Finite Group Databasebeta

← #19143973

Subgroups — #19143973

Order 5504.

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

normal subgroups only

This group has 704 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
2C2×#13063356C212yesmaximal, complemented
3#14583081C214yesmaximal, complemented
4#14583076C214yesmaximal, complemented
5#14582917C212yesmaximal, complemented
6#14582883C211yesmaximal, characteristic, complemented
7C43×#536C211yesmaximal, characteristic, complemented
8#13063373C2212yes—
9#13063372—22no—
10#13063371—22no—
11#13063370—22no—
12#13063370C2212yes—
13C2×#10621898—22no—
14C2×#10621898C2212yes—
15#41×D43—24no—
16#13063356—24no—
17#13063356C2218yescomplemented
18C2×#10621885—22no—
19C2×#10621884—22no—
20C2×#10621877C2212yescomplemented
21#13063253C2218yescomplemented
22#13063244C2212yescomplemented
23#13063242C2212yescomplemented
24C2×#41×C43C2211yescharacteristic
25C22×Q8×C43C2211yescharacteristic, complemented
26C4⋊Q8×C43C2212yescomplemented
27#124×C43C2212yescomplemented
28#123×C43—22no—
29C22⋊Q8×C43—22no—
30#117×C43C2211yescharacteristic, complemented
31#41×C43—22no—
32#41×C43—22no—
33C2×Q8×C43—21no—
34C2×Q8×C43—22no—
35C2×Q8×C43—22no—
36C2×Q8×C43C2312yes—
37C2×Q8×C43C2311yescharacteristic
38C2×D4×C43D412yes—
39C2×D4×C43C2311yescharacteristic
40C22×C4×C43—21no—
41C22×C4×C43D412yes—
42C22×C4×C43C2311yescharacteristic
43C2×C22⋊D43—24no—
44C2×C22⋊D43—22no—
45#10621902—24no—
46Q8×D43—24no—
47#10621900—28no—
48#10621900—24no—
49D4×D43—28no—
50D4×D43—24no—
51#10621898—24no—
52#10621898—24no—
53#10621898—22no—
54#10621898—24no—
55#10621898—22no—
56#10621898D414yes—
57#10621898C2314yes—
58C2×D172—22no—
59C2×D172D412yes—
60C2×C4×D43—24no—
61C2×C4×D43—22no—
62C2×Dic86—22no—
63C2×Dic86D412yes—
64C4⋊C4×C43—21no—
65C4⋊C4×C43—24no—
66C22⋊C4×C43—21no—
67C22⋊C4×C43—24no—
68C42×C43C2312yes—
69#10621885—28no—
70#10621884—28no—
71#10621877—22no—
72#10621877C2314yes—
73C2×C43⋊C8—22no—
74C23×C43C2×D412yes—
75C23×C43C2×D411yescharacteristic
76C22×D43—24no—
77C22×D43—22no—
78Q8×C43—22no—
79Q8×C43—24no—
80Q8×C43—41no—
81Q8×C43—41no—
82D4×C43—24no—
83D4×C43—22no—
84C2×C4×C43—22no—
85C2×C4×C43—22no—
86C2×C4×C43—22no—
87C2×C4×C43—22no—
88C2×C4×C43—42no—
89C2×C4×C43C2×D412yes—
90C2×C4×C43C2×D412yes—
91C2×C4×C43C2×D412yes—
92C2×C4×C43C2×D411yescharacteristic
93C2×C4×C43C2411yescharacteristic
94C22⋊D43—28no—
95C22⋊D43—24no—
96C22⋊D43—24no—
97C2×Dic43—24no—
98C2×Dic43—22no—
99D172—24no—
100D172—22no—
101D172C2×D414yes—
102C4×D43—28no—
103C4×D43—24no—
104C4×D43—24no—
105Dic86—24no—
106Dic86—22no—
107Dic86C2×D414yes—
108C43⋊C8—24no—
109C22×C43—22no—
110C22×C43—41no—
111C22×C43—42no—
112C22×C43#12012yes—
113C22×C43C22×D411yescharacteristic
114C2×D43—24no—
115C2×D43—82no—
116C2×D43—84no—
117C4×C43—22no—
118C4×C43—42no—
119C4×C43—42no—
120C4×C43#12011yescharacteristic
121C4×C43#12011yescharacteristic
122C4×C43C22×D412yes—
123Dic43—24no—
124#3022—431nomaximal, Sylow
125C2×C43—21no—
126C2×C43—22no—
127C2×C43—42no—
128C2×C43C2×#12011yescharacteristic
129D43—44no—
130#585—431no—
131C2×#137—432no—
132#536D4311yescharacteristic, complemented
133#456—434no—
