Finite Group Databasebeta

← #19143986

Subgroups — #19143986

Order 5504.

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

normal subgroups only

This group has 766 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
1#14583735C211yesmaximal, characteristic, complemented
2#14583410C212yesmaximal, complemented
3#14583088C214yesmaximal, complemented
4#14583085C214yesmaximal, complemented
5#14582873C211yesmaximal, characteristic, complemented
6#14582872C212yesmaximal, complemented
7C43×#525C211yesmaximal, characteristic, complemented
8#13063373—21no—
9#13063372—22no—
10#13063371—22no—
11#13063370—21no—
12#13063370C2214yes—
13C2×#10621898—22no—
14C2×#10621898C2212yes—
15#41×D43—24no—
16#13063308C2214yescomplemented
17#13063304—22no—
18#13063299—22no—
19#13063294C2214yescomplemented
20C4×C22⋊D43—24no—
21#13063284C2212yescomplemented
22#13063240C2214yescomplemented
23#13063239C2218yescomplemented
24C2×#41×C43C2211yescharacteristic
25C23×C4×C43C2211yescharacteristic, complemented
26#123×C43C2212yescomplemented
27C22⋊Q8×C43—21no—
28#121×C43—21no—
29#120×C43C2212yescomplemented
30C4×D4×C43—22no—
31#117×C43C2211yescharacteristic, complemented
32C24×C43C2311yescharacteristic
33#41×C43—22no—
34#41×C43—22no—
35C2×Q8×C43D411yescharacteristic, complemented
36C2×D4×C43—22no—
37C2×D4×C43D411yescharacteristic, complemented
38C2×D4×C43C2312yes—
39C22×C4×C43—21no—
40C22×C4×C43—21no—
41C22×C4×C43—22no—
42C22×C4×C43—22no—
43C22×C4×C43D412yescomplemented
44C22×C4×C43C2311yescharacteristic
45C22×C4×C43C2311yescharacteristic
46C2×C22⋊D43—24no—
47C2×C22⋊D43—24no—
48C2×C22⋊D43C2314yes—
49#10621902—24no—
50Q8×D43—24no—
51#10621900—24no—
52#10621900—28no—
53D4×D43—24no—
54D4×D43—28no—
55#10621898—24no—
56#10621898—24no—
57#10621898—22no—
58#10621898—22no—
59#10621898—24no—
60#10621898—22no—
61#10621898—22no—
62C2×D172—22no—
63C2×D172D412yescomplemented
64C2×C4×D43—24no—
65C2×C4×D43D414yescomplemented
66C2×Dic86—22no—
67C2×Dic86D412yescomplemented
68C4⋊C4×C43—21no—
69C4⋊C4×C43—22no—
70C22⋊C4×C43—22no—
71C22⋊C4×C43—22no—
72C22⋊C4×C43C2312yes—
73C42×C43—21no—
74C22⋊Dic43—24no—
75C22⋊Dic43C2314yes—
76(C2×D43)⋊C4—24no—
77C4⋊Dic43—22no—
78Dic43⋊C4—24no—
79C4×Dic43—22no—
80C23×C43—22no—
81C23×C43—21no—
82C23×C43—42no—
83C23×C43C2×D412yes—
84C23×C43C2411yescharacteristic
85C22×D43—24no—
86C22×D43C2×D414yes—
87Q8×C43—22no—
88D4×C43—24no—
89D4×C43—22no—
90D4×C43—44no—
91C2×C4×C43—22no—
92C2×C4×C43—22no—
93C2×C4×C43—22no—
94C2×C4×C43—22no—
95C2×C4×C43—21no—
96C2×C4×C43—41no—
97C2×C4×C43—42no—
98C2×C4×C43—41no—
99C2×C4×C43C2×D412yes—
100C2×C4×C43C2×D412yes—
101C2×C4×C43C2×D412yes—
102C2×C4×C43C2×D411yescharacteristic
103C2×C4×C43C2×D411yescharacteristic
104C22⋊D43—28no—
105C22⋊D43—24no—
106C22⋊D43—24no—
107C22⋊D43—48no—
108C2×Dic43—24no—
109C2×Dic43—24no—
110C2×Dic43C2×D414yes—
111D172—24no—
112D172—22no—
113D172—22no—
114C4×D43—28no—
115C4×D43—24no—
116C4×D43—24no—
117Dic86—24no—
118Dic86—22no—
119Dic86—22no—
120C22×C43—24no—
121C22×C43—22no—
122C22×C43—22no—
123C22×C43—41no—
124C22×C43—42no—
125C22×C43—42no—
126C22×C43—41no—
127C22×C43—41no—
128C22×C43C22×D412yes—
129C22×C43C22×D411yescharacteristic
130C2×D43—24no—
131C2×D43—44no—
132C2×D43—84no—
133C4×C43—22no—
134C4×C43—42no—
135C4×C43—41no—
136C4×C43#12011yescharacteristic
137C4×C43#12012yes—
138C4×C43#12011yescharacteristic
139Dic43—24no—
140Dic43—44no—
141#3025—431nomaximal, Sylow
142C2×C43—22no—
143C2×C43—21no—
144C2×C43—42no—
145C2×C43—42no—
146C2×C43C2×#12011yescharacteristic
147D43—44no—
148#585—431no—
149#525—432no—
150#525D4311yescharacteristic, complemented
151#458—434no—
152B(4,𝔽2)—434no—
153#410—431no—
