Finite Group Databasebeta

← #36362829

Subgroups — #36362829

Order 22144.

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

normal subgroups only

This group has 784 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
1C2×#19149725C211yesmaximal, characteristic, complemented
2C2×#19149705C211yesmaximal, characteristic, complemented
3C22×#14583811C211yesmaximal, characteristic, complemented
4#26011634C214yesmaximal, complemented
5#26011624C214yesmaximal, complemented
6#26011495C211yesmaximal, characteristic, complemented
7#26011485C211yesmaximal, characteristic, complemented
8#26011458C211yesmaximal, characteristic, complemented
9#535×C173C211yesmaximal, characteristic, complemented
10#19149725—24no—
11#19149725—24no—
12C2×#14583827—21no—
13C2×#14583826C2211yescharacteristic
14C2×#14583826C2212yescomplemented
15C22×D692C2211yescharacteristic
16C22×D692C2212yescomplemented
17C2×D4×D173—21no—
18C2×D4×D173—22no—
19#19149705—24no—
20C2×#14583814C2212yescomplemented
21C2×#14583811—24no—
22C2×#14583811C2214yescomplemented
23C2×#14583811C2212yescomplemented
24C2×#14583810C2211yescharacteristic, complemented
25C22×#13063436C2211yescharacteristic, complemented
26#19149602C2212yescomplemented
27#19149599C2212yescomplemented
28#19149599C2214yescomplemented
29C22⋊#13063436C2214yescomplemented
30C2×#41×C173C2211yescharacteristic
31C22×D4×C173C2211yescharacteristic, complemented
32#127×C173—21no—
33#124×C173—21no—
34#121×C173C2214yescomplemented
35#120×C173—22no—
36#117×C173C2211yescharacteristic, complemented
37C23×D173—21no—
38C23×D173—22no—
39#14583827—28no—
40#14583826—24no—
41#14583826—24no—
42#14583826—22no—
43#14583826D414yescomplemented
44C2×Dic346C2311yescharacteristic
45C2×D692—24no—
46C2×D692—24no—
47C2×D692—21no—
48C2×D692—22no—
49C2×D692—22no—
50C2×D692D414yescomplemented
51C2×D692C2312yes—
52C2×D692C2311yescharacteristic
53C2×C22⋊D173—21no—
54C2×C22⋊D173—22no—
55D4×D173—216no—
56D4×D173—28no—
57C2×C4×D173—21no—
58C2×C4×D173—22no—
59#14583814—24no—
60#14583811—28no—
61#14583811—24no—
62#14583810—22no—
63C2×#13063436—22no—
64C2×#13063436C2312yes—
65C2×#13063436C2312yes—
66C24×C173—21no—
67#41×C173—22no—
68#41×C173—22no—
69C2×Q8×C173D411yescharacteristic
70C2×D4×C173—24no—
71C2×D4×C173—22no—
72C2×D4×C173—22no—
73C2×D4×C173—21no—
74C2×D4×C173D412yes—
75C2×D4×C173D411yescharacteristic
76C2×D4×C173C2312yes—
77C22×C4×C173C2312yes—
78C22×C4×C173C2311yescharacteristic
79C4⋊C4×C173C2312yes—
80C22⋊C4×C173—21no—
81C22⋊C4×C173—24no—
82C42×C173—21no—
83C22×D173—21no—
84C22×D173—22no—
85C22×D173—28no—
86C22×D173—24no—
87C22×D173—82no—
88C22×D173—84no—
89Dic346—21no—
90Dic346C2×D412yes—
91D692—28no—
92D692—24no—
93D692—22no—
94D692—21no—
95D692C2×D412yes—
96D692C2×D414yes—
97C22⋊D173—28no—
98C22⋊D173—24no—
99C2×Dic173—21no—
100C4×D173—28no—
101C4×D173—24no—
102#13063436—22no—
103#13063436—22no—
104C23×C173—22no—
105C23×C173—42no—
106C23×C173—82no—
107C23×C173C2×D412yes—
108C23×C173C2×D411yescharacteristic
109Q8×C173—22no—
110D4×C173—24no—
111D4×C173—22no—
112D4×C173—22no—
113D4×C173—22no—
114D4×C173—42no—
115D4×C173—84no—
116C2×C4×C173—22no—
117C2×C4×C173—22no—
118C2×C4×C173—22no—
119C2×C4×C173C2×D412yes—
120C2×C4×C173C2×D412yes—
121C2×C4×C173C2×D412yes—
122C2×C4×C173C2×D411yescharacteristic
123C2×C4×C173C2411yescharacteristic
124C22×C173—22no—
125C22×C173—22no—
126C22×C173—24no—
127C22×C173—42no—
128C22×C173—44no—
129C22×C173—84no—
130C22×C173#12012yes—
131C22×C173C22×D411yescharacteristic
132C2×D173—24no—
133C2×D173—22no—
134C2×D173—44no—
135C2×D173—42no—
136C2×D173—84no—
137C2×D173—88no—
138C4×C173—22no—
139C4×C173—42no—
140C4×C173#12012yes—
141C4×C173C22×D411yescharacteristic
142C4×C173C22×D411yescharacteristic
143Dic173—22no—
144C2×C173—22no—
145C2×C173—42no—
146C2×C173—82no—
147C2×C173C2×#12011yescharacteristic
148C2×C173#57612yes—
149D173—44no—
150D173—42no—
151C173#300811yescharacteristic, complemented, Sylow
152#3008—1731nomaximal, Sylow
153C2×H(5,𝔽2)—1731no—
154C2×AGL(1,ℤ/8)—1731no—
