Finite Group Databasebeta

← #41007794

Subgroups — #41007794

Order 26973.

2102 conjugacy classes of subgroups; only the 298 normal subgroups are shown.

normal subgroups only

This group has 2102 conjugacy classes of subgroups — too many to draw. Only its 298 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
1C32×C32⋊(C37⋊C3)C312yesmaximal, complemented
2C3×H(3,𝔽3)×C37⋊C3C316yesmaximal, complemented
3#22582550C311yesmaximal, characteristic, complemented
4(C3×C37⋊C3)⋊H(3,𝔽3)C3112yesmaximal, complemented
5(C3×C37⋊C3)⋊H(3,𝔽3)C3118yesmaximal, complemented
6C37×C32⋊H(3,𝔽3)C311yesmaximal, characteristic, complemented
7C33×C37⋊C3C3219yescomplemented
8C33×C37⋊C3C3216yescomplemented
9C3×C32⋊(C37⋊C3)—318no—
10C3×C32⋊(C37⋊C3)C3212yes—
11C3×C32⋊(C37⋊C3)C32136yescomplemented
12C3×C32⋊(C37⋊C3)C3214yescomplemented
13C3×C32⋊(C37⋊C3)C32112yescomplemented
14C3×C32⋊(C37⋊C3)C3216yescomplemented
15C3×C32⋊(C37⋊C3)C32136yescomplemented
16C3×C32⋊(C37⋊C3)C3216yescomplemented
17H(3,𝔽3)×C37⋊C3—354no—
18C34×C37C3211yescharacteristic, complemented
19C3×H(3,𝔽3)×C37C3216yescomplemented
20C3×H(3,𝔽3)×C37C3216yescomplemented
21C33×C37—36no—
22C33×C37—92no—
23C33×C37C3319yes—
24C33×C37C3312yes—
25C33×C37C3312yes—
26C32×C37⋊C3—318no—
27C32×C37⋊C3—354no—
28C32×C37⋊C3—318no—
29C32×C37⋊C3—918no—
30C32×C37⋊C3—912no—
31C32×C37⋊C3C3319yes—
32C32×C37⋊C3C33112yes—
33C32×C37⋊C3C3316yes—
34H(3,𝔽3)×C37—318no—
35H(3,𝔽3)×C37—96no—
36C32⋊(C37⋊C3)—318no—
37C32⋊(C37⋊C3)—312no—
38C32⋊(C37⋊C3)—3108no—
39C32⋊(C37⋊C3)—336no—
40C32⋊(C37⋊C3)—96no—
41C32⋊(C37⋊C3)—936no—
42C32⋊(C37⋊C3)H(3,𝔽3)118yescomplemented
43H(3,𝔽3)2—371nomaximal, Sylow
44C32×C37—32no—
45C32×C37—34no—
46C32×C37—34no—
47C32×C37—318no—
48C32×C37—32no—
49C32×C37—318no—
50C32×C37—32no—
51C32×C37—99no—
52C32×C37C3×H(3,𝔽3)16yescomplemented
53C32×C37C3411yescharacteristic
54C3×C37⋊C3—36no—
55C3×C37⋊C3—312no—
56C3×C37⋊C3—318no—
57C3×C37⋊C3—312no—
58C3×C37⋊C3—312no—
59C3×C37⋊C3—318no—
60C3×C37⋊C3—324no—
61C3×C37⋊C3—981no—
62C3×C37⋊C3—954no—
63C3×C37⋊C3C3×H(3,𝔽3)118yescomplemented
64C32×H(3,𝔽3)—372no—
65C32×H(3,𝔽3)—376no—
66C32⋊H(3,𝔽3)—371no—
67C32⋊H(3,𝔽3)—3712no—
68C32⋊H(3,𝔽3)—3718no—
69C32⋊H(3,𝔽3)C37⋊C311yescharacteristic, complemented
70C3×C37—36no—
71C3×C37—99no—
72C3×C37—92no—
73C3×C37C32×H(3,𝔽3)12yes—
74C3×C37H(5,𝔽3)12yes—
75C37⋊C3—318no—
76C37⋊C3—99no—
77C37⋊C3—912no—
78C34—379no—
79C34—376no—
80C34C3×C37⋊C311yescharacteristic, complemented
81C3×H(3,𝔽3)—372no—
82C3×H(3,𝔽3)—376no—
83C3×H(3,𝔽3)—3736no—
84C3×H(3,𝔽3)—3712no—
85C3×H(3,𝔽3)—3736no—
86C3×H(3,𝔽3)—374no—
87C3×H(3,𝔽3)—376no—
88C3×H(3,𝔽3)—11154no—
89C3×H(3,𝔽3)—11118no—
90C3×H(3,𝔽3)C3×C37⋊C316yescomplemented
91C3×H(3,𝔽3)C3×C37⋊C316yescomplemented
92C37H(3,𝔽3)211yescharacteristic, complemented, Sylow
93C33—36no—
94C33—92no—
95C33—376no—
96C33—3712no—
97C33—379no—
98C33—33312no—
99C33—33318no—
100C33—11118no—
101C33—11154no—
102C33—11118no—
103C33C32×C37⋊C312yes—
104C33C32×C37⋊C319yes—
105C33C32×C37⋊C312yes—
106H(3,𝔽3)—318no—
107H(3,𝔽3)—96no—
108H(3,𝔽3)—3718no—
109H(3,𝔽3)—33336no—
110H(3,𝔽3)—3336no—
111H(3,𝔽3)—111108no—
112H(3,𝔽3)—11136no—
113H(3,𝔽3)—11118no—
114H(3,𝔽3)—11112no—
115C32—32no—
116C32—34no—
117C32—318no—
118C32—34no—
119C32—32no—
120C32—318no—
121C32—32no—
122C32—99no—
123C32—3718no—
124C32—33354no—
125C32—33381no—
126C32—11112no—
127C32—11112no—
128C32—11118no—
129C32—11124no—
130C32—11118no—
131C32—1116no—
132C32—11112no—
133C32H(3,𝔽3)×C37⋊C316yescomplemented
134C32C33×C37⋊C311yescharacteristic
135C3—36no—
136C3—92no—
137C3—99no—
138C3—33312no—
139C3—3339no—
140C3—11118no—
141C3C3×H(3,𝔽3)×C37⋊C312yes—
142C3C37⋊H(5,𝔽3)12yes—
subgroupG/H
center, FrattiniC32C33×C37⋊C3
commutator, socle, cosocleC32×C37C34
FittingC37×C32⋊H(3,𝔽3)C3
hypercenterC32⋊H(3,𝔽3)C37⋊C3
hypocenterC37H(3,𝔽3)2
LCSG ▷ C32×C37 ▷ C37
UCSC1 ◁ C32 ◁ C32⋊H(3,𝔽3)

