Finite Group Databasebeta

← #19140309

Subgroups — #19140309

Order 5504.

1134 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 1134 conjugacy classes of subgroups — too many to draw. Only its 165 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
1C23×C22⋊D43C211yesmaximal, characteristic, complemented
2C2×#13063306C211yesmaximal, characteristic, complemented
3C2×#13063293C211yesmaximal, characteristic, complemented
4#14582814C214yesmaximal, complemented
5#14582807C214yesmaximal, complemented
6#14582760C211yesmaximal, characteristic, complemented
7#14582755C211yesmaximal, characteristic, complemented
8#14582754C211yesmaximal, characteristic, complemented
9C2×#120×C43C211yesmaximal, characteristic, complemented
10C24×D43C2211yescharacteristic
11C22×C22⋊D43—24no—
12C22×C22⋊D43—24no—
13C22×C22⋊D43—24no—
14C22×C22⋊D43C2214yes—
15C22×C22⋊D43C2212yescomplemented
16C23×Dic43C2211yescharacteristic
17#13063306—24no—
18#13063293—24no—
19C2×Dic43⋊C4C2212yescomplemented
20C2×C22⋊Dic43C2211yescharacteristic
21C2×C22⋊Dic43C2211yescharacteristic
22C2×C22⋊Dic43C2212yescomplemented
23C2×C22⋊Dic43C2214yescomplemented
24C2×(C2×D43)⋊C4C2212yescomplemented
25C2×(C2×D43)⋊C4C2214yescomplemented
26C43⋊#95C2214yescomplemented
27C25×C43C2211yescharacteristic
28C22×D4×C43C2211yescharacteristic
29C22×D4×C43C2212yescomplemented
30#120×C43—24no—
31C2×C22⋊C4×C43C2211yescharacteristic
32C2×C22⋊C4×C43C2212yescomplemented
33C24×C43—22no—
34C24×C43—22no—
35C24×C43—44no—
36C24×C43D414yescomplemented
37C24×C43C2311yescharacteristic
38C24×C43C2312yes—
39C24×C43C2311yescharacteristic
40C23×D43—22no—
41C23×D43—22no—
42C23×D43—44no—
43C23×D43D414yescomplemented
44C23×D43C2312yes—
45C2×D4×C43—22no—
46C2×D4×C43—24no—
47C2×D4×C43—22no—
48C2×D4×C43—22no—
49C2×D4×C43—44no—
50C22×C4×C43C2312yes—
51C22×C4×C43C2311yescharacteristic
52C2×C22⋊D43—232no—
53C2×C22⋊D43—28no—
54C2×C22⋊D43—28no—
55C2×C22⋊D43—28no—
56C2×C22⋊D43—24no—
57C2×C22⋊D43—44no—
58C22×Dic43—22no—
59C22×Dic43—22no—
60C22×Dic43D414yescomplemented
61C22×Dic43C2312yes—
62C22×Dic43C2312yes—
63C22×Dic43C2312yes—
64C22⋊C4×C43—22no—
65C22⋊C4×C43—24no—
66C22⋊Dic43—24no—
67C22⋊Dic43—22no—
68C22⋊Dic43—22no—
69C22⋊Dic43—44no—
70(C2×D43)⋊C4—24no—
71(C2×D43)⋊C4—44no—
72Dic43⋊C4—24no—
73C23×C43—24no—
74C23×C43—28no—
75C23×C43—22no—
76C23×C43—22no—
77C23×C43—24no—
78C23×C43—24no—
79C23×C43—21no—
80C23×C43—21no—
81C23×C43—21no—
82C23×C43—24no—
83C23×C43—48no—
84C23×C43—48no—
85C23×C43—48no—
86C23×C43C2×D412yes—
87C23×C43C2×D412yes—
88C23×C43C2×D414yes—
89C23×C43C2×D412yes—
90C23×C43C2411yescharacteristic
91C22×D43—28no—
92C22×D43—22no—
93C22×D43—22no—
94C22×D43—24no—
95C22×D43—24no—
96C22×D43—48no—
97C22×D43—48no—
98C22×D43—48no—
99C22×D43C2×D414yes—
100D4×C43—48no—
101D4×C43—44no—
102C2×C4×C43—21no—
103C2×C4×C43—21no—
104C2×C4×C43—21no—
105C2×C4×C43—22no—
106C2×C4×C43—24no—
107C22⋊D43—232no—
108C22⋊D43—48no—
109C2×Dic43—28no—
110C2×Dic43—22no—
111C2×Dic43—22no—
112C2×Dic43—22no—
113C2×Dic43—22no—
114C2×Dic43—24no—
115C2×Dic43—24no—
116C2×Dic43C2×D414yes—
117C22×C43—24no—
118C22×C43—24no—
119C22×C43—24no—
120C22×C43—24no—
121C22×C43—24no—
122C22×C43—24no—
123C22×C43—416no—
124C22×C43—44no—
125C22×C43—44no—
126C22×C43—41no—
127C22×C43—42no—
128C22×C43—42no—
129C22×C43—42no—
130C22×C43#12014yes—
131C22×C43C22×D411yescharacteristic
132C22×C43C22×D411yescharacteristic
133C22×C43C22×D411yescharacteristic
134C22×C43H(5,𝔽2)14yes—
135C2×D43—24no—
136C2×D43—24no—
137C2×D43—24no—
138C2×D43—416no—
139C2×D43—44no—
140C2×D43—44no—
141C4×C43—41no—
142C4×C43—42no—
143Dic43—24no—
144Dic43—42no—
145Dic43—42no—
146#2613—431nomaximal, Sylow
147C2×C43—24no—
148C2×C43—44no—
149C2×C43—81no—
150C2×C43C2×#12011yescharacteristic
151C2×C43#53412yes—
152C2×C43#53412yes—
