Finite Group Databasebeta

← #19143994

Subgroups — #19143994

Order 5504.

800 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 145 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×#13063370C211yesmaximal, characteristic, complemented
2#14583461C212yesmaximal, complemented
3#14583458C211yesmaximal, characteristic, complemented
4#14583090C214yesmaximal, complemented
5#14583087C214yesmaximal, complemented
6C2×#13063239C212yesmaximal, complemented
7C2×#99×C43C211yesmaximal, characteristic, complemented
8#13063370—22no—
9#13063370—24no—
10#13063370C2214yes—
11C2×#10621898—21no—
12C2×#10621900—22no—
13C22×C22⋊D43—21no—
14C22×C22⋊D43C2212yes—
15C22×C22⋊D43C2211yescharacteristic, complemented
16C2×D4×D43—22no—
17#13063309—22no—
18#13063309C2214yescomplemented
19#13063308—21no—
20#13063306C2212yescomplemented
21#13063299—24no—
22#13063297—22no—
23#13063295—22no—
24#13063295C2214yescomplemented
25#13063294—21no—
26#13063293C2212yescomplemented
27C2×C22⋊Dic43C2211yescharacteristic, complemented
28#13063239C2218yescomplemented
29C22×D4×C43C2211yescharacteristic
30C2×C22⋊C4×C43C2212yescomplemented
31#99×C43C2214yescomplemented
32C24×C43D411yescharacteristic, complemented
33C24×C43C2311yescharacteristic
34C23×D43—21no—
35C23×D43D412yescomplemented
36C23×D43C2311yescharacteristic, complemented
37C2×D4×C43—22no—
38C2×D4×C43—22no—
39C2×D4×C43D412yescomplemented
40C2×D4×C43C2312yes—
41C22×C4×C43—22no—
42C22×C4×C43D411yescharacteristic, complemented
43C2×C22⋊D43—22no—
44C2×C22⋊D43—24no—
45C2×C22⋊D43—22no—
46C2×C22⋊D43—22no—
47C2×C22⋊D43—24no—
48C2×C22⋊D43—22no—
49C2×C22⋊D43—22no—
50C2×C22⋊D43—21no—
51C2×C22⋊D43—22no—
52C2×C22⋊D43—21no—
53C2×C22⋊D43—22no—
54C2×C22⋊D43—22no—
55C2×C22⋊D43D414yescomplemented
56C2×C22⋊D43C2314yes—
57C22×Dic43—21no—
58C22×Dic43—21no—
59C22×Dic43D412yescomplemented
60C22×Dic43C2311yescharacteristic, complemented
61#10621900—28no—
62#10621900—28no—
63D4×D43—28no—
64D4×D43—28no—
65#10621898—24no—
66#10621898—24no—
67C2×D172—21no—
68C2×D172—24no—
69C2×C4×D43—22no—
70C2×Dic86—21no—
71C22⋊C4×C43—22no—
72C22⋊C4×C43C2314yescomplemented
73C22⋊Dic43—21no—
74C22⋊Dic43—22no—
75C22⋊Dic43C2312yescomplemented
76(C2×D43)⋊C4—22no—
77(C2×D43)⋊C4—24no—
78(C2×D43)⋊C4—24no—
79C4⋊Dic43—24no—
80Dic43⋊C4—22no—
81C23×C43—21no—
82C23×C43—21no—
83C23×C43—81no—
84C23×C43—81no—
85C23×C43C2×D412yes—
86C23×C43C2×D411yescharacteristic
87C23×C43C2×D411yescharacteristic
88C23×C43C2×D412yes—
89C23×C43C2×D412yes—
90C23×C43C2411yescharacteristic
91C22×D43—22no—
92C22×D43—24no—
93C22×D43—22no—
94C22×D43—22no—
95C22×D43—21no—
96C22×D43—81no—
97C22×D43—82no—
98C22×D43—82no—
99C22×D43C2×D414yes—
100D4×C43—24no—
101D4×C43—24no—
102C2×C4×C43—21no—
103C2×C4×C43—21no—
104C2×C4×C43—24no—
105C2×C4×C43—82no—
106C2×C4×C43C2×D412yes—
107C22⋊D43—28no—
108C22⋊D43—24no—
109C22⋊D43—24no—
110C22⋊D43—24no—
111C22⋊D43—24no—
112C22⋊D43—41no—
113C22⋊D43—42no—
114C22⋊D43—41no—
115C22⋊D43—82no—
116C2×Dic43—24no—
117C2×Dic43—22no—
118C2×Dic43—22no—
119C2×Dic43—21no—
120C2×Dic43—22no—
121C2×Dic43—22no—
122C2×Dic43—81no—
123C2×Dic43C2×D414yes—
124D172—24no—
125D172—84no—
126C4×D43—28no—
127Dic86—24no—
128C22×C43—22no—
129C22×C43—42no—
130C22×C43—41no—
131C22×C43—41no—
132C22×C43—41no—
133C22×C43—42no—
134C22×C43—42no—
135C22×C43—42no—
136C22×C43—81no—
137C22×C43#12011yescharacteristic
138C22×C43#12012yes—
139C22×C43#12011yescharacteristic
140C22×C43C22×D412yes—
141C22×C43C22×D411yescharacteristic
142C2×D43—24no—
143C2×D43—44no—
144C2×D43—44no—
