Finite Group Databasebeta

← #19137924

Subgroups — #19137924

Order 5504.

1116 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 1116 conjugacy classes of subgroups — too many to draw. Only its 169 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×D172C211yesmaximal, characteristic
2C2×#13063310C212yesmaximal, complemented
3C2×C22⋊C4×D43C211yesmaximal, characteristic, complemented
4#14582817C212yesmaximal, complemented
5#14582816C212yesmaximal, complemented
6#14582777C212yesmaximal, complemented
7#14582772C211yesmaximal, characteristic, complemented
8#14582754C211yesmaximal, characteristic, complemented
9#14582754C211yesmaximal, characteristic, complemented
10#14582754C211yesmaximal, characteristic, complemented
11C43×#385C211yesmaximal, characteristic, complemented
12C24×D43C2211yescharacteristic
13C24×D43C2211yescharacteristic
14C22×D172—24no—
15C22×D172—28no—
16C22×D172C2212yes—
17C22×D172C2212yes—
18C22×C4×D43C2211yescharacteristic, complemented
19C22×C4×D43C2211yescharacteristic, complemented
20#13063310—28no—
21C2×Dic43⋊C4C2212yescomplemented
22C2×C22⋊Dic43C2211yescharacteristic
23C2×C22⋊Dic43C2211yescharacteristic
24C2×(C2×D43)⋊C4C2211yescharacteristic, complemented
25C2×(C2×D43)⋊C4C2212yescomplemented
26C2×(C2×D43)⋊C4C2211yescharacteristic, complemented
27C2×(C2×D43)⋊C4C2212yescomplemented
28C2×(C2×D43)⋊C4C2212yescomplemented
29C2×(C2×D43)⋊C4C2212yescomplemented
30C2×(C2×D43)⋊C4C2212yescomplemented
31C2×(C2×D43)⋊C4C2212yescomplemented
32C22⋊C4×D43—24no—
33C43⋊#95C2212yescomplemented
34C23×C4×C43C2211yescharacteristic
35C2×C4⋊C4×C43C2212yescomplemented
36C2×C22⋊C4×C43C2211yescharacteristic
37C2×C22⋊C4×C43C2211yescharacteristic
38#95×C43C2212yescomplemented
39C24×C43C2311yescharacteristic
40C23×D43—22no—
41C23×D43—22no—
42C23×D43—22no—
43C23×D43—24no—
44C23×D43—24no—
45C23×D43—22no—
46C23×D43—22no—
47C23×D43—42no—
48C23×D43—42no—
49C23×D43D414yes—
50C23×D43C2311yescharacteristic
51C23×D43C2312yes—
52C23×D43C2311yescharacteristic
53C22×C4×C43—24no—
54C22×C4×C43—22no—
55C22×C4×C43C2311yescharacteristic
56C22×C4×C43C2312yes—
57C22×C4×C43C2311yescharacteristic
58C22×C4×C43C2312yes—
59C22×Dic43C2311yescharacteristic
60C22×Dic43C2312yes—
61C22×Dic43C2311yescharacteristic
62C2×D172—232no—
63C2×D172—24no—
64C2×D172—24no—
65C2×D172—24no—
66C2×D172—24no—
67C2×D172—24no—
68C2×D172D418yes—
69C2×C4×D43—22no—
70C2×C4×D43—22no—
71C2×C4×D43—24no—
72C2×C4×D43—42no—
73C4⋊C4×C43—24no—
74C22⋊C4×C43—22no—
75C22⋊C4×C43—22no—
76C22⋊Dic43—22no—
77C22⋊Dic43—22no—
78(C2×D43)⋊C4—22no—
79(C2×D43)⋊C4—22no—
80(C2×D43)⋊C4—24no—
81(C2×D43)⋊C4—24no—
82(C2×D43)⋊C4—24no—
83(C2×D43)⋊C4—42no—
84(C2×D43)⋊C4—42no—
85(C2×D43)⋊C4—42no—
86Dic43⋊C4—24no—
87C23×C43—22no—
88C23×C43—22no—
89C23×C43C2×D412yes—
90C23×C43#4112yes—
91C23×C43#4112yes—
92C23×C43C2411yescharacteristic
93C22×D43—21no—
94C22×D43—21no—
95C22×D43—22no—
96C22×D43—28no—
97C22×D43—28no—
98C22×D43—22no—
99C22×D43—22no—
100C22×D43—22no—
101C22×D43—21no—
