Finite Group Databasebeta

← #36368344

Subgroups — #36368344

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
2C2×#19149725C211yesmaximal, characteristic, complemented
3#26012303C218yesmaximal, complemented
4#26012296C218yesmaximal, complemented
5#26012291C216yesmaximal, complemented
6C2×#19149707C216yesmaximal, complemented
7#568×C173C211yesmaximal, characteristic, complemented
8#19149727—21no—
9#19149726C2214yescomplemented
10#19149725—24no—
11#19149725C2218yescomplemented
12#19149725C2214yescomplemented
13#19149724—23no—
14C2×#14583827—21no—
15C2×#14583826C2213yescomplemented
16C2×#14583826C2216yescomplemented
17#41×D173—24no—
18C22×D692C2213yescomplemented
19C2×D4×D173—23no—
20#19149712C2214yescomplemented
21#19149712C22112yescomplemented
22#19149707C22112yescomplemented
23#19149707C22124yescomplemented
24#19149707C22112yescomplemented
25#19149701C22124yescomplemented
26C2×#14583815C22112yescomplemented
27C2×D1384C22112yescomplemented
28C2×#41×C173C2211yescharacteristic
29#131×C173C2218yescomplemented
30C2×M16×C173C2216yescomplemented
31C23×D173—23no—
32#14583827—24no—
33#14583827—28no—
34#14583826—26no—
35#14583826—26no—
36#14583826—26no—
37#14583826C23112yes—
38#14583826C2314yes—
39#14583826C23112yescomplemented
40#14583825—212no—
41C2×Dic346C2311yescharacteristic
42C2×Dic346C2316yescomplemented
43Q8×D173—24no—
44C2×D692—23no—
45C2×D692—26no—
46C2×D692C23112yes—
47C2×D692C2313yes—
48C2×D692C2316yescomplemented
49C2×C22⋊D173—23no—
50C2×C22⋊D173—26no—
51D4×D173—224no—
52D4×D173—212no—
53C2×C4×D173—23no—
54C2×C4×D173—26no—
55Dic692C2318yescomplemented
56#14583815C23124yescomplemented
57#14583815C2318yescomplemented
58D1384C23124yescomplemented
59#41×C173D414yes—
60#41×C173C2314yes—
61C2×Q8×C173D411yescharacteristic
62C2×D4×C173D413yes—
63C22×C4×C173C2313yes—
64M16×C173C2314yes—
65M16×C173C23112yescomplemented
66C2×C8×C173C23112yescomplemented
67C22×D173—23no—
68C22×D173—26no—
69C22×D173—212no—
70C22×D173—86no—
71Dic346—26no—
72Dic346—21no—
73Dic346C2414yes—
74Dic346C2412yes—
75D692—212no—
76D692—26no—
77D692—23no—
78D692C2414yes—
79D692C2416yes—
80C22⋊D173—212no—
81C22⋊D173—212no—
82C22⋊D173—24no—
83C2×Dic173—21no—
84C2×Dic173—26no—
85C4×D173—24no—
86C4×D173—212no—
87C4×D173—212no—
88C23×C173C2×D413yes—
89Q8×C173C2×D414yes—
90D4×C173C2×D4112yes—
91C2×C4×C173C2×D413yes—
92C2×C4×C173C2×D416yes—
93C2×C4×C173C2411yescharacteristic
94C2×C4×C173C2416yes—
95C8×C173C2418yes—
96C22×C173—43no—
97C22×C173C22×D411yescharacteristic
98C22×C173C22×D416yes—
99C2×D173—24no—
100C2×D173—26no—
101C2×D173—46no—
102C2×D173—86no—
103C2×D173—812no—
104C4×C173C22×D411yescharacteristic
105C4×C173C22×D416yes—
106C4×C173C2511yescharacteristic
107Dic173—22no—
108Dic173—24no—
109C2×C173—21no—
110C2×C173—26no—
111C2×C173C23×D411yescharacteristic
112D173—46no—
113D173—44no—
114C173#358511yescharacteristic, complemented, Sylow
115#3585—1731nomaximal, Sylow
116#585—1731no—
117C2×H(5,𝔽2)—1731no—
118#577—1738no—
119#576—1738no—
120#575—1736no—
121C2×AGL(1,ℤ/8)—1736no—
122#568D17311yescharacteristic, complemented
1232−1+4—3461no—
1242−1+4—1734no—
125H(5,𝔽2)—3464no—
126H(5,𝔽2)—3463no—
127H(5,𝔽2)—1734no—
128H(5,𝔽2)—1738no—
129C2×#41—3464no—
130C2×#41—3461no—
131C2×#41—1736no—
132C2×#41—1733no—
133C2×#41C2×D17311yescharacteristic
134C22×D4—3463no—
135C22×D4—1733no—
136#137—1734no—
137#137—17312no—
138AGL(1,ℤ/8)—17312no—
139AGL(1,ℤ/8)—17324no—
140AGL(1,ℤ/8)—17312no—
141#135—17324no—
142C2×SD16—17312no—
143C2×D8—17312no—
144#131C2×D17318yescomplemented
145C2×M16C2×D17316yescomplemented
146C24—3463no—
147#41—3468no—
148#41—3464no—
149#41—34612no—
150#41—3466no—
151#41—3466no—
152#41—3466no—
153#41—17312no—
154#41—1734no—
155#41—17312no—
156#41D69214yes—
157#41C22×D17314yes—
158C2×Q8—3464no—
159C2×Q8—1731no—
160C2×Q8—1736no—
161C2×Q8D69211yescharacteristic
162C2×D4—3463no—
163C2×D4—3466no—
164C2×D4—34624no—
165C2×D4—34612no—
166C2×D4—3463no—
167C2×D4—3466no—
168C2×D4—17312no—
169C2×D4—1733no—
170C2×D4—1736no—
171C2×D4D69213yes—
172C22×C4—3463no—
173C22×C4—3466no—
174C22×C4C22×D17313yes—
175Q16—1738no—
176SD16—17324no—
177SD16—1738no—
178D8—17324no—
179M16C22×D17314yes—
180M16C22×D173112yescomplemented
181C2×C8C22×D173112yescomplemented
182C23—3466no—
183C23—34612no—
184C23—3463no—
185C23—13846no—
186C23C2×D69213yes—
187Q8—3466no—
188Q8—3461no—
189Q8—1734no—
190Q8—1732no—
191Q8C2×D69214yes—
192D4—34612no—
193D4—3464no—
194D4—34612no—
195D4—3463no—
196D4—34612no—
197D4—3466no—
198D4—1734no—
199D4—1736no—
200D4C2×D692112yes—
201C2×C4—3461no—
202C2×C4—3466no—
203C2×C4—3464no—
204C2×C4—34612no—
205C2×C4—34612no—
206C2×C4C2×D69213yes—
207C2×C4C2×D69216yes—
208C2×C4C23×D17316yes—
209C2×C4C23×D17311yescharacteristic
210C8C23×D17318yes—
211C22—43no—
212C22—3464no—
213C22—3466no—
214C22—138412no—
215C22—13846no—
216C22—6926no—
217C22C22×D69211yescharacteristic
218C22C22×D69216yes—
219C4—3462no—
220C4—3464no—
221C4C22×D69211yescharacteristic
222C4C22×D69216yes—
223C4C24×D17311yescharacteristic
224C2—21no—
225C2—26no—
226C2—6924no—
227C2—6926no—
228C2C23×D69211yescharacteristic
subgroupG/H
centerC2C23×D692
commutator, cosocleC4×C173C25
FrattiniC4C24×D173
Fitting#568×C173C2
socleC2×C173C23×D4
hypercenter#568D173
hypocenterC173#3585
LCSG ▷ C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C2×#41 ◁ #568