134#455—434no—
135#421—432no—
136#413—431no—
137C43#302211yescharacteristic, complemented, Sylow
1382−1+4—432no—
1392−1+4—862no—
140H(5,𝔽2)—432no—
141H(5,𝔽2)—862no—
142H(5,𝔽2)—862no—
143C2×#41—432no—
144C2×#41—864no—
145C2×#41—862no—
146C2×#41C2×D4311yescharacteristic
147C22×Q8C2×D4311yescharacteristic, complemented
148#137—438no—
149#137—864no—
150C2×Q16—862no—
151C2×SD16—862no—
152C2×M16—432no—
153C4⋊Q8C2×D4312yescomplemented
154#124C2×D4312yescomplemented
155#123—22no—
156C22⋊Q8—22no—
157#117C2×D4311yescharacteristic, complemented
158#104—438no—
159#101—432no—
160#100—432no—
161#41—22no—
162#41—22no—
163#41—434no—
164#41—434no—
165#41—864no—
166#41—864no—
167#41—864no—
168#41—868no—
169#41—864no—
170#41—862no—
171#41—862no—
172#41—864no—
173C2×Q8—21no—
174C2×Q8—22no—
175C2×Q8—22no—
176C2×Q8—432no—
177C2×Q8—862no—
178C2×Q8—864no—
179C2×Q8C22×D4312yes—
180C2×Q8C22×D4311yescharacteristic
181C2×D4—432no—
182C2×D4—862no—
183C2×D4—864no—
184C2×D4—868no—
185C2×D4—862no—
186C2×D4—864no—
187C2×D4C22⋊D4312yes—
188C2×D4C22×D4311yescharacteristic
189C22×C4—21no—
190C22×C4—864no—
191C22×C4—862no—
192C22×C4C22⋊D4312yes—
193C22×C4C22×D4311yescharacteristic
194Q16—868no—
195SD16—868no—
196M16—434no—
197M16—862no—
198C2×C8—862no—
199C4⋊C4—21no—
200C4⋊C4—24no—
201C22⋊C4—21no—
202C22⋊C4—24no—
203C42C22×D4312yes—
204C23—864no—
205C23—862no—
206C23C2×C22⋊D4312yes—
207C23C2×C22⋊D4311yescharacteristic
208Q8—22no—
209Q8—24no—
210Q8—41no—
211Q8—41no—
212Q8—434no—
213Q8—864no—
214Q8—862no—
215D4—24no—
216D4—22no—
217D4—434no—
218D4—864no—
219D4—868no—
220D4—864no—
221D4—864no—
222D4—862no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—42no—
228C2×C4—868no—
229C2×C4—864no—
230C2×C4—862no—
231C2×C4—864no—
232C2×C4—864no—
233C2×C4D4×D4312yes—
234C2×C4D4×D4312yes—
235C2×C4C2×C22⋊D4312yes—
236C2×C4C2×C22⋊D4311yescharacteristic
237C2×C4C23×D4311yescharacteristic
238C8—864no—
239C22—22no—
240C22—41no—
241C22—42no—
242C22—864no—
243C22—3442no—
244C22—3444no—
245C22#1306329312yes—
246C22C22×C22⋊D4311yescharacteristic
247C4—22no—
248C4—42no—
249C4—42no—
250C4—864no—
251C4#1306329311yescharacteristic
252C4#1306329311yescharacteristic
253C4C2×D4×D4312yes—
254C2—21no—
255C2—22no—
256C2—42no—
257C2—1724no—
258C2C2×#1306329311yescharacteristic
subgroupG/H
centerC2C2×#13063293
commutatorC2×C4×C43C24
FrattiniC2×C4C23×D43
FittingC43×#536C2
socleC2×C43C2×#120
hypercenter#536D43
hypocenterC43#3022
LCSG ▷ C2×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C2 ◁ C22×C4 ◁ #536

Complements

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

NH ≅ G/Ntype
2. C2×#13063356C2semidirect
3. #14583081C2semidirect
4. #14583076C2semidirect
5. #14582917C2semidirect
6. #14582883C2semidirect
7. C43×#536C2semidirect
17. #13063356C22semidirect
20. C2×#10621877C22semidirect
21. #13063253C22semidirect
22. #13063244C22semidirect
23. #13063242C22semidirect
25. C22×Q8×C43C22semidirect
26. C4⋊Q8×C43C22semidirect
27. #124×C43C22semidirect
30. #117×C43C22semidirect
132. #536D43semidirect
137. C43#3022semidirect
147. C22×Q8C2×D43semidirect
153. C4⋊Q8C2×D43semidirect
154. #124C2×D43semidirect
157. #117C2×D43semidirect