Finite Group Databasebeta

← C22×#17984134 #31795591

Subgroups — C22×#17984134 #31795591

Order 17536.

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×#17984134C2148yesmaximal, direct product factor
2C22×#14555511C212yesmaximal, complemented
3C22×#14555510C213yesmaximal, complemented
4C23×D548C213yesmaximal, complemented
5C22×D4×D137C216yesmaximal, complemented
6C22×#41×C137C211yesmaximal, characteristic, complemented
7C24×D137C2216yescomplemented
8#17984134C221256yesdirect product factor
9C2×#14555511C22148yescomplemented
10C2×#14555510C22172yescomplemented
11C22×Dic274C2211yescharacteristic, complemented
12C22×D548C2216yescomplemented
13C22×D548C2213yescomplemented
14C22×D548C22172yescomplemented
15C22×C22⋊D137C2216yescomplemented
16C2×D4×D137C221144yescomplemented
17C22×C4×D137C2216yescomplemented
18C2×#41×C137C22124yescomplemented
19C22×Q8×C137C2211yescharacteristic, complemented
20C22×D4×C137C2213yescomplemented
21C23×C4×C137C2213yescomplemented
22C23×D137—424no—
23C23×D137C2316yes—
24C23×D137C23172yescomplemented
25#14555511C231128yescomplemented
26#14555510C231192yescomplemented
27C2×Dic274C23112yescomplemented
28C2×D548C23172yescomplemented
29C2×D548C23136yescomplemented
30C2×D548C231192yescomplemented
31C2×C22⋊D137C23172yescomplemented
32D4×D137C231384yescomplemented
33C22×Dic137C2312yes—
34C2×C4×D137C23172yescomplemented
35C24×C137C2313yes—
36#41×C137C23164yescomplemented
37C2×Q8×C137C23112yescomplemented
38C2×D4×C137C23136yescomplemented
39C22×C4×C137C2313yes—
40C22×C4×C137C2311yescharacteristic
41C22×C4×C137C23136yescomplemented
42C22×D137—224no—
43C22×D137—4144no—
44C22×D137C24136yes—
45C22×D137C24196yescomplemented
46Dic274C24116yescomplemented
47D548C24196yescomplemented
48D548C24148yescomplemented
49C22⋊D137C24196yescomplemented
50C2×Dic137C24112yes—
51C4×D137C24196yescomplemented
52C23×C137—212no—
53C23×C137C24118yes—
54C23×C137C2411yescharacteristic
55Q8×C137C24116yescomplemented
56D4×C137C24148yescomplemented
57C2×C4×C137C2416yes—
58C2×C4×C137C24118yes—
59C2×C4×C137C24148yescomplemented
60C22×C137—236no—
61C22×C137H(5,𝔽2)14yescomplemented
62C22×C137C25112yes—
63C22×C137C2513yes—
64C2×D137—272no—
65C2×D137—496no—
66C2×D137C25124yes—
67C4×C137C2514yes—
68C4×C137C25112yes—
69Dic137C2518yes—
70C2×C137—212no—
71C2×C137C2×H(5,𝔽2)16yescomplemented
72C2×C137C2611yescharacteristic
73D137—224no—
74C137C22×H(5,𝔽2)11yescharacteristic, complemented, Sylow
75C22×H(5,𝔽2)—1371nomaximal, Sylow
76C2×H(5,𝔽2)—13748no—
77C22×#41—1372no—
78C22×#41—1373no—
79C22×#41D13711yescharacteristic, complemented
80C23×D4—1376no—
81C23×D4—1373no—
82C25—1376no—
83H(5,𝔽2)—137256no—
84C2×#41—13748no—
85C2×#41—13772no—
86C2×#41C2×D137124yescomplemented
87C22×Q8—1371no—
88C22×Q8C2×D13711yescharacteristic, complemented
89C22×D4—1376no—
90C22×D4—137144no—
91C22×D4—1376no—
92C22×D4—1373no—
93C22×D4—13772no—
94C22×D4C2×D13713yescomplemented
95C23×C4—1376no—
96C23×C4C2×D13713yescomplemented
97C24—54824no—
98C24—1376no—
99C24—13772no—
100C24C22×D13713yes—
101#41—137128no—
102#41—137192no—
103#41C22×D137164yescomplemented
