Finite Group Databasebeta

← C23×C23×#120 #19210281

Subgroups — C23×C23×#120 #19210281

Order 5888.

42862 conjugacy classes of subgroups; only the 7902 normal subgroups are shown.

normal subgroups only

This group has 42862 conjugacy classes of subgroups — too many to draw. Only its 7902 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
1C27×C23C211yesmaximal, characteristic, complemented
2C24×D4×C23C213yesmaximal, complemented
3C22×C23×#120C2156yesmaximal, direct product factor
4C23×C22⋊C4×C23C213yesmaximal, complemented
5C26×C23—248no—
6C26×C23C2213yes—
7C26×C23C22128yescomplemented
8C26×C23C2211yescharacteristic, complemented
9C23×D4×C23—248no—
10C23×D4×C23C22184yescomplemented
11C24×C4×C23C2213yescomplemented
12C2×C23×#120C221448yesdirect product factor
13C22×C22⋊C4×C23C22184yescomplemented
14C25×C23—2512no—
15C25×C23—248no—
16C25×C23—2672no—
17C25×C23—224no—
18C25×C23D4148yescomplemented
19C25×C23C23142yes—
20C25×C23C2311yescharacteristic
21C25×C23C231112yescomplemented
22C25×C23C23114yescomplemented
23C22×D4×C23—2672no—
24C22×D4×C23C231336yescomplemented
25C23×C4×C23—224no—
26C23×C4×C23C23142yescomplemented
27C23×#120C231512yesdirect product factor
28C2×C22⋊C4×C23C231336yescomplemented
29C24×C23—23584no—
30C24×C23—2384no—
31C24×C23—2336no—
32C24×C23—21344no—
33C24×C23—2168no—
34C24×C23—464no—
35C24×C23C2×D4124yes—
36C24×C23C2×D41336yescomplemented
37C24×C23C24184yes—
38C24×C23C2417yes—
39C24×C23C24164yescomplemented
40C24×C23C24128yescomplemented
41C2×D4×C23—21344no—
42C2×D4×C23C241192yescomplemented
43C22×C4×C23—2168no—
44C22×C4×C23C24184yescomplemented
45C22⋊C4×C23C241192yescomplemented
46C23×#120C2311yesmaximal, characteristic, direct product factor, Sylow
47C23×C23—23584no—
48C23×C23—21344no—
49C23×C23—2336no—
50C23×C23—2384no—
51C23×C23—2168no—
52C23×C23—4224no—
53C23×C23#120164yesdirect product factor
54C23×C23C22×D4184yes—
55C23×C23C22×D41336yescomplemented
56C23×C23C25124yes—
57C23×C23C2517yes—
58C23×C23C2518yescomplemented
59D4×C23—2384no—
60C2×C4×C23—2168no—
61C2×C4×C23C25124yescomplemented
62C27C2×C2311yescharacteristic, complemented
63C24×D4C2×C2313yescomplemented
64C22×#120C2×C23156yesdirect product factor
65C23×C22⋊C4C2×C2313yescomplemented
66C22×C23—2512no—
67C22×C23—248no—
68C22×C23—2672no—
69C22×C23—224no—
70C22×C23—4112no—
71C22×C23C2×#1201112yesdirect product factor
72C22×C23C23×D4142yes—
73C22×C23C23×D4148yescomplemented
74C22×C23C2611yescharacteristic
75C4×C23—224no—
76C26—248no—
77C26C22×C2313yes—
78C26C22×C23128yescomplemented
79C26C22×C2311yescharacteristic, complemented
80C23×D4—248no—
81C23×D4C22×C23184yescomplemented
82C24×C4C22×C2313yescomplemented
83C2×#120C22×C231448yesdirect product factor
84C22×C22⋊C4C22×C23184yescomplemented
85C2×C23—248no—
86C2×C23—48no—
87C2×C23C22×#120128yesdirect product factor
88C2×C23C24×D413yes—
89C25—2512no—
90C25—248no—
91C25—2672no—
92C25—224no—
93C25D4×C23148yescomplemented
94C25C23×C23142yes—
95C25C23×C2311yescharacteristic
96C25C23×C231112yescomplemented
97C25C23×C23114yescomplemented
98C22×D4—2672no—
99C22×D4C23×C231336yescomplemented
100C23×C4—224no—
101C23×C4C23×C23142yescomplemented
102#120C23×C231512yesdirect product factor
103C2×C22⋊C4C23×C231336yescomplemented
104C23C23×#12011yescharacteristic, direct product factor, Sylow
105C24—23584no—
106C24—2384no—
107C24—2336no—
108C24—21344no—
109C24—2168no—
110C24—464no—
111C24C2×D4×C23124yes—
112C24C2×D4×C231336yescomplemented
113C24C24×C23184yes—
114C24C24×C2317yes—
115C24C24×C23164yescomplemented
116C24C24×C23128yescomplemented
117C2×D4—21344no—
118C2×D4C24×C231192yescomplemented
119C22×C4—2168no—
120C22×C4C24×C23184yescomplemented
121C22⋊C4C24×C231192yescomplemented
122C23—23584no—
123C23—2336no—
124C23—2384no—
125C23—21344no—
126C23—2168no—
127C23—4224no—
128C23C23×#120164yesdirect product factor
129C23C22×D4×C23184yes—
130C23C22×D4×C231336yescomplemented
131C23C25×C23124yes—
132C23C25×C2317yes—
133C23C25×C2318yescomplemented
134D4—2384no—
135C2×C4—2168no—
136C2×C4C25×C23124yescomplemented
137C22—2512no—
138C22—2672no—
139C22—248no—
140C22—224no—
141C22—4112no—
142C22C2×C23×#1201112yesdirect product factor
143C22C23×D4×C23142yes—
144C22C23×D4×C23148yescomplemented
145C22C26×C2311yescharacteristic
146C4—224no—
147C2—248no—
148C2—48no—
149C2C22×C23×#120128yesdirect product factor
150C2C24×D4×C2313yes—
subgroupG/H
center, socleC25×C23C23
commutator, Frattini, cosocleC22C26×C23
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C25×C23 ◁ G