155C22×D8—1731no—
156#535D17311yescharacteristic, complemented
157#449—1734no—
158#447—1734no—
159#418—1731no—
160#416—1731no—
161#408—1731no—
162H(5,𝔽2)—3464no—
163H(5,𝔽2)—3464no—
164C2×#41—3461no—
165C2×#41—1732no—
166C2×#41—1731no—
167C2×#41C2×D17311yescharacteristic
168C22×D4—3462no—
169C22×D4—3461no—
170C22×D4—1732no—
171C22×D4—1731no—
172C22×D4C2×D17311yescharacteristic, complemented
173AGL(1,ℤ/8)—3464no—
174C2×SD16—1732no—
175C2×D8—3464no—
176C2×D8—1734no—
177C2×D8—1732no—
178C2×M16—1731no—
179C22×C8—1731no—
180#127—21no—
181#124—21no—
182#121C2×D17314yescomplemented
183#120—22no—
184#117C2×D17311yescharacteristic, complemented
185#103—1732no—
186#102—1732no—
187#102—1734no—
188C22⋊C8—1734no—
189C24—21no—
190C24—3462no—
191C24—3461no—
192#41—22no—
193#41—22no—
194#41—3464no—
195#41—3464no—
196#41—3468no—
197#41—3462no—
198#41—1734no—
199C2×Q8—1731no—
200C2×Q8C22⋊D17311yescharacteristic
201C2×D4—24no—
202C2×D4—22no—
203C2×D4—22no—
204C2×D4—21no—
205C2×D4—3464no—
206C2×D4—3464no—
207C2×D4—34616no—
208C2×D4—3468no—
209C2×D4—3462no—
210C2×D4—3461no—
211C2×D4—3462no—
212C2×D4—3462no—
213C2×D4—3461no—
214C2×D4—1734no—
215C2×D4—1731no—
216C2×D4—1732no—
217C2×D4C22⋊D17312yes—
218C2×D4C22⋊D17311yescharacteristic
219C2×D4C22×D17312yes—
220C22×C4—3462no—
221C22×C4—3461no—
222C22×C4C22×D17312yes—
223C22×C4C22×D17311yescharacteristic
224SD16—3464no—
225D8—3468no—
226D8—3464no—
227M16—3462no—
228C2×C8—3462no—
229C2×C8—1732no—
230C2×C8—1732no—
231C4⋊C4C22×D17312yes—
232C22⋊C4—21no—
233C22⋊C4—24no—
234C42—21no—
235C23—22no—
236C23—42no—
237C23—82no—
238C23—3468no—
239C23—3464no—
240C23—3462no—
241C23—3461no—
242C23—13842no—
243C23—13844no—
244C23C2×C22⋊D17312yes—
245C23C2×C22⋊D17311yescharacteristic
246Q8—22no—
247Q8—3461no—
248Q8—1732no—
249D4—24no—
250D4—22no—
251D4—22no—
252D4—22no—
253D4—42no—
254D4—84no—
255D4—3468no—
256D4—3468no—
257D4—3464no—
258D4—3464no—
259D4—3462no—
260D4—3461no—
261D4—1732no—
262D4—1734no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—3468no—
267C2×C4—3461no—
268C2×C4—3464no—
269C2×C4D4×D17312yes—
270C2×C4D4×D17312yes—
271C2×C4C2×C22⋊D17312yes—
272C2×C4C2×C22⋊D17311yescharacteristic
273C2×C4C23×D17311yescharacteristic
274C8—3462no—
275C8—3462no—
276C22—22no—
277C22—22no—
278C22—24no—
279C22—42no—
280C22—44no—
281C22—84no—
282C22—3464no—
283C22—3462no—
284C22—13844no—
285C22—13848no—
286C22—6924no—
287C22—6922no—
288C22#1914964712yes—
289C22C22×C22⋊D17311yescharacteristic
290C4—22no—
291C4—42no—
292C4—3462no—
293C4#1914964712yes—
294C4C2×D4×D17311yescharacteristic
295C4C2×D4×D17311yescharacteristic
296C2—22no—
297C2—42no—
298C2—82no—
299C2—6924no—
300C2—6922no—
301C2C2×#1914964711yescharacteristic
302C2#2601229412yes—
subgroupG/H
centerC22C22×C22⋊D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#535×C173C2
socleC22×C173C22×D4
hypercenter#535D173
hypocenterC173#3008
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C2×#41 ◁ #535

Complements

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

NH ≅ G/Ntype
1. C2×#19149725C2semidirect
2. C2×#19149705C2semidirect
3. C22×#14583811C2semidirect
4. #26011634C2semidirect
5. #26011624C2semidirect
6. #26011495C2semidirect
7. #26011485C2semidirect
8. #26011458C2semidirect
9. #535×C173C2semidirect
14. C2×#14583826C22semidirect
16. C22×D692C22semidirect
20. C2×#14583814C22semidirect
22. C2×#14583811C22semidirect
23. C2×#14583811C22semidirect
24. C2×#14583810C22semidirect
25. C22×#13063436C22semidirect
26. #19149602C22semidirect
27. #19149599C22semidirect
28. #19149599C22semidirect
29. C22⋊#13063436C22semidirect
31. C22×D4×C173C22semidirect
34. #121×C173C22semidirect
36. #117×C173C22semidirect
43. #14583826D4semidirect
50. C2×D692D4semidirect
151. C173#3008semidirect
156. #535D173semidirect
172. C22×D4C2×D173semidirect
182. #121C2×D173semidirect
184. #117C2×D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497252
C2×#191497052
C22×#145838112
#260116342
#260116242
#260114952
#260114852
#260114582
#535×C1732
#3008173