Finite Group Databasebeta

← #19138241

Subgroups — #19138241

Order 5504.

1088 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 1088 conjugacy classes of subgroups — too many to draw. Only its 161 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
1C22×D4×D43C211yesmaximal, characteristic
2C2×#13063311C212yesmaximal, complemented
3C2×#13063303C211yesmaximal, characteristic, complemented
4#14582817C211yesmaximal, characteristic, complemented
5#14582816C211yesmaximal, characteristic, complemented
6#14582813C211yesmaximal, characteristic, complemented
7#14582799C212yesmaximal, complemented
8#14582754C212yesmaximal, complemented
9#14582754C212yesmaximal, complemented
10#14582746C211yesmaximal, characteristic, complemented
11C43×#394C211yesmaximal, characteristic, complemented
12C24×D43C2212yes—
13C22×Dic86C2211yescharacteristic, complemented
14C22×D172C2211yescharacteristic
15C22×C22⋊D43C2212yes—
16C2×D4×D43—28no—
17C2×D4×D43—24no—
18C22×C4×D43C2211yescharacteristic
19#13063311—28no—
20#13063303—24no—
21C2×Dic43⋊C4C2211yescharacteristic, complemented
22C2×Dic43⋊C4C2211yescharacteristic, complemented
23C2×C22⋊Dic43C2212yescomplemented
24C2×C22⋊Dic43C2212yescomplemented
25C2×(C2×D43)⋊C4C2211yescharacteristic
26C2×(C2×D43)⋊C4C2211yescharacteristic
27C2×(C2×D43)⋊C4C2212yes—
28C2×(C2×D43)⋊C4C2212yescomplemented
29C2×(C2×D43)⋊C4C2212yescomplemented
30C2×(C2×D43)⋊C4C2212yescomplemented
31C2×(C2×D43)⋊C4C2212yescomplemented
32C2×C4×Dic43C2212yescomplemented
33C43⋊#95C2211yescharacteristic, complemented
34C22×D4×C43C2211yescharacteristic
35C2×C4⋊C4×C43C2211yescharacteristic, complemented
36C2×C22⋊C4×C43C2212yescomplemented
37C2×C22⋊C4×C43C2212yescomplemented
38#95×C43C2211yescharacteristic, complemented
39C24×C43C2312yes—
40C23×D43—24no—
41C23×D43—24no—
42C23×D43—24no—
43C23×D43—24no—
44C23×D43—24no—
45C23×D43—42no—
46C23×D43—42no—
47C23×D43C2312yes—
48C23×D43C2311yescharacteristic
49C2×D4×C43—22no—
50C2×D4×C43—24no—
51C22×C4×C43C2312yes—
52C22×C4×C43C2311yescharacteristic
53C22×C4×C43C2311yescharacteristic
54C22×C4×C43C2311yescharacteristic
55C2×C22⋊D43—24no—
56C2×C22⋊D43—24no—
57C2×C22⋊D43D418yes—
58C22×Dic43C2312yes—
59C22×Dic43C2311yescharacteristic
60C22×Dic43C2311yescharacteristic
61C22×Dic43C2311yescharacteristic
62D4×D43—232no—
63C2×D172—22no—
64C2×D172—24no—
65C2×C4×D43—24no—
66C2×C4×D43D414yes—
67C2×Dic86—22no—
68C2×Dic86—24no—
69C4⋊C4×C43—22no—
70C22⋊C4×C43—24no—
71C22⋊C4×C43—42no—
72C22⋊Dic43—24no—
73C22⋊Dic43—42no—
74(C2×D43)⋊C4—22no—
75(C2×D43)⋊C4—22no—
76(C2×D43)⋊C4—24no—
77(C2×D43)⋊C4—24no—
78(C2×D43)⋊C4—24no—
79(C2×D43)⋊C4—42no—
80(C2×D43)⋊C4—42no—
81Dic43⋊C4—22no—
82Dic43⋊C4—22no—
83C4×Dic43—24no—
84C23×C43—22no—
85C23×C43—22no—
86C23×C43—22no—
87C23×C43—22no—
88C23×C43—22no—
89C23×C43—22no—
90C23×C43C2×D414yes—
91C23×C43C2411yescharacteristic
92C22×D43—22no—
93C22×D43—21no—
94C22×D43—21no—
95C22×D43—216no—
96C22×D43—22no—