Complements

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

NH ≅ G/Ntype
1. C27×C23C2semidirect
2. C24×D4×C23C2semidirect
3. C22×C23×#120C2direct (N × H)
4. C23×C22⋊C4×C23C2semidirect
7. C26×C23C22semidirect
8. C26×C23C22semidirect
10. C23×D4×C23C22semidirect
11. C24×C4×C23C22semidirect
12. C2×C23×#120C22direct (N × H)
13. C22×C22⋊C4×C23C22semidirect
18. C25×C23D4semidirect
21. C25×C23C23semidirect
22. C25×C23C23semidirect
24. C22×D4×C23C23semidirect
26. C23×C4×C23C23semidirect
27. C23×#120C23direct (N × H)
28. C2×C22⋊C4×C23C23semidirect
36. C24×C23C2×D4semidirect
39. C24×C23C24semidirect
40. C24×C23C24semidirect
42. C2×D4×C23C24semidirect
44. C22×C4×C23C24semidirect
45. C22⋊C4×C23C24semidirect
46. C23×#120C23direct (N × H)
53. C23×C23#120direct (N × H)
55. C23×C23C22×D4semidirect
58. C23×C23C25semidirect
61. C2×C4×C23C25semidirect
62. C27C2×C23semidirect
63. C24×D4C2×C23semidirect
64. C22×#120C2×C23direct (N × H)
65. C23×C22⋊C4C2×C23semidirect
71. C22×C23C2×#120direct (N × H)
73. C22×C23C23×D4semidirect
78. C26C22×C23semidirect
79. C26C22×C23semidirect
81. C23×D4C22×C23semidirect
82. C24×C4C22×C23semidirect
83. C2×#120C22×C23direct (N × H)
84. C22×C22⋊C4C22×C23semidirect
87. C2×C23C22×#120direct (N × H)
93. C25D4×C23semidirect
96. C25C23×C23semidirect
97. C25C23×C23semidirect
99. C22×D4C23×C23semidirect
101. C23×C4C23×C23semidirect
102. #120C23×C23direct (N × H)
103. C2×C22⋊C4C23×C23semidirect
104. C23C23×#120direct (N × H)
112. C24C2×D4×C23semidirect
115. C24C24×C23semidirect
116. C24C24×C23semidirect
118. C2×D4C24×C23semidirect
120. C22×C4C24×C23semidirect
121. C22⋊C4C24×C23semidirect
128. C23C23×#120direct (N × H)
130. C23C22×D4×C23semidirect
133. C23C25×C23semidirect
136. C2×C4C25×C23semidirect
142. C22C2×C23×#120direct (N × H)
144. C22C23×D4×C23semidirect
149. C2C22×C23×#120direct (N × H)

Maximal subgroups

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

subgroupindex
C27×C232
C24×D4×C232
C22×C23×#1202
C23×C22⋊C4×C232
C23×#12023