Finite Group Databasebeta

← C2×#26012296 #36368316

Subgroups — C2×#26012296 #36368316

Order 22144.

1512 conjugacy classes of subgroups; only the 549 normal subgroups are shown.

normal subgroups only

This group has 1512 conjugacy classes of subgroups — too many to draw. Only its 549 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×#19149725C212yesmaximal, complemented
2#26012296C2116yesmaximal, direct product factor
3C2×#19149707C216yesmaximal, complemented
4C2×#19149701C213yesmaximal, complemented
5C22×D1384C213yesmaximal, complemented
6C2×#131×C173C211yesmaximal, characteristic, complemented
7#19149725—28no—
8#19149725C22116yescomplemented
9C2×#14583827—22no—
10C2×#14583826C2216yescomplemented
11C22×D692C2216yescomplemented
12C2×D4×D173—26no—
13#19149707C22148yescomplemented
14#19149701C22124yescomplemented
15C2×Dic692C2211yescharacteristic, complemented
16C2×#14583815C2216yescomplemented
17C2×D1384C2213yescomplemented
18C2×D1384C2216yescomplemented
19C2×D1384C22124yescomplemented
20C2×#41×C173C2211yescharacteristic
21#131×C173C2218yescomplemented
22C2×M16×C173C2213yescomplemented
23C22×C8×C173C2213yescomplemented
24C23×D173—26no—
25#14583827—216no—
26#14583826—212no—
27#14583826C23124yescomplemented
28C2×Dic346C2312yes—
29C2×D692—26no—
30C2×D692—212no—
31C2×D692C2316yes—
32C2×D692C23124yescomplemented
33C2×C22⋊D173—26no—
34D4×D173—248no—
35C2×C4×D173—26no—
36Dic692C2314yescomplemented
37#14583815C23124yescomplemented
38D1384C23112yescomplemented
39D1384C23124yescomplemented
40#41×C173D414yes—
41#41×C173C2314yes—
42C2×Q8×C173D411yescharacteristic
43C2×D4×C173D413yes—
44C22×C4×C173C2313yes—
45M16×C173C23112yescomplemented
46C2×C8×C173C2311yescharacteristic
47C2×C8×C173C2313yes—
48C2×C8×C173C23112yescomplemented
49C22×D173—26no—
50C22×D173—224no—
51C22×D173—812no—
52Dic346—22no—
53Dic346C2414yes—
54D692—224no—
55D692—26no—
56D692C24112yes—
57C22⋊D173—224no—
58C2×Dic173—22no—
59C4×D173—224no—
60C23×C173C2×D413yes—
61Q8×C173C2×D414yes—
62D4×C173C2×D4112yes—
63C2×C4×C173C2×D413yes—
64C2×C4×C173C2×D416yes—
65C2×C4×C173C2411yescharacteristic
66C2×C4×C173C2416yes—
67C8×C173C2412yes—
68C8×C173C2416yes—
69C22×C173—26no—
70C22×C173C22×D416yes—
71C22×C173C22×D411yescharacteristic
72C2×D173—212no—
73C2×D173—412no—
74C2×D173—824no—
75C4×C173C22×D411yescharacteristic
76C4×C173C22×D416yes—
77C4×C173C2511yescharacteristic
78Dic173—24no—
79C2×C173—26no—
80C2×C173#57612yescomplemented
81C2×C173C23×D411yescharacteristic
82D173—412no—
83C173C2×#57611yescharacteristic, complemented, Sylow
84C2×#576—1731nomaximal, Sylow
85C2×H(5,𝔽2)—1732no—
86#576—17316no—
87C2×AGL(1,ℤ/8)—1736no—
88C2×#135—1733no—
89C22×D8—1733no—
90C2×#131D17311yescharacteristic, complemented
91H(5,𝔽2)—3468no—
92H(5,𝔽2)—17316no—
93C2×#41—3462no—
94C2×#41—1736no—
95C2×#41C2×D17311yescharacteristic
96C22×D4—3466no—
97C22×D4—1736no—
98AGL(1,ℤ/8)—17348no—
99#135—17324no—
100C2×Q16—1731no—
101C2×SD16—1736no—
102C2×D8—1733no—
103C2×D8—1736no—
104C2×D8—17324no—
105#131C2×D17318yescomplemented
106C2×M16C2×D17313yescomplemented
107C22×C8C2×D17313yescomplemented
108C24—3466no—
109#41—34616no—
110#41—34612no—
111#41—17324no—
112#41D69214yes—
113#41C22×D17314yes—
114C2×Q8—1732no—
115C2×Q8D69211yescharacteristic
116C2×D4—3466no—
117C2×D4—34648no—
118C2×D4—3466no—
119C2×D4—34612no—
120C2×D4—1736no—
121C2×D4—17324no—
122C2×D4D69213yes—
123C22×C4—3466no—
124C22×C4C22×D17313yes—
125Q16—1734no—
126SD16—17324no—
127D8—17312no—
128D8—17324no—
129M16C22×D173112yescomplemented
130C2×C8C22×D17311yescharacteristic
131C2×C8C22×D17313yes—
132C2×C8C22×D173112yescomplemented
133C23—34624no—
134C23—3466no—
135C23—138412no—
136C23C2×D69213yes—
137Q8—3462no—
138Q8—1734no—
139Q8C2×D69214yes—
140D4—34624no—
141D4—34624no—
142D4—3466no—
143D4—17312no—
144D4C2×D692112yes—
145C2×C4—3462no—
146C2×C4—34624no—
147C2×C4C2×D69216yes—
148C2×C4C2×D69213yes—
149C2×C4C23×D17316yes—
150C2×C4C23×D17311yescharacteristic
151C8C23×D17312yes—
152C8C23×D17316yes—
153C22—26no—
154C22—34612no—
155C22—138424no—
156C22—69212no—
157C22C22×D69216yes—
158C22C22×D69211yescharacteristic
159C4—3464no—
160C4C22×D69216yes—
161C4C22×D69211yescharacteristic
162C4C24×D17311yescharacteristic
163C2—26no—
164C2—69212no—
165C2#2601229612yesdirect product factor
166C2C23×D69211yescharacteristic
subgroupG/H
centerC22C22×D692
commutator, cosocleC4×C173C25
FrattiniC4C24×D173
FittingC2×#131×C173C2
socleC22×C173C22×D4
hypercenterC2×#131D173
hypocenterC173C2×#576
LCSG ▷ C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C2×#41 ◁ C2×#131

