Finite Group Databasebeta

← #19140928

Subgroups — #19140928

Order 5504.

1000 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 1000 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C2×#13063295C212yesmaximal, complemented
2C2×#13063295C212yesmaximal, complemented
3C2×#13063293C212yesmaximal, complemented
4#14582817C212yesmaximal, complemented
5#14582816C211yesmaximal, characteristic, complemented
6#14582814C212yesmaximal, complemented
7#14582800C211yesmaximal, characteristic, complemented
8#14582799C212yesmaximal, complemented
9C43×#394C211yesmaximal, characteristic, complemented
10C24×D43C2212yescomplemented
11C22×D172C2211yescharacteristic
12C22×D172C2212yes—
13C22×D172C2211yescharacteristic
14C22×C22⋊D43C2212yes—
15C22×C22⋊D43C2212yes—
16#13063295—28no—
17#13063295—28no—
18#13063293—28no—
19C2×Dic43⋊C4C2212yescomplemented
20C2×C22⋊Dic43C2212yescomplemented
21C2×(C2×D43)⋊C4C2212yescomplemented
22C2×(C2×D43)⋊C4C2212yescomplemented
23C2×(C2×D43)⋊C4C2212yescomplemented
24C2×(C2×D43)⋊C4C2212yescomplemented
25C2×(C2×D43)⋊C4C2212yescomplemented
26C2×(C2×D43)⋊C4C2212yescomplemented
27C2×C4×Dic43C2211yescharacteristic, complemented
28C43⋊#95C2211yescharacteristic, complemented
29C22×D4×C43C2211yescharacteristic
30C2×C4⋊C4×C43C2211yescharacteristic, complemented
31C2×C22⋊C4×C43C2212yescomplemented
32C2×C22⋊C4×C43C2212yescomplemented
33#95×C43C2211yescharacteristic, complemented
34C24×C43C2312yes—
35C23×D43—24no—
36C23×D43—24no—
37C23×D43—24no—
38C23×D43—42no—
39C23×D43—42no—
40C23×D43—42no—
41C23×D43—42no—
42C23×D43C2312yes—
43C23×D43C2312yes—
44C2×D4×C43—24no—
45C2×D4×C43—41no—
46C22×C4×C43C2312yes—
47C22×C4×C43C2311yescharacteristic
48C22×C4×C43C2311yescharacteristic
49C22×C4×C43C2311yescharacteristic
50C2×C22⋊D43—24no—
51C2×C22⋊D43—24no—
52C2×C22⋊D43—42no—
53C2×C22⋊D43—42no—
54C2×C22⋊D43—42no—
55C2×C22⋊D43D418yescomplemented
56C22×Dic43C2312yes—
57C22×Dic43C2312yes—
58C2×D172—24no—
59C2×D172—24no—
60C2×D172—24no—
61C2×D172—41no—
62C2×D172—41no—
63C2×D172—42no—
64C2×D172—41no—
65C2×D172D414yescomplemented
66C4⋊C4×C43—41no—
67C22⋊C4×C43—24no—
68C22⋊C4×C43—24no—
69C22⋊Dic43—24no—
70(C2×D43)⋊C4—24no—
71(C2×D43)⋊C4—24no—
72(C2×D43)⋊C4—24no—
73(C2×D43)⋊C4—24no—
74(C2×D43)⋊C4—24no—
75(C2×D43)⋊C4—24no—
76Dic43⋊C4—42no—
77C4×Dic43—41no—
78C23×C43—22no—
79C23×C43—22no—
80C23×C43—22no—
81C23×C43—22no—
82C23×C43—22no—
83C23×C43—22no—
84C23×C43C2×D414yes—
85C23×C43C2411yescharacteristic
86C22×D43—22no—
87C22×D43—22no—
88C22×D43—22no—
89C22×D43—22no—
90C22×D43—22no—
91C22×D43—22no—
92C22×D43—22no—
93C22×D43—22no—
94C22×D43—22no—
95C22×D43—22no—
96C22×D43—22no—
97C22×D43—22no—
98C22×D43—48no—
99C22×D43—48no—
100C22×D43—48no—
101C22×D43—48no—
