Finite Group Databasebeta

← C22×#11924174 #72246409

Subgroups — C22×#11924174 #72246409

Order 4704.

1346 conjugacy classes of subgroups; only the 306 normal subgroups are shown.

normal subgroups only

This group has 1346 conjugacy classes of subgroups — too many to draw. Only its 306 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
1C22×#10596873C212yesmaximal, complemented
2C2×#11924174C2112yesmaximal, direct product factor
3C23×#64424C211yesmaximal, characteristic
4C22×#87639C311yesmaximal, characteristic, complemented, Hall
5C23×#3861C2211yescharacteristic
6C22×#64424C418yescomplemented
7C22×#64424C2216yes—
8C2×#10596873C22112yescomplemented
9#11924174C22116yesdirect product factor
10C23×C7⋊D7C2×C311yescharacteristic
11C22×C7×Dic7C2×C312yescomplemented
12C2×#87639C2×C3112yescomplemented
13C22×C4×AGL(1,𝔽7)—72nomaximal
14C22×#3861C2×C414yescomplemented
15C22×#3861C2313yes—
16C2×#64424C2×C4124yescomplemented
17C2×#64424C2314yes—
18#10596873C2318yescomplemented
19C23×C72C22×C311yescharacteristic
20C22×C7⋊D7C3×C418yescomplemented
21C22×C7⋊D7C22×C316yes—
22C2×C7×Dic7C22×C3112yescomplemented
23#87639C22×C3116yescomplemented
24C23×AGL(1,𝔽7)—72no—
25C23×AGL(1,𝔽7)—143no—
26C22×C4×ΓL(1,𝔽8)—72no—
27C22×ΓL(1,𝔽8)⋊C4—142no—
28C2×C4×AGL(1,𝔽7)—724no—
29C2×#3861C22×C416yescomplemented
30C2×#3861C2411yescharacteristic
31#64424C22×C418yescomplemented
32C22×C4×D7—72no—
33C22×C72C2×C3×C414yescomplemented
34C22×C72C23×C313yes—
35C2×C7⋊D7C2×C3×C4124yescomplemented
36C2×C7⋊D7C23×C314yes—
37C7×Dic7C23×C318yescomplemented
38C23×ΓL(1,𝔽8)—142no—
39C23×ΓL(1,𝔽8)—283no—
40C22×AGL(1,𝔽7)—712no—
41C22×AGL(1,𝔽7)—716no—
42C22×AGL(1,𝔽7)—1424no—
43C22×AGL(1,𝔽7)—1418no—
44C2×C4×ΓL(1,𝔽8)—712no—
45C2×ΓL(1,𝔽8)⋊C4—1412no—
46C4×AGL(1,𝔽7)—732no—
47#3861C23×C411yescharacteristic, complemented, Hall
48C23×D7—72no—
49C23×D7—143no—
50C22×C4×C7—72no—
51C22×Dic7AGL(1,𝔽7)12yescomplemented
52C2×C4×D7—724no—
53C2×C72C22×C3×C416yescomplemented
54C2×C72C24×C311yescharacteristic
55C7⋊D7C22×C3×C418yescomplemented
56C23×C3×C4—491noHall
57C22×ΓL(1,𝔽8)—148no—
58C22×ΓL(1,𝔽8)—146no—
59C22×ΓL(1,𝔽8)—2812no—
60C22×ΓL(1,𝔽8)—289no—
61C2×AGL(1,𝔽7)—748no—
62C2×AGL(1,𝔽7)—78no—
63C2×AGL(1,𝔽7)—1472no—
64C2×AGL(1,𝔽7)—1412no—
65C4×ΓL(1,𝔽8)—78no—
66ΓL(1,𝔽8)⋊C4—148no—
67C23×C7—23no—
68C23×C7C2×AGL(1,𝔽7)12yes—
69C22×D7—716no—
70C22×D7—712no—
71C22×D7—1424no—
72C22×D7—1418no—
73C2×C4×C7—712no—
74C2×Dic7C2×AGL(1,𝔽7)112yescomplemented
75C4×D7—732no—
76C72C23×C3×C411yescharacteristic, complemented, Sylow