102C22×D43—22no—
103C22×D43—21no—
104C22×D43—22no—
105C22×D43—22no—
106C22×D43—22no—
107C22×D43—22no—
108C22×D43—28no—
109C22×D43—44no—
110C22×D43—48no—
111C22×D43—48no—
112C22×D43—48no—
113C22×D43—48no—
114C22×D43—44no—
115C22×D43—44no—
116C22×D43C2×D412yes—
117C22×D43C2×D414yes—
118C22×D43C2×D414yes—
119C22×D43C2×D412yes—
120C2×C4×C43—28no—
121C2×C4×C43—22no—
122C2×C4×C43—22no—
123C2×C4×C43—22no—
124C2×C4×C43—21no—
125C2×C4×C43—21no—
126C2×C4×C43—22no—
127C2×C4×C43—22no—
128C2×C4×C43—22no—
129C2×C4×C43—22no—
130C2×C4×C43C2×D414yes—
131C2×Dic43—22no—
132C2×Dic43—22no—
133C2×Dic43—21no—
134C2×Dic43—21no—
135C2×Dic43—22no—
136C2×Dic43—22no—
137C2×Dic43—22no—
138D172—216no—
139D172—216no—
140C4×D43—44no—
141C4×D43—44no—
142C22×C43—22no—
143C22×C43—22no—
144C22×C43—24no—
145C22×C43—24no—
146C22×C43#12314yescomplemented
147C22×C43C22×D411yescharacteristic
148C22×C43C22×D412yes—
149C22×C43C2×#4111yescharacteristic
150C22×C43C2×#4111yescharacteristic
151C22×C43H(5,𝔽2)12yes—
152C2×D43—22no—
153C2×D43—22no—
154C2×D43—24no—
155C2×D43—24no—
156C2×D43—24no—
157C2×D43—22no—
158C2×D43—22no—
159C2×D43—22no—
160C2×D43—24no—
161C2×D43—24no—
162C2×D43—416no—
163C2×D43—416no—
164C2×D43—42no—
165C2×D43—42no—
166C2×D43—42no—
167C2×D43—42no—
168C2×D43—42no—
169C2×D43—42no—
170C2×D43#12014yes—
171C4×C43—24no—
172C4×C43—41no—
173C4×C43—41no—
174C4×C43—42no—
175Dic43—41no—
176Dic43—41no—
177Dic43—42no—
178#2405—431nomaximal, Sylow
179C2×C43—24no—
180C2×C43C2×#12011yescharacteristic
181C2×C43C2×#12311yescharacteristic
182C2×C43#53411yescharacteristic
183C2×C43#54612yes—
184C2×C43#54612yes—
185D43—24no—
186D43—42no—
187D43—44no—
188C23×D4—431no—
189C2×#123—432no—
190C22×C22⋊C4—431no—
191#394—432no—
192#394—432no—
193#386—432no—
194#385—431no—
195#385D4311yescharacteristic, complemented
196#379—431no—
197#379—431no—
198#379—431no—
199C43#240511yescharacteristic, complemented, Sylow
200C25—431no—
201C25—431no—
202C22×D4—432no—
203C22×D4—432no—
204C22×D4—868no—
205C22×D4—864no—
206C23×C4—431no—
207C23×C4—431no—
208C23×C4C2×D4311yescharacteristic
209#123—868no—
210C2×C4⋊C4—432no—
211C2×C4⋊C4C2×D4312yescomplemented
212C2×C22⋊C4—431no—
213C2×C22⋊C4—431no—
214C2×C22⋊C4—431no—
215C2×C22⋊C4—431no—
216C2×C22⋊C4—432no—
217C2×C22⋊C4—432no—
218C2×C22⋊C4—432no—
219C2×C22⋊C4—432no—
220C2×C22⋊C4—432no—
221C2×C22⋊C4—432no—
222C2×C22⋊C4—864no—
223C2×C22⋊C4C2×D4311yescharacteristic
224C2×C22⋊C4C2×D4311yescharacteristic
225#95—432no—
226#95C2×D4312yescomplemented
227C24—431no—
228C24—432no—
229C24—431no—
230C24—434no—
231C24—864no—
232C24—864no—
233C24—862no—
234C24—862no—
235C24—862no—
236C24—862no—
237C24—862no—
238C24—1722no—
239C24—1722no—
240C24C22×D4311yescharacteristic
241C2×D4—438no—
242C2×D4—864no—
243C2×D4—864no—
244C2×D4—864no—
245C2×D4—864no—
246C2×D4—8632no—
247C2×D4—864no—
248C22×C4—22no—
249C22×C4—24no—
250C22×C4—431no—
251C22×C4—432no—
252C22×C4—431no—
253C22×C4—862no—
254C22×C4—862no—
255C22×C4—864no—
256C22×C4—1722no—