145C2×D43—44no—
146C2×D43—42no—
147C2×D43—41no—
148C2×D43—82no—
149C2×D43—84no—
150C4×C43—22no—
151C4×C43—44no—
152Dic43—24no—
153Dic43—42no—
154Dic43—41no—
155#3026—431nomaximal, Sylow
156C2×C43—21no—
157C2×C43—22no—
158C2×C43—21no—
159C2×C43—22no—
160C2×C43—21no—
161C2×C43—42no—
162C2×C43C2×#12011yescharacteristic
163D43—44no—
164D43—81no—
165C2×H(5,𝔽2)—431no—
166#534—431no—
167#534—432no—
168#458—434no—
169B(4,𝔽2)—434no—
170C2×#99—432no—
171C2×#99D4311yescharacteristic, complemented
172C43#302611yescharacteristic, complemented, Sylow
173H(5,𝔽2)—434no—
174H(5,𝔽2)—864no—
175H(5,𝔽2)—862no—
176C2×#41—862no—
177C2×#41—861no—
178C22×D4—432no—
179C22×D4—431no—
180C22×D4—862no—
181C22×D4—861no—
182C22×D4C2×D4311yescharacteristic
183#123—432no—
184#123—434no—
185#123—862no—
186#123—861no—
187#121—864no—
188#121—862no—
189#120—432no—
190#120—434no—
191#120—862no—
192#120—861no—
193C2×C22⋊C4—431no—
194C2×C22⋊C4C2×D4312yescomplemented
195#99—438no—
196#99C2×D4314yescomplemented
197C24—432no—
198C24—431no—
199C24—861no—
200C24D17211yescharacteristic, complemented
201C24C22×D4311yescharacteristic
202#41—864no—
203#41—868no—
204#41—868no—
205#41—864no—
206C2×Q8—861no—
207C2×D4—22no—
208C2×D4—22no—
209C2×D4—434no—
210C2×D4—434no—
211C2×D4—861no—
212C2×D4—864no—
213C2×D4—868no—
214C2×D4—868no—
215C2×D4—862no—
216C2×D4—864no—
217C2×D4—862no—
218C2×D4—862no—
219C2×D4—864no—
220C2×D4—862no—
221C2×D4—862no—
222C2×D4—862no—
223C2×D4—861no—
224C2×D4—861no—
225C2×D4—862no—
226C2×D4—862no—
227C2×D4D17212yescomplemented
228C2×D4C22×D4312yes—
229C22×C4—22no—
230C22×C4—432no—
231C22×C4—431no—
232C22×C4—862no—
233C22×C4—861no—
234C22×C4—861no—
235C22×C4D17211yescharacteristic, complemented
236C4⋊C4—864no—
237C4⋊C4—862no—
238C22⋊C4—22no—
239C22⋊C4—432no—
240C22⋊C4—864no—
241C22⋊C4—864no—
242C22⋊C4—861no—
243C22⋊C4—862no—
244C22⋊C4—862no—
245C22⋊C4C22×D4314yescomplemented
246C23—21no—
247C23—21no—
248C23—81no—
249C23—81no—
250C23—434no—
251C23—864no—
252C23—862no—
253C23—862no—
254C23—862no—
255C23—861no—
256C23—3442no—
257C23—3442no—
258C23—3441no—
259C23C2×D17212yes—
260C23C2×D17211yescharacteristic
261C23C2×D17211yescharacteristic
262C23D4×D4312yes—
263C23D4×D4312yes—
264C23C23×D4311yescharacteristic
265Q8—864no—
266D4—24no—
267D4—24no—
268D4—864no—
269D4—868no—
270D4—864no—
271D4—864no—
272D4—864no—
273D4—864no—
274D4—3444no—
275D4—3442no—
276D4—1721no—
277D4—1721no—
278D4—1722no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—24no—
282C2×C4—82no—
283C2×C4—434no—
284C2×C4—868no—
285C2×C4—864no—
286C2×C4—862no—
287C2×C4—862no—
288C2×C4—862no—
289C2×C4—862no—
290C2×C4—861no—
291C2×C4—3441no—
292C2×C4C2×D17212yes—
293C22—22no—
294C22—42no—
295C22—41no—
296C22—41no—
297C22—42no—
298C22—41no—
299C22—42no—
300C22—42no—
301C22—81no—
302C22—864no—
303C22—3442no—
304C22—3444no—
305C22—1724no—
306C22—1722no—
307C22—1724no—
308C22—1724no—
309C22—1721no—
310C22#1306329511yescharacteristic
311C22#1306329511yescharacteristic
312C22#1306329512yes—
313C22C2×D4×D4312yes—
314C22C22×D17211yescharacteristic
315C4—22no—
316C4—44no—
317C4—864no—
318C4—1722no—
319C4—1721no—
320C2—22no—
321C2—21no—
322C2—21no—
323C2—21no—
324C2—22no—
325C2—42no—
326C2—3441no—
327C2—1724no—
328C2C2×#1306329511yescharacteristic
subgroupG/H
centerC2C2×#13063295
commutatorC23×C43C24
FrattiniC23C23×D43
FittingC2×#99×C43C2
socleC2×C43C2×#120
hypercenterC2×#99D43
hypocenterC43#3026
LCSG ▷ C23×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C2 ◁ C24 ◁ C2×#99