97C22×D43—22no—
98C22×D43—22no—
99C22×D43—22no—
100C22×D43—22no—
101C22×D43—22no—
102C22×D43—48no—
103C22×D43—48no—
104C22×D43—48no—
105C22×D43—48no—
106C22×D43—48no—
107C22×D43—48no—
108C22×D43C2×D414yes—
109C22×D43C2×D412yes—
110C22×D43#4114yes—
111D4×C43—28no—
112C2×C4×C43—22no—
113C2×C4×C43—22no—
114C2×C4×C43—21no—
115C2×C4×C43—22no—
116C2×C4×C43—21no—
117C2×C4×C43—22no—
118C2×C4×C43—21no—
119C2×C4×C43—22no—
120C2×C4×C43—21no—
121C2×C4×C43C2×D412yes—
122C22⋊D43—216no—
123C2×Dic43—22no—
124C2×Dic43—21no—
125C2×Dic43—21no—
126C2×Dic43—22no—
127C2×Dic43—22no—
128C2×Dic43—21no—
129C2×Dic43—22no—
130C2×Dic43—21no—
131C2×Dic43C2×D414yes—
132C2×Dic43C2×D412yes—
133D172—28no—
134C4×D43—28no—
135Dic86—44no—
136C22×C43—24no—
137C22×C43—24no—
138C22×C43—24no—
139C22×C43—42no—
140C22×C43—42no—
141C22×C43—42no—
142C22×C43—42no—
143C22×C43C22×D411yescharacteristic
144C22×C43C22×D412yes—
145C22×C43C2×#4112yes—
146C22×C43H(5,𝔽2)11yescharacteristic
147C22×C43H(5,𝔽2)11yescharacteristic
148C2×D43—22no—
149C2×D43—22no—
150C2×D43—24no—
151C2×D43—24no—
152C2×D43—24no—
153C2×D43—24no—
154C2×D43—24no—
155C2×D43—432no—
156C2×D43—42no—
157C2×D43—42no—
158C2×D43—42no—
159C2×D43—42no—
160C2×D43#12414yescomplemented
161C4×C43—22no—
162C4×C43—41no—
163C4×C43—41no—
164C4×C43—42no—
165Dic43—41no—
166Dic43—41no—
167Dic43—42no—
168Dic43#12014yes—
169#2435—431nomaximal, Sylow
170C2×C43—44no—
171C2×C43C2×#12011yescharacteristic
172C2×C43C2×#12411yescharacteristic
173C2×C43#53511yescharacteristic
174C2×C43#54612yes—
175C2×C43#54612yes—
176D43—24no—
177D43—44no—
178C23×D4—431no—
179C2×#124—432no—
180C2×C22⋊Q8—431no—
181#394—431no—
182#394—431no—
183#394—431no—
184#394D4311yescharacteristic, complemented
185#390—432no—
186#379—432no—
187#379—432no—
188C4×C22⋊C4—431no—
189C43#243511yescharacteristic, complemented, Sylow
190C25—432no—
191C22×Q8—431no—
192C22×D4—432no—
193C22×D4—431no—
194C22×D4—864no—
195C22×D4—868no—
196C22×D4C2×D4311yescharacteristic
197C23×C4—431no—
198#124—868no—
199C22⋊Q8—864no—
200C2×C4⋊C4—431no—
201C2×C4⋊C4—431no—
202C2×C4⋊C4C2×D4311yescharacteristic, complemented
203C2×C22⋊C4—432no—
204C2×C22⋊C4—431no—
205C2×C22⋊C4—431no—
206C2×C22⋊C4—432no—
207C2×C22⋊C4—432no—
208C2×C22⋊C4—432no—
209C2×C22⋊C4—432no—
210C2×C22⋊C4—432no—
211C2×C22⋊C4—432no—
212C2×C22⋊C4C2×D4312yescomplemented
213C2×C22⋊C4C2×D4312yescomplemented
214C2×C42—432no—
215#95—431no—
216#95C2×D4311yescharacteristic, complemented
217C24—431no—
218C24—432no—
219C24—864no—
220C24—864no—
221C24—864no—
222C24—864no—
223C24—864no—
224C24—1722no—
225C24—1722no—
226C24C22×D4312yes—
227C2×Q8—862no—
228C2×Q8—864no—
229C2×D4—24no—
230C2×D4—22no—
231C2×D4—438no—
232C2×D4—864no—
233C2×D4—864no—
234C2×D4—864no—
235C2×D4—8632no—
236C2×D4—862no—
237C22×C4—431no—
238C22×C4—434no—
239C22×C4—432no—
240C22×C4—431no—
241C22×C4—431no—
242C22×C4—864no—
243C22×C4C22×D4312yes—
244C22×C4C22×D4311yescharacteristic
