Finite Group Databasebeta

← C22×#14581838 #24922265

Subgroups — C22×#14581838 #24922265

Order 10624.

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×#14581838C2148yesmaximal, direct product factor
2C22×#13050591C212yesmaximal, complemented
3C22×#13050590C213yesmaximal, complemented
4C23×D332C213yesmaximal, complemented
5C22×D4×D83C216yesmaximal, complemented
6C22×#41×C83C211yesmaximal, characteristic, complemented
7C24×D83C2216yescomplemented
8#14581838C221256yesdirect product factor
9C2×#13050591C22148yescomplemented
10C2×#13050590C22172yescomplemented
11C22×Dic166C2211yescharacteristic, complemented
12C22×D332C2216yescomplemented
13C22×D332C2213yescomplemented
14C22×D332C22172yescomplemented
15C22×C22⋊D83C2216yescomplemented
16C2×D4×D83C221144yescomplemented
17C22×C4×D83C2216yescomplemented
18C2×#41×C83C22124yescomplemented
19C22×Q8×C83C2211yescharacteristic, complemented
20C22×D4×C83C2213yescomplemented
21C23×C4×C83C2213yescomplemented
22C23×D83—424no—
23C23×D83C2316yes—
24C23×D83C23172yescomplemented
25#13050591C231128yescomplemented
26#13050590C231192yescomplemented
27C2×Dic166C23112yescomplemented
28C2×D332C23172yescomplemented
29C2×D332C23136yescomplemented
30C2×D332C231192yescomplemented
31C2×C22⋊D83C23172yescomplemented
32D4×D83C231384yescomplemented
33C22×Dic83C2312yes—
34C2×C4×D83C23172yescomplemented
35C24×C83C2313yes—
36#41×C83C23164yescomplemented
37C2×Q8×C83C23112yescomplemented
38C2×D4×C83C23136yescomplemented
39C22×C4×C83C2313yes—
40C22×C4×C83C2311yescharacteristic
41C22×C4×C83C23136yescomplemented
42C23×C83—212no—
43C23×C83C24118yes—
44C23×C83C2411yescharacteristic
45C22×D83—224no—
46C22×D83—4144no—
47C22×D83C24136yes—
48C22×D83C24196yescomplemented
49Q8×C83C24116yescomplemented
50D4×C83C24148yescomplemented
51C2×C4×C83C2416yes—
52C2×C4×C83C24118yes—
53C2×C4×C83C24148yescomplemented
54C22⋊D83C24196yescomplemented
55C2×Dic83C24112yes—
56D332C24196yescomplemented
57D332C24148yescomplemented
58C4×D83C24196yescomplemented
59Dic166C24116yescomplemented
60C22×C83—236no—
61C22×C83H(5,𝔽2)14yescomplemented
62C22×C83C25112yes—
63C22×C83C2513yes—
64C2×D83—272no—
65C2×D83—496no—
66C2×D83C25124yes—
67C4×C83C2514yes—
68C4×C83C25112yes—
69Dic83C2518yes—
70C2×C83—212no—
71C2×C83C2×H(5,𝔽2)16yescomplemented
72C2×C83C2611yescharacteristic
73D83—224no—
74C22×H(5,𝔽2)—831nomaximal, Sylow
75C83C22×H(5,𝔽2)11yescharacteristic, complemented, Sylow
76C2×H(5,𝔽2)—8348no—
77C22×#41—832no—
78C22×#41—833no—
79C22×#41D8311yescharacteristic, complemented
80C23×D4—836no—
81C23×D4—833no—
82C25—836no—
83H(5,𝔽2)—83256no—
84C2×#41—8348no—
85C2×#41—8372no—
86C2×#41C2×D83124yescomplemented
87C22×Q8—831no—
88C22×Q8C2×D8311yescharacteristic, complemented
89C22×D4—836no—
90C22×D4—83144no—
91C22×D4—836no—
92C22×D4—833no—
93C22×D4—8372no—
94C22×D4C2×D8313yescomplemented
95C23×C4—836no—
96C23×C4C2×D8313yescomplemented
97C24—33224no—
98C24—8372no—
99C24—836no—
100C24C22×D8313yes—
101#41—83128no—
102#41—83192no—
103#41C22×D83164yescomplemented
