42no—
278C22—41no—
279C22—231no—
280C22—231no—
281C22—462no—
282C22—462no—
283C22—461no—
284C22—461no—
285C22—921no—
286C22—921no—
287C22—921no—
288C22—922no—
289C22—922no—
290C22—924no—
291C22—922no—
292C22—924no—
293C22—922no—
294C22—922no—
295C22—928no—
296C22—921no—
297C22—1841no—
298C22C22⋊C4×D2311yescharacteristic
299C4—21no—
300C4—41no—
301C4—81no—
302C4—461no—
303C4—921no—
304C4—1841no—
305C2—21no—
306C2—41no—
307C2—42no—
308C2—41no—
309C2—232no—
310C2—461no—
311C2—922no—
312C2—921no—
313C2—921no—
314C2#99×D2311yescharacteristic
subgroupG/H
centerC2#99×D23
commutatorC23×C23C22×C4
FrattiniC2×D4C22×D23
FittingC23×AΣL(1,𝔽16)C2
socleC2×C23C2×#99
prime radical#120×D23C2
cosocleC2×D4×C23C23
hypercenterAΣL(1,𝔽16)D23
hypocenterC23C2×AΣL(1,𝔽16)
LCSG ▷ C23×C23 ▷ C22×C23 ▷ C2×C23 ▷ C23
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ AΣL(1,𝔽16)

Complements

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

NH ≅ G/Ntype
2. #100×D23C2semidirect
3. #99×D23C2semidirect
4. #13092205C2semidirect
5. #13092204C2semidirect
6. #13092203C2semidirect
7. C23×AΣL(1,𝔽16)C2semidirect
8. C24×D23C4semidirect
12. #10625124C4semidirect
17. #10625053C4semidirect
20. C22⋊C4×D23C4semidirect
21. C23×#100C22semidirect
22. C23×#99C22semidirect
23. #10625016C22semidirect
24. #10625015C22semidirect
27. C24×C23C2×C4semidirect
34. C23×D23C2×C4semidirect
53. C22⋊C4×C23C2×C4semidirect
56. C22⋊Dic23C2×C4semidirect
154. AΣL(1,𝔽16)D23direct (N × H)
164. D23AΣL(1,𝔽16)direct (N × H)
180. #100C2×D23semidirect
183. #99C2×D23semidirect
184. C23C2×AΣL(1,𝔽16)semidirect
193. C24C4×D23semidirect
221. C22⋊C4C4×D23semidirect

Maximal subgroups

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

subgroupindex
#120×D232
#100×D232
#99×D232
#130922052
#130922042
#130922032
C23×AΣL(1,𝔽16)2
C2×AΣL(1,𝔽16)23