102C22×D43—48no—
103C22×D43—48no—
104C22×D43—48no—
105C22×D43C2×D414yes—
106C22×D43C2×D414yes—
107D4×C43—44no—
108C2×C4×C43—21no—
109C2×C4×C43—21no—
110C2×C4×C43—22no—
111C2×C4×C43—22no—
112C2×C4×C43—22no—
113C2×C4×C43—22no—
114C2×C4×C43—22no—
115C2×C4×C43—21no—
116C2×C4×C43—21no—
117C2×C4×C43C2×D412yes—
118C22⋊D43—48no—
119C22⋊D43—48no—
120C2×Dic43—22no—
121C2×Dic43—22no—
122C2×Dic43—22no—
123C2×Dic43—22no—
124C2×Dic43—22no—
125C2×Dic43C2×D414yes—
126D172—44no—
127D172—44no—
128D172—48no—
129C22×C43—24no—
130C22×C43—24no—
131C22×C43—24no—
132C22×C43—42no—
133C22×C43—42no—
134C22×C43—42no—
135C22×C43—42no—
136C22×C43C22×D412yes—
137C22×C43C22×D411yescharacteristic
138C22×C43H(5,𝔽2)11yescharacteristic
139C22×C43H(5,𝔽2)11yescharacteristic
140C22×C43H(5,𝔽2)12yes—
141C2×D43—24no—
142C2×D43—24no—
143C2×D43—24no—
144C2×D43—24no—
145C2×D43—24no—
146C2×D43—24no—
147C2×D43—432no—
148C2×D43—42no—
149C2×D43—42no—
150C2×D43—42no—
151C2×D43—42no—
152C2×D43—42no—
153C2×D43—42no—
154C2×D43—42no—
155C2×D43—42no—
156C4×C43—22no—
157C4×C43—41no—
158C4×C43—42no—
159C4×C43—41no—
160Dic43—24no—
161Dic43—42no—
162#2679—431nomaximal, Sylow
163C2×C43—44no—
164C2×C43#53411yescharacteristic
165C2×C43#53412yes—
166C2×C43D4211yescharacteristic
167C2×C43D4212yes—
168C2×C43H(3,𝔽4)11yescharacteristic
169D43—44no—
170D43—44no—
171C2×#120—432no—
172C2×#120—432no—
173C2×#120—432no—
174#394—431no—
175#394—432no—
176#394—432no—
177#394D4311yescharacteristic, complemented
178#390—431no—
179#390—432no—
180C43#267911yescharacteristic, complemented, Sylow
181C25—432no—
182C22×D4—432no—
183C22×D4—432no—
184C22×D4—431no—
185C22×D4—431no—
186C22×D4—432no—
187C22×D4C2×D4311yescharacteristic
188#120—868no—
189#120—868no—
190#120—868no—
191C2×C4⋊C4—432no—
192C2×C4⋊C4C2×D4311yescharacteristic, complemented
193C2×C22⋊C4—432no—
194C2×C22⋊C4—432no—
195C2×C22⋊C4—432no—
196C2×C22⋊C4—432no—
197C2×C22⋊C4—432no—
198C2×C22⋊C4—432no—
199C2×C22⋊C4—432no—
200C2×C22⋊C4C2×D4312yescomplemented
201C2×C22⋊C4C2×D4312yescomplemented
202C2×C42—431no—
203#95—431no—
204#95C2×D4311yescharacteristic, complemented
205C24—432no—
206C24—432no—
207C24—864no—
208C24—864no—
209C24—864no—
210C24—1722no—
211C24—1722no—
212C24—1722no—
213C24—1722no—
214C24C22×D4312yes—
215C2×D4—24no—
216C2×D4—41no—
217C2×D4—438no—
218C2×D4—434no—
219C2×D4—864no—
220C2×D4—864no—
221C2×D4—864no—
222C2×D4—864no—
223C2×D4—864no—
224C2×D4—1721no—
225C2×D4—1721no—
226C2×D4—1721no—
227C2×D4—1722no—
228C2×D4—1722no—
229C2×D4—1722no—
230C2×D4—1722no—
231C22×C4—432no—
232C22×C4—432no—
233C22×C4C22×D4311yescharacteristic
234C22×C4C22×D4311yescharacteristic
235C22×C4C22×D4312yes—
236C22×C4C22×D4311yescharacteristic
237C4⋊C4—41no—