Complements

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

NH ≅ G/Ntype
1. C2×#19149725C2semidirect
2. #26012296C2direct (N × H)
3. C2×#19149707C2semidirect
4. C2×#19149701C2semidirect
5. C22×D1384C2semidirect
6. C2×#131×C173C2semidirect
8. #19149725C22semidirect
10. C2×#14583826C22semidirect
11. C22×D692C22semidirect
13. #19149707C22semidirect
14. #19149701C22semidirect
15. C2×Dic692C22semidirect
16. C2×#14583815C22semidirect
17. C2×D1384C22semidirect
18. C2×D1384C22semidirect
19. C2×D1384C22semidirect
21. #131×C173C22semidirect
22. C2×M16×C173C22semidirect
23. C22×C8×C173C22semidirect
27. #14583826C23semidirect
32. C2×D692C23semidirect
36. Dic692C23semidirect
37. #14583815C23semidirect
38. D1384C23semidirect
39. D1384C23semidirect
45. M16×C173C23semidirect
48. C2×C8×C173C23semidirect
80. C2×C173#576semidirect
83. C173C2×#576semidirect
90. C2×#131D173semidirect
105. #131C2×D173semidirect
106. C2×M16C2×D173semidirect
107. C22×C8C2×D173semidirect
129. M16C22×D173semidirect
132. C2×C8C22×D173semidirect
165. C2#26012296direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#191497252
#260122962
C2×#191497072
C2×#191497012
C22×D13842
C2×#131×C1732
C2×#576173