Finite Group Databasebeta

← #36368343

Subgroups — #36368343

Order 22144.

1464 conjugacy classes of subgroups; only the 545 normal subgroups are shown.

normal subgroups only

This group has 1464 conjugacy classes of subgroups — too many to draw. Only its 545 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#26012327C211yesmaximal, characteristic, complemented
2#26012302C218yesmaximal, complemented
3#26012294C218yesmaximal, complemented
4#26012289C216yesmaximal, complemented
5C2×#19149705C216yesmaximal, complemented
6#26012258C211yesmaximal, characteristic, complemented
7C2×H(5,𝔽2)×C173C211yesmaximal, characteristic, complemented
8#19149727—21no—
9#19149726C2214yescomplemented
10#19149725C2214yescomplemented
11#19149724—23no—
12C2×#14583826C2216yescomplemented
13#41×D173—24no—
14#19149710C2214yescomplemented
15#19149709C22112yescomplemented
16#19149705C22112yescomplemented
17#19149705C22124yescomplemented
18#19149704C22112yescomplemented
19#19149699C22124yescomplemented
20C2×#14583814C22112yescomplemented
21C2×#14583811C22112yescomplemented
22#19149686C2218yescomplemented
23C2×#14583810C2216yescomplemented
24H(5,𝔽2)×C173—24no—
25H(5,𝔽2)×C173C2218yescomplemented
26C2×#41×C173—21no—
27C2×#41×C173C2211yescharacteristic
28C2×#41×C173C2213yescomplemented
29C22×D4×C173—23no—
30C22×D4×C173C2213yescomplemented
31#14583827—24no—
32#14583826—26no—
33#14583826—26no—
34#14583826C2314yes—
35#14583826C23112yescomplemented
36#14583825—212no—
37C2×Dic346C2316yescomplemented
38Q8×D173—24no—
39C2×D692C2316yescomplemented
40C2×C22⋊D173—26no—
41D4×D173—212no—
42C2×C4×D173—26no—
43#14583816C2318yescomplemented
44#14583814C23124yescomplemented
45#14583813C2318yescomplemented
46#14583811C23124yescomplemented
47#14583810C2314yes—
48#14583810C23112yescomplemented
49C2×#13063436C23112yescomplemented
50C24×C173—23no—
51#41×C173—28no—
52#41×C173—26no—
53#41×C173D414yes—
54#41×C173C23112yes—
55#41×C173C2314yes—
56C2×Q8×C173D411yescharacteristic
57C2×Q8×C173C2311yescharacteristic
58C2×D4×C173—23no—
59C2×D4×C173—224no—
60C2×D4×C173—23no—
61C2×D4×C173—26no—
62C2×D4×C173D413yes—
63C2×D4×C173C2313yes—
64C2×D4×C173C23112yes—
65C22×C4×C173—23no—
66C22×C4×C173C2313yes—
67C22×D173—26no—
68Dic346—26no—
69Dic346C2414yes—
70D692—26no—
71D692C2414yes—
72C22⋊D173—212no—
73C22⋊D173—24no—
74C2×Dic173—26no—
75C4×D173—24no—
76C4×D173—212no—
77#13063436C2418yes—
78C23×C173—23no—
79C23×C173—212no—
80C23×C173—86no—
81C23×C173C2×D413yes—
82Q8×C173—21no—
83Q8×C173C2×D414yes—
84Q8×C173C2412yes—
85D4×C173—212no—
86D4×C173—212no—
87D4×C173—23no—
88D4×C173C2×D4112yes—
89D4×C173C2416yes—
90C2×C4×C173—21no—
91C2×C4×C173—212no—
92C2×C4×C173C2×D416yes—
93C2×C4×C173C2×D413yes—
94C2×C4×C173C2416yes—
95C2×C4×C173C2411yescharacteristic
96C22×C173—26no—
97C22×C173—43no—
98C22×C173—46no—
99C22×C173—812no—
100C22×C173C22×D411yescharacteristic
101C22×C173C22×D416yes—
102C2×D173—24no—
103C2×D173—86no—
104C4×C173—22no—
105C4×C173C22×D416yes—
106C4×C173C22×D411yescharacteristic
107C4×C173C2511yescharacteristic
108Dic173—24no—
109C2×C173—26no—
110C2×C173—21no—
111C2×C173—46no—
112C2×C173C23×D411yescharacteristic
113D173—44no—
114C173#358511yescharacteristic, complemented, Sylow
115#3585—1731nomaximal, Sylow
116#585—1731no—
117C2×H(5,𝔽2)D17311yescharacteristic, complemented
118#577—1738no—
119#576—1738no—
120#575—1736no—
121C2×AGL(1,ℤ/8)—1736no—
122#568—1731no—
1232−1+4—3461no—
1242−1+4—1734no—
125H(5,𝔽2)—24no—