104C2×Q8—13712no—
105C2×Q8C22×D137112yescomplemented
106C2×D4—13772no—
107C2×D4—137384no—
108C2×D4—13772no—
109C2×D4—13736no—
110C2×D4—137192no—
111C2×D4C22×D137136yescomplemented
112C22×C4—1372no—
113C22×C4—13772no—
114C22×C4C22×D13713yes—
115C22×C4C22×D13711yescharacteristic
116C22×C4C22×D137136yescomplemented
117C23—212no—
118C23—27424no—
119C23—548144no—
120C23—13736no—
121C23—13796no—
122C23C23×D137118yes—
123C23C23×D13711yescharacteristic
124Q8—13716no—
125Q8C23×D137116yescomplemented
126D4—13796no—
127D4—13796no—
128D4—13748no—
129D4C23×D137148yescomplemented
130C2×C4—13712no—
131C2×C4—13796no—
132C2×C4C23×D137118yes—
133C2×C4C23×D13716yes—
134C2×C4C23×D137148yescomplemented
135C22—236no—
136C22—27472no—
137C22—54896no—
138C22—13724no—
139C22#1798413414yesdirect product factor
140C22C24×D137112yes—
141C22C24×D13713yes—
142C4—1378no—
143C4C24×D137112yes—
144C4C24×D13714yes—
145C2—212no—
146C2—27424no—
147C2C2×#1798413416yesdirect product factor
148C2C25×D13711yescharacteristic
subgroupG/H
centerC23C23×D137
commutator, cosocleC2×C137C26
FrattiniC2C25×D137
FittingC22×#41×C137C2
socleC23×C137C24
hypercenterC22×#41D137
hypocenterC137C22×H(5,𝔽2)
LCSG ▷ C2×C137 ▷ C137
UCSC1 ◁ C23 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C2×#17984134C2direct (N × H)
2. C22×#14555511C2semidirect
3. C22×#14555510C2semidirect
4. C23×D548C2semidirect
5. C22×D4×D137C2semidirect
6. C22×#41×C137C2semidirect
7. C24×D137C22semidirect
8. #17984134C22direct (N × H)
9. C2×#14555511C22semidirect
10. C2×#14555510C22semidirect
11. C22×Dic274C22semidirect
12. C22×D548C22semidirect
13. C22×D548C22semidirect
14. C22×D548C22semidirect
15. C22×C22⋊D137C22semidirect
16. C2×D4×D137C22semidirect
17. C22×C4×D137C22semidirect
18. C2×#41×C137C22semidirect
19. C22×Q8×C137C22semidirect
20. C22×D4×C137C22semidirect
21. C23×C4×C137C22semidirect
24. C23×D137C23semidirect
25. #14555511C23semidirect
26. #14555510C23semidirect
27. C2×Dic274C23semidirect
28. C2×D548C23semidirect
29. C2×D548C23semidirect
30. C2×D548C23semidirect
31. C2×C22⋊D137C23semidirect
32. D4×D137C23semidirect
34. C2×C4×D137C23semidirect
36. #41×C137C23semidirect
37. C2×Q8×C137C23semidirect
38. C2×D4×C137C23semidirect
41. C22×C4×C137C23semidirect
45. C22×D137C24semidirect
46. Dic274C24semidirect
47. D548C24semidirect
48. D548C24semidirect
49. C22⋊D137C24semidirect
51. C4×D137C24semidirect
55. Q8×C137C24semidirect
56. D4×C137C24semidirect
59. C2×C4×C137C24semidirect
61. C22×C137H(5,𝔽2)semidirect
71. C2×C137C2×H(5,𝔽2)semidirect
74. C137C22×H(5,𝔽2)semidirect
79. C22×#41D137semidirect
86. C2×#41C2×D137semidirect
88. C22×Q8C2×D137semidirect
94. C22×D4C2×D137semidirect
96. C23×C4C2×D137semidirect
103. #41C22×D137semidirect
105. C2×Q8C22×D137semidirect
111. C2×D4C22×D137semidirect
116. C22×C4C22×D137semidirect
125. Q8C23×D137semidirect
129. D4C23×D137semidirect
134. C2×C4C23×D137semidirect
139. C22#17984134direct (N × H)
147. C2C2×#17984134direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#179841342
C22×#145555112
C22×#145555102
C23×D5482
C22×D4×D1372
C22×#41×C1372
C22×H(5,𝔽2)137