Finite Group Databasebeta

← C22×#19149725 #36368365

Subgroups — C22×#19149725 #36368365

Order 22144.

6560 conjugacy classes of subgroups; only the 3221 normal subgroups are shown.

normal subgroups only

This group has 6560 conjugacy classes of subgroups — too many to draw. Only its 3221 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×#19149725C2148yesmaximal, direct product factor
2C22×#14583827C212yesmaximal, complemented
3C22×#14583826C213yesmaximal, complemented
4C23×D692C213yesmaximal, complemented
5C22×D4×D173C216yesmaximal, complemented
6C22×#41×C173C211yesmaximal, characteristic, complemented
7C24×D173C2216yescomplemented
8#19149725C221256yesdirect product factor
9C2×#14583827C22148yescomplemented
10C2×#14583826C22172yescomplemented
11C22×Dic346C2211yescharacteristic, complemented
12C22×D692C2216yescomplemented
13C22×D692C2213yescomplemented
14C22×D692C22172yescomplemented
15C22×C22⋊D173C2216yescomplemented
16C2×D4×D173C221144yescomplemented
17C22×C4×D173C2216yescomplemented
18C2×#41×C173C22124yescomplemented
19C22×Q8×C173C2211yescharacteristic, complemented
20C22×D4×C173C2213yescomplemented
21C23×C4×C173C2213yescomplemented
22C23×D173—424no—
23C23×D173C2316yes—
24C23×D173C23172yescomplemented
25#14583827C231128yescomplemented
26#14583826C231192yescomplemented
27C2×Dic346C23112yescomplemented
28C2×D692C23172yescomplemented
29C2×D692C23136yescomplemented
30C2×D692C231192yescomplemented
31C2×C22⋊D173C23172yescomplemented
32D4×D173C231384yescomplemented
33C22×Dic173C2312yes—
34C2×C4×D173C23172yescomplemented
35C24×C173C2313yes—
36#41×C173C23164yescomplemented
37C2×Q8×C173C23112yescomplemented
38C2×D4×C173C23136yescomplemented
39C22×C4×C173C2311yescharacteristic
40C22×C4×C173C2313yes—
41C22×C4×C173C23136yescomplemented
42C22×D173—224no—
43C22×D173—4144no—
44C22×D173C24136yes—
45C22×D173C24196yescomplemented
46Dic346C24116yescomplemented
47D692C24196yescomplemented
48D692C24148yescomplemented
49C22⋊D173C24196yescomplemented
50C2×Dic173C24112yes—
51C4×D173C24196yescomplemented
52C23×C173—212no—
53C23×C173C24118yes—
54C23×C173C2411yescharacteristic
55Q8×C173C24116yescomplemented
56D4×C173C24148yescomplemented
57C2×C4×C173C24118yes—
58C2×C4×C173C2416yes—
59C2×C4×C173C24148yescomplemented
60C22×C173—236no—
61C22×C173H(5,𝔽2)14yescomplemented
62C22×C173C25112yes—
63C22×C173C2513yes—
64C2×D173—272no—
65C2×D173—496no—
66C2×D173C25124yes—
67C4×C173C2514yes—
68C4×C173C25112yes—
69Dic173C2518yes—
70C2×C173—212no—
71C2×C173C2×H(5,𝔽2)16yescomplemented
72C2×C173C2611yescharacteristic
73D173—224no—
74C173C22×H(5,𝔽2)11yescharacteristic, complemented, Sylow
75C22×H(5,𝔽2)—1731nomaximal, Sylow
76C2×H(5,𝔽2)—17348no—
77C22×#41—1732no—
78C22×#41—1733no—
79C22×#41D17311yescharacteristic, complemented
80C23×D4—1736no—
81C23×D4—1733no—
82C25—1736no—
83H(5,𝔽2)—173256no—
84C2×#41—17348no—
85C2×#41—17372no—
86C2×#41C2×D173124yescomplemented
87C22×Q8—1731no—
88C22×Q8C2×D17311yescharacteristic, complemented
89C22×D4—1736no—
90C22×D4—173144no—
91C22×D4—1736no—
92C22×D4—1733no—
93C22×D4—17372no—
94C22×D4C2×D17313yescomplemented
95C23×C4—1736no—
96C23×C4C2×D17313yescomplemented
97C24—1736no—
98C24—17372no—
99C24—69224no—