153C2×C43#53512yes—
154D43—44no—
155C23×D4—431no—
156C2×#123—431no—
157C2×#120—431no—
158C2×#120D4311yescharacteristic, complemented
159#394—434no—
160#392—434no—
161#381—431no—
162#379—431no—
163#379—431no—
164C43#261311yescharacteristic, complemented, Sylow
165C25—431no—
166C25C2×D4311yescharacteristic
167C22×D4—434no—
168C22×D4—432no—
169C22×D4—864no—
170C22×D4—864no—
171C22×D4—864no—
172C22×D4C2×D4311yescharacteristic
173C22×D4C2×D4312yescomplemented
174C23×C4—431no—
175#123—864no—
176#120—24no—
177#120—864no—
178C2×C4⋊C4—432no—
179C2×C22⋊C4—431no—
180C2×C22⋊C4—431no—
181C2×C22⋊C4—432no—
182C2×C22⋊C4—432no—
183C2×C22⋊C4—434no—
184C2×C22⋊C4—434no—
185C2×C22⋊C4C2×D4311yescharacteristic
186C2×C22⋊C4C2×D4312yescomplemented
187#95—434no—
188C24—22no—
189C24—22no—
190C24—44no—
191C24—432no—
192C24—434no—
193C24—862no—
194C24—862no—
195C24—1724no—
196C24C22⋊D4314yescomplemented
197C24C22×D4311yescharacteristic
198C24C22×D4312yes—
199C24C22×D4311yescharacteristic
200C2×D4—22no—
201C2×D4—22no—
202C2×D4—22no—
203C2×D4—24no—
204C2×D4—44no—
205C2×D4—868no—
206C2×D4—868no—
207C2×D4—8632no—
208C2×D4—868no—
209C2×D4—864no—
210C2×D4—1724no—
211C22×C4—434no—
212C22×C4—432no—
213C22×C4—432no—
214C22×C4—432no—
215C22×C4—862no—
216C22×C4—862no—
217C22×C4C22×D4312yes—
218C22×C4C22×D4311yescharacteristic
219C4⋊C4—864no—
220C22⋊C4—22no—
221C22⋊C4—24no—
222C22⋊C4—864no—
223C22⋊C4—864no—
224C22⋊C4—862no—
225C22⋊C4—862no—
226C22⋊C4—1724no—
227C22⋊C4—1724no—
228C23—24no—
229C23—28no—
230C23—22no—
231C23—22no—
232C23—24no—
233C23—24no—
234C23—21no—
235C23—21no—
236C23—21no—
237C23—24no—
238C23—48no—
239C23—48no—
240C23—48no—
241C23—434no—
242C23—868no—
243C23—862no—
244C23—862no—
245C23—864no—
246C23—864no—
247C23—1728no—
248C23—1728no—
249C23—1728no—
250C23D4×D4312yes—
251C23D4×D4312yes—
252C23C2×C22⋊D4314yes—
253C23C2×C22⋊D4312yes—
254C23C23×D4311yescharacteristic
255D4—48no—
256D4—44no—
257D4—8632no—
258D4—1728no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—24no—
264C2×C4—434no—
265C2×C4—868no—
266C2×C4—862no—
267C2×C4—862no—
268C2×C4—862no—
269C2×C4—862no—
270C2×C4—864no—
271C2×C4—864no—
272C22—24no—
273C22—24no—
274C22—24no—
275C22—24no—
276C22—24no—
277C22—24no—
278C22—416no—
279C22—44no—
280C22—44no—
281C22—41no—
282C22—42no—
283C22—42no—
284C22—42no—
285C22—864no—
286C22—864no—
287C22—864no—
288C22—17216no—
289C22—1724no—
290C22—1724no—
291C22#1306329314yes—
292C22C2×D4×D4311yescharacteristic
293C22C2×D4×D4311yescharacteristic
294C22C22×C22⋊D4311yescharacteristic
295C22#1306337014yes—
296C4—41no—
297C4—42no—
298C4—864no—
299C4—1722no—
300C4—1722no—
301C2—24no—
302C2—44no—
303C2—81no—
304C2—1724no—
305C2C2×#1306329311yescharacteristic
306C2#1458345812yes—
307C2#1458345912yes—
308C2#1458346412yes—
subgroupG/H
center, FrattiniC23C23×D43
commutator, socleC23×C43C24
FittingC2×#120×C43C2
hypercenterC2×#120D43
hypocenterC43#2613
LCSG ▷ C23×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×#120

Complements

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

NH ≅ G/Ntype
1. C23×C22⋊D43C2semidirect
2. C2×#13063306C2semidirect
3. C2×#13063293C2semidirect
4. #14582814C2semidirect
5. #14582807C2semidirect
6. #14582760C2semidirect
7. #14582755C2semidirect
8. #14582754C2semidirect
9. C2×#120×C43C2semidirect
15. C22×C22⋊D43C22semidirect
19. C2×Dic43⋊C4C22semidirect
22. C2×C22⋊Dic43C22semidirect
23. C2×C22⋊Dic43C22semidirect
24. C2×(C2×D43)⋊C4C22semidirect
25. C2×(C2×D43)⋊C4C22semidirect
26. C43⋊#95C22semidirect
29. C22×D4×C43C22semidirect
32. C2×C22⋊C4×C43C22semidirect
36. C24×C43D4semidirect
43. C23×D43D4semidirect
60. C22×Dic43D4semidirect
158. C2×#120D43semidirect
164. C43#2613semidirect
173. C22×D4C2×D43semidirect
186. C2×C22⋊C4C2×D43semidirect
196. C24C22⋊D43semidirect