Complements

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

NH ≅ G/Ntype
1. #26012327C2semidirect
2. C2×#19149725C2semidirect
3. #26012303C2semidirect
4. #26012296C2semidirect
5. #26012291C2semidirect
6. C2×#19149707C2semidirect
7. #568×C173C2semidirect
9. #19149726C22semidirect
11. #19149725C22semidirect
12. #19149725C22semidirect
15. C2×#14583826C22semidirect
16. C2×#14583826C22semidirect
18. C22×D692C22semidirect
20. #19149712C22semidirect
21. #19149712C22semidirect
22. #19149707C22semidirect
23. #19149707C22semidirect
24. #19149707C22semidirect
25. #19149701C22semidirect
26. C2×#14583815C22semidirect
27. C2×D1384C22semidirect
29. #131×C173C22semidirect
30. C2×M16×C173C22semidirect
39. #14583826C23semidirect
42. C2×Dic346C23semidirect
48. C2×D692C23semidirect
55. Dic692C23semidirect
56. #14583815C23semidirect
57. #14583815C23semidirect
58. D1384C23semidirect
65. M16×C173C23semidirect
66. C2×C8×C173C23semidirect
114. C173#3585semidirect
122. #568D173semidirect
144. #131C2×D173semidirect
145. C2×M16C2×D173semidirect
180. M16C22×D173semidirect
181. C2×C8C22×D173semidirect

Maximal subgroups

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

subgroupindex
#260123272
C2×#191497252
#260123032
#260122962
#260122912
C2×#191497072
#568×C1732
#3585173