Finite Group Databasebeta

← #36362847

Subgroups — #36362847

Order 22144.

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

normal subgroups only

This group has 732 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×#19149726C211yesmaximal, characteristic, complemented
2C2×#19149705C211yesmaximal, characteristic, complemented
3C22×#14583814C211yesmaximal, characteristic, complemented
4#26011634C214yesmaximal, complemented
5#26011629C214yesmaximal, complemented
6#26011499C211yesmaximal, characteristic, complemented
7#26011485C211yesmaximal, characteristic, complemented
8#26011458C211yesmaximal, characteristic, complemented
9#535×C173C211yesmaximal, characteristic, complemented
10#19149726—24no—
11#19149726—24no—
12C2×#14583826C2211yescharacteristic
13C2×#14583826C2212yescomplemented
14C2×#14583825—21no—
15C2×#14583825—22no—
16C22×Dic346C2211yescharacteristic
17C22×Dic346C2212yescomplemented
18C2×Q8×D173—21no—
19#19149705—24no—
20C2×#14583814—24no—
21C2×#14583814C2214yes—
22C2×#14583814C2212yes—
23C2×#14583811C2212yes—
24C2×#14583810C2211yescharacteristic
25C22×#13063436C2211yescharacteristic
26#19149602C2212yescomplemented
27#19149602C2214yescomplemented
28#19149599C2212yescomplemented
29C22⋊#13063436C2214yescomplemented
30C2×#41×C173C2211yescharacteristic
31C22×D4×C173C2211yescharacteristic
32#127×C173—21no—
33#124×C173—21no—
34#121×C173C2214yes—
35#120×C173—22no—
36#117×C173C2211yescharacteristic
37#14583826—24no—
38#14583826—24no—
39#14583826—22no—
40#14583826D414yescomplemented
41#14583825—216no—
42#14583825—28no—
43C2×Dic346—24no—
44C2×Dic346—24no—
45C2×Dic346—21no—
46C2×Dic346—22no—
47C2×Dic346—22no—
48C2×Dic346D414yescomplemented
49C2×Dic346C2312yes—
50C2×Dic346C2311yescharacteristic
51Q8×D173—28no—
52C2×D692C2311yescharacteristic
53C2×C22⋊D173—21no—
54C2×C22⋊D173—22no—
55C22×Dic173—21no—
56C22×Dic173—22no—
57C2×C4×D173—21no—
58C2×C4×D173—22no—
59#14583814—28no—
60#14583814—24no—
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—
84Dic346—28no—
85Dic346—24no—
86Dic346—21no—
87Dic346—22no—
88Dic346C2×D412yes—
89Dic346C2×D414yes—
90D692—21no—
91D692C2×D412yes—
92C22⋊D173—28no—
93C22⋊D173—24no—
94C2×Dic173—28no—
95C2×Dic173—24no—
96C2×Dic173—21no—
97C2×Dic173—22no—
98C4×D173—28no—
99C4×D173—24no—
100#13063436—22no—
101#13063436—22no—
102C23×C173—22no—
103C23×C173—42no—
104C23×C173—82no—
105C23×C173C2×D412yes—
106C23×C173C2×D411yescharacteristic
107Q8×C173—22no—
108D4×C173—24no—
109D4×C173—22no—
110D4×C173—22no—
111D4×C173—22no—
112D4×C173—42no—
113D4×C173—84no—
114C2×C4×C173—22no—
115C2×C4×C173—22no—
116C2×C4×C173—22no—
117C2×C4×C173C2×D412yes—
118C2×C4×C173C2×D412yes—
119C2×C4×C173C2×D412yes—
120C2×C4×C173C2×D411yescharacteristic
121C2×C4×C173C2411yescharacteristic
122C22×C173—22no—
123C22×C173—22no—
124C22×C173—24no—
125C22×C173—42no—
126C22×C173—44no—
127C22×C173—84no—
128C22×C173#12012yes—
129C22×C173C22×D411yescharacteristic
130C2×D173—22no—
131C2×D173—42no—
132C4×C173—22no—
133C4×C173—42no—
134C4×C173#12012yes—
135C4×C173C22×D411yescharacteristic
136C4×C173C22×D411yescharacteristic
137Dic173—24no—
138Dic173—22no—
139C2×C173—22no—
140C2×C173—42no—
141C2×C173—82no—
142C2×C173C2×#12011yescharacteristic
143C2×C173#57712yes—
144D173—42no—
145C173#301011yescharacteristic, complemented, Sylow
146#3010—1731nomaximal, Sylow
147C2×2−1+4—1731no—
148C2×AGL(1,ℤ/8)—1731no—
149C22×SD16—1731no—
150#535D17311yescharacteristic, complemented
