Finite Group Databasebeta

← #36362809

Subgroups — #36362809

Order 22144.

758 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 758 conjugacy classes of subgroups — too many to draw. Only its 149 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×#14583827C211yesmaximal, characteristic, complemented
2C2×#19149710C211yesmaximal, characteristic, complemented
3C2×#19149707C211yesmaximal, characteristic, complemented
4#26012003C211yesmaximal, characteristic, complemented
5#26011656C214yesmaximal, complemented
6#26011639C214yesmaximal, complemented
7#26011495C211yesmaximal, characteristic, complemented
8#26011454C211yesmaximal, characteristic, complemented
9#419×C173C211yesmaximal, characteristic, complemented
10C2×#14583827—24no—
11C2×#14583827—24no—
12C2×#14583827—22no—
13C2×#14583827C2214yes—
14C2×#14583826C2211yescharacteristic
15C22×D692—21no—
16C22×D692C2211yescharacteristic
17C22×C4×D173—21no—
18C22×C4×D173C2211yescharacteristic
19#19149710—24no—
20#19149707—24no—
21C2×#14583816C2212yescomplemented
22C2×#14583815C2212yescomplemented
23C2×#14583813C2212yescomplemented
24C2×D1384C2212yescomplemented
25C2×#14583810C2211yescharacteristic
26#19149663—21no—
27#19149659C2212yescomplemented
28#19149654C2212yescomplemented
29#19149649—21no—
30C4×D692—22no—
31#19149599C2214yescomplemented
32#19149591C2214yescomplemented
33C22×Q8×C173C2211yescharacteristic
34C2×M16×C173C2211yescharacteristic
35#117×C173C2211yescharacteristic
36#103×C173C2214yescomplemented
37C23×D173—21no—
38C23×D173D411yescharacteristic
39#14583827—216no—
40#14583827—28no—
41#14583827—28no—
42#14583826—22no—
43#14583826—22no—
44C2×Dic346C2311yescharacteristic
45C2×D692—28no—
46C2×D692—24no—
47C2×D692—22no—
48C2×D692—21no—
49C2×D692—22no—
50C2×D692D414yescomplemented
51C2×D692C2311yescharacteristic
52C2×D692C2312yes—
53C2×C22⋊D173—21no—
54C22×Dic173D411yescharacteristic
55C2×C4×D173—28no—
56C2×C4×D173—24no—
57C2×C4×D173—22no—
58C2×C4×D173—22no—
59C2×C4×D173—21no—
60C2×C4×D173D412yes—
61C2×C4×D173C2312yes—
62#14583816—24no—
63#14583815—24no—
64#14583813—24no—
65D1384—24no—
66#14583810—22no—
67C2×#13063436C2312yes—
68C4⋊Dic173—22no—
69#14583803—22no—
70#14583803—22no—
71C2×Q8×C173—22no—
72C2×Q8×C173—21no—
73C2×Q8×C173D414yescomplemented
74C2×Q8×C173C2312yes—
75C22×C4×C173—21no—
76C22×C4×C173C2311yescharacteristic
77M16×C173—22no—
78C2×C8×C173C2312yes—
79C4⋊C4×C173C2312yes—
80C22⋊C4×C173—21no—
81C42×C173—21no—
82C22×D173—22no—
83C22×D173—24no—
84C22×D173—22no—
85C22×D173—21no—
86C22×D173—42no—
87C22×D173—44no—
88C22×D173C2×D412yes—
89Dic346—21no—
90Dic346—21no—
91D692—216no—
92D692—24no—
93D692—22no—
94D692—21no—
95D692—21no—
96D692—44no—
97D692C2×D414yes—
98C22⋊D173—42no—
99C2×Dic173—22no—
100C2×Dic173—21no—
101C2×Dic173C2×D412yes—
102C4×D173—216no—
103C4×D173—24no—
104C4×D173—22no—
105C4×D173—22no—
106C4×D173—42no—
107#13063436—22no—
108C23×C173C2×D411yescharacteristic
109Q8×C173—24no—
110Q8×C173—22no—
111Q8×C173C2×D414yes—
112C2×C4×C173—24no—
113C2×C4×C173—22no—
114C2×C4×C173—22no—
115C2×C4×C173C2×D411yescharacteristic
116C2×C4×C173C2×D412yes—
117C2×C4×C173C2×D412yes—
118C2×C4×C173C2411yescharacteristic
119C8×C173—22no—
120C22×C173—22no—
121C22×C173#12012yes—
122C22×C173C22×D411yescharacteristic
123C2×D173—22no—
124C2×D173—24no—
125C2×D173—22no—
126C2×D173—22no—
127C2×D173—44no—
128C2×D173—48no—
129C2×D173—44no—
130C2×D173—41no—
131C2×D173—42no—
132C4×C173—24no—
133C4×C173—42no—
134C4×C173#12012yes—
135C4×C173C22×D411yescharacteristic
136C4×C173C22×D411yescharacteristic
137Dic173—22no—
138Dic173—41no—
139C2×C173—22no—
140C2×C173C2×#12011yescharacteristic
141C2×C173#57512yes—
142D173—42no—
143D173—44no—
144D173—81no—