Complements

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

NH ≅ G/Ntype
1. C32×C32⋊(C37⋊C3)C3semidirect
2. C3×H(3,𝔽3)×C37⋊C3C3semidirect
3. #22582550C3semidirect
4. (C3×C37⋊C3)⋊H(3,𝔽3)C3semidirect
5. (C3×C37⋊C3)⋊H(3,𝔽3)C3semidirect
6. C37×C32⋊H(3,𝔽3)C3semidirect
7. C33×C37⋊C3C32semidirect
8. C33×C37⋊C3C32semidirect
11. C3×C32⋊(C37⋊C3)C32semidirect
12. C3×C32⋊(C37⋊C3)C32semidirect
13. C3×C32⋊(C37⋊C3)C32semidirect
14. C3×C32⋊(C37⋊C3)C32semidirect
15. C3×C32⋊(C37⋊C3)C32semidirect
16. C3×C32⋊(C37⋊C3)C32semidirect
18. C34×C37C32semidirect
19. C3×H(3,𝔽3)×C37C32semidirect
20. C3×H(3,𝔽3)×C37C32semidirect
42. C32⋊(C37⋊C3)H(3,𝔽3)semidirect
52. C32×C37C3×H(3,𝔽3)semidirect
63. C3×C37⋊C3C3×H(3,𝔽3)semidirect
69. C32⋊H(3,𝔽3)C37⋊C3semidirect
80. C34C3×C37⋊C3semidirect
90. C3×H(3,𝔽3)C3×C37⋊C3semidirect
91. C3×H(3,𝔽3)C3×C37⋊C3semidirect
92. C37H(3,𝔽3)2semidirect
133. C32H(3,𝔽3)×C37⋊C3semidirect

Maximal subgroups

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

subgroupindex
C32×C32⋊(C37⋊C3)3
C3×H(3,𝔽3)×C37⋊C33
#225825503
(C3×C37⋊C3)⋊H(3,𝔽3)3
C37×C32⋊H(3,𝔽3)3
H(3,𝔽3)237