Finite Group Databasebeta

← C26×#120 #72268590

Subgroups — C26×#120 #72268590

Order 2048.

136608424 conjugacy classes of subgroups; only the 9782632 normal subgroups are shown.

normal subgroups only

This group has 136608424 conjugacy classes of subgroups — too many to draw. Only its 9782632 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
1C26×C22⋊C4C213yesmaximal, complemented
2C27×D4C213yesmaximal, complemented
3C25×#120C21504yesmaximal, direct product factor
4C210C211yesmaximal, characteristic, complemented
5C29—2384no—
6C29C2213yes—
7C29C221252yescomplemented
8C29C2211yescharacteristic, complemented
9C26×D4—2384no—
10C26×D4C221756yescomplemented
11C27×C4C2213yescomplemented
12C24×#120C22141664yesdirect product factor
13C25×C22⋊C4C221756yescomplemented
14C28—232768no—
15C28—2384no—
16C28—248384no—
17C28—2192no—
18C28D41384yescomplemented
19C28C231378yes—
20C28C2311yescharacteristic
21C28C23110416yescomplemented
22C28C231126yescomplemented
23C25×D4—248384no—
24C25×D4C23131248yescomplemented
25C26×C4—2192no—
26C26×C4C231378yescomplemented
27C23×#120C231714240yesdirect product factor
28C24×C22⋊C4C23131248yescomplemented
29C27—22064384no—
30C27—224576no—
31C27—224192no—
32C27—2999936no—
33C27—212096no—
34C27—44096no—
35C27C2×D41192yes—
36C27C2×D4124192yescomplemented
37C27C2417812yes—
38C27C24163yes—
39C27C24189280yescomplemented
40C27C2412604yescomplemented
41C24×D4—2999936no—
42C24×D4C241267840yescomplemented
43C25×C4—212096no—
44C25×C4C2417812yescomplemented
45C22×#120C2412666496yesdirect product factor
46C23×C22⋊C4C241267840yescomplemented
47C26—221331968no—
48C26—2774144no—
49C26—2249984no—
50C26—24285440no—
51C26—2124992no—
52C26—4129024no—
53C26#12014096yesdirect product factor
54C26C22×D416048yes—
55C26C22×D41249984yescomplemented
56C26C25133480yes—
57C26C251651yes—
58C26C251166656yescomplemented
59C26C25111160yescomplemented
60C23×D4—24285440no—
61C23×D4C251499968yescomplemented
62C24×C4—2124992no—
63C24×C4C25133480yescomplemented
64C2×#120C2512064384yesdirect product factor
65C22×C22⋊C4C251499968yescomplemented
66C25—245711360no—
67C25—2535680no—
68C25—23999744no—
69C25—23999744no—
70C25—2267840no—
71C25—4666624no—
72C25C2×#120164512yesdirect product factor
73C25C23×D4131248yes—
74C25C23×D41535680yescomplemented
75C25C26131248yes—
76C25C2611395yes—
77C25C26164512yescomplemented
78C25C26110416yescomplemented
79C22×D4—23999744no—
80C22×D4C261193536yescomplemented
81C23×C4—2267840no—
82C23×C4C26131248yescomplemented
83#120C261262144yesdirect product factor
84C2×C22⋊C4C261193536yescomplemented
85C24—221331968no—
86C24—2249984no—
87C24—2774144no—
88C24—24285440no—
89C24—2124992no—
90C24—4714240no—
91C24C22×#1201166656yesdirect product factor
92C24C24×D4133480yes—
93C24C24×D41249984yescomplemented
94C24C2716048yes—
95C24C271651yes—
96C24C2714096yescomplemented
97C24C2712016yescomplemented
98C2×D4—2774144no—
99C2×D4C27112288yescomplemented
100C22×C4—2124992no—
101C22×C4C2716048yescomplemented
102C22⋊C4C27112288yescomplemented
103C23—22064384no—
104C23—2999936no—
105C23—224192no—
106C23—224576no—
107C23—212096no—
108C23—4166656no—
109C23C23×#120189280yesdirect product factor
110C23C25×D417812yes—
111C23C25×D4124192yescomplemented
112C23C281192yes—
113C23C28163yes—
114C23C28164yescomplemented
115D4—224576no—
116C2×C4—212096no—
117C2×C4C281192yescomplemented
118C22—232768no—
119C22—248384no—
120C22—2384no—
121C22—2192no—
122C22—48064no—
123C22C24×#120110416yesdirect product factor
124C22C26×D41378yes—
125C22C26×D41384yescomplemented
126C22C2911yescharacteristic
127C4—2192no—
128C2—2384no—
129C2—464no—
130C2C25×#1201252yesdirect product factor
131C2C27×D413yes—
subgroupG/H
center, socleC28C23
commutator, Frattini, cosocleC22C29
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C28 ◁ G
LECSG ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C26×C22⋊C4C2semidirect
2. C27×D4C2semidirect
3. C25×#120C2direct (N × H)
4. C210C2semidirect
7. C29C22semidirect
8. C29C22semidirect
10. C26×D4C22semidirect
11. C27×C4C22semidirect
12. C24×#120C22direct (N × H)
13. C25×C22⋊C4C22semidirect
18. C28D4semidirect
21. C28C23semidirect
22. C28C23semidirect
24. C25×D4C23semidirect
26. C26×C4C23semidirect
27. C23×#120C23direct (N × H)
28. C24×C22⋊C4C23semidirect
36. C27C2×D4semidirect
39. C27C24semidirect
40. C27C24semidirect
42. C24×D4C24semidirect
44. C25×C4C24semidirect
45. C22×#120C24direct (N × H)
46. C23×C22⋊C4C24semidirect
53. C26#120direct (N × H)
55. C26C22×D4semidirect
58. C26C25semidirect
59. C26C25semidirect
61. C23×D4C25semidirect
63. C24×C4C25semidirect
64. C2×#120C25direct (N × H)
65. C22×C22⋊C4C25semidirect
72. C25C2×#120direct (N × H)
74. C25C23×D4semidirect
77. C25C26semidirect
78. C25C26semidirect
80. C22×D4C26semidirect
82. C23×C4C26semidirect
83. #120C26direct (N × H)
84. C2×C22⋊C4C26semidirect
91. C24C22×#120direct (N × H)
93. C24C24×D4semidirect
96. C24C27semidirect
97. C24C27semidirect
99. C2×D4C27semidirect
101. C22×C4C27semidirect
102. C22⋊C4C27semidirect
109. C23C23×#120direct (N × H)
111. C23C25×D4semidirect
114. C23C28semidirect
117. C2×C4C28semidirect
123. C22C24×#120direct (N × H)
125. C22C26×D4semidirect
130. C2C25×#120direct (N × H)

Maximal subgroups

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

subgroupindex
C26×C22⋊C42
C27×D42
C25×#1202
C2102