145C173#300511yescharacteristic, complemented, Sylow
146#3005—1731nomaximal, Sylow
147C22×#41—1731no—
148C2×#137—1731no—
149C2×AGL(1,ℤ/8)—1731no—
150#525—1731no—
151#452—1734no—
152#449—1734no—
153#419D17311yescharacteristic, complemented
154#418—1731no—
155#407—1731no—
156C2×#41—3464no—
157C2×#41—3464no—
158C2×#41—3462no—
159C2×#41—1734no—
160C2×#41—1731no—
161C22×Q8C2×D17311yescharacteristic
162C22×D4—3461no—
163C22×D4—1731no—
164C23×C4—3461no—
165C23×C4—1731no—
166#137—3464no—
167AGL(1,ℤ/8)—3464no—
168C2×Q16—1732no—
169C2×SD16—1732no—
170C2×SD16—1732no—
171C2×D8—1732no—
172C2×M16—1731no—
173C2×M16C2×D17311yescharacteristic
174#123—3461no—
175C22⋊Q8—1732no—
176#121—1732no—
177#120—3461no—
178C4×D4—3462no—
179#117C2×D17311yescharacteristic
180#103C2×D17314yescomplemented
181#102—1734no—
182C22⋊C8—1734no—
183C24—3461no—
184C24—1731no—
185#41—34616no—
186#41—3468no—
187#41—3468no—
188#41—3462no—
189#41—3462no—
190C2×Q8—22no—
191C2×Q8—21no—
192C2×Q8—1731no—
193C2×Q8D69214yescomplemented
194C2×Q8C22×D17312yes—
195C2×D4—3468no—
196C2×D4—3464no—
197C2×D4—3462no—
198C2×D4—3461no—
199C2×D4—3461no—
200C2×D4—3462no—
201C2×D4—1734no—
202C2×D4—1731no—
203C2×D4—1732no—
204C22×C4—21no—
205C22×C4—3468no—
206C22×C4—3464no—
207C22×C4—3462no—
208C22×C4—3462no—
209C22×C4—3461no—
210C22×C4—1731no—
211C22×C4—1732no—
212C22×C4—1732no—
213C22×C4C22×D17311yescharacteristic
214Q16—3464no—
215SD16—3464no—
216SD16—3464no—
217D8—3464no—
218M16—22no—
219M16—3462no—
220C2×C8—1732no—
221C2×C8C22×D17312yes—
222C4⋊C4—3462no—
223C4⋊C4C22×D17312yes—
224C22⋊C4—21no—
225C22⋊C4—3462no—
226C22⋊C4—3462no—
227C42—21no—
228C23—3462no—
229C23—3464no—
230C23—3462no—
231C23—3461no—
232C23—1732no—
233C23—6922no—
234C23—6924no—
235C23D4×D17311yescharacteristic
236Q8—24no—
237Q8—22no—
238Q8—3461no—
239Q8—3461no—
240Q8C2×D69214yes—
241D4—34616no—
242D4—3464no—
243D4—3461no—
244D4—3462no—
245D4—3461no—
246D4—1734no—
247D4—6924no—
248D4—6922no—
249C2×C4—24no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—34616no—
253C2×C4—3462no—
254C2×C4—3464no—
255C2×C4—3462no—
256C2×C4—3462no—
257C2×C4—3461no—
258C2×C4—1732no—
259C2×C4—6922no—
260C2×C4D4×D17311yescharacteristic
261C2×C4D4×D17312yes—
262C2×C4C2×D69212yes—
263C2×C4C23×D17311yescharacteristic
264C8—22no—
265C8—3462no—
266C22—22no—
267C22—3462no—
268C22—3464no—
269C22—3462no—
270C22—3462no—
271C22—6924no—
272C22—6928no—
273C22—6921no—
274C22—6922no—
275C22—6924no—
276C22#1914964912yes—
277C22C2×D4×D17311yescharacteristic
278C4—24no—
279C4—42no—
280C4—3462no—
281C4—6921no—
282C4#1914964912yes—
283C4C2×D4×D17311yescharacteristic
284C4C22×D69211yescharacteristic
285C2—22no—
286C2—13841no—
287C2—6922no—
288C2—6924no—
289C2C2×#1914964911yescharacteristic
290C2#2601229012yes—
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#419×C173C2
socleC22×C173C22×D4
hypercenter#419D173
hypocenterC173#3005
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
1. C22×#14583827C2semidirect
2. C2×#19149710C2semidirect
3. C2×#19149707C2semidirect
4. #26012003C2semidirect
5. #26011656C2semidirect
6. #26011639C2semidirect
7. #26011495C2semidirect
8. #26011454C2semidirect
9. #419×C173C2semidirect
21. C2×#14583816C22semidirect
22. C2×#14583815C22semidirect
23. C2×#14583813C22semidirect
24. C2×D1384C22semidirect
27. #19149659C22semidirect
28. #19149654C22semidirect
31. #19149599C22semidirect
32. #19149591C22semidirect
36. #103×C173C22semidirect
50. C2×D692D4semidirect
73. C2×Q8×C173D4semidirect
145. C173#3005semidirect
153. #419D173semidirect
180. #103C2×D173semidirect
193. C2×Q8D692semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C22×#145838272
C2×#191497102
C2×#191497072
#260120032
#260116562
#260116392
#260114952
#260114542
#419×C1732
#3005173