Complements

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

NH ≅ G/Ntype
1. C2×#13063370C2semidirect
2. #14583461C2semidirect
3. #14583458C2semidirect
4. #14583090C2semidirect
5. #14583087C2semidirect
6. C2×#13063239C2semidirect
7. C2×#99×C43C2semidirect
15. C22×C22⋊D43C22semidirect
18. #13063309C22semidirect
20. #13063306C22semidirect
24. #13063295C22semidirect
26. #13063293C22semidirect
27. C2×C22⋊Dic43C22semidirect
28. #13063239C22semidirect
30. C2×C22⋊C4×C43C22semidirect
31. #99×C43C22semidirect
32. C24×C43D4semidirect
35. C23×D43D4semidirect
36. C23×D43C23semidirect
39. C2×D4×C43D4semidirect
42. C22×C4×C43D4semidirect
55. C2×C22⋊D43D4semidirect
59. C22×Dic43D4semidirect
60. C22×Dic43C23semidirect
72. C22⋊C4×C43C23semidirect
75. C22⋊Dic43C23semidirect
171. C2×#99D43semidirect
172. C43#3026semidirect
194. C2×C22⋊C4C2×D43semidirect
196. #99C2×D43semidirect
200. C24D172semidirect
227. C2×D4D172semidirect
235. C22×C4D172semidirect
245. C22⋊C4C22×D43semidirect