238C4⋊C4—1722no—
239C22⋊C4—24no—
240C22⋊C4—24no—
241C22⋊C4—864no—
242C22⋊C4—864no—
243C22⋊C4—864no—
244C22⋊C4—864no—
245C22⋊C4—864no—
246C22⋊C4—864no—
247C22⋊C4—864no—
248C42—1721no—
249C23—22no—
250C23—22no—
251C23—22no—
252C23—22no—
253C23—22no—
254C23—22no—
255C23—434no—
256C23—434no—
257C23—862no—
258C23—862no—
259C23—862no—
260C23—862no—
261C23—862no—
262C23—862no—
263C23—862no—
264C23—862no—
265C23—862no—
266C23—862no—
267C23—862no—
268C23—862no—
269C23—1728no—
270C23—1728no—
271C23—1728no—
272C23—1728no—
273C23—1728no—
274C23—1728no—
275C23—1728no—
276C23D4×D4314yes—
277C23C23×D4311yescharacteristic
278D4—44no—
279D4—1724no—
280D4—1724no—
281D4—1728no—
282D4—1728no—
283D4—1728no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—22no—
293C2×C4—434no—
294C2×C4—862no—
295C2×C4—862no—
296C2×C4—862no—
297C2×C4—862no—
298C2×C4—862no—
299C2×C4D4×D4312yes—
300C22—24no—
301C22—24no—
302C22—24no—
303C22—42no—
304C22—42no—
305C22—42no—
306C22—42no—
307C22—864no—
308C22—864no—
309C22—864no—
310C22—864no—
311C22—864no—
312C22—864no—
313C22—17232no—
314C22—1722no—
315C22—1722no—
316C22—1722no—
317C22—1722no—
318C22—1722no—
319C22—1722no—
320C22—1722no—
321C22—1722no—
322C22C2×D4×D4311yescharacteristic
323C22C2×D4×D4312yes—
324C22#1306337011yescharacteristic
325C22#1306337112yes—
326C22#1306337111yescharacteristic
327C4—22no—
328C4—41no—
329C4—41no—
330C4—42no—
331C4—864no—
332C4—1722no—
333C2—44no—
334C2—1724no—
335C2—1724no—
336C2#1458345711yescharacteristic
337C2#1458346012yes—
338C2#1458353111yescharacteristic
339C2#1458353312yes—
340C2#1458364111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D43
commutator, socleC23×C43C24
FittingC43×#394C2
hypercenter#394D43
hypocenterC43#2679
LCSG ▷ C23×C43 ▷ C43
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×#13063295C2semidirect
2. C2×#13063295C2semidirect
3. C2×#13063293C2semidirect
4. #14582817C2semidirect
5. #14582816C2semidirect
6. #14582814C2semidirect
7. #14582800C2semidirect
8. #14582799C2semidirect
9. C43×#394C2semidirect
10. C24×D43C22semidirect
19. C2×Dic43⋊C4C22semidirect
20. C2×C22⋊Dic43C22semidirect
21. C2×(C2×D43)⋊C4C22semidirect
22. C2×(C2×D43)⋊C4C22semidirect
23. C2×(C2×D43)⋊C4C22semidirect
24. C2×(C2×D43)⋊C4C22semidirect
25. C2×(C2×D43)⋊C4C22semidirect
26. C2×(C2×D43)⋊C4C22semidirect
27. C2×C4×Dic43C22semidirect
28. C43⋊#95C22semidirect
30. C2×C4⋊C4×C43C22semidirect
31. C2×C22⋊C4×C43C22semidirect
32. C2×C22⋊C4×C43C22semidirect
33. #95×C43C22semidirect
55. C2×C22⋊D43D4semidirect
65. C2×D172D4semidirect
177. #394D43semidirect
180. C43#2679semidirect
192. C2×C4⋊C4C2×D43semidirect
200. C2×C22⋊C4C2×D43semidirect
201. C2×C22⋊C4C2×D43semidirect
204. #95C2×D43semidirect