151#449—1734no—
152#448—1734no—
153#419—1731no—
154#416—1731no—
155#408—1731no—
1562−1+4—3464no—
1572−1+4—3464no—
158C2×#41—3462no—
159C2×#41—3461no—
160C2×#41—1732no—
161C2×#41—1731no—
162C2×#41C2×D17311yescharacteristic
163C22×Q8—3461no—
164C22×Q8—1732no—
165C22×Q8—1731no—
166C22×D4C2×D17311yescharacteristic
167AGL(1,ℤ/8)—3464no—
168C2×SD16—3464no—
169C2×SD16—1734no—
170C2×SD16—1732no—
171C2×D8—1732no—
172C2×M16—1731no—
173C22×C8—1731no—
174#127—21no—
175#124—21no—
176#121C2×D17314yes—
177#120—22no—
178#117C2×D17311yescharacteristic
179#103—1732no—
180#103—1734no—
181#102—1732no—
182C22⋊C8—1734no—
183C24—21no—
184#41—22no—
185#41—22no—
186#41—3464no—
187#41—3464no—
188#41—34616no—
189#41—3468no—
190#41—3462no—
191#41—1734no—
192C2×Q8—3464no—
193C2×Q8—3464no—
194C2×Q8—3468no—
195C2×Q8—3462no—
196C2×Q8—3462no—
197C2×Q8—3461no—
198C2×Q8—1734no—
199C2×Q8—1731no—
200C2×Q8—1732no—
201C2×Q8C22⋊D17311yescharacteristic
202C2×D4—24no—
203C2×D4—22no—
204C2×D4—22no—
205C2×D4—21no—
206C2×D4—3462no—
207C2×D4—3461no—
208C2×D4—1731no—
209C2×D4C22⋊D17312yes—
210C2×D4C22⋊D17311yescharacteristic
211C2×D4C22×D17312yes—
212C22×C4—3462no—
213C22×C4—3461no—
214C22×C4—3462no—
215C22×C4—3461no—
216C22×C4C22×D17312yes—
217C22×C4C22×D17311yescharacteristic
218SD16—3468no—
219SD16—3464no—
220D8—3464no—
221M16—3462no—
222C2×C8—3462no—
223C2×C8—1732no—
224C2×C8—1732no—
225C4⋊C4C22×D17312yes—
226C22⋊C4—21no—
227C22⋊C4—24no—
228C42—21no—
229C23—22no—
230C23—42no—
231C23—82no—
232C23—3461no—
233C23C2×C22⋊D17312yes—
234C23C2×C22⋊D17311yescharacteristic
235Q8—22no—
236Q8—3468no—
237Q8—3464no—
238Q8—3461no—
239Q8—3462no—
240Q8—1732no—
241Q8—1734no—
242D4—24no—
243D4—22no—
244D4—22no—
245D4—22no—
246D4—42no—
247D4—84no—
248D4—3468no—
249D4—3464no—
250D4—3461no—
251D4—1732no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—3468no—
256C2×C4—3468no—
257C2×C4—3464no—
258C2×C4—3462no—
259C2×C4—3461no—
260C2×C4—3464no—
261C2×C4D4×D17312yes—
262C2×C4D4×D17312yes—
263C2×C4C2×C22⋊D17312yes—
264C2×C4C2×C22⋊D17311yescharacteristic
265C2×C4C23×D17311yescharacteristic
266C8—3462no—
267C8—3462no—
268C22—22no—
269C22—22no—
270C22—24no—
271C22—42no—
272C22—44no—
273C22—84no—
274C22—3462no—
275C22—6922no—
276C22#1914964712yes—
277C22C22×C22⋊D17311yescharacteristic
278C4—22no—
279C4—42no—
280C4—3464no—
281C4—3462no—
282C4#1914964712yes—
283C4C2×D4×D17311yescharacteristic
284C4C2×D4×D17311yescharacteristic
285C2—22no—
286C2—42no—
287C2—82no—
288C2—6922no—
289C2C2×#1914964711yescharacteristic
290C2#2601230212yes—
subgroupG/H
centerC22C22×C22⋊D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#535×C173C2
socleC22×C173C22×D4
hypercenter#535D173
hypocenterC173#3010
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×#19149726C2semidirect
2. C2×#19149705C2semidirect
3. C22×#14583814C2semidirect
4. #26011634C2semidirect
5. #26011629C2semidirect
6. #26011499C2semidirect
7. #26011485C2semidirect
8. #26011458C2semidirect
9. #535×C173C2semidirect
13. C2×#14583826C22semidirect
17. C22×Dic346C22semidirect
26. #19149602C22semidirect
27. #19149602C22semidirect
28. #19149599C22semidirect
29. C22⋊#13063436C22semidirect
40. #14583826D4semidirect
48. C2×Dic346D4semidirect
145. C173#3010semidirect
150. #535D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497262
C2×#191497052
C22×#145838142
#260116342
#260116292
#260114992
#260114852
#260114582
#535×C1732
#3010173