257C22×C4C22×D4311yescharacteristic
258C22×C4C22×D4312yes—
259C22×C4C22×D4311yescharacteristic
260C22×C4C22×D4312yes—
261C4⋊C4—24no—
262C4⋊C4—864no—
263C22⋊C4—22no—
264C22⋊C4—22no—
265C22⋊C4—862no—
266C22⋊C4—862no—
267C22⋊C4—864no—
268C22⋊C4—864no—
269C22⋊C4—864no—
270C22⋊C4—862no—
271C22⋊C4—862no—
272C22⋊C4—1722no—
273C22⋊C4—1722no—
274C22⋊C4—1722no—
275C23—22no—
276C23—22no—
277C23—434no—
278C23—432no—
279C23—434no—
280C23—432no—
281C23—868no—
282C23—868no—
283C23—861no—
284C23—861no—
285C23—862no—
286C23—868no—
287C23—862no—
288C23—862no—
289C23—862no—
290C23—861no—
291C23—862no—
292C23—861no—
293C23—862no—
294C23—862no—
295C23—862no—
296C23—862no—
297C23—1724no—
298C23—1728no—
299C23—1728no—
300C23—1728no—
301C23—1728no—
302C23—1724no—
303C23—1724no—
304C23#1062189812yes—
305C23D4×D4312yes—
306C23#1062190212yes—
307C23C23×D4311yescharacteristic
308D4—8616no—
309D4—8616no—
310C2×C4—28no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—862no—
321C2×C4—861no—
322C2×C4—861no—
323C2×C4—862no—
324C2×C4—862no—
325C2×C4—862no—
326C2×C4—862no—
327C2×C4—1724no—
328C2×C4—1724no—
329C2×C4D4×D4314yes—
330C22—22no—
331C22—22no—
332C22—24no—
333C22—24no—
334C22—434no—
335C22—862no—
336C22—862no—
337C22—864no—
338C22—864no—
339C22—864no—
340C22—862no—
341C22—862no—
342C22—862no—
343C22—864no—
344C22—864no—
345C22—17216no—
346C22—17216no—
347C22—1722no—
348C22—1722no—
349C22—1722no—
350C22—1722no—
351C22—1722no—
352C22—1722no—
353C22#1306331014yescomplemented
354C22C2×D4×D4311yescharacteristic
355C22C2×D4×D4312yes—
356C22C2×#1062189811yescharacteristic
357C22C2×#1062190211yescharacteristic
358C22#1306337112yes—
359C4—24no—
360C4—41no—
361C4—41no—
362C4—42no—
363C4—1721no—
364C4—1721no—
365C4—1722no—
366C2—24no—
367C2—864no—
368C2—1722no—
369C2—1724no—
370C2#120×D4311yescharacteristic
371C2C2×#1306331011yescharacteristic
372C2#1458346011yescharacteristic
373C2#1458354012yes—
374C2#1458354512yes—
subgroupG/H
center, FrattiniC23C23×D43
commutator, socleC23×C43C24
FittingC43×#385C2
hypercenter#385D43
hypocenterC43#2405
LCSG ▷ C23×C43 ▷ C43
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
2. C2×#13063310C2semidirect
3. C2×C22⋊C4×D43C2semidirect
4. #14582817C2semidirect
5. #14582816C2semidirect
6. #14582777C2semidirect
7. #14582772C2semidirect
8. #14582754C2semidirect
9. #14582754C2semidirect
10. #14582754C2semidirect
11. C43×#385C2semidirect
18. C22×C4×D43C22semidirect
19. C22×C4×D43C22semidirect
21. C2×Dic43⋊C4C22semidirect
24. C2×(C2×D43)⋊C4C22semidirect
25. C2×(C2×D43)⋊C4C22semidirect
26. C2×(C2×D43)⋊C4C22semidirect
27. C2×(C2×D43)⋊C4C22semidirect
28. C2×(C2×D43)⋊C4C22semidirect
29. C2×(C2×D43)⋊C4C22semidirect
30. C2×(C2×D43)⋊C4C22semidirect
31. C2×(C2×D43)⋊C4C22semidirect
33. C43⋊#95C22semidirect
35. C2×C4⋊C4×C43C22semidirect
38. #95×C43C22semidirect
146. C22×C43#123semidirect
195. #385D43semidirect
199. C43#2405semidirect
211. C2×C4⋊C4C2×D43semidirect
226. #95C2×D43semidirect
353. C22#13063310semidirect