245C22×C4C22×D4311yescharacteristic
246C22×C4C22×D4311yescharacteristic
247C4⋊C4—22no—
248C4⋊C4—862no—
249C4⋊C4—862no—
250C22⋊C4—24no—
251C22⋊C4—42no—
252C22⋊C4—862no—
253C22⋊C4—862no—
254C22⋊C4—864no—
255C22⋊C4—864no—
256C22⋊C4—864no—
257C22⋊C4—864no—
258C22⋊C4—1722no—
259C22⋊C4—1722no—
260C22⋊C4—1722no—
261C42—864no—
262C23—22no—
263C23—22no—
264C23—22no—
265C23—22no—
266C23—22no—
267C23—22no—
268C23—432no—
269C23—434no—
270C23—434no—
271C23—8616no—
272C23—862no—
273C23—861no—
274C23—861no—
275C23—862no—
276C23—862no—
277C23—862no—
278C23—862no—
279C23—862no—
280C23—862no—
281C23—1728no—
282C23—1728no—
283C23—1728no—
284C23—1728no—
285C23—1728no—
286C23—1728no—
287C23D4×D4314yes—
288C23C23×D4311yescharacteristic
289Q8—1724no—
290D4—28no—
291D4—868no—
292D4—8616no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—22no—
299C2×C4—21no—
300C2×C4—22no—
301C2×C4—21no—
302C2×C4—434no—
303C2×C4—432no—
304C2×C4—868no—
305C2×C4—862no—
306C2×C4—862no—
307C2×C4—861no—
308C2×C4—861no—
309C2×C4—862no—
310C2×C4—861no—
311C2×C4—861no—
312C2×C4—862no—
313C2×C4D4×D4312yes—
314C22—24no—
315C22—24no—
316C22—24no—
317C22—42no—
318C22—42no—
319C22—42no—
320C22—42no—
321C22—434no—
322C22—862no—
323C22—862no—
324C22—864no—
325C22—864no—
326C22—864no—
327C22—864no—
328C22—864no—
329C22—17232no—
330C22—1722no—
331C22—1722no—
332C22—1722no—
333C22—1722no—
334C22C2×D4×D4311yescharacteristic
335C22C2×D4×D4312yes—
336C22#41×D4312yes—
337C22#1306337011yescharacteristic
338C22#1306337111yescharacteristic
339C4—22no—
340C4—41no—
341C4—41no—
342C4—42no—
343C4—434no—
344C4—1721no—
345C4—1721no—
346C4—1722no—
347C2—44no—
348C2—864no—
349C2—1724no—
350C2#120×D4311yescharacteristic
351C2#124×D4311yescharacteristic
352C2#1458346511yescharacteristic
353C2#1458353812yes—
354C2#1458354112yes—
subgroupG/H
center, FrattiniC23C23×D43
commutator, socleC23×C43C24
FittingC43×#394C2
hypercenter#394D43
hypocenterC43#2435
LCSG ▷ C23×C43 ▷ C43
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
2. C2×#13063311C2semidirect
3. C2×#13063303C2semidirect
4. #14582817C2semidirect
5. #14582816C2semidirect
6. #14582813C2semidirect
7. #14582799C2semidirect
8. #14582754C2semidirect
9. #14582754C2semidirect
10. #14582746C2semidirect
11. C43×#394C2semidirect
13. C22×Dic86C22semidirect
21. C2×Dic43⋊C4C22semidirect
22. C2×Dic43⋊C4C22semidirect
23. C2×C22⋊Dic43C22semidirect
24. C2×C22⋊Dic43C22semidirect
28. C2×(C2×D43)⋊C4C22semidirect
29. C2×(C2×D43)⋊C4C22semidirect
30. C2×(C2×D43)⋊C4C22semidirect
31. C2×(C2×D43)⋊C4C22semidirect
32. C2×C4×Dic43C22semidirect
33. C43⋊#95C22semidirect
35. C2×C4⋊C4×C43C22semidirect
36. C2×C22⋊C4×C43C22semidirect
37. C2×C22⋊C4×C43C22semidirect
38. #95×C43C22semidirect
160. C2×D43#124semidirect
184. #394D43semidirect
189. C43#2435semidirect
202. C2×C4⋊C4C2×D43semidirect
212. C2×C22⋊C4C2×D43semidirect
213. C2×C22⋊C4C2×D43semidirect
216. #95C2×D43semidirect