77C24×C3—491no—
78C22×C3×C4—4912no—
79C22×C3×C4—982no—
80C2×ΓL(1,𝔽8)—1412no—
81C2×ΓL(1,𝔽8)—142no—
82C2×ΓL(1,𝔽8)—2818no—
83C2×ΓL(1,𝔽8)—283no—
84AGL(1,𝔽7)—716no—
85AGL(1,𝔽7)—1424no—
86C23×C4—491noSylow
87C22×C7—29no—
88C22×C7—212no—
89C22×C7C4×AGL(1,𝔽7)18yescomplemented
90C22×C7C22×AGL(1,𝔽7)16yes—
91C2×D7—748no—
92C2×D7—78no—
93C2×D7—1472no—
94C2×D7—1412no—
95C4×C7—78no—
96Dic7C22×AGL(1,𝔽7)18yescomplemented
97C23×C3—496no—
98C23×C3—498no—
99C23×C3—1961no—
100C2×C3×C4—4916no—
101C2×C3×C4—9812no—
102ΓL(1,𝔽8)—142no—
103ΓL(1,𝔽8)—283no—
104C24—491no—
105C22×C4—72no—
106C22×C4—4912no—
107C2×C7—218no—
108C2×C7—23no—
109C2×C7C2×C4×AGL(1,𝔽7)112yescomplemented
110C2×C7C23×AGL(1,𝔽7)12yes—
111D7—716no—
112D7—1424no—
113C22×C3—494no—
114C22×C3—4924no—
115C22×C3—1963no—
116C22×C3—1964no—
117C3×C4—988no—
118C23—496no—
119C23—498no—
120C23#64423⋊C211yescharacteristic
121C2×C4—712no—
122C2×C4—4916no—
123C7—23no—
124C7C22×C4×AGL(1,𝔽7)12yescomplemented
125C2×C3—498no—
126C2×C3—1966no—
127C2×C3—1961no—
128C22—4924no—
129C22—494no—
130C22#1192417414yesdirect product factor
131C22C2×#64423⋊C213yes—
132C4—78no—
133C3—1961noSylow
134C2—498no—
135C2C22×#64423⋊C211yescharacteristic
136C2C2×#1192417416yesdirect product factor
subgroupG/H
center, hypercenterC23#64423⋊C2
commutator, hypocenterC72C23×C3×C4
FrattiniC2C22×#64423⋊C2
Fitting, socleC23×C72C22×C3
prime radicalC23×#64424C2
cosocleC2×C72C24×C3
LCSG ▷ C72
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C22×#10596873C2semidirect
2. C2×#11924174C2direct (N × H)
4. C22×#87639C3semidirect
6. C22×#64424C4semidirect
8. C2×#10596873C22semidirect
9. #11924174C22direct (N × H)
11. C22×C7×Dic7C2×C3semidirect
12. C2×#87639C2×C3semidirect
14. C22×#3861C2×C4semidirect
16. C2×#64424C2×C4semidirect
18. #10596873C23semidirect
20. C22×C7⋊D7C3×C4semidirect
22. C2×C7×Dic7C22×C3semidirect
23. #87639C22×C3semidirect
29. C2×#3861C22×C4semidirect
31. #64424C22×C4semidirect
33. C22×C72C2×C3×C4semidirect
35. C2×C7⋊D7C2×C3×C4semidirect
37. C7×Dic7C23×C3semidirect
47. #3861C23×C4semidirect
51. C22×Dic7AGL(1,𝔽7)semidirect
53. C2×C72C22×C3×C4semidirect
55. C7⋊D7C22×C3×C4semidirect
74. C2×Dic7C2×AGL(1,𝔽7)semidirect
76. C72C23×C3×C4semidirect
89. C22×C7C4×AGL(1,𝔽7)semidirect
96. Dic7C22×AGL(1,𝔽7)semidirect
109. C2×C7C2×C4×AGL(1,𝔽7)semidirect
124. C7C22×C4×AGL(1,𝔽7)semidirect
130. C22#11924174direct (N × H)
136. C2C2×#11924174direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#105968732
C2×#119241742
C23×#644242
C22×#876393
C22×C4×AGL(1,𝔽7)7