126H(5,𝔽2)—3463no—
127H(5,𝔽2)—1734no—
128H(5,𝔽2)C2×D17318yescomplemented
129C2×#41—21no—
130C2×#41—3464no—
131C2×#41—1736no—
132C2×#41C2×D17311yescharacteristic
133C2×#41C2×D17313yescomplemented
134C22×D4—23no—
135C22×D4C2×D17313yescomplemented
136#137—17312no—
137#137—1734no—
138AGL(1,ℤ/8)—17312no—
139AGL(1,ℤ/8)—17312no—
140AGL(1,ℤ/8)—17324no—
141#135—17324no—
142C2×SD16—17312no—
143C2×D8—17312no—
144#131—1738no—
145C2×M16—1736no—
146C24—23no—
147#41—28no—
148#41—26no—
149#41—3464no—
150#41—34612no—
151#41—3466no—
152#41—3466no—
153#41—1734no—
154#41—17312no—
155#41C22⋊D17314yes—
156#41C22×D173112yes—
157#41C22×D17314yes—
158C2×Q8—3464no—
159C2×Q8—1736no—
160C2×Q8C22⋊D17311yescharacteristic
161C2×Q8C22×D17311yescharacteristic
162C2×D4—23no—
163C2×D4—224no—
164C2×D4—23no—
165C2×D4—26no—
166C2×D4—3466no—
167C2×D4—34612no—
168C2×D4—1736no—
169C2×D4C22⋊D17313yes—
170C2×D4C22×D173112yes—
171C2×D4C22×D17313yes—
172C22×C4—23no—
173C22×C4—3466no—
174C22×C4C22×D17313yes—
175Q16—1738no—
176SD16—1738no—
177SD16—17324no—
178D8—17324no—
179M16—1734no—
180M16—17312no—
181C2×C8—17312no—
182C23—23no—
183C23—212no—
184C23—86no—
185C23—3466no—
186C23C2×C22⋊D17313yes—
187Q8—21no—
188Q8—3466no—
189Q8—1734no—
190Q8C2×C22⋊D17314yes—
191Q8C23×D17312yes—
192D4—212no—
193D4—212no—
194D4—23no—
195D4—3464no—
196D4—34612no—
197D4—3466no—
198D4—1734no—
199D4C2×C22⋊D173112yes—
200D4C23×D17316yes—
201C2×C4—21no—
202C2×C4—212no—
203C2×C4—3466no—
204C2×C4—3464no—
205C2×C4—34612no—
206C2×C4C2×C22⋊D17313yes—
207C2×C4C2×C22⋊D17316yes—
208C2×C4C23×D17316yes—
209C2×C4C23×D17311yescharacteristic
210C8—1738no—
211C22—26no—
212C22—43no—
213C22—46no—
214C22—812no—
215C22—3464no—
216C22—13846no—
217C22C22×C22⋊D17311yescharacteristic
218C22C22×C22⋊D17316yes—
219C4—22no—
220C4—3464no—
221C4C22×C22⋊D17316yes—
222C4C22×C22⋊D17311yescharacteristic
223C4C24×D17311yescharacteristic
224C2—21no—
225C2—26no—
226C2—46no—
227C2—6924no—
228C2C23×C22⋊D17311yescharacteristic
subgroupG/H
centerC2C23×C22⋊D173
commutator, cosocleC4×C173C25
FrattiniC4C24×D173
FittingC2×H(5,𝔽2)×C173C2
socleC2×C173C23×D4
hypercenterC2×H(5,𝔽2)D173
hypocenterC173#3585
LCSG ▷ C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C2×#41 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #26012327C2semidirect
2. #26012302C2semidirect
3. #26012294C2semidirect
4. #26012289C2semidirect
5. C2×#19149705C2semidirect
6. #26012258C2semidirect
7. C2×H(5,𝔽2)×C173C2semidirect
9. #19149726C22semidirect
10. #19149725C22semidirect
12. C2×#14583826C22semidirect
14. #19149710C22semidirect
15. #19149709C22semidirect
16. #19149705C22semidirect
17. #19149705C22semidirect
18. #19149704C22semidirect
19. #19149699C22semidirect
20. C2×#14583814C22semidirect
21. C2×#14583811C22semidirect
22. #19149686C22semidirect
23. C2×#14583810C22semidirect
25. H(5,𝔽2)×C173C22semidirect
28. C2×#41×C173C22semidirect
30. C22×D4×C173C22semidirect
35. #14583826C23semidirect
37. C2×Dic346C23semidirect
39. C2×D692C23semidirect
43. #14583816C23semidirect
44. #14583814C23semidirect
45. #14583813C23semidirect
46. #14583811C23semidirect
48. #14583810C23semidirect
49. C2×#13063436C23semidirect
114. C173#3585semidirect
117. C2×H(5,𝔽2)D173semidirect
128. H(5,𝔽2)C2×D173semidirect
133. C2×#41C2×D173semidirect
135. C22×D4C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260123272
#260123022
#260122942
#260122892
C2×#191497052
#260122582
C2×H(5,𝔽2)×C1732
#3585173