100C24C22×D17313yes—
101#41—173128no—
102#41—173192no—
103#41C22×D173164yescomplemented
104C2×Q8—17312no—
105C2×Q8C22×D173112yescomplemented
106C2×D4—17372no—
107C2×D4—173384no—
108C2×D4—17372no—
109C2×D4—17336no—
110C2×D4—173192no—
111C2×D4C22×D173136yescomplemented
112C22×C4—1732no—
113C22×C4—17372no—
114C22×C4C22×D17313yes—
115C22×C4C22×D17311yescharacteristic
116C22×C4C22×D173136yescomplemented
117C23—212no—
118C23—34624no—
119C23—17336no—
120C23—17396no—
121C23—692144no—
122C23C23×D173118yes—
123C23C23×D17311yescharacteristic
124Q8—17316no—
125Q8C23×D173116yescomplemented
126D4—17396no—
127D4—17396no—
128D4—17348no—
129D4C23×D173148yescomplemented
130C2×C4—17312no—
131C2×C4—17396no—
132C2×C4C23×D173118yes—
133C2×C4C23×D17316yes—
134C2×C4C23×D173148yescomplemented
135C22—236no—
136C22—34672no—
137C22—17324no—
138C22—69296no—
139C22#1914972514yesdirect product factor
140C22C24×D173112yes—
141C22C24×D17313yes—
142C4—1738no—
143C4C24×D173112yes—
144C4C24×D17314yes—
145C2—212no—
146C2—34624no—
147C2C2×#1914972516yesdirect product factor
148C2C25×D17311yescharacteristic
subgroupG/H
centerC23C23×D173
commutator, cosocleC2×C173C26
FrattiniC2C25×D173
FittingC22×#41×C173C2
socleC23×C173C24
hypercenterC22×#41D173
hypocenterC173C22×H(5,𝔽2)
LCSG ▷ C2×C173 ▷ C173
UCSC1 ◁ C23 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C2×#19149725C2direct (N × H)
2. C22×#14583827C2semidirect
3. C22×#14583826C2semidirect
4. C23×D692C2semidirect
5. C22×D4×D173C2semidirect
6. C22×#41×C173C2semidirect
7. C24×D173C22semidirect
8. #19149725C22direct (N × H)
9. C2×#14583827C22semidirect
10. C2×#14583826C22semidirect
11. C22×Dic346C22semidirect
12. C22×D692C22semidirect
13. C22×D692C22semidirect
14. C22×D692C22semidirect
15. C22×C22⋊D173C22semidirect
16. C2×D4×D173C22semidirect
17. C22×C4×D173C22semidirect
18. C2×#41×C173C22semidirect
19. C22×Q8×C173C22semidirect
20. C22×D4×C173C22semidirect
21. C23×C4×C173C22semidirect
24. C23×D173C23semidirect
25. #14583827C23semidirect
26. #14583826C23semidirect
27. C2×Dic346C23semidirect
28. C2×D692C23semidirect
29. C2×D692C23semidirect
30. C2×D692C23semidirect
31. C2×C22⋊D173C23semidirect
32. D4×D173C23semidirect
34. C2×C4×D173C23semidirect
36. #41×C173C23semidirect
37. C2×Q8×C173C23semidirect
38. C2×D4×C173C23semidirect
41. C22×C4×C173C23semidirect
45. C22×D173C24semidirect
46. Dic346C24semidirect
47. D692C24semidirect
48. D692C24semidirect
49. C22⋊D173C24semidirect
51. C4×D173C24semidirect
55. Q8×C173C24semidirect
56. D4×C173C24semidirect
59. C2×C4×C173C24semidirect
61. C22×C173H(5,𝔽2)semidirect
71. C2×C173C2×H(5,𝔽2)semidirect
74. C173C22×H(5,𝔽2)semidirect
79. C22×#41D173semidirect
86. C2×#41C2×D173semidirect
88. C22×Q8C2×D173semidirect
94. C22×D4C2×D173semidirect
96. C23×C4C2×D173semidirect
103. #41C22×D173semidirect
105. C2×Q8C22×D173semidirect
111. C2×D4C22×D173semidirect
116. C22×C4C22×D173semidirect
125. Q8C23×D173semidirect
129. D4C23×D173semidirect
134. C2×C4C23×D173semidirect
139. C22#19149725direct (N × H)
147. C2C2×#19149725direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#191497252
C22×#145838272
C22×#145838262
C23×D6922
C22×D4×D1732
C22×#41×C1732
C22×H(5,𝔽2)173