Finite Group Databasebeta

← C2×C23×#2846 #19184236

Subgroups — C2×C23×#2846 #19184236

Order 5888.

32890 conjugacy classes of subgroups; only the 1886 normal subgroups are shown.

normal subgroups only

This group has 32890 conjugacy classes of subgroups — too many to draw. Only its 1886 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
2C22×C23×#120C217yesmaximal, complemented
3C23×#2846C2116yesmaximal, direct product factor
4C2×C23×#379C217yesmaximal, complemented
5C26×C23—256no—
6C26×C23C2217yes—
7C26×C23C2218yescomplemented
8C23×D4×C23C2217yescomplemented
9C2×C23×#120—256no—
10C2×C23×#120C22156yescomplemented
11C22×C22⋊C4×C23C22121yescomplemented
12C23×#379C22156yescomplemented
13C25×C23—2896no—
14C25×C23—2168no—
15C25×C23—2224no—
16C25×C23D4156yescomplemented
17C25×C23C23128yes—
18C25×C23C2317yes—
19C25×C23C2311yescharacteristic, complemented
20C22×D4×C23—284no—
21C22×D4×C23C23128yescomplemented
22C23×C4×C23C2317yescomplemented
23C23×#120—2224no—
24C2×C22⋊C4×C23—284no—
25C2×C22⋊C4×C23C23184yescomplemented
26C24×C23—22048no—
27C24×C23—2896no—
28C24×C23—21792no—
29C24×C23—2672no—
30C24×C23—2336no—
31C24×C23—256no—
32C24×C23—214no—
33C24×C23C2×D4184yes—
34C24×C23C2×D41112yescomplemented
35C24×C23C24114yes—
36C24×C23C2411yescharacteristic
37C24×C23C2412yescomplemented
38C2×D4×C23—2168no—
39C2×D4×C23—4112no—
40C22×C4×C23—242no—
41C22×C4×C23C24114yescomplemented
42C22⋊C4×C23—2168no—
43C2×#2846C2311yesmaximal, characteristic, direct product factor, Sylow
44C23×C23—22048no—
45C23×C23—21792no—
46C23×C23—2896no—
47C23×C23—2336no—
48C23×C23—2672no—
49C23×C23—256no—
50C23×C23—214no—
51C23×C23—428no—
52C23×C23#1201112yescomplemented
53C23×C23C22×D4184yes—
54C23×C23C22×D4114yes—
55C23×C23C2511yescharacteristic
56D4×C23—4112no—
57C2×C4×C23—242no—
58C2×C4×C23—428no—
59C27C2×C2311yescharacteristic, complemented
60C22×#120C2×C2317yescomplemented
61#2846C2×C23116yesdirect product factor
62C2×#379C2×C2317yescomplemented
63C22×C23—2896no—
64C22×C23—2224no—
65C22×C23—2168no—
66C22×C23—414no—
67C22×C23—88no—
68C22×C23C2×#120128yes—
69C22×C23C2×#120156yescomplemented
70C22×C23C23×D417yes—
71C4×C23—414no—
72C26—256no—
73C26C22×C2317yes—
74C26C22×C2318yescomplemented
75C23×D4C22×C2317yescomplemented
76C2×#120—256no—
77C2×#120C22×C23156yescomplemented
78C22×C22⋊C4C22×C23121yescomplemented
79#379C22×C23156yescomplemented
80C2×C23—256no—
81C2×C23—82no—
82C2×C23#284618yesdirect product factor
83C2×C23C22×#12017yes—
84C25—2896no—
85C25—2168no—
86C25—2224no—
87C25D4×C23156yescomplemented
88C25C23×C23128yes—
89C25C23×C2317yes—
90C25C23×C2311yescharacteristic, complemented
91C22×D4—284no—
92C22×D4C23×C23128yescomplemented
93C23×C4C23×C2317yescomplemented
94#120—2224no—
95C2×C22⋊C4—284no—
96C2×C22⋊C4C23×C23184yescomplemented
97C23C2×#284611yescharacteristic, direct product factor, Sylow
98C24—22048no—
99C24—2896no—
100C24—21792no—
101C24—2672no—
102C24—2336no—
103C24—256no—
104C24—214no—
105C24C2×D4×C23184yes—
106C24C2×D4×C231112yescomplemented
107C24C24×C23114yes—
108C24C24×C2311yescharacteristic
109C24C24×C2312yescomplemented
110C2×D4—2168no—
111C2×D4—4112no—
112C22×C4—242no—
113C22×C4C24×C23114yescomplemented
114C22⋊C4—2168no—
115C23—22048no—
116C23—2896no—
117C23—21792no—
118C23—256no—
119C23—2336no—
120C23—2672no—
121C23—214no—
122C23—428no—
123C23C23×#1201112yescomplemented
124C23C22×D4×C23184yes—
125C23C22×D4×C23114yes—
126C23C25×C2311yescharacteristic
127D4—4112no—
128C2×C4—242no—
129C2×C4—428no—
130C22—2896no—
131C22—2168no—
132C22—2224no—
133C22—414no—
134C22—88no—
135C22C2×C23×#120128yes—
136C22C2×C23×#120156yescomplemented
137C22C23×D4×C2317yes—
138C4—414no—
139C2—256no—
140C2—82no—
141C2C23×#284618yesdirect product factor
142C2C22×C23×#12017yes—
subgroupG/H
center, socleC24×C23C24
commutator, Frattini, cosocleC23C25×C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24×C23 ◁ G

Complements

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

NH ≅ G/Ntype
1. C27×C23C2semidirect
2. C22×C23×#120C2semidirect
3. C23×#2846C2direct (N × H)
4. C2×C23×#379C2semidirect
7. C26×C23C22semidirect
8. C23×D4×C23C22semidirect
10. C2×C23×#120C22semidirect
11. C22×C22⋊C4×C23C22semidirect
12. C23×#379C22semidirect
16. C25×C23D4semidirect
19. C25×C23C23semidirect
21. C22×D4×C23C23semidirect
22. C23×C4×C23C23semidirect
25. C2×C22⋊C4×C23C23semidirect
34. C24×C23C2×D4semidirect
37. C24×C23C24semidirect
41. C22×C4×C23C24semidirect
43. C2×#2846C23direct (N × H)
52. C23×C23#120semidirect
59. C27C2×C23semidirect
60. C22×#120C2×C23semidirect
61. #2846C2×C23direct (N × H)
62. C2×#379C2×C23semidirect
69. C22×C23C2×#120semidirect
74. C26C22×C23semidirect
75. C23×D4C22×C23semidirect
77. C2×#120C22×C23semidirect
78. C22×C22⋊C4C22×C23semidirect
79. #379C22×C23semidirect
82. C2×C23#2846direct (N × H)
87. C25D4×C23semidirect
90. C25C23×C23semidirect
92. C22×D4C23×C23semidirect
93. C23×C4C23×C23semidirect
96. C2×C22⋊C4C23×C23semidirect
97. C23C2×#2846direct (N × H)
106. C24C2×D4×C23semidirect
109. C24C24×C23semidirect
113. C22×C4C24×C23semidirect
123. C23C23×#120semidirect
136. C22C2×C23×#120semidirect
141. C2C23×#2846direct (N × H)

Maximal subgroups

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

subgroupindex
C27×C232
C22×C23×#1202
C23×#28462
C2×C23×#3792
C2×#284623