154#410—432no—
155C43#302511yescharacteristic, complemented, Sylow
1562−1+4—862no—
1572−1+4—861no—
158H(5,𝔽2)—434no—
159H(5,𝔽2)—862no—
160H(5,𝔽2)—861no—
161C2×#41—432no—
162C2×#41—864no—
163C2×#41—862no—
164C2×#41C2×D4311yescharacteristic
165C23×C4C2×D4311yescharacteristic, complemented
166#123—434no—
167#123C2×D4312yescomplemented
168C22⋊Q8—21no—
169C22⋊Q8—862no—
170#121—21no—
171#121—862no—
172#120—434no—
173#120C2×D4312yescomplemented
174C4×D4—22no—
175C4×D4—864no—
176#117—432no—
177#117C2×D4311yescharacteristic, complemented
178#99—434no—
179#99—438no—
180C24C22×D4311yescharacteristic
181#41—22no—
182#41—22no—
183#41—864no—
184#41—864no—
185#41—868no—
186#41—864no—
187#41—864no—
188#41—862no—
189#41—862no—
190#41—862no—
191#41—862no—
192#41—864no—
193C2×Q8—432no—
194C2×Q8—862no—
195C2×Q8—864no—
196C2×Q8C22⋊D4311yescharacteristic, complemented
197C2×D4—22no—
198C2×D4—434no—
199C2×D4—432no—
200C2×D4—862no—
201C2×D4—868no—
202C2×D4—864no—
203C2×D4—864no—
204C2×D4—864no—
205C2×D4C22⋊D4311yescharacteristic, complemented
206C2×D4C22×D4312yes—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—434no—
212C22×C4—864no—
213C22×C4C22⋊D4312yescomplemented
214C22×C4C22×D4311yescharacteristic
215C22×C4C22×D4311yescharacteristic
216C4⋊C4—21no—
217C4⋊C4—22no—
218C4⋊C4—864no—
219C4⋊C4—862no—
220C22⋊C4—22no—
221C22⋊C4—22no—
222C22⋊C4—434no—
223C22⋊C4—864no—
224C22⋊C4—864no—
225C22⋊C4C22×D4312yes—
226C42—21no—
227C42—862no—
228C23—22no—
229C23—21no—
230C23—42no—
231C23—434no—
232C23—864no—
233C23C2×C22⋊D4312yes—
234C23C23×D4311yescharacteristic
235Q8—22no—
236Q8—864no—
237Q8—862no—
238Q8—862no—
239D4—24no—
240D4—22no—
241D4—44no—
242D4—864no—
243D4—868no—
244D4—864no—
245D4—864no—
246D4—862no—
247D4—862no—
248D4—1728no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—41no—
255C2×C4—42no—
256C2×C4—41no—
257C2×C4—434no—
258C2×C4—868no—
259C2×C4—864no—
260C2×C4—864no—
261C2×C4—864no—
262C2×C4—864no—
263C2×C4D4×D4312yes—
264C2×C4D4×D4312yes—
265C2×C4C2×C22⋊D4312yes—
266C2×C4C2×C22⋊D4311yescharacteristic
267C2×C4C2×C22⋊D4311yescharacteristic
268C22—22no—
269C22—24no—
270C22—22no—
271C22—41no—
272C22—42no—
273C22—41no—
274C22—42no—
275C22—41no—
276C22—864no—
277C22—3444no—
278C22—1724no—
279C22C2×D4×D4312yes—
280C22C22×C22⋊D4311yescharacteristic
281C4—22no—
282C4—42no—
283C4—41no—
284C4—864no—
285C4—1724no—
286C4#1306329311yescharacteristic
287C4#1306329312yes—
288C4#1306329311yescharacteristic
289C2—21no—
290C2—22no—
291C2—42no—
292C2—42no—
293C2—1724no—
294C2C2×#1306329311yescharacteristic
subgroupG/H
centerC4#13063293
commutatorC23×C43C24
FrattiniC23C23×D43
FittingC43×#525C2
socleC2×C43C2×#120
hypercenter#525D43
hypocenterC43#3025
LCSG ▷ C23×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C4 ◁ C22×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. #14583735C2semidirect
2. #14583410C2semidirect
3. #14583088C2semidirect
4. #14583085C2semidirect
5. #14582873C2semidirect
6. #14582872C2semidirect
7. C43×#525C2semidirect
16. #13063308C22semidirect
19. #13063294C22semidirect
21. #13063284C22semidirect
22. #13063240C22semidirect
23. #13063239C22semidirect
25. C23×C4×C43C22semidirect
26. #123×C43C22semidirect
29. #120×C43C22semidirect
31. #117×C43C22semidirect
35. C2×Q8×C43D4semidirect
37. C2×D4×C43D4semidirect
43. C22×C4×C43D4semidirect
63. C2×D172D4semidirect
65. C2×C4×D43D4semidirect
67. C2×Dic86D4semidirect
150. #525D43semidirect
155. C43#3025semidirect
165. C23×C4C2×D43semidirect
167. #123C2×D43semidirect
173. #120C2×D43semidirect
177. #117C2×D43semidirect
196. C2×Q8C22⋊D43semidirect
205. C2×D4C22⋊D43semidirect
213. C22×C4C22⋊D43semidirect