104C2×Q8—8312no—
105C2×Q8C22×D83112yescomplemented
106C2×D4—8372no—
107C2×D4—83384no—
108C2×D4—8372no—
109C2×D4—8336no—
110C2×D4—83192no—
111C2×D4C22×D83136yescomplemented
112C22×C4—832no—
113C22×C4—8372no—
114C22×C4C22×D8313yes—
115C22×C4C22×D8311yescharacteristic
116C22×C4C22×D83136yescomplemented
117C23—212no—
118C23—332144no—
119C23—8396no—
120C23—8336no—
121C23—16624no—
122C23C23×D83118yes—
123C23C23×D8311yescharacteristic
124Q8—8316no—
125Q8C23×D83116yescomplemented
126D4—8396no—
127D4—8396no—
128D4—8348no—
129D4C23×D83148yescomplemented
130C2×C4—8312no—
131C2×C4—8396no—
132C2×C4C23×D83118yes—
133C2×C4C23×D8316yes—
134C2×C4C23×D83148yescomplemented
135C22—236no—
136C22—33296no—
137C22—8324no—
138C22—16672no—
139C22#1458183814yesdirect product factor
140C22C24×D83112yes—
141C22C24×D8313yes—
142C4—838no—
143C4C24×D83112yes—
144C4C24×D8314yes—
145C2—212no—
146C2—16624no—
147C2C2×#1458183816yesdirect product factor
148C2C25×D8311yescharacteristic
subgroupG/H
centerC23C23×D83
commutator, cosocleC2×C83C26
FrattiniC2C25×D83
FittingC22×#41×C83C2
socleC23×C83C24
hypercenterC22×#41D83
hypocenterC83C22×H(5,𝔽2)
LCSG ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C2×#14581838C2direct (N × H)
2. C22×#13050591C2semidirect
3. C22×#13050590C2semidirect
4. C23×D332C2semidirect
5. C22×D4×D83C2semidirect
6. C22×#41×C83C2semidirect
7. C24×D83C22semidirect
8. #14581838C22direct (N × H)
9. C2×#13050591C22semidirect
10. C2×#13050590C22semidirect
11. C22×Dic166C22semidirect
12. C22×D332C22semidirect
13. C22×D332C22semidirect
14. C22×D332C22semidirect
15. C22×C22⋊D83C22semidirect
16. C2×D4×D83C22semidirect
17. C22×C4×D83C22semidirect
18. C2×#41×C83C22semidirect
19. C22×Q8×C83C22semidirect
20. C22×D4×C83C22semidirect
21. C23×C4×C83C22semidirect
24. C23×D83C23semidirect
25. #13050591C23semidirect
26. #13050590C23semidirect
27. C2×Dic166C23semidirect
28. C2×D332C23semidirect
29. C2×D332C23semidirect
30. C2×D332C23semidirect
31. C2×C22⋊D83C23semidirect
32. D4×D83C23semidirect
34. C2×C4×D83C23semidirect
36. #41×C83C23semidirect
37. C2×Q8×C83C23semidirect
38. C2×D4×C83C23semidirect
41. C22×C4×C83C23semidirect
48. C22×D83C24semidirect
49. Q8×C83C24semidirect
50. D4×C83C24semidirect
53. C2×C4×C83C24semidirect
54. C22⋊D83C24semidirect
56. D332C24semidirect
57. D332C24semidirect
58. C4×D83C24semidirect
59. Dic166C24semidirect
61. C22×C83H(5,𝔽2)semidirect
71. C2×C83C2×H(5,𝔽2)semidirect
75. C83C22×H(5,𝔽2)semidirect
79. C22×#41D83semidirect
86. C2×#41C2×D83semidirect
88. C22×Q8C2×D83semidirect
94. C22×D4C2×D83semidirect
96. C23×C4C2×D83semidirect
103. #41C22×D83semidirect
105. C2×Q8C22×D83semidirect
111. C2×D4C22×D83semidirect
116. C22×C4C22×D83semidirect
125. Q8C23×D83semidirect
129. D4C23×D83semidirect
134. C2×C4C23×D83semidirect
139. C22#14581838direct (N × H)
147. C2C2×#14581838direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#145818382
C22×#130505912
C22×#130505902
C23×D3322
C22×D4×D832
C22×#41×C832
C